  
  [1X1 [33X[0;0YMaintenance Issues for the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YThis  chapter collects examples of computations that arose in the context of
  maintaining  the  [5XGAP[105X  Character Table Library. The sections have been added
  when  the  issues in question arose; the dates of the additions are shown in
  the section titles.[133X
  
  
  [1X1.1 [33X[0;0YDisproving Possible Character Tables (November 2006)[133X[101X
  
  [33X[0;0YI  do  not  know a necessary and sufficient criterion for checking whether a
  given  matrix  together  with  a  list of power maps describes the character
  table  of  a finite group. Examples of [13Xpseudo character tables[113X (tables which
  satisfy certain necessary conditions but for which actually no group exists)
  have   been   given   in [Gag86].  Another  such  example  is  described  in
  Section [14X2.4-17[114X.  The  tables  in the [5XGAP[105X Character Table Library satisfy the
  usual  tests.  However, there are table candidates for which these tests are
  not  good  enough. Another question would be whether a given character table
  belongs  to  the  group for which it is claimed to belong, see Section [14X1.1-4[114X
  for an example.[133X
  
  
  [1X1.1-1 [33X[0;0YA Perfect Pseudo Character Table (November 2006)[133X[101X
  
  [33X[0;0Y(This example arose from a discussion with Jack Schmidt.)[133X
  
  [33X[0;0YUp  to  version 1.1.3  of  the  [5XGAP[105X  Character Table Library, the table with
  identifier  [10X"P41/G1/L1/V4/ext2"[110X  was not correct. The problem occurs already
  in the microfiches that are attached to [HP89].[133X
  
  [33X[0;0YIn  the  following,  we show that this table is not the character table of a
  finite  group,  using the [5XGAP[105X library of perfect groups. Currently we do not
  know how to prove this inconsistency alone from the table.[133X
  
  [33X[0;0YWe  start  with  the  construction  of  the inconsistent table; apart from a
  little  editing,  the following input equals the data formerly stored in the
  file [11Xdata/ctoholpl.tbl[111X of the [5XGAP[105X Character Table Library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= rec([127X[104X
    [4X[25X>[125X [27X  Identifier:= "P41/G1/L1/V4/ext2",[127X[104X
    [4X[25X>[125X [27X  InfoText:= Concatenation( [[127X[104X
    [4X[25X>[125X [27X    "origin: Hanrath library,\n",[127X[104X
    [4X[25X>[125X [27X    "structure is 2^7.L2(8),\n",[127X[104X
    [4X[25X>[125X [27X    "characters sorted with permutation (12,14,15,13)(19,20)" ] ),[127X[104X
    [4X[25X>[125X [27X  UnderlyingCharacteristic:= 0,[127X[104X
    [4X[25X>[125X [27X  SizesCentralizers:= [64512,1024,1024,64512,64,64,64,64,128,128,64,[127X[104X
    [4X[25X>[125X [27X    64,128,128,18,18,14,14,14,14,14,14,18,18,18,18,18,18],[127X[104X
    [4X[25X>[125X [27X  ComputedPowerMaps:= [,[1,1,1,1,2,3,3,2,3,2,2,1,3,2,16,16,20,20,22,[127X[104X
    [4X[25X>[125X [27X    22,18,18,26,26,27,27,23,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,4,[127X[104X
    [4X[25X>[125X [27X    1,21,22,17,18,19,20,16,15,15,16,16,15],,,,[1,2,3,4,5,6,7,8,9,10,[127X[104X
    [4X[25X>[125X [27X    11,12,13,14,15,16,4,1,4,1,4,1,26,25,28,27,23,24]],[127X[104X
    [4X[25X>[125X [27X  Irr:= 0,[127X[104X
    [4X[25X>[125X [27X  AutomorphismsOfTable:= Group( [(23,26,27)(24,25,28),(9,13)(10,14),[127X[104X
    [4X[25X>[125X [27X    (17,19,21)(18,20,22)] ),[127X[104X
    [4X[25X>[125X [27X  ConstructionInfoCharacterTable:= ["ConstructClifford",[[[1,2,3,4,[127X[104X
    [4X[25X>[125X [27X    5,6,7,8,9],[1,7,8,3,9,2],[1,4,5,6,2],[1,2,2,2,2,2,2,2]],[127X[104X
    [4X[25X>[125X [27X    [["L2(8)"],["Dihedral",18],["Dihedral",14],["2^3"]],[[[1,2,3,4],[127X[104X
    [4X[25X>[125X [27X    [1,1,1,1],["elab",4,25]],[[1,2,3,4,4,4,4,4,4,4],[2,6,5,2,3,4,5,[127X[104X
    [4X[25X>[125X [27X    6,7,8],["elab",10,17]],[[1,2],[3,4],[[1,1],[-1,1]]],[[1,3],[4,[127X[104X
    [4X[25X>[125X [27X    2],[[1,1],[-1,1]]],[[1,3],[5,3],[[1,1],[-1,1]]],[[1,3],[6,4],[127X[104X
    [4X[25X>[125X [27X    [[1,1],[-1,1]]],[[1,2],[7,2],[[1,1],[1,-1]]],[[1,2],[8,3],[[1,[127X[104X
    [4X[25X>[125X [27X    1],[-1,1]]],[[1,2],[9,5],[[1,1],[1,-1]]]]]],[127X[104X
    [4X[25X>[125X [27X  );;[127X[104X
    [4X[25Xgap>[125X [27XConstructClifford( tbl, tbl.ConstructionInfoCharacterTable[2] );[127X[104X
    [4X[25Xgap>[125X [27XConvertToLibraryCharacterTableNC( tbl );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSuppose  that  there  is  a group [22XG[122X, say, with this table. Then [22XG[122X is perfect
  since the table has only one linear character.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength( LinearCharacters( tbl ) );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XIsPerfectCharacterTable( tbl );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe table satisfies the orthogonality relations, the structure constants are
  nonnegative integers, and symmetrizations of the irreducibles decompose into
  the irreducibles, with nonnegative integral coefficients.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsInternallyConsistent( tbl );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xirr:= Irr( tbl );;[127X[104X
    [4X[25Xgap>[125X [27Xtest:= Concatenation( List( [ 2 .. 7 ],[127X[104X
    [4X[25X>[125X [27X              n -> Symmetrizations( tbl, irr, n ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XAppend( test, Set( Tensored( irr, irr ) ) );[127X[104X
    [4X[25Xgap>[125X [27Xfail in Decomposition( irr, test, "nonnegative" );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xif ForAny( Tuples( [ 1 .. NrConjugacyClasses( tbl ) ], 3 ),[127X[104X
    [4X[25X>[125X [27X     t -> not ClassMultiplicationCoefficient( tbl, t[1], t[2], t[3] )[127X[104X
    [4X[25X>[125X [27X              in NonnegativeIntegers ) then[127X[104X
    [4X[25X>[125X [27X     Error( "contradiction" );[127X[104X
    [4X[25X>[125X [27Xfi;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [5XGAP[105X  Library  of  Perfect  Groups  contains representatives of the four
  isomorphism types of perfect groups of order [22X|G| = 64512[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:= Size( tbl );[127X[104X
    [4X[28X64512[128X[104X
    [4X[25Xgap>[125X [27XNumberPerfectGroups( n );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xgrps:= List( [ 1 .. 4 ], i -> PerfectGroup( IsPermGroup, n, i ) );[127X[104X
    [4X[28X[ L2(8) 2^6 E 2^1, L2(8) N 2^6 E 2^1 I, L2(8) N 2^6 E 2^1 II, [128X[104X
    [4X[28X  L2(8) N 2^6 E 2^1 III ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  we  believe  that  the  classification of perfect groups of order [22X|G|[122X is
  correct  then all we have to do is to show that none of the character tables
  of these four groups is equivalent to the given table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbls:= List( grps, CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27XList( tbls,[127X[104X
    [4X[25X>[125X [27X         x -> TransformingPermutationsCharacterTables( x, tbl ) );[127X[104X
    [4X[28X[ fail, fail, fail, fail ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  fact,  already the matrices of irreducible characters of the four groups
  do not fit to the given table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( tbls,[127X[104X
    [4X[25X>[125X [27X         t -> TransformingPermutations( Irr( t ), Irr( tbl ) ) );[127X[104X
    [4X[28X[ fail, fail, fail, fail ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  us  look  closer  at  the tables in question. Each character table of a
  perfect group of order [22X64512[122X has exactly one irreducible character of degree
  [22X63[122X  that  takes  exactly  the values [22X-1[122X, [22X0[122X, [22X7[122X, and [22X63[122X; moreover, the value [22X7[122X
  occurs in exactly two classes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtestchars:= List( tbls,[127X[104X
    [4X[25X>[125X [27X  t -> Filtered( Irr( t ),[127X[104X
    [4X[25X>[125X [27X         x -> x[1] = 63 and Set( x ) = [ -1, 0, 7, 63 ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( testchars, Length );[127X[104X
    [4X[28X[ 1, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XList( testchars, l -> Number( l[1], x -> x = 7 ) );[127X[104X
    [4X[28X[ 2, 2, 2, 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Another way to state this is that in each of the four tables [22Xt[122X in question,
  there  are ten preimage classes of the involution class in the simple factor
  group  [22XL_2(8)[122X,  there are eight preimage classes of this class in the factor
  group  [22X2^6.L_2(8)[122X,  and that the unique class in which an irreducible degree
  [22X63[122X character of this factor group takes the value [22X7[122X splits in [22Xt[122X.)[133X
  
  [33X[0;0YIn the erroneous table, however, there is only one class with the value [22X7[122X in
  this character.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtestchars:= List( [ tbl ],[127X[104X
    [4X[25X>[125X [27X  t -> Filtered( Irr( t ),[127X[104X
    [4X[25X>[125X [27X         x -> x[1] = 63 and Set( x ) = [ -1, 0, 7, 63 ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( testchars, Length );[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27XList( testchars, l -> Number( l[1], x -> x = 7 ) );[127X[104X
    [4X[28X[ 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  property can be checked easily for the displayed table stored in fiche
  [22X2[122X,  row [22X4[122X, column [22X7[122X of [HP89], with the name [10X6L1<>Z^7<>L2(8); V4; MOD 2[110X, and
  it turns out that this table is not correct.[133X
  
  [33X[0;0YNote  that  these  microfiches  contain [13Xtwo[113X tables of order [22X64512[122X, and there
  were [13Xthree[113X tables of groups of that order in the [5XGAP[105X Character Table Library
  that contain [10Xorigin: Hanrath library[110X in their [2XInfoText[102X ([14XReference: InfoText[114X)
  value.  Besides  the incorrect table, these library tables are the character
  tables  of the groups [10XPerfectGroup( 64512, 1 )[110X and [10XPerfectGroup( 64512, 3 )[110X,
  respectively.  (The  matrices  of irreducible characters of these tables are
  equivalent.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFiltered( [ 1 .. 4 ], i ->[127X[104X
    [4X[25X>[125X [27X       TransformingPermutationsCharacterTables( tbls[i],[127X[104X
    [4X[25X>[125X [27X           CharacterTable( "P41/G1/L1/V1/ext2" ) ) <> fail );[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27XFiltered( [ 1 .. 4 ], i ->[127X[104X
    [4X[25X>[125X [27X       TransformingPermutationsCharacterTables( tbls[i],[127X[104X
    [4X[25X>[125X [27X           CharacterTable( "P41/G1/L1/V2/ext2" ) ) <> fail );[127X[104X
    [4X[28X[ 3 ][128X[104X
    [4X[25Xgap>[125X [27XTransformingPermutations( Irr( tbls[1] ), Irr( tbls[3] ) ) <> fail;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  version 1.2  of  the [5XGAP[105X Character Table Library, the character table
  with  the  [2XIdentifier[102X  ([14XReference:  Identifier  for  tables  of marks[114X) value
  [10X"P41/G1/L1/V4/ext2"[110X  corresponds  to the group [10XPerfectGroup( 64512, 4 )[110X. The
  choice of this group was somewhat arbitrary since the vector system [10XV4[110X seems
  to  be  not  defined in [HP89]; anyhow, this group and the remaining perfect
  group, [10XPerfectGroup( 64512, 2 )[110X, have equivalent matrices of irreducibles.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFiltered( [ 1 .. 4 ], i ->[127X[104X
    [4X[25X>[125X [27X       TransformingPermutationsCharacterTables( tbls[i],[127X[104X
    [4X[25X>[125X [27X           CharacterTable( "P41/G1/L1/V4/ext2" ) ) <> fail );[127X[104X
    [4X[28X[ 4 ][128X[104X
    [4X[25Xgap>[125X [27XTransformingPermutations( Irr( tbls[2] ), Irr( tbls[4] ) ) <> fail;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.1-2 [33X[0;0YAn Error in the Character Table of [22XE_6(2)[122X[101X[1X (March 2016)[133X[101X
  
  [33X[0;0YIn  March  2016, Bill Unger computed the character table of the simple group
  [22XE_6(2)[122X  with  Magma  (see  [CP96])  and  compared it with the table that was
  contained  in the [5XGAP[105X Character Table Library since 2000. It turned out that
  the two tables did not coincide.[133X
  
  [33X[0;0YThe  differences  concern  irrational character values on classes of element
  order  [22X91[122X  and  power map values on these classes. (The character values and
  power  maps  fit  to  each  other  in  both  tables; thus it may be that the
  assumption  of a wrong power has implied the wrong character values, or vice
  versa.) Specifically, the [22X11[122Xth power map in the [5XGAP[105X table fixed all elements
  of  order  [22X91[122X.  Using  the smallest matrix representation of [22XE_6(2)[122X over the
  field  with  two elements, one can easily find an element [22Xg[122X of order [22X91[122X, and
  show  that  the characteristic polynomials of [22Xg[122X and [22Xg^11[122X differ. Hence these
  two  elements  cannot  be conjugate in [22XE_6(2)[122X. In other words, the [5XGAP[105X table
  was wrong.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "E6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= PseudoRandom( g ); until Order( x ) = 91;[127X[104X
    [4X[25Xgap>[125X [27XCharacteristicPolynomial( x ) = CharacteristicPolynomial( x^11 );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe wrong [5XGAP[105X table has been corrected in version 1.3.0 of the [5XGAP[105X Character
  Table Library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "E6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xord91:= Positions( OrdersClassRepresentatives( t ), 91 );[127X[104X
    [4X[28X[ 163, 164, 165, 166, 167, 168 ][128X[104X
    [4X[25Xgap>[125X [27XPowerMap( t, 11 ){ ord91 };[127X[104X
    [4X[28X[ 167, 168, 163, 164, 165, 166 ][128X[104X
  [4X[32X[104X
  
  
  [1X1.1-3 [33X[0;0YAn Error in a Power Map of the Character Table of [22X2.F_4(2).2[122X[101X[1X (November[101X
  [1X2015)[133X[101X
  
  [33X[0;0YAs  a part of the computations for [BMO17], the character table of the group
  [22X2.F_4(2).2[122X  was  computed  automatically from a representation of the group,
  using  Magma  (see  [CP96]).  It turned out that the [22X2[122X-nd power map that had
  been stored on the library character table of [22X2.F_4(2).2[122X had been wrong.[133X
  
  [33X[0;0YIn  fact,  this  was the one and only case of a power map for an [5XAtlas[105X group
  which   was  not  determined  by  the  character  table,  and  the  [2XInfoText[102X
  ([14XReference:  InfoText[114X)  value  of  the character table had mentioned the two
  alternatives.[133X
  
  [33X[0;0YNote  that  the  ambiguity  is  not present in the table of the factor group
  [22XF_4(2).2[122X,  and  only  four  faithful  irreducible  characters  of [22X2.F_4(2).2[122X
  distinguish the four relevant conjugacy classes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "2.F4(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xf:= CharacterTable( "F4(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xmap:= PowerMap( t, 2 );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 11, 11, 3, 3, 3, 5, 5, 5, 3, 6, 6, 5, [128X[104X
    [4X[28X  5, 7, 7, 5, 8, 7, 29, 29, 9, 9, 9, 9, 11, 11, 9, 9, 9, 9, 11, 11, [128X[104X
    [4X[28X  43, 43, 20, 20, 20, 14, 14, 13, 13, 20, 21, 24, 28, 28, 57, 57, 29, [128X[104X
    [4X[28X  29, 29, 29, 33, 33, 35, 37, 37, 37, 37, 33, 33, 37, 37, 35, 41, 41, [128X[104X
    [4X[28X  42, 42, 79, 79, 43, 43, 83, 83, 45, 45, 47, 47, 53, 53, 91, 91, 57, [128X[104X
    [4X[28X  57, 61, 61, 61, 98, 98, 70, 70, 63, 63, 81, 81, 83, 83, 1, 6, 7, [128X[104X
    [4X[28X  11, 16, 17, 24, 24, 21, 27, 27, 25, 26, 29, 41, 53, 53, 53, 46, 56, [128X[104X
    [4X[28X  56, 56, 56, 62, 75, 75, 78, 78, 77, 77, 79, 79, 86, 86, 85, 85, 88, [128X[104X
    [4X[28X  88, 88, 88, 95, 95, 96, 96 ][128X[104X
    [4X[25Xgap>[125X [27XPositionSublist( map, [ 86, 86, 85, 85 ] );[127X[104X
    [4X[28X140[128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t ){ [ 140 .. 143 ] };[127X[104X
    [4X[28X[ 32, 32, 32, 32 ][128X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( t ){ [ 140 .. 143 ] };[127X[104X
    [4X[28X[ 64, 64, 64, 64 ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( t, f ){ [ 140 ..143 ] };[127X[104X
    [4X[28X[ 86, 86, 87, 87 ][128X[104X
    [4X[25Xgap>[125X [27XPowerMap( f, 2 ){ [ 86, 87 ] };[127X[104X
    [4X[28X[ 50, 50 ][128X[104X
    [4X[25Xgap>[125X [27Xpos:= PositionsProperty( Irr( t ),[127X[104X
    [4X[25X>[125X [27X   x -> x[1] <> x[2] and Length( Set( x{ [ 140 .. 143 ] } ) ) > 1 );[127X[104X
    [4X[28X[ 144, 145, 146, 147 ][128X[104X
    [4X[25Xgap>[125X [27XList( pos, i -> Irr(t)[i]{ [ 140 .. 143 ] } );[127X[104X
    [4X[28X[ [ 2*E(16)-2*E(16)^7, -2*E(16)+2*E(16)^7, 2*E(16)^3-2*E(16)^5, [128X[104X
    [4X[28X      -2*E(16)^3+2*E(16)^5 ], [128X[104X
    [4X[28X  [ -2*E(16)+2*E(16)^7, 2*E(16)-2*E(16)^7, -2*E(16)^3+2*E(16)^5, [128X[104X
    [4X[28X      2*E(16)^3-2*E(16)^5 ], [128X[104X
    [4X[28X  [ -2*E(16)^3+2*E(16)^5, 2*E(16)^3-2*E(16)^5, 2*E(16)-2*E(16)^7, [128X[104X
    [4X[28X      -2*E(16)+2*E(16)^7 ], [128X[104X
    [4X[28X  [ 2*E(16)^3-2*E(16)^5, -2*E(16)^3+2*E(16)^5, -2*E(16)+2*E(16)^7, [128X[104X
    [4X[28X      2*E(16)-2*E(16)^7 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YI  had  not  found  a  suitable subgroup of [22X2.F_4(2).2[122X whose character table
  could  be  used  to decide the question which of the two alternatives is the
  correct one.[133X
  
  
  [1X1.1-4 [33X[0;0YA Character Table with a Wrong Name (May 2017)[133X[101X
  
  [33X[0;0Y(This example is much older.)[133X
  
  [33X[0;0YThe character table that is shown in [Ost86, p. 126 f.] is claimed to be the
  table  of  a  Sylow [22X2[122X subgroup [22XP[122X of the sporadic simple Lyons group [22XLy[122X. This
  table  had  been  contained in the character table library of the [5XCAS[105X system
  (see [NPP84]), which was one of the predecessors of [5XGAP[105X.[133X
  
  [33X[0;0YIt  is  easy  to  see  that no subgroup of [22XLy[122X can have this character table.
  Namely,  the  group  of  that  table  contains  elements of order eight with
  centralizer order [22X2^6[122X, and this does not occur in [22XLy[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "Ly" );;[127X[104X
    [4X[25Xgap>[125X [27Xorders:= OrdersClassRepresentatives( tbl );;[127X[104X
    [4X[25Xgap>[125X [27Xorder8:= Filtered( [ 1 .. Length( orders ) ], x -> orders[x] = 8 );[127X[104X
    [4X[28X[ 12, 13 ][128X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( tbl ){ order8 } / 2^6;[127X[104X
    [4X[28X[ 15/2, 3/2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  table  of  [22XP[122X  has  been  computed  in  [Bre91] with character theoretic
  methods.   Nowadays   it   would   be  no  problem  to  take  a  permutation
  representation of [22XLy[122X, to compute its Sylow [22X2[122X subgroup, and use this group to
  compute  its  character table. However, the task is even easier if we assume
  that  [22XLy[122X  has  a  subgroup of the structure [22X3.McL.2[122X. This subgroup is of odd
  index,  hence it contains a conjugate of [22XP[122X. Clearly the Sylow [22X2[122X subgroups in
  the  factor  group  [22XMcL.2[122X  are  isomorphic  with [22XP[122X. Thus we can start with a
  rather small permutation representation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "McL.2" );;[127X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( g );[127X[104X
    [4X[28X275[128X[104X
    [4X[25Xgap>[125X [27Xsyl:= SylowSubgroup( g, 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xpc:= Image( IsomorphismPcGroup( syl ) );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( pc );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  character table coincides with the one which is stored in the Character
  Table Library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsRecord( TransformingPermutationsCharacterTables( t,[127X[104X
    [4X[25X>[125X [27X                 CharacterTable( "LyN2" ) ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2 [33X[0;0YSome finite factor groups of perfect space groups (February 2014)[133X[101X
  
  [33X[0;0YIf  one  wants to find a group to which a given character table from the [5XGAP[105X
  Character    Table    Library    belongs,   one   can   try   the   function
  [2XGroupInfoForCharacterTable[102X ([14XCTblLib: GroupInfoForCharacterTable[114X). For a long
  time,  this  was  not successful in the case of [22X16[122X character tables that had
  been  computed  by W. Hanrath (see Section [21XOrdinary and Brauer Tables in the
  [5XGAP[105X Character Table Library[121X in the [5XCTblLib[105X manual).[133X
  
  [33X[0;0YUsing  the  information from [HP89], it is straightforward to construct such
  groups  as  factor  groups  of  infinite  groups. Since version 1.3.0 of the
  [5XCTblLib[105X     package,     calling     [2XGroupInfoForCharacterTable[102X    ([14XCTblLib:
  GroupInfoForCharacterTable[114X)  for  the  [22X16[122X  library tables in question yields
  nonempty  lists  and  thus  allows  one  to  access  the  results  of  these
  constructions, via the function [10XCTblLib.FactorGroupOfPerfectSpaceGroup[110X. This
  is  an  undocumented auxiliary function that becomes available automatically
  when  [2XGroupInfoForCharacterTable[102X  ([14XCTblLib:  GroupInfoForCharacterTable[114X) has
  been called for the first time.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGroupInfoForCharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27XIsBound( CTblLib.FactorGroupOfPerfectSpaceGroup );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YBelow  we list the [22X16[122X group constructions. In each case, an epimorphism from
  the space group in question is defined by mapping the generators returned by
  by   the   function   [10XgeneratorsOfPerfectSpaceGroup[110X  defined  below  to  the
  generators   stored   in   the   attribute   [2XGeneratorsOfGroup[102X   ([14XReference:
  GeneratorsOfGroup[114X)        of        the        group       returned       by
  [10XCTblLib.FactorGroupOfPerfectSpaceGroup[110X.[133X
  
  
  [1X1.2-1 [33X[0;0YConstructing the space groups in question[133X[101X
  
  [33X[0;0YIn [HP89],  a space group [22XS[122X is described as a subgroup [22X{ M(g, t); g ∈ P, t ∈
  T }[122X of GL[22X(d+1, ℤ)[122X, where[133X
  
                            M(g, t) = ⌈      g  0 ⌉
                                      ⌊ V(g)+t  1 ⌋,
  
  [33X[0;0Ythe  [13Xpoint  group[113X  [22XP[122X  of [22XS[122X is a finite subgroup of GL[22X(d, ℤ)[122X, the [13Xtranslation
  lattice[113X [22XT[122X of [22XS[122X is a sublattice of [22Xℤ^d[122X, and the [13Xvector system[113X [22XV[122X of [22XS[122X is a map
  from [22XP[122X to [22Xℤ^d[122X. Note that [22XV[122X maps the identity matrix [22XI ∈[122X GL[22X(d, ℤ)[122X to the zero
  vector,  and  [22XM(T):=  {  M(I,  t); t ∈ T }[122X is a normal subgroup of [22XS[122X that is
  isomorphic with [22XT[122X. More generally, [22XM(n T)[122X is a normal subgroup of [22XS[122X, for any
  positive integer [22Xn[122X.[133X
  
  [33X[0;0YSpecifically,  [22XP[122X  is given by generators [22Xg_1, g_2, ..., g_k[122X, [22XT[122X is given by a
  [22Xℤ[122X-basis  [22XB  =  {  b_1,  b_2,  ..., b_d }[122X of [22XT[122X, and [22XV[122X is given by the vectors
  [22XV(g_1), V(g_2), ..., V(g_k)[122X.[133X
  
  [33X[0;0YIn  the examples below, the matrix representation of [22XP[122X is irreducible, so we
  need just the following [22Xk+1[122X elements to generate [22XS[122X:[133X
  
          ⌈    g_1  0 ⌉  ⌈    g_2  0 ⌉       ⌈    g_k  0 ⌉  ⌈   I  0 ⌉
          ⌊ V(g_1)  1 ⌋, ⌊ V(g_2)  1 ⌋, ..., ⌊ V(g_k)  1 ⌋, ⌊ b_1  1 ⌋.
  
  [33X[0;0YThese generators are returned by the function [10XgeneratorsOfPerfectSpaceGroup[110X,
  when the inputs are [22X[ g_1, g_2, ..., g_k ][122X, [22X[ V(g_1), V(g_2), ..., V(g_k) ][122X,
  and [22Xb_1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XgeneratorsOfPerfectSpaceGroup:= function( Pgens, V, t )[127X[104X
    [4X[25X>[125X [27X    local d, result, i, m;[127X[104X
    [4X[25X>[125X [27X    d:= Length( Pgens[1] );[127X[104X
    [4X[25X>[125X [27X    result:= [];[127X[104X
    [4X[25X>[125X [27X    for i in [ 1 .. Length( Pgens ) ] do[127X[104X
    [4X[25X>[125X [27X      m:= IdentityMat( d+1 );[127X[104X
    [4X[25X>[125X [27X      m{ [ 1 .. d ] }{ [ 1 .. d ] }:= Pgens[i];[127X[104X
    [4X[25X>[125X [27X      m[ d+1 ]{ [ 1 .. d ] }:= V[i];[127X[104X
    [4X[25X>[125X [27X      result[i]:= m;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X    m:= IdentityMat( d+1 );[127X[104X
    [4X[25X>[125X [27X    m[ d+1 ]{ [ 1 .. d ] }:= t;[127X[104X
    [4X[25X>[125X [27X    Add( result, m );[127X[104X
    [4X[25X>[125X [27X    return result;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  
  [1X1.2-2 [33X[0;0YConstructing the factor groups in question[133X[101X
  
  [33X[0;0YThe  space  group  [22XS[122X  acts  on [22Xℤ^d[122X, via [22Xv ⋅ M(g, t) = v g + V(g) + t[122X. A (not
  necessarily  faithful)  representation  of [22XS/M(n T)[122X can be obtained from the
  corresponding  action  of [22XS[122X on [22Xℤ^d/(n ℤ^d)[122X, that is, by reducing the vectors
  modulo  [22Xn[122X.  For the [5XGAP[105X computations, we work instead with vectors of length
  [22Xd+1[122X,  extending  each vector in [22Xℤ^d[122X by [22X1[122X in the last position, and acting on
  these  vectors  by  right  multiplicaton  with elements of [22XS[122X. Multiplication
  followed  by  reduction  modulo  [22Xn[122X  is  implemented  by  the action function
  returned by [10XmultiplicationModulo[110X when this is called with argument [22Xn[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XmultiplicationModulo:= n -> function( v, g )[127X[104X
    [4X[25X>[125X [27X       return List( v * g, x -> x mod n ); end;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  some  of  the  examples,  the representation of [22XP[122X given in [HP89] is the
  action  on  the factor of a permutation module modulo its trivial submodule.
  For that, we provide the function [10XdeletedPermutationMat[110X, cf. [HP89, p. 269].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XdeletedPermutationMat:= function( pi, n )[127X[104X
    [4X[25X>[125X [27X    local mat, j, i;[127X[104X
    [4X[25X>[125X [27X    mat:= PermutationMat( pi, n );[127X[104X
    [4X[25X>[125X [27X    mat:= mat{ [ 1 .. n-1 ] }{ [ 1 .. n-1 ] };[127X[104X
    [4X[25X>[125X [27X    j:= n ^ pi;[127X[104X
    [4X[25X>[125X [27X    if j <> n then[127X[104X
    [4X[25X>[125X [27X      for i in [ 1 .. n-1 ] do[127X[104X
    [4X[25X>[125X [27X        mat[i][j]:= -1;[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    return mat;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAfter  constructing permutation generators for the example groups, we verify
  that  the  groups  fit  to the character tables from the [5XGAP[105X Character Table
  Library and to the permutation generators stored for the construction of the
  group via [10XCTblLib.FactorGroupOfPerfectSpaceGroup[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup:= function( gens, id )[127X[104X
    [4X[25X>[125X [27X    local sm, act, stored, hom;[127X[104X
    [4X[25X>[125X [27X    sm:= SmallerDegreePermutationRepresentation( Group( gens ) );[127X[104X
    [4X[25X>[125X [27X    gens:= List( gens, x -> x^sm );[127X[104X
    [4X[25X>[125X [27X    act:= Images( sm );[127X[104X
    [4X[25X>[125X [27X    if not IsRecord( TransformingPermutationsCharacterTables([127X[104X
    [4X[25X>[125X [27X                         CharacterTable( act ),[127X[104X
    [4X[25X>[125X [27X                         CharacterTable( id ) ) ) then[127X[104X
    [4X[25X>[125X [27X      return "wrong character table";[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    GroupInfoForCharacterTable( id );[127X[104X
    [4X[25X>[125X [27X    stored:= CTblLib.FactorGroupOfPerfectSpaceGroup( id );[127X[104X
    [4X[25X>[125X [27X    hom:= GroupHomomorphismByImages( stored, act,[127X[104X
    [4X[25X>[125X [27X              GeneratorsOfGroup( stored ), gens );[127X[104X
    [4X[25X>[125X [27X    if hom = fail or not IsBijective( hom ) then[127X[104X
    [4X[25X>[125X [27X      return "wrong group";[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    return true;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  
  [1X1.2-3 [33X[0;0YExamples with point group [22XA_5[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  are two examples with [22Xd = 5[122X. The generators of the point group are as
  follows (see [HP89, p. 272]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= deletedPermutationMat( (1,3)(2,4), 6 );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= deletedPermutationMat( (1,2,3)(4,5,6), 6 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn both cases, the vector system is [22XV_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= [ [ 2, 2, 0, 0, 1 ], 0 * b[1] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn the first example, the translation lattice is the sublattice [22XL = 2 L_1[122X of
  the full lattice [22XL_1 = ℤ^d[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= [ 2, 0, 0, 0, 0 ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character  table with identifier [10X"P1/G2/L1/V2/ext4"[110X belongs to
  the  factor  group of [22XS[122X modulo the normal subgroup [22XM(4 L)[122X, so we compute the
  action on an orbit modulo [22X8[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 8 );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, [ 1, 0, 0, 0, 0, 1 ], fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P1/G2/L1/V2/ext4" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  second  example, the translation lattice is the sublattice [22X2 L_2[122X of
  [22Xℤ^d[122X where [22XL_2[122X has the following basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbas:= [ [-1,-1, 1, 1, 1 ],[127X[104X
    [4X[25X>[125X [27X           [-1, 1,-1, 1, 1 ],[127X[104X
    [4X[25X>[125X [27X           [ 1, 1, 1,-1,-1 ],[127X[104X
    [4X[25X>[125X [27X           [ 1, 1,-1,-1, 1 ],[127X[104X
    [4X[25X>[125X [27X           [-1, 1, 1,-1, 1 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the sake of simplicity, we rewrite the action of the point group to one
  on [22XL_2[122X, and we adjust also the vector system.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB:= Basis( Rationals^Length( bas ), bas );;[127X[104X
    [4X[25Xgap>[125X [27Xabas:= List( bas, x -> Coefficients( B, x * a ) );;[127X[104X
    [4X[25Xgap>[125X [27Xbbas:= List( bas, x -> Coefficients( B, x * b ) );;[127X[104X
    [4X[25Xgap>[125X [27Xvbas:= List( v, x -> Coefficients( B, x ) );[127X[104X
    [4X[28X[ [ 3/2, 1, 2, 3/2, -1 ], [ 0, 0, 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order  to  work  with  integral  matrices  (which  is  necessary because
  [10XmultiplicationModulo[110X  uses  [5XGAP[105X's  [10Xmod[110X  operator), we double both the vector
  system and the translation lattice.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xvbas:= vbas * 2;[127X[104X
    [4X[28X[ [ 3, 2, 4, 3, -2 ], [ 0, 0, 0, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xt:= 2 * t;[127X[104X
    [4X[28X[ 4, 0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character  table with identifier [10X"P1/G2/L2/V2/ext4"[110X belongs to
  the  factor  group  of  [22XS[122X modulo the normal subgroup [22XM(8 L_2)[122X; since we have
  doubled the lattice, we compute the action on an orbit modulo [22X16[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ abas, bbas ], vbas, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 16 );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, [ 0, 0, 0, 0, 0, 1 ], fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P1/G2/L2/V2/ext4" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-4 [33X[0;0YExamples with point group [22XL_3(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  are  three  examples  with  [22Xd  =  6[122X  and  one example with [22Xd = 8[122X. The
  generators  of  the  point group for the first three examples are as follows
  (see [HP89, p. 290]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= [ [ 0, 1, 0, 1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0, 1, 1, 1, 1 ],[127X[104X
    [4X[25X>[125X [27X         [-1,-1,-1,-1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0,-1,-1,-1,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 1, 1, 1, 0, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 1, 0, 1, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xb:= [ [-1, 0, 0, 0, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0,-1, 0,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 1, 1, 1, 1, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 1, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [-1,-1,-1, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0, 0, 0, 0, 0 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe first vector system is the trivial vector system [22XV_1[122X (that is, the space
  group  [22XS[122X  is  a  split  extension  of  the  point  group and the translation
  lattice), and the translation lattice is the full lattice [22XL_1 = ℤ^d[122X.[133X
  
  [33X[0;0YThe  library  character table with identifier [10X"P11/G1/L1/V1/ext4"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(4 L_1)[122X, so we compute the
  action on an orbit modulo [22X4[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= List( [ 1, 2 ], i -> 0 * a[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 1, 0, 0, 0, 0, 0 ];;[127X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xseed:= [ 1, 0, 0, 0, 0, 0, 1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, seed, fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P11/G1/L1/V1/ext4" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe second vector system is [22XV_2[122X, and the translation lattice is [22X2 L_1[122X.[133X
  
  [33X[0;0YThe  library  character table with identifier [10X"P11/G1/L1/V2/ext4"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(8 L_1)[122X, so we compute the
  action on an orbit modulo [22X8[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= [ [ 1, 0, 1, 0, 0, 0 ], 0 * a[1] ];;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 2, 0, 0, 0, 0, 0 ];;[127X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 8 );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P11/G1/L1/V2/ext4" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe third vector system is [22XV_3[122X, and the translation lattice is [22X2 L_1[122X.[133X
  
  [33X[0;0YThe  library  character table with identifier [10X"P11/G1/L1/V3/ext4"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(8 L_1)[122X, so we compute the
  action on an orbit modulo [22X8[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= [ [ 0, 1, 0, 0, 1, 0 ], 0 * a[1] ];;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 2, 0, 0, 0, 0, 0 ];;[127X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 8 );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P11/G1/L1/V3/ext4" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  generators  of  the  point  group for the fourth example are as follows
  (see [HP89, p. 293]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= [ [ 1, 0, 0, 1, 0,-1, 0, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0,-1, 1, 0,-1, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0, 0, 1, 0,-1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0,-1, 0,-1, 0, 1, 1,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0,-1, 1, 1,-1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 1,-1,-1, 0, 0, 0, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0,-1, 1, 0,-1, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0,-1, 0, 0, 0, 0, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xb:= [ [ 1, 0,-2, 0, 1,-1, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0,-1, 0, 0, 0, 0, 1,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0,-1, 0, 1,-1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [-1,-1, 1,-1,-1, 2, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0,-1, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0,-1, 0,-1,-1, 1, 1,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 1,-1, 0, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0, 0, 0, 0, 0, 0, 0 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  vector  system  is  the  trivial vector system [22XV_1[122X, and the translation
  lattice is the full lattice [22XL_1 = ℤ^d[122X.[133X
  
  [33X[0;0YThe  library  character table with identifier [10X"P11/G4/L1/V1/ext3"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(3 L_1)[122X, so we compute the
  action on an orbit modulo [22X3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= List( [ 1, 2 ], i -> 0 * a[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 1, 0, 0, 0, 0, 0, 0, 0 ];;[127X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xseed:= [ 1, 0, 0, 0, 0, 0, 0, 0, 1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, seed, fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P11/G4/L1/V1/ext3" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-5 [33X[0;0YExample with point group SL[22X_2(7)[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  is  one  example with [22Xd = 8[122X. The generators of the point group are as
  follows (see [HP89, p. 295]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= KroneckerProduct( IdentityMat( 4 ), [ [ 0, 1 ], [ -1, 0 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= [ [ 0,-1, 0, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 1, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [-1, 0, 0, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0,-1, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 1, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 0, 1 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  vector  system  is  the  trivial vector system [22XV_1[122X, and the translation
  lattice  is the sublattice [22XL_2[122X of [22Xℤ^d[122X that has the following basis, which is
  called [22XB(2,8)[122X in [HP89, p. 269].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbas:= [ [ 1, 1, 0, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 1, 1, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 1, 1, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 1, 1, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 0, 1, 1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 0, 0, 1, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 0, 0, 0, 1, 1 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 0, 0, 0,-1, 1 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor the sake of simplicity, we rewrite the action to one on [22XL_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB:= Basis( Rationals^Length( bas ), bas );;[127X[104X
    [4X[25Xgap>[125X [27Xabas:= List( bas, x -> Coefficients( B, x * a ) );;[127X[104X
    [4X[25Xgap>[125X [27Xbbas:= List( bas, x -> Coefficients( B, x * b ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P12/G1/L2/V1/ext2"[110X belongs to
  the  factor group of [22XS[122X modulo the normal subgroup [22XM(2 L_2)[122X. The action on an
  orbit  modulo  [22X2[122X is not faithful, its kernel contains the centre of SL[22X(2,7)[122X.
  We  can  compute  a faithful representation by acting on pairs: One entry is
  the usual vector and the other entry carries the action of the point group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= List( [ 1, 2 ], i -> 0 * a[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 1, 0, 0, 0, 0, 0, 0, 0 ];;[127X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ abas, bbas ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xfunpairs:= function( pair, g )[127X[104X
    [4X[25X>[125X [27X   return [ fun( pair[1], g ), pair[2] * g ];[127X[104X
    [4X[25X>[125X [27X   end;;[127X[104X
    [4X[25Xgap>[125X [27Xseed:= [ [ 1, 0, 0, 0, 0, 0, 0, 0, 1 ],[127X[104X
    [4X[25X>[125X [27X            [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, seed, funpairs );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, funpairs ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P12/G1/L2/V1/ext2" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-6 [33X[0;0YExample with point group [22X2^3.L_3(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  is  one  example with [22Xd = 7[122X. The generators of the point group are as
  follows (see [HP89, p. 297]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= PermutationMat( (2,4)(5,7), 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= PermutationMat( (1,3,2)(4,6,5), 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xc:= DiagonalMat( [ -1, -1, 1, 1, -1, -1, 1 ] );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  vector  system  is  the  trivial vector system [22XV_1[122X, and the translation
  lattice  is the sublattice [22XL_2[122X of [22Xℤ^d[122X that has the following basis, which is
  called [22XB(2,7)[122X in [HP89, p. 269].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbas:= [ [ 1, 1, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 1, 1, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 1, 1, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 1, 1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 0, 1, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 0, 0, 1, 1 ],[127X[104X
    [4X[25X>[125X [27X           [ 0, 0, 0, 0, 0,-1, 1 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor the sake of simplicity, we rewrite the action to one on [22XL_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB:= Basis( Rationals^Length( bas ), bas );;[127X[104X
    [4X[25Xgap>[125X [27Xabas:= List( bas, x -> Coefficients( B, x * a ) );;[127X[104X
    [4X[25Xgap>[125X [27Xbbas:= List( bas, x -> Coefficients( B, x * b ) );;[127X[104X
    [4X[25Xgap>[125X [27Xcbas:= List( bas, x -> Coefficients( B, x * c ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P13/G1/L2/V1/ext2"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(2 L_2)[122X, so we compute the
  action on an orbit modulo [22X2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= List( [ 1 .. 3 ], i -> 0 * a[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 1, 0, 0, 0, 0, 0, 0 ];;[127X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ abas,bbas,cbas ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 0, 1 ], fun );;[127X[104X
    [4X[25Xgap>[125X [27Xact:= Action( g, orb, fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P13/G1/L2/V1/ext2" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-7 [33X[0;0YExamples with point group [22XA_6[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  are  two  examples  with [22Xd = 10[122X. In both cases, the generators of the
  point group are as follows (see [HP89, p. 307]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= [ [ 0,-1, 0, 0, 0, 0, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0,-1, 0, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xc:= [ [ 0, 0, 0, 0, 0, 0, 0,-1, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 0,-1, 1,-1 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0,-1, 1, 0,-1, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0,-1, 1, 0, 0, 0, 0,-1, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 0, 0, 0,-1 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 1,-1, 0, 0, 1 ], [127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 1,-1, 0, 0, 0, 0, 0, 1 ], [127X[104X
    [4X[25X>[125X [27X         [-1, 0, 1, 0, 0,-1, 0, 0, 0, 0 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  both  examples,  the vector system is the trivial vector system [22XV_1[122X, and
  the  translation  lattices are the lattices [22XL_2[122X and [22XL_5[122X, respectively, which
  have the following bases.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbas2:= [ [ 0, 1,-1, 0, 0, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 0, 1,-1, 0, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 0, 0, 0, 1,-1, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 0, 0, 0, 0, 1,-1, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 0, 0, 0, 0, 1, 0,-1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 0, 0, 0, 0, 0, 0, 1,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 0, 0, 0, 0, 0, 0, 0, 1,-1 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 0, 0, 1, 0, 0, 0, 0, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 1, 0, 0, 0, 0, 0, 1, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xbas5:= [ [ 0,-1, 1, 1,-1, 1, 1,-1,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 1, 0,-1,-1,-1, 1, 1,-1,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 1, 1,-1, 1, 1,-1, 0, 1, 1 ],[127X[104X
    [4X[25X>[125X [27X            [ 1, 1, 0,-1, 0,-1, 1,-1, 1,-1 ],[127X[104X
    [4X[25X>[125X [27X            [-1, 0,-1, 1, 1, 0,-1,-1, 1,-1 ],[127X[104X
    [4X[25X>[125X [27X            [ 0, 1,-1, 1, 1,-1, 1, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X            [-1,-1, 1, 1, 0,-1,-1,-1,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X            [ 1,-1, 0,-1, 1,-1, 1, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X            [-1, 1,-1, 1,-1, 0,-1, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X            [ 1,-1,-1, 1, 1, 1, 0, 0,-1,-1 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor the sake of simplicity, we rewrite the action to actions on [22XL_2[122X and [22XL_5[122X,
  respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB2:= Basis( Rationals^Length( bas2 ), bas2 );;[127X[104X
    [4X[25Xgap>[125X [27Xbbas2:= List( bas2, x -> Coefficients( B2, x * b ) );;[127X[104X
    [4X[25Xgap>[125X [27Xcbas2:= List( bas2, x -> Coefficients( B2, x * c ) );;[127X[104X
    [4X[25Xgap>[125X [27XB5:= Basis( Rationals^Length( bas5 ), bas5 );;[127X[104X
    [4X[25Xgap>[125X [27Xbbas5:= List( bas5, x -> Coefficients( B5, x * b ) );;[127X[104X
    [4X[25Xgap>[125X [27Xcbas5:= List( bas5, x -> Coefficients( B5, x * c ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P21/G3/L2/V1/ext2"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(2 L_2)[122X, so we compute the
  action on an orbit modulo [22X2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= List( [ 1, 2 ], i -> 0 * bbas2[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];;[127X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ bbas2, cbas2 ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xseed:= [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, seed, fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P21/G3/L2/V1/ext2" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P21/G3/L5/V1/ext2"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(2 L_5)[122X, so we compute the
  action on an orbit modulo [22X2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ bbas5, cbas5 ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, seed, fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P21/G3/L5/V1/ext2" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-8 [33X[0;0YExamples with point group [22XL_2(8)[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  are  two  examples  with  [22Xd = 7[122X. In both cases, the generators of the
  point group are as follows (see [HP89, p. 327]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= [ [ 0,-1, 0, 1, 0,-1, 1],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0,-1, 0, 1,-1, 0],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0,-1, 1, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0,-1, 0, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 1,-1, 0, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [ 0,-1, 1, 0,-1, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [ 1,-1, 0, 1, 0,-1, 0] ];;[127X[104X
    [4X[25Xgap>[125X [27Xb:= [ [-1, 0, 1, 0,-1, 1, 0],[127X[104X
    [4X[25X>[125X [27X         [ 0,-1, 0, 1,-1, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0,-1, 1, 0, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0,-1, 0, 0, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [ 0, 1,-1, 0, 0, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [-1, 1, 0,-1, 0, 0, 0],[127X[104X
    [4X[25X>[125X [27X         [-1, 0, 1, 0,-1, 0, 1] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  both  examples, the vector system is [22XV_2[122X. The translation lattice in the
  first example is the lattice [22XL = 3 ℤ^d[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= [ [ 2, 1, 0, 0, 0, 1, 4 ],[127X[104X
    [4X[25X>[125X [27X         [ 2, 0, 0, 0, 0, 0, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 3, 0, 0, 0, 0, 0, 0 ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P41/G1/L1/V3/ext3"[110X belongs to
  the  factor  group of [22XS[122X modulo the normal subgroup [22XM(3 L)[122X, so we compute the
  action on an orbit modulo [22X9[122X.[133X
  
  [33X[0;0YThe  orbits  in this action are quite long. we choose a seed vector from the
  fixed space of an element of order [22X7[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xaa:= sgens[1];;[127X[104X
    [4X[25Xgap>[125X [27Xbb:= sgens[2];;[127X[104X
    [4X[25Xgap>[125X [27Xelm:= aa*bb;;[127X[104X
    [4X[25Xgap>[125X [27XOrder( elm );[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27Xfixed:= NullspaceMat( elm - aa^0 );[127X[104X
    [4X[28X[ [ 1, 1, 1, 1, 1, 1, 1, 0 ], [ -4, 1, 1, -5, -5, 2, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 9 );;[127X[104X
    [4X[25Xgap>[125X [27Xseed:= fun( fixed[2], aa^0 );[127X[104X
    [4X[28X[ 5, 1, 1, 4, 4, 2, 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, seed, fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P41/G1/L1/V3/ext3" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe translation lattice in the second example is the lattice [22XL = 6 ℤ^d[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= [ 6, 0, 0, 0, 0, 0, 0 ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P41/G1/L1/V4/ext3"[110X belongs to
  the  factor  group of [22XS[122X modulo the normal subgroup [22XM(6 L)[122X, so we compute the
  action on an orbit modulo [22X18[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 18 );;[127X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xseed:= fun( fixed[2], aa^0 );[127X[104X
    [4X[28X[ 14, 1, 1, 13, 13, 2, 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, seed, fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P41/G1/L1/V4/ext3" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-9 [33X[0;0YExample with point group [22XM_11[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  is  one example with [22Xd = 10[122X. The generators of the point group are as
  follows (see [HP89, p. 334]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= deletedPermutationMat( (1,9)(3,5)(7,11)(8,10), 11 );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= deletedPermutationMat( (1,4,3,2)(5,8,7,6), 11 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe vector system is [22XV_2[122X, and the translation lattice is [22XL = 2 ℤ^d[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= [ 0 * a[1],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P48/G1/L1/V2/ext2"[110X belongs to
  the  factor  group of [22XS[122X modulo the normal subgroup [22XM(2 L)[122X, so we compute the
  action on an orbit modulo [22X4[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P48/G1/L1/V2/ext2" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-10 [33X[0;0YExample with point group [22XU_3(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  is  one  example with [22Xd = 7[122X. The generators of the point group are as
  follows (see [HP89, p. 335]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= [ [ 0, 0,-1, 1, 0,-1, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0,-1, 1, 1,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 1,-1, 0, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 1, 0,-1, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [-1, 1, 1,-1, 0, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [-1, 0, 1,-1, 0, 0, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 0, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xb:= [ [ 0, 0, 0, 0, 0, 0, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0,-1, 1, 0,-1, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0,-1, 1, 1,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 1,-1, 0, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 1, 0,-1, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [-1, 1, 1,-1, 0, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [-1, 0, 1,-1, 0, 0, 1 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe vector system is [22XV_2[122X, and the translation lattice is [22XL = 3 ℤ^d[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= [ [ 2, 1, 0, 0, 2, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X         0 * b[1] ];;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 3, 0, 0, 0, 0, 0, 0 ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P49/G1/L1/V2/ext3"[110X belongs to
  the  factor  group of [22XS[122X modulo the normal subgroup [22XM(3 L)[122X, so we compute the
  action on an orbit modulo [22X9[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 9 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  orbits  in this action are quite long. we choose a seed vector from the
  fixed space of an element of order [22X12[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xaa:= sgens[1];;[127X[104X
    [4X[25Xgap>[125X [27Xbb:= sgens[2];;[127X[104X
    [4X[25Xgap>[125X [27Xelm:= aa*bb^4;;[127X[104X
    [4X[25Xgap>[125X [27XOrder( elm );[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27Xfixed:= NullspaceMat( elm - aa^0 );[127X[104X
    [4X[28X[ [ -1, -1, 1, 1, -1, -1, 1, 0 ], [ 0, -3, 1, 1, -1, -2, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xseed:= fun( fixed[2], aa^0 );[127X[104X
    [4X[28X[ 0, 6, 1, 1, 8, 7, 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, seed, fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P49/G1/L1/V2/ext3" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-11 [33X[0;0YExamples with point group [22XU_4(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YThere  are  two  examples  with  [22Xd = 6[122X. In both cases, the generators of the
  point group are as follows (see [HP89, p. 336]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= [ [ 0, 1, 0,-1,-1, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0,-1, 0, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0,-1, 0, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0,-1, 0, 0, 1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0, 0, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xb:= [ [ 0,-1, 0, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 1, 0,-1,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 1, 1, 0,-1 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 0, 0, 0,-1, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 0, 1, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X         [ 1, 0, 0, 0, 0, 0 ] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  both  examples,  the vector system is the trivial vector system [22XV_1[122X, and
  the translation lattice is the full lattice [22XL_1 = ℤ^d[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv:= List( [ 1, 2 ], i -> 0 * a[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= [ 1, 0, 0, 0, 0, 0 ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P50/G1/L1/V1/ext3"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(3 L_1)[122X, so we compute the
  action on an orbit modulo [22X3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P50/G1/L1/V1/ext3" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library  character table with identifier [10X"P50/G1/L1/V1/ext4"[110X belongs to
  the factor group of [22XS[122X modulo the normal subgroup [22XM(4 L_1)[122X, so we compute the
  action on an orbit modulo [22X4[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( sgens );;[127X[104X
    [4X[25Xgap>[125X [27Xfun:= multiplicationModulo( 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;[127X[104X
    [4X[25Xgap>[125X [27Xpermgens:= List( sgens, x -> Permutation( x, orb, fun ) );;[127X[104X
    [4X[25Xgap>[125X [27XverifyFactorGroup( permgens, "P50/G1/L1/V1/ext4" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.2-12 [33X[0;0YA remark on one of the example groups[133X[101X
  
  [33X[0;0YThe  (perfect)  character  table  with identifier [10X"P1/G2/L2/V2/ext4"[110X has the
  property that its character degrees are exactly the divisors of [22X60[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdegrees:= CharacterDegrees( CharacterTable( "P1/G2/L2/V2/ext4" ) );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 2, 2 ], [ 3, 2 ], [ 4, 2 ], [ 5, 1 ], [ 6, 5 ], [128X[104X
    [4X[28X  [ 10, 4 ], [ 12, 4 ], [ 15, 20 ], [ 20, 2 ], [ 30, 29 ], [ 60, 8 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( degrees, x -> x[1] ) = DivisorsInt( 60 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere  are  nilpotent  groups  with  the  same set of character degrees, for
  example  the  direct  product of four extraspecial groups of the orders [22X2^3[122X,
  [22X2^3[122X,  [22X3^3[122X,  and  [22X5^3[122X,  respectively.  This  phenomenon  has  been  described
  in [NR14].[133X
  
  
  [1X1.3 [33X[0;0YGenerality problems (December 2004/October 2015)[133X[101X
  
  [33X[0;0YThe  term  [21Xgenerality  problem[121X  is  used  for problems concerning consistent
  choices  of  conjugacy  classes  of  Brauer  tables  for  the same group, in
  different  characteristics.  The  definition  and some examples are given in
  [JLPW95, p. x].[133X
  
  [33X[0;0YSection  [14X1.3-1[114X  shows  how to detect generality problems and lists the known
  generality problems, and Section [14X1.3-2[114X gives an example that actually arose.[133X
  
  
  [1X1.3-1 [33X[0;0YListing possible generality problems[133X[101X
  
  [33X[0;0YWe  use  the  following  idea for finding character tables which may involve
  generality problems. (The functions shown in this section are based on [5XGAP[105X 3
  code that was originally written by Jürgen Müller.)[133X
  
  [33X[0;0YIf  the  [22Xp[122X-modular  Brauer  table  [22Xmtbl[122X,  say,  of  a group contributes to a
  generality  problem  then  some  choice of conjugacy classes is necessary in
  order  to  write  down  this  table,  in the sense that some symmetry of the
  corresponding ordinary table [22Xtbl[122X, say, is broken in [22Xmtbl[122X. This situation can
  be detected as follows. We assume that the class fusion from [22Xmtbl[122X to [22Xtbl[122X has
  been  fixed.  All  possible  class fusions are obtained as the orbit of this
  class  fusion  under  the actions of table automorphisms of [22Xtbl[122X, via mapping
  the  images  of  the  class  fusion  (with the function [2XOnTuples[102X ([14XReference:
  OnTuples[114X)),  and  of  the  table  automorphisms  of  [22Xmtbl[122X, via permuting the
  preimages.  The  case  of broken symmetries occurs if and only if this orbit
  splits  into  several orbits when only the action of the table automorphisms
  of  [22Xmtbl[122X  is  considered. Equivalently, symmetries are broken if and only if
  the  orbit  under table automorphisms of [22Xmtbl[122X is not closed under the action
  of table automorphisms of [22Xtbl[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBrokenSymmetries:= function( ordtbl, modtbl )[127X[104X
    [4X[25X>[125X [27X    local taut, maut, triv, fus, orb;[127X[104X
    [4X[25X>[125X [27X    taut:= AutomorphismsOfTable( ordtbl );[127X[104X
    [4X[25X>[125X [27X    maut:= AutomorphismsOfTable( modtbl );[127X[104X
    [4X[25X>[125X [27X    triv:= TrivialSubgroup( taut );[127X[104X
    [4X[25X>[125X [27X    fus:= GetFusionMap( modtbl, ordtbl );[127X[104X
    [4X[25X>[125X [27X    orb:= MakeImmutable( Set( OrbitFusions( maut, fus, triv ) ) );[127X[104X
    [4X[25X>[125X [27X    return ForAny( GeneratorsOfGroup( taut ),[127X[104X
    [4X[25X>[125X [27X               x -> ForAny( orb,[127X[104X
    [4X[25X>[125X [27X                        fus -> not OnTuples( fus, x ) in orb ) );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XRemark:[113X (Thanks to Klaus Lux for discussions on this topic.)[133X
  
  [30X    [33X[0;6YIt may happen that some symmetry [22Xσ_m[122X of a Brauer table does not belong
        to  a  symmetry  [22Xσ_o[122X of the corresponding ordinary table, in the sense
        that  permuting  the  preimage  classes  of a fusion [22Xf[122X between the two
        tables with [22Xσ_m[122X and permuting the image classes with [22Xσ_o[122X yields [22Xf[122X.[133X
  
        [33X[0;6YFor example, consider the group [22XG = 2.A_6.2_1[122X, the double cover of the
        symmetric  group  [22XS_6[122X  on six points. The [22X2[122X-modular Brauer table of [22XG[122X,
        which  is  essentially  equal to that of [22XS_6[122X, has a table automorphism
        group  order  two,  and  the  nonidentity  element in it swaps the two
        classes of element order three. The automorphism group of the ordinary
        character  table of [22XG[122X, however, fixes the two classes of element order
        three;  note  that exactly one of these classes possesses square roots
        in the [21Xouter half[121X [22XG ∖ G'[122X.[133X
  
        [33X[0;6YThus  it  is  not  sufficient  to compare the orbit of the fixed class
        fusion under the automorphisms of the ordinary table with the orbit of
        the same fusion under the automorphisms of the Brauer table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "2.A6.2_1" );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= t mod 2;;[127X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( m, t );[127X[104X
    [4X[28X[ 1, 4, 6, 9 ][128X[104X
    [4X[25Xgap>[125X [27XAutomorphismsOfTable( t );[127X[104X
    [4X[28XGroup([ (16,17), (14,15), (14,15)(16,17) ])[128X[104X
    [4X[25Xgap>[125X [27XAutomorphismsOfTable( m );[127X[104X
    [4X[28XGroup([ (2,3) ])[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X2.A6.2_1mod2[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  5  2  2  1[128X[104X
    [4X[28X     3  2  2  2  .[128X[104X
    [4X[28X     5  1  .  .  1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 3a 3b 5a[128X[104X
    [4X[28X    2P 1a 3a 3b 5a[128X[104X
    [4X[28X    3P 1a 1a 1a 5a[128X[104X
    [4X[28X    5P 1a 3a 3b 1a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1     1  1  1  1[128X[104X
    [4X[28XX.2     4  1 -2 -1[128X[104X
    [4X[28XX.3     4 -2  1 -1[128X[104X
    [4X[28XX.4    16 -2 -2  1[128X[104X
    [4X[25Xgap>[125X [27XDisplay( t );[127X[104X
    [4X[28X2.A6.2_1[128X[104X
    [4X[28X[128X[104X
    [4X[28X      2  5   5  4  2  2  2  2  3  1   1  4  4  3  2  2   2   2[128X[104X
    [4X[28X      3  2   2  .  2  2  2  2  .  .   .  1  1  .  1  1   1   1[128X[104X
    [4X[28X      5  1   1  .  .  .  .  .  .  1   1  .  .  .  .  .   .   .[128X[104X
    [4X[28X[128X[104X
    [4X[28X        1a  2a 4a 3a 6a 3b 6b 8a 5a 10a 2b 4b 8b 6c 6d 12a 12b[128X[104X
    [4X[28X     2P 1a  1a 2a 3a 3a 3b 3b 4a 5a  5a 1a 2a 4a 3a 3a  6b  6b[128X[104X
    [4X[28X     3P 1a  2a 4a 1a 2a 1a 2a 8a 5a 10a 2b 4b 8b 2b 2b  4b  4b[128X[104X
    [4X[28X     5P 1a  2a 4a 3a 6a 3b 6b 8a 1a  2a 2b 4b 8b 6d 6c 12b 12a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1      1   1  1  1  1  1  1  1  1   1  1  1  1  1  1   1   1[128X[104X
    [4X[28XX.2      1   1  1  1  1  1  1  1  1   1 -1 -1 -1 -1 -1  -1  -1[128X[104X
    [4X[28XX.3      5   5  1  2  2 -1 -1 -1  .   .  3 -1  1  .  .  -1  -1[128X[104X
    [4X[28XX.4      5   5  1  2  2 -1 -1 -1  .   . -3  1 -1  .  .   1   1[128X[104X
    [4X[28XX.5      5   5  1 -1 -1  2  2 -1  .   . -1  3  1 -1 -1   .   .[128X[104X
    [4X[28XX.6      5   5  1 -1 -1  2  2 -1  .   .  1 -3 -1  1  1   .   .[128X[104X
    [4X[28XX.7     16  16  . -2 -2 -2 -2  .  1   1  .  .  .  .  .   .   .[128X[104X
    [4X[28XX.8      9   9  1  .  .  .  .  1 -1  -1  3  3 -1  .  .   .   .[128X[104X
    [4X[28XX.9      9   9  1  .  .  .  .  1 -1  -1 -3 -3  1  .  .   .   .[128X[104X
    [4X[28XX.10    10  10 -2  1  1  1  1  .  .   .  2 -2  . -1 -1   1   1[128X[104X
    [4X[28XX.11    10  10 -2  1  1  1  1  .  .   . -2  2  .  1  1  -1  -1[128X[104X
    [4X[28XX.12     4  -4  . -2  2  1 -1  . -1   1  .  .  .  .  .   B  -B[128X[104X
    [4X[28XX.13     4  -4  . -2  2  1 -1  . -1   1  .  .  .  .  .  -B   B[128X[104X
    [4X[28XX.14     4  -4  .  1 -1 -2  2  . -1   1  .  .  .  A -A   .   .[128X[104X
    [4X[28XX.15     4  -4  .  1 -1 -2  2  . -1   1  .  .  . -A  A   .   .[128X[104X
    [4X[28XX.16    16 -16  . -2  2 -2  2  .  1  -1  .  .  .  .  .   .   .[128X[104X
    [4X[28XX.17    20 -20  .  2 -2  2 -2  .  .   .  .  .  .  .  .   .   .[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(3)-E(3)^2[128X[104X
    [4X[28X  = Sqrt(-3) = i3[128X[104X
    [4X[28XB = -E(12)^7+E(12)^11[128X[104X
    [4X[28X  = Sqrt(3) = r3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWhen considering several characteristics in parallel, one argues as follows.
  The  possible  class  fusions from a Brauer table [22Xmtbl[122X to its ordinary table
  [22Xtbl[122X  are  given by the orbit of a fixed class fusion under the action of the
  table  automorphisms of [22Xtbl[122X. If there are several orbits under the action of
  the automorphisms of [22Xmtbl[122X then we choose one orbit. Due to this choice, only
  those  table  automorphisms  of [22Xtbl[122X are admissible for other characteristics
  that  stabilize  the  chosen  orbit.  For the second characteristic, we take
  again  the  set of all class fusions from the Brauer table to [22Xtbl[122X, and split
  it  into orbits under the table automorphisms of the Brauer table. Now there
  are  two  possibilities.  Either  the  action  of the admissible subgroup of
  automorphisms of [22Xtbl[122X joins these orbits into one orbit or not. In the former
  case,  we  choose  again  one of the orbits, replace the group of admissible
  automorphisms  of  [22Xtbl[122X by the stabilizer of this orbit, and proceed with the
  next characteristic. In the latter case, we have found a generality problem,
  since  we  are  not free to choose an arbitrary class fusion from the set of
  possibilities.[133X
  
  [33X[0;0YThe  following  function  returns the set of primes which may be involved in
  generality  problems  for  the given ordinary character table. Note that the
  procedure  sketched  above  does not tell which characteristics are actually
  involved or which classes are affected by the choices; for example, we could
  argue   that   one  is  always  free  to  choose  a  fusion  for  the  first
  characteristics,  and that only the other ones cause problems. We return [13Xall[113X
  those  primes  [22Xp[122X for which broken symmetries between the [22Xp[122X-modular table and
  the ordinary table have been detected.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrimesOfGeneralityProblems:= function( ordtbl )[127X[104X
    [4X[25X>[125X [27X    local consider, p, modtbl, taut, triv, admiss, fusion, maut,[127X[104X
    [4X[25X>[125X [27X          allfusions, orbits, orbit, reps;[127X[104X
    [4X[25X>[125X [27X    # Find the primes for which symmetries are broken.[127X[104X
    [4X[25X>[125X [27X    consider:= [];[127X[104X
    [4X[25X>[125X [27X    for p in Filtered( PrimeDivisors( Size( ordtbl ) ), IsPrimeInt ) do[127X[104X
    [4X[25X>[125X [27X      modtbl:= ordtbl mod p;[127X[104X
    [4X[25X>[125X [27X      if modtbl <> fail and BrokenSymmetries( ordtbl, modtbl ) then[127X[104X
    [4X[25X>[125X [27X        Add( consider, p );[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X    # Compute the choices and detect generality problems.[127X[104X
    [4X[25X>[125X [27X    taut:= AutomorphismsOfTable( ordtbl );[127X[104X
    [4X[25X>[125X [27X    triv:= TrivialSubgroup( taut );[127X[104X
    [4X[25X>[125X [27X    admiss:= taut;[127X[104X
    [4X[25X>[125X [27X    for p in consider do[127X[104X
    [4X[25X>[125X [27X      modtbl:= ordtbl mod p;[127X[104X
    [4X[25X>[125X [27X      fusion:= GetFusionMap( modtbl, ordtbl );[127X[104X
    [4X[25X>[125X [27X      maut:= AutomorphismsOfTable( modtbl );[127X[104X
    [4X[25X>[125X [27X      # - We need not apply the action of 'maut' here,[127X[104X
    [4X[25X>[125X [27X      #   since 'maut' will later be used to get representatives.[127X[104X
    [4X[25X>[125X [27X      # - We need not apply all elements in 'taut' but only[127X[104X
    [4X[25X>[125X [27X      #   representatives of left cosets of 'admiss' in 'taut',[127X[104X
    [4X[25X>[125X [27X      #   since 'admiss' will later be used to get representatives.[127X[104X
    [4X[25X>[125X [27X      # allfusions:= OrbitFusions( maut, fusion, taut );[127X[104X
    [4X[25X>[125X [27X      allfusions:= Set( RightTransversal( taut, admiss ),[127X[104X
    [4X[25X>[125X [27X                        x -> OnTuples( fusion, x^-1 ) );[127X[104X
    [4X[25X>[125X [27X      # For computing representatives, 'RepresentativesFusions' is not[127X[104X
    [4X[25X>[125X [27X      # suitable because 'allfusions' is in generally not closed[127X[104X
    [4X[25X>[125X [27X      # under the actions.[127X[104X
    [4X[25X>[125X [27X      # reps:= RepresentativesFusions( maut, allfusions, admiss );[127X[104X
    [4X[25X>[125X [27X      orbits:= [];[127X[104X
    [4X[25X>[125X [27X      while not IsEmpty( allfusions ) do[127X[104X
    [4X[25X>[125X [27X        orbit:= OrbitFusions( maut, allfusions[1], admiss );[127X[104X
    [4X[25X>[125X [27X        Add( orbits, orbit );[127X[104X
    [4X[25X>[125X [27X        SubtractSet( allfusions, orbit );[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X      reps:= List( orbits, x -> x[1] );[127X[104X
    [4X[25X>[125X [27X      if Length( reps ) = 1 then[127X[104X
    [4X[25X>[125X [27X        # Reduce the symmetries that are still available.[127X[104X
    [4X[25X>[125X [27X        admiss:= Stabilizer( admiss,[127X[104X
    [4X[25X>[125X [27X                             Set( OrbitFusions( maut, fusion, triv ) ),[127X[104X
    [4X[25X>[125X [27X                             OnSetsTuples );[127X[104X
    [4X[25X>[125X [27X      else[127X[104X
    [4X[25X>[125X [27X        # We have found a generality problem.[127X[104X
    [4X[25X>[125X [27X        return consider;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X    # There is no generality problem for this table.[127X[104X
    [4X[25X>[125X [27X    return [];[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  us  look at a small example, the [22X5[122X-modular character table of the group
  [22X2.A_5.2[122X.  The  irreducible characters of degree [22X2[122X have the values [22X± sqrt{-2}[122X
  on  the  classes  [10X8a[110X and [10X8b[110X, and the values [22X± sqrt{-3}[122X on the classes [10X6b[110X and
  [10X6c[110X. When we define which of the two classes of element order [22X8[122X is called [10X8a[110X,
  this will also define which class is called [10X6b[110X. The ordinary character table
  does  not  relate  the  two  pairs of classes, there are table automorphisms
  which  interchange  each pair independently. This symmetry is thus broken in
  the [22X5[122X-modular character table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "2.A5.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= t mod 5;;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X2.A5.2mod5[128X[104X
    [4X[28X[128X[104X
    [4X[28X      2  4  4  3  2  2  2  3  3  2  2[128X[104X
    [4X[28X      3  1  1  .  1  1  1  .  .  1  1[128X[104X
    [4X[28X      5  1  1  .  .  .  .  .  .  .  .[128X[104X
    [4X[28X[128X[104X
    [4X[28X        1a 2a 4a 3a 6a 2b 8a 8b 6b 6c[128X[104X
    [4X[28X     2P 1a 1a 2a 3a 3a 1a 4a 4a 3a 3a[128X[104X
    [4X[28X     3P 1a 2a 4a 1a 2a 2b 8a 8b 2b 2b[128X[104X
    [4X[28X     5P 1a 2a 4a 3a 6a 2b 8b 8a 6c 6b[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1      1  1  1  1  1  1  1  1  1  1[128X[104X
    [4X[28XX.2      1  1  1  1  1 -1 -1 -1 -1 -1[128X[104X
    [4X[28XX.3      3  3 -1  .  .  1 -1 -1 -2 -2[128X[104X
    [4X[28XX.4      3  3 -1  .  . -1  1  1  2  2[128X[104X
    [4X[28XX.5      5  5  1 -1 -1  1 -1 -1  1  1[128X[104X
    [4X[28XX.6      5  5  1 -1 -1 -1  1  1 -1 -1[128X[104X
    [4X[28XX.7      2 -2  . -1  1  .  A -A  B -B[128X[104X
    [4X[28XX.8      2 -2  . -1  1  . -A  A -B  B[128X[104X
    [4X[28XX.9      4 -4  .  1 -1  .  .  .  B -B[128X[104X
    [4X[28XX.10     4 -4  .  1 -1  .  .  . -B  B[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(8)+E(8)^3[128X[104X
    [4X[28X  = Sqrt(-2) = i2[128X[104X
    [4X[28XB = E(3)-E(3)^2[128X[104X
    [4X[28X  = Sqrt(-3) = i3[128X[104X
    [4X[25Xgap>[125X [27XAutomorphismsOfTable( t );[127X[104X
    [4X[28XGroup([ (11,12), (9,10) ])[128X[104X
    [4X[25Xgap>[125X [27XAutomorphismsOfTable( m );[127X[104X
    [4X[28XGroup([ (7,8)(9,10) ])[128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( m, t );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 8, 9, 10, 11, 12 ][128X[104X
    [4X[25Xgap>[125X [27XBrokenSymmetries( t, m );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XBrokenSymmetries( t, t mod 2 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XBrokenSymmetries( t, t mod 3 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XPrimesOfGeneralityProblems( t );[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  no  symmetry is broken in the [22X2[122X- and [22X3[122X-modular character tables of [22XG[122X,
  there is no generality problem in this case.[133X
  
  [33X[0;0YFor  an example of a generality problem, we look at the smallest Janko group
  [22XJ_1[122X.  As  is  mentioned in [JLPW95, p. x], the unique irreducible [22X11[122X-modular
  Brauer character of degree [22X7[122X distinguishes the two (algebraically conjugate)
  classes  of  element  order  [22X5[122X. Since also the unique irreducible [22X19[122X-modular
  Brauer character of degree [22X22[122X distinguishes these classes, we have to choose
  these classes consistently.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "J1" );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= t mod 11;;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( m, rec( chars:= Filtered( Irr( m ), x -> x[1] = 7 ) ) );[127X[104X
    [4X[28XJ1mod11[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  3  3  1  1  1  1  .   1   1   .   .   .   .   .[128X[104X
    [4X[28X     3  1  1  1  1  1  1  .   .   .   1   1   .   .   .[128X[104X
    [4X[28X     5  1  1  1  1  1  .  .   1   1   1   1   .   .   .[128X[104X
    [4X[28X     7  1  .  .  .  .  .  1   .   .   .   .   .   .   .[128X[104X
    [4X[28X    11  1  .  .  .  .  .  .   .   .   .   .   .   .   .[128X[104X
    [4X[28X    19  1  .  .  .  .  .  .   .   .   .   .   1   1   1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 3a 5a 5b 6a 7a 10a 10b 15a 15b 19a 19b 19c[128X[104X
    [4X[28X    2P 1a 1a 3a 5b 5a 3a 7a  5b  5a 15b 15a 19b 19c 19a[128X[104X
    [4X[28X    3P 1a 2a 1a 5b 5a 2a 7a 10b 10a  5b  5a 19b 19c 19a[128X[104X
    [4X[28X    5P 1a 2a 3a 1a 1a 6a 7a  2a  2a  3a  3a 19b 19c 19a[128X[104X
    [4X[28X    7P 1a 2a 3a 5b 5a 6a 1a 10b 10a 15b 15a 19a 19b 19c[128X[104X
    [4X[28X   11P 1a 2a 3a 5a 5b 6a 7a 10a 10b 15a 15b 19a 19b 19c[128X[104X
    [4X[28X   19P 1a 2a 3a 5a 5b 6a 7a 10a 10b 15a 15b  1a  1a  1a[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     7 -1  1  A *A -1  .   B  *B   C  *C   D   E   F[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(5)+E(5)^4[128X[104X
    [4X[28X  = (-1+Sqrt(5))/2 = b5[128X[104X
    [4X[28XB = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4[128X[104X
    [4X[28X  = (3+Sqrt(5))/2 = 2+b5[128X[104X
    [4X[28XC = -2*E(5)-2*E(5)^4[128X[104X
    [4X[28X  = 1-Sqrt(5) = 1-r5[128X[104X
    [4X[28XD = -E(19)-E(19)^2-E(19)^3-E(19)^5-E(19)^7-E(19)^8-E(19)^11-E(19)^12-E\[128X[104X
    [4X[28X(19)^14-E(19)^16-E(19)^17-E(19)^18[128X[104X
    [4X[28XE = -E(19)^2-E(19)^3-E(19)^4-E(19)^5-E(19)^6-E(19)^9-E(19)^10-E(19)^13\[128X[104X
    [4X[28X-E(19)^14-E(19)^15-E(19)^16-E(19)^17[128X[104X
    [4X[28XF = -E(19)-E(19)^4-E(19)^6-E(19)^7-E(19)^8-E(19)^9-E(19)^10-E(19)^11-E\[128X[104X
    [4X[28X(19)^12-E(19)^13-E(19)^15-E(19)^18[128X[104X
    [4X[25Xgap>[125X [27Xm:= t mod 19;;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( m, rec( chars:= Filtered( Irr( m ), x -> x[1] = 22 ) ) );[127X[104X
    [4X[28XJ1mod19[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  3  3  1  1  1  1  .   1   1   .   .   .[128X[104X
    [4X[28X     3  1  1  1  1  1  1  .   .   .   .   1   1[128X[104X
    [4X[28X     5  1  1  1  1  1  .  .   1   1   .   1   1[128X[104X
    [4X[28X     7  1  .  .  .  .  .  1   .   .   .   .   .[128X[104X
    [4X[28X    11  1  .  .  .  .  .  .   .   .   1   .   .[128X[104X
    [4X[28X    19  1  .  .  .  .  .  .   .   .   .   .   .[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 3a 5a 5b 6a 7a 10a 10b 11a 15a 15b[128X[104X
    [4X[28X    2P 1a 1a 3a 5b 5a 3a 7a  5b  5a 11a 15b 15a[128X[104X
    [4X[28X    3P 1a 2a 1a 5b 5a 2a 7a 10b 10a 11a  5b  5a[128X[104X
    [4X[28X    5P 1a 2a 3a 1a 1a 6a 7a  2a  2a 11a  3a  3a[128X[104X
    [4X[28X    7P 1a 2a 3a 5b 5a 6a 1a 10b 10a 11a 15b 15a[128X[104X
    [4X[28X   11P 1a 2a 3a 5a 5b 6a 7a 10a 10b  1a 15a 15b[128X[104X
    [4X[28X   19P 1a 2a 3a 5a 5b 6a 7a 10a 10b 11a 15a 15b[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1    22 -2  1  A *A  1  1  -A -*A   .   B  *B[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(5)+E(5)^4[128X[104X
    [4X[28X  = (-1+Sqrt(5))/2 = b5[128X[104X
    [4X[28XB = -2*E(5)-2*E(5)^4[128X[104X
    [4X[28X  = 1-Sqrt(5) = 1-r5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that the degree [22X7[122X character above also distinguishes the three classes
  of element order [22X19[122X, and the same holds for the unique irreducible degree [22X31[122X
  character from characteristic [22X7[122X. Thus also the prime [22X7[122X occurs in the list of
  candidates for generality problems.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrimesOfGeneralityProblems( t );[127X[104X
    [4X[28X[ 7, 11, 19 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we list the candidates for generality problems from [5XGAP[105X's Character
  Table Library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlist:= [];;[127X[104X
    [4X[25Xgap>[125X [27XisGeneralityProblem:= function( ordtbl )[127X[104X
    [4X[25X>[125X [27X    local res;[127X[104X
    [4X[25X>[125X [27X    res:= PrimesOfGeneralityProblems( ordtbl );[127X[104X
    [4X[25X>[125X [27X    if res = [] then[127X[104X
    [4X[25X>[125X [27X      return false;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    Add( list, [ Identifier( ordtbl ), res ] );[127X[104X
    [4X[25X>[125X [27X    return true;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27XAllCharacterTableNames( IsDuplicateTable, false,[127X[104X
    [4X[25X>[125X [27X       isGeneralityProblem, true );;[127X[104X
    [4X[25Xgap>[125X [27XPrintArray( SortedList( list ) );[127X[104X
    [4X[28X[ [           (2.A4x2.G2(4)).2,            [ 2, 5, 7, 13 ] ],[128X[104X
    [4X[28X  [          (2^2x3).L3(4).2_1,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [               (2x12).L3(4),                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [              (4^2x3).L3(4),                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [                 (7:3xHe):2,               [ 5, 7, 17 ] ],[128X[104X
    [4X[28X  [                 (A5xA12):2,                   [ 2, 3 ] ],[128X[104X
    [4X[28X  [                 (D10xHN).2,     [ 2, 3, 5, 7, 11, 19 ] ],[128X[104X
    [4X[28X  [              (S3x2.Fi22).2,              [ 3, 11, 13 ] ],[128X[104X
    [4X[28X  [                     12.M22,            [ 2, 5, 7, 11 ] ],[128X[104X
    [4X[28X  [                   12.M22.2,            [ 2, 5, 7, 11 ] ],[128X[104X
    [4X[28X  [             12_1.L3(4).2_1,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [                 12_2.L3(4),                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [             12_2.L3(4).2_1,                [ 3, 5, 7 ] ],[128X[104X
    [4X[28X  [             12_2.L3(4).2_2,                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [             12_2.L3(4).2_3,                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [             2.(A4xG2(4)).2,            [ 2, 5, 7, 13 ] ],[128X[104X
    [4X[28X  [                   2.2E6(2),                 [ 13, 19 ] ],[128X[104X
    [4X[28X  [                 2.2E6(2).2,                 [ 13, 19 ] ],[128X[104X
    [4X[28X  [                      2.A10,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [                      2.A11,                [ 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                    2.A11.2,               [ 5, 7, 11 ] ],[128X[104X
    [4X[28X  [                      2.A12,             [ 2, 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                    2.A12.2,               [ 5, 7, 11 ] ],[128X[104X
    [4X[28X  [                      2.A13,         [ 2, 3, 5, 7, 11 ] ],[128X[104X
    [4X[28X  [                    2.A13.2,               [ 5, 7, 13 ] ],[128X[104X
    [4X[28X  [                  2.Alt(14),             [ 2, 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                  2.Alt(15),                [ 2, 5, 7 ] ],[128X[104X
    [4X[28X  [                  2.Alt(16),             [ 2, 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                  2.Alt(17),             [ 2, 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                  2.Alt(18),             [ 2, 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                        2.B,                 [ 17, 23 ] ],[128X[104X
    [4X[28X  [                    2.F4(2),           [ 2, 7, 13, 17 ] ],[128X[104X
    [4X[28X  [                   2.Fi22.2,                 [ 11, 13 ] ],[128X[104X
    [4X[28X  [                    2.G2(4),                   [ 2, 7 ] ],[128X[104X
    [4X[28X  [                  2.G2(4).2,               [ 5, 7, 13 ] ],[128X[104X
    [4X[28X  [                       2.HS,            [ 3, 5, 7, 11 ] ],[128X[104X
    [4X[28X  [                     2.HS.2,                  [ 3, 11 ] ],[128X[104X
    [4X[28X  [                2.L3(4).2_1,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [                       2.Ru,           [ 5, 7, 13, 29 ] ],[128X[104X
    [4X[28X  [                      2.Suz,               [ 2, 5, 11 ] ],[128X[104X
    [4X[28X  [                    2.Suz.2,               [ 3, 7, 13 ] ],[128X[104X
    [4X[28X  [                  2.Sym(15),                [ 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                  2.Sym(16),                [ 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                  2.Sym(17),                [ 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                  2.Sym(18),                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [                    2.Sz(8),               [ 2, 5, 13 ] ],[128X[104X
    [4X[28X  [                 2^2.2E6(2),                 [ 13, 19 ] ],[128X[104X
    [4X[28X  [               2^2.2E6(2).2,                 [ 13, 19 ] ],[128X[104X
    [4X[28X  [                 2^2.Fi22.2,              [ 3, 11, 13 ] ],[128X[104X
    [4X[28X  [              2^2.L3(4).2^2,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [              2^2.L3(4).2_1,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [                  2^2.Sz(8),               [ 2, 5, 13 ] ],[128X[104X
    [4X[28X  [                  2x2.F4(2),           [ 2, 7, 13, 17 ] ],[128X[104X
    [4X[28X  [                   2x3.Fi22,                [ 2, 3, 5 ] ],[128X[104X
    [4X[28X  [                   2x6.Fi22,                [ 2, 3, 5 ] ],[128X[104X
    [4X[28X  [                    2x6.M22,               [ 2, 5, 11 ] ],[128X[104X
    [4X[28X  [                   2xFi22.2,                 [ 11, 13 ] ],[128X[104X
    [4X[28X  [                     2xFi23,              [ 3, 17, 23 ] ],[128X[104X
    [4X[28X  [                     3.Fi22,                [ 2, 3, 5 ] ],[128X[104X
    [4X[28X  [                   3.Fi22.2,           [ 2, 5, 11, 13 ] ],[128X[104X
    [4X[28X  [                       3.J3,              [ 2, 17, 19 ] ],[128X[104X
    [4X[28X  [                     3.J3.2,           [ 2, 5, 17, 19 ] ],[128X[104X
    [4X[28X  [                3.L3(4).2_3,                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [              3.L3(4).3.2_3,                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [                  3.L3(7).2,               [ 3, 7, 19 ] ],[128X[104X
    [4X[28X  [                 3.L3(7).S3,               [ 3, 7, 19 ] ],[128X[104X
    [4X[28X  [                      3.McL,               [ 2, 5, 11 ] ],[128X[104X
    [4X[28X  [                    3.McL.2,            [ 2, 3, 5, 11 ] ],[128X[104X
    [4X[28X  [                       3.ON,       [ 3, 7, 11, 19, 31 ] ],[128X[104X
    [4X[28X  [                     3.ON.2,    [ 3, 5, 7, 11, 19, 31 ] ],[128X[104X
    [4X[28X  [                    3.Suz.2,               [ 2, 3, 13 ] ],[128X[104X
    [4X[28X  [                  3x2.F4(2),           [ 2, 7, 13, 17 ] ],[128X[104X
    [4X[28X  [                 3x2.Fi22.2,                 [ 11, 13 ] ],[128X[104X
    [4X[28X  [                  3x2.G2(4),                   [ 2, 7 ] ],[128X[104X
    [4X[28X  [                     3xFi23,              [ 3, 17, 23 ] ],[128X[104X
    [4X[28X  [                       3xJ1,              [ 7, 11, 19 ] ],[128X[104X
    [4X[28X  [                  3xL3(7).2,               [ 3, 7, 19 ] ],[128X[104X
    [4X[28X  [                     4.HS.2,               [ 5, 7, 11 ] ],[128X[104X
    [4X[28X  [                      4.M22,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [              4_1.L3(4).2_1,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [              4_2.L3(4).2_1,                [ 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                     6.Fi22,                [ 2, 3, 5 ] ],[128X[104X
    [4X[28X  [                   6.Fi22.2,           [ 2, 5, 11, 13 ] ],[128X[104X
    [4X[28X  [                6.L3(4).2_1,                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [                      6.M22,               [ 2, 5, 11 ] ],[128X[104X
    [4X[28X  [                    6.O7(3),               [ 3, 5, 13 ] ],[128X[104X
    [4X[28X  [                  6.O7(3).2,               [ 3, 5, 13 ] ],[128X[104X
    [4X[28X  [                      6.Suz,               [ 2, 5, 11 ] ],[128X[104X
    [4X[28X  [                    6.Suz.2,         [ 2, 3, 5, 7, 13 ] ],[128X[104X
    [4X[28X  [                  6x2.F4(2),           [ 2, 7, 13, 17 ] ],[128X[104X
    [4X[28X  [                        A12,                   [ 2, 3 ] ],[128X[104X
    [4X[28X  [                        A14,                [ 2, 5, 7 ] ],[128X[104X
    [4X[28X  [                        A17,                   [ 2, 7 ] ],[128X[104X
    [4X[28X  [                        A18,             [ 2, 3, 5, 7 ] ],[128X[104X
    [4X[28X  [                          B,         [ 13, 17, 23, 31 ] ],[128X[104X
    [4X[28X  [                        F3+,             [ 17, 23, 29 ] ],[128X[104X
    [4X[28X  [                      F3+.2,             [ 17, 23, 29 ] ],[128X[104X
    [4X[28X  [                     Fi22.2,                 [ 11, 13 ] ],[128X[104X
    [4X[28X  [                       Fi23,              [ 3, 17, 23 ] ],[128X[104X
    [4X[28X  [                         HN,           [ 2, 3, 11, 19 ] ],[128X[104X
    [4X[28X  [                       HN.2,           [ 5, 7, 11, 19 ] ],[128X[104X
    [4X[28X  [                         He,                  [ 5, 17 ] ],[128X[104X
    [4X[28X  [                       He.2,               [ 5, 7, 17 ] ],[128X[104X
    [4X[28X  [        Isoclinic(12.M22.2),            [ 2, 5, 7, 11 ] ],[128X[104X
    [4X[28X  [  Isoclinic(12_1.L3(4).2_1),                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [  Isoclinic(12_2.L3(4).2_1),                [ 3, 5, 7 ] ],[128X[104X
    [4X[28X  [  Isoclinic(12_2.L3(4).2_3),                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [         Isoclinic(2.A11.2),               [ 5, 7, 11 ] ],[128X[104X
    [4X[28X  [         Isoclinic(2.A12.2),               [ 5, 7, 11 ] ],[128X[104X
    [4X[28X  [         Isoclinic(2.A13.2),               [ 5, 7, 13 ] ],[128X[104X
    [4X[28X  [        Isoclinic(2.Fi22.2),                 [ 11, 13 ] ],[128X[104X
    [4X[28X  [       Isoclinic(2.G2(4).2),               [ 5, 7, 13 ] ],[128X[104X
    [4X[28X  [          Isoclinic(2.HS.2),                  [ 3, 11 ] ],[128X[104X
    [4X[28X  [          Isoclinic(2.HSx2),            [ 3, 5, 7, 11 ] ],[128X[104X
    [4X[28X  [     Isoclinic(2.L3(4).2_1),                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [         Isoclinic(2.Suz.2),               [ 3, 7, 13 ] ],[128X[104X
    [4X[28X  [   Isoclinic(4_1.L3(4).2_1),                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [   Isoclinic(4_2.L3(4).2_1),                [ 3, 5, 7 ] ],[128X[104X
    [4X[28X  [        Isoclinic(6.Fi22.2),           [ 2, 5, 11, 13 ] ],[128X[104X
    [4X[28X  [     Isoclinic(6.L3(4).2_1),                   [ 5, 7 ] ],[128X[104X
    [4X[28X  [       Isoclinic(6.O7(3).2),               [ 3, 5, 13 ] ],[128X[104X
    [4X[28X  [         Isoclinic(6.Suz.2),         [ 2, 3, 5, 7, 13 ] ],[128X[104X
    [4X[28X  [                         J1,              [ 7, 11, 19 ] ],[128X[104X
    [4X[28X  [                       J1x2,              [ 7, 11, 19 ] ],[128X[104X
    [4X[28X  [                         J3,              [ 2, 17, 19 ] ],[128X[104X
    [4X[28X  [                       J3.2,           [ 2, 5, 17, 19 ] ],[128X[104X
    [4X[28X  [                  L3(4).2_3,                   [ 3, 7 ] ],[128X[104X
    [4X[28X  [                L3(4).3.2_3,                [ 2, 3, 7 ] ],[128X[104X
    [4X[28X  [                    L3(7).2,               [ 3, 7, 19 ] ],[128X[104X
    [4X[28X  [                   L3(7).S3,               [ 3, 7, 19 ] ],[128X[104X
    [4X[28X  [                  L3(9).2_1,               [ 3, 7, 13 ] ],[128X[104X
    [4X[28X  [                    L5(2).2,               [ 2, 7, 31 ] ],[128X[104X
    [4X[28X  [                         Ly,              [ 7, 37, 67 ] ],[128X[104X
    [4X[28X  [                        M23,               [ 2, 3, 23 ] ],[128X[104X
    [4X[28X  [                         ON,       [ 3, 7, 11, 19, 31 ] ],[128X[104X
    [4X[28X  [                       ON.2,    [ 3, 5, 7, 11, 19, 31 ] ],[128X[104X
    [4X[28X  [                         Ru,           [ 5, 7, 13, 29 ] ],[128X[104X
    [4X[28X  [                  S3xFi22.2,                 [ 11, 13 ] ],[128X[104X
    [4X[28X  [                      Suz.2,                  [ 3, 13 ] ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote that this list may become longer as new Brauer tables become available.
  (For  example, the prime [22X2[122X was added to the entries for extensions of [22XF_4(2)[122X
  when the [22X2[122X-modular table of [22XF_4(2)[122X became available.)[133X
  
  
  [1X1.3-2 [33X[0;0YA generality problem concerning the group [22XJ_3[122X[101X[1X (April 2015)[133X[101X
  
  [33X[0;0YIn  March 2015, Klaus Lux reported an inconsistency in the character data of
  [5XGAP[105X:[133X
  
  [33X[0;0YThe  sporadic  simple  Janko  group  [22XJ_3[122X has a unique [22X19[122X-modular irreducible
  Brauer  character  of  degree [22X110[122X. In the character table that is printed in
  the  [5XAtlas[105X of Brauer characters [JLPW95, p. 219], the Brauer character value
  on the class [10X17A[110X is [22Xb_17[122X. The [5XAtlas[105X of Group Representations [WWT+] provides
  a  straight  line  program for computing class representatives of [22XJ_3[122X. If we
  compute  the  Brauer character value in question, we do not get [22Xb_17[122X but its
  algebraic conjugate, [22X-1-b_17[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "J3" );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= t mod 19;;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( Irr( m ), x -> x[1] = 110 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( cand );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xslp:= AtlasProgram( "J3", "classes" );;[127X[104X
    [4X[25Xgap>[125X [27X17a:= Position( slp.outputs, "17A" );[127X[104X
    [4X[28X18[128X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "J3", Characteristic, 19,[127X[104X
    [4X[25X>[125X [27X              Dimension, 110 );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info );;[127X[104X
    [4X[25Xgap>[125X [27Xreps:= ResultOfStraightLineProgram( slp.program,[127X[104X
    [4X[25X>[125X [27X              gens.generators );;[127X[104X
    [4X[25Xgap>[125X [27XQuadratic( BrauerCharacterValue( reps[ 17a ] ) );[127X[104X
    [4X[28Xrec( ATLAS := "-1-b17", a := -1, b := -1, d := 2, [128X[104X
    [4X[28X  display := "(-1-Sqrt(17))/2", root := 17 )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHow  shall  we  resolve  this  inconsistency, by replacing the straight line
  program  or  by  swapping  the  classes  [10X17A[110X and [10X17B[110X in the character table?
  Before we decide this, we look at related information.[133X
  
  [33X[0;0YThe  following  table  lists  the  [22Xp[122X-modular  irreducible characters of [22XJ_3[122X,
  according  to [JLPW95],  that can be used to define which of the two classes
  of  element order [22X17[122X shall be called [10X17A[110X; a [22X+[122X sign in the last column of the
  table  indicates  that the representation is available in the [5XAtlas[105X of Group
  Representations.[133X
  
      ┌────┬──────┬─────────┬─────────┬────────┐
      │  [22Xp[122X │ [22Xφ(1)[122X │  [22Xφ([122X[10X17A[110X[22X)[122X │  [22Xφ([122X[10X17B[110X[22X)[122X │ [5XAtlas[105X? │ 
      ├────┼──────┼─────────┼─────────┼────────┤
      ├────┼──────┼─────────┼─────────┼────────┤
      │  [22X2[122X │   [22X78[122X │  [22X1-b_17[122X │  [22X2+b_17[122X │   [22X+[122X    │ 
      │  [22X2[122X │   [22X80[122X │  [22X3-b_17[122X │  [22X4+b_17[122X │   [22X+[122X    │ 
      │  [22X2[122X │  [22X244[122X │  [22Xb_17-2[122X │ [22X-3-b_17[122X │   [22X+[122X    │ 
      │  [22X2[122X │  [22X966[122X │  [22Xr_17-3[122X │ [22X-3-r_17[122X │   [22X+[122X    │ 
      │ [22X19[122X │  [22X110[122X │    [22Xb_17[122X │ [22X-1-b_17[122X │   [22X+[122X    │ 
      │ [22X19[122X │  [22X214[122X │  [22X1-b_17[122X │  [22X2+b_17[122X │   [22X+[122X    │ 
      │ [22X19[122X │  [22X706[122X │   [22X-b_17[122X │  [22X1+b_17[122X │   [22X+[122X    │ 
      │ [22X19[122X │ [22X1214[122X │ [22X-1+b_17[122X │ [22X-2-b_17[122X │   [22X-[122X    │ 
      └────┴──────┴─────────┴─────────┴────────┘
  
       [1XTable:[101X Representations of [22XJ_3[122X that may define [10X17A[110X
  
  
  [33X[0;0YNote  that  the irreducible Brauer characters in characteristic [22X3[122X and [22X5[122X that
  distinguish  the  two classes [10X17A[110X and [10X17B[110X occur in pairs of Galois conjugate
  characters.[133X
  
  [33X[0;0YThe  following  computations  show  that  the given straight line program is
  compatible  with  the  four  characters  in  characteristic  [22X2[122X  but  is  not
  compatible with the three available characters in characteristic [22X19[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtable:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor pair in [ [  2, [ 78, 80, 244, 966 ] ],[127X[104X
    [4X[25X>[125X [27X                 [ 19, [ 110, 214, 706 ] ] ] do[127X[104X
    [4X[25X>[125X [27X     p:= pair[1];[127X[104X
    [4X[25X>[125X [27X     for d in pair[2] do[127X[104X
    [4X[25X>[125X [27X       info:= OneAtlasGeneratingSetInfo( "J3", Characteristic, p,[127X[104X
    [4X[25X>[125X [27X                  Dimension, d );[127X[104X
    [4X[25X>[125X [27X       gens:= AtlasGenerators( info );[127X[104X
    [4X[25X>[125X [27X       reps:= ResultOfStraightLineProgram( slp.program,[127X[104X
    [4X[25X>[125X [27X                  gens.generators );[127X[104X
    [4X[25X>[125X [27X       val:= BrauerCharacterValue( reps[ 17a ] );[127X[104X
    [4X[25X>[125X [27X       Add( table, [ p, d, Quadratic( val ).ATLAS,[127X[104X
    [4X[25X>[125X [27X                           Quadratic( StarCyc( val ) ).ATLAS ] );[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XPrintArray( table );[127X[104X
    [4X[28X[ [       2,      78,   1-b17,   2+b17 ],[128X[104X
    [4X[28X  [       2,      80,   3-b17,   4+b17 ],[128X[104X
    [4X[28X  [       2,     244,  -2+b17,  -3-b17 ],[128X[104X
    [4X[28X  [       2,     966,  -3+r17,  -3-r17 ],[128X[104X
    [4X[28X  [      19,     110,  -1-b17,     b17 ],[128X[104X
    [4X[28X  [      19,     214,   2+b17,   1-b17 ],[128X[104X
    [4X[28X  [      19,     706,   1+b17,    -b17 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  the problem is an inconsistency between the [22X2[122X-modular and the
  [22X19[122X-modular  character  table of [22XJ_3[122X in [JLPW95]. In particular, changing the
  straight line program would not help to resolve the problem.[133X
  
  [33X[0;0YHow  shall we proceed in order to fix the problem? We can decide to keep the
  [22X19[122X-modular  table of [22XJ_3[122X, and to swap the two classes of element order [22X17[122X in
  the  [22X2[122X-modular table; then also the straight line program has to be changed,
  and the classes of element orders [22X17[122X and [22X51[122X in the [22X2[122X-modular character table
  of  the triple cover [22X3.J_3[122X of [22XJ_3[122X have to be adjusted. Alternatively, we can
  keep  the  [22X2[122X-modular  table of [22XJ_3[122X and the straight line program, and adjust
  the  conjugacy  classes  of element orders divisible by [22X17[122X in the [22X19[122X-modular
  character tables of [22XJ_3[122X, [22X3.J_3[122X, [22XJ_3.2[122X, and [22X3.J_3.2[122X.[133X
  
  [33X[0;0YWe  decide  to  change  the  [22X19[122X-modular  character  tables.  Note that these
  character  tables ---or equivalently, the corresponding Brauer trees--- have
  been  described in [HL89], where explicit choices are mentioned that lead to
  the  shown  Brauer trees. Questions about the consistency with Brauer tables
  in  other  characteristic  had  not  been  an  issue in this book. (Only the
  consistency  of the Brauer trees among the [22X19[122X-blocks of [22X3.J_3[122X is mentioned.)
  In fact, the book mentions that the [22X19[122X-modular Brauer trees for [22XJ_3[122X had been
  computed already by W. Feit. The inconsistency of Brauer character tables in
  different  characteristic  has  apparently  been  overlooked  when  the data
  for [JLPW95] have been put together, and had not been detected until now.[133X
  
  [33X[0;0Y[13XRemarks:[113X[133X
  
  [30X    [33X[0;6YSuch  a  change  of  a  Brauer  table  can in general affect the class
        fusions  from  and  to  this table. Note that Brauer tables may impose
        conditions on the choice of the fusion among possible fusions that are
        equivalent  w. r. t. the table automorphisms of the ordinary table. In
        this  particular  case, in fact no class fusion had to be changed, see
        the sections [14X9.6-1[114X and Section [14X9.6-3[114X.[133X
  
  [30X    [33X[0;6YThe change of the character tables affects the decomposition matrices.
        Thus  the  PDF  files containing the [22X19[122X-modular decomposition matrices
        had             to             be             updated,             see
        [7Xhttp://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/dec/tex/J3/index.html[107X.[133X
  
  [30X    [33X[0;6YJürgen  Müller  has  checked  that the conjugacy classes of all Brauer
        tables  of  [22XJ_3[122X,  [22X3.J_3[122X,  [22XJ_3.2[122X,  [22X3.J_3.2[122X are consistent after the fix
        described above.[133X
  
  
  [1X1.3-3 [33X[0;0YA generality problem concerning the group [22XHN[122X[101X[1X (August 2022)[133X[101X
  
  [33X[0;0YThe classes [10X20A[110X, [10X20B[110X of the Harada-Norton group [22XHN[122X in the [22X11[122X- and [22X19[122X-modular
  character  tables  are  determined  by  unique  Brauer  characters that have
  different values on these classes. Once we have [13Xdefined[113X these classes in one
  characteristic,  the  two  Brauer  characters  tell  us  how  to [13Xchoose[113X them
  consistently  in  the other characteristic. Thus the question is whether the
  two Brauer tables are consistent w.r.t. this property or not.[133X
  
  [33X[0;0Y(Note  that  this  question  can  be  answered  independently  of  all other
  questions  of  this  kind for [22XHN[122X, because the permutation that swaps exactly
  the  classes  [10X20A[110X  and [10X20B[110X is a table automorphism of the ordinary character
  table of [22XHN[122X.)[133X
  
  [33X[0;0YWe  start  with  the  ordinary  character table of [22XHN[122X. There are exactly two
  ordinary  irreducible  characters  that take different values on the classes
  [10X20A[110X, [10X20B[110X, these are [22Xχ_51[122X and [22Xχ_52[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "HN" );;[127X[104X
    [4X[25Xgap>[125X [27Xpos20:= Positions( OrdersClassRepresentatives( t ), 20 );[127X[104X
    [4X[28X[ 39, 40, 41, 42, 43 ][128X[104X
    [4X[25Xgap>[125X [27Xdiff:= Filtered( Irr( t ), x -> x[39] <> x[40] );;[127X[104X
    [4X[25Xgap>[125X [27XList( diff, x -> Position( Irr( t ), x ) );[127X[104X
    [4X[28X[ 51, 52 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThese  values  are  irrational and lie in the field that is generated by the
  square root of [22X5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( diff, x -> List( x{ [ 1, 39, 40 ] },[127X[104X
    [4X[25X>[125X [27X                          CTblLib.StringOfAtlasIrrationality ) );[127X[104X
    [4X[28X[ [ "5103000", "2r5+1", "-2r5+1" ], [ "5103000", "-2r5+1", "2r5+1" ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  each  of the characteristics [22Xp ∈ { 11, 19 }[122X, the [22Xp[122X-modular reductions of
  [22Xχ_51[122X  and [22Xχ_52[122X decompose differently into irreducibles. Note that the Galois
  automorphism  of the ordinary character table that maps [22Xsqrt5[122X to [22X-sqrt5[122X does
  not live in the [22X11[122X- and [22X19[122X-modular Brauer tables.[133X
  
  [33X[0;0YFor [22Xp = 11[122X, the reduction of [22Xχ_51[122X is [22Xφ_40 + φ_48[122X, with [22Xφ_40(1) = 1575176[122X and
  [22Xφ_48(1)  = 3527824[122X, and the reduction of [22Xχ_52[122X is [22Xφ_39 + φ_49[122X, with [22Xφ_39(1) =
  1361919[122X and [22Xφ_49(1) = 3741081[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt11:= t mod 11;;[127X[104X
    [4X[25Xgap>[125X [27Xrest11:= RestrictedClassFunctions( diff, t11 );;[127X[104X
    [4X[25Xgap>[125X [27Xdec11:= Decomposition( Irr( t11 ), rest11, "nonnegative" );;[127X[104X
    [4X[25Xgap>[125X [27XList( dec11, Set );[127X[104X
    [4X[28X[ [ 0, 1 ], [ 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( dec11, x -> Positions( x, 1 ) );[127X[104X
    [4X[28X[ [ 40, 48 ], [ 39, 49 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( Irr( t11 ){ [ 40, 48 ] }, x -> x[1] );[127X[104X
    [4X[28X[ 1575176, 3527824 ][128X[104X
    [4X[25Xgap>[125X [27XList( Irr( t11 ){ [ 39, 49 ] }, x -> x[1] );[127X[104X
    [4X[28X[ 1361919, 3741081 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor [22Xp = 19[122X, the reduction of [22Xχ_51[122X is [22Xφ_42 + φ_45[122X, with [22Xφ_42(1) = 2125925[122X and
  [22Xφ_45(1)  = 2977075[122X, and the reduction of [22Xχ_52[122X is [22Xφ_33 + φ_48[122X, with [22Xφ_33(1) =
  1197330[122X and [22Xφ_48(1) = 3905670[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt19:= t mod 19;;[127X[104X
    [4X[25Xgap>[125X [27Xrest19:= RestrictedClassFunctions( diff, t19 );;[127X[104X
    [4X[25Xgap>[125X [27Xdec19:= Decomposition( Irr( t19 ), rest19, "nonnegative" );;[127X[104X
    [4X[25Xgap>[125X [27XList( dec19, Set );[127X[104X
    [4X[28X[ [ 0, 1 ], [ 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( dec19, x -> Positions( x, 1 ) );[127X[104X
    [4X[28X[ [ 42, 45 ], [ 33, 48 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( Irr( t19 ){ [ 42, 45 ] }, x -> x[1] );[127X[104X
    [4X[28X[ 2125925, 2977075 ][128X[104X
    [4X[25Xgap>[125X [27XList( Irr( t19 ){ [ 33, 48 ] }, x -> x[1] );[127X[104X
    [4X[28X[ 1197330, 3905670 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe Frobenius-Schur indicators of all involved [22Xp[122X-modular constituents are [22X+[122X.
  This  implies  that  [22Xχ_51[122X  reduces  orthogonally  stably  modulo  [22X11[122X but not
  orthogonally  stably  modulo  [22X19[122X,  whereas  [22Xχ_52[122X reduces orthogonally stably
  modulo [22X19[122X but not orthogonally stably modulo [22X11[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIndicator( t11, 2 ){ [ 39, 40, 48, 49 ] };[127X[104X
    [4X[28X[ 1, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XIndicator( t19, 2 ){ [ 33, 42, 45, 48 ] };[127X[104X
    [4X[28X[ 1, 1, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  version  up  to 1.3.4 of the character table library, this condition was
  not  satisfied: The reduction of [22Xχ_51[122X modulo both [22X11[122X and [22X19[122X was orthogonally
  stable, and the reduction of [22Xχ_52[122X modulo both [22X11[122X and [22X19[122X was not orthogonally
  stable.  However,  this  cannot  happen,  due  to  theoretical results about
  orthogonal  discriminants  of  the involved characters. Thus we have found a
  way  to decide the consistency of the classes [10X20A[110X and [10X20B[110X in characteristics
  [22X11[122X and [22X19[122X: Either the [22X11[122X-modular character table or the [22X19[122X-modular character
  table of [22XHN[122X had to be changed for version 1.3.5, by swapping the classes [10X20A[110X
  and [10X20B[110X.[133X
  
  [33X[0;0YWe  decided  to  change  the  [22X11[122X-modular  table,  because there are no other
  generality  problems for [22XHN[122X involving the [22X19[122X-modular table, and hence we are
  sure that this table will not need to be changed because of new solutions to
  generality problems.[133X
  
  [33X[0;0YFor  [22XHN.2[122X,  the  situation is similar. There are additionally two classes of
  element order [22X40[122X that have to be swapped if [10X20A[110X and [10X20B[110X get swapped. Thus we
  have to change the [22X11[122X-modular table of [22XHN.2[122X accordingly.[133X
  
  [33X[0;0YChanges  of  this  kind  may  affect  also  derived  character tables in the
  library. In this case, the [22X11[122X-modular table with identifier [10X"(D10xHN).2"[110X was
  changed  as  well. Note that this table is not stored explicitly in the data
  files,  it  gets  constructed  from  ordinary and modular library tables via
  [2XConstructIndexTwoSubdirectProduct[102X                                  ([14XCTblLib:
  ConstructIndexTwoSubdirectProduct[114X).[133X
  
  
  [1X1.4 [33X[0;0YBrauer Tables that can be derived from Known Tables[133X[101X
  
  [33X[0;0YIn  a few situations, one can derive the [22Xp[122X-modular Brauer character table of
  a group from known character theoretic information.[133X
  
  [33X[0;0YFor  quite  some time, a method is available in [5XGAP[105X that computes the Brauer
  characters   of   [22Xp[122X-solvable   groups   (see  [14X'Reference:  BrauerTable'[114X  and
  [14X'Reference: IsPSolubleCharacterTable'[114X).[133X
  
  [33X[0;0YThe  following  sections  list  other  situations where Brauer tables can be
  computed by [5XGAP[105X.[133X
  
  
  [1X1.4-1 [33X[0;0YBrauer Tables via Construction Information[133X[101X
  
  [33X[0;0YIf  a given ordinary character table [22Xt[122X, say, has been constructed from other
  ordinary  character  tables  then  [5XGAP[105X  may  be able to create the [22Xp[122X-modular
  Brauer  table of [22Xt[122X from the [22Xp[122X-modular Brauer tables of the [21Xingredients[121X. This
  happens currently in the following cases.[133X
  
  [30X    [33X[0;6Y[22Xt[122X  has  been  constructed with [2XCharacterTableDirectProduct[102X ([14XReference:
        CharacterTableDirectProduct[114X), and [5XGAP[105X can compute the [22Xp[122X-modular Brauer
        tables of the direct factors.[133X
  
  [30X    [33X[0;6Y[22Xt[122X   has  been  constructed  with  [2XCharacterTableIsoclinic[102X  ([14XReference:
        CharacterTableIsoclinic[114X),  and  [5XGAP[105X  can  compute the [22Xp[122X-modular Brauer
        table  of  the table that is stored in [22Xt[122X as the value of the attribute
        [2XSourceOfIsoclinicTable[102X ([14XReference: SourceOfIsoclinicTable[114X).[133X
  
  [30X    [33X[0;6Y[22Xt[122X   has   the   attribute   [2XConstructionInfoCharacterTable[102X   ([14XCTblLib:
        ConstructionInfoCharacterTable[114X)  set,  the first entry of this list [22Xl[122X,
        say,   is   one   of   the   strings   [10X"ConstructGS3"[110X   (see   [14X2.3-2[114X),
        [10X"ConstructIndexTwoSubdirectProduct"[110X  (see  [14X2.3-6[114X), [10X"ConstructMGA"[110X (see
        [14X2.3-1[114X),  [10X"ConstructPermuted"[110X,  [10X"ConstructV4G"[110X (see [14X2.3-4[114X), and [5XGAP[105X can
        construct  the  [22Xp[122X-modular  Brauer  table(s)  of  the relevant ordinary
        character table(s), which are library tables whose names occur in [22Xl[122X.[133X
  
  
  [1X1.4-2 [33X[0;0YLiftable Brauer Characters (May 2017)[133X[101X
  
  [33X[0;0YLet  [22XB[122X  be a [22Xp[122X-block of cyclic defect for the finite group [22XG[122X. It can be read
  off  from the set Irr[22X(B)[122X of ordinary irreducible characters of [22XB[122X whether all
  irreducible  Brauer  characters in [22XB[122X are restrictions of ordinary characters
  to the [22Xp[122X-regular classes of [22XG[122X, as follows.[133X
  
  [33X[0;0YIf  [22XB[122X has only one irreducible Brauer character then all ordinary characters
  in  [22XB[122X restrict to this Brauer character. So let us assume that [22XB[122X contains at
  least  two  irreducible  Brauer  characters, and consider the set [22XS[122X, say, of
  restrictions of Irr[22X(B)[122X to the [22Xp[122X-regular classes of [22XG[122X.[133X
  
  [33X[0;0YThe  block [22XB[122X contains exactly [22X|S| - 1[122X irreducible Brauer characters, and the
  decomposition  of  the  characters  in  [22XS[122X  into  these  Brauer characters is
  described  by  an  [22X|S|[122X  by [22X|S| - 1[122X matrix [22XM[122X, say, whose entries are zero and
  one,  such  that  exactly two nonzero entries occur in each column. (See for
  example [HL89, Theorem 2.1.5], which refers to [Dad66].)[133X
  
  [33X[0;0YIf  all  irreducible  Brauer  characters  of  [22XB[122X occur in [22XS[122X then the matrix [22XM[122X
  contains  [22X|S|  -  1[122X  rows  that contain exactly one nonzero entry, hence the
  remaining  row  consists  only of [22X1[122Xs. This means that the element of largest
  degree  in  [22XS[122X is equal to the sum of all other elements in [22XS[122X. Conversely, if
  the element of largest degree in [22XS[122X is equal to the sum of all other elements
  in  [22XS[122X  then  the  matrix  [22XM[122X  has  the  structure  as stated above, hence all
  irreducible Brauer characters of [22XB[122X occur in [22XS[122X.[133X
  
  [33X[0;0YAlternatively,  one  could state that all irreducible Brauer characters of [22XB[122X
  are  restricted ordinary characters if and only if the Brauer tree of [22XB[122X is a
  [13Xstar[113X  (see  [HL89,  p.  2].  If  [22XB[122X  contains at least two irreducible Brauer
  characters  then  this happens if and only if one of the types [22X×[122X or [22X∘[122X occurs
  for exactly one node in the Brauer graph of [22XB[122X, see [HL89, Lemma 2.1.13], and
  the distribution to types is determined by Irr[22X(B)[122X.[133X
  
  [33X[0;0YThe  default  method  for  [2XBrauerTableOp[102X  ([14XReference: BrauerTableOp[114X) that is
  contained  in the [5XGAP[105X library has been extended in version 4.11 such that it
  checks whether the Sylow [22Xp[122X-subgroups of the given group [22XG[122X are cyclic and, if
  yes,  whether  all  [22Xp[122X-blocks  of  [22XG[122X have the property discussed above. (This
  feature arose from a discussion with Klaus Lux.)[133X
  
  [33X[0;0YExamples  where  this  method is successful for all blocks are the [22Xp[122X-modular
  character tables of the groups PSL[22X(2, q)[122X, where [22Xp[122X is odd and does not divide
  [22Xq[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( PSL( 2, 11 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmodt:= t mod 5;;[127X[104X
    [4X[25Xgap>[125X [27Xmodt <> fail;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XInfoText( modt );[127X[104X
    [4X[28X"computed using that all Brauer characters lift to char. zero"[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAnother such example is the [22X5[122X-modular table of the Mathieu group [22XM_11[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlib:= CharacterTable( "M11" );;[127X[104X
    [4X[25Xgap>[125X [27Xfromgroup:= CharacterTable( MathieuGroup( 11 ) );;[127X[104X
    [4X[25Xgap>[125X [27XDecompositionMatrix( lib mod 5 );[127X[104X
    [4X[28X[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 0, 1, 0, 0 ], [ 1, 0, 0, 0, 1, 1, 1, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfromgroup mod 5 <> fail;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere   are   cases   where  all  Brauer  characters  of  a  block  lift  to
  characteristic  zero  but  the defect group of the block is not cyclic, thus
  the  method cannot be used. An example is the [22X2[122X-modular table of the Mathieu
  group [22XM_11[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDecompositionMatrix( lib mod 2 );[127X[104X
    [4X[28X[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 1, 0, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ], [ 1, 0, 0, 0, 1 ], [128X[104X
    [4X[28X  [ 1, 1, 0, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfromgroup mod 2;[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  
  [1X1.5 [33X[0;0YInformation about certain subgroups of the Monster group[133X[101X
  
  
  [1X1.5-1  [33X[0;0YThe  Monster  group  does  not contain subgroups of the type [22X2.U_4(2)[122X[101X[1X[101X
  [1X(August 2023)[133X[101X
  
  [33X[0;0YIn  the  context  of  a question about decomposition numbers of the sporadic
  simple  Monster  group  [22XMM[122X,  Benjamin  Sambale  was  interested  in possible
  embeddings  of certain groups [22XG[122X into [22XMM[122X such that the decomposition matrices
  of  [22XG[122X are known. For a given [22XG[122X, the first steps were to compute the possible
  class  fusions  of  [22XG[122X  in  [22XMM[122X  and  then  to check whether the corresponding
  embeddings would be interesting.[133X
  
  [33X[0;0YApparently,  calling  [2XPossibleClassFusions[102X ([14XReference: PossibleClassFusions[114X)
  with its default parameters often runs very long and requires a lot of space
  when  [22XG[122X  is  a small group such as [22X2.U_4(2)[122X. We can do better by calling the
  function  with the parameter [10Xdecompose:= false[110X. This has the effect that one
  criterion   is   omitted  that  checks  the  decomposability  of  restricted
  characters  of  [22XMM[122X  as  an  integral linear combination of characters of the
  subgroup.  As  a  rule of thumb, if the number of classes of the subgroup is
  small  compared  to  the  number  of  classes of the group and if the result
  consists   of   many  candidates  then  it  might  be  faster  to  omit  the
  decomposability criterion.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "2.U4(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xsfusm:= PossibleClassFusions( s, m, rec( decompose:= false ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( sfusm );[127X[104X
    [4X[28X2332[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLooking at the (many) candidates, we see that all map the central involution
  of  [22X2.U_4(2)[122X  to  the class [10X2B[110X of [22XMM[122X, thus any subgroup of the type [22X2.U_4(2)[122X
  lies  inside  the [10X2B[110X normalizer in [22XMM[122X. We compute the possible class fusions
  into this subgroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSet( List( sfusm, x -> x[2] ) );[127X[104X
    [4X[28X[ 3 ][128X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "MN2B" );[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1" )[128X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t, rec( decompose:= false ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( sfust );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus we have shown that [22XMM[122X does not contain subgroups of the type [22X2.U_4(2)[122X.[133X
  
  
  [1X1.5-2 [33X[0;0YPerfect central extensions of [22XL_3(4)[122X[101X[1X (August 2023)[133X[101X
  
  [33X[0;0YThere         was         the         question        in        MathOverflow
  ([7Xhttps://mathoverflow.net/questions/450255[107X) which perfect central extensions
  of  the simple group [22XG = L_3(4)[122X are subgroups of the sporadic simple Monster
  group [22XMM[122X.[133X
  
  [33X[0;0YFirst we get the list of perfect central extensions of [22XG[122X (asuming that their
  character tables are contained in the character table library).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsimp:= CharacterTable( "L3(4)" );;[127X[104X
    [4X[25Xgap>[125X [27Xextnames:= AllCharacterTableNames( Identifier,[127X[104X
    [4X[25X>[125X [27X                  x -> EndsWith(x, "L3(4)" ) );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= List( extnames, CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= Filtered( ext, x -> Length( ClassPositionsOfCentre( x ) ) =[127X[104X
    [4X[25X>[125X [27X                             Size( x ) / Size( simp ) );;[127X[104X
    [4X[25Xgap>[125X [27XSortBy( ext, Size );[127X[104X
    [4X[25Xgap>[125X [27Xnames:= List( ext, Identifier );[127X[104X
    [4X[28X[ "L3(4)", "2.L3(4)", "3.L3(4)", "2^2.L3(4)", "4_1.L3(4)", [128X[104X
    [4X[28X  "4_2.L3(4)", "6.L3(4)", "(2x4).L3(4)", "(2^2x3).L3(4)", [128X[104X
    [4X[28X  "12_1.L3(4)", "12_2.L3(4)", "4^2.L3(4)", "(2x12).L3(4)", [128X[104X
    [4X[28X  "(4^2x3).L3(4)" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe fact that [22XG[122X is [13Xnot[113X isomorphic to a subgroup of [22XMM[122X is shown in [HW08] (at
  the end of this paper).[133X
  
  [33X[0;0YAnd  the  following  embeddings  of  central  extensions  of  [22XG[122X in [22XMM[122X can be
  established using known subgroups of [22XMM[122X.[133X
  
  [30X    [33X[0;6Y[22X2.G < 2.U_4(3) < 2^2.U_6(2) < Fi_23 < 3.Fi_24' < MM[122X.[133X
  
  [30X    [33X[0;6Y[22X2^2.G < He < 3.Fi_24' < MM[122X.[133X
  
  [30X    [33X[0;6Y[22X6.G < 2.G_2(4) < 6.Suz < 3^1+12_+.2Suz < MM[122X.[133X
  
  [33X[0;0YNote  that  [22XG[122X is a subgroup of [22XU_4(3)[122X but not of [22X2.U_4(3)[122X, [22X3.G[122X is a subgroup
  of  [22XG_2(4)[122X  but  not of [22X2.G_2(4)[122X, and [22XG_2(4)[122X is a subgroup of [22XSuz[122X but not of
  [22X2.Suz[122X. The positive statements follow from [CCN+85, pp. 52, 97, 131] and the
  negative ones from the following computations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "L3(4)" ),[127X[104X
    [4X[25X>[125X [27X                                 CharacterTable( "2.U4(3)" ) ) );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "3.L3(4)" ),[127X[104X
    [4X[25X>[125X [27X                                 CharacterTable( "2.G2(4)" ) ) );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "G2(4)" ),[127X[104X
    [4X[25X>[125X [27X                                 CharacterTable( "2.Suz" ) ) );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe group [22X3.G[122X centralizes an element of order three. If [22X3.G[122X is a subgroup of
  [22XMM[122X  then  it is contained in a [10X3A[110X centralizer (of the structure [22X3.Fi_24'[122X), a
  [10X3B[110X  centralizer (of the structure [22X3^1+12_+.2Suz[122X) or a [10X3C[110X centralizer (of the
  structure  [22X3  × Th[122X). Clearly the case [22X3C[122X cannot occur, and [10X3B[110X is excluded by
  the   fact   that  no  class  fusion  between  [22X3.G[122X  and  the  [22X3B[122X  normalizer
  [22X3^1+12_+.2Suz.2[122X is possible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "MN3B" );[127X[104X
    [4X[28XCharacterTable( "3^(1+12).2.Suz.2" )[128X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "3.L3(4)" ), t ) );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf [22X3.G[122X is contained in the [10X3A[110X centralizer then this embedding induces one of
  [22XG[122X  into some maximal subgroup of [22XFi_24'[122X. Using the known character tables of
  these  maximal  subgroups  in  GAP's character table library, one shows that
  only  [22XFi_23[122X  admits  a class fusion, but this subgroup lifts to [22X3 × Fi_23[122X in
  [22X3.Fi_24'[122X and thus cannot lead to a subgroup of type [22X3.G[122X..[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( CharacterTable( "Fi24'" ) ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L3(4)" );;[127X[104X
    [4X[25Xgap>[125X [27XFiltered( mx, x -> Length( PossibleClassFusions( s, x ) ) > 0 );[127X[104X
    [4X[28X[ CharacterTable( "Fi23" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  other candidates [22Xm.G[122X contain at least one central involution. If [22Xm.G[122X is
  a  subgroup of [22XMM[122X then it is contained in a [10X2A[110X centralizer (of the structure
  [22X2.B[122X)  or  a  [10X2B[110X centralizer (of the structure [10X2^{1+24}_+.Co_1[110X). Again we use
  [2XPossibleClassFusions[102X   ([14XReference:   PossibleClassFusions[114X)   to   list   all
  candidates  for  the  class  fusion,  but  here  we  prescribe  the  central
  involution of the [10X2A[110X or [10X2B[110X centralizer as an image of one central involution
  in [22Xm.G[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdone:= [ "L3(4)", "2.L3(4)", "3.L3(4)", "2^2.L3(4)", "6.L3(4)" ];;[127X[104X
    [4X[25Xgap>[125X [27Xnames:= Filtered( names, x -> not x in done );[127X[104X
    [4X[28X[ "4_1.L3(4)", "4_2.L3(4)", "(2x4).L3(4)", "(2^2x3).L3(4)", [128X[104X
    [4X[28X  "12_1.L3(4)", "12_2.L3(4)", "4^2.L3(4)", "(2x12).L3(4)", [128X[104X
    [4X[28X  "(4^2x3).L3(4)" ][128X[104X
    [4X[25Xgap>[125X [27Xinvcent:= List( [ "MN2A", "MN2B" ], CharacterTable );[127X[104X
    [4X[28X[ CharacterTable( "2.B" ), CharacterTable( "2^1+24.Co1" ) ][128X[104X
    [4X[25Xgap>[125X [27XForAll( invcent, x -> ClassPositionsOfCentre( x ) = [ 1, 2 ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xcand:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xords:= "dummy";;   #  Avoid a message about an unbound variable ...[127X[104X
    [4X[25Xgap>[125X [27Xfor name in names do[127X[104X
    [4X[25X>[125X [27X     s:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     ords:= OrdersClassRepresentatives( s );[127X[104X
    [4X[25X>[125X [27X     invpos:= Filtered( ClassPositionsOfCentre( s ), i -> ords[i] = 2 );[127X[104X
    [4X[25X>[125X [27X     for i in invpos do[127X[104X
    [4X[25X>[125X [27X       for t in invcent do[127X[104X
    [4X[25X>[125X [27X         init:= InitFusion( s, t );[127X[104X
    [4X[25X>[125X [27X         if init = fail then[127X[104X
    [4X[25X>[125X [27X           continue;[127X[104X
    [4X[25X>[125X [27X         fi;[127X[104X
    [4X[25X>[125X [27X         init[i]:= 2;[127X[104X
    [4X[25X>[125X [27X         fus:= PossibleClassFusions( s, t, rec( fusionmap:= init,[127X[104X
    [4X[25X>[125X [27X                                                decompose:= false ) );[127X[104X
    [4X[25X>[125X [27X         if fus <> [] then[127X[104X
    [4X[25X>[125X [27X           Add( cand, [ s, t, i, fus ] );[127X[104X
    [4X[25X>[125X [27X         fi;[127X[104X
    [4X[25X>[125X [27X       od;[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XList( cand, x -> x{ [ 1 .. 3 ] } );[127X[104X
    [4X[28X[ [ CharacterTable( "4_1.L3(4)" ), CharacterTable( "2^1+24.Co1" ), 3 ][128X[104X
    [4X[28X    , [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 2 ],[128X[104X
    [4X[28X  [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 3 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Note    that    we    have    called    [2XPossibleClassFusions[102X    ([14XReference:
  PossibleClassFusions[114X)  with  the  option [10Xdecompose:= false[110X, in order to save
  space and time. See Section [14X1.5-1[114X for more details.)[133X
  
  [33X[0;0YConcerning  the  candidate  [22X(2 × 4).G[122X, we see that only fusions are possible
  for  which  the  central involution in question is mapped to a [10X2A[110X element of
  [22XMM[122X.  Since  we  get  candidates  only  for  two  out  of  the  three central
  involutions, we see that [22X(2 × 4).G[122X does not embed into [22XMM[122X.[133X
  
  [33X[0;0YThus  it  turns  out that exactly one group [22Xm.G[122X cannot be excluded this way.
  Namely,  these  character-theoretical  criteria  leave  the possibility that
  [22X4_1.G[122X may occur as a subgroup of [22X2^1+24_+.Co_1[122X.[133X
  
  [33X[0;0YMoreover, we see that if this happens then the centre [22XC[122X of [22X4_1.G[122X lies inside
  the  normal subgroup [22XN = 2^1+24_+[122X. The centralizer of [22XC[122X in [22XN[122X has order [22X2^24[122X,
  and  the  centralizer  of [22XC[122X in [22X2^1+24_+.Co_1[122X has order [22X2^24 ⋅ |Co_3|[122X. We see
  that [22X4_1.G[122X, if it exists as a subgroup of [22X2^1+24_+.Co_1[122X, must lie inside the
  subgroup [22X[2^24].Co_3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= cand[1][1];[127X[104X
    [4X[28XCharacterTable( "4_1.L3(4)" )[128X[104X
    [4X[25Xgap>[125X [27Xt:= cand[1][2];[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1" )[128X[104X
    [4X[25Xgap>[125X [27Xfus:= cand[1][4];[127X[104X
    [4X[28X[ [ 1, 5, 2, 5, 9, 8, 23, 27, 24, 27, 49, 49, 50, 50, 70, 74, 71, 74, [128X[104X
    [4X[28X      70, 74, 71, 74, 114, 119, 115, 119, 114, 119, 115, 119 ] ][128X[104X
    [4X[25Xgap>[125X [27XClassPositionsOfCentre( s );[127X[104X
    [4X[28X[ 1, 2, 3, 4 ][128X[104X
    [4X[25Xgap>[125X [27X5 in ClassPositionsOfPCore( t, 2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xsiz:= SizesCentralizers( t )[5] / 2^24;[127X[104X
    [4X[28X495766656000[128X[104X
    [4X[25Xgap>[125X [27Xmx:= Filtered( List( Maxes( CharacterTable( "Co1" ) ),[127X[104X
    [4X[25X>[125X [27X                        CharacterTable ),[127X[104X
    [4X[25X>[125X [27X                  x -> Size( x ) mod siz = 0 );[127X[104X
    [4X[28X[ CharacterTable( "Co3" ) ][128X[104X
    [4X[25Xgap>[125X [27XSize( mx[1] ) = siz;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YI  do  not  see  a  character-theoretic  argument  that  could  disprove the
  existence of such an [22X4_1.G[122X type subgroup.[133X
  
  
  [1X1.5-3 [33X[0;0YThe character table of [22X(2 × O_8^+(3)).S_4 ≤ 2.B[122X[101X[1X (October 2023)[133X[101X
  
  [33X[0;0YConsider a maximal subgroup [22XH[122X of type [22X(3^2:2 × O_8^+(3)).S_4[122X in the sporadic
  simple  Monster  group. The character table of [22XH[122X has been contributed by Tim
  Burness.  We  can  view [22XH[122X as [22XO_8^+(3).(3^2:2S_4) = O_8^+(3).F[122X. The character
  table  of  [22XH[122X determines that of [22XF[122X, and this table determines the isomorphism
  type of [22XF[122X as [10XSmallGroup( 432, 734 )[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XtblH:= CharacterTable( "(3^2:2xO8+(3)).S4" );[127X[104X
    [4X[28XCharacterTable( "(3^2:2xO8+(3)).S4" )[128X[104X
    [4X[25Xgap>[125X [27XN:= ClassPositionsOfSolvableResiduum( tblH );;[127X[104X
    [4X[25Xgap>[125X [27XtblF:= tblH / N;;[127X[104X
    [4X[25Xgap>[125X [27XSize( tblF );[127X[104X
    [4X[28X432[128X[104X
    [4X[25Xgap>[125X [27Xknown:= NamesOfEquivalentLibraryCharacterTables( tblF );[127X[104X
    [4X[28X[ "3^2.2.S4", "M12M7" ][128X[104X
    [4X[25Xgap>[125X [27XFiltered( GroupInfoForCharacterTable( known[1] ),[127X[104X
    [4X[25X>[125X [27X             x -> x[1] = "SmallGroup" );[127X[104X
    [4X[28X[ [ "SmallGroup", [ 432, 734 ] ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Note    that    the    precomputed   [2XGroupInfoForCharacterTable[102X   ([14XCTblLib:
  GroupInfoForCharacterTable[114X)  information  about [5XGAP[105X library character tables
  means  that  exactly  one  isomorphism  type of groups fits to the character
  table of [22XF[122X.)[133X
  
  [33X[0;0YWe  compute  that [22XO_3(F) ≅ 3^2[122X has complements in [22XF[122X, thus [22XH[122X has a subgroup [22XV[122X
  of  the  type [22XO_8^+(3).2S_4[122X, which is a complement of [22XO_3(H)[122X in [22XH[122X, thus [22XV[122X is
  isomorphic with [22XH / O_3(H)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:= SmallGroup( 432, 734 );;[127X[104X
    [4X[25Xgap>[125X [27XP:= PCore( G, 3 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ComplementClassesRepresentatives( G, P ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  can derive the character table of [22XV[122X from that of [22XH[122X, and compute that the
  group  structure  of  [22XV[122X  is  [22X(2  ×  O_8^+(3)).S_4[122X. For that, we consider the
  element  orders of the unique normal subgroup of order [22X2 |O_8^+(3)|[122X in [22XV[122X. If
  this normal subgroup would not be isomorphic with [22X2 × O_8^+(3)[122X then it would
  have  one  of the structures [22XO_8^+(3).2_1[122X or [22XO_8^+(3).2_2[122X, but then it would
  contain elements of the orders [22X24[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XtblV:= tblH / ClassPositionsOfPCore( tblH, 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xord:= 2 * Size( tblH ) / Size( tblF );[127X[104X
    [4X[28X9904359628800[128X[104X
    [4X[25Xgap>[125X [27Xclasses:= SizesConjugacyClasses( tblV );;[127X[104X
    [4X[25Xgap>[125X [27X2N:= Filtered( ClassPositionsOfNormalSubgroups( tblV ),[127X[104X
    [4X[25X>[125X [27X                  l -> Sum( classes{ l } ) = ord );;[127X[104X
    [4X[25Xgap>[125X [27XLength( 2N );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XSet( OrdersClassRepresentatives( tblV ){ 2N[1] } );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 26, 30 ][128X[104X
    [4X[25Xgap>[125X [27XSet( OrdersClassRepresentatives( CharacterTable( "O8+(3)" ) ) );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20 ][128X[104X
    [4X[25Xgap>[125X [27XSet( OrdersClassRepresentatives( CharacterTable( "O8+(3).2_1" ) ) );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 24, 26, 28, [128X[104X
    [4X[28X  30, 36, 40 ][128X[104X
    [4X[25Xgap>[125X [27XSet( OrdersClassRepresentatives( CharacterTable( "O8+(3).2_2" ) ) );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 24, 26, 28, [128X[104X
    [4X[28X  30, 36 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  class  fusion of [22XV[122X into the Monster group shows that [22XV[122X centralizes a [10X2A[110X
  element  in  the Monster, hence [22XV[122X is a subgroup of a maximal subgroup of the
  type [22X2.B[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XtblM:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27XVfusM:= PossibleClassFusions( tblV, tblM );;[127X[104X
    [4X[25Xgap>[125X [27XLength( VfusM );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XZV:= ClassPositionsOfCentre( tblV );[127X[104X
    [4X[28X[ 1, 2 ][128X[104X
    [4X[25Xgap>[125X [27XSet( List( VfusM, l -> l{ ZV } ) );[127X[104X
    [4X[28X[ [ 1, 2 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFrom  the  list of maximal subgroups of [22XB[122X, we see that either [22XV[122X is contained
  in  the  preimage of [22XFi_23[122X under the natural epimorphism from [22X2.B[122X to [22XB[122X, or [22XV[122X
  is equal to the preimage of [22XO_8^+(3).S_4[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XtblB:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27XmxB:= List( Maxes( tblB ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( mxB, s -> Size( s ) mod ( Size( tblV ) / 2 ) = 0 );[127X[104X
    [4X[28X[ CharacterTable( "Fi23" ), CharacterTable( "O8+(3).S4" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe former possibility is excluded from the fact that the factor of [22XV[122X by its
  center does not admit a class fusion into [22XFi_23[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( tblV / ZV, CharacterTable( "Fi23" ) ) );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe conclude that [22XV[122X is a maximal subgroup of [22X2.B[122X.[133X
  
  [33X[0;0YThus  we have used the character table of [22XH[122X to construct the character table
  of a maximal subgroup of [22X2.B[122X, with very little effort.[133X
  
  [33X[0;0YThis  table is meanwhile available in the table library, with the identifier
  [10X"(2xO8+(3)).S4"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlib:= CharacterTable( "(2xO8+(3)).S4" );;[127X[104X
    [4X[25Xgap>[125X [27XTransformingPermutationsCharacterTables( tblV, lib ) <> fail;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIrr( lib ) = Irr( tblV );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order to add the table to the library, we have to provide also the class
  fusions  from [22XV[122X to [22X2.B[122X, to the maximal subgroup [22X(2 × O_8^+(3)).S_4[122X of [22XB[122X, and
  to  the  maximal  subgroup  [22X(3^2:2  ×  O_8^+(3)).S_4[122X  of  [22XM[122X,  such  that the
  compositions of fusions from [22XV[122X to [22XB[122X via [22XO_8^+(3).S_4[122X and [22X2.B[122X are compatible,
  [22X...[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XtblU:= CharacterTable( "O8+(3).S4" );;[127X[104X
    [4X[25Xgap>[125X [27Xtbl2B:= CharacterTable( "2.B" );;[127X[104X
    [4X[25Xgap>[125X [27XCompositionMaps( GetFusionMap( tblU, tblB ),[127X[104X
    [4X[25X>[125X [27X                    GetFusionMap( lib, tblU ) ) =[127X[104X
    [4X[25X>[125X [27X   CompositionMaps( GetFusionMap( tbl2B, tblB ),[127X[104X
    [4X[25X>[125X [27X                    GetFusionMap( lib, tbl2B ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22X...[122X  and  that  the  compositions  of  fusions  from  [22XV[122X  to  [22XM[122X  via [22X(3^2:2 ×
  O_8^+(3)).S_4[122X and [22X2.B[122X are compatible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XtblH:= CharacterTable( "(3^2:2xO8+(3)).S4" );;[127X[104X
    [4X[25Xgap>[125X [27XCompositionMaps( GetFusionMap( tblH, tblM ),[127X[104X
    [4X[25X>[125X [27X                    GetFusionMap( lib, tblH ) ) =[127X[104X
    [4X[25X>[125X [27X   CompositionMaps( GetFusionMap( tbl2B, tblM ),[127X[104X
    [4X[25X>[125X [27X                    GetFusionMap( lib, tbl2B ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
