  
  [1X2 [33X[0;0YTutorial for the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YThis  chapter  gives  an overview of the basic functionality provided by the
  [5XAtlasRep[105X package. The main concepts and interface functions are presented in
  the first sections, and Section [14X2.4[114X shows a few small examples.[133X
  
  
  [1X2.1 [33X[0;0YAccessing a Specific Group in [5XAtlasRep[105X[101X[1X[133X[101X
  
  [33X[0;0YThe  [5XAtlasRep[105X  package  gives  access  to  a  database,  the  [5XATLAS[105X of Group
  Representations  [ATLAS],  that  contains  generators  and  related data for
  several  groups,  mainly for extensions of simple groups (see Section [14X2.1-1[114X)
  and for their maximal subgroups (see Section [14X2.1-2[114X).[133X
  
  [33X[0;0YNote  that the data are not part of the package. They are fetched from a web
  server as soon as they are needed for the first time, see Section [14X4.3-1[114X.[133X
  
  [33X[0;0YFirst  of  all,  we  load the [5XAtlasRep[105X package. Some of the examples require
  also the [5XGAP[105X packages [5XCTblLib[105X and [5XTomLib[105X, so we load also these packages.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "AtlasRep" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "CTblLib" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "TomLib" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X2.1-1 [33X[0;0YAccessing a Group in [5XAtlasRep[105X[101X[1X via its Name[133X[101X
  
  [33X[0;0YEach  group  that occurs in this database is specified by a [13Xname[113X, which is a
  string similar to the name used in the [5XATLAS[105X of Finite Groups [CCNPW85]. For
  those groups whose character tables are contained in the [5XGAP[105X Character Table
  Library [Bre04],   the   names  are  equal  to  the  [2XIdentifier[102X  ([14XReference:
  Identifier  (for  character  tables)[114X)  values  of  these  character  tables.
  Examples  of such names are [10X"M24"[110X for the Mathieu group [22XM_24[122X, [10X"2.A6"[110X for the
  double  cover  of  the  alternating group [22XA_6[122X, and [10X"2.A6.2_1"[110X for the double
  cover  of  the symmetric group [22XS_6[122X. The names that actually occur are listed
  in  the  first  column of the overview table that is printed by the function
  [2XDisplayAtlasInfo[102X  ([14X3.5-1[114X),  called  without  arguments, see below. The other
  columns of the table describe the data that are available in the database.[133X
  
  [33X[0;0YFor  example,  [2XDisplayAtlasInfo[102X  ([14X3.5-1[114X)  may  print  the  following  lines.
  Omissions are indicated with [21X[10X...[110X[121X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo();[127X[104X
    [4X[28Xgroup                    |  # | maxes | cl | cyc | out | fnd | chk | prs[128X[104X
    [4X[28X-------------------------+----+-------+----+-----+-----+-----+-----+----[128X[104X
    [4X[28X...[128X[104X
    [4X[28X2.A5                     | 26 |     3 |    |     |     |     |  +  |  + [128X[104X
    [4X[28X2.A5.2                   | 11 |     4 |    |     |     |     |  +  |  + [128X[104X
    [4X[28X2.A6                     | 18 |     5 |    |     |     |     |     |    [128X[104X
    [4X[28X2.A6.2_1                 |  3 |     6 |    |     |     |     |     |    [128X[104X
    [4X[28X2.A7                     | 24 |       |    |     |     |     |     |    [128X[104X
    [4X[28X2.A7.2                   |  7 |       |    |     |     |     |     |    [128X[104X
    [4X[28X...[128X[104X
    [4X[28XM22                      | 58 |     8 |  + |  +  |     |  +  |  +  |  + [128X[104X
    [4X[28XM22.2                    | 46 |     7 |  + |  +  |     |  +  |  +  |  + [128X[104X
    [4X[28XM23                      | 66 |     7 |  + |  +  |     |  +  |  +  |  + [128X[104X
    [4X[28XM24                      | 62 |     9 |  + |  +  |     |  +  |  +  |  + [128X[104X
    [4X[28XMcL                      | 46 |    12 |  + |  +  |     |  +  |  +  |  + [128X[104X
    [4X[28XMcL.2                    | 27 |    10 |    |  +  |     |  +  |  +  |  + [128X[104X
    [4X[28XO7(3)                    | 28 |       |    |     |     |     |     |    [128X[104X
    [4X[28XO7(3).2                  |  3 |       |    |     |     |     |     |    [128X[104X
    [4X[28X...[128X[104X
  [4X[32X[104X
  
  [33X[0;0YCalled  with  a  group  name  as  the only argument, the function [2XAtlasGroup[102X
  ([14X3.5-7[114X)  returns  a  group  isomorphic  to the group with the given name. If
  permutation  generators  are  available  in  the database then a permutation
  group (of smallest available degree) is returned, otherwise a matrix group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "M24" );[127X[104X
    [4X[28XGroup([ (1,4)(2,7)(3,17)(5,13)(6,9)(8,15)(10,19)(11,18)(12,21)(14,16)(20,[128X[104X
    [4X[28X    24)(22,23), (1,4,6)(2,21,14)(3,9,15)(5,18,10)(13,17,16)(19,24,23) ])[128X[104X
    [4X[25Xgap>[125X [27XIsPermGroup( g );  NrMovedPoints( g );  Size( g );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X24[128X[104X
    [4X[28X244823040[128X[104X
  [4X[32X[104X
  
  
  [1X2.1-2 [33X[0;0YAccessing a Maximal Subgroup of a Group in [5XAtlasRep[105X[101X[1X[133X[101X
  
  [33X[0;0YMany  maximal  subgroups  of  extensions of simple groups can be constructed
  using the function [2XAtlasSubgroup[102X ([14X3.5-8[114X). Given the name of the extension of
  the simple group and the number of the conjugacy class of maximal subgroups,
  this function returns a representative from this class.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasSubgroup( "M24", 1 );[127X[104X
    [4X[28XGroup([ (2,10)(3,12)(4,14)(6,9)(8,16)(15,18)(20,22)(21,24), [128X[104X
    [4X[28X  (1,7,2,9)(3,22,10,23)(4,19,8,12)(5,14)(6,18)(13,16,17,24) ])[128X[104X
    [4X[25Xgap>[125X [27XIsPermGroup( g );  NrMovedPoints( g );  Size( g );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X23[128X[104X
    [4X[28X10200960[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  classes  of  maximal subgroups are ordered w. r. t. decreasing subgroup
  order. So the first class contains the largest maximal subgroups.[133X
  
  [33X[0;0YNote  that groups obtained by [2XAtlasSubgroup[102X ([14X3.5-8[114X) may be not very suitable
  for  computations  in  the  sense that much nicer representations exist. For
  example,  the  sporadic simple O'Nan group [22XO'N[122X contains a maximal subgroup [22XS[122X
  isomorphic with the Janko group [22XJ_1[122X; the smallest permutation representation
  of  [22XO'N[122X  has degree [22X122760[122X, so restricting this representation to [22XS[122X yields a
  representation   of  [22XJ_1[122X  of  that  degree.  However,  [22XJ_1[122X  has  a  faithful
  permutation  representation  of degree [22X266[122X, which admits much more efficient
  computations.  If  you  are  just interested in [22XJ_1[122X and not in its embedding
  into  [22XO'N[122X  then one possibility to get a [21Xnicer[121X faithful representation is to
  call            [2XSmallerDegreePermutationRepresentation[102X           ([14XReference:
  SmallerDegreePermutationRepresentation[114X). In the abovementioned example, this
  works  well;  in  general,  we cannot expect that we get a representation of
  smallest degree in this way.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= AtlasSubgroup( "ON", 3 );[127X[104X
    [4X[28X<permutation group of size 175560 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( s );  Size( s );[127X[104X
    [4X[28X122760[128X[104X
    [4X[28X175560[128X[104X
    [4X[25Xgap>[125X [27Xhom:= SmallerDegreePermutationRepresentation( s );;[127X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( Image( hom ) );[127X[104X
    [4X[28X266[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn this particular case, one could of course also ask directly for the group
  [22XJ_1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xj1:= AtlasGroup( "J1" );[127X[104X
    [4X[28X<permutation group of size 175560 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( j1 );[127X[104X
    [4X[28X266[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  you  have a group [22XG[122X, say, and you are really interested in the embedding
  of  a  maximal  subgroup  of  [22XG[122X  into  [22XG[122X  then an easy way to get compatible
  generators  is  to  create  [22XG[122X  with  [2XAtlasGroup[102X  ([14X3.5-7[114X)  and  then  to call
  [2XAtlasSubgroup[102X ([14X3.5-8[114X) with first argument the group [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "ON" );[127X[104X
    [4X[28X<permutation group of size 460815505920 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xs:= AtlasSubgroup( g, 3 );[127X[104X
    [4X[28X<permutation group of size 175560 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubset( g, s );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSubset( g, j1 );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YAccessing Specific Generators in [5XAtlasRep[105X[101X[1X[133X[101X
  
  [33X[0;0YThe  function  [2XDisplayAtlasInfo[102X ([14X3.5-1[114X), called with an admissible name of a
  group as the only argument, lists the [5XATLAS[105X data available for this group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5" );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[128X[104X
    [4X[28X 1: G <= Sym(5)                  3-trans., on cosets of A4 (1st max.)[128X[104X
    [4X[28X 2: G <= Sym(6)                  2-trans., on cosets of D10 (2nd max.)[128X[104X
    [4X[28X 3: G <= Sym(10)                 rank 3, on cosets of S3 (3rd max.)[128X[104X
    [4X[28X 4: G <= GL(4a,2)[128X[104X
    [4X[28X 5: G <= GL(4b,2)[128X[104X
    [4X[28X 6: G <= GL(4,3)[128X[104X
    [4X[28X 7: G <= GL(6,3)[128X[104X
    [4X[28X 8: G <= GL(2a,4)[128X[104X
    [4X[28X 9: G <= GL(2b,4)[128X[104X
    [4X[28X10: G <= GL(3,5)[128X[104X
    [4X[28X11: G <= GL(5,5)[128X[104X
    [4X[28X12: G <= GL(3a,9)[128X[104X
    [4X[28X13: G <= GL(3b,9)[128X[104X
    [4X[28X14: G <= GL(4,Z)[128X[104X
    [4X[28X15: G <= GL(5,Z)[128X[104X
    [4X[28X16: G <= GL(6,Z)[128X[104X
    [4X[28X17: G <= GL(3a,Field([Sqrt(5)]))[128X[104X
    [4X[28X18: G <= GL(3b,Field([Sqrt(5)]))[128X[104X
    [4X[28X[128X[104X
    [4X[28XPrograms for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X--------------------[128X[104X
    [4X[28Xpresentation[128X[104X
    [4X[28Xstd. gen. checker[128X[104X
    [4X[28Xmaxes (all 3):[128X[104X
    [4X[28X  1:  A4[128X[104X
    [4X[28X  2:  D10[128X[104X
    [4X[28X  3:  S3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn order to fetch one of the listed permutation groups or matrix groups, you
  can  call  [2XAtlasGroup[102X  ([14X3.5-7[114X)  with  second  argument the function [2XPosition[102X
  ([14XReference: Position[114X) and third argument the position in the list.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasGroup( "A5", Position, 1 );[127X[104X
    [4X[28XGroup([ (1,2)(3,4), (1,3,5) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  this approach may yield a different group after an update of the
  database, if new data for the group become available.[133X
  
  [33X[0;0YAlternatively, you can describe the desired group by conditions, such as the
  degree  in  the  case of a permutation group, and the dimension and the base
  ring in the case of a matrix group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasGroup( "A5", NrMovedPoints, 10 );[127X[104X
    [4X[28XGroup([ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ])[128X[104X
    [4X[25Xgap>[125X [27XAtlasGroup( "A5", Dimension, 4, Ring, GF(2) );[127X[104X
    [4X[28X<matrix group of size 60 with 2 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  same  holds for the restriction to maximal subgroups: Use [2XAtlasSubgroup[102X
  ([14X3.5-8[114X)   with  the  same  arguments  as  [2XAtlasGroup[102X  ([14X3.5-7[114X),  except  that
  additionally  the number of the class of maximal subgroups is entered as the
  last  argument.  Note  that  the  conditions  refer to the group, not to the
  subgroup;  it  may  happen that the subgroup moves fewer points than the big
  group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasSubgroup( "A5", Dimension, 4, Ring, GF(2), 1 );[127X[104X
    [4X[28X<matrix group of size 12 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasSubgroup( "A5", NrMovedPoints, 10, 3 );[127X[104X
    [4X[28XGroup([ (2,4)(3,5)(6,8)(7,10), (1,4)(3,8)(5,7)(6,10) ])[128X[104X
    [4X[25Xgap>[125X [27XSize( g );  NrMovedPoints( g );[127X[104X
    [4X[28X6[128X[104X
    [4X[28X9[128X[104X
  [4X[32X[104X
  
  
  [1X2.3 [33X[0;0YBasic Concepts used in [5XAtlasRep[105X[101X[1X[133X[101X
  
  
  [1X2.3-1 [33X[0;0YGroups, Generators, and Representations[133X[101X
  
  [33X[0;0YUp  to  now,  we  have  talked only about groups and subgroups. The [5XAtlasRep[105X
  package  provides  access  to [13Xgroup generators[113X, and in fact these generators
  have  the  property  that  mapping  one  set of generators to another set of
  generators  for  the same group defines an isomorphism. These generators are
  called [13Xstandard generators[113X, see Section [14X3.3[114X.[133X
  
  [33X[0;0YSo  instead  of thinking about several generating sets of a group [22XG[122X, say, we
  can  think about one abstract group [22XG[122X, with one fixed set of generators, and
  mapping  these  generators  to  any  set  of generators provided by [5XAtlasRep[105X
  defines  a  representation  of [22XG[122X. This viewpoint motivates the name [21X[5XATLAS[105X of
  Group Representations[121X for the database.[133X
  
  [33X[0;0YIf you are interested in the generators provided by the database rather than
  in    the    groups    they    generate,    you   can   use   the   function
  [2XOneAtlasGeneratingSetInfo[102X  ([14X3.5-5[114X)  instead  of [2XAtlasGroup[102X ([14X3.5-7[114X), with the
  same  arguments.  This will yield a record that describes the representation
  in  question.  Calling the function [2XAtlasGenerators[102X ([14X3.5-2[114X) with this record
  will  then  yield  a  record with the additional component [10Xgenerators[110X, which
  holds the list of generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 10 );[127X[104X
    [4X[28Xrec( groupname := "A5", id := "",[128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1, 10 ],[128X[104X
    [4X[28X  isPrimitive := true, maxnr := 3, p := 10, rankAction := 3,[128X[104X
    [4X[28X  repname := "A5G1-p10B0", repnr := 3, size := 60, stabilizer := "S3",[128X[104X
    [4X[28X  standardization := 1, transitivity := 1, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27Xinfo2:= AtlasGenerators( info );[127X[104X
    [4X[28Xrec( generators := [ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ], [128X[104X
    [4X[28X  groupname := "A5", id := "", [128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1, 10 ], [128X[104X
    [4X[28X  isPrimitive := true, maxnr := 3, p := 10, rankAction := 3, [128X[104X
    [4X[28X  repname := "A5G1-p10B0", repnr := 3, size := 60, stabilizer := "S3", [128X[104X
    [4X[28X  standardization := 1, transitivity := 1, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27Xinfo2.generators;[127X[104X
    [4X[28X[ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ][128X[104X
  [4X[32X[104X
  
  
  [1X2.3-2 [33X[0;0YStraight Line Programs[133X[101X
  
  [33X[0;0YFor  computing  certain  group  elements  from  standard generators, such as
  generators  of  a  subgroup or class representatives, [5XAtlasRep[105X uses [13Xstraight
  line  programs[113X,  see  [14X'Reference:  Straight Line Programs'[114X. Essentially this
  means to evaluate words in the generators, similar to [2XMappedWord[102X ([14XReference:
  MappedWord[114X) but more efficiently.[133X
  
  [33X[0;0YIt can be useful to deal with these straight line programs, see [2XAtlasProgram[102X
  ([14X3.5-3[114X).  For  example, an automorphism [22Xα[122X, say, of the group [22XG[122X, if available
  in  [5XAtlasRep[105X, is given by a straight line program that defines the images of
  standard generators of [22XG[122X. This way, one can for example compute the image of
  a  subgroup [22XU[122X of [22XG[122X under [22Xα[122X by first applying the straight line program for [22Xα[122X
  to standard generators of [22XG[122X, and then applying the straight line program for
  the restriction from [22XG[122X to [22XU[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprginfo:= AtlasProgramInfo( "A5", "maxes", 1 );[127X[104X
    [4X[28Xrec( groupname := "A5", identifier := [ "A5", "A5G1-max1W1", 1 ], size := 12, [128X[104X
    [4X[28X  standardization := 1, subgroupname := "A4" )[128X[104X
    [4X[25Xgap>[125X [27Xprg:= AtlasProgram( prginfo.identifier );[127X[104X
    [4X[28Xrec( groupname := "A5", identifier := [ "A5", "A5G1-max1W1", 1 ], [128X[104X
    [4X[28X  program := <straight line program>, size := 12, standardization := 1, [128X[104X
    [4X[28X  subgroupname := "A4" )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( prg.program );[127X[104X
    [4X[28X# input:[128X[104X
    [4X[28Xr:= [ g1, g2 ];[128X[104X
    [4X[28X# program:[128X[104X
    [4X[28Xr[3]:= r[1]*r[2];[128X[104X
    [4X[28Xr[4]:= r[2]*r[1];[128X[104X
    [4X[28Xr[5]:= r[3]*r[3];[128X[104X
    [4X[28Xr[1]:= r[5]*r[4];[128X[104X
    [4X[28X# return values:[128X[104X
    [4X[28X[ r[1], r[2] ][128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineProgram( prg.program, info2.generators );[127X[104X
    [4X[28X[ (1,10)(2,3)(4,9)(7,8), (1,2,3)(4,6,7)(5,8,9) ][128X[104X
  [4X[32X[104X
  
  
  [1X2.4 [33X[0;0YExamples of Using the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  
  [1X2.4-1 [33X[0;0YExample: Class Representatives[133X[101X
  
  [33X[0;0YFirst  we show the computation of class representatives of the Mathieu group
  [22XM_11[122X,  in  a [22X2[122X-modular matrix representation. We start with the ordinary and
  Brauer character tables of this group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "M11" );;[127X[104X
    [4X[25Xgap>[125X [27Xmodtbl:= tbl mod 2;;[127X[104X
    [4X[25Xgap>[125X [27XCharacterDegrees( modtbl );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 10, 1 ], [ 16, 2 ], [ 44, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  output of [2XCharacterDegrees[102X ([14XReference: CharacterDegrees[114X) means that the
  [22X2[122X-modular irreducibles of [22XM_11[122X have degrees [22X1[122X, [22X10[122X, [22X16[122X, [22X16[122X, and [22X44[122X.[133X
  
  [33X[0;0YUsing  [2XDisplayAtlasInfo[102X  ([14X3.5-1[114X), we find out that matrix generators for the
  irreducible [22X10[122X-dimensional representation are available in the database.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "M11", Characteristic, 2 );[127X[104X
    [4X[28XRepresentations for G = M11:    (all refer to std. generators 1)[128X[104X
    [4X[28X----------------------------[128X[104X
    [4X[28X 6: G <= GL(10,2)  character 10a[128X[104X
    [4X[28X 7: G <= GL(32,2)  character 16ab[128X[104X
    [4X[28X 8: G <= GL(44,2)  character 44a[128X[104X
    [4X[28X16: G <= GL(16a,4) character 16a[128X[104X
    [4X[28X17: G <= GL(16b,4) character 16b[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  decide to work with this representation. We fetch the generators and
  compute the list of class representatives of [22XM_11[122X in the representation. The
  ordering of class representatives is the same as that in the character table
  of the [5XATLAS[105X of Finite Groups ([CCNPW85]), which coincides with the ordering
  of columns in the [5XGAP[105X table we have fetched above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "M11", Characteristic, 2,[127X[104X
    [4X[25X>[125X [27X                                            Dimension, 10 );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info.identifier );;[127X[104X
    [4X[25Xgap>[125X [27Xccls:= AtlasProgram( "M11", gens.standardization, "classes" );[127X[104X
    [4X[28Xrec( groupname := "M11", identifier := [ "M11", "M11G1-cclsW1", 1 ],[128X[104X
    [4X[28X  outputs := [ "1A", "2A", "3A", "4A", "5A", "6A", "8A", "8B", "11A", "11B" ],[128X[104X
    [4X[28X  program := <straight line program>, standardization := 1 )[128X[104X
    [4X[25Xgap>[125X [27Xreps:= ResultOfStraightLineProgram( ccls.program, gens.generators );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  we  would  need  only  a few class representatives, we could use the [5XGAP[105X
  library  function  [2XRestrictOutputsOfSLP[102X ([14XReference: RestrictOutputsOfSLP[114X) to
  create a straight line program that computes only specified outputs. Here is
  an example where only the class representatives of order eight are computed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord8prg:= RestrictOutputsOfSLP( ccls.program,[127X[104X
    [4X[25X>[125X [27X                  Filtered( [ 1 .. 10 ], i -> ccls.outputs[i][1] = '8' ) );[127X[104X
    [4X[28X<straight line program>[128X[104X
    [4X[25Xgap>[125X [27Xord8reps:= ResultOfStraightLineProgram( ord8prg, gens.generators );;[127X[104X
    [4X[25Xgap>[125X [27XList( ord8reps, m -> Position( reps, m ) );[127X[104X
    [4X[28X[ 7, 8 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet us check that the class representatives have the right orders.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( reps, Order ) = OrdersClassRepresentatives( tbl );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFrom  the  class representatives, we can compute the Brauer character we had
  started  with.  This  Brauer  character  is  defined  on  all classes of the
  [22X2[122X-modular  table. So we first pick only those representatives, using the [5XGAP[105X
  function  [2XGetFusionMap[102X  ([14XReference:  GetFusionMap[114X);  in  this  situation, it
  returns the class fusion from the Brauer table into the ordinary table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfus:= GetFusionMap( modtbl, tbl );[127X[104X
    [4X[28X[ 1, 3, 5, 9, 10 ][128X[104X
    [4X[25Xgap>[125X [27Xmodreps:= reps{ fus };;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThen   we   call   the   [5XGAP[105X   function   [2XBrauerCharacterValue[102X   ([14XReference:
  BrauerCharacterValue[114X),  which  computes  the Brauer character value from the
  matrix given.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xchar:= List( modreps, BrauerCharacterValue );[127X[104X
    [4X[28X[ 10, 1, 0, -1, -1 ][128X[104X
    [4X[25Xgap>[125X [27XPosition( Irr( modtbl ), char );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X2.4-2 [33X[0;0YExample: Permutation and Matrix Representations[133X[101X
  
  [33X[0;0YThe  second  example  shows  the computation of a permutation representation
  from a matrix representation. We work with the [22X10[122X-dimensional representation
  used  above,  and  consider the action on the [22X2^10[122X vectors of the underlying
  row space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgrp:= Group( gens.generators );;[127X[104X
    [4X[25Xgap>[125X [27Xv:= GF(2)^10;;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= Orbits( grp, AsList( v ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 1, 396, 55, 330, 66, 165, 11 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  there  are  six  nontrivial  orbits,  and  we can compute the
  permutation actions on these orbits directly using [2XAction[102X ([14XReference: Action
  homomorphisms[114X).  However,  for  larger  examples,  one cannot write down all
  orbits  on  the  row  space,  so  one  has to use another strategy if one is
  interested in a particular orbit.[133X
  
  [33X[0;0YLet  us  assume  that we are interested in the orbit of length [22X11[122X. The point
  stabilizer  is  the  first maximal subgroup of [22XM_11[122X, thus the restriction of
  the representation to this subgroup has a nontrivial fixed point space. This
  restriction can be computed using the [5XAtlasRep[105X package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( "M11", 6, 1 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow computing the fixed point space is standard linear algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xid:= IdentityMat( 10, GF(2) );;[127X[104X
    [4X[25Xgap>[125X [27Xsub1:= Subspace( v, NullspaceMat( gens.generators[1] - id ) );;[127X[104X
    [4X[25Xgap>[125X [27Xsub2:= Subspace( v, NullspaceMat( gens.generators[2] - id ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfix:= Intersection( sub1, sub2 );[127X[104X
    [4X[28X<vector space of dimension 1 over GF(2)>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe final step is of course the computation of the permutation action on the
  orbit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( grp, Basis( fix )[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xact:= Action( grp, orb );;  Print( act, "\n" );[127X[104X
    [4X[28XGroup( [ ( 1, 2)( 4, 6)( 5, 8)( 7,10), ( 1, 3, 5, 9)( 2, 4, 7,11) ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  this  group  is  [13Xnot[113X equal to the group obtained by fetching the
  permutation  representation  from  the  database. This is due to a different
  numbering of the points, so the groups are permutation isomorphic.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpermgrp:= Group( AtlasGenerators( "M11", 1 ).generators );;[127X[104X
    [4X[25Xgap>[125X [27XPrint( permgrp, "\n" );[127X[104X
    [4X[28XGroup( [ ( 2,10)( 4,11)( 5, 7)( 8, 9), ( 1, 4, 3, 8)( 2, 5, 6, 9) ] )[128X[104X
    [4X[25Xgap>[125X [27Xpermgrp = act;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsConjugate( SymmetricGroup(11), permgrp, act );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X2.4-3 [33X[0;0YExample: Outer Automorphisms[133X[101X
  
  [33X[0;0YThe  straight  line  programs  for  applying outer automorphisms to standard
  generators  can  of course be used to define the automorphisms themselves as
  [5XGAP[105X mappings.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "G2(3)", IsStraightLineProgram );[127X[104X
    [4X[28XPrograms for G = G2(3):    (all refer to std. generators 1)[128X[104X
    [4X[28X-----------------------[128X[104X
    [4X[28Xclass repres.[128X[104X
    [4X[28Xpresentation[128X[104X
    [4X[28Xrepr. cyc. subg.[128X[104X
    [4X[28Xstd. gen. checker[128X[104X
    [4X[28Xautomorphisms:[128X[104X
    [4X[28X  2[128X[104X
    [4X[28Xmaxes (all 10):[128X[104X
    [4X[28X   1:  U3(3).2[128X[104X
    [4X[28X   2:  U3(3).2[128X[104X
    [4X[28X   3:  (3^(1+2)+x3^2):2S4[128X[104X
    [4X[28X   4:  (3^(1+2)+x3^2):2S4[128X[104X
    [4X[28X   5:  L3(3).2[128X[104X
    [4X[28X   6:  L3(3).2[128X[104X
    [4X[28X   7:  L2(8).3[128X[104X
    [4X[28X   8:  2^3.L3(2)[128X[104X
    [4X[28X   9:  L2(13)[128X[104X
    [4X[28X  10:  2^(1+4)+:3^2.2[128X[104X
    [4X[25Xgap>[125X [27Xprog:= AtlasProgram( "G2(3)", "automorphism", "2" ).program;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "G2(3)", Dimension, 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info ).generators;;[127X[104X
    [4X[25Xgap>[125X [27Ximgs:= ResultOfStraightLineProgram( prog, gens );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIf we are not suspicious whether the script really describes an automorphism
  then  we  should tell this to [5XGAP[105X, in order to avoid the expensive checks of
  the    properties    of    being   a   homomorphism   and   bijective   (see
  Section [14X'Reference: Creating Group Homomorphisms'[114X). This looks as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= Group( gens );;[127X[104X
    [4X[25Xgap>[125X [27Xaut:= GroupHomomorphismByImagesNC( g, g, gens, imgs );;[127X[104X
    [4X[25Xgap>[125X [27XSetIsBijective( aut, true );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  we  are  suspicious whether the script describes an automorphism then we
  might have the idea to check it with [5XGAP[105X, as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xaut:= GroupHomomorphismByImages( g, g, gens, imgs );;[127X[104X
    [4X[25Xgap>[125X [27XIsBijective( aut );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Note  that  even for a comparatively small group such as [22XG_2(3)[122X, this was a
  difficult task for [5XGAP[105X before version 4.3.)[133X
  
  [33X[0;0YOften one can form images under an automorphism [22Xα[122X, say, without creating the
  homomorphism  object.  This  is  obvious  for the standard generators of the
  group [22XG[122X themselves, but also for generators of a maximal subgroup [22XM[122X computed
  from  standard  generators of [22XG[122X, provided that the straight line programs in
  question  refer to the same standard generators. Note that the generators of
  [22XM[122X  are  given  by evaluating words in terms of standard generators of [22XG[122X, and
  their  images  under  [22Xα[122X  can be obtained by evaluating the same words at the
  images under [22Xα[122X of the standard generators of [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmax1:= AtlasProgram( "G2(3)", 1 ).program;;[127X[104X
    [4X[25Xgap>[125X [27Xmgens:= ResultOfStraightLineProgram( max1, gens );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= CompositionOfStraightLinePrograms( max1, prog );;[127X[104X
    [4X[25Xgap>[125X [27Xmimgs:= ResultOfStraightLineProgram( comp, gens );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  list  [10Xmgens[110X  is the list of generators of the first maximal subgroup of
  [22XG_2(3)[122X,  [10Xmimgs[110X  is  the  list  of images under the automorphism given by the
  straight  line  program  [10Xprog[110X.  Note  that  applying the program returned by
  [2XCompositionOfStraightLinePrograms[102X                                ([14XReference:
  CompositionOfStraightLinePrograms[114X)  means to apply first [10Xprog[110X and then [10Xmax1[110X.
  Since   we   have  already  constructed  the  [5XGAP[105X  object  representing  the
  automorphism, we can check whether the results are equal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmimgs = List( mgens, x -> x^aut );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHowever, it should be emphasized that using [10Xaut[110X requires a huge machinery of
  computations  behind the scenes, whereas applying the straight line programs
  [10Xprog[110X  and  [10Xmax1[110X involves only elementary operations with the generators. The
  latter  is  feasible  also for larger groups, for which constructing the [5XGAP[105X
  automorphism might be too hard.[133X
  
  
  [1X2.4-4 [33X[0;0YExample: Using Semi-presentations and Black Box Programs[133X[101X
  
  [33X[0;0YLet  us  suppose  that  we  want to restrict a representation of the Mathieu
  group  [22XM_12[122X  to a non-maximal subgroup of the type [22XL_2(11)[122X. The idea is that
  this  subgroup  can  be found as a maximal subgroup of a maximal subgroup of
  the  type  [22XM_11[122X,  which  is  itself  maximal  in  [22XM_12[122X. For that, we fetch a
  representation of [22XM_12[122X and use a straight line program for restricting it to
  the first maximal subgroup, which has the type [22XM_11[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "M12", NrMovedPoints, 12 );[127X[104X
    [4X[28Xrec( charactername := "1a+11a", groupname := "M12", id := "a",[128X[104X
    [4X[28X  identifier := [ "M12", [ "M12G1-p12aB0.m1", "M12G1-p12aB0.m2" ], 1, 12 ],[128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 12, rankAction := 2,[128X[104X
    [4X[28X  repname := "M12G1-p12aB0", repnr := 1, size := 95040, stabilizer := "M11",[128X[104X
    [4X[28X  standardization := 1, transitivity := 5, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27XgensM12:= AtlasGenerators( info.identifier );;[127X[104X
    [4X[25Xgap>[125X [27XrestM11:= AtlasProgram( "M12", "maxes", 1 );;[127X[104X
    [4X[25Xgap>[125X [27XgensM11:= ResultOfStraightLineProgram( restM11.program,[127X[104X
    [4X[25X>[125X [27X                                          gensM12.generators );[127X[104X
    [4X[28X[ (3,9)(4,12)(5,10)(6,8), (1,4,11,5)(2,10,8,3) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  [13Xcannot[113X  simply  apply  a  straight line program for a group to some
  generators, since they are not necessarily [13Xstandard[113X generators of the group.
  We check this property using a semi-presentation for [22XM_11[122X, see [14X6.1-7[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XcheckM11:= AtlasProgram( "M11", "check" );[127X[104X
    [4X[28Xrec( groupname := "M11", identifier := [ "M11", "M11G1-check1", 1, 1 ],[128X[104X
    [4X[28X  program := <straight line decision>, standardization := 1 )[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( checkM11.program, gensM11 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we are lucky that applying the appropriate program for [22XM_11[122X will give us
  the required generators for [22XL_2(11)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XrestL211:= AtlasProgram( "M11", "maxes", 2 );;[127X[104X
    [4X[25Xgap>[125X [27XgensL211:= ResultOfStraightLineProgram( restL211.program, gensM11 );[127X[104X
    [4X[28X[ (3,9)(4,12)(5,10)(6,8), (1,11,9)(2,12,8)(3,6,10) ][128X[104X
    [4X[25Xgap>[125X [27XG:= Group( gensL211 );;  Size( G );  IsSimple( G );[127X[104X
    [4X[28X660[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YUsually  representations  are not given in terms of standard generators. For
  example,  let  us  take  the  [22XM_11[122X  type  group returned by the [5XGAP[105X function
  [2XMathieuGroup[102X ([14XReference: MathieuGroup[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:= MathieuGroup( 11 );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfGroup( G );[127X[104X
    [4X[28X[ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ][128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( checkM11.program, gens );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  we  want to compute an [22XL_2(11)[122X type subgroup of this group, we can use a
  black  box  program  for  computing  standard generators, and then apply the
  straight line program for computing the restriction.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfind:= AtlasProgram( "M11", "find" );[127X[104X
    [4X[28Xrec( groupname := "M11", identifier := [ "M11", "M11G1-find1", 1, 1 ],[128X[104X
    [4X[28X  program := <black box program>, standardization := 1 )[128X[104X
    [4X[25Xgap>[125X [27Xstdgens:= ResultOfBBoxProgram( find.program, Group( gens ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( stdgens, Order );[127X[104X
    [4X[28X[ 2, 4 ][128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( checkM11.program, stdgens );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XgensL211:= ResultOfStraightLineProgram( restL211.program, stdgens );;[127X[104X
    [4X[25Xgap>[125X [27XList( gensL211, Order );[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XG:= Group( gensL211 );;  Size( G );  IsSimple( G );[127X[104X
    [4X[28X660[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X2.4-5 [33X[0;0YExample: Using the [5XGAP[105X[101X[1X Library of Tables of Marks[133X[101X
  
  [33X[0;0YThe  [5XGAP[105X  Library  of  Tables  of  Marks  (the  [5XGAP[105X package [5XTomLib[105X, [NMP11])
  provides,  for  many  almost  simple  groups,  information  for constructing
  representatives  of  all conjugacy classes of subgroups. If this information
  is   compatible   with  the  standard  generators  of  the  [5XATLAS[105X  of  Group
  Representations  then  we can use it to restrict any representation from the
  [5XATLAS[105X  to  prescribed  subgroups.  This  is  useful  in particular for those
  subgroups  for  which  the  [5XATLAS[105X  of  Group Representations itself does not
  contain a straight line program.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "A5" );[127X[104X
    [4X[28XTableOfMarks( "A5" )[128X[104X
    [4X[25Xgap>[125X [27Xinfo:= StandardGeneratorsInfo( tom );[127X[104X
    [4X[28X[ rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5", [128X[104X
    [4X[28X      generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] [128X[104X
    [4X[28X         ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [9Xtrue[109X value of the component [10XATLAS[110X indicates that the information stored
  on  [10Xtom[110X  refers  to  the standard generators of type [22X1[122X in the [5XATLAS[105X of Group
  Representations.[133X
  
  [33X[0;0YWe  want  to  restrict  a  [22X4[122X-dimensional integral representation of [22XA_5[122X to a
  Sylow [22X2[122X subgroup of [22XA_5[122X, and use [2XRepresentativeTomByGeneratorsNC[102X ([14XReference:
  RepresentativeTomByGeneratorsNC[114X) for that.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "A5", Ring, Integers, Dimension, 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xstdgens:= AtlasGenerators( info.identifier );[127X[104X
    [4X[28Xrec( dim := 4,[128X[104X
    [4X[28X  generators := [ [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [128X[104X
    [4X[28X          [ -1, -1, -1, -1 ] ], [128X[104X
    [4X[28X      [ [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ] ] ], [128X[104X
    [4X[28X  groupname := "A5", id := "", identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ], [128X[104X
    [4X[28X  repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60, [128X[104X
    [4X[28X  standardization := 1, type := "matint" )[128X[104X
    [4X[25Xgap>[125X [27Xorders:= OrdersTom( tom );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 10, 12, 60 ][128X[104X
    [4X[25Xgap>[125X [27Xpos:= Position( orders, 4 );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xsub:= RepresentativeTomByGeneratorsNC( tom, pos, stdgens.generators );[127X[104X
    [4X[28X<matrix group of size 4 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup( sub );[127X[104X
    [4X[28X[ [ [ 1, 0, 0, 0 ], [ -1, -1, -1, -1 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ], [128X[104X
    [4X[28X  [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ -1, -1, -1, -1 ] ] ][128X[104X
  [4X[32X[104X
  
  
  [1X2.4-6 [33X[0;0YExample: Index [22X770[122X[101X[1X Subgroups in [22XM_22[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  sporadic simple Mathieu group [22XM_22[122X contains a unique class of subgroups
  of  index  [22X770[122X  (and  order  [22X576[122X).  This can be seen for example using [5XGAP[105X's
  Library of Tables of Marks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "M22" );[127X[104X
    [4X[28XTableOfMarks( "M22" )[128X[104X
    [4X[25Xgap>[125X [27Xsubord:= Size( UnderlyingGroup( tom ) ) / 770;[127X[104X
    [4X[28X576[128X[104X
    [4X[25Xgap>[125X [27Xord:= OrdersTom( tom );;[127X[104X
    [4X[25Xgap>[125X [27Xtomstabs:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = subord );[127X[104X
    [4X[28X[ 144 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  permutation  representation  of  [22XM_22[122X  on  the  right  cosets of such a
  subgroup [22XS[122X is contained in the [5XATLAS[105X of Group Representations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "M22", NrMovedPoints, 770 );[127X[104X
    [4X[28XRepresentations for G = M22:    (all refer to std. generators 1)[128X[104X
    [4X[28X----------------------------[128X[104X
    [4X[28X12: G <= Sym(770) rank 9, on cosets of (A4xA4):4 < 2^4:A6[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe now verify the information shown about the point stabilizer and about the
  maximal overgroups of [22XS[122X in [22XM_22[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxtom:= MaximalSubgroupsTom( tom );[127X[104X
    [4X[28X[ [ 155, 154, 153, 152, 151, 150, 146, 145 ], [128X[104X
    [4X[28X  [ 22, 77, 176, 176, 231, 330, 616, 672 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( tomstabs, i -> List( maxtom[1], j -> ContainedTom( tom, i, j ) ) );[127X[104X
    [4X[28X[ [ 0, 10, 0, 0, 0, 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see that the only maximal subgroups of [22XM_22[122X that contain [22XS[122X have index [22X77[122X
  in  [22XM_22[122X.  According  to the [5XATLAS[105X of Finite Groups, these maximal subgroups
  have  the  structure  [22X2^4:A_6[122X.  From  that and from the structure of [22XA_6[122X, we
  conclude that [22XS[122X has the structure [22X2^4:(3^2:4)[122X.[133X
  
  [33X[0;0YAlternatively,  we  look at the permutation representation of degree [22X770[122X. We
  fetch  it  from  the  [5XATLAS[105X  of  Group Representations. There is exactly one
  nontrivial  block  system  for this representation, with [22X77[122X blocks of length
  [22X10[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "M22", NrMovedPoints, 770 );[127X[104X
    [4X[28X<permutation group of size 443520 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xallbl:= AllBlocks( g );;[127X[104X
    [4X[25Xgap>[125X [27XList( allbl, Length );[127X[104X
    [4X[28X[ 10 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFurthermore, [5XGAP[105X computes that the point stabilizer [22XS[122X has the structure [22X(A_4
  × A_4):4[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xstab:= Stabilizer( g, 1 );[127X[104X
    [4X[28X<permutation group of size 576 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XStructureDescription( stab );[127X[104X
    [4X[28X"(A4 x A4) : C4"[128X[104X
    [4X[25Xgap>[125X [27Xblocks:= Orbit( g, allbl[1], OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xact:= Action( g, blocks, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription( Stabilizer( act, 1 ) );[127X[104X
    [4X[28X"(C2 x C2 x C2 x C2) : A6"[128X[104X
  [4X[32X[104X
  
  
  [1X2.4-7 [33X[0;0YExample: Index [22X462[122X[101X[1X Subgroups in [22XM_22[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  [5XATLAS[105X  of  Group  Representations contains three degree [22X462[122X permutation
  representations of the group [22XM_22[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "M22", NrMovedPoints, 462 );[127X[104X
    [4X[28XRepresentations for G = M22:    (all refer to std. generators 1)[128X[104X
    [4X[28X----------------------------[128X[104X
    [4X[28X7: G <= Sym(462a) rank 5, on cosets of 2^4:A5 < 2^4:A6[128X[104X
    [4X[28X8: G <= Sym(462b) rank 8, on cosets of 2^4:A5 < L3(4), 2^4:S5[128X[104X
    [4X[28X9: G <= Sym(462c) rank 8, on cosets of 2^4:A5 < L3(4), 2^4:A6[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  point  stabilizers  in  these  three representations have the structure
  [22X2^4:A_5[122X.  Using  [5XGAP[105X's  Library  of  Tables of Marks, we can show that these
  stabilizers are exactly the three classes of subgroups of order [22X960[122X in [22XM_22[122X.
  For that, we first verify that the group generators stored in [5XGAP[105X's table of
  marks  coincide  with  the  standard  generators  used by the [5XATLAS[105X of Group
  Representations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "M22" );[127X[104X
    [4X[28XTableOfMarks( "M22" )[128X[104X
    [4X[25Xgap>[125X [27Xgenstom:= GeneratorsOfGroup( UnderlyingGroup( tom ) );;[127X[104X
    [4X[25Xgap>[125X [27XcheckM22:= AtlasProgram( "M22", "check" );[127X[104X
    [4X[28Xrec( groupname := "M22", identifier := [ "M22", "M22G1-check1", 1, 1 ],[128X[104X
    [4X[28X  program := <straight line decision>, standardization := 1 )[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( checkM22.program, genstom );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere are indeed three classes of subgroups of order [22X960[122X in [22XM_22[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord:= OrdersTom( tom );;[127X[104X
    [4X[25Xgap>[125X [27Xtomstabs:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 960 );[127X[104X
    [4X[28X[ 147, 148, 149 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  representatives  of  these  three  classes  in  the  three
  representations  [10X462a[110X, [10X462b[110X, and [10X462c[110X. We see that each of the three classes
  occurs as a point stabilizer in exactly one of the three representations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xatlasreps:= AllAtlasGeneratingSetInfos( "M22", NrMovedPoints, 462 );[127X[104X
    [4X[28X[ rec( charactername := "1a+21a+55a+154a+231a", groupname := "M22",[128X[104X
    [4X[28X      id := "a",[128X[104X
    [4X[28X      identifier := [ "M22", [ "M22G1-p462aB0.m1", "M22G1-p462aB0.m2" ], 1,[128X[104X
    [4X[28X          462 ], isPrimitive := false, p := 462, rankAction := 5,[128X[104X
    [4X[28X      repname := "M22G1-p462aB0", repnr := 7, size := 443520,[128X[104X
    [4X[28X      stabilizer := "2^4:A5 < 2^4:A6", standardization := 1,[128X[104X
    [4X[28X      transitivity := 1, type := "perm" ),[128X[104X
    [4X[28X  rec( charactername := "1a+21a^2+55a+154a+210a", groupname := "M22",[128X[104X
    [4X[28X      id := "b",[128X[104X
    [4X[28X      identifier := [ "M22", [ "M22G1-p462bB0.m1", "M22G1-p462bB0.m2" ], 1,[128X[104X
    [4X[28X          462 ], isPrimitive := false, p := 462, rankAction := 8,[128X[104X
    [4X[28X      repname := "M22G1-p462bB0", repnr := 8, size := 443520,[128X[104X
    [4X[28X      stabilizer := "2^4:A5 < L3(4), 2^4:S5", standardization := 1,[128X[104X
    [4X[28X      transitivity := 1, type := "perm" ),[128X[104X
    [4X[28X  rec( charactername := "1a+21a^2+55a+154a+210a", groupname := "M22",[128X[104X
    [4X[28X      id := "c",[128X[104X
    [4X[28X      identifier := [ "M22", [ "M22G1-p462cB0.m1", "M22G1-p462cB0.m2" ], 1,[128X[104X
    [4X[28X          462 ], isPrimitive := false, p := 462, rankAction := 8,[128X[104X
    [4X[28X      repname := "M22G1-p462cB0", repnr := 9, size := 443520,[128X[104X
    [4X[28X      stabilizer := "2^4:A5 < L3(4), 2^4:A6", standardization := 1,[128X[104X
    [4X[28X      transitivity := 1, type := "perm" ) ][128X[104X
    [4X[25Xgap>[125X [27Xatlasreps:= List( atlasreps, AtlasGroup );;[127X[104X
    [4X[25Xgap>[125X [27Xtomstabreps:= List( atlasreps, G -> List( tomstabs,[127X[104X
    [4X[25X>[125X [27Xi -> RepresentativeTomByGenerators( tom, i, GeneratorsOfGroup( G ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( tomstabreps, x -> List( x, NrMovedPoints ) );[127X[104X
    [4X[28X[ [ 462, 462, 461 ], [ 460, 462, 462 ], [ 462, 461, 462 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YMore   precisely,   we   see   that  the  point  stabilizers  in  the  three
  representations  [10X462a[110X, [10X462b[110X, [10X462c[110X lie in the subgroup classes [22X149[122X, [22X147[122X, [22X148[122X,
  respectively, of the table of marks.[133X
  
  [33X[0;0YThe  point  stabilizers in the representations [10X462b[110X and [10X462c[110X are isomorphic,
  but not isomorphic with the point stabilizer in [10X462a[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xstabs:= List( atlasreps, G -> Stabilizer( G, 1 ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( stabs, IdGroup );[127X[104X
    [4X[28X[ [ 960, 11358 ], [ 960, 11357 ], [ 960, 11357 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( stabs, PerfectIdentification );[127X[104X
    [4X[28X[ [ 960, 2 ], [ 960, 1 ], [ 960, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  three  representations  are  imprimitive.  The containment of the point
  stabilizers  in maximal subgroups of [22XM_22[122X can be computed using the table of
  marks of [22XM_22[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxtom:= MaximalSubgroupsTom( tom );[127X[104X
    [4X[28X[ [ 155, 154, 153, 152, 151, 150, 146, 145 ], [128X[104X
    [4X[28X  [ 22, 77, 176, 176, 231, 330, 616, 672 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( tomstabs, i -> List( maxtom[1], j -> ContainedTom( tom, i, j ) ) );[127X[104X
    [4X[28X[ [ 21, 0, 0, 0, 1, 0, 0, 0 ], [ 21, 6, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 6, 0, 0, 0, 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe see:[133X
  
  [30X    [33X[0;6YThe point stabilizers in [10X462a[110X (subgroups in the class [22X149[122X of the table
        of  marks) are contained only in maximal subgroups in class [22X154[122X; these
        groups have the structure [22X2^4:A_6[122X.[133X
  
  [30X    [33X[0;6YThe  point  stabilizers  in  [10X462b[110X  (subgroups  in  the  class [22X147[122X) are
        contained  in  maximal  subgroups  in  the  classes [22X155[122X and [22X151[122X; these
        groups have the structures [22XL_3(4)[122X and [22X2^4:S_5[122X, respectively.[133X
  
  [30X    [33X[0;6YThe  point  stabilizers  in  [10X462c[110X  (subgroups  in  the  class [22X148[122X) are
        contained in maximal subgroups in the classes [22X155[122X and [22X154[122X.[133X
  
  [33X[0;0YWe  identify the supergroups of the point stabilizers by computing the block
  systems.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbl:= List( atlasreps, AllBlocks );;[127X[104X
    [4X[25Xgap>[125X [27XList( bl, Length );[127X[104X
    [4X[28X[ 1, 3, 2 ][128X[104X
    [4X[25Xgap>[125X [27XList( bl, l -> List( l, Length ) );[127X[104X
    [4X[28X[ [ 6 ], [ 21, 21, 2 ], [ 21, 6 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that the two block systems with blocks of length [22X21[122X for [10X462b[110X belong to
  the same supergroups (of the type [22XL_3(4)[122X); each of these subgroups fixes two
  different subsets of [22X21[122X points.[133X
  
  [33X[0;0YThe  representation [10X462a[110X is [13Xmultiplicity-free[113X, that is, it splits into a sum
  of pairwise nonisomorphic irreducible representations. This can be seen from
  the  fact  that  the  rank  of this permutation representation (that is, the
  number  of  orbits  of  the  point  stabilizer)  is  five;  each permutation
  representation with this property is multiplicity-free.[133X
  
  [33X[0;0YThe other two representations have rank eight. We have seen the ranks in the
  overview that was shown by [2XDisplayAtlasInfo[102X ([14X3.5-1[114X) in the beginning. Now we
  compute the ranks from the permutation groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( atlasreps, RankAction );[127X[104X
    [4X[28X[ 5, 8, 8 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  fact  the  two  representations  [10X462b[110X and [10X462c[110X have the same permutation
  character. We check this by computing the possible permutation characters of
  degree [22X462[122X for [22XM_22[122X, and decomposing them into irreducible characters, using
  the character table from [5XGAP[105X's Character Table Library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M22" );;[127X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( t, 462 );[127X[104X
    [4X[28X[ Character( CharacterTable( "M22" ), [ 462, 30, 3, 2, 2, 2, 3, 0, 0, 0, 0, 0 [128X[104X
    [4X[28X     ] ), Character( CharacterTable( "M22" ), [128X[104X
    [4X[28X    [ 462, 30, 12, 2, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XMatScalarProducts( t, Irr( t ), perms );[127X[104X
    [4X[28X[ [ 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 1, 2, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn   particular,   we   see   that   the   rank  eight  characters  are  not
  multiplicity-free.[133X
  
