  
  [1X3 [33X[0;0YThe User Interface of the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YThe  [13Xuser  interface[113X  is  the  part  of the [5XGAP[105X interface that allows one to
  display information about the current contents of the database and to access
  individual  data  (perhaps  from  a  remote  server, see Section [14X4.3-1[114X). The
  corresponding  functions  are described in this chapter. See Section [14X2.4[114X for
  some small examples how to use the functions of the interface.[133X
  
  [33X[0;0YExtensions  of  the [5XAtlasRep[105X package are regarded as another part of the [5XGAP[105X
  interface,  they  are described in Chapter [14X5[114X. Finally, the low level part of
  the interface are described in Chapter [14X7[114X.[133X
  
  [33X[0;0YFor  some  of  the  examples  in  this chapter, the [5XGAP[105X packages [5XCTblLib[105X and
  [5XTomLib[105X are needed, so we load them.[133X
  
  [4X[32X  Example  [32X[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
  
  
  [1X3.1 [33X[0;0YAccessing vs. Constructing Representations[133X[101X
  
  [33X[0;0YNote  that  [13Xaccessing[113X the data means in particular that it is [13Xnot[113X the aim of
  this  package  to [13Xconstruct[113X representations from known ones. For example, if
  at  least  one  permutation  representation  for  a group [22XG[122X is stored but no
  matrix   representation   in   a   positive   characteristic  [22Xp[122X,  say,  then
  [2XOneAtlasGeneratingSetInfo[102X  ([14X3.5-5[114X)  returns  [9Xfail[109X  when  it  is  asked for a
  description of an available set of matrix generators for [22XG[122X in characteristic
  [22Xp[122X,  although  such a representation can be obtained by reduction modulo [22Xp[122X of
  an integral matrix representation, which in turn can be constructed from any
  permutation representation.[133X
  
  
  [1X3.2 [33X[0;0YGroup Names Used in the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YWhen  you  access  data  via  the [5XAtlasRep[105X package, you specify the group in
  question  by  an  admissible [13Xname[113X. Thus it is essential to know these names,
  which are called [13Xthe [5XGAP[105X names[113X of the group in the following.[133X
  
  [33X[0;0YFor  a  group  [22XG[122X, say, whose character table is available in [5XGAP[105X's Character
  Table  Library,  the  admissible names of [22XG[122X are the admissible names of this
  character  table.  If  [22XG[122X  is  almost simple, one such name is the [2XIdentifier[102X
  ([14XReference: Identifier (for character tables)[114X) value of the character table,
  see [2XAccess to Library Character Tables[102X ([14XCTblLib: Access to Library Character
  Tables[114X).  This name is usually very similar to the name used in the [5XATLAS[105X of
  Finite  Groups  [CCNPW85].  For  example, [10X"M22"[110X is a [5XGAP[105X name of the Mathieu
  group [22XM_22[122X, [10X"12_1.U4(3).2_1"[110X is a [5XGAP[105X name of [22X12_1.U_4(3).2_1[122X, the two names
  [10X"S5"[110X  and [10X"A5.2"[110X are [5XGAP[105X names of the symmetric group [22XS_5[122X, and the two names
  [10X"F3+"[110X and [10X"Fi24'"[110X are [5XGAP[105X names of the simple Fischer group [22XFi_24^'[122X.[133X
  
  [33X[0;0YWhen a [5XGAP[105X name is required as an input of a package function, this input is
  case  insensitive.  For  example,  both [10X"A5"[110X and [10X"a5"[110X are valid arguments of
  [2XDisplayAtlasInfo[102X ([14X3.5-1[114X).[133X
  
  [33X[0;0YInternally,  for example as part of filenames (see Section [14X7.6[114X), the package
  uses  names  that  may  differ  from  the  [5XGAP[105X names; these names are called
  [13X[5XATLAS[105X-file names[113X. For example, [10X"A5"[110X, [10X"TE62"[110X, and [10X"F24"[110X are [5XATLAS[105X-file names.
  Of  these,  only  [10X"A5"[110X  is  also  a  [5XGAP[105X  name,  but  the other two are not;
  corresponding [5XGAP[105X names are [10X"2E6(2)"[110X and [10X"Fi24'"[110X, respectively.[133X
  
  
  [1X3.3 [33X[0;0YStandard Generators Used in the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YFor the general definition of [13Xstandard generators[113X of a group, see [Wil96].[133X
  
  [33X[0;0YSeveral  [13Xdifferent[113X  standard  generators  may  be  defined  for a group, the
  definitions can be found at[133X
  
  [33X[0;0Y[7Xhttp://brauer.maths.qmul.ac.uk/Atlas[107X[133X
  
  [33X[0;0YWhen  one specifies the standardization, the [22Xi[122X-th set of standard generators
  is  denoted  by  the  number [22Xi[122X. Note that when more than one set of standard
  generators  is  defined  for  a group, one must be careful to use [13Xcompatible
  standardization[113X.  For  example,  the  straight  line programs, straight line
  decisions  and  black  box  programs  in  the  database  refer to a specific
  standardization  of  their  inputs.  That  is,  a  straight line program for
  computing  generators of a certain subgroup of a group [22XG[122X is defined only for
  a  specific  set  of  standard  generators of [22XG[122X, and applying the program to
  matrix or permutation generators of [22XG[122X but w.r.t. a different standardization
  may  yield  unpredictable  results.  Therefore  the  results returned by the
  functions   described   in   this  chapter  contain  information  about  the
  standardizations they refer to.[133X
  
  
  [1X3.4 [33X[0;0YClass Names Used in the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YFor  each  straight  line program (see [2XAtlasProgram[102X ([14X3.5-3[114X)) that is used to
  compute  lists  of  class  representatives,  it is essential to describe the
  classes  in  which these elements lie. Therefore, in these cases the records
  returned  by  the  function [2XAtlasProgram[102X ([14X3.5-3[114X) contain a component [10Xoutputs[110X
  with value a list of [13Xclass names[113X.[133X
  
  [33X[0;0YCurrently  we  define  these  class names only for simple groups and certain
  extensions of simple groups, see Section [14X3.4-1[114X. The function [2XAtlasClassNames[102X
  ([14X3.4-2[114X)  can  be  used to compute the list of class names from the character
  table in the [5XGAP[105X Library.[133X
  
  
  [1X3.4-1 [33X[0;0YDefinition of [5XATLAS[105X[101X[1X Class Names[133X[101X
  
  [33X[0;0YFor  the definition of class names of an almost simple group, we assume that
  the  ordinary  character tables of all nontrivial normal subgroups are shown
  in the [5XATLAS[105X of Finite Groups [CCNPW85].[133X
  
  [33X[0;0YEach  class name is a string consisting of the element order of the class in
  question  followed  by  a  combination  of  capital letters, digits, and the
  characters  [10X'[110X and [10X-[110X (starting with a capital letter). For example, [10X1A[110X, [10X12A1[110X,
  and  [10X3B'[110X  denote  the  class  that contains the identity element, a class of
  element order [22X12[122X, and a class of element order [22X3[122X, respectively.[133X
  
  [31X1[131X   [33X[0;6YFor  the  table  of  a  [13Xsimple[113X  group, the class names are the same as
        returned  by  the  two argument version of the [5XGAP[105X function [2XClassNames[102X
        ([14XReference:  ClassNames[114X),  cf. [CCNPW85,  Chapter  7,  Section 5]: The
        classes  are  arranged  w.r.t. increasing  element  order and for each
        element  order  w.r.t. decreasing  centralizer  order,  the  conjugacy
        classes  that  contain  elements of order [22Xn[122X are named [22Xn[122X[10XA[110X, [22Xn[122X[10XB[110X, [22Xn[122X[10XC[110X, [22X...[122X;
        the  alphabet  used  here  is potentially infinite, and reads [10XA[110X, [10XB[110X, [10XC[110X,
        [22X...[122X, [10XZ[110X, [10XA1[110X, [10XB1[110X, [22X...[122X, [10XA2[110X, [10XB2[110X, [22X...[122X.[133X
  
        [33X[0;6YFor  example,  the classes of the alternating group [22XA_5[122X have the names
        [10X1A[110X, [10X2A[110X, [10X3A[110X, [10X5A[110X, and [10X5B[110X.[133X
  
  [31X2[131X   [33X[0;6YNext we consider the case of an [13Xupward extension[113X [22XG.A[122X of a simple group
        [22XG[122X by a [13Xcyclic[113X group of order [22XA[122X. The [5XATLAS[105X defines class names for each
        element  [22Xg[122X of [22XG.A[122X only w.r.t. the group [22XG.a[122X, say, that is generated by
        [22XG[122X  and  [22Xg[122X; namely, there is a power of [22Xg[122X (with the exponent coprime to
        the order of [22Xg[122X) for which the class has a name of the same form as the
        class  names  for  simple  groups,  and  the  name  of  the class of [22Xg[122X
        w.r.t. [22XG.a[122X  is  then  obtained  from this name by appending a suitable
        number  of  dashes [10X'[110X.  So  dashed  class  names refer exactly to those
        classes that are not printed in the [5XATLAS[105X.[133X
  
        [33X[0;6YFor  example, those classes of the symmetric group [22XS_5[122X that do not lie
        in  [22XA_5[122X  have the names [10X2B[110X, [10X4A[110X, and [10X6A[110X. The outer classes of the group
        [22XL_2(8).3[122X  have  the  names  [10X3B[110X,  [10X6A[110X,  [10X9D[110X, and [10X3B'[110X, [10X6A'[110X, [10X9D'[110X. The outer
        elements  of  order  [22X5[122X  in  the group [22XSz(32).5[122X lie in the classes with
        names [10X5B[110X, [10X5B'[110X, [10X5B''[110X, and [10X5B'''[110X.[133X
  
        [33X[0;6YIn the group [22XG.A[122X, the class of [22Xg[122X may fuse with other classes. The name
        of  the  class  of [22Xg[122X in [22XG.A[122X is obtained from the names of the involved
        classes of [22XG.a[122X by concatenating their names after removing the element
        order part from all of them except the first one.[133X
  
        [33X[0;6YFor  example,  the  elements  of  order  [22X9[122X  in the group [22XL_2(27).6[122X are
        contained  in the subgroup [22XL_2(27).3[122X but not in [22XL_2(27)[122X. In [22XL_2(27).3[122X,
        they  lie  in  the  classes  [10X9A[110X, [10X9A'[110X, [10X9B[110X, and [10X9B'[110X; in [22XL_2(27).6[122X, these
        classes fuse to [10X9AB[110X and [10X9A'B'[110X.[133X
  
  [31X3[131X   [33X[0;6YNow  we  define  class  names  for  [13Xgeneral upward extensions[113X [22XG.A[122X of a
        simple  group  [22XG[122X.  Each  element  [22Xg[122X  of such a group lies in an upward
        extension  [22XG.a[122X  by  a cyclic group, and the class names w.r.t. [22XG.a[122X are
        already  defined.  The  name  of  the class of [22Xg[122X in [22XG.A[122X is obtained by
        concatenating  the  names  of  the  classes in the orbit of [22XG.A[122X on the
        classes  of  cyclic  upward  extensions of [22XG[122X, after ordering the names
        lexicographically and removing the element order part from all of them
        except  the  first one. An [13Xexception[113X is the situation where dashed and
        non-dashed  class  names  appear in an orbit; in this case, the dashed
        names are omitted.[133X
  
        [33X[0;6YFor  example,  the  classes  [10X21A[110X and [10X21B[110X of the group [22XU_3(5).3[122X fuse in
        [22XU_3(5).S_3[122X  to the class [10X21AB[110X, and the class [10X2B[110X of [22XU_3(5).2[122X fuses with
        the  involution  classes  [10X2B'[110X,  [10X2B''[110X  in  the  groups  [22XU_3(5).2^'[122X  and
        [22XU_3(5).2^{''}[122X to the class [10X2B[110X of [22XU_3(5).S_3[122X.[133X
  
        [33X[0;6YIt  may  happen  that  some names in the [10Xoutputs[110X component of a record
        returned by [2XAtlasProgram[102X ([14X3.5-3[114X) do not uniquely determine the classes
        of   the  corresponding  elements.  For  example,  the  (algebraically
        conjugate)  classes  [10X39A[110X  and  [10X39B[110X  of  the  group  [22XCo_1[122X have not been
        distinguished  yet. In such cases, the names used contain a minus sign
        [10X-[110X,  and  mean  [21Xone  of  the classes in the range described by the name
        before  and  the  name after the minus sign[121X; the element order part of
        the  name  does not appear after the minus sign. So the name [10X39A-B[110X for
        the  group  [22XCo_1[122X means [10X39A[110X or [10X39B[110X, and the name [10X20A-B'''[110X for the group
        [22XSz(32).5[122X  means  one  of the classes of element order [22X20[122X in this group
        (these classes lie outside the simple group [22XSz[122X).[133X
  
  [31X4[131X   [33X[0;6YFor  a  [13Xdownward  extension[113X  [22Xm.G.A[122X  of an almost simple group [22XG.A[122X by a
        cyclic  group  of  order  [22Xm[122X, let [22Xπ[122X denote the natural epimorphism from
        [22Xm.G.A[122X onto [22XG.A[122X. Each class name of [22Xm.G.A[122X has the form [10XnX_0[110X, [10XnX_1[110X etc.,
        where  [10XnX[110X is the class name of the image under [22Xπ[122X, and the indices [10X0[110X, [10X1[110X
        etc.  are chosen according to the position of the class in the lifting
        order  rows for [22XG[122X, see [CCNPW85, Chapter 7, Section 7, and the example
        in Section 8]).[133X
  
        [33X[0;6YFor example, if [22Xm = 6[122X then [10X1A_1[110X and [10X1A_5[110X denote the classes containing
        the  generators of the kernel of [22Xπ[122X, that is, central elements of order
        [22X6[122X.[133X
  
  [1X3.4-2 AtlasClassNames[101X
  
  [29X[2XAtlasClassNames[102X( [3Xtbl[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya list of class names.[133X
  
  [33X[0;0YLet  [3Xtbl[103X  be the ordinary or modular character table of a group [22XG[122X, say, that
  is  almost simple or a downward extension of an almost simple group and such
  that  [3Xtbl[103X  is an [5XATLAS[105X table from the [5XGAP[105X Character Table Library, according
  to  its  [2XInfoText[102X  ([14XReference: InfoText[114X) value. Then [2XAtlasClassNames[102X returns
  the  list of class names for [22XG[122X, as defined in Section [14X3.4-1[114X. The ordering of
  class names is the same as the ordering of the columns of [3Xtbl[103X.[133X
  
  [33X[0;0Y(The  function may work also for character tables that are not [5XATLAS[105X tables,
  but then clearly the class names returned are somewhat arbitrary.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( CharacterTable( "L3(4).3" ) );[127X[104X
    [4X[28X[ "1A", "2A", "3A", "4ABC", "5A", "5B", "7A", "7B", "3B", "3B'", "3C", "3C'", [128X[104X
    [4X[28X  "6B", "6B'", "15A", "15A'", "15B", "15B'", "21A", "21A'", "21B", "21B'" ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( CharacterTable( "U3(5).2" ) );[127X[104X
    [4X[28X[ "1A", "2A", "3A", "4A", "5A", "5B", "5CD", "6A", "7AB", "8AB", "10A", "2B", [128X[104X
    [4X[28X  "4B", "6D", "8C", "10B", "12B", "20A", "20B" ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( CharacterTable( "L2(27).6" ) );[127X[104X
    [4X[28X[ "1A", "2A", "3AB", "7ABC", "13ABC", "13DEF", "14ABC", "2B", "4A", "26ABC", [128X[104X
    [4X[28X  "26DEF", "28ABC", "28DEF", "3C", "3C'", "6A", "6A'", "9AB", "9A'B'", "6B", [128X[104X
    [4X[28X  "6B'", "12A", "12A'" ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( CharacterTable( "L3(4).3.2_2" ) );[127X[104X
    [4X[28X[ "1A", "2A", "3A", "4ABC", "5AB", "7A", "7B", "3B", "3C", "6B", "15A", [128X[104X
    [4X[28X  "15B", "21A", "21B", "2C", "4E", "6E", "8D", "14A", "14B" ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( CharacterTable( "3.A6" ) );[127X[104X
    [4X[28X[ "1A_0", "1A_1", "1A_2", "2A_0", "2A_1", "2A_2", "3A_0", "3B_0", "4A_0", [128X[104X
    [4X[28X  "4A_1", "4A_2", "5A_0", "5A_1", "5A_2", "5B_0", "5B_1", "5B_2" ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( CharacterTable( "2.A5.2" ) );[127X[104X
    [4X[28X[ "1A_0", "1A_1", "2A_0", "3A_0", "3A_1", "5AB_0", "5AB_1", "2B_0", "4A_0", [128X[104X
    [4X[28X  "4A_1", "6A_0", "6A_1" ][128X[104X
  [4X[32X[104X
  
  [1X3.4-3 AtlasCharacterNames[101X
  
  [29X[2XAtlasCharacterNames[102X( [3Xtbl[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya list of character names.[133X
  
  [33X[0;0YLet  [3Xtbl[103X  be  the  ordinary  or  modular  character table of a simple group.
  [2XAtlasCharacterNames[102X returns a list of strings, the [22Xi[122X-th entry being the name
  of  the  [22Xi[122X-th irreducible character of [3Xtbl[103X; this name consists of the degree
  of this character followed by distinguishing lowercase letters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasCharacterNames( CharacterTable( "A5" ) );                   [127X[104X
    [4X[28X[ "1a", "3a", "3b", "4a", "5a" ][128X[104X
  [4X[32X[104X
  
  
  [1X3.5 [33X[0;0YAccessing Data of the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YNote  that  the  output  of the examples in this section refers to a perhaps
  outdated  table of contents; the current version of the database may contain
  more information than is shown here.[133X
  
  [1X3.5-1 DisplayAtlasInfo[101X
  
  [29X[2XDisplayAtlasInfo[102X( [[3Xlistofnames[103X, ][[3Xstd[103X, ][[3X"contents"[103X, [3Xsources[103X, ][[3X...[103X] ) [32X function
  [29X[2XDisplayAtlasInfo[102X( [3Xgapname[103X[, [3Xstd[103X][, [3X...[103X] ) [32X function
  
  [33X[0;0YThis  function lists the information available via the [5XAtlasRep[105X package, for
  the  given input. Depending on whether remote access to data is enabled (see
  Section  [14X4.3-1[114X), all the data provided by the [5XATLAS[105X of Group Representations
  or only those in the local installation are considered.[133X
  
  [33X[0;0YAn   interactive   alternative   to   [2XDisplayAtlasInfo[102X   is   the   function
  [2XBrowseAtlasInfo[102X ([14XBrowse: BrowseAtlasInfo[114X), see [BL11].[133X
  
  [33X[0;0YCalled   without   arguments,   [2XDisplayAtlasInfo[102X  prints  an  overview  what
  information the [5XATLAS[105X of Group Representations provides. One line is printed
  for each group [22XG[122X, with the following columns.[133X
  
  [8X[10Xgroup[110X[108X
        [33X[0;6Ythe [5XGAP[105X name of [22XG[122X (see Section [14X3.2[114X),[133X
  
  [8X[10X#[110X[108X
        [33X[0;6Ythe  number  of faithful representations stored for [22XG[122X that satisfy the
        additional conditions given (see below),[133X
  
  [8X[10Xmaxes[110X[108X
        [33X[0;6Ythe   number   of  available  straight  line  programs  for  computing
        generators of maximal subgroups of [22XG[122X,[133X
  
  [8X[10Xcl[110X[108X
        [33X[0;6Ya  [10X+[110X  sign  if  at  least one program for computing representatives of
        conjugacy classes of elements of [22XG[122X is stored,[133X
  
  [8X[10Xcyc[110X[108X
        [33X[0;6Ya  [10X+[110X  sign  if  at  least one program for computing representatives of
        classes of maximally cyclic subgroups of [22XG[122X is stored,[133X
  
  [8X[10Xout[110X[108X
        [33X[0;6Ydescriptions  of  outer  automorphisms  of  [22XG[122X  for  which at least one
        program is stored,[133X
  
  [8X[10Xfnd[110X[108X
        [33X[0;6Ya  [10X+[110X  sign  if  at least one program is available for finding standard
        generators,[133X
  
  [8X[10Xchk[110X[108X
        [33X[0;6Ya  [10X+[110X  sign if at least one program is available for checking whether a
        set of generators is a set of standard generators, and[133X
  
  [8X[10Xprs[110X[108X
        [33X[0;6Ya  [10X+[110X  sign  if  at  least  one  program  is  available  that encodes a
        presentation.[133X
  
  [33X[0;0Y(The  list  can  be  printed  to  the screen or can be fed into a pager, see
  Section [14X4.3-5[114X.)[133X
  
  [33X[0;0YCalled  with  a  list  [3Xlistofnames[103X of strings that are [5XGAP[105X names for a group
  from  the  [5XATLAS[105X  of  Group  Representations,  [2XDisplayAtlasInfo[102X  prints  the
  overview described above but restricted to the groups in this list.[133X
  
  [33X[0;0YIn  addition  to  or  instead  of  [3Xlistofnames[103X,  the string [10X"contents"[110X and a
  description  [3Xsources[103X  of  the  data may be given about which the overview is
  formed. See below for admissible values of [3Xsources[103X.[133X
  
  [33X[0;0YCalled  with  a string [3Xgapname[103X that is a [5XGAP[105X name for a group from the [5XATLAS[105X
  of  Group  Representations,  [2XDisplayAtlasInfo[102X  prints  an  overview  of  the
  information  that  is available for this group. One line is printed for each
  faithful  representation,  showing  the number of this representation (which
  can be used in calls of [2XAtlasGenerators[102X ([14X3.5-2[114X)), and a string of one of the
  following forms; in both cases, [3Xid[103X is a (possibly empty) string.[133X
  
  [8X[10XG <= Sym([3Xn[103X[10X[3Xid[103X[10X)[110X[108X
        [33X[0;6Ydenotes  a  permutation  representation  of degree [3Xn[103X, for example [10XG <=
        Sym(40a)[110X  and [10XG <= Sym(40b)[110X denote two (nonequivalent) representations
        of degree [22X40[122X.[133X
  
  [8X[10XG <= GL([3Xn[103X[10X[3Xid[103X[10X,[3Xdescr[103X[10X)[110X[108X
        [33X[0;6Ydenotes a matrix representation of dimension [3Xn[103X over a coefficient ring
        described  by  [3Xdescr[103X, which can be a prime power, [10Xℤ[110X (denoting the ring
        of  integers),  a  description  of  an  algebraic  extension  field, [10Xℂ[110X
        (denoting  an  unspecified  algebraic extension field), or [10Xℤ/[3Xm[103X[10Xℤ[110X for an
        integer  [3Xm[103X  (denoting  the  ring of residues mod [3Xm[103X); for example, [10XG <=
        GL(2a,4)[110X  and [10XG <= GL(2b,4)[110X denote two (nonequivalent) representations
        of dimension [22X2[122X over the field with four elements.[133X
  
  [33X[0;0YAfter the representations, the programs available for [3Xgapname[103X are listed.[133X
  
  [33X[0;0YThe following optional arguments can be used to restrict the overviews.[133X
  
  [8X[3Xstd[103X[108X
        [33X[0;6Ymust  be  a  positive integer or a list of positive integers; if it is
        given then only those representations are considered that refer to the
        [3Xstd[103X-th  set  of  standard  generators  or  the  [22Xi[122X-th  set  of standard
        generators, for [22Xi[122X in [3Xstd[103X (see Section [14X3.3[114X),[133X
  
  [8X[10X"contents"[110X and [3Xsources[103X[108X
        [33X[0;6Yfor  a  string  or  a list of strings [3Xsources[103X, restrict the data about
        which  the  overview is formed; if [3Xsources[103X is the string [10X"public"[110X then
        only  non-private data (see Chapter [14X5[114X) are considered, if [3Xsources[103X is a
        string  that  denotes  a  private  extension  in  the sense of a [3Xdirid[103X
        argument  of [2XAtlasOfGroupRepresentationsNotifyPrivateDirectory[102X ([14X5.1-1[114X)
        then  only  the  data  that  belong  to  this  private  extension  are
        considered;  also  a list of such strings may be given, then the union
        of these data is considered,[133X
  
  [8X[10XIdentifier[110X and [3Xid[103X[108X
        [33X[0;6Yrestrict  to  representations with [10Xidentifier[110X component in the list [3Xid[103X
        (note  that this component is itself a list, entering this list is not
        admissible), or satisfying the function [3Xid[103X,[133X
  
  [8X[10XIsPermGroup[110X and [9Xtrue[109X[108X
        [33X[0;6Yrestrict to permutation representations,[133X
  
  [8X[10XNrMovedPoints[110X and [3Xn[103X[108X
        [33X[0;6Yfor  a positive integer, a list of positive integers, or a property [3Xn[103X,
        restrict  to  permutation  representations of degree equal to [3Xn[103X, or in
        the list [3Xn[103X, or satisfying the function [3Xn[103X,[133X
  
  [8X[10XNrMovedPoints[110X and the string [10X"minimal"[110X[108X
        [33X[0;6Yrestrict to faithful permutation representations of minimal degree (if
        this information is available),[133X
  
  [8X[10XIsTransitive[110X and [9Xtrue[109X or [9Xfalse[109X[108X
        [33X[0;6Yrestrict to transitive or intransitive permutation representations (if
        this information is available),[133X
  
  [8X[10XIsPrimitive[110X and [9Xtrue[109X or [9Xfalse[109X[108X
        [33X[0;6Yrestrict  to  primitive or imprimitive permutation representations (if
        this information is available),[133X
  
  [8X[10XTransitivity[110X and [3Xn[103X[108X
        [33X[0;6Yfor  a  nonnegative  integer,  a  list  of  nonnegative integers, or a
        property  [3Xn[103X,  restrict  to permutation representations of transitivity
        equal  to  [3Xn[103X,  or in the list [3Xn[103X, or satisfying the function [3Xn[103X (if this
        information is available),[133X
  
  [8X[10XRankAction[110X and [3Xn[103X[108X
        [33X[0;6Yfor  a  nonnegative  integer,  a  list  of  nonnegative integers, or a
        property  [3Xn[103X,  restrict to permutation representations of rank equal to
        [3Xn[103X, or in the list [3Xn[103X, or satisfying the function [3Xn[103X (if this information
        is available),[133X
  
  [8X[10XIsMatrixGroup[110X and [9Xtrue[109X[108X
        [33X[0;6Yrestrict to matrix representations,[133X
  
  [8X[10XCharacteristic[110X and [3Xp[103X[108X
        [33X[0;6Yfor  a  prime  integer,  a  list  of  prime integers, or a property [3Xp[103X,
        restrict to matrix representations over fields of characteristic equal
        to  [3Xp[103X, or in the list [3Xp[103X, or satisfying the function [3Xp[103X (representations
        over  residue  class  rings  that  are  not fields can be addressed by
        entering [9Xfail[109X as the value of [3Xp[103X),[133X
  
  [8X[10XDimension[110X and [3Xn[103X[108X
        [33X[0;6Yfor  a positive integer, a list of positive integers, or a property [3Xn[103X,
        restrict  to matrix representations of dimension equal to [3Xn[103X, or in the
        list [3Xn[103X, or satisfying the function [3Xn[103X,[133X
  
  [8X[10XCharacteristic[110X, [3Xp[103X, [10XDimension[110X,
        and the string [10X"minimal"[110X[108X
        [33X[0;6Yfor  a  prime  integer  [3Xp[103X, restrict to faithful matrix representations
        over  fields  of characteristic [3Xp[103X that have minimal dimension (if this
        information is available),[133X
  
  [8X[10XRing[110X and [3XR[103X[108X
        [33X[0;6Yfor  a  ring  or a property [3XR[103X, restrict to matrix representations over
        this  ring  or  satisfying this function (note that the representation
        might be defined over a proper subring of [3XR[103X),[133X
  
  [8X[10XRing[110X, [3XR[103X, [10XDimension[110X,
        and the string [10X"minimal"[110X[108X
        [33X[0;6Yfor  a  ring  [3XR[103X, restrict to faithful matrix representations over this
        ring that have minimal dimension (if this information is available),[133X
  
  [8X[10XCharacter[110X and [3Xchi[103X[108X
        [33X[0;6Yfor  a  class  function  or a list of class functions [3Xchi[103X, restrict to
        matrix representations with these characters (note that the underlying
        characteristic   of   the   class  function,  see  Section [14X'Reference:
        UnderlyingCharacteristic'[114X,   determines   the  characteristic  of  the
        matrices), and[133X
  
  [8X[10XIsStraightLineProgram[110X and [9Xtrue[109X[108X
        [33X[0;6Yrestrict  to  straight  line  programs,  straight  line decisions (see
        Section [14X6.1[114X), and black box programs (see Section [14X6.2[114X).[133X
  
  [33X[0;0YNote  that  the  above  conditions  refer  only  to  the information that is
  available  without  accessing the representations. For example, if it is not
  stored  in  the  table  of  contents whether a permutation representation is
  primitive  then  this representation does not match an [10XIsPrimitive[110X condition
  in [2XDisplayAtlasInfo[102X.[133X
  
  [33X[0;0YIf  [21Xminimality[121X  information  is  requested  and  no available representation
  matches this condition then either no minimal representation is available or
  the     information     about     the    minimality    is    missing.    See
  [2XMinimalRepresentationInfo[102X   ([14X6.3-1[114X)  for  checking  whether  the  minimality
  information  is  available for the group in question. Note that in the cases
  where  the string [10X"minimal"[110X occurs as an argument, [2XMinimalRepresentationInfo[102X
  ([14X6.3-1[114X)  is  called with third argument [10X"lookup"[110X; this is because the stored
  information  was  precomputed  just  for  the  groups  in the [5XATLAS[105X of Group
  Representations,  so  trying  to  compute  non-stored minimality information
  (using other available databases) will hardly be successful.[133X
  
  [33X[0;0YThe representations are ordered as follows. Permutation representations come
  first   (ordered   according   to   their   degrees),   followed  by  matrix
  representations  over  finite  fields  (ordered first according to the field
  size and second according to the dimension), matrix representations over the
  integers,  and  then  matrix representations over algebraic extension fields
  (both  kinds  ordered  according to the dimension), the last representations
  are matrix representations over residue class rings (ordered first according
  to the modulus and second according to the dimension).[133X
  
  [33X[0;0YThe  maximal  subgroups are ordered according to decreasing group order. For
  an extension [22XG.p[122X of a simple group [22XG[122X by an outer automorphism of prime order
  [22Xp[122X,  this  means  that  [22XG[122X  is  the  first  maximal subgroup and then come the
  extensions  of  the  maximal  subgroups  of [22XG[122X and the novelties; so the [22Xn[122X-th
  maximal  subgroup  of  [22XG[122X and the [22Xn[122X-th maximal subgroup of [22XG.p[122X are in general
  not related. (This coincides with the numbering used for the [2XMaxes[102X ([14XCTblLib:
  Maxes[114X) attribute for character tables.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( [ "M11", "A5" ] );[127X[104X
    [4X[28Xgroup |  # | maxes | cl | cyc | out | fnd | chk | prs[128X[104X
    [4X[28X------+----+-------+----+-----+-----+-----+-----+----[128X[104X
    [4X[28XM11   | 42 |     5 |  + |  +  |     |  +  |  +  |  +[128X[104X
    [4X[28XA5    | 18 |     3 |    |     |     |     |  +  |  +[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  output means that the [5XATLAS[105X of Group Representations contains [22X42[122X
  representations  of  the  Mathieu  group  [22XM_11[122X,  straight  line programs for
  computing  generators  of  representatives  of  all  five classes of maximal
  subgroups,  for  computing  representatives  of  the  conjugacy  classes  of
  elements  and  of  generators  of  maximally  cyclic  subgroups, contains no
  straight  line  program for applying outer automorphisms (well, in fact [22XM_11[122X
  admits  no  nontrivial  outer  automorphism),  and  contains  straight  line
  decisions  that  check  a  set  of generators or a set of group elements for
  being  a  set of standard generators. Analogously, [22X18[122X representations of the
  alternating  group  [22XA_5[122X  are available, straight line programs for computing
  generators of representatives of all three classes of maximal subgroups, and
  no  straight  line  programs  for computing representatives of the conjugacy
  classes of elements, of generators of maximally cyclic subgroups, and no for
  computing  images  under  outer  automorphisms;  straight line decisions for
  checking the standardization of generators or group elements are available.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", IsPermGroup, true );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[128X[104X
    [4X[28X1: G <= Sym(5)  3-trans., on cosets of A4 (1st max.)[128X[104X
    [4X[28X2: G <= Sym(6)  2-trans., on cosets of D10 (2nd max.)[128X[104X
    [4X[28X3: G <= Sym(10) rank 3, on cosets of S3 (3rd max.)[128X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", NrMovedPoints, [ 4 .. 9 ] );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[128X[104X
    [4X[28X1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)[128X[104X
    [4X[28X2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  first  three  representations  stored  for  [22XA_5[122X are (in fact primitive)
  permutation representations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", Dimension, [ 1 .. 3 ] );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[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[28X12: G <= GL(3a,9)[128X[104X
    [4X[28X13: G <= GL(3b,9)[128X[104X
    [4X[28X17: G <= GL(3a,Field([Sqrt(5)]))[128X[104X
    [4X[28X18: G <= GL(3b,Field([Sqrt(5)]))[128X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", Characteristic, 0 );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[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[32X[104X
  
  [33X[0;0YThe  representations  with number between [22X4[122X and [22X13[122X are (in fact irreducible)
  matrix  representations over various finite fields, those with numbers [22X14[122X to
  [22X16[122X  are  integral  matrix  representations,  and  the  last  two  are matrix
  representations over the field generated by [22Xsqrt{5}[122X over the rational number
  field.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", Identifier, "a" );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[128X[104X
    [4X[28X 4: G <= GL(4a,2)[128X[104X
    [4X[28X 8: G <= GL(2a,4)[128X[104X
    [4X[28X12: G <= GL(3a,9)[128X[104X
    [4X[28X17: G <= GL(3a,Field([Sqrt(5)]))[128X[104X
  [4X[32X[104X
  
  [33X[0;0YEach  of  the  representations  with the numbers [22X4, 8, 12[122X, and [22X17[122X is labeled
  with the distinguishing letter [10Xa[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", NrMovedPoints, IsPrimeInt );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[128X[104X
    [4X[28X1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)[128X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", Characteristic, IsOddInt );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[128X[104X
    [4X[28X 6: G <= GL(4,3)[128X[104X
    [4X[28X 7: G <= GL(6,3)[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[25Xgap>[125X [27XDisplayAtlasInfo( "A5", Dimension, IsPrimeInt );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[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[28X15: G <= GL(5,Z)[128X[104X
    [4X[28X17: G <= GL(3a,Field([Sqrt(5)]))[128X[104X
    [4X[28X18: G <= GL(3b,Field([Sqrt(5)]))[128X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", Ring, IsFinite and IsPrimeField );[127X[104X
    [4X[28XRepresentations for G = A5:    (all refer to std. generators 1)[128X[104X
    [4X[28X---------------------------[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[28X10: G <= GL(3,5)[128X[104X
    [4X[28X11: G <= GL(5,5)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe above examples show how the output can be restricted using a property (a
  unary function that returns either [9Xtrue[109X or [9Xfalse[109X) that follows [2XNrMovedPoints[102X
  ([14XReference:  NrMovedPoints  (for a permutation)[114X), [2XCharacteristic[102X ([14XReference:
  Characteristic[114X), [2XDimension[102X ([14XReference: Dimension[114X), or [2XRing[102X ([14XReference: Ring[114X)
  in the argument list of [2XDisplayAtlasInfo[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayAtlasInfo( "A5", IsStraightLineProgram, true );[127X[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;0YStraight   line   programs   are   available  for  computing  generators  of
  representatives  of  the  three  classes  of maximal subgroups of [22XA_5[122X, and a
  straight  line  decision  for  checking whether given generators are in fact
  standard  generators  is  available  as  well  as a presentation in terms of
  standard generators, see [2XAtlasProgram[102X ([14X3.5-3[114X).[133X
  
  [1X3.5-2 AtlasGenerators[101X
  
  [29X[2XAtlasGenerators[102X( [3Xgapname[103X, [3Xrepnr[103X[, [3Xmaxnr[103X] ) [32X function
  [29X[2XAtlasGenerators[102X( [3Xidentifier[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya record containing generators for a representation, or [9Xfail[109X.[133X
  
  [33X[0;0YIn  the  first  form,  [3Xgapname[103X  must  be  a  string denoting a [5XGAP[105X name (see
  Section [14X3.2[114X) of a group, and [3Xrepnr[103X a positive integer. If the [5XATLAS[105X of Group
  Representations  contains  at least [3Xrepnr[103X representations for the group with
  [5XGAP[105X  name  [3Xgapname[103X then [2XAtlasGenerators[102X, when called with [3Xgapname[103X and [3Xrepnr[103X,
  returns   an   immutable  record  describing  the  [3Xrepnr[103X-th  representation;
  otherwise  [9Xfail[109X  is returned. If a third argument [3Xmaxnr[103X, a positive integer,
  is given then an immutable record describing the restriction of the [3Xrepnr[103X-th
  representation to the [3Xmaxnr[103X-th maximal subgroup is returned.[133X
  
  [33X[0;0YThe result record has at least the following components.[133X
  
  [8X[10Xgenerators[110X[108X
        [33X[0;6Ya list of generators for the group,[133X
  
  [8X[10Xgroupname[110X[108X
        [33X[0;6Ythe [5XGAP[105X name of the group (see Section [14X3.2[114X),[133X
  
  [8X[10Xidentifier[110X[108X
        [33X[0;6Ya  [5XGAP[105X  object  (a list of filenames plus additional information) that
        uniquely  determines  the  representation;  the  value  can be used as
        [3Xidentifier[103X argument of [2XAtlasGenerators[102X.[133X
  
  [8X[10Xrepnr[110X[108X
        [33X[0;6Ythe  number of the representation in the current session, equal to the
        argument [3Xrepnr[103X if this is given.[133X
  
  [8X[10Xstandardization[110X[108X
        [33X[0;6Ythe positive integer denoting the underlying standard generators,[133X
  
  [33X[0;0YAdditionally,  the  group  order  may  be  stored in the component [10Xsize[110X, and
  describing  components  may be available that depend on the data type of the
  representation:  For permutation representations, these are [10Xp[110X for the number
  of  moved  points,  [10Xid[110X  for  the  distinguishing  string  as  described  for
  [2XDisplayAtlasInfo[102X   ([14X3.5-1[114X),   and   information   about  primitivity,  point
  stabilizers etc. if available; for matrix representations, these are [10Xdim[110X for
  the dimension of the matrices, [10Xring[110X (if known) for the ring generated by the
  matrix  entries, [10Xid[110X for the distinguishing string, and information about the
  character if available.[133X
  
  [33X[0;0YIt  should  be  noted  that  the  number [3Xrepnr[103X refers to the number shown by
  [2XDisplayAtlasInfo[102X  ([14X3.5-1[114X)  [13Xin  the current session[113X; it may be that after the
  addition of new representations, [3Xrepnr[103X refers to another representation.[133X
  
  [33X[0;0YThe  alternative form of [2XAtlasGenerators[102X, with only argument [3Xidentifier[103X, can
  be  used  to  fetch  the  result  record  with  [10Xidentifier[110X  value  equal  to
  [3Xidentifier[103X. The purpose of this variant is to access the [13Xsame[113X representation
  also in [13Xdifferent[113X [5XGAP[105X sessions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgens1:= AtlasGenerators( "A5", 1 );[127X[104X
    [4X[28Xrec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5", id := "", [128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], [128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, [128X[104X
    [4X[28X  repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", [128X[104X
    [4X[28X  standardization := 1, transitivity := 3, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27Xgens8:= AtlasGenerators( "A5", 8 );[127X[104X
    [4X[28Xrec( dim := 2, [128X[104X
    [4X[28X  generators := [ [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ], [ [ 0*Z(2), [128X[104X
    [4X[28X               Z(2)^0 ], [ Z(2)^0, Z(2)^0 ] ] ], groupname := "A5", [128X[104X
    [4X[28X  id := "a", identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], [128X[104X
    [4X[28X      1, 4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2), [128X[104X
    [4X[28X  size := 60, standardization := 1, type := "matff" )[128X[104X
    [4X[25Xgap>[125X [27Xgens17:= AtlasGenerators( "A5", 17 );[127X[104X
    [4X[28Xrec( dim := 3, [128X[104X
    [4X[28X  generators := [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, [128X[104X
    [4X[28X              1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ], [128X[104X
    [4X[28X  groupname := "A5", id := "a", identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], [128X[104X
    [4X[28X  repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]), size := 60, [128X[104X
    [4X[28X  standardization := 1, type := "matalg" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YEach of the above pairs of elements generates a group isomorphic to [22XA_5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgens1max2:= AtlasGenerators( "A5", 1, 2 );[127X[104X
    [4X[28Xrec( generators := [ (1,2)(3,4), (2,3)(4,5) ], groupname := "D10",[128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5, 2 ],[128X[104X
    [4X[28X  repnr := 1, size := 10, standardization := 1 )[128X[104X
    [4X[25Xgap>[125X [27Xid:= gens1max2.identifier;;[127X[104X
    [4X[25Xgap>[125X [27Xgens1max2 = AtlasGenerators( id );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xmax2:= Group( gens1max2.generators );;[127X[104X
    [4X[25Xgap>[125X [27XSize( max2 );[127X[104X
    [4X[28X10[128X[104X
    [4X[25Xgap>[125X [27XIdGroup( max2 ) = IdGroup( DihedralGroup( 10 ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  elements stored in [10Xgens1max2.generators[110X describe the restriction of the
  first  representation  of  [22XA_5[122X  to  a  group  in the second class of maximal
  subgroups   of   [22XA_5[122X   according   to  the  list  in  the  [5XATLAS[105X  of  Finite
  Groups [CCNPW85]; this subgroup is isomorphic to the dihedral group [22XD_10[122X.[133X
  
  [1X3.5-3 AtlasProgram[101X
  
  [29X[2XAtlasProgram[102X( [3Xgapname[103X[, [3Xstd[103X], [3X...[103X ) [32X function
  [29X[2XAtlasProgram[102X( [3Xidentifier[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya record containing a program, or [9Xfail[109X.[133X
  
  [33X[0;0YIn the first form, [3Xgapname[103X must be a string denoting a [5XGAP[105X name (see Section
   [14X3.2[114X)  of  a  group [22XG[122X, say. If the [5XATLAS[105X of Group Representations contains a
  straight  line  program (see Section [14X'Reference: Straight Line Programs'[114X) or
  straight   line  decision  (see  Section [14X6.1[114X)  or  black  box  program  (see
  Section [14X6.2[114X)  as  described  by  the  remaining  arguments  (see below) then
  [2XAtlasProgram[102X  returns an immutable record containing this program. Otherwise
  [9Xfail[109X is returned.[133X
  
  [33X[0;0YIf   the   optional   argument  [3Xstd[103X  is  given,  only  those  straight  line
  programs/decisions  are  considered that take generators from the [3Xstd[103X-th set
  of standard generators of [22XG[122X as input, see Section [14X3.3[114X.[133X
  
  [33X[0;0YThe result record has the following components.[133X
  
  [8X[10Xprogram[110X[108X
        [33X[0;6Ythe required straight line program/decision, or black box program,[133X
  
  [8X[10Xstandardization[110X[108X
        [33X[0;6Ythe positive integer denoting the underlying standard generators of [22XG[122X,[133X
  
  [8X[10Xidentifier[110X[108X
        [33X[0;6Ya  [5XGAP[105X  object  (a list of filenames plus additional information) that
        uniquely  determines  the program; the value can be used as [3Xidentifier[103X
        argument of [2XAtlasProgram[102X (see below).[133X
  
  [33X[0;0YIn the first form, the last arguments must be as follows.[133X
  
  [8X(the string [10X"maxes"[110X and) a positive integer [3Xmaxnr[103X
  [108X
        [33X[0;6Ythe  required  program  computes  generators  of  the [3Xmaxnr[103X-th maximal
        subgroup of the group with [5XGAP[105X name [3Xgapname[103X.[133X
  
        [33X[0;6YIn  this  case,  the  result record of [2XAtlasProgram[102X also may contain a
        component  [10Xsize[110X,  whose  value is the order of the maximal subgroup in
        question.[133X
  
  [8Xone of the strings [10X"classes"[110X or [10X"cyclic"[110X[108X
        [33X[0;6Ythe  required program computes representatives of conjugacy classes of
        elements   or   representatives  of  generators  of  maximally  cyclic
        subgroups of [22XG[122X, respectively.[133X
  
        [33X[0;6YSee [BSWW01]  and [SWW00] for the background concerning these straight
        line  programs. In these cases, the result record of [2XAtlasProgram[102X also
        contains  a component [10Xoutputs[110X, whose value is a list of class names of
        the outputs, as described in Section [14X3.4[114X.[133X
  
  [8Xthe strings [10X"automorphism"[110X and [3Xautname[103X[108X
        [33X[0;6Ythe  required program computes images of standard generators under the
        outer automorphism of [22XG[122X that is given by this string.[133X
  
        [33X[0;6YNote  that  a  value  [10X"2"[110X  of  [3Xautname[103X  means  that  the square of the
        automorphism  is  an  inner  automorphism  of  [22XG[122X  (not necessarily the
        identity mapping) but the automorphism itself is not.[133X
  
  [8Xthe string [10X"check"[110X[108X
        [33X[0;6Ythe  required  result is a straight line decision that takes a list of
        generators  for  [22XG[122X  and  returns [9Xtrue[109X if these generators are standard
        generators of [22XG[122X w.r.t. the standardization [3Xstd[103X, and [9Xfalse[109X otherwise.[133X
  
  [8Xthe string [10X"presentation"[110X[108X
        [33X[0;6Ythe  required  result is a straight line decision that takes a list of
        group  elements  and  returns  [9Xtrue[109X  if  these  elements  are standard
        generators of [22XG[122X w.r.t. the standardization [3Xstd[103X, and [9Xfalse[109X otherwise.[133X
  
        [33X[0;6YSee [2XStraightLineProgramFromStraightLineDecision[102X ([14X6.1-9[114X) for an example
        how  to  derive  defining  relators  for  [22XG[122X  in  terms of the standard
        generators from such a straight line decision.[133X
  
  [8Xthe string [10X"find"[110X[108X
        [33X[0;6Ythe  required result is a black box program that takes [22XG[122X and returns a
        list of standard generators of [22XG[122X, w.r.t. the standardization [3Xstd[103X.[133X
  
  [8Xthe string [10X"restandardize"[110X and an integer [3Xstd2[103X[108X
        [33X[0;6Ythe  required result is a straight line program that computes standard
        generators of [22XG[122X w.r.t. the [3Xstd2[103X-th set of standard generators of [22XG[122X; in
        this case, the argument [3Xstd[103X must be given.[133X
  
  [8Xthe strings [10X"other"[110X and [3Xdescr[103X[108X
        [33X[0;6Ythe required program is described by [3Xdescr[103X.[133X
  
  [33X[0;0YThe second form of [2XAtlasProgram[102X, with only argument the list [3Xidentifier[103X, can
  be  used  to  fetch  the  result  record  with  [10Xidentifier[110X  value  equal  to
  [3Xidentifier[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprog:= AtlasProgram( "A5", 2 );[127X[104X
    [4X[28Xrec( groupname := "A5", identifier := [ "A5", "A5G1-max2W1", 1 ],[128X[104X
    [4X[28X  program := <straight line program>, size := 10, standardization := 1,[128X[104X
    [4X[28X  subgroupname := "D10" )[128X[104X
    [4X[25Xgap>[125X [27XStringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] );[127X[104X
    [4X[28X"[ a, bbab ]"[128X[104X
    [4X[25Xgap>[125X [27Xgens1:= AtlasGenerators( "A5", 1 );[127X[104X
    [4X[28Xrec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5", id := "", [128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], [128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, [128X[104X
    [4X[28X  repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", [128X[104X
    [4X[28X  standardization := 1, transitivity := 3, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27Xmaxgens:= ResultOfStraightLineProgram( prog.program, gens1.generators );[127X[104X
    [4X[28X[ (1,2)(3,4), (2,3)(4,5) ][128X[104X
    [4X[25Xgap>[125X [27Xmaxgens = gens1max2.generators;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  example  shows  that  for  restricting  representations given by
  standard  generators  to  a  maximal  subgroup of [22XA_5[122X, we can also fetch and
  apply the appropriate straight line program. Such a program (see [14X'Reference:
  Straight  Line  Programs'[114X)  takes  standard  generators of a group --in this
  example  [22XA_5[122X--  as  its  input, and returns a list of elements in this group
  --in  this  example generators of the [22XD_10[122X subgroup we had met above-- which
  are  computed  essentially  by  evaluating  structured words in terms of the
  standard generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprog:= AtlasProgram( "J1", "cyclic" );[127X[104X
    [4X[28Xrec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ],[128X[104X
    [4X[28X  outputs := [ "6A", "7A", "10B", "11A", "15B", "19A" ],[128X[104X
    [4X[28X  program := <straight line program>, standardization := 1 )[128X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfGroup( FreeGroup( "x", "y" ) );;[127X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineProgram( prog.program, gens );[127X[104X
    [4X[28X[ x*y*x*y^2*x*y*x*y^2*x*y*x*y*x*y^2*x*y^2, x*y, x*y*x*y^2*x*y*x*y*x*y^2*x*y^2,[128X[104X
    [4X[28X  x*y*x*y*x*y^2*x*y^2*x*y*x*y^2*x*y*x*y*x*y^2*x*y^2*x*y*x*y^2*x*y*x*y*x*y^[128X[104X
    [4X[28X    2*x*y^2, x*y*x*y*x*y^2*x*y^2, x*y*x*y^2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  example  shows  how  to fetch and use straight line programs for
  computing  generators  of representatives of maximally cyclic subgroups of a
  given group.[133X
  
  [1X3.5-4 AtlasProgramInfo[101X
  
  [29X[2XAtlasProgramInfo[102X( [3Xgapname[103X[, [3Xstd[103X][, [3X"contents"[103X, [3Xsources[103X][, [3X...[103X] ) [32X function
  [6XReturns:[106X  [33X[0;10Ya record describing a program, or [9Xfail[109X.[133X
  
  [33X[0;0Y[2XAtlasProgramInfo[102X  takes  the  same  arguments  as  [2XAtlasProgram[102X ([14X3.5-3[114X), and
  returns  a  similar result. The only difference is that the records returned
  by [2XAtlasProgramInfo[102X have no components [10Xprogram[110X and [10Xoutputs[110X. The idea is that
  one  can use [2XAtlasProgramInfo[102X for testing whether the program in question is
  available  at  all,  but  without  transferring it from a remote server. The
  [10Xidentifier[110X  component  of the result of [2XAtlasProgramInfo[102X can then be used to
  fetch the program with [2XAtlasProgram[102X ([14X3.5-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasProgramInfo( "J1", "cyclic" );[127X[104X
    [4X[28Xrec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ], [128X[104X
    [4X[28X  standardization := 1 )[128X[104X
  [4X[32X[104X
  
  [1X3.5-5 OneAtlasGeneratingSetInfo[101X
  
  [29X[2XOneAtlasGeneratingSetInfo[102X( [[3Xgapname[103X, ][[3Xstd[103X, ][[3X...[103X] ) [32X function
  [6XReturns:[106X  [33X[0;10Ya   record   describing   a   representation  that  satisfies  the
            conditions, or [9Xfail[109X.[133X
  
  [33X[0;0YLet [3Xgapname[103X be a string denoting a [5XGAP[105X name (see Section  [14X3.2[114X) of a group [22XG[122X,
  say.   If   the  [5XATLAS[105X  of  Group  Representations  contains  at  least  one
  representation     for    [22XG[122X    with    the    required    properties    then
  [2XOneAtlasGeneratingSetInfo[102X  returns  a record [3Xr[103X whose components are the same
  as those of the records returned by [2XAtlasGenerators[102X ([14X3.5-2[114X), except that the
  component  [10Xgenerators[110X is not contained; the component [10Xidentifier[110X of [3Xr[103X can be
  used  as input for [2XAtlasGenerators[102X ([14X3.5-2[114X) in order to fetch the generators.
  If  no representation satisfying the given conditions is available then [9Xfail[109X
  is returned.[133X
  
  [33X[0;0YIf the argument [3Xstd[103X is given then it must be a positive integer or a list of
  positive integers, denoting the sets of standard generators w.r.t. which the
  representation shall be given (see Section [14X3.3[114X).[133X
  
  [33X[0;0YThe  argument  [3Xgapname[103X  can  be  missing  (then  all  available  groups  are
  considered), or a list of group names can be given instead.[133X
  
  [33X[0;0YFurther  restrictions  can be entered as arguments, with the same meaning as
  described     for     [2XDisplayAtlasInfo[102X     ([14X3.5-1[114X).     The     result    of
  [2XOneAtlasGeneratingSetInfo[102X  describes  the  first  generating  set for [22XG[122X that
  matches the restrictions, in the ordering shown by [2XDisplayAtlasInfo[102X ([14X3.5-1[114X).[133X
  
  [33X[0;0YNote that even in the case that the user parameter [21Xremote[121X has the value [9Xtrue[109X
  (see  Section [14X4.3-1[114X), [2XOneAtlasGeneratingSetInfo[102X does [13Xnot[113X attempt to [13Xtransfer[113X
  remote data files, just the table of contents is evaluated. So this function
  (as well as [2XAllAtlasGeneratingSetInfos[102X ([14X3.5-6[114X)) can be used to check for the
  availability  of  certain  representations,  and  afterwards  one  can  call
  [2XAtlasGenerators[102X ([14X3.5-2[114X) for those representations one wants to work with.[133X
  
  [33X[0;0YIn  the  following  example,  we try to access information about permutation
  representations for the alternating group [22XA_5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "A5" );[127X[104X
    [4X[28Xrec( groupname := "A5", id := "",[128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],[128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,[128X[104X
    [4X[28X  repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",[128X[104X
    [4X[28X  standardization := 1, transitivity := 3, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info.identifier );[127X[104X
    [4X[28Xrec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5", id := "", [128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], [128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, [128X[104X
    [4X[28X  repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", [128X[104X
    [4X[28X  standardization := 1, transitivity := 3, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27Xinfo = OneAtlasGeneratingSetInfo( "A5", IsPermGroup, true );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinfo = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, "minimal" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinfo = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, [ 1 .. 10 ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 20 );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  a  permutation  representation of degree [22X20[122X could be obtained by
  taking  twice  the primitive representation on [22X10[122X points; however, the [5XATLAS[105X
  of Group Representations does not store this imprimitive representation (cf.
  Section [14X3.1[114X).[133X
  
  [33X[0;0YWe  continue this example a little. Next we access matrix representations of
  [22XA_5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "A5", IsMatrixGroup, true );[127X[104X
    [4X[28Xrec( dim := 4, groupname := "A5", id := "a",[128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1, 2 ],[128X[104X
    [4X[28X  repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2), size := 60,[128X[104X
    [4X[28X  standardization := 1, type := "matff" )[128X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info.identifier );[127X[104X
    [4X[28Xrec( dim := 4, [128X[104X
    [4X[28X  generators := [ <an immutable 4x4 matrix over GF2>, <an immutable 4x[128X[104X
    [4X[28X        4 matrix over GF2> ], groupname := "A5", id := "a", [128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1, 2 ], [128X[104X
    [4X[28X  repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2), size := 60, [128X[104X
    [4X[28X  standardization := 1, type := "matff" )[128X[104X
    [4X[25Xgap>[125X [27Xinfo = OneAtlasGeneratingSetInfo( "A5", Dimension, 4 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinfo = OneAtlasGeneratingSetInfo( "A5", Characteristic, 2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinfo = OneAtlasGeneratingSetInfo( "A5", Ring, GF(2) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 2 );[127X[104X
    [4X[28Xrec( dim := 2, groupname := "A5", id := "a",[128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1, 4 ],[128X[104X
    [4X[28X  repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2), size := 60,[128X[104X
    [4X[28X  standardization := 1, type := "matff" )[128X[104X
    [4X[25Xgap>[125X [27XOneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 1 );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "A5", Characteristic, 0, Dimension, 4 );[127X[104X
    [4X[28Xrec( dim := 4, groupname := "A5", id := "",[128X[104X
    [4X[28X  identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ], repname := "A5G1-Zr4B0",[128X[104X
    [4X[28X  repnr := 14, ring := Integers, size := 60, standardization := 1,[128X[104X
    [4X[28X  type := "matint" )[128X[104X
    [4X[25Xgap>[125X [27Xgens:= 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 [27Xinfo = OneAtlasGeneratingSetInfo( "A5", Ring, Integers );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinfo = OneAtlasGeneratingSetInfo( "A5", Ring, CF(37) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOneAtlasGeneratingSetInfo( "A5", Ring, Integers mod 77 );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(5), Dimension, 3 );[127X[104X
    [4X[28Xrec( dim := 3, groupname := "A5", id := "a",[128X[104X
    [4X[28X  identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], repname := "A5G1-Ar3aB0",[128X[104X
    [4X[28X  repnr := 17, ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,[128X[104X
    [4X[28X  type := "matalg" )[128X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info.identifier );[127X[104X
    [4X[28Xrec( dim := 3, [128X[104X
    [4X[28X  generators := [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, [128X[104X
    [4X[28X              1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ], [128X[104X
    [4X[28X  groupname := "A5", id := "a", identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], [128X[104X
    [4X[28X  repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]), size := 60, [128X[104X
    [4X[28X  standardization := 1, type := "matalg" )[128X[104X
    [4X[25Xgap>[125X [27XOneAtlasGeneratingSetInfo( "A5", Ring, GF(17) );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X3.5-6 AllAtlasGeneratingSetInfos[101X
  
  [29X[2XAllAtlasGeneratingSetInfos[102X( [[3Xgapname[103X, ][[3Xstd[103X, ][[3X...[103X] ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe  list  of  all records describing representations that satisfy
            the conditions.[133X
  
  [33X[0;0Y[2XAllAtlasGeneratingSetInfos[102X  is similar to [2XOneAtlasGeneratingSetInfo[102X ([14X3.5-5[114X).
  The  difference  is  that  the  list of [13Xall[113X records describing the available
  representations  with  the  given properties is returned instead of just one
  such  component.  In  particular  an  empty  list  is  returned  if  no such
  representation is available.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAllAtlasGeneratingSetInfos( "A5", IsPermGroup, true );[127X[104X
    [4X[28X[ rec( groupname := "A5", id := "",[128X[104X
    [4X[28X      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],[128X[104X
    [4X[28X      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,[128X[104X
    [4X[28X      repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",[128X[104X
    [4X[28X      standardization := 1, transitivity := 3, type := "perm" ),[128X[104X
    [4X[28X  rec( groupname := "A5", id := "",[128X[104X
    [4X[28X      identifier := [ "A5", [ "A5G1-p6B0.m1", "A5G1-p6B0.m2" ], 1, 6 ],[128X[104X
    [4X[28X      isPrimitive := true, maxnr := 2, p := 6, rankAction := 2,[128X[104X
    [4X[28X      repname := "A5G1-p6B0", repnr := 2, size := 60, stabilizer := "D10",[128X[104X
    [4X[28X      standardization := 1, transitivity := 2, type := "perm" ),[128X[104X
    [4X[28X  rec( 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[32X[104X
  
  [33X[0;0YNote  that  a matrix representation in any characteristic can be obtained by
  reducing  a permutation representation or an integral matrix representation;
  however,   the  [5XATLAS[105X  of  Group  Representations  does  not  [13Xstore[113X  such  a
  representation (cf. Section [14X3.1[114X).[133X
  
  
  [1X3.5-7 [33X[0;0YAtlasGroup[133X[101X
  
  [29X[2XAtlasGroup[102X( [[3Xgapname[103X[, [3Xstd[103X, ]][[3X...[103X] ) [32X function
  [29X[2XAtlasGroup[102X( [3Xidentifier[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya group that satisfies the conditions, or [9Xfail[109X.[133X
  
  [33X[0;0Y[2XAtlasGroup[102X  takes  the  same arguments as [2XOneAtlasGeneratingSetInfo[102X ([14X3.5-5[114X),
  and  returns  the  group generated by the [10Xgenerators[110X component of the record
  that  is returned by [2XOneAtlasGeneratingSetInfo[102X ([14X3.5-5[114X) with these arguments;
  if  [2XOneAtlasGeneratingSetInfo[102X  ([14X3.5-5[114X)  returns  [9Xfail[109X  then  also [2XAtlasGroup[102X
  returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "A5" );[127X[104X
    [4X[28XGroup([ (1,2)(3,4), (1,3,5) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlternatively,  it  is  possible  to  enter  exactly  one argument, a record
  [3Xidentifier[103X    as    returned   by   [2XOneAtlasGeneratingSetInfo[102X   ([14X3.5-5[114X)   or
  [2XAllAtlasGeneratingSetInfos[102X  ([14X3.5-6[114X),  or  the [10Xidentifier[110X component of such a
  record.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "A5" );[127X[104X
    [4X[28Xrec( groupname := "A5", id := "",[128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], [128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, [128X[104X
    [4X[28X  repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", [128X[104X
    [4X[28X  standardization := 1, transitivity := 3, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27XAtlasGroup( info );[127X[104X
    [4X[28XGroup([ (1,2)(3,4), (1,3,5) ])[128X[104X
    [4X[25Xgap>[125X [27XAtlasGroup( info.identifier );[127X[104X
    [4X[28XGroup([ (1,2)(3,4), (1,3,5) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn   the   groups  returned  by  [2XAtlasGroup[102X,  the  value  of  the  attribute
  [2XAtlasRepInfoRecord[102X  ([14X3.5-9[114X)  is set. This information is used for example by
  [2XAtlasSubgroup[102X  ([14X3.5-8[114X)  when  this function is called with second argument a
  group created by [2XAtlasGroup[102X.[133X
  
  
  [1X3.5-8 [33X[0;0YAtlasSubgroup[133X[101X
  
  [29X[2XAtlasSubgroup[102X( [3Xgapname[103X[, [3Xstd[103X][, [3X...[103X], [3Xmaxnr[103X ) [32X function
  [29X[2XAtlasSubgroup[102X( [3Xidentifier[103X, [3Xmaxnr[103X ) [32X function
  [29X[2XAtlasSubgroup[102X( [3XG[103X, [3Xmaxnr[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya group that satisfies the conditions, or [9Xfail[109X.[133X
  
  [33X[0;0YThe  arguments of [2XAtlasSubgroup[102X, except the last argument [3Xmaxn[103X, are the same
  as  for [2XAtlasGroup[102X ([14X3.5-7[114X). If the [5XATLAS[105X of Group Representations provides a
  straight line program for restricting representations of the group with name
  [3Xgapname[103X  (given  w.r.t.  the  [3Xstd[103X-th  standard  generators)  to the [3Xmaxnr[103X-th
  maximal  subgroup  and  if  a representation with the required properties is
  available,  in  the  sense  that  calling  [2XAtlasGroup[102X  ([14X3.5-7[114X) with the same
  arguments  except  [3Xmaxnr[103X  yields  a  group,  then  [2XAtlasSubgroup[102X returns the
  restriction of this representation to the [3Xmaxnr[103X-th maximal subgroup.[133X
  
  [33X[0;0YIn all other cases, [9Xfail[109X is returned.[133X
  
  [33X[0;0YNote  that the conditions refer to the group and not to the subgroup. It may
  happen  that  in  the  restriction  of  a  permutation  representation  to a
  subgroup,  fewer  points  are  moved,  or  that  the restriction of a matrix
  representation  turns  out  to  be  defined  over a smaller ring. Here is an
  example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasSubgroup( "A5", NrMovedPoints, 5, 1 );[127X[104X
    [4X[28XGroup([ (1,5)(2,3), (1,3,5) ])[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( g );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlternatively,  it  is  possible  to  enter exactly two arguments, the first
  being  a  record [3Xidentifier[103X as returned by [2XOneAtlasGeneratingSetInfo[102X ([14X3.5-5[114X)
  or [2XAllAtlasGeneratingSetInfos[102X ([14X3.5-6[114X), or the [10Xidentifier[110X component of such a
  record, or a group [3XG[103X constructed with [2XAtlasGroup[102X ([14X3.5-7[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "A5" );[127X[104X
    [4X[28Xrec( groupname := "A5", id := "", [128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], [128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, [128X[104X
    [4X[28X  repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", [128X[104X
    [4X[28X  standardization := 1, transitivity := 3, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27XAtlasSubgroup( info, 1 );[127X[104X
    [4X[28XGroup([ (1,5)(2,3), (1,3,5) ])[128X[104X
    [4X[25Xgap>[125X [27XAtlasSubgroup( info.identifier, 1 );[127X[104X
    [4X[28XGroup([ (1,5)(2,3), (1,3,5) ])[128X[104X
    [4X[25Xgap>[125X [27XAtlasSubgroup( AtlasGroup( "A5" ), 1 );[127X[104X
    [4X[28XGroup([ (1,5)(2,3), (1,3,5) ])[128X[104X
  [4X[32X[104X
  
  [1X3.5-9 AtlasRepInfoRecord[101X
  
  [29X[2XAtlasRepInfoRecord[102X( [3XG[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ythe  record  stored  in the group [3XG[103X when this was constructed with
            [2XAtlasGroup[102X ([14X3.5-7[114X).[133X
  
  [33X[0;0YFor  a  group [3XG[103X that has been constructed with [2XAtlasGroup[102X ([14X3.5-7[114X), the value
  of  this  attribute  is  the info record that describes [3XG[103X, in the sense that
  this  record was the first argument of the call to [2XAtlasGroup[102X ([14X3.5-7[114X), or it
  is  the  result  of  the  call to [2XOneAtlasGeneratingSetInfo[102X ([14X3.5-5[114X) with the
  conditions that were listed in the call to [2XAtlasGroup[102X ([14X3.5-7[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasRepInfoRecord( AtlasGroup( "A5" ) );[127X[104X
    [4X[28Xrec( groupname := "A5", id := "", [128X[104X
    [4X[28X  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], [128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, [128X[104X
    [4X[28X  repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", [128X[104X
    [4X[28X  standardization := 1, transitivity := 3, type := "perm" )[128X[104X
  [4X[32X[104X
  
  
  [1X3.6 [33X[0;0Y[5XBrowse[105X[101X[1X Applications Provided by [5XAtlasRep[105X[101X[1X[133X[101X
  
  [33X[0;0YThe functions [2XBrowseMinimalDegrees[102X ([14X3.6-1[114X), [2XBrowseBibliographySporadicSimple[102X
  ([14X3.6-2[114X),  and  [2XBrowseAtlasInfo[102X  ([14XBrowse: BrowseAtlasInfo[114X) (an alternative to
  [2XDisplayAtlasInfo[102X  ([14X3.5-1[114X)) are available only if the [5XGAP[105X package [5XBrowse[105X (see
  [BL11]) is loaded.[133X
  
  [1X3.6-1 BrowseMinimalDegrees[101X
  
  [29X[2XBrowseMinimalDegrees[102X( [[3Xgroupnames[103X] ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe list of info records for the clicked representations.[133X
  
  [33X[0;0YIf  the  [5XGAP[105X  package  [5XBrowse[105X  (see  [BL11]) is loaded then this function is
  available.  It  opens a browse table whose rows correspond to the groups for
  which  the  [5XATLAS[105X  of  Group Representations contains some information about
  minimal degrees, whose columns correspond to the characteristics that occur,
  and whose entries are the known minimal degrees.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xif IsBound( BrowseMinimalDegrees ) then[127X[104X
    [4X[25X>[125X [27X  down:= NCurses.keys.DOWN;;  DOWN:= NCurses.keys.NPAGE;;[127X[104X
    [4X[25X>[125X [27X  right:= NCurses.keys.RIGHT;;  END:= NCurses.keys.END;;[127X[104X
    [4X[25X>[125X [27X  enter:= NCurses.keys.ENTER;;  nop:= [ 14, 14, 14 ];;[127X[104X
    [4X[25X>[125X [27X  # just scroll in the table[127X[104X
    [4X[25X>[125X [27X  BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN,[127X[104X
    [4X[25X>[125X [27X         right, right, right ], "sedddrrrddd", nop, nop, "Q" ) );[127X[104X
    [4X[25X>[125X [27X  BrowseMinimalDegrees();;[127X[104X
    [4X[25X>[125X [27X  # restrict the table to the groups with minimal ordinary degree 6[127X[104X
    [4X[25X>[125X [27X  BrowseData.SetReplay( Concatenation( "scf6",[127X[104X
    [4X[25X>[125X [27X       [ down, down, right, enter, enter ] , nop, nop, "Q" ) );[127X[104X
    [4X[25X>[125X [27X  BrowseMinimalDegrees();;[127X[104X
    [4X[25X>[125X [27X  BrowseData.SetReplay( false );[127X[104X
    [4X[25X>[125X [27Xfi;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  an argument [3Xgroupnames[103X is given then it must be a list of group names of
  the  [5XATLAS[105X  of Group Representations; the browse table is then restricted to
  the  rows  corresponding  to  these  group names and to the columns that are
  relevant  for  these  groups.  A perhaps interesting example is the subtable
  with  the  data concerning sporadic simple groups and their covering groups,
  which has been published in [Jan05]. This table can be shown as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xif IsBound( BrowseMinimalDegrees ) then[127X[104X
    [4X[25X>[125X [27X  # just scroll in the table[127X[104X
    [4X[25X>[125X [27X  BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN, END ],[127X[104X
    [4X[25X>[125X [27X         "rrrrrrrrrrrrrr", nop, nop, "Q" ) );[127X[104X
    [4X[25X>[125X [27X  BrowseMinimalDegrees( BibliographySporadicSimple.groupNamesJan05 );;[127X[104X
    [4X[25X>[125X [27Xfi;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  browse  table  does  not  contain  rows for the groups [22X6.M_22[122X, [22X12.M_22[122X,
  [22X6.Fi_22[122X.  Note that in spite of the title of [Jan05], the entries in Table 1
  of  this  paper  are  in  fact  the  minimal degrees of faithful [13Xirreducible[113X
  representations, and in the above three cases, these degrees are larger than
  the  minimal degrees of faithful representations. The underlying data of the
  browse table is about the minimal faithful (but not necessarily irreducible)
  degrees.[133X
  
  [33X[0;0YThe    return    value    of    [2XBrowseMinimalDegrees[102X    is   the   list   of
  [2XOneAtlasGeneratingSetInfo[102X ([14X3.5-5[114X) values for those representations that have
  been [21Xclicked[121X in visual mode.[133X
  
  [33X[0;0YThe variant without arguments of this function is also available in the menu
  shown by [2XBrowseGapData[102X ([14XBrowse: BrowseGapData[114X).[133X
  
  [1X3.6-2 BrowseBibliographySporadicSimple[101X
  
  [29X[2XBrowseBibliographySporadicSimple[102X(  ) [32X function
  [6XReturns:[106X  [33X[0;10Ya    record   as   returned   by   [2XParseBibXMLExtString[102X   ([14XGAPDoc:
            ParseBibXMLextString[114X).[133X
  
  [33X[0;0YIf  the  [5XGAP[105X  package  [5XBrowse[105X  (see  [BL11]) is loaded then this function is
  available.  It  opens a browse table whose rows correspond to the entries of
  the  bibliographies in the [5XATLAS[105X of Finite Groups [CCNPW85] and in the [5XATLAS[105X
  of Brauer Characters [JLPW95].[133X
  
  [33X[0;0YThe  function  is  based on [2XBrowseBibliography[102X ([14XBrowse: BrowseBibliography[114X),
  see  the  documentation of this function for details, e.g., about the return
  value.[133X
  
  [33X[0;0YThe  returned record encodes the bibliography entries corresponding to those
  rows of the table that are [21Xclicked[121X in visual mode, in the same format as the
  return value of [2XParseBibXMLExtString[102X ([14XGAPDoc: ParseBibXMLextString[114X), see the
  manual of the [5XGAP[105X package [5XGAPDoc[105X [LN11] for details.[133X
  
  [33X[0;0Y[2XBrowseBibliographySporadicSimple[102X  can  be  called also via the menu shown by
  [2XBrowseGapData[102X ([14XBrowse: BrowseGapData[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xif IsBound( BrowseBibliographySporadicSimple ) then[127X[104X
    [4X[25X>[125X [27X  enter:= NCurses.keys.ENTER;;  nop:= [ 14, 14, 14 ];;[127X[104X
    [4X[25X>[125X [27X  BrowseData.SetReplay( Concatenation([127X[104X
    [4X[25X>[125X [27X    # choose the application[127X[104X
    [4X[25X>[125X [27X    "/Bibliography of Sporadic Simple Groups", [ enter, enter ],[127X[104X
    [4X[25X>[125X [27X    # search in the title column for the Atlas of Finite Groups[127X[104X
    [4X[25X>[125X [27X    "scr/Atlas of finite groups", [ enter,[127X[104X
    [4X[25X>[125X [27X    # and quit[127X[104X
    [4X[25X>[125X [27X    nop, nop, nop, nop ], "Q" ) );[127X[104X
    [4X[25X>[125X [27X  BrowseGapData();;[127X[104X
    [4X[25X>[125X [27X  BrowseData.SetReplay( false );[127X[104X
    [4X[25X>[125X [27Xfi;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  bibliographies contained in the [5XATLAS[105X of Finite Groups [CCNPW85] and in
  the [5XATLAS[105X of Brauer Characters [JLPW95] are available online in HTML format,
  see [7Xhttp://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/bibl/index.html[107X.[133X
  
  [33X[0;0YThe    source    data    in    BibXMLext   format,   which   are   used   by
  [2XBrowseBibliographySporadicSimple[102X,  is  part of the [5XAtlasRep[105X package, in four
  files with suffix [11Xxml[111X in the package's [11Xbibl[111X directory. Note that each of the
  two books contains two bibliographies.[133X
  
  [33X[0;0YDetails  about  the BibXMLext format, including information how to transform
  the  data into other formats such as BibTeX, can be found in the [5XGAP[105X package
  [5XGAPDoc[105X (see [LN11]).[133X
  
