  
  [1X7 [33X[0;0YTechnicalities of the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YThis  chapter  describes  those  parts  of the [5XGAP[105X interface to the [5XATLAS[105X of
  Group   Representations   that   do   not   belong  to  the  user  interface
  (cf. Chapter [14X3[114X).[133X
  
  [33X[0;0YBesides   global   variables   used   for   administrational  purposes  (see
  Section [14X7.1[114X)  and  several  sanity  checks  (see  Section [14X7.9[114X),  they can be
  regarded  as  the interface between the data actually contained in the files
  and  the corresponding [5XGAP[105X objects (see Section [14X7.2[114X, [14X7.3[114X, [14X7.4[114X, and [14X7.5[114X), and
  the  interface between the remote and the local version of the database (see
  Section [14X7.6[114X  and [14X7.8[114X).  The  former interface contains functions to read and
  write  files  in [5XMeatAxe[105X format, which may be interesting for users familiar
  with  [5XMeatAxe[105X standalones (see for example [Rin]). Other low level functions
  may be undocumented in the sense that they are not described in this manual.
  Users  interested  in  them may look at the actual implementation in the [11Xgap[111X
  directory  of  the  package,  but it may happen that this will be changed in
  future versions of the package.[133X
  
  
  [1X7.1 [33X[0;0YGlobal Variables Used by the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YFor  debugging  purposes,  [5XAtlasRep[105X functions print information depending on
  the  info  level  of  the  info  classes  [2XInfoAtlasRep[102X ([14X7.1-1[114X), [2XInfoCMeatAxe[102X
  ([14X7.1-2[114X), and [2XInfoBBox[102X ([14X7.1-3[114X) (cf. [14X'Reference: Info Functions'[114X).[133X
  
  [33X[0;0YThe  info  level  of  an  info  class  can  be  changed  using  [2XSetInfoLevel[102X
  ([14XReference:  InfoLevel[114X). For example, the info level of [2XInfoAtlasRep[102X ([14X7.1-1[114X)
  can  be set to the nonnegative integer [22Xn[122X using [10XSetInfoLevel( InfoAtlasRep, [110X[22Xn[122X[10X
  )[110X.[133X
  
  [1X7.1-1 InfoAtlasRep[101X
  
  [33X[1;0Y[29X[2XInfoAtlasRep[102X [32X info class[133X
  
  [33X[0;0YIf  the info level of [2XInfoAtlasRep[102X is at least [22X1[122X then information about [9Xfail[109X
  results  of  [5XAtlasRep[105X  functions is printed. If the info level is at least [22X2[122X
  then  also  information  about  calls  to  external programs is printed. The
  default level is [22X0[122X, no information is printed on this level.[133X
  
  [1X7.1-2 InfoCMeatAxe[101X
  
  [33X[1;0Y[29X[2XInfoCMeatAxe[102X [32X info class[133X
  
  [33X[0;0YIf  the info level of [2XInfoCMeatAxe[102X is at least [22X1[122X then information about [9Xfail[109X
  results  of  [10XC[110X-[5XMeatAxe[105X  functions  (see Section [14X7.3[114X) is printed. The default
  level is zero, no information is printed on this level.[133X
  
  [1X7.1-3 InfoBBox[101X
  
  [33X[1;0Y[29X[2XInfoBBox[102X [32X info class[133X
  
  [33X[0;0YIf  the  info  level  of  [2XInfoBBox[102X is at least [22X1[122X then information about [9Xfail[109X
  results  of  functions  dealing with black box programs (see Section [14X6.2[114X) is
  printed. The default level is [22X0[122X, no information is printed on this level.[133X
  
  [1X7.1-4 AGR[101X
  
  [33X[1;0Y[29X[2XAGR[102X [32X global variable[133X
  
  [33X[0;0Yis  a  record  whose  components are functions and data that are used by the
  high  level  interface functions. Some of the components are documented, see
  for example the index of the package manual.[133X
  
  [1X7.1-5 AtlasOfGroupRepresentationsInfo[101X
  
  [33X[1;0Y[29X[2XAtlasOfGroupRepresentationsInfo[102X [32X global variable[133X
  
  [33X[0;0YThis  is  a  record  that is defined in the file [11Xgap/types.g[111X of the package,
  with the following components.[133X
  
  [8X[10XGAPnames[110X[8X[108X
        [33X[0;6Ya  list of pairs, each containing the [5XGAP[105X name and the [5XATLAS[105X-file name
        of a group, see Section [14X3.3[114X,[133X
  
  [8X[10Xnotified[110X[8X[108X
        [33X[0;6Ya  list  used  for  administrating  extensions  of  the  database (see
        Chapter [14X5[114X);        the        value        is        changed        by
        [2XAtlasOfGroupRepresentationsNotifyData[102X            ([14X5.1-1[114X)           and
        [2XAtlasOfGroupRepresentationsForgetData[102X ([14X5.1-2[114X),[133X
  
  [8X[10Xcharacterinfo[110X[8X, [10Xpermrepinfo[110X[8X, [10Xringinfo[110X[8X[108X
        [33X[0;6Yadditional  information about representations, concerning the afforded
        characters,  the point stabilizers of permutation representations, and
        the rings of definition of matrix representations; this information is
        used by [2XDisplayAtlasInfo[102X ([14X3.6-1[114X),[133X
  
  [8X[10XTableOfContents[110X[8X[108X
        [33X[0;6Ya  record  with  at most the components [10Xcore[110X, [10Xinternal[110X, [10Xlocal[110X, [10Xmerged[110X,
        [10Xtypes[110X,  and  the  identifiers of database extensions. The value of the
        component  [10Xtypes[110X is set in [2XAGR.DeclareDataType[102X ([14X7.5-1[114X), and the values
        of       the       other      components      are      created      by
        [2XAtlasOfGroupRepresentationsNotifyData[102X ([14X5.1-1[114X).[133X
  
  [8X[10XaccessFunctions[110X[8X[108X
        [33X[0;6Ya  list  of records, each describing how to access the data files, see
        Sections [14X4.2-5[114X and [14X7.2[114X, and[133X
  
  
  [1X7.2 [33X[0;0YHow to Customize the Access to Data files[133X[101X
  
  [33X[0;0YBy   default,   locally  available  data  files  are  stored  in  prescribed
  directories,  and  the  files  are  exactly  the  text  files that have been
  downloaded from appropriate places in the internet. However, a more flexible
  approach may be useful.[133X
  
  [33X[0;0YFirst,  one  may  want  to  use  [13Xdifferent file formats[113X, for example [5XMeatAxe[105X
  binary files may be provided parallel to [5XMeatAxe[105X text files. Second, one may
  want  to use [13Xa different directory structure[113X, for example the same structure
  as  used on some server –this makes sense for example if a local mirror of a
  server  is  available,  because then one can read the server files directly,
  without transferring/copying them to another directory.[133X
  
  [33X[0;0YIn  order  to  achieve  this  (and  perhaps more), we admit to customize the
  meaning of the following three access steps.[133X
  
  [8XAre the required data locally available?[108X
        [33X[0;6YThere  may be different file formats available, such as text or binary
        files,  and  it  may happen that the data are available in one file or
        are distributed to several files.[133X
  
  [8XHow can a file be made locally available?[108X
        [33X[0;6YA  different remote file may be fetched, or some postprocessing may be
        required.[133X
  
  [8XHow is the data of a file accessed by [5XGAP[105X?[108X
        [33X[0;6YA different function may be needed to evaluate the file contents.[133X
  
  [33X[0;0YFor  creating  an overview of the locally available data, the first of these
  steps  must  be  available  independent  of  actually  accessing the file in
  question.  For updating the local copy of the server data, the second of the
  above  steps  must be available independent of the third one. Therefore, the
  package  provides  the possibility to extend the default behaviour by adding
  new      records      to      the      [10XaccessFunctions[110X      component     of
  [2XAtlasOfGroupRepresentationsInfo[102X  ([14X7.1-5[114X). The relevant record components are
  as follows.[133X
  
  [8X[10Xdescription[110X[8X[108X
        [33X[0;6YThis  must  be  a short string that describes for which kinds of files
        the  functions  in the current record are intended, which file formats
        are  supported  etc.  The  value is used as key in the user preference
        [10XFileAccessFunctions[110X, see Section [14X4.2-5[114X.[133X
  
  [8X [10Xlocation( [110X[8X[22Xfiles, type[122X[10X )[110X[8X [108X
        [33X[0;6YLet  [22Xfiles[122X  be  a list of pairs [10X[ dirname, filename ][110X, and [22Xtype[122X be the
        data type (see [2XAGR.DeclareDataType[102X ([14X7.5-1[114X)) to which the files belong.
        This  function  must  return  either  the  absolute  paths  where  the
        mechanism  implemented by the current record expects the local version
        of the given files, or [9Xfail[109X if this function does not feel responsible
        for these files.[133X
  
        [33X[0;6YThe  files  are  regarded  as  not  locally available if all installed
        [10Xlocation[110X  functions  return either [9Xfail[109X or paths of nonexisting files,
        in the sense of [2XIsExistingFile[102X ([14XReference: IsExistingFile[114X).[133X
  
  [8X [10Xfetch( [110X[8X[22Xfilepath, filename, dirname, type[122X[10X )[110X[8X [108X
        [33X[0;6YThis  function is called if a file is not locally available and if the
        [10Xlocation[110X  function in the current record has returned a list of paths.
        The  argument  [22Xtype[122X must be the same as for the [10Xlocation[110X function, and
        [22Xfilepath[122X and [22Xfilename[122X must be strings ([13Xnot[113X lists of strings).[133X
  
        [33X[0;6YThe  return  value  must be [9Xtrue[109X if the function succeeded with making
        the file locally available (including postprocessing if applicable), a
        string  with  the  contents  of  the data file if the remote data were
        directly  loaded  into  the  [5XGAP[105X  session  (if  no  local  caching  is
        possible), and [9Xfalse[109X otherwise.[133X
  
  [8X[10Xcontents( [110X[8X[22Xfiles, type, filepaths[122X[10X )[110X[8X[108X
        [33X[0;6YThis  function  is  called  when  the [10Xlocation[110X function in the current
        record  has  returned  the  path(s)  [22Xfilepath[122X, and if either these are
        paths  of  existing  files or the [10Xfetch[110X function in the current record
        has  been  called  for these paths, and the return value was [9Xtrue[109X. The
        first three arguments must be the same as for the [10Xlocation[110X function.[133X
  
        [33X[0;6YThe  return  value  must  be the contents of the file(s), in the sense
        that  the  [5XGAP[105X  matrix, matrix list, permutation, permutation list, or
        program  described by the file(s) is returned. This means that besides
        reading  the  file(s)  via  the appropriate function, interpreting the
        contents may be necessary.[133X
  
  [33X[0;0YIn  [2XAGR.FileContents[102X ([14X7.6-2[114X), those records in the [10XaccessFunctions[110X component
  of  [2XAtlasOfGroupRepresentationsInfo[102X  ([14X7.1-5[114X)  are  considered  –in  reversed
  order–   whose   [10Xdescription[110X   component   occurs  in  the  user  preference
  [10XFileAccessFunctions[110X, see Section [14X4.2-5[114X.[133X
  
  
  [1X7.3 [33X[0;0YReading and Writing MeatAxe Format Files[133X[101X
  
  [1X7.3-1 ScanMeatAxeFile[101X
  
  [33X[1;0Y[29X[2XScanMeatAxeFile[102X( [3Xfilename[103X[, [3Xq[103X][, [3X"string"[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe  matrix  or list of permutations stored in the file or encoded
            by the string.[133X
  
  [33X[0;0YLet  [3Xfilename[103X be the name of a [5XGAP[105X readable file (see [14X'Reference: Filename'[114X)
  that contains a matrix or a permutation or a list of permutations in [5XMeatAxe[105X
  text  format  (see  the  section  about  the  program  [11Xzcv[111X  in the [10XC[110X-[5XMeatAxe[105X
  documentation [Rin]),  and  let  [3Xq[103X be a prime power. [2XScanMeatAxeFile[102X returns
  the corresponding [5XGAP[105X matrix or list of permutations, respectively.[133X
  
  [33X[0;0YIf the file contains a matrix then the way how it is read by [2XScanMeatAxeFile[102X
  depends  on  the value of the user preference [10XHowToReadMeatAxeTextFiles[110X, see
  Section [14X4.2-7[114X.[133X
  
  [33X[0;0YIf  the  parameter [3Xq[103X is given then the result matrix is represented over the
  field  with  [3Xq[103X  elements,  the default for [3Xq[103X is the field size stored in the
  file.[133X
  
  [33X[0;0YIf  the file contains a list of permutations then it is read with [2XStringFile[102X
  ([14XGAPDoc: StringFile[114X); the parameter [3Xq[103X, if given, is ignored in this case.[133X
  
  [33X[0;0YIf  the  string  [10X"string"[110X  is  entered  as the third argument then the first
  argument  must  be  a  string  as obtained by reading a file in [5XMeatAxe[105X text
  format as a text stream (see [2XInputTextFile[102X ([14XReference: InputTextFile[114X)). Also
  in  this  case, [2XScanMeatAxeFile[102X returns the corresponding [5XGAP[105X matrix or list
  of permutations, respectively.[133X
  
  [1X7.3-2 MeatAxeString[101X
  
  [33X[1;0Y[29X[2XMeatAxeString[102X( [3Xmat[103X, [3Xq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMeatAxeString[102X( [3Xperms[103X, [3Xdegree[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMeatAxeString[102X( [3Xperm[103X, [3Xq[103X, [3Xdims[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMeatAxeString[102X( [3Xintmat[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya string encoding the [5XGAP[105X objects given as input in [10XC[110X-[5XMeatAxe[105X text
            format, see [Rin].[133X
  
  [33X[0;0YIn  the  first  form, for a matrix [3Xmat[103X whose entries lie in the finite field
  with [3Xq[103X elements, [2XMeatAxeString[102X returns a string that encodes [3Xmat[103X as a matrix
  over [10XGF([3Xq[103X[10X)[110X.[133X
  
  [33X[0;0YIn the second form, for a nonempty list [3Xperms[103X of permutations that move only
  points  up  to  the  positive integer [3Xdegree[103X, [2XMeatAxeString[102X returns a string
  that encodes [3Xperms[103X as permutations of degree [3Xdegree[103X.[133X
  
  [33X[0;0YIn the third form, for a permutation [3Xperm[103X with largest moved point [22Xn[122X, say, a
  prime  power  [3Xq[103X,  and  a  list  [3Xdims[103X  of  length two containing two positive
  integers  larger  than  or  equal  to [22Xn[122X, [2XMeatAxeString[102X returns a string that
  encodes  [3Xperm[103X as a matrix over [10XGF([3Xq[103X[10X)[110X, of dimensions [3Xdims[103X, whose first [22Xn[122X rows
  and  columns  describe the permutation matrix corresponding to [3Xperm[103X, and the
  remaining rows and columns are zero.[133X
  
  [33X[0;0YIn the fourth form, for a matrix [3Xintmat[103X of integers, [2XMeatAxeString[102X returns a
  string that encodes [3Xintmat[103X as an integer matrix.[133X
  
  [33X[0;0YWhen  strings  are  printed  to  files using [2XPrintTo[102X ([14XReference: PrintTo[114X) or
  [2XAppendTo[102X  ([14XReference: AppendTo[114X) then line breaks are inserted whenever lines
  exceed  the  number  of  characters  given  by  the second entry of the list
  returned  by  [2XSizeScreen[102X ([14XReference: SizeScreen[114X), see [14X'Reference: Operations
  for  Output  Streams'[114X.  This  behaviour  is  not desirable for creating data
  files. So the recommended functions for printing the result of [2XMeatAxeString[102X
  to  a  file  are  [2XFileString[102X  ([14XGAPDoc:  FileString[114X) and [2XWriteAll[102X ([14XReference:
  WriteAll[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:= [ [ 1, -1 ], [ 0, 1 ] ] * Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xstr:= MeatAxeString( mat, 3 );[127X[104X
    [4X[28X"1 3 2 2\n12\n01\n"[128X[104X
    [4X[25Xgap>[125X [27Xmat = ScanMeatAxeFile( str, "string" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xstr:= MeatAxeString( mat, 9 );[127X[104X
    [4X[28X"1 9 2 2\n12\n01\n"[128X[104X
    [4X[25Xgap>[125X [27Xmat = ScanMeatAxeFile( str, "string" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xperms:= [ (1,2,3)(5,6) ];;[127X[104X
    [4X[25Xgap>[125X [27Xstr:= MeatAxeString( perms, 6 );[127X[104X
    [4X[28X"12 1 6 1\n2\n3\n1\n4\n6\n5\n"[128X[104X
    [4X[25Xgap>[125X [27Xperms = ScanMeatAxeFile( str, "string" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xstr:= MeatAxeString( perms, 8 );[127X[104X
    [4X[28X"12 1 8 1\n2\n3\n1\n4\n6\n5\n7\n8\n"[128X[104X
    [4X[25Xgap>[125X [27Xperms = ScanMeatAxeFile( str, "string" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the  output of [2XMeatAxeString[102X in the case of permutation matrices
  depends on the user preference [10XWriteMeatAxeFilesOfMode2[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xperm:= (1,2,4);;[127X[104X
    [4X[25Xgap>[125X [27Xstr:= MeatAxeString( perm, 3, [ 5, 6 ] );[127X[104X
    [4X[28X"2 3 5 6\n2\n4\n3\n1\n5\n"[128X[104X
    [4X[25Xgap>[125X [27Xmat:= ScanMeatAxeFile( str, "string" );;  Print( mat, "\n" );[127X[104X
    [4X[28X[ [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [128X[104X
    [4X[28X  [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [128X[104X
    [4X[28X  [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [128X[104X
    [4X[28X  [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [128X[104X
    [4X[28X  [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ] ][128X[104X
    [4X[25Xgap>[125X [27Xpref:= UserPreference( "AtlasRep", "WriteMeatAxeFilesOfMode2" );;[127X[104X
    [4X[25Xgap>[125X [27XSetUserPreference( "AtlasRep", "WriteMeatAxeFilesOfMode2", true );[127X[104X
    [4X[25Xgap>[125X [27XMeatAxeString( mat, 3 ) = str;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSetUserPreference( "AtlasRep", "WriteMeatAxeFilesOfMode2", false );[127X[104X
    [4X[25Xgap>[125X [27XMeatAxeString( mat, 3 );[127X[104X
    [4X[28X"1 3 5 6\n010000\n000100\n001000\n100000\n000010\n"[128X[104X
    [4X[25Xgap>[125X [27XSetUserPreference( "AtlasRep", "WriteMeatAxeFilesOfMode2", pref );[127X[104X
  [4X[32X[104X
  
  [1X7.3-3 FFList[101X
  
  [33X[1;0Y[29X[2XFFList[102X( [3XF[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya list of elements in the given finite field.[133X
  
  [33X[1;0Y[29X[2XFFLists[102X [32X global variable[133X
  
  [33X[0;0Y[2XFFList[102X  is a utility program for the conversion of vectors and matrices from
  [5XMeatAxe[105X  format  to [5XGAP[105X format and vice versa. It is used by [2XScanMeatAxeFile[102X
  ([14X7.3-1[114X) and [2XMeatAxeString[102X ([14X7.3-2[114X).[133X
  
  [33X[0;0YFor  a  finite  field  [3XF[103X,  [2XFFList[102X returns a list [22Xl[122X giving the correspondence
  between the [5XMeatAxe[105X numbering and the [5XGAP[105X numbering of the elements in [3XF[103X.[133X
  
  [33X[0;0YThe  element  of  [3XF[103X  corresponding  to [5XMeatAxe[105X number [22Xn[122X is [22Xl[ n+1 ][122X, and the
  [5XMeatAxe[105X number of the field element [22Xz[122X is [10XPosition( [110X[22Xl, z[122X[10X ) - 1[110X.[133X
  
  [33X[0;0YThe global variable [2XFFLists[102X is used to store the information about [3XF[103X once it
  has been computed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFFList( GF(4) );[127X[104X
    [4X[28X[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2 ][128X[104X
    [4X[25Xgap>[125X [27XIsBound( FFLists[4] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe [5XMeatAxe[105X defines the bijection between the elements in the field with [22Xq =
  p^d[122X  elements  and  the  set [22X{ 0, 1, ..., q-1 }[122X of integers by assigning the
  field  element  [22X∑_{i=0}^{d-1}  c_i z^i[122X to the integer [22X∑_{i=0}^{d-1} c_i p^i[122X,
  where  the [22Xc_i[122X are in the set [22X{ 0, 1, ..., p-1 }[122X and [22Xz[122X is the primitive root
  of  the  field  with [22Xq[122X elements that corresponds to the residue class of the
  indeterminate, modulo the ideal spanned by the Conway polynomial of degree [22Xd[122X
  over the field with [22Xp[122X elements.[133X
  
  [33X[0;0YThe finite fields introduced by the [5XStandardFF[105X package [Lüb23] are supported
  by  [2XFFList[102X  and  [2XFFLists[102X,  in  the  sense  that  the  bijection  defined  by
  [2XStandardIsomorphismGF[102X  ([14XStandardIsomorphismGF???[114X)  is  applied automatically
  when    [3XF[103X    is    a    field    in    the    filter   [2XIsStandardFiniteField[102X
  ([14XIsStandardFiniteField???[114X).[133X
  
  [1X7.3-4 CMtxBinaryFFMatOrPerm[101X
  
  [33X[1;0Y[29X[2XCMtxBinaryFFMatOrPerm[102X( [3Xelm[103X, [3Xdef[103X, [3Xoutfile[103X[, [3Xbase[103X] ) [32X function[133X
  
  [33X[0;0YLet  the  pair  [22X([3Xelm[103X,  [3Xdef[103X)[122X be either of the form [22X(M, q)[122X where [22XM[122X is a matrix
  over  a  finite  field  [22XF[122X, say, with [22Xq ≤ 256[122X elements, or of the form [22X(π, n)[122X
  where  [22Xπ[122X is a permutation with largest moved point at most [22Xn[122X. Let [3Xoutfile[103X be
  a  string.  [2XCMtxBinaryFFMatOrPerm[102X  writes  the [10XC[110X-[5XMeatAxe[105X binary format of [22XM[122X,
  viewed as a matrix over [22XF[122X, or of [22Xπ[122X, viewed as a permutation on the points up
  to [22Xn[122X, to the file with name [3Xoutfile[103X.[133X
  
  [33X[0;0YIn  the  case  of  a  permutation  [22Xπ[122X,  the optional argument [3Xbase[103X prescribes
  whether  the  binary  file  contains  the  points from [22X0[122X to [3Xdeg[103X[22X- 1[122X ([3Xbase[103X[22X= 0[122X,
  supported  by  version 2.4  of  the  [10XC[110X-[5XMeatAxe[105X)  or the points from [22X1[122X to [3Xdeg[103X
  ([3Xbase[103X[22X=  1[122X,  supported  by  older versions of the [10XC[110X-[5XMeatAxe[105X). The default for
  [3Xbase[103X  is given by the value of the user preference [10XBaseOfMeatAxePermutation[110X,
  see Section [14X4.2-10[114X.[133X
  
  [33X[0;0Y(The binary format is described in the [10XC[110X-[5XMeatAxe[105X manual [Rin].)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtmpdir:= DirectoryTemporary();;[127X[104X
    [4X[25Xgap>[125X [27Xmat:= Filename( tmpdir, "mat" );;[127X[104X
    [4X[25Xgap>[125X [27Xq:= 4;;[127X[104X
    [4X[25Xgap>[125X [27Xmats:= GeneratorsOfGroup( GL(10,q) );;[127X[104X
    [4X[25Xgap>[125X [27XCMtxBinaryFFMatOrPerm( mats[1], q, Concatenation( mat, "1" ) );[127X[104X
    [4X[25Xgap>[125X [27XCMtxBinaryFFMatOrPerm( mats[2], q, Concatenation( mat, "2" ) );[127X[104X
    [4X[25Xgap>[125X [27Xprm:= Filename( tmpdir, "prm" );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 200;;[127X[104X
    [4X[25Xgap>[125X [27Xperms:= GeneratorsOfGroup( SymmetricGroup( n ) );;[127X[104X
    [4X[25Xgap>[125X [27XCMtxBinaryFFMatOrPerm( perms[1], n, Concatenation( prm, "1" ) );[127X[104X
    [4X[25Xgap>[125X [27XCMtxBinaryFFMatOrPerm( perms[2], n, Concatenation( prm, "2" ) );[127X[104X
    [4X[25Xgap>[125X [27XCMtxBinaryFFMatOrPerm( perms[1], n, Concatenation( prm, "1a" ), 0 );[127X[104X
    [4X[25Xgap>[125X [27XCMtxBinaryFFMatOrPerm( perms[2], n, Concatenation( prm, "2b" ), 1 );[127X[104X
  [4X[32X[104X
  
  [1X7.3-5 FFMatOrPermCMtxBinary[101X
  
  [33X[1;0Y[29X[2XFFMatOrPermCMtxBinary[102X( [3Xfname[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe matrix or permutation stored in the file.[133X
  
  [33X[0;0YLet [3Xfname[103X be the name of a file that contains the [10XC[110X-[5XMeatAxe[105X binary format of
  a  matrix over a finite field or of a permutation, as is described in [Rin].
  [2XFFMatOrPermCMtxBinary[102X returns the corresponding [5XGAP[105X matrix or permutation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFFMatOrPermCMtxBinary( Concatenation( mat, "1" ) ) = mats[1];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XFFMatOrPermCMtxBinary( Concatenation( mat, "2" ) ) = mats[2];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XFFMatOrPermCMtxBinary( Concatenation( prm, "1" ) ) = perms[1];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XFFMatOrPermCMtxBinary( Concatenation( prm, "2" ) ) = perms[2];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XFFMatOrPermCMtxBinary( Concatenation( prm, "1a" ) ) = perms[1];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XFFMatOrPermCMtxBinary( Concatenation( prm, "2b" ) ) = perms[2];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X7.4 [33X[0;0YReading and Writing [5XATLAS[105X[101X[1X Straight Line Programs[133X[101X
  
  [1X7.4-1 ScanStraightLineProgram[101X
  
  [33X[1;0Y[29X[2XScanStraightLineProgram[102X( [3Xfilename[103X[, [3X"string"[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya record containing the straight line program, or [9Xfail[109X.[133X
  
  [33X[0;0YLet  [3Xfilename[103X be the name of a file that contains a straight line program in
  the sense that it consists only of lines in the following form.[133X
  
  [8X[10X#[110X[8X[22Xanything[122X[108X
        [33X[0;6Ylines starting with a hash sign [10X#[110X are ignored,[133X
  
  [8X[10Xecho [110X[8X[22Xanything[122X[108X
        [33X[0;6Ylines  starting with [10Xecho[110X are ignored for the [10Xprogram[110X component of the
        result  record  (see  below),  they  are  used to set up the bijection
        between  the  labels  used in the program and conjugacy class names in
        the case that the program computes dedicated class representatives,[133X
  
  [8X[10Xinp [110X[8X[22Xn[122X[108X
        [33X[0;6Ymeans  that  there are [22Xn[122X inputs, referred to via the labels [10X1[110X, [10X2[110X, [22X...[122X,
        [22Xn[122X,[133X
  
  [8X[10Xinp [110X[8X[22Xk[122X [22Xa1[122X [22Xa2[122X ... [22Xak[122X[108X
        [33X[0;6Ymeans  that  the  next [22Xk[122X inputs are referred to via the labels [22Xa1[122X, [22Xa2[122X,
        ..., [22Xak[122X,[133X
  
  [8X[10Xcjr [110X[8X[22Xa[122X [22Xb[122X[108X
        [33X[0;6Ymeans that [22Xa[122X is replaced by [22Xb[122X[10X^(-1) * [110X[22Xa[122X[10X * [110X[22Xb[122X,[133X
  
  [8X[10Xcj [110X[8X[22Xa[122X [22Xb[122X [22Xc[122X[108X
        [33X[0;6Ymeans that [22Xc[122X is defined as [22Xb[122X[10X^(-1) * [110X[22Xa[122X[10X * [110X[22Xb[122X,[133X
  
  [8X[10Xcom [110X[8X[22Xa[122X [22Xb[122X [22Xc[122X[108X
        [33X[0;6Ymeans that [22Xc[122X is defined as [22Xa[122X[10X^(-1) * [110X[22Xb[122X^(-1)[10X * [110X[22Xa[122X[10X * [110X[22Xb[122X,[133X
  
  [8X[10Xiv [110X[8X[22Xa[122X [22Xb[122X[108X
        [33X[0;6Ymeans that [22Xb[122X is defined as [22Xa[122X[10X^(-1)[110X,[133X
  
  [8X[10Xmu [110X[8X[22Xa[122X [22Xb[122X [22Xc[122X[108X
        [33X[0;6Ymeans that [22Xc[122X is defined as [22Xa[122X[10X * [110X[22Xb[122X,[133X
  
  [8X[10Xpwr [110X[8X[22Xa[122X [22Xb[122X [22Xc[122X[108X
        [33X[0;6Ymeans that [22Xc[122X is defined as [22Xb[122X[10X^[110X[22Xa[122X,[133X
  
  [8X[10Xcp [110X[8X[22Xa[122X [22Xb[122X[108X
        [33X[0;6Ymeans that [22Xb[122X is defined as a copy of [22Xa[122X,[133X
  
  [8X[10Xoup [110X[8X[22Xl[122X[108X
        [33X[0;6Ymeans that there are [22Xl[122X outputs, stored in the labels [10X1[110X, [10X2[110X, [22X...[122X, [22Xl[122X, and[133X
  
  [8X[10Xoup [110X[8X[22Xl[122X [22Xb1[122X [22Xb2[122X ... [22Xbl[122X[108X
        [33X[0;6Ymeans that the next [22Xl[122X outputs are stored in the labels [22Xb1[122X, [22Xb2[122X, ... [22Xbl[122X.[133X
  
  [33X[0;0YEach  of  the  labels  [22Xa[122X,  [22Xb[122X,  [22Xc[122X  can be any nonempty sequence of digits and
  alphabet  characters,  except  that the first argument of [10Xpwr[110X must denote an
  integer.[133X
  
  [33X[0;0YIf  the  [10Xinp[110X  or  [10Xoup[110X  statements  are  missing  then  the  input or output,
  respectively,  is  assumed  to  be given by the labels [10X1[110X and [10X2[110X. There can be
  multiple [10Xinp[110X lines at the beginning of the program and multiple [10Xoup[110X lines at
  the end of the program. Only the first [10Xinp[110X or [10Xoup[110X line may omit the names of
  the  elements. For example, an empty file [3Xfilename[103X or an empty string [3Xstring[103X
  represent  a  straight  line  program  with  two inputs that are returned as
  outputs.[133X
  
  [33X[0;0YNo  command except [10Xcjr[110X may overwrite its own input. For example, the line [10Xmu
  a b a[110X is not legal. (This is not checked.)[133X
  
  [33X[0;0Y[2XScanStraightLineProgram[102X  returns  a  record  containing  as the value of its
  component    [10Xprogram[110X   the   corresponding   [5XGAP[105X   straight   line   program
  (see [2XIsStraightLineProgram[102X  ([14XReference: IsStraightLineProgram[114X)) if the input
  string  satisfies the syntax rules stated above, and returns [9Xfail[109X otherwise.
  In  the  latter  case,  information  about  the  first corrupted line of the
  program is printed if the info level of [2XInfoCMeatAxe[102X ([14X7.1-2[114X) is at least [22X1[122X.[133X
  
  [33X[0;0YIf  the  string  [10X"string"[110X  is  entered as the second argument then the first
  argument  must  be  a  string  as obtained by reading a file in [5XMeatAxe[105X text
  format as a text stream (see [2XInputTextFile[102X ([14XReference: InputTextFile[114X)). Also
  in  this  case,  [2XScanStraightLineProgram[102X  returns  either  a record with the
  corresponding [5XGAP[105X straight line program or [9Xfail[109X.[133X
  
  [33X[0;0YIf  the  input describes a straight line program that computes certain class
  representatives  of  the  group  in  question  then  the  result record also
  contains the component [10Xoutputs[110X. Its value is a list of strings, the entry at
  position  [22Xi[122X  denoting  the  name  of  the class in which the [22Xi[122X output of the
  straight  line program lies; see Section [14X3.5[114X for the definition of the class
  names that occur.[133X
  
  [33X[0;0YSuch   straight   line   programs   must  end  with  a  sequence  of  output
  specifications of the following form.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xecho "Classes 1A 2A 3A 5A 5B"[128X[104X
    [4X[28Xoup 5 3 1 2 4 5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis example means that the list of outputs of the program contains elements
  of  the  classes [10X1A[110X, [10X2A[110X, [10X3A[110X, [10X5A[110X, and [10X5B[110X (in this order), and that inside the
  program, these elements are referred to by the five names [10X3[110X, [10X1[110X, [10X2[110X, [10X4[110X, and [10X5[110X.[133X
  
  [1X7.4-2 AtlasStringOfProgram[101X
  
  [33X[1;0Y[29X[2XAtlasStringOfProgram[102X( [3Xprog[103X[, [3Xoutputnames[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XAtlasStringOfProgram[102X( [3Xprog[103X, [3X"mtx"[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya string encoding the straight line program/decision in the format
            used in [5XATLAS[105X files.[133X
  
  [33X[0;0YFor   a   straight   line   program   or   straight   line   decision   [3Xprog[103X
  (see [2XIsStraightLineProgram[102X     ([14XReference:     IsStraightLineProgram[114X)    and
  [2XIsStraightLineDecision[102X  ([14X6.1-1[114X)),  this function returns a string describing
  the  input  format  of  an equivalent straight line program or straight line
  decision  as  used  in  the  data  files, that is, the lines are of the form
  described in [2XScanStraightLineProgram[102X ([14X7.4-1[114X).[133X
  
  [33X[0;0YA  list of strings that is given as the optional second argument [3Xoutputnames[103X
  is  interpreted  as  the  class  names  corresponding  to  the outputs; this
  argument  has  the  effect  that  appropriate  [10Xecho[110X statements appear in the
  result string.[133X
  
  [33X[0;0YIf  the string [10X"mtx"[110X is given as the second argument then the result has the
  format  used  in  the [10XC[110X-[5XMeatAxe[105X (see [Rin]) rather than the format described
  for [2XScanStraightLineProgram[102X  ([14X7.4-1[114X).  (Note  that the [10XC[110X-[5XMeatAxe[105X format does
  not  make  sense  if the argument [3Xoutputnames[103X is given, and that this format
  does not support [10Xinp[110X and [10Xoup[110X statements.)[133X
  
  [33X[0;0YThe  argument  [3Xprog[103X  must  not  be  a  black  box program (see [2XIsBBoxProgram[102X
  ([14X6.2-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xstr:= "inp 2\nmu 1 2 3\nmu 3 1 2\niv 2 1\noup 2 1 2";;[127X[104X
    [4X[25Xgap>[125X [27Xprg:= ScanStraightLineProgram( str, "string" );[127X[104X
    [4X[28Xrec( program := <straight line program> )[128X[104X
    [4X[25Xgap>[125X [27Xprg:= prg.program;;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( prg );[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[2]:= r[3]*r[1];[128X[104X
    [4X[28Xr[1]:= r[2]^-1;[128X[104X
    [4X[28X# return values:[128X[104X
    [4X[28X[ r[1], r[2] ][128X[104X
    [4X[25Xgap>[125X [27XStringOfResultOfStraightLineProgram( prg, [ "a", "b" ] );[127X[104X
    [4X[28X"[ (aba)^-1, aba ]"[128X[104X
    [4X[25Xgap>[125X [27XAtlasStringOfProgram( prg );[127X[104X
    [4X[28X"inp 2\nmu 1 2 3\nmu 3 1 2\niv 2 1\noup 2\n"[128X[104X
    [4X[25Xgap>[125X [27Xprg:= StraightLineProgram( "(a^2b^3)^-1", [ "a", "b" ] );[127X[104X
    [4X[28X<straight line program>[128X[104X
    [4X[25Xgap>[125X [27XPrint( AtlasStringOfProgram( prg ) );[127X[104X
    [4X[28Xinp 2[128X[104X
    [4X[28Xpwr 2 1 4[128X[104X
    [4X[28Xpwr 3 2 5[128X[104X
    [4X[28Xmu 4 5 3[128X[104X
    [4X[28Xiv 3 4[128X[104X
    [4X[28Xoup 1 4[128X[104X
    [4X[25Xgap>[125X [27Xprg:= StraightLineProgram( [ [2,3], [ [3,1,1,4], [1,2,3,1] ] ], 2 );[127X[104X
    [4X[28X<straight line program>[128X[104X
    [4X[25Xgap>[125X [27XPrint( AtlasStringOfProgram( prg ) );[127X[104X
    [4X[28Xinp 2[128X[104X
    [4X[28Xpwr 3 2 3[128X[104X
    [4X[28Xpwr 4 1 5[128X[104X
    [4X[28Xmu 3 5 4[128X[104X
    [4X[28Xpwr 2 1 6[128X[104X
    [4X[28Xmu 6 3 5[128X[104X
    [4X[28Xoup 2 4 5[128X[104X
    [4X[25Xgap>[125X [27XPrint( AtlasStringOfProgram( prg, "mtx" ) );[127X[104X
    [4X[28X# inputs are expected in 1 2[128X[104X
    [4X[28Xzsm pwr3 2 3[128X[104X
    [4X[28Xzsm pwr4 1 5[128X[104X
    [4X[28Xzmu 3 5 4[128X[104X
    [4X[28Xzsm pwr2 1 6[128X[104X
    [4X[28Xzmu 6 3 5[128X[104X
    [4X[28Xecho "outputs are in 4 5"[128X[104X
    [4X[25Xgap>[125X [27Xstr:= "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5";;[127X[104X
    [4X[25Xgap>[125X [27Xprg:= ScanStraightLineDecision( str );;[127X[104X
    [4X[25Xgap>[125X [27XAtlasStringOfProgram( prg.program );[127X[104X
    [4X[28X"inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5\n"[128X[104X
  [4X[32X[104X
  
  
  [1X7.5 [33X[0;0YData Types Used in the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YEach representation or program that is administrated by the [5XAtlasRep[105X package
  belongs  to  a  unique  [13Xdata  type[113X.  Informally,  examples of data types are
  [21Xpermutation  representation[121X,  [21Xmatrix  representation  over  the integers[121X, or
  [21Xstraight line program for computing class representatives[121X.[133X
  
  [33X[0;0YThe idea is that for each data type, there can be[133X
  
  [30X    [33X[0;6Ya column of its own in the output produced by [2XDisplayAtlasInfo[102X ([14X3.6-1[114X)
        when  called  without  arguments or with only argument a list of group
        names,[133X
  
  [30X    [33X[0;6Ya  line  format of its own for the output produced by [2XDisplayAtlasInfo[102X
        ([14X3.6-1[114X) when called with first argument a group name,[133X
  
  [30X    [33X[0;6Yan input format of its own for [2XAtlasProgram[102X ([14X3.6-4[114X),[133X
  
  [30X    [33X[0;6Yan input format of its own for [2XOneAtlasGeneratingSetInfo[102X ([14X3.6-6[114X), and[133X
  
  [30X    [33X[0;6Yspecific  tests  for  the  data of this data type; these functions are
        used by the global tests described in Section [14X7.9[114X.[133X
  
  [33X[0;0YFormally,  a  data  type is defined by a record whose components are used by
  the interface functions. The details are described in the following.[133X
  
  [1X7.5-1 AGR.DeclareDataType[101X
  
  [33X[1;0Y[29X[2XAGR.DeclareDataType[102X( [3Xkind[103X, [3Xname[103X, [3Xrecord[103X ) [32X function[133X
  
  [33X[0;0YLet  [3Xkind[103X  be  one of the strings [10X"rep"[110X or [10X"prg"[110X, and [3Xrecord[103X be a record. If
  [3Xkind[103X  is  [10X"rep"[110X  then  [2XAGR.DeclareDataType[102X  declares  a  new  data  type  of
  representations,  if  [3Xkind[103X  is  [10X"prg"[110X  then  it  declares a new data type of
  programs.  The  string  [3Xname[103X  is  the  name of the type, for example [10X"perm"[110X,
  [10X"matff"[110X, or [10X"classes"[110X. [5XAtlasRep[105X stores the data for each group internally in
  a record whose component [3Xname[103X holds the list of the data about the type with
  this name.[133X
  
  [33X[0;0Y[13XMandatory components[113X of [3Xrecord[103X are[133X
  
  [8X[10XFilenameFormat[110X[8X[108X
        [33X[0;6YThis  defines  the format of the filenames containing data of the type
        in  question.  The value must be a list that can be used as the second
        argument  of [2XAGR.ParseFilenameFormat[102X ([14X7.6-1[114X), such that only filenames
        of  the  type  in  question  match.  (It  is  not checked whether this
        [21Xdetection  function[121X  matches exactly one type, so declaring a new type
        needs care.)[133X
  
  [8X[10XAddFileInfo[110X[8X[108X
        [33X[0;6YThis  defines  the information stored in the table of contents for the
        data  of  the  type.  The  value  must  be a function that takes three
        arguments  (the current list of data for the type and the given group,
        a list returned by [2XAGR.ParseFilenameFormat[102X ([14X7.6-1[114X) for the given type,
        and  a  filename).  This function adds the necessary parts of the data
        entry  to  the list, and returns [9Xtrue[109X if the data belongs to the type,
        otherwise  [9Xfalse[109X  is returned; note that the latter case occurs if the
        filename  matches  the format description but additional conditions on
        the parts of the name are not satisfied (for example integer parts may
        be required to be positive or prime powers).[133X
  
  [8X[10XReadAndInterpretDefault[110X[8X[108X
        [33X[0;6YThis is the function that does the work for the default [10Xcontents[110X value
        of  the  [10XaccessFunctions[110X  component of [2XAtlasOfGroupRepresentationsInfo[102X
        ([14X7.1-5[114X),  see  Section [14X7.2[114X.  This function must take a path and return
        the [5XGAP[105X object given by this file.[133X
  
  [8X[10XAddDescribingComponents[110X[8X (for [10Xrep[110X[8X only)[108X
        [33X[0;6YThis  function takes two arguments, a record (that will be returned by
        [2XAtlasGenerators[102X   ([14X3.6-3[114X),   [2XOneAtlasGeneratingSetInfo[102X   ([14X3.6-6[114X),   or
        [2XAllAtlasGeneratingSetInfos[102X  ([14X3.6-7[114X))  and  the  type record [3Xrecord[103X. It
        sets  the components [10Xp[110X, [10Xdim[110X, [10Xid[110X, and [10Xring[110X that are promised for return
        values of the abovementioned three functions.[133X
  
  [8X[10XDisplayGroup[110X[8X (for [10Xrep[110X[8X only)[108X
        [33X[0;6YThis  defines  the  format  of  the  lines printed by [2XDisplayAtlasInfo[102X
        ([14X3.6-1[114X)  for  a given group. The value must be a function that takes a
        list  as  returned by the function given in the component [10XAddFileInfo[110X,
        and  returns  the  string  to  be  printed  for  the representation in
        question.[133X
  
  [33X[0;0Y[13XOptional components[113X of [3Xrecord[103X are[133X
  
  [8X[10XDisplayOverviewInfo[110X[8X[108X
        [33X[0;6YThis   is   used   to   introduce  a  new  column  in  the  output  of
        [2XDisplayAtlasInfo[102X ([14X3.6-1[114X) when this is called without arguments or with
        a  list  of group names as its only argument. The value must be a list
        of length three, containing at its first position a string used as the
        header of the column, at its second position one of the strings [10X"r"[110X or
        [10X"l"[110X,  denoting  right or left aligned column entries, and at its third
        position  a  function  that  takes  two arguments (a list of tables of
        contents of the [5XAtlasRep[105X package and a group name), and returns a list
        of length two, containing the string to be printed as the column value
        and  [9Xtrue[109X  or  [9Xfalse[109X, depending on whether private data is involved or
        not.  (The  default  is  [9Xfail[109X,  indicating that no new column shall be
        printed.)[133X
  
  [8X[10XDisplayPRG[110X[8X (for [10Xprg[110X[8X only)[108X
        [33X[0;6YThis is used in [2XDisplayAtlasInfo[102X ([14X3.6-1[114X) for [5XATLAS[105X programs. The value
        must  be  a  function  that  takes four arguments (a list of tables of
        contents to examine, a list containing the [5XGAP[105X name and the [5XATLAS[105X name
        of  the  given  group,  a  list  of  integers or [9Xtrue[109X for the required
        standardization,  and  a  list of all available standardizations), and
        returns  the  list of lines (strings) to be printed as the information
        about  the  available  programs  of the current type and for the given
        group. (The default is to return an empty list.)[133X
  
  [8X[10XAccessGroupCondition[110X[8X (for [10Xrep[110X[8X only)[108X
        [33X[0;6YThis is used in [2XDisplayAtlasInfo[102X ([14X3.6-1[114X) and [2XOneAtlasGeneratingSetInfo[102X
        ([14X3.6-6[114X). The value must be a function that takes two arguments (a list
        as  returned  by  [2XOneAtlasGeneratingSetInfo[102X  ([14X3.6-6[114X),  and  a  list of
        conditions), and returns [9Xtrue[109X or [9Xfalse[109X, depending on whether the first
        argument  satisfies  the conditions. (The default value is [2XReturnFalse[102X
        ([14XReference: ReturnFalse[114X).)[133X
  
        [33X[0;6YThe function must support conditions such as [10X[ IsPermGroup, true ][110X and
        [10X[  NrMovedPoints,  [ 5, 6 ] ][110X, in general a list of functions followed
        by  a  prescribed  value, a list of prescribed values, another (unary)
        function,  or the string [10X"minimal"[110X. For an overview of the interesting
        functions, see [2XDisplayAtlasInfo[102X ([14X3.6-1[114X).[133X
  
  [8X[10XAccessPRG[110X[8X (for [10Xprg[110X[8X only)[108X
        [33X[0;6YThis  is  used  in  [2XAtlasProgram[102X ([14X3.6-4[114X). The value must be a function
        that  takes  four  arguments (the current table of contents, the group
        name,  an  integer  or  a  list  of  integers or [9Xtrue[109X for the required
        standardization,  and  a  list  of  conditions  given  by the optional
        arguments  of [2XAtlasProgram[102X ([14X3.6-4[114X)), and returns either [9Xfail[109X or a list
        that  together  with  the group name forms the identifier of a program
        that   matches  the  conditions.  (The  default  value  is  [2XReturnFail[102X
        ([14XReference: ReturnFail[114X).)[133X
  
  [8X[10XAtlasProgram[110X[8X (for [10Xprg[110X[8X only)[108X
        [33X[0;6YThis  is  used in [2XAtlasProgram[102X ([14X3.6-4[114X) to create the result value from
        the identifier. (The default value is [10XAtlasProgramDefault[110X, which works
        whenever  the  second entry of the identifier is the filename; this is
        not  the case for example if the program is the composition of several
        programs.)[133X
  
  [8X[10XAtlasProgramInfo[110X[8X (for [10Xprg[110X[8X only)[108X
        [33X[0;6YThis  is  used  in [2XAtlasProgramInfo[102X ([14X3.6-5[114X) to create the result value
        from the identifier. (The default value is [10XAtlasProgramDefault[110X.)[133X
  
  [8X[10XTOCEntryString[110X[8X[108X
        [33X[0;6YThis  is  used in [2XStringOfAtlasTableOfContents[102X ([14X5.1-3[114X). The value must
        be  a function that takes two or three arguments (the name [3Xname[103X of the
        type,  a  list  as  returned  by  [2XAGR.ParseFilenameFormat[102X ([14X7.6-1[114X), and
        optionally  a  string  that indicates the [21Xremote[121X format) and returns a
        string  that describes the appropriate data format. (The default value
        is [10XTOCEntryStringDefault[110X.)[133X
  
  [8X[10XPostprocessFileInfo[110X[8X[108X
        [33X[0;6YThis is used in the construction of a table of contents for testing or
        rearranging  the data of the current table of contents. The value must
        be  a  function that takes two arguments, the table of contents record
        and  the  record  in  it that belongs to one fixed group. (The default
        function does nothing.)[133X
  
  [8X[10XSortTOCEntries[110X[8X[108X
        [33X[0;6YThis  is  used  in the construction of a table of contents for sorting
        the  entries  after  they  have  been added and after the value of the
        component  [10XPostprocessFileInfo[110X  has  been  called. The value must be a
        function  that  takes  a  list  as returned by [2XAGR.ParseFilenameFormat[102X
        ([14X7.6-1[114X),  and  returns  the  sorting  key. (There is no default value,
        which means that no sorting is needed.)[133X
  
  [8X[10XTestFileHeaders[110X[8X (for [10Xrep[110X[8X only)[108X
        [33X[0;6YThis is used in the function [10XAGR.Test.FileHeaders[110X. The value must be a
        function  that  takes  the  same  four  arguments  as [2XAGR.FileContents[102X
        ([14X7.6-2[114X),  except  that  the  third  argument  is a list as returned by
        [2XAGR.ParseFilenameFormat[102X  ([14X7.6-1[114X).  (The  default  value  is [2XReturnTrue[102X
        ([14XReference: ReturnTrue[114X).)[133X
  
  [8X[10XTestFiles[110X[8X (for [10Xrep[110X[8X only)[108X
        [33X[0;6YThis  is  used in the function [10XAGR.Test.Files[110X. The format of the value
        and the default are the same as for the component [10XTestFileHeaders[110X.[133X
  
  [8X[10XTestWords[110X[8X (for [10Xprg[110X[8X only)[108X
        [33X[0;6YThis  is  used  in  the  function  [10XAGR.Test.Words[110X. The value must be a
        function  that  takes five arguments where the first four are the same
        arguments  as  for  [2XAGR.FileContents[102X  ([14X7.6-2[114X),  except  that the fifth
        argument is [9Xtrue[109X or [9Xfalse[109X, indicating verbose mode or not.[133X
  
  
  [1X7.6 [33X[0;0YFilenames Used in the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0Y[5XAtlasRep[105X  expects that the filename of each data file describes the contents
  of  the file. This section lists the definitions of the supported structures
  of filenames.[133X
  
  [33X[0;0YEach  filename consists of two parts, separated by a minus sign [10X-[110X. The first
  part is always of the form [22Xgroupname[122X[10XG[110X[22Xi[122X, where the integer [22Xi[122X denotes the [22Xi[122X-th
  set  of  standard  generators  for  the  group  [22XG[122X, say, with [5XATLAS[105X-file name
  [22Xgroupname[122X  (see [14X3.3[114X).  The translations of the name [22Xgroupname[122X to the name(s)
  used    within    [5XGAP[105X    is    given    by   the   component   [10XGAPnames[110X   of
  [2XAtlasOfGroupRepresentationsInfo[102X ([14X7.1-5[114X).[133X
  
  [33X[0;0YThe  names  of  files  that  contain straight line programs or straight line
  decisions  have  one  of  the  following  forms. In each of these cases, the
  suffix [10XW[110X[22Xn[122X means that [22Xn[122X is the version number of the program.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-cycW[110X[8X[22Xn[122X[108X
        [33X[0;6YIn this case, the file contains a straight line program that returns a
        list of representatives of generators of maximally cyclic subgroups of
        [22XG[122X. An example is [10XCo1G1-cycW1[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-cclsW[110X[8X[22Xn[122X[108X
        [33X[0;6YIn this case, the file contains a straight line program that returns a
        list   of   conjugacy  class  representatives  of  [22XG[122X.  An  example  is
        [10XRuG1-cclsW1[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10XcycW[110X[8X[22Xn[122X[10X-cclsW[110X[8X[22Xm[122X[108X
        [33X[0;6YIn this case, the file contains a straight line program that takes the
        return value of the program in the file [22Xgroupname[122X[10XG[110X[22Xi[122X[10X-cycW[110X[22Xn[122X (see above),
        and returns a list of conjugacy class representatives of [22XG[122X. An example
        is [10XM11G1cycW1-cclsW1[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-max[110X[8X[22Xk[122X[10XW[110X[8X[22Xn[122X[108X
        [33X[0;6YIn  this  case,  the  file contains a straight line program that takes
        generators  of  [22XG[122X  w. r. t. the  [22Xi[122X-th  set of standard generators, and
        returns  a list of generators (in general [13Xnot[113X standard generators) for
        a  subgroup  [22XU[122X in the [22Xk[122X-th class of maximal subgroups of [22XG[122X. An example
        is [10XJ1G1-max7W1[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10Xmax[110X[8X[22Xk[122X[10XW[110X[8X[22Xn[122X[10X-[110X[8X[22Xsubgroupname[122X[10XG[110X[8X[22Xj[122X[10XW[110X[8X[22Xm[122X[108X
        [33X[0;6YIn this case, the file contains a straight line program that takes the
        return  value  of  the  program  in  the  file [22Xgroupname[122X[10XG[110X[22Xi[122X[10X-max[110X[22Xk[122X[10XW[110X[22Xn[122X (see
        above),  which  are  generators  for a group [22XU[122X, say; [22Xsubgroupname[122X is a
        name  for [22XU[122X, and the return value is a list of standard generators for
        [22XU[122X,  w. r. t. the  [22Xj[122X-th  set  of  standard  generators. (Of course this
        implies  that  the  groups in the [22Xk[122X-th class of maximal subgroups of [22XG[122X
        are  isomorphic  to  the  group with name [22Xsubgroupname[122X.) An example is
        [10XJ1G1max1W1-L211G1W1[110X; the first class of maximal subgroups of the Janko
        group  [22XJ_1[122X  consists of groups isomorphic to the linear group [22XL_2(11)[122X,
        for which standard generators are defined.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-a[110X[8X[22Xoutname[122X[10XW[110X[8X[22Xn[122X[108X
        [33X[0;6YIn  this  case,  the  file contains a straight line program that takes
        generators  of  [22XG[122X  w. r. t. the  [22Xi[122X-th  set of standard generators, and
        returns  the  list of their images under the outer automorphism [22Xα[122X of [22XG[122X
        given  by the name [22Xoutname[122X; if this name is empty then [22Xα[122X is the unique
        nontrivial outer automorphism of [22XG[122X; if it is a positive integer [22Xk[122X then
        [22Xα[122X  is  a  generator of the unique cyclic order [22Xk[122X subgroup of the outer
        automorphism  group  of  [22XG[122X; if it is of the form [10X2_1[110X or [10X2a[110X, [10X4_2[110X or [10X4b[110X,
        [10X3_3[110X  or  [10X3c[110X  [22X...[122X  then  [22Xα[122X  generates the cyclic group of automorphisms
        induced on [22XG[122X by [22XG.2_1[122X, [22XG.4_2[122X, [22XG.3_3[122X [22X...[122X; finally, if it is of the form
        [22Xk[122X[10Xp[110X[22Xd[122X, with [22Xk[122X one of the above forms and [22Xd[122X an integer then [22Xd[122X denotes the
        number of dashes appended to the automorphism described by [22Xk[122X; if [22Xd = 1[122X
        then   [22Xd[122X   can   be  omitted.  Examples  are  [10XA5G1-aW1[110X,  [10XL34G1-a2_1W1[110X,
        [10XU43G1-a2_3pW1[110X,  and  [10XO8p3G1-a2_2p5W1[110X;  these  file  names describe the
        outer  order  [22X2[122X automorphism of [22XA_5[122X (induced by the action of [22XS_5[122X) and
        the  order  [22X2[122X automorphisms of [22XL_3(4)[122X, [22XU_4(3)[122X, and [22XO_8^+(3)[122X induced by
        the  actions  of  [22XL_3(4).2_1[122X,  [22XU_4(3).2_2^'[122X, and [22XO_8^+(3).2_2^{'''''}[122X,
        respectively.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-ker[110X[8X[22Xfactgroupname[122X[10XW[110X[8X[22Xn[122X[108X
        [33X[0;6YIn  this  case,  the  file contains a straight line program that takes
        generators  of  [22XG[122X  w. r. t. the  [22Xi[122X-th  set of standard generators, and
        returns  generators  of  the kernel of an epimorphism that maps [22XG[122X to a
        group with [5XATLAS[105X-file name [22Xfactgroupname[122X. An example is [10X2A5G1-kerA5W1[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-G[110X[8X[22Xj[122X[10XW[110X[8X[22Xn[122X[108X
        [33X[0;6YIn  this  case,  the  file contains a straight line program that takes
        generators  of  [22XG[122X  w. r. t. the  [22Xi[122X-th  set of standard generators, and
        returns  standard  generators  of  [22XG[122X w. r. t. the [22Xj[122X-th set of standard
        generators. An example is [10XL35G1-G2W1[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-check[110X[8X[22Xn[122X[108X
        [33X[0;6YIn  this  case,  the file contains a straight line decision that takes
        generators  of  [22XG[122X,  and  returns [9Xtrue[109X if these generators are standard
        generators w. r. t. the [22Xi[122X-th standardization, and [9Xfalse[109X otherwise.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-P[110X[8X[22Xn[122X[108X
        [33X[0;6YIn  this  case,  the file contains a straight line decision that takes
        some  group  elements, and returns [9Xtrue[109X if these elements are standard
        generators   for  [22XG[122X,  w. r. t. the  [22Xi[122X-th  standardization,  and  [9Xfalse[109X
        otherwise.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-find[110X[8X[22Xn[122X[108X
        [33X[0;6YIn  this  case,  the  file  contains  a black box program that takes a
        group,  and returns (if it is successful) a set of standard generators
        for [22XG[122X, w. r. t. the [22Xi[122X-th standardization.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-X[110X[8X[22Xdescr[122X[10XW[110X[8X[22Xn[122X[108X
        [33X[0;6YIn  this  case,  the  file contains a straight line program that takes
        generators  of  [22XG[122X  w. r. t. the  [22Xi[122X-th  set of standard generators, and
        whose  return  value corresponds to [22Xdescr[122X. This format is used only in
        private extensions (see Chapter [14X5[114X), such a script can be accessed with
        [22Xdescr[122X as the third argument of [2XAtlasProgram[102X ([14X3.6-4[114X).[133X
  
  [33X[0;0YThe  names  of files that contain group generators have one of the following
  forms.  In  each of these cases, [22Xid[122X is a (possibly empty) string that starts
  with   a   lowercase   alphabet   letter  (see [2XIsLowerAlphaChar[102X  ([14XReference:
  IsLowerAlphaChar[114X)),  and  [22Xm[122X  is  a  nonnegative  integer,  meaning  that the
  generators  are  written  w. r. t. the [22Xm[122X-th basis (the meaning is defined by
  the [5XATLAS[105X developers).[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-f[110X[8X[22Xq[122X[10Xr[110X[8X[22Xdim[122X[22Xid[122X[10XB[110X[8X[22Xm[122X[10X.m[110X[8X[22Xnr[122X[108X
        [33X[0;6Ya file in [5XMeatAxe[105X text file format containing the [22Xnr[122X-th generator of a
        matrix  representation  over  the  field with [22Xq[122X elements, of dimension
        [22Xdim[122X. An example is [10XS5G1-f2r4aB0.m1[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-p[110X[8X[22Xn[122X[22Xid[122X[10XB[110X[8X[22Xm[122X[10X.m[110X[8X[22Xnr[122X[108X
        [33X[0;6Ya file in [5XMeatAxe[105X text file format containing the [22Xnr[122X-th generator of a
        permutation representation on [22Xn[122X points. An example is [10XM11G1-p11B0.m1[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-Ar[110X[8X[22Xdim[122X[22Xid[122X[10XB[110X[8X[22Xm[122X[10X.g[110X[8X[108X
        [33X[0;6Ya   [5XGAP[105X   readable   file   containing  all  generators  of  a  matrix
        representation  of  dimension  [22Xdim[122X  over an algebraic number field not
        specified further. An example is [10XA5G1-Ar3aB0.g[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-Zr[110X[8X[22Xdim[122X[22Xid[122X[10XB[110X[8X[22Xm[122X.g[108X
        [33X[0;6Ya   [5XGAP[105X   readable   file   containing  all  generators  of  a  matrix
        representation  over  the  integers,  of  dimension [22Xdim[122X. An example is
        [10XA5G1-Zr4B0.g[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-Hr[110X[8X[22Xdim[122X[22Xid[122X[10XB[110X[8X[22Xm[122X[10X.g[110X[8X[108X
        [33X[0;6Ya   [5XGAP[105X   readable   file   containing  all  generators  of  a  matrix
        representation  over  a  quaternion  algebra  over an algebraic number
        field, of dimension [22Xdim[122X. An example is [10X2A6G1-Hr2aB0.g[110X.[133X
  
  [8X[22Xgroupname[122X[10XG[110X[8X[22Xi[122X[10X-Z[110X[8X[22Xn[122X[10Xr[110X[8X[22Xdim[122X[22Xid[122X[10XB[110X[8X[22Xm[122X[10X.g[110X[8X[108X
        [33X[0;6Ya   [5XGAP[105X   readable   file   containing  all  generators  of  a  matrix
        representation  of  dimension  [22Xdim[122X over the ring of integers mod [22Xn[122X. An
        example is [10X2A8G1-Z4r4aB0.g[110X.[133X
  
  [1X7.6-1 AGR.ParseFilenameFormat[101X
  
  [33X[1;0Y[29X[2XAGR.ParseFilenameFormat[102X( [3Xstring[103X, [3Xformat[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya  list of strings and integers if [3Xstring[103X matches [3Xformat[103X, and [9Xfail[109X
            otherwise.[133X
  
  [33X[0;0YLet  [3Xstring[103X  be a filename, and [3Xformat[103X be a list [22X[ [ c_1, c_2, ..., c_n ], [
  f_1,  f_2, ..., f_n ] ][122X such that each entry [22Xc_i[122X is a list of strings and of
  functions  that take a character as their argument and return [11Xtrue[111X or [11Xfalse[111X,
  and  such  that each entry [22Xf_i[122X is a function for parsing a filename, such as
  the currently undocumented functions [10XParseForwards[110X and [10XParseBackwards[110X.[133X
  
  [33X[0;0Y[2XAGR.ParseFilenameFormat[102X returns a list of strings and integers such that the
  concatenation  of  their  [2XString[102X ([14XReference: String[114X) values yields [3Xstring[103X if
  [3Xstring[103X  matches  [3Xformat[103X, and [9Xfail[109X otherwise. Matching is defined as follows.
  Splitting  [3Xstring[103X  at each minus character ([10X-[110X) yields [22Xm[122X parts [22Xs_1, s_2, ...,
  s_m[122X.  The string [3Xstring[103X matches [3Xformat[103X if [22Xs_i[122X matches the conditions in [22Xc_i[122X,
  for  [22X1  ≤  i  ≤  n[122X,  in  the sense that applying [22Xf_i[122X to [22Xs_i[122X and [22Xc_i[122X yields a
  non-[9Xfail[109X result.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xformat:= [ [ [ IsChar, "G", IsDigitChar ],[127X[104X
    [4X[25X>[125X [27X                [ "p", IsDigitChar, AGR.IsLowerAlphaOrDigitChar,[127X[104X
    [4X[25X>[125X [27X                  "B", IsDigitChar, ".m", IsDigitChar ] ],[127X[104X
    [4X[25X>[125X [27X              [ ParseBackwards, ParseForwards ] ];;[127X[104X
    [4X[25Xgap>[125X [27XAGR.ParseFilenameFormat( "A6G1-p10B0.m1", format );[127X[104X
    [4X[28X[ "A6", "G", 1, "p", 10, "", "B", 0, ".m", 1 ][128X[104X
    [4X[25Xgap>[125X [27XAGR.ParseFilenameFormat( "A6G1-p15aB0.m1", format );[127X[104X
    [4X[28X[ "A6", "G", 1, "p", 15, "a", "B", 0, ".m", 1 ][128X[104X
    [4X[25Xgap>[125X [27XAGR.ParseFilenameFormat( "A6G1-f2r16B0.m1", format );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X7.6-2 AGR.FileContents[101X
  
  [33X[1;0Y[29X[2XAGR.FileContents[102X( [3Xfiles[103X, [3Xtype[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe  [5XGAP[105X object obtained from reading and interpreting the file(s)
            given by [3Xfiles[103X.[133X
  
  [33X[0;0YLet  [3Xfiles[103X  be  a  list  of  pairs  of the form [10X[ dirname, filename ][110X, where
  [10Xdirname[110X  and  [10Xfilename[110X  are  strings,  and  let  [3Xtype[103X  be  a  data type (see
  [2XAGR.DeclareDataType[102X  ([14X7.5-1[114X)).  Each  [10Xdirname[110X  must  be  one  of [10X"datagens"[110X,
  [10X"dataword"[110X,    or    the    [10Xdirid[110X   value   of   a   data   extension   (see
  [2XAtlasOfGroupRepresentationsNotifyData[102X  ([14X5.1-1[114X)).  If the contents of each of
  the  files  in question is accessible and their data belong to the data type
  [10Xtype[110X then [2XAGR.FileContents[102X returns the contents of the files; otherwise [9Xfail[109X
  is returned.[133X
  
  [33X[0;0YNote  that  if  some  file  is  already stored in the [3Xdirname[103X directory then
  [2XAGR.FileContents[102X  does  [13Xnot[113X  check  whether  the  relevant table of contents
  actually contains [3Xfilename[103X.[133X
  
  
  [1X7.7 [33X[0;0YThe record component [10Xidentifier[110X[101X[1X used by the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YThe     functions    [2XAtlasGenerators[102X    ([14X3.6-3[114X),    [2XAtlasProgram[102X    ([14X3.6-4[114X),
  [2XAtlasProgramInfo[102X    ([14X3.6-5[114X),    [2XOneAtlasGeneratingSetInfo[102X    ([14X3.6-6[114X),    and
  [2XAllAtlasGeneratingSetInfos[102X  ([14X3.6-7[114X)  return  records  which have a component
  [10Xidentifier[110X.  The  value  of this component describes the record in the sense
  that  one can reconstruct the whole record from it, and the [10Xidentifier[110X value
  can  be  used as an input for [2XAtlasGenerators[102X ([14X3.6-3[114X), [2XAtlasProgram[102X ([14X3.6-4[114X),
  [2XAtlasProgramInfo[102X ([14X3.6-5[114X), [2XAtlasGroup[102X ([14X3.6-8[114X), and [2XAtlasSubgroup[102X ([14X3.6-9[114X).[133X
  
  [33X[0;0YThe [10Xidentifier[110X component has the following format.[133X
  
  [30X    [33X[0;6YFor  records  describing  representations,  it is a list of the form [10X[
        gapname, files, std, info ][110X.[133X
  
  [30X    [33X[0;6YFor  records  describing  straight  line  programs  and  straight line
        decisions, it is a list of the form [10X[ gapname, files, std ][110X.[133X
  
  [33X[0;0YHere  [10Xgapname[110X  is  the  [5XGAP[105X name of the group in question, [10Xfiles[110X defines the
  data  files,  [10Xstd[110X is the standardization of its generators, and [10Xinfo[110X is some
  information  that depends on the type of the representation, for example the
  number of moved points in the case of a permutation representation.[133X
  
  [33X[0;0YThe [10Xfiles[110X entry has one of the following formats:[133X
  
  [30X    [33X[0;6Ya  string,  in  the case that exactly one file is needed that does not
        belong  to a private extension; an example of such an [10Xidentifier[110X value
        is [10X[ "J1", "J1G1-cycW1", 1 ][110X[133X
  
  [30X    [33X[0;6Ya  list  whose entries are strings (which refer to files from the core
        part  of  the  database)  and pairs of the form [10X[ tocid, file ][110X (which
        refer  to  files  from  the  extension  given  by  [10Xtocid[110X); examples of
        [10Xidentifier[110X values are [10X[ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5
        ][110X,  [10X[  "2.M12",  [  [  "mfer",  "2M12G1-cclsW1" ] ], 1 ][110X, [10X[ "2.M12", [
        "M12G1-max1W1", [ "internal", "2M12G1-kerM12W1" ] ], 1 ][110X, [10X[ "2.M12", [
        [  "mfer",  "2M12G1-p24bB0.m1" ], [ "mfer", "2M12G1-p24bB0.m2" ] ], 1,
        24 ][110X.[133X
  
  [33X[0;0YUp to version 1.5 of the [5XAtlasRep[105X package, a different [10Xidentifier[110X format was
  used  for  files from extensions of the database. Namely, the first entry of
  the  list was a pair [10X[ tocid, groupname ][110X, and the second entry was either a
  string  or  a  list  of  strings. Note that with that old format, it was not
  possible  to  describe a combination of several files from different sources
  (core   part   and   extension,   or  different  extensions).  The  function
  [2XAtlasRepIdentifier[102X ([14X7.7-1[114X) can be used to convert between the two formats.[133X
  
  
  [1X7.7-1 [33X[0;0YAtlasRepIdentifier[133X[101X
  
  [33X[1;0Y[29X[2XAtlasRepIdentifier[102X( [3Xoldid[103X ) [32X function[133X
  [33X[1;0Y[29X[2XAtlasRepIdentifier[102X( [3Xid[103X, [3X"old"[103X ) [32X function[133X
  
  [33X[0;0YThis  function  converts  between the [21Xold format[121X (the one used up to version
  1.5.1 of the package) and the [21Xnew format[121X (the one used since version 2.0) of
  the [10Xidentifier[110X component of the records returned by [5XAtlasRep[105X functions. Note
  that  the  two  formats  differ only for [10Xidentifier[110X components that describe
  data from non-core parts of the database.[133X
  
  [33X[0;0YIf  the  only  argument  is a list [3Xoldid[103X that is an [10Xidentifier[110X in old format
  then  the  function  returns  the corresponding [10Xidentifier[110X in new format. If
  there  are  two arguments, a list [3Xid[103X that is an [10Xidentifier[110X in new format and
  the  string [3X"old"[103X, then the function returns the corresponding [10Xidentifier[110X in
  old format if this is possible, and [9Xfail[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xid:= [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ];;[127X[104X
    [4X[25Xgap>[125X [27XAtlasRepIdentifier( id ) = id;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xid:= [ "L2(8)", "L28G1-check1", 1, 1 ];;[127X[104X
    [4X[25Xgap>[125X [27XAtlasRepIdentifier( id ) = id;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xoldid:= [ [ "priv", "C4" ], [ "C4G1-p4B0.m1" ], 1, 4 ];;[127X[104X
    [4X[25Xgap>[125X [27Xnewid:= AtlasRepIdentifier( oldid );[127X[104X
    [4X[28X[ "C4", [ [ "priv", "C4G1-p4B0.m1" ] ], 1, 4 ][128X[104X
    [4X[25Xgap>[125X [27Xoldid = AtlasRepIdentifier( newid, "old" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xoldid:= [ [ "priv", "C4" ], "C4G1-max1W1", 1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xnewid:= AtlasRepIdentifier( oldid );[127X[104X
    [4X[28X[ "C4", [ [ "priv", "C4G1-max1W1" ] ], 1 ][128X[104X
    [4X[25Xgap>[125X [27Xoldid = AtlasRepIdentifier( newid, "old" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xoldid:= [ [ "priv", "C4" ], "C4G1-Ar1aB0.g", 1, 1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xnewid:= AtlasRepIdentifier( oldid );[127X[104X
    [4X[28X[ "C4", [ [ "priv", "C4G1-Ar1aB0.g" ] ], 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xoldid = AtlasRepIdentifier( newid, "old" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xoldid:= [ [ "priv", "C4" ], "C4G1-XtestW1", 1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xnewid:= AtlasRepIdentifier( oldid );[127X[104X
    [4X[28X[ "C4", [ [ "priv", "C4G1-XtestW1" ] ], 1 ][128X[104X
    [4X[25Xgap>[125X [27Xoldid = AtlasRepIdentifier( newid, "old" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xoldid:= [ [ "mfer", "2.M12" ],[127X[104X
    [4X[25X>[125X [27X [ "2M12G1-p264aB0.m1", "2M12G1-p264aB0.m2" ], 1, 264 ];;[127X[104X
    [4X[25Xgap>[125X [27Xnewid:= AtlasRepIdentifier( oldid );[127X[104X
    [4X[28X[ "2.M12", [128X[104X
    [4X[28X  [ [ "mfer", "2M12G1-p264aB0.m1" ], [ "mfer", "2M12G1-p264aB0.m2" ] ][128X[104X
    [4X[28X    , 1, 264 ][128X[104X
    [4X[25Xgap>[125X [27Xoldid = AtlasRepIdentifier( newid, "old" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X7.8 [33X[0;0YThe Tables of Contents of the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YThe list of [5XAtlasRep[105X data is stored in several [13Xtables of contents[113X, which are
  given  essentially by JSON documents, one for the core data and one for each
  data  extension  in  the  sense  of  Chapter  [14X5[114X. The only exception are data
  extensions  by  locally  available  files  in  a  given directory, where the
  contents  of  this  directory itself describes the data in question. One can
  create such a JSON document for the contents of a given local data directory
  with the function [2XStringOfAtlasTableOfContents[102X ([14X5.1-3[114X).[133X
  
  [33X[0;0YHere  are  the  administrational  functions  that  are  called  when  a data
  extension  gets notified with [2XAtlasOfGroupRepresentationsNotifyData[102X ([14X5.1-1[114X).
  In  each  case,  [22Xgapname[122X  and [22Xatlasname[122X denote the [5XGAP[105X and [5XATLAS[105X name of the
  group in question (see Section [14X3.3[114X), and [22Xdirid[122X denotes the identifier of the
  data extension.[133X
  
  [33X[0;0YThe  following  functions define group names, available representations, and
  straight line programs.[133X
  
  [8X[10XAGR.GNAN( [110X[8X[22Xgapname, atlasname[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YCalled  with  two  strings  [22Xgapname[122X  (the  [5XGAP[105X  name of the group) and
        [22Xatlasname[122X   (the  [5XATLAS[105X  name  of  the  group),  [10XAGR.GNAN[110X  stores  the
        information   in  the  list  [10XAtlasOfGroupRepresentationsInfo.GAPnames[110X,
        which  defines  the name mapping between the [5XATLAS[105X names and [5XGAP[105X names
        of the groups.[133X
  
        [33X[0;6YAn example of a valid call is [10XAGR.GNAN("A5.2","S5")[110X.[133X
  
  [8X[10XAGR.TOC( [110X[8X[22Xtypename, filename, crc[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6Y[10XAGR.TOC[110X  notifies  an entry to the [10XTableOfContents.( [110X[22Xdirid[122X[10X )[110X component
        of  [2XAtlasOfGroupRepresentationsInfo[102X  ([14X7.1-5[114X). The string [22Xtypename[122X must
        be  the  name  of the data type to which the entry belongs, the string
        [22Xfilename[122X  must  be  the  prefix of the data file(s), and [22Xcrc[122X must be a
        list  that  contains the checksums of the data files, which are either
        integers   (see   [2XCrcFile[102X   ([14XReference:   CrcFile[114X))  or  strings  (see
        [10XHexSHA256[110X).  In  particular, the number of files that are described by
        the entry equals the length of [22Xcrc[122X.[133X
  
        [33X[0;6YThe  optional  argument  [22Xdirid[122X  is equal to the argument with the same
        name         in         the        corresponding        call        of
        [2XAtlasOfGroupRepresentationsNotifyData[102X ([14X5.1-1[114X). If no [22Xdirid[122X argument is
        given  then  the  current  value of [10XAGR.DIRID[110X is taken as the default;
        this  value is set automatically before a [11Xtoc.json[111X file gets evaluated
        by   [2XAtlasOfGroupRepresentationsNotifyData[102X   ([14X5.1-1[114X),   and  is  reset
        afterwards. If [10XAGR.DIRID[110X is not bound and [22Xdirid[122X is not given then this
        function has no effect.[133X
  
        [33X[0;6YAn example of a valid call is [10XAGR.TOC("perm","alt/A5/mtx/S5G1-p5B0.m",
        [-3581724,115937465])[110X.[133X
  
  [33X[0;0YThe  following  functions  add  data  about  the  groups  and their standard
  generators.  The  function  calls  must  be executed after the corresponding
  [10XAGR.GNAN[110X calls.[133X
  
  [8X[10XAGR.GRS( [110X[8X[22Xgapname, size[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YThe  integer  [22Xsize[122X  is  stored as the order of the group with [5XGAP[105X name
        [22Xgapname[122X, in [10XAtlasOfGroupRepresentationsInfo.GAPnames[110X.[133X
  
        [33X[0;6YAn example of a valid call is [10XAGR.GRS("A5.2",120)[110X.[133X
  
  [8X[10XAGR.MXN( [110X[8X[22Xgapname, nrMaxes[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YThe  integer  [22XnrMaxes[122X  is  stored  as the number of classes of maximal
        subgroups    of    the    group    with    [5XGAP[105X    name   [22Xgapname[122X,   in
        [10XAtlasOfGroupRepresentationsInfo.GAPnames[110X.[133X
  
        [33X[0;6YAn example of a valid call is [10XAGR.MXN("A5.2",4)[110X.[133X
  
  [8X[10XAGR.MXO( [110X[8X[22Xgapname, sizesMaxes[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YThe  list  [22XsizesMaxes[122X  of  subgroup  orders  of the classes of maximal
        subgroups  of  the group with [5XGAP[105X name [22Xgapname[122X (not necessarily dense,
        in        non-increasing        order)        is       stored       in
        [10XAtlasOfGroupRepresentationsInfo.GAPnames[110X.[133X
  
        [33X[0;6YAn example of a valid call is [10XAGR.MXO("A5.2",[60,24,20,12])[110X.[133X
  
  [8X[10XAGR.MXS( [110X[8X[22Xgapname, structureMaxes[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YThe  list  [22XstructureMaxes[122X  of strings describing the structures of the
        maximal  subgroups of the group with [5XGAP[105X name [22Xgapname[122X (not necessarily
        dense), is stored in [10XAtlasOfGroupRepresentationsInfo.GAPnames[110X.[133X
  
        [33X[0;6YAn        example        of        a        valid        call       is
        [10XAGR.MXS("A5.2",["A5","S4","5:4","S3x2"])[110X.[133X
  
  [8X[10XAGR.STDCOMP( [110X[8X[22Xgapname, factorCompatibility[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YThe  list [22XfactorCompatibility[122X (with entries the standardization of the
        group  with  [5XGAP[105X  name  [22Xgapname[122X  , the [5XGAP[105X name of a factor group, the
        standardization  of  this  factor group, and [9Xtrue[109X or [9Xfalse[109X, indicating
        whether  mapping  the  standard  generators  for  [22Xgapname[122X  to those of
        [22Xfactgapname[122X     defines     an     epimorphism)     is    stored    in
        [10XAtlasOfGroupRepresentationsInfo.GAPnames[110X.[133X
  
        [33X[0;6YAn example of a valid call is [10XAGR.STDCOMP("2.A5.2",[1,"A5.2",1,true])[110X.[133X
  
  [33X[0;0YThe  following  functions  add  data  about representations or straight line
  programs  that  are already known. The function calls must be executed after
  the corresponding [10XAGR.TOC[110X calls.[133X
  
  [8X[10XAGR.RNG( [110X[8X[22Xrepname, descr[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YCalled  with  two  strings  [22Xrepname[122X  (denoting  the  name  of  a  file
        containing  the generators of a matrix representation over a ring that
        is  not determined by the filename) and [22Xdescr[122X (describing this ring [22XR[122X,
        say), [10XAGR.RNG[110X adds the triple [22X[ repname, descr, R ][122X to the list stored
        in the [10Xringinfo[110X component of [2XAtlasOfGroupRepresentationsInfo[102X ([14X7.1-5[114X).[133X
  
        [33X[0;6YAn        example        of        a        valid        call       is
        [10XAGR.RNG("A5G1-Ar3aB0","Field([Sqrt(5)])")[110X.[133X
  
  [8X[10XAGR.TOCEXT( [110X[8X[22Xatlasname, std, maxnr, files[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YCalled with [22Xatlasname[122X, the positive integers [22Xstd[122X (the standardization)
        and [22Xmaxnr[122X (the number of the class of maximal subgroups), and the list
        [22Xfiles[122X (of filenames of straight line programs for computing generators
        of  the [22Xmaxnr[122X-th maximal subgroup, using a straight line program for a
        factor  group  plus  perhaps  some straight line program for computing
        kernel    generators),    [10XAGR.TOCEXT[110X   stores   the   information   in
        [10XAtlasOfGroupRepresentationsInfo.GAPnames[110X.[133X
  
        [33X[0;6YAn example of a valid call is [10XAGR.TOCEXT("2A5",1,3,["A5G1-max3W1"])[110X.[133X
  
  [8X[10XAGR.API( [110X[8X[22Xrepname, info[, dirid][122X[10X )[110X[8X[108X
        [33X[0;6YCalled  with  the  string  [22Xrepname[122X (denoting the name of a permutation
        representation)  and the list [22Xinfo[122X (describing the point stabilizer of
        this  representation),  [10XAGR.API[110X  binds  the  component  [22Xrepname[122X of the
        record  [10XAtlasOfGroupRepresentationsInfo.permrepinfo[110X  to  a record that
        describes the contents of [22Xinfo[122X.[133X
  
        [33X[0;6Y[22Xinfo[122X has the following entries.[133X
  
        [30X    [33X[0;12YAt position [22X1[122X, the transitivity is stored.[133X
  
        [30X    [33X[0;12YIf  the  transitivity  is zero then [22Xinfo[122X has length two, and the
              second entry is the list of orbit lengths.[133X
  
        [30X    [33X[0;12YIf  the  transitivity  is  positive then [22Xinfo[122X has length four or
              five, and the second entry is the rank of the action.[133X
  
        [30X    [33X[0;12YIf  the  transitivity is positive then the third entry is one of
              the strings [10X"prim"[110X, [10X"imprim"[110X, denoting primitivity or not.[133X
  
        [30X    [33X[0;12YIf  the transitivity is positive then the fourth entry is either
              the string [10X"???"[110X or a string that describes the structure of the
              point  stabilizer.  If  the  third  entry  is [10X"imprim"[110X then this
              description  consists  of a subgroup part and a maximal subgroup
              part, separated by [10X" < "[110X.[133X
  
        [30X    [33X[0;12YIf  the third entry is [10X"prim"[110X then the fifth entry is either the
              string  [10X"???"[110X  or  the  number of the class of maximal subgroups
              that are the point stabilizers.[133X
  
        [33X[0;6YAn        example        of        a        valid        call       is
        [10XAGR.API("A5G1-p5B0",[3,2,"prim","A4",1])[110X.[133X
  
  [8X[10XAGR.CHAR( [110X[8X[22Xgapname, repname, char, pos[, charname[, dirid]][122X[10X )[110X[8X[108X
        [33X[0;6YCalled  with the strings [22Xgapname[122X and [22Xrepname[122X (denoting the name of the
        representation),   the   integer   [22Xchar[122X  (the  characteristic  of  the
        representation),  and  [22Xpos[122X  (the  position or list of positions of the
        irreducible   constituent(s)),  [10XAGR.CHAR[110X  stores  the  information  in
        [10XAtlasOfGroupRepresentationsInfo.characterinfo[110X.[133X
  
        [33X[0;6YA string describing the character can be entered as [22Xcharname[122X.[133X
  
        [33X[0;6YIf  [22Xdirid[122X is given but no [22Xcharname[122X is known then one can enter [9Xfail[109X as
        the fifth argument.[133X
  
        [33X[0;6YAn        example        of        a        valid        call       is
        [10XAGR.CHAR("M11","M11G1-p11B0",0,[1,2],"1a+10a")[110X.[133X
  
  
  [1X7.9 [33X[0;0YSanity Checks for the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YThe  file  [11Xtst/testall.g[111X  of  the  package  contains  [2XTest[102X ([14XReference: Test[114X)
  statements for checking whether the [5XAtlasRep[105X functions behave as documented.
  One  can run these tests by calling [10XReadPackage( "AtlasRep", "tst/testall.g"
  )[110X.  The  examples  in  the package manual form a part of the tests, they are
  collected in the file [11Xtst/docxpl.tst[111X of the package.[133X
  
  [33X[0;0YThe remainder of this section deals with consistency checks of the data. The
  tests  described in Section [14X7.9-1[114X can be used for data from any extension of
  the  database (see Chapter [14X5[114X), Section [14X7.9-2[114X lists tests which apply only to
  the core part of the database.[133X
  
  [33X[0;0YAll  these tests apply only to [13Xlocally[113X available files (see Section [14X7.8[114X), no
  files  are downloaded during the tests. Thus the required space and time for
  running these tests depend on the amount of locally available data.[133X
  
  [33X[0;0YSome  of  the  tests compute and verify additional data, such as information
  about  point  stabilizers  of  permutation  representations. In these cases,
  output   lines   starting   with   [10X#E[110X  are  error  messages  that  point  to
  inconsistencies,  whereas  output  lines  starting with [10X#I[110X inform about data
  that  have  been computed and were not yet stored, or about stored data that
  were  not  verified.  These  tests  are  experimental in the sense that they
  involve several heuristics. Depending on the data to which they are applied,
  it may happen that the tests run out of space or do not finish in acceptable
  time. Please inform the package maintainer if you run into such problems.[133X
  
  
  [1X7.9-1 [33X[0;0YSanity Checks for a Table of Contents[133X[101X
  
  [33X[0;0YThe  following  tests  can  be used to check the data that belong to a given
  part  of the database (core data or extension). Each of these tests is given
  by  a function with optional argument [22Xtocid[122X, the identifying string that had
  been entered as the second argument of [2XAtlasOfGroupRepresentationsNotifyData[102X
  ([14X5.1-1[114X).  The  contents  of the core part can be checked by entering [10X"core"[110X,
  which  is also the default for [22Xtocid[122X. The function returns [9Xfalse[109X if an error
  occurs,  otherwise  [9Xtrue[109X.  Currently  the  following  tests of this kind are
  available.  (For  some  of  them,  the  global option [10XTryToExtendData[110X can be
  entered in order to try the computation of not yet stored data.)[133X
  
  [8X[10XAGR.Test.GroupOrders()[110X[8X[108X
        [33X[0;6Ychecks  whether  the  group orders stored in the [10XGAPnames[110X component of
        [2XAtlasOfGroupRepresentationsInfo[102X ([14X7.1-5[114X) coincide with the group orders
        computed  from  an  [5XATLAS[105X  permutation  representation of degree up to
        [10XAGR.Test.MaxTestDegree[110X, from the available character table or table of
        marks  with  the given name, or from the structure of the name, in the
        sense  that  splitting  the name at the first dot ([10X.[110X) or colon ([10X:[110X) and
        applying  the  same  criteria  to  derive the group order from the two
        parts may yield enough information.[133X
  
  [8X[10XAGR.Test.Words( [[110X[8X[22Xtocid[122X[10X] )[110X[8X[108X
        [33X[0;6Yprocesses  the  straight line programs that belong to [22Xtocid[122X, using the
        function  stored  in  the  [10XTestWords[110X  component  of  the  data type in
        question.[133X
  
        [33X[0;6YThe    straight    line    programs    for   the   cases   listed   in
        [10XAGR.Test.HardCases.TestWords[110X are omitted.[133X
  
  [8X[10XAGR.Test.ClassScripts( [[110X[8X[22Xtocid[122X[10X] )[110X[8X[108X
        [33X[0;6Ychecks  whether  the  straight  line programs that belong to [22Xtocid[122X and
        that   compute   representatives  of  certain  conjugacy  classes  are
        consistent  with  information stored on the [5XGAP[105X character table of the
        group  in  question,  in  the  sense that the given class names really
        occur  in  the  character  table  and  that  the  element  orders  and
        centralizer orders for the classes are correct.[133X
  
  [8X[10XAGR.Test.CycToCcls( [[110X[8X[22Xtocid[122X[10X][:TryToExtendData] )[110X[8X[108X
        [33X[0;6Ychecks  whether  all  straight  line programs that belong to [22Xtocid[122X and
        that  compute  class  representatives  from  representatives of cyclic
        subgroups  possess  a corresponding straight line program ([13Xanywhere[113X in
        the database) for computing representatives of cyclic subgroups.[133X
  
  [8X[10XAGR.Test.FileHeaders( [[110X[8X[22Xtocid[122X[10X] )[110X[8X[108X
        [33X[0;6Ychecks  whether  the  [5XMeatAxe[105X  text  files that belong to [22Xtocid[122X have a
        header  line  that  is  consistent  with the filename, and whether the
        contents  of  all  [5XGAP[105X  format  data  files  that  belong  to [22Xtocid[122X is
        consistent with the filename.[133X
  
  [8X[10XAGR.Test.Files( [[110X[8X[22Xtocid[122X[10X] )[110X[8X[108X
        [33X[0;6Ychecks whether the [5XMeatAxe[105X text files that belong to [22Xtocid[122X can be read
        with  [2XScanMeatAxeFile[102X  ([14X7.3-1[114X)  such  that the result is not [9Xfail[109X. The
        function  does not check whether the first line of a [5XMeatAxe[105X text file
        is  consistent  with  the  filename,  since  this  can  be tested with
        [10XAGR.Test.FileHeaders[110X.[133X
  
  [8X[10XAGR.Test.BinaryFormat( [[110X[8X[22Xtocid[122X[10X] )[110X[8X[108X
        [33X[0;6Ychecks  whether  all  [5XMeatAxe[105X  text files that belong to [22Xtocid[122X satisfy
        that   applying   first   [2XCMtxBinaryFFMatOrPerm[102X   ([14X7.3-4[114X)   and   then
        [2XFFMatOrPermCMtxBinary[102X ([14X7.3-5[114X) yields the same object.[133X
  
  [8X[10XAGR.Test.Primitivity( [[110X[8X[22Xtocid[122X[10X][:TryToExtendData] )[110X[8X[108X
        [33X[0;6Ychecks   the   stored  primitivity  information  for  the  permutation
        representations  that  belong to [22Xtocid[122X. That is, the number of orbits,
        in  case  of  a  transitive  action  the  transitivity,  the rank, the
        information  about the point stabilizers are computed if possible, and
        compared with the stored information.[133X
  
  [8X[10XAGR.Test.Characters( [[110X[8X[22Xtocid[122X[10X][:TryToExtendData] )[110X[8X[108X
        [33X[0;6Ychecks  the  character  information  (that  belongs  to [22Xtocid[122X) for the
        matrix and permutation representations.[133X
  
  [8X[10XAGR.Test.StdCompatibility( [[110X[8X[22Xtocid[122X[10X][:TryToExtendData] )[110X[8X[108X
        [33X[0;6Ychecks  whether  the  information  about the compatibility of standard
        generators  of  a  group  and  its factor groups that is stored in the
        [10XGAPnames[110X  component  of  [2XAtlasOfGroupRepresentationsInfo[102X  ([14X7.1-5[114X)  and
        belongs to [22Xtocid[122X coincides with computed values.[133X
  
        [33X[0;6YThe following criterion is used for computing the value for a group [22XG[122X.
        Use  the [5XGAP[105X Character Table Library to determine factor groups [22XF[122X of [22XG[122X
        for  which standard generators are defined and moreover a presentation
        in  terms of these standard generators is known. Evaluate the relators
        of  the presentation in the standard generators of [22XG[122X, and let [22XN[122X be the
        normal  closure  of  these  elements  in  [22XG[122X. Then mapping the standard
        generators  of [22XF[122X to the [22XN[122X-cosets of the standard generators of [22XG[122X is an
        epimorphism.  If  [22X|G/N| = |F|[122X holds then [22XG/N[122X and [22XF[122X are isomorphic, and
        the  standard  generators  of [22XG[122X and [22XF[122X are compatible in the sense that
        mapping the standard generators of [22XG[122X to their [22XN[122X-cosets yields standard
        generators of [22XF[122X.[133X
  
  [8X[10XAGR.Test.KernelGenerators( [[110X[8X[22Xtocid[122X[10X][:TryToExtendData] )[110X[8X[108X
        [33X[0;6Ychecks  whether  the straight line programs (that belong to [22Xtocid[122X) for
        computing  generators of kernels of natural epimorphisms between [5XATLAS[105X
        groups  compute  generators  of  normal  subgroups  of the right group
        orders. If it is known that the given standard generators of the given
        group are compatible with some standard generators of the factor group
        in  question (see the section about [10XAGR.Test.StdCompatibility[110X) then it
        is  also checked whether evaluating the straight line program at these
        standard generators of the factor group yields only the identity.[133X
  
        [33X[0;6YNote that the verification of normal subgroups of matrix groups may be
        [13Xvery[113X  time  and  space  consuming if the package [5Xrecog[105X [NSA+25] is not
        available.[133X
  
        [33X[0;6YThe  function also tries to [13Xfind[113X words for computing kernel generators
        of  those epimorphisms for which no straight line programs are stored;
        the  candidates  are  given  by  stored  factor  fusions  between  the
        character tables from the [5XGAP[105X Character Table Library.[133X
  
  [8X[10XAGR.Test.MaxesOrders( [[110X[8X[22Xtocid[122X[10X] )[110X[8X[108X
        [33X[0;6Ychecks whether the orders of maximal subgroups stored in the component
        [10XGAPnames[110X  of [2XAtlasOfGroupRepresentationsInfo[102X ([14X7.1-5[114X) coincide with the
        orders   computed   from  the  restriction  of  an  [5XATLAS[105X  permutation
        representation   of  degree  up  to  [10XAGR.Test.MaxTestDegree[110X  (using  a
        straight  line  program  that  belongs  to  [22Xtocid[122X), from the character
        table,  or  the  table  of  marks  with  the  given  name, or from the
        information  about  maximal  subgroups  of  the  factor group modulo a
        normal subgroup that is contained in the Frattini subgroup.[133X
  
  [8X[10XAGR.Test.MaxesStructure()[110X[8X[108X
        [33X[0;6Ychecks  whether the names of maximal subgroups stored in the component
        [10XGAPnames[110X  of [2XAtlasOfGroupRepresentationsInfo[102X ([14X7.1-5[114X) coincide with the
        names computed from the [5XGAP[105X character table with the given name.[133X
  
  [8X[10XAGR.Test.MaxesStandardization( [[110X[8X[22Xtocid[122X[10X] )[110X[8X[108X
        [33X[0;6Ychecks  whether  the straight line programs (that belong to [22Xtocid[122X) for
        standardizing  the  generators  of maximal subgroups are correct: If a
        semi-presentation  is  available  for  the  maximal  subgroup  and the
        standardization  in  question  then  it is used, otherwise an explicit
        isomorphism is tried.[133X
  
  [8X[10XAGR.Test.CompatibleMaxes( [[110X[8X[22Xtocid[122X[10X][:TryToExtendData] )[110X[8X[108X
        [33X[0;6Ychecks  whether  the information about deriving straight line programs
        for  restricting  to subgroups from straight line programs that belong
        to a factor group coincide with computed values.[133X
  
        [33X[0;6YThe following criterion is used for computing the value for a group [22XG[122X.
        If [22XF[122X is a factor group of [22XG[122X such that the standard generators of [22XG[122X and
        [22XF[122X are compatible (see the test function [10XAGR.Test.StdCompatibility[110X) and
        if  there are a presentation for [22XF[122X and a permutation representation of
        [22XG[122X  then  it is checked whether the [10X"maxes"[110X type straight line programs
        for  [22XF[122X  can be used to compute generators for the maximal subgroups of
        [22XG[122X;  if  not  then  generators of the kernel of the natural epimorphism
        from [22XG[122X to [22XF[122X, must be added.[133X
  
  
  [1X7.9-2 [33X[0;0YOther Sanity Checks[133X[101X
  
  [33X[0;0YThe  tests  described in this section are intended for checking data that do
  not  belong  to  a  particular  part of the [5XAtlasRep[105X database. Therefore [13Xall[113X
  locally  available  data are used in these tests. Each of the tests is given
  by  a  function  without arguments that returns [9Xfalse[109X if a contradiction was
  found  during  the  test, and [9Xtrue[109X otherwise. Additionally, certain messages
  are  printed when contradictions between stored and computed data are found,
  when   stored   data   cannot  be  verified  computationally,  or  when  the
  computations  yield improvements of the stored data. Currently the following
  tests of this kind are available.[133X
  
  [8X[10XAGR.Test.Standardization()[110X[8X[108X
        [33X[0;6Ychecks  whether  all  generating sets corresponding to the same set of
        standard  generators  have  the same element orders; for the case that
        straight line programs for computing certain class representatives are
        available,  also  the  orders  of  these  representatives  are checked
        w. r. t. all generating sets.[133X
  
  [8X[10XAGR.Test.StdTomLib()[110X[8X[108X
        [33X[0;6Ychecks  whether the standard generators are compatible with those that
        occur in the [5XTomLib[105X package.[133X
  
  [8X[10XAGR.Test.MinimalDegrees()[110X[8X[108X
        [33X[0;6Ychecks  that the (permutation and matrix) representations available in
        the  database  do  not have smaller degree than the minimum claimed in
        Section [14X6.3[114X.[133X
  
  [33X[0;0YFinally,  we reset the user preference and the info level which had been set
  at the beginning of Chapter [14X2[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetUserPreference( "AtlasRep", "DisplayFunction", origpref );[127X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel( InfoAtlasRep, globallevel );[127X[104X
  [4X[32X[104X
  
