[AtlasRep] 4 Technicalities of the AtlasRep Package

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# 4 Technicalities of the AtlasRep Package

### Sections

1. Global Variables Used by the AtlasRep Package
2. Reading and Writing MeatAxe Format Files
3. Reading and Writing Atlas Straight Line Programs
4. Data Types Used in the Atlas of Group Representations
5. Filenames Used in the Atlas of Group Representations
6. The Tables of Contents of the Atlas of Group Representations
7. Sanity Checks for the Atlas of Group Representations

This chapter describes those parts of the GAP interface to the ATLAS of Group Representations that do not belong to the user interface (cf. Chapter The User Interface of the AtlasRep Package). Besides global variables used for administrational purposes (see Global Variables Used by the AtlasRep Package) and some sanity checks (see Sanity Checks for the Atlas of Group Representations), they can be regarded as the interface between the data actually contained in the files and the corresponding GAP objects (see Reading and Writing MeatAxe Format Files and Reading and Writing Atlas Straight Line Programs), and the interface between the remote and the local version of the database (see Filenames Used in the Atlas of Group Representations and The Tables of Contents of the Atlas of Group Representations). The former interface contains functions to read and write files in MeatAxe format, which may be interesting for users familiar with MeatAxe standalones (see for example Rin98). 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 `gap` directory of the package, but it may happen that they are changed in future versions of the package.

## 4.1 Global Variables Used by the AtlasRep Package

For debugging purposes, the functions from the GAP interface to the ATLAS of Group Representations print information depending on the info level of the info classes `InfoAtlasRep` and `InfoCMeatAxe` (cf. Info Functions in the GAP Reference Manual).

The info level of an info class can be changed using `SetInfoLevel` (see SetInfoLevel in the GAP Reference Manual). For example, the info level of `InfoAtlasRep` can be set to the nonnegative integer n using `SetInfoLevel( InfoAtlasRep,` n`)`.

Information about files being read can be obtained by setting the value of the global variable `InfoRead1` to `Print`.

- `InfoAtlasRep V`

  If the info level of `InfoAtlasRep` is at least 1 then information about `fail` results of functions in the AtlasRep package is printed. If the info level is at least 2 then information about calls to external programs is printed. The default level is 0, no information is printed on this level.

- `InfoCMeatAxe V`

  If the info level of `InfoCMeatAxe` is at least 1 then information about `fail` results of C-MeatAxe functions is printed. The default level is 0, no information is printed on this level.

- `CMeatAxe.FastRead V`

  If this component is bound and has the value `true` then `ScanMeatAxeFile` (see ScanMeatAxeFile) reads matrices via `StringFile` (see gapdoc:StringFile in the GAP Package GapDoc). Otherwise each file containing a matrix over a finite field is read line by line via `ReadLine` (see ReadLine in the GAP Reference Manual), and the GAP matrix is constructed line by line, in a compressed representation (see Row Vectors over Finite Fields and Matrices over Finite Fields in the GAP Reference Manual), which makes it possible to read large matrices in a reasonable amount of space. The `StringFile` approach is faster but needs more intermediate space when matrices over finite fields are read.

- `AtlasOfGroupRepresentationsInfo V`

  `AtlasOfGroupRepresentationsInfo` is a record that is set in the file `gap/types.g` of the package. It has the following components.

  `remote`
    a boolean that controls what files are available; if the value is `true` then GAP is allowed to try remotely accessing any ATLAS file from the servers (see below) and thus all files listed in the global table of contents are available, if the value is `false` then GAP may access only those files that are stored in the database directories of the local GAP installation (see Local or Remote Installation of the AtlasRep Package),

  `compress`
    indexgzip compressed form; if the value is `true` then these files are compressed with `gzip` after they have been fetched from a server (see User Parameters for the AtlasRep Package),

  `wget` (optional)
    indexperl files, otherwise `perl` is used (see the manual index for references),

  `servers`
    indexftp where server is a string denoting the `ftp` address of a server where files can be fetched that are not stored in the local database, path is a string describing the path where the data directories on the server reside (relative to the directory entered via `ftp` login), login is the login name for the server access, and password is the corresponding password; only this list must be adjusted if new servers are added or if the addresses of servers change,

  `dirnames`
    a list of the names of the directories on the servers that contain subdirectories corresponding to the individual groups,

  `GAPnames`
    a list of pairs, each containing the GAP name and the ATLAS-file name of a group (see Group Names Used in the AtlasRep Package),

  `groupnames`
    a list of triples, each containing at the first position the name of the directory on each server that contains data about the group G in question, at the second position the name of the (usually simple) group for which a subdirectory exists that contains the data about G, and at the third position the ATLAS-file name used for G (see Filenames used in the Atlas of Group Representations),

  `fieldinfo`
    a list of triples, each containing at the first position the name of the file with the matrices of a characteristic zero representation, at the second position a string describing the field generated by the matrix entries, and at the third position this field itself; `DisplayAtlasInfo` (see DisplayAtlasInfo) displays this information for representations over proper extensions of the rational number field only if the representation is mentioned in the `fieldinfo` list,

  `private`
    a list of pairs of strings used for administrating private data (see Chapter Private Extensions of the AtlasRep Package),

  `markprivate`
    a string used to mark private data (see The Effect of Private Extensions on the User Interface), and

  `TableOfContents`
    a record with at most the components `local`, `remote`, `types`, `data`, and the names of private data directories; the values of the components `local` and `remote` can be computed automatically by `ReloadAtlasTableOfContents` (see ReloadAtlasTableOfContents), and the values of the components for local data directories are created by `AtlasOfGroupRepresentationsNotifyPrivateDirectory` (see AtlasOfGroupRepresentationsNotifyPrivateDirectory).

  ## 4.2 Reading and Writing MeatAxe Format Files

- `ScanMeatAxeFile(` filename`[,` q`] ) F`
- `ScanMeatAxeFile(` string`[,` q`], "string" ) F`

  Let filename be the name of a GAP readable file (see Filename in the GAP Reference Manual) that contains a matrix or a permutation or a list of permutations in MeatAxe text format (see the section about the program `zcv` in the MeatAxe manual Rin98), and let q be a prime power. `ScanMeatAxeFile` returns the corresponding GAP matrix or list of permutations, respectively.

  If the file contains a matrix then the way how it is read depends on the value of the global variable `CMeatAxe.FastRead` (see Global Variables Used by the AtlasRep Package). If the parameter q is given then the result matrix is represented over the field with q elements.

  If the file contains a list of permutations then it is read with `StringFile` (see gapdoc:StringFile in the GAP package GapDoc); no parameter q is needed in this case.

  In the second form, the first argument string must be a string as obtained by reading a file in MeatAxe text format as a text stream (see InputTextFile in the GAP Reference Manual), and the third argument the string `"string"`. Also in this case, `ScanMeatAxeFile` returns the corresponding GAP matrix or list of permutations, respectively.

- `MeatAxeString(` mat`,` q`) O`
- `MeatAxeString(` perms`,` degree`) O`
- `MeatAxeString(` perm`,` q`,` dims`) O`

  In the first form, for a matrix mat whose entries lie in the finite field with q elements, `MeatAxeString` returns a string that encodes mat as a matrix over `GF(`q`)`, in MeatAxe text format.

  In the second form, for a nonempty list perms of permutations that move only points up to the positive integer degree, `MeatAxeString` returns a string that encodes perms as permutations of degree degree, in MeatAxe text format (see Rin98).

  In the third form, for a permutation perm with largest moved point n, say, a prime power q, and a list dims of length 2 containing two positive integers larger than or equal to n, `MeatAxeString` returns a string that encodes perm as a matrix over `GF(`q`)`, of dimensions dims, whose first n rows and columns describe the permutation matrix corresponding to perm, and the remaining rows and columns are zero.

  When strings are printed to files using `PrintTo` or `AppendTo` then line breaks are inserted after `SizeScreen()[1]` characters not containing newline characters (see Operations for Output Streams in the GAP Reference Manual). This behaviour is not desirable for creating MeatAxe files. So the recommended functions for printing the result of `MeatAxeString` to a file are `FileString` (see gapdoc:FileString in the GAP Package GapDoc) and `WriteAll`.

      gap> mat:= [ [ 1, -1 ], [ 0, 1 ] ] * Z(3)^0;;
      gap> str:= MeatAxeString( mat, 3 );
      "1 3 2 2\n12\n01\n"
      gap> mat = ScanMeatAxeFile( str, "string" );
      true
      gap> str:= MeatAxeString( mat, 9 );
      "1 9 2 2\n12\n01\n"
      gap> mat = ScanMeatAxeFile( str, "string" );
      true
      gap> perms:= [ (1,2,3)(5,6) ];;
      gap> str:= MeatAxeString( perms, 6 );
      "12 1 6 1\n2\n3\n1\n4\n6\n5\n"
      gap> perms = ScanMeatAxeFile( str, "string" );
      true
      gap> str:= MeatAxeString( perms, 8 );
      "12 1 8 1\n2\n3\n1\n4\n6\n5\n7\n8\n"
      gap> perms = ScanMeatAxeFile( str, "string" );
      true
      gap> perm:= (1,2,4);;
      gap> str:= MeatAxeString( perm, 3, [ 5, 6 ] );
      "2 3 5 6\n2\n4\n3\n1\n5\n"
      gap> mat:= ScanMeatAxeFile( str, "string" );;  Print( mat, "\n" );
      [ [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],
        [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ],
        [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ],
        [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],
        [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ] ]
      gap> MeatAxeString( mat, 3 ) = str;
      true

- `FFList(` F`) F`
- `FFLists V`

  `FFList` is a utility program for the conversion of vectors and matrices from MeatAxe format to GAP format and vice versa. It is used by `ScanMeatAxeFile` (see ScanMeatAxeFile) and `MeatAxeString` (see MeatAxeString).

  For a finite field F, `FFList` returns a list l giving the correspondence between the MeatAxe numbering and the GAP numbering of the elements in F.

  The element of F corresponding to MeatAxe number n is ``l`[` n`+1 ]`, and the MeatAxe number of the field element z is `Position(` l`,` z`) - 1`.

  The global variable `FFLists` is used to store the information about F once it has been computed.

      gap> FFList( GF(4) );
      [ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2 ]
      gap> IsBound( FFLists[4] );
      true

  ## 4.3 Reading and Writing Atlas Straight Line Programs

- `ScanStraightLineProgram(` filename`) F`
- `ScanStraightLineProgram(` string`, "string" ) F`

  Let filename 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.

  `#` anything``
    lines starting with a hash sign `#` are ignored,

  `echo` anything``
    lines starting with `echo` are ignored for the `program` 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,

  `inp` n``
    means that there are n inputs, referred to via the labels `1`, `2`, ¼, n,

  `inp` k`` a1`` a2`...` ak``
    means that the next k inputs are referred to via the labels a1, a2, ..., ak,

  `cjr` a`` b``
    means that a is replaced by ``b`^(-1)` a`` b``,

  `cj` a`` b`` c``
    means that c is defined as ``b`^(-1)` a`` b``,

  `com` a`` b`` c``
    means that c is defined as ``a`^(-1)` b`^(-1)` a`` b``,

  `iv` a`` b``
    means that b is defined as ``a`^(-1)`,

  `mu` a`` b`` c``
    means that c is defined as ``a`*`b``,

  `pwr` a`` b`` c``
    means that c is defined to be ``b`^`a``, and

  `cp` a`` b``
    means that b is defined as a copy of a,

  `oup` l``
    means that there are l outputs, stored in the labels `1`, `2`, ¼, l,

  `oup` l`` b1`` b2`...` bl``
    means that the next l outputs are stored in the labels b1, b2, ... bl.

  Each of the labels a, b, c can be any nonempty sequence of digits and alphabet characters, except that the first argument of `pwr` must denote an integer.

  If the `inp` or `oup` statements are missing then the input or output, respectively, is assumed to be given by the labels `1` and `2`. There can be multiple `inp` lines at the beginning of the program and multiple `oup` lines at the end of the program. Only the first `inp` or `oup` line may omit the names of the elements. For example, an empty file filename or an empty string string represent a straight line program with two inputs that are returned as outputs.

  No command (except `cjr`) may overwrite its own input. For example, the line `mu a b a` is not legal. (This is not checked.)

  `ScanStraightLineProgram` returns a record containing as the value of its component `program` the corresponding GAP straight line program (see IsStraightLineProgram in the GAP Reference Manual) if the input string satisfies the syntax rules stated above, and returns `fail` otherwise. In the latter case, information about the first corrupted line of the program is printed if the info level of `InfoCMeatAxe` is at least 1 (see InfoCMeatAxe).

  In the second form, the first argument string must be a string as obtained by reading a file in MeatAxe text format as a text stream (see InputTextFile in the GAP Reference Manual), and the second argument the string `"string"`. Also in this case, `ScanStraightLineProgram` returns either a record with the corresponding GAP straight line program or `fail`.

  If 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 `outputs`. Its value is a list of strings, the entry at position i denoting the name of the class in which the i output of the straight line program lies; see Class Names Used in the AtlasRep Package for the definition of the class names that occur.

  Such straight line programs must end with a sequence of output specifications of the following form.

      echo "Classes 1A 2A 3A 5A 5B"
      oup 5 3 1 2 4 5

  This example means that the list of outputs of the program contains elements of the classes `1A`, `2A`, `3A`, `5A`, and `5B` (in this order), and that inside the program, these elements are referred to by the names `3`, `1`, `2`, `4`, and `5`.

- `AtlasStringOfStraightLineProgram(` prog`[,` outputnames`] ) F`

  For a straight line program prog (see IsStraightLineProgram in the GAP Reference Manual), `AtlasStringOfStraightLineProgram` returns a string describing the format of an equivalent straight line program as used in the ATLAS of Group Representations, that is, the lines are of the form described in ScanStraightLineProgram.

  A list of strings that is given as the optional second argument outputnames is interpreted as the class names corresponding to the outputs; this argument has the effect that appropriate `echo` statements appear in the result string.

      gap> str:= "inp 2\nmu 1 2 3\nmu 3 1 2\niv 2 1\noup 2 1 2";;
      gap> prg:= ScanStraightLineProgram( str, "string" );
      rec( program := <straight line program> )
      gap> prg:= prg.program;;
      gap> Display( prg );
      # input:
      r:= [ g1, g2 ];
      # program:
      r[3]:= r[1]*r[2];
      r[2]:= r[3]*r[1];
      r[1]:= r[2]^-1;
      # return values:
      [ r[1], r[2] ]
      gap> StringOfResultOfStraightLineProgram( prg, [ "a", "b" ] );
      "[ (aba)^-1, aba ]"
      gap> AtlasStringOfStraightLineProgram( prg );
      "inp 2\nmu 1 2 3\nmu 3 1 2\niv 2 1\noup 2\n"
      gap> prg:= StraightLineProgram( "(a^2b^3)^-1", [ "a", "b" ] );
      <straight line program>
      gap> Print( AtlasStringOfStraightLineProgram( prg ) );
      inp 2
      pwr 2 1 4
      pwr 3 2 5
      mu 4 5 3
      iv 3 4
      oup 1 4
      gap> prg:= StraightLineProgram( [ [2,3], [ [3,1,1,4], [1,2,3,1] ] ], 2 );
      <straight line program>
      gap> Print( AtlasStringOfStraightLineProgram( prg ) );
      inp 2
      pwr 3 2 3
      pwr 4 1 5
      mu 3 5 4
      pwr 2 1 6
      mu 6 3 5
      oup 2 4 5

  ## 4.4 Data Types Used in the Atlas of Group Representations

- `AGRDeclareDataType(` kind`,` name`,` record`) F`

  Let kind be one of the strings `"rep"` or `"slp"`, and record be a record. `AGRDeclareDataType` declares a new data type of representations (if kind is `"rep"`) or of straight line programs (if kind is `"slp"`). For each group used in the AtlasRep package, the record that contains the information about the data will have a component name whose value is a list with the data about the new type. Examples of name are `"perm"`, `"matff"`, and `"classes"`.

  Compulsory components of record are

  `FilenameFormat`
    This 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 `AGRParseFilenameFormat` (see AGRParseFilenameFormat), such that only filenames of the type in question match.

  `AddFileInfo`
    This 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 `AGRParseFilenameFormat` for the given type, see AGRParseFilenameFormat, and a filename). This function adds the necessary parts of the data entry to the list, and returns `true` if the data belongs to the type, otherwise `false` 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).

  `DisplayGroup` (for `rep` only)
    This defines the format of the lines printed by `DisplayAtlasInfo` (see DisplayAtlasInfo) for a given group. The value must be a function that takes a list as returned by the function given in the component `AddFileInfo`, and returns the string to be printed for the representation in question.

  Optional components of record are

  `DisplayOverviewInfo`
    This is used to introduce a new column in the output of `DisplayAtlasInfo` (see DisplayAtlasInfo) 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 `"r"` or `"l"`, 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 AtlasRep package and a group name), and returns a list of length two, containing the string to be printed as the column value and `true` or `false`, depending on whether private data is involved or not. (The default is to print no column for the data type.)

  `DisplaySLP` (for `slp` only)
    This is used in `DisplayAtlasInfo` for straight line programs. The value must be a function that takes four arguments (a list of tables of contents to examine, the name of the given group, a list of integers or `true` 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 straight line programs of the current type and for the given group. (The default is to return an empty list.)

  `AccessGroupCondition` (for `rep` only)
    This is used in `DisplayAtlasInfo` and `OneAtlasGeneratingSet`. The value must be a function that takes two arguments (a list as returned by `AGRParseFilenameFormat`, see AGRParseFilenameFormat, and a list of conditions --see below), and returns `true` or `false`, depending on whether the first argument satisfies the conditions. (The default value is `ReturnFalse`.)

  `AccessSLP` (for `slp` only)
    This is used in `AtlasStraightLineProgram` (see AtlasStraightLineProgram). The value must be a function that takes three arguments (the record with the information about the given group in the current table of contents, an integer or a list of integers or `true` for the required standardization, and a list of conditions --see below), and returns either `fail` or a list that together with the group name forms the identifier of a straight line program that matches the conditions. (The default value is `ReturnFail`.)

  `TOCEntryString`
    This is used in `StringOfAtlasTableOfContents`, so it is needed only for those data types for which data is stored in the official part. The value must be a function that takes two arguments (the name name of the type and a list as returned by `AGRParseFilenameFormat`, see AGRParseFilenameFormat) and returns a string that describes the appropriate function call. (The default value is `TOCEntryStringDefault`.)

  `PostprocessFileInfo`
    This is used in the construction of a table of contents via `ReloadAtlasTableOfContents`, 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 value is `Ignore`.)

  `SortTOCEntries`
    This is used in the construction of a table of contents via `AtlasTableOfContents`, for sorting the entries after they have been added and after `PostprocessFileInfo` has been called. The value must be a function that takes a list as returned by `AGRParseFilenameFormat` (see AGRParseFilenameFormat), and returns the sorting key. (There is no default value, which means that no sorting is needed.)

  `TestFileHeaders` (for `rep` only)
    This is used in the function `AtlasOfGroupRepresentationsTestFileHeaders` (see AtlasOfGroupRepresentationsTestFileHeaders). The value must be a function that takes the same three arguments as `FilenameAtlas` (see FilenameAtlas), except that the first argument `"datagens"` can be replaced by `"local"` and that the third argument is a list as returned by `AGRParseFilenameFormat` (see AGRParseFilenameFormat). (The default value is `ReturnTrue`.)

  `TestFiles` (for `rep` only)
    This is used in the function `AtlasOfGroupRepresentationsTestFiles`. (see AtlasOfGroupRepresentationsTestFiles). The format of the value and the default are the same as for `TestFileHeaders`.

  `TestWords` (for `slp` only)
    This is used in the function `AtlasOfGroupRepresentationsTestWords` (see AtlasOfGroupRepresentationsTestWords). The value must be a function that takes the same three arguments as `FilenameAtlas` (see FilenameAtlas), except that the first argument `"dataword"` can be replaced by `"local"`. (The default is to call `AGRTestWordsDefault` with fourth argument `false`.)

  The lists of conditions used as arguments of `AccessGroupCondition` and `AccessSLP` usually contain functions such as `IsPermGroup`, which are just checked for presence, and functions such as `NrMovedPoints` followed by a prescribed value or a list of prescribed values. Examples of such conditions are `[ IsPermGroup ]` and `[ NrMovedPoints, [ 5, 6 ] ]`.

  Note that it is not checked whether the ``detection function'' in the component `FilenameFormat` matches exactly one type, so one must be very careful when declaring a new type.

  ## 4.5 Filenames Used in the Atlas of Group Representations

  The data of each local GAP version of the ATLAS of Group Representations is either private (see Chapter Private Extensions of the AtlasRep Package) or is stored in the two directories `datagens` and `dataword`. In the following, we describe the format of filenames in the latter two directories, as a reference of the ``official'' part of the ATLAS.

  In the directory `datagens`, the generators for the representations available are stored, the directory `dataword` contains the straight line programs to compute conjugacy class representatives, generators of maximal subgroups, or images of generators under automorphisms of a given group G from standard generators of G (see Standard Generators Used in the AtlasRep Package).

  The name of each data file in the ATLAS of Group Representations describes the contents of the file. This section lists the definitions of the filenames used.

  Each filename consists of two parts, separated by a minus sign `-`. The first part is always of the form ``groupname`G`i``, where the integer i denotes the i-th set of standard generators for the group G, say, with ATLAS-file name groupname (see Group Names Used in the AtlasRep Package). The translations of the name groupname to the name(s) used within GAP is given by the list `AtlasOfGroupRepresentationsInfo.GAPnames` (see AtlasOfGroupRepresentationsInfo).

  The filenames in the directory `dataword` have one of the following forms. In each of these cases, the suffix `W`n`` means that n is the version number of the straight line program.

  ``groupname`G`i`-cycW`n``
    In this case, the file contains a straight line program that returns a list of representatives of generators of maximally cyclic subgroups of G. An example is `Co1G1-cycW1`.

  ``groupname`G`i`-cclsW`n``
    In this case, the file contains a straight line program that returns a list of conjugacy class representatives of G. An example is `RuG1-cclsW1`.

  ``groupname`G`i`cycW`n`-cclsW`m``
    In this case, the file contains a straight line program that takes the return value of the program in the file ``groupname`G`i`-cycW`n`` (see above), and returns a list of conjugacy class representatives of G. An example is `M11G1cycW1-cclsW1`.

  ``groupname`G`i`-max`k`W`n``
    In this case, the file contains a straight line program that takes generators of G w.r.t. the i-th set of standard generators, and returns a list of generators (in general not standard generators) for a subgroup U in the k-th class of maximal subgroups of G. An example is `J1G1-max7W1`.

  ``groupname`G`i`max`k`W`n`-`subgroupname`G`j`W`m``
    In this case, the file contains a straight line program that takes the return value of the program in the file ``groupname`G`i`-max`k`W`n`` (see above), which are generators for a group U, say; subgroupname is a name for U, and the return value is a list of standard generators for U, w.r.t. the j-th set of standard generators. (Of course this implies that the groups in the k-th class of maximal subgroups of G are isomorphic to the group with name subgroupname.) An example is `J1G1max1W1-L211G1W1`; the first class of maximal subgroups of the Janko group J1 consists of groups isomorphic to the linear group L2(11), for which standard generators are defined.

  ``groupname`G`i`-a`outname`W`n``
    In this case, the file contains a straight line program that takes generators of G w.r.t. the i-th set of standard generators, and returns the list of their images under the outer automorphism a of G given by the name outname; if this name is empty then a is the unique nontrivial outer automorphism of G; if it is a positive integer k then a is a generator of the unique cyclic order k subgroup of the outer automorphism group of G; if it is of the form 21 or 2a, 42 or 4b, 33 or 3c ¼ then a generates the cyclic group of automorphisms induced on G by G.21, G.42, G.33 ¼; finally, if it is of the form ``k`p`d``, with k one of the above forms and d an integer then d denotes the number of dashes appended to the automorphism described by k; if d = 1 then d can be omitted. Examples are `A5G1-aW1`, `L34G1-a2_1W1`, `U43G1-a2_3pW1`, and `O8p3G1-a2_2p5W1`; these file names describe the outer order 2 automorphism of A5 (induced by the action of S5) and the order 2 automorphisms of L3(4), U4(3), and O8+(3) induced by the actions of L3(4).21, U4(3).22¢, and O8+(3).22¢¢¢¢¢, respectively.

  ``groupname`G`i`-G`j`W`n``
    In this case, the file contains a straight line program that takes generators of G w.r.t. the i-th set of standard generators, and returns standard generators of G w.r.t. the j-th set of standard generators. An example is `L35G1-G2W1`.

  ``groupname`G`i`-X`descr`W`n``
    In this case, the file contains a straight line program that takes generators of G w.r.t. the i-th set of standard generators, and whose return value corresponds to descr. This format is used only in private extensions (see Chapter Private Extensions of the AtlasRep Package), such a script can be accessed with descr as the third argument of `AtlasStraightLineProgram` (see AtlasStraightLineProgram).

  The filenames in the directory `datagens` have one of the following forms. In each of these cases, id is a (possibly empty) string that starts with a lowercase alphabet letter (see IsLowerAlphaChar in the GAP Reference Manual), and m is a nonnegative integer, meaning that the generators are written w.r.t. the m-th basis (the meaning is defined by the ATLAS developers).

  ``groupname`G`i`-f`q`r`dim``id`B`m`.m`nr``
    a file in MeatAxe textfile format containing the nr-th generator of a matrix representation over the field with q elements, of dimension dim. An example is `S5G1-f2r4aB0.m1`.

  ``groupname`G`i`-p`n``id`B`m`.m`nr``
    a file in MeatAxe textfile format containing the nr-th generator of a permutation representation on n points. An example is `M11G1-p11B0.m1`.

  ``groupname`G`i`-Ar`dim``id`B`m`.g`
    a GAP readable file containing all generators of a matrix representation of dimension dim over an algebraic number field not specified further. An example is `A5G1-Ar3aB0.g`.

  ``groupname`G`i`-Zr`dim``id`B`m`.g`
    a GAP readable file containing all generators of a matrix representation over the integers, of dimension dim. An example is `A5G1-Zr4B0.g`.

  ``groupname`G`i`-Hr`dim``id`B`m`.g`
    a GAP readable file containing all generators of a matrix representation over a quaternion algebra over an algebraic number field, of dimension dim. An example is `2A6G1-Hr2aB0.g`.

  ``groupname`G`i`-Z`n`r`dim``id`B`m`.g`
    a GAP readable file containing all generators of a matrix representation of dimension dim over the ring of integers mod n. An example is `2A8G1-Z4r4aB0.g`.

- `AGRParseFilenameFormat(` string`,` format`) F`

  Let string be a filename, and format be a list [ [ c1, c2, ¼, cn ], [ f1, f2, ¼, fn ] ] such that each entry ci is a list of strings and of functions that take a character as their argument and return `true` or `false`, and such that each entry fi is a function for parsing a filename, such as `ParseForwards` and `ParseBackwards`.

  `AGRParseFilenameFormat` returns a list of strings and integers such that the concatenation of their `String` values yields string if string matches format, and `fail` otherwise. Matching is defined as follows. Splitting string at each `-` character yields m parts s1, s2, ¼, sm. The string string matches format if si matches the conditions in ci, for 1 £ i £ n, in the sense that applying fi to si and ci yields a non-fail result.

- `FilenameAtlas(` dirname`,` groupname`,` filename`) F`

  Let dirname, groupname, and filename be strings, where dirname is one of `"datagens"`, `"dataword"`. If groupname is the ATLAS-file name of a group G (see Group Names Used in the AtlasRep Package), and filename is the name of an accessible file in the dirname directory of the ATLAS, with data concerning G, then `FilenameAtlas` returns the corresponding filename (see Filename in the GAP Reference Manual), otherwise `fail` is returned.

  A file is regarded as accessible either if it is already stored in the dirname directory of the local installation or if the `remote` component of the record `AtlasOfGroupRepresentationsInfo` has value `true` (see AtlasOfGroupRepresentationsInfo) and the required file is found on one of the relevant servers; in the latter case, the file is transferred to the local directory before `FilenameAtlas` returns the filename, such that afterwards the file can be read into GAP using `Read` (see Read in the GAP Reference Manual).

  Note that if a file with name filename is already stored in the dirname directory then `FilenameAtlas` does not check whether the table of contents of the ATLAS of Group Representations actually contains filename.

  ## 4.6 The Tables of Contents of the Atlas of Group Representations

  The list of data currently available is stored in two tables of contents, one for the local GAP installation, one for the remote servers. They are created and accessed automatically when `ReloadAtlasTableOfContents` (see ReloadAtlasTableOfContents) is called.

  The low level functions used by `ReloadAtlasTableOfContents` are `AtlasTableOfContents`, which actually fetches and composes the necessary information, and `StringOfAtlasTableOfContents`, which translates the filenames into appropriate function calls for notifying them, and creates the string that is to be printed to a file. These functions and the utilities for them can be found in the files `access.gd` and `access.gi` in the `gap` directory of the package.

  Note that it is assumed that the local data directories contain only files that are also available on servers. Private extensions to the database (cf. Extending the Atlas Database and Chapter Private Extensions of the AtlasRep Package) cannot be handled by putting the data files into the local directories.

  The following two administrational functions may be useful for private extensions of the package (see Chapter Private Extensions of the AtlasRep Package).

- `AGRGNAN(` gapname`,` atlasname`) F`

  Let gapname be a string denoting a GAP group name, and atlasname be a string denoting the corresponding ATLAS-file name used in filenames of the ATLAS of Group Representations. `AGRGNAN` adds the pair `[` gapname`,` atlasname`]` to the list `AtlasOfGroupRepresentationsInfo.GAPnames` (see AtlasOfGroupRepresentationsInfo), making the ATLAS data involving atlasname accessible for the group with name gapname.

  An example of a valid call is `AGRGNAN("A6.2_2","PGL29")`.

- `AGRFLD(` filename`,` descr`) F`

  Let filename be a string denoting the name of a file containing the generators of a representation written over a proper extension of the rational number field, and descr be a string describing the field generated by the matrix entries. `AGRFLD` adds the triple `[` filename`,` descr`,` F`]` to the list `AtlasOfGroupRepresentationsInfo.fieldinfo` (see AtlasOfGroupRepresentationsInfo), where F is the field given by descr.

  An example of a valid call is `AGRFLD("A5G1-Ar3aB0","Field([Sqrt(5)])")`.

  ## 4.7 Sanity Checks for the Atlas of Group Representations

  The fact that the ATLAS of Group Representations is designed as an open database (see Local or Remote Installation of the AtlasRep Package) makes it especially desirable to have consistency checks available which can be run automatically whenever new data are added by the developers of the ATLAS. These checks can also be used for the private extensions of the package (see Chapter Private Extensions of the AtlasRep Package).

  The file `testall.g` in the `tst` directory of the package contains `ReadTest` (see ReadTest in the GAP Reference Manual) statements for executing a collection of such sanity checks; one can run them by starting GAP in the `tst` directory, and then calling `Read( "testall.g" )`. If no problem occurs then GAP prints only lines starting with one of the following.

      + $Id:
      + GAP4stones:

  The functions available to perform these tests and detailed information about them can be found in the files `test.gd` and `test.gi` in the `gap` directory of the package.

  All these tests apply only to the local table of contents (see The Tables of Contents of the Atlas of Group Representations), that is, only those data files are checked that are actually available in the local GAP installation. No files are fetched from servers during these tests.

  The examples in this manual form a part of these tests, they are collected in the file `docxpl.tst` in the `tst` directory of the package.

- `AtlasOfGroupRepresentationsTestFileHeaders( [`tocid`[,` groupname`]] ) F`

  First suppose that `AtlasOfGroupRepresentationsTestFileHeaders` is called with two arguments tocid, the identifier of a directory (see AtlasOfGroupRepresentationsNotifyPrivateDirectory), and groupname, a component name in the table of contents of the directory. The function checks for those data files for groupname in the tocid directory that are in MeatAxe text format whether the filename and the header line are consistent; it checks the data file in GAP format whether the file name is consistent with the contents of the file.

  If only one argument tocid is given then all representations available for groupname are checked with the three argument version.

  If only one argument tocid is given then all available groups in the directory with identifier tocid are checked; the contents of the local `dataword` directory can be checked by entering `"local"`, which is also the default for tocid.

  In all cases, the return value is `false` if an error occurred, and `true` otherwise.

- `AtlasOfGroupRepresentationsTestFiles( [`tocid`[,` groupname`]] ) F`

  `AtlasOfGroupRepresentationsTestFiles` is an analogue of `AtlasOfGroupRepresentationsTestFileHeaders` (see AtlasOfGroupRepresentationsTestFileHeaders). The function does not check if the first line of a MeatAxe file is consistent with the filename but checks whether reading the files with `ScanMeatAxeFile` returns non-`fail` results.

  In all cases, the return value is `false` if an error occurred, and `true` otherwise.

- `AtlasOfGroupRepresentationsTestWords(` tocid`) F`

  Called with one argument tocid, a string, `AtlasOfGroupRepresentationsTestWords` reads all straight line programs that are stored in the directory with identifier tocid (see AtlasOfGroupRepresentationsNotifyPrivateDirectory), using `ScanStraightLineProgram` (see ScanStraightLineProgram), and then runs the program on group generators. The contents of the local `dataword` directory can be checked by entering `"local"`. The return value is `false` if an error occurs, otherwise `true`.

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