  
  [1X6 [33X[0;0YNew [5XGAP[105X[101X[1X Objects and Utility Functions Provided by the [5XAtlasRep[105X[101X[1X Package[133X[101X
  
  [33X[0;0YThis  chapter  describes  [5XGAP[105X objects and functions that are provided by the
  [5XAtlasRep[105X package but that might be of general interest.[133X
  
  [33X[0;0YThe  new objects are straight line decisions (see Section [14X6.1[114X) and black box
  programs (see Section [14X6.2[114X).[133X
  
  [33X[0;0YThe  new functions are concerned with representations of minimal degree, see
  Section [14X6.3[114X.[133X
  
  
  [1X6.1 [33X[0;0YStraight Line Decisions[133X[101X
  
  [33X[0;0Y[13XStraight   line  decisions[113X  are  similar  to  straight  line  programs  (see
  Section [14X'Reference:  Straight  Line  Programs'[114X)  but return [9Xtrue[109X or [9Xfalse[109X. A
  straight  line  decisions  checks  a  property  for its inputs. An important
  example  is  to  check whether a given list of group generators is in fact a
  list of standard generators (cf. Section[14X3.3[114X) for this group.[133X
  
  [33X[0;0YA  straight  line  decision in [5XGAP[105X is represented by an object in the filter
  [2XIsStraightLineDecision[102X ([14X6.1-1[114X) that stores a list of [21Xlines[121X each of which has
  one of the following three forms.[133X
  
  [31X1[131X   [33X[0;6Ya nonempty dense list [22Xl[122X of integers,[133X
  
  [31X2[131X   [33X[0;6Ya  pair  [22X[  l,  i  ][122X  where [22Xl[122X is a list of form 1. and [22Xi[122X is a positive
        integer,[133X
  
  [31X3[131X   [33X[0;6Ya list [22X[[122X[10X"Order"[110X[22X, i, n ][122X where [22Xi[122X and [22Xn[122X are positive integers.[133X
  
  [33X[0;0YThe first two forms have the same meaning as for straight line programs (see
  Section [14X'Reference:  Straight  Line  Programs'[114X), the last form means a check
  whether the element stored at the label [22Xi[122X-th has the order [22Xn[122X.[133X
  
  [33X[0;0YFor  the  meaning  of  the  list  of lines, see [2XResultOfStraightLineDecision[102X
  ([14X6.1-6[114X).[133X
  
  [33X[0;0YStraight  line  decisions  can  be  constructed  using  [2XStraightLineDecision[102X
  ([14X6.1-5[114X),    defining    attributes   for   straight   line   decisions   are
  [2XNrInputsOfStraightLineDecision[102X   ([14X6.1-3[114X)   and   [2XLinesOfStraightLineDecision[102X
  ([14X6.1-2[114X),     an     operation     for    straight    line    decisions    is
  [2XResultOfStraightLineDecision[102X ([14X6.1-6[114X).[133X
  
  [33X[0;0YSpecial  methods applicable to straight line decisions are installed for the
  operations  [2XDisplay[102X ([14XReference: Display[114X), [2XIsInternallyConsistent[102X ([14XReference:
  IsInternallyConsistent[114X),   [2XPrintObj[102X   ([14XReference:   PrintObj[114X),  and  [2XViewObj[102X
  ([14XReference: ViewObj[114X).[133X
  
  [33X[0;0YFor  a straight line decision [3Xprog[103X, the default [2XDisplay[102X ([14XReference: Display[114X)
  method  prints  the  interpretation  of [3Xprog[103X as a sequence of assignments of
  associative  words  and  of order checks; a record with components [10Xgensnames[110X
  (with  value  a  list  of strings) and [10Xlistname[110X (a string) may be entered as
  second  argument  of  [2XDisplay[102X ([14XReference: Display[114X), in this case these names
  are  used,  the  default  for  [10Xgensnames[110X is [10X[ g1, g2, [110X[22X...[122X[10X ][110X, the default for
  [3Xlistname[103X is [22Xr[122X.[133X
  
  [1X6.1-1 IsStraightLineDecision[101X
  
  [29X[2XIsStraightLineDecision[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YEach    straight    line    decision    in    [5XGAP[105X   lies   in   the   filter
  [2XIsStraightLineDecision[102X.[133X
  
  [1X6.1-2 LinesOfStraightLineDecision[101X
  
  [29X[2XLinesOfStraightLineDecision[102X( [3Xprog[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ythe list of lines that define the straight line decision.[133X
  
  [33X[0;0YThis   defining   attribute   for  the  straight  line  decision  [3Xprog[103X  (see
  [2XIsStraightLineDecision[102X  ([14X6.1-1[114X))  corresponds  to [2XLinesOfStraightLineProgram[102X
  ([14XReference: LinesOfStraightLineProgram[114X) for straight line programs.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],[127X[104X
    [4X[25X>[125X [27X[ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );[127X[104X
    [4X[28X<straight line decision>[128X[104X
    [4X[25Xgap>[125X [27XLinesOfStraightLineDecision( dec );[127X[104X
    [4X[28X[ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ], [128X[104X
    [4X[28X  [ "Order", 3, 5 ] ][128X[104X
  [4X[32X[104X
  
  [1X6.1-3 NrInputsOfStraightLineDecision[101X
  
  [29X[2XNrInputsOfStraightLineDecision[102X( [3Xprog[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ythe number of inputs required for the straight line decision.[133X
  
  [33X[0;0YThis   defining   attribute   corresponds  to  [2XNrInputsOfStraightLineProgram[102X
  ([14XReference: NrInputsOfStraightLineProgram[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNrInputsOfStraightLineDecision( dec );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X6.1-4 ScanStraightLineDecision[101X
  
  [29X[2XScanStraightLineDecision[102X( [3Xstring[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya record containing the straight line decision, or [9Xfail[109X.[133X
  
  [33X[0;0YLet  [3Xstring[103X  be  a string that encodes a straight line decision in the sense
  that  it  consists  of the lines listed for [2XScanStraightLineProgram[102X ([14X7.4-1[114X),
  except  that  [10Xoup[110X  lines are not allowed, and instead lines of the following
  form may occur.[133X
  
  [8X[10Xchor [3Xa[103X[10X [3Xb[103X[10X[110X[108X
        [33X[0;6Ymeans  that  it is checked whether the order of the element at label [3Xa[103X
        is [3Xb[103X.[133X
  
  [33X[0;0Y[2XScanStraightLineDecision[102X  returns  a  record  containing as the value of its
  component   [10Xprogram[110X   the   corresponding   [5XGAP[105X   straight   line   decision
  (see [2XIsStraightLineDecision[102X  ([14X6.1-1[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
  
  [4X[32X  Example  [32X[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[28Xrec( program := <straight line decision> )[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[28Xif Order( r[1] ) <> 2 then  return false;  fi;[128X[104X
    [4X[28Xif Order( r[2] ) <> 3 then  return false;  fi;[128X[104X
    [4X[28Xr[3]:= r[1]*r[2];[128X[104X
    [4X[28Xif Order( r[3] ) <> 5 then  return false;  fi;[128X[104X
    [4X[28X# return value:[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.1-5 StraightLineDecision[101X
  
  [29X[2XStraightLineDecision[102X( [3Xlines[103X[, [3Xnrgens[103X] ) [32X function
  [29X[2XStraightLineDecisionNC[102X( [3Xlines[103X[, [3Xnrgens[103X] ) [32X function
  [6XReturns:[106X  [33X[0;10Ythe straight line decision given by the list of lines.[133X
  
  [33X[0;0YLet  [3Xlines[103X  be  a list of lists that defines a unique straight line decision
  (see [2XIsStraightLineDecision[102X  ([14X6.1-1[114X));  in  this  case  [2XStraightLineDecision[102X
  returns this program, otherwise an error is signalled. The optional argument
  [3Xnrgens[103X specifies the number of input generators of the program; if a list of
  integers  (a  line  of form 1. in the definition above) occurs in [3Xlines[103X then
  this  number  is  not determined by [3Xlines[103X and therefore [13Xmust[113X be specified by
  the argument [3Xnrgens[103X; if not then [2XStraightLineDecision[102X returns [9Xfail[109X.[133X
  
  [33X[0;0Y[2XStraightLineDecisionNC[102X  does  the  same as [2XStraightLineDecision[102X, except that
  the internal consistency of the program is not checked.[133X
  
  [1X6.1-6 ResultOfStraightLineDecision[101X
  
  [29X[2XResultOfStraightLineDecision[102X( [3Xprog[103X, [3Xgens[103X[, [3Xorderfunc[103X] ) [32X operation
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if all checks succeed, otherwise [9Xfalse[109X.[133X
  
  [33X[0;0Y[2XResultOfStraightLineDecision[102X    evaluates   the   straight   line   decision
  (see [2XIsStraightLineDecision[102X  ([14X6.1-1[114X)) [3Xprog[103X at the group elements in the list
  [3Xgens[103X.[133X
  
  [33X[0;0YThe  function for computing the order of a group element can be given as the
  optional  argument [3Xorderfunc[103X. For example, this may be a function that gives
  up  at  a  certain  limit if one has to be aware of extremely huge orders in
  failure cases.[133X
  
  [33X[0;0YThe  [13Xresult[113X  of  a straight line decision with lines [22Xp_1, p_2, ..., p_k[122X when
  applied to [3Xgens[103X is defined as follows.[133X
  
  [8X(a)[108X
        [33X[0;6YFirst  a  list  [22Xr[122X of intermediate values is initialized with a shallow
        copy of [3Xgens[103X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xi  ≤ k[122X, before the [22Xi[122X-th step, let [22Xr[122X be of length [22Xn[122X. If [22Xp_i[122X is the
        external  representation  of  an  associative  word  in  the  first  [22Xn[122X
        generators  then the image of this word under the homomorphism that is
        given  by  mapping [22Xr[122X to these first [22Xn[122X generators is added to [22Xr[122X. If [22Xp_i[122X
        is  a  pair [22X[ l, j ][122X, for a list [22Xl[122X, then the same element is computed,
        but  instead  of being added to [22Xr[122X, it replaces the [22Xj[122X-th entry of [22Xr[122X. If
        [22Xp_i[122X  is a triple [22X[[122X[10X"Order"[110X[22X, i, n ][122X then it is checked whether the order
        of [22Xr[i][122X is [22Xn[122X; if not then [9Xfalse[109X is returned immediately.[133X
  
  [8X(c)[108X
        [33X[0;6YIf  all [22Xk[122X lines have been processed and no order check has failed then
        [9Xtrue[109X is returned.[133X
  
  [33X[0;0YHere are some examples.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdec:= StraightLineDecision( [ ], 1 );[127X[104X
    [4X[28X<straight line decision>[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( dec, [ () ] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  straight  line  decision  [10Xdec[110X returns [9Xtrue[109X –for [13Xany[113X input of the
  right length.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],[127X[104X
    [4X[25X>[125X [27X      [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );[127X[104X
    [4X[28X<straight line decision>[128X[104X
    [4X[25Xgap>[125X [27XLinesOfStraightLineDecision( dec );[127X[104X
    [4X[28X[ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ], [128X[104X
    [4X[28X  [ "Order", 3, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( dec, [ (), () ] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,4,5) ] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  straight  line  decision admits two inputs; it tests whether the
  orders of the inputs are [22X2[122X and [22X3[122X, and the order of their product is [22X5[122X.[133X
  
  
  [1X6.1-7 [33X[0;0YSemi-Presentations and Presentations[133X[101X
  
  [33X[0;0YWe  can  associate  a  [13Xfinitely  presented group[113X [22XF / R[122X to each straight line
  decision  [3Xdec[103X,  say, as follows. The free generators of the free group [22XF[122X are
  in  bijection  with  the inputs, and the defining relators generating [22XR[122X as a
  normal  subgroup  of [22XF[122X are given by those words [22Xw^k[122X for which [3Xdec[103X contains a
  check whether the order of [22Xw[122X equals [22Xk[122X.[133X
  
  [33X[0;0YSo  if  [3Xdec[103X  returns  [9Xtrue[109X  for  the  input list [22X[ g_1, g_2, ..., g_n ][122X then
  mapping the free generators of [22XF[122X to the inputs defines an epimorphism [22XΦ[122X from
  [22XF[122X  to  the  group  [22XG[122X, say, that is generated by these inputs, such that [22XR[122X is
  contained in the kernel of [22XΦ[122X.[133X
  
  [33X[0;0Y(Note  that  [21Xsatisfying  [3Xdec[103X[121X  is  a  stronger  property  than  [21Xsatisfying  a
  presentation[121X.  For  example, [22X⟨ x ∣ x^2 = x^3 = 1 ⟩[122X is a presentation for the
  trivial  group, but the straight line decision that checks whether the order
  of [22Xx[122X is both [22X2[122X and [22X3[122X clearly always returns [9Xfalse[109X.)[133X
  
  [33X[0;0YThe  [5XATLAS[105X  of  Group  Representations  contains  the following two kinds of
  straight line decisions.[133X
  
  [30X    [33X[0;6YA  [13Xpresentation[113X  is a straight line decision [3Xdec[103X that is defined for a
        set  of  standard generators of a group [22XG[122X and that returns [9Xtrue[109X if and
        only  if  the  list  of  inputs is in fact a sequence of such standard
        generators  for [22XG[122X. In other words, the relators derived from the order
        checks  in  the  way  described above are defining relators for [22XG[122X, and
        moreover these relators are words in terms of standard generators. (In
        particular  the  kernel  of  the  map  [22XΦ[122X equals [22XR[122X whenever [3Xdec[103X returns
        [9Xtrue[109X.)[133X
  
  [30X    [33X[0;6YA  [13Xsemi-presentation[113X  is  a straight line decision [3Xdec[103X that is defined
        for  a  set  of standard generators of a group [22XG[122X and that returns [9Xtrue[109X
        for a list of inputs [13Xthat is known to generate a group isomorphic with
        [22XG[122X[113X  if  and  only  if  these inputs form in fact a sequence of standard
        generators  for [22XG[122X. In other words, the relators derived from the order
        checks  in  the  way  described  above  are  [13Xnot  necessarily defining
        relators[113X for [22XG[122X, but if we assume that the [22Xg_i[122X generate [22XG[122X then they are
        standard  generators. (In particular, [22XF / R[122X may be a larger group than
        [22XG[122X  but  in  this  case  [22XΦ[122X  maps  the  free generators of [22XF[122X to standard
        generators of [22XG[122X.)[133X
  
        [33X[0;6YMore about semi-presentations can be found in [NW05].[133X
  
  [33X[0;0YAvailable    presentations    and    semi-presentations    are   listed   by
  [2XDisplayAtlasInfo[102X  ([14X3.5-1[114X),  they  can  be accessed via [2XAtlasProgram[102X ([14X3.5-3[114X).
  (Clearly   each   presentation   is   also   a   semi-presentation.   So   a
  semi-presentation  for  some  standard  generators of a group is regarded as
  available  whenever  a  presentation  for these standard generators and this
  group is available.)[133X
  
  [33X[0;0YNote   that  different  groups  can  have  the  same  semi-presentation.  We
  illustrate  this  with  an  example  that is mentioned in [NW05]. The groups
  [22XL_2(7)  ≅  L_3(2)[122X and [22XL_2(8)[122X are generated by elements of the orders [22X2[122X and [22X3[122X
  such that their product has order [22X7[122X, and no further conditions are necessary
  to define standard generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcheck:= AtlasProgram( "L2(8)", "check" );[127X[104X
    [4X[28Xrec( groupname := "L2(8)", identifier := [ "L2(8)", "L28G1-check1", 1, 1 ], [128X[104X
    [4X[28X  program := <straight line decision>, standardization := 1 )[128X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( "L2(8)", 1 );[127X[104X
    [4X[28Xrec( charactername := "1a+8a", [128X[104X
    [4X[28X  generators := [ (1,2)(3,4)(6,7)(8,9), (1,3,2)(4,5,6)(7,8,9) ], [128X[104X
    [4X[28X  groupname := "L2(8)", id := "", [128X[104X
    [4X[28X  identifier := [ "L2(8)", [ "L28G1-p9B0.m1", "L28G1-p9B0.m2" ], 1, 9 ], [128X[104X
    [4X[28X  isPrimitive := true, maxnr := 1, p := 9, rankAction := 2, [128X[104X
    [4X[28X  repname := "L28G1-p9B0", repnr := 1, size := 504, stabilizer := "2^3:7", [128X[104X
    [4X[28X  standardization := 1, transitivity := 3, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( check.program, gens.generators );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( "L3(2)", 1 );[127X[104X
    [4X[28Xrec( generators := [ (2,4)(3,5), (1,2,3)(5,6,7) ], groupname := "L3(2)", [128X[104X
    [4X[28X  id := "a", identifier := [ "L3(2)", [ "L27G1-p7aB0.m1", "L27G1-p7aB0.m2" ], [128X[104X
    [4X[28X      1, 7 ], isPrimitive := true, maxnr := 1, p := 7, rankAction := 2, [128X[104X
    [4X[28X  repname := "L27G1-p7aB0", repnr := 1, size := 168, stabilizer := "S4", [128X[104X
    [4X[28X  standardization := 1, transitivity := 2, type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( check.program, gens.generators );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.1-8 AsStraightLineDecision[101X
  
  [29X[2XAsStraightLineDecision[102X( [3Xbbox[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan  equivalent  straight  line  decision  for  the given black box
            program, or [9Xfail[109X.[133X
  
  [33X[0;0YFor    a    black    box   program   (see   [2XIsBBoxProgram[102X   ([14X6.2-1[114X))   [3Xbbox[103X,
  [2XAsStraightLineDecision[102X    returns    a    straight    line   decision   (see
  [2XIsStraightLineDecision[102X  ([14X6.1-1[114X))  with the same output as [3Xbbox[103X, in the sense
  of  [2XAsBBoxProgram[102X ([14X6.2-5[114X), if such a straight line decision exists, and [9Xfail[109X
  otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ],[127X[104X
    [4X[25X>[125X [27X             [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xdec:= StraightLineDecision( lines, 2 );[127X[104X
    [4X[28X<straight line decision>[128X[104X
    [4X[25Xgap>[125X [27Xbboxdec:= AsBBoxProgram( dec );[127X[104X
    [4X[28X<black box program>[128X[104X
    [4X[25Xgap>[125X [27Xasdec:= AsStraightLineDecision( bboxdec );[127X[104X
    [4X[28X<straight line decision>[128X[104X
    [4X[25Xgap>[125X [27XLinesOfStraightLineDecision( asdec );[127X[104X
    [4X[28X[ [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ [ 1, 1, 2, 1 ], 3 ], [128X[104X
    [4X[28X  [ "Order", 3, 5 ] ][128X[104X
  [4X[32X[104X
  
  [1X6.1-9 StraightLineProgramFromStraightLineDecision[101X
  
  [29X[2XStraightLineProgramFromStraightLineDecision[102X( [3Xdec[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ythe  straight  line  program associated to the given straight line
            decision.[133X
  
  [33X[0;0YFor  a  straight  line  decision  [3Xdec[103X  (see  [2XIsStraightLineDecision[102X ([14X6.1-1[114X),
  [2XStraightLineProgramFromStraightLineDecision[102X   returns   the   straight  line
  program   (see   [2XIsStraightLineProgram[102X   ([14XReference:  IsStraightLineProgram[114X)
  obtained  by  replacing  each  line of type 3. (i.e, each order check) by an
  assignment of the power in question to a new slot, and by declaring the list
  of these elements as the return value.[133X
  
  [33X[0;0YThis  means that the return value describes exactly the defining relators of
  the  presentation  that  is  associated  to  the straight line decision, see
  [14X6.1-7[114X.[133X
  
  [33X[0;0YFor  example,  one  can  use the return value for printing the relators with
  [2XStringOfResultOfStraightLineProgram[102X                              ([14XReference:
  StringOfResultOfStraightLineProgram[114X),  or  for  explicitly  constructing the
  relators   as   words   in   terms   of   free   generators,   by   applying
  [2XResultOfStraightLineProgram[102X  ([14XReference: ResultOfStraightLineProgram[114X) to the
  program and to these generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],[127X[104X
    [4X[25X>[125X [27X[ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );[127X[104X
    [4X[28X<straight line decision>[128X[104X
    [4X[25Xgap>[125X [27Xprog:= StraightLineProgramFromStraightLineDecision( dec );[127X[104X
    [4X[28X<straight line program>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( prog );[127X[104X
    [4X[28X# input:[128X[104X
    [4X[28Xr:= [ g1, g2 ];[128X[104X
    [4X[28X# program:[128X[104X
    [4X[28Xr[3]:= r[1]*r[2];[128X[104X
    [4X[28Xr[4]:= r[1]^2;[128X[104X
    [4X[28Xr[5]:= r[2]^3;[128X[104X
    [4X[28Xr[6]:= r[3]^5;[128X[104X
    [4X[28X# return values:[128X[104X
    [4X[28X[ r[4], r[5], r[6] ][128X[104X
    [4X[25Xgap>[125X [27XStringOfResultOfStraightLineProgram( prog, [ "a", "b" ] );[127X[104X
    [4X[28X"[ a^2, b^3, (ab)^5 ]"[128X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfGroup( FreeGroup( "a", "b" ) );[127X[104X
    [4X[28X[ a, b ][128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineProgram( prog, gens );[127X[104X
    [4X[28X[ a^2, b^3, a*b*a*b*a*b*a*b*a*b ][128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YBlack Box Programs[133X[101X
  
  [33X[0;0Y[13XBlack  box  programs[113X  formalize the idea that one takes some group elements,
  forms  arithmetic  expressions  in  terms of them, tests properties of these
  expressions,  executes  conditional  statements  (including jumps inside the
  program)  depending  on  the  results of these tests, and eventually returns
  some result.[133X
  
  [33X[0;0YA specification of the language can be found in [Nic06], see also[133X
  
  [33X[0;0Y[7Xhttp://brauer.maths.qmul.ac.uk/Atlas/info/blackbox.html[107X.[133X
  
  [33X[0;0YThe  [13Xinputs[113X  of  a black box program may be explicit group elements, and the
  program  may  also  ask  for random elements from a given group. The [13Xprogram
  steps[113X  form  products,  inverses,  conjugates,  commutators,  etc.  of known
  elements,  [13Xtests[113X  concern essentially the orders of elements, and the [13Xresult[113X
  is a list of group elements or [9Xtrue[109X or [9Xfalse[109X or [9Xfail[109X.[133X
  
  [33X[0;0YExamples that can be modeled by black box programs are[133X
  
  [8X[13Xstraight line programs[113X,[108X
        [33X[0;6Ywhich  require  a  fixed  number of input elements and form arithmetic
        expressions  of  elements  but  do  not  use  random  elements, tests,
        conditional statements and jumps; the return value is always a list of
        elements; these programs are described in Section [14X'Reference: Straight
        Line Programs'[114X.[133X
  
  [8X[13Xstraight line decisions[113X,[108X
        [33X[0;6Ywhich  differ  from straight line programs only in the sense that also
        order  tests  are admissible, and that the return value is [9Xtrue[109X if all
        these  tests  are  satisfied, and [9Xfalse[109X as soon as the first such test
        fails; they are described in Section [14X6.1[114X.[133X
  
  [8X[13Xscripts for finding standard generators[113X,[108X
        [33X[0;6Ywhich take a group and a function to generate a random element in this
        group  but  no  explicit input elements, admit all control structures,
        and  return  either  a  list  of  standard  generators  or  [9Xfail[109X;  see
        [2XResultOfBBoxProgram[102X ([14X6.2-4[114X) for examples.[133X
  
  [33X[0;0YIn  the  case of general black box programs, currently [5XGAP[105X provides only the
  possibility  to read an existing program via [2XScanBBoxProgram[102X ([14X6.2-2[114X), and to
  run  the  program  using  [2XRunBBoxProgram[102X ([14X6.2-3[114X). It is not our aim to write
  such programs in [5XGAP[105X.[133X
  
  [33X[0;0YThe  special  case of the [21Xfind[121X scripts mentioned above is also admissible as
  an  argument of [2XResultOfBBoxProgram[102X ([14X6.2-4[114X), which returns either the set of
  generators or [9Xfail[109X.[133X
  
  [33X[0;0YContrary  to  the  general  situation, more support is provided for straight
  line  programs  and  straight line decisions in [5XGAP[105X, see Section [14X'Reference:
  Straight  Line  Programs'[114X  for  functions  that  manipulate  them  (compose,
  restrict etc.).[133X
  
  [33X[0;0YThe   functions  [2XAsStraightLineProgram[102X  ([14X6.2-6[114X)  and  [2XAsStraightLineDecision[102X
  ([14X6.1-8[114X)  can  be used to transform a general black box program object into a
  straight line program or a straight line decision if this is possible.[133X
  
  [33X[0;0YConversely,  one  can  create an equivalent general black box program from a
  straight  line  program  or from a straight line decision with [2XAsBBoxProgram[102X
  ([14X6.2-5[114X).[133X
  
  [33X[0;0Y(Computing a straight line program related to a given straight line decision
  is  supported  in  the  sense of [2XStraightLineProgramFromStraightLineDecision[102X
  ([14X6.1-9[114X).)[133X
  
  [33X[0;0YNote that none of these three kinds of objects is a special case of another:
  Running  a  black  box  program with [2XRunBBoxProgram[102X ([14X6.2-3[114X) yields a record,
  running a straight line program with [2XResultOfStraightLineProgram[102X ([14XReference:
  ResultOfStraightLineProgram[114X)  yields  a  list  of  elements,  and  running a
  straight line decision with [2XResultOfStraightLineDecision[102X ([14X6.1-6[114X) yields [9Xtrue[109X
  or [9Xfalse[109X.[133X
  
  [1X6.2-1 IsBBoxProgram[101X
  
  [29X[2XIsBBoxProgram[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YEach black box program in [5XGAP[105X lies in the filter [2XIsBBoxProgram[102X.[133X
  
  [1X6.2-2 ScanBBoxProgram[101X
  
  [29X[2XScanBBoxProgram[102X( [3Xstring[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  record  containing  the  black box program encoded by the input
            string, or [9Xfail[109X.[133X
  
  [33X[0;0YFor  a  string  [3Xstring[103X  that describes a black box program, e.g., the return
  value  of  [2XStringFile[102X  ([14XGAPDoc:  StringFile[114X),  [2XScanBBoxProgram[102X computes this
  black  box  program. If this is successful then the return value is a record
  containing  as  the  value  of  its  component [10Xprogram[110X the corresponding [5XGAP[105X
  object that represents the program, otherwise [9Xfail[109X is returned.[133X
  
  [33X[0;0YAs  the  first  example, we construct a black box program that tries to find
  standard generators for the alternating group [22XA_5[122X; these standard generators
  are  any  pair  of  elements  of the orders [22X2[122X and [22X3[122X, respectively, such that
  their product has order [22X5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfindstr:= "\[127X[104X
    [4X[25X>[125X [27X  set V 0\n\[127X[104X
    [4X[25X>[125X [27Xlbl START1\n\[127X[104X
    [4X[25X>[125X [27X  rand 1\n\[127X[104X
    [4X[25X>[125X [27X  ord 1 A\n\[127X[104X
    [4X[25X>[125X [27X  incr V\n\[127X[104X
    [4X[25X>[125X [27X  if V gt 100 then timeout\n\[127X[104X
    [4X[25X>[125X [27X  if A notin 1 2 3 5 then fail\n\[127X[104X
    [4X[25X>[125X [27X  if A noteq 2 then jmp START1\n\[127X[104X
    [4X[25X>[125X [27Xlbl START2\n\[127X[104X
    [4X[25X>[125X [27X  rand 2\n\[127X[104X
    [4X[25X>[125X [27X  ord 2 B\n\[127X[104X
    [4X[25X>[125X [27X  incr V\n\[127X[104X
    [4X[25X>[125X [27X  if V gt 100 then timeout\n\[127X[104X
    [4X[25X>[125X [27X  if B notin 1 2 3 5 then fail\n\[127X[104X
    [4X[25X>[125X [27X  if B noteq 3 then jmp START2\n\[127X[104X
    [4X[25X>[125X [27X  # The elements 1 and 2 have the orders 2 and 3, respectively.\n\[127X[104X
    [4X[25X>[125X [27X  set X 0\n\[127X[104X
    [4X[25X>[125X [27Xlbl CONJ\n\[127X[104X
    [4X[25X>[125X [27X  incr X\n\[127X[104X
    [4X[25X>[125X [27X  if X gt 100 then timeout\n\[127X[104X
    [4X[25X>[125X [27X  rand 3\n\[127X[104X
    [4X[25X>[125X [27X  cjr 2 3\n\[127X[104X
    [4X[25X>[125X [27X  mu 1 2 4   # ab\n\[127X[104X
    [4X[25X>[125X [27X  ord 4 C\n\[127X[104X
    [4X[25X>[125X [27X  if C notin 2 3 5 then fail\n\[127X[104X
    [4X[25X>[125X [27X  if C noteq 5 then jmp CONJ\n\[127X[104X
    [4X[25X>[125X [27X  oup 2 1 2";;[127X[104X
    [4X[25Xgap>[125X [27Xfind:= ScanBBoxProgram( findstr );[127X[104X
    [4X[28Xrec( program := <black box program> )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe second example is a black box program that checks whether its two inputs
  are standard generators for [22XA_5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcheckstr:= "\[127X[104X
    [4X[25X>[125X [27Xchor 1 2\n\[127X[104X
    [4X[25X>[125X [27Xchor 2 3\n\[127X[104X
    [4X[25X>[125X [27Xmu 1 2 3\n\[127X[104X
    [4X[25X>[125X [27Xchor 3 5";;[127X[104X
    [4X[25Xgap>[125X [27Xcheck:= ScanBBoxProgram( checkstr );[127X[104X
    [4X[28Xrec( program := <black box program> )[128X[104X
  [4X[32X[104X
  
  [1X6.2-3 RunBBoxProgram[101X
  
  [29X[2XRunBBoxProgram[102X( [3Xprog[103X, [3XG[103X, [3Xinput[103X, [3Xoptions[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  record  describing the result and the statistics of running the
            black box program [3Xprog[103X, or [9Xfail[109X, or the string [10X"timeout"[110X.[133X
  
  [33X[0;0YFor a black box program [3Xprog[103X, a group [3XG[103X, a list [3Xinput[103X of group elements, and
  a record [3Xoptions[103X, [2XRunBBoxProgram[102X applies [3Xprog[103X to [3Xinput[103X, where [3XG[103X is used only
  to compute random elements.[133X
  
  [33X[0;0YThe  return value is [9Xfail[109X if a syntax error or an explicit [10Xfail[110X statement is
  reached  at  runtime,  and  the  string  [10X"timeout"[110X if a [10Xtimeout[110X statement is
  reached.  (The  latter  might  mean  that  the random choices were unlucky.)
  Otherwise a record with the following components is returned.[133X
  
  [8X[10Xgens[110X[108X
        [33X[0;6Ya list of group elements, bound if an [10Xoup[110X statement was reached,[133X
  
  [8X[10Xresult[110X[108X
        [33X[0;6Y[9Xtrue[109X  if  a  [10Xtrue[110X  statement  was  reached,  [9Xfalse[109X  if  either a [10Xfalse[110X
        statement or a failed order check was reached,[133X
  
  [33X[0;0YThe  other  components serve as statistical information about the numbers of
  the  various  operations ([10Xmultiply[110X, [10Xinvert[110X, [10Xpower[110X, [10Xorder[110X, [10Xrandom[110X, [10Xconjugate[110X,
  [10Xconjugateinplace[110X, [10Xcommutator[110X), and the runtime in milliseconds ([10Xtimetaken[110X).[133X
  
  [33X[0;0YThe following components of [3Xoptions[103X are supported.[133X
  
  [8X[10Xrandomfunction[110X[108X
        [33X[0;6Ythe  function  called  with  argument  [3XG[103X  in order to compute a random
        element of [3XG[103X (default [2XPseudoRandom[102X ([14XReference: PseudoRandom[114X))[133X
  
  [8X[10Xorderfunction[110X[108X
        [33X[0;6Ythe  function  for  computing  element  orders  (the  default is [2XOrder[102X
        ([14XReference: Order[114X)),[133X
  
  [8X[10Xquiet[110X[108X
        [33X[0;6Yif [9Xtrue[109X then ignore [10Xecho[110X statements (default [9Xfalse[109X),[133X
  
  [8X[10Xverbose[110X[108X
        [33X[0;6Yif  [9Xtrue[109X  then  print  information  about  the  line that is currently
        processed, and about order checks (default [9Xfalse[109X),[133X
  
  [8X[10Xallowbreaks[110X[108X
        [33X[0;6Yif  [9Xtrue[109X  then call [2XError[102X ([14XReference: Error[114X) when a [10Xbreak[110X statement is
        reached, otherwise ignore [10Xbreak[110X statements (default [9Xtrue[109X).[133X
  
  [33X[0;0YAs  an example, we run the black box programs constructed in the example for
  [2XScanBBoxProgram[102X ([14X6.2-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AlternatingGroup( 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xres:= RunBBoxProgram( find.program, g, [], rec() );;[127X[104X
    [4X[25Xgap>[125X [27XIsBound( res.gens );  IsBound( res.result );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XList( res.gens, Order );[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XOrder( Product( res.gens ) );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27Xres:= RunBBoxProgram( check.program, "dummy", res.gens, rec() );;[127X[104X
    [4X[25Xgap>[125X [27XIsBound( res.gens );  IsBound( res.result );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xres.result;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xothergens:= GeneratorsOfGroup( g );;[127X[104X
    [4X[25Xgap>[125X [27Xres:= RunBBoxProgram( check.program, "dummy", othergens, rec() );;[127X[104X
    [4X[25Xgap>[125X [27Xres.result;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-4 ResultOfBBoxProgram[101X
  
  [29X[2XResultOfBBoxProgram[102X( [3Xprog[103X, [3XG[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya  list  of  group  elements  or  [9Xtrue[109X, [9Xfalse[109X, [9Xfail[109X, or the string
            [10X"timeout"[110X.[133X
  
  [33X[0;0YThis  function  calls [2XRunBBoxProgram[102X ([14X6.2-3[114X) with the black box program [3Xprog[103X
  and  second argument either a group or a list of group elements; the default
  options  are  assumed.  The  return  value is [9Xfail[109X if this call yields [9Xfail[109X,
  otherwise  the  [10Xgens[110X  component  of  the  result,  if  bound,  or the [10Xresult[110X
  component if not.[133X
  
  [33X[0;0YAs  an example, we run the black box programs constructed in the example for
  [2XScanBBoxProgram[102X ([14X6.2-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AlternatingGroup( 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xres:= ResultOfBBoxProgram( find.program, g );;[127X[104X
    [4X[25Xgap>[125X [27XList( res, Order );[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XOrder( Product( res ) );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27Xres:= ResultOfBBoxProgram( check.program, res );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xothergens:= GeneratorsOfGroup( g );;[127X[104X
    [4X[25Xgap>[125X [27Xres:= ResultOfBBoxProgram( check.program, othergens );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-5 AsBBoxProgram[101X
  
  [29X[2XAsBBoxProgram[102X( [3Xslp[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan  equivalent  black  box  program  for  the  given straight line
            program or straight line decision.[133X
  
  [33X[0;0YLet  [3Xslp[103X  be  a straight line program (see [2XIsStraightLineProgram[102X ([14XReference:
  IsStraightLineProgram[114X))     or     a    straight    line    decision    (see
  [2XIsStraightLineDecision[102X  ([14X6.1-1[114X)).  Then  [2XAsBBoxProgram[102X  returns  a black box
  program [3Xbbox[103X (see [2XIsBBoxProgram[102X ([14X6.2-1[114X)) with the [21Xsame[121X output as [3Xslp[103X, in the
  sense  that  [2XResultOfBBoxProgram[102X  ([14X6.2-4[114X) yields the same result for [3Xbbox[103X as
  [2XResultOfStraightLineProgram[102X   ([14XReference:   ResultOfStraightLineProgram[114X)  or
  [2XResultOfStraightLineDecision[102X ([14X6.1-6[114X), respectively, for [3Xslp[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= FreeGroup( "x", "y" );;  gens:= GeneratorsOfGroup( f );;[127X[104X
    [4X[25Xgap>[125X [27Xslp:= StraightLineProgram( [ [1,2,2,3], [3,-1] ], 2 );[127X[104X
    [4X[28X<straight line program>[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineProgram( slp, gens );[127X[104X
    [4X[28Xy^-3*x^-2[128X[104X
    [4X[25Xgap>[125X [27Xbboxslp:= AsBBoxProgram( slp );[127X[104X
    [4X[28X<black box program>[128X[104X
    [4X[25Xgap>[125X [27XResultOfBBoxProgram( bboxslp, gens );[127X[104X
    [4X[28X[ y^-3*x^-2 ][128X[104X
    [4X[25Xgap>[125X [27Xlines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ],[127X[104X
    [4X[25X>[125X [27X             [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xdec:= StraightLineDecision( lines, 2 );[127X[104X
    [4X[28X<straight line decision>[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,5) ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,4) ] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xbboxdec:= AsBBoxProgram( dec );[127X[104X
    [4X[28X<black box program>[128X[104X
    [4X[25Xgap>[125X [27XResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,5) ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,4) ] );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-6 AsStraightLineProgram[101X
  
  [29X[2XAsStraightLineProgram[102X( [3Xbbox[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan  equivalent  straight  line  program  for  the  given black box
            program, or [9Xfail[109X.[133X
  
  [33X[0;0YFor    a    black    box   program   (see   [2XAsBBoxProgram[102X   ([14X6.2-5[114X))   [3Xbbox[103X,
  [2XAsStraightLineProgram[102X    returns    a    straight    line    program    (see
  [2XIsStraightLineProgram[102X  ([14XReference:  IsStraightLineProgram[114X))  with  the  same
  output as [3Xbbox[103X if such a straight line program exists, and [9Xfail[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplay( AsStraightLineProgram( bboxslp ) );[127X[104X
    [4X[28X# input:[128X[104X
    [4X[28Xr:= [ g1, g2 ];[128X[104X
    [4X[28X# program:[128X[104X
    [4X[28Xr[3]:= r[1]^2;[128X[104X
    [4X[28Xr[4]:= r[2]^3;[128X[104X
    [4X[28Xr[5]:= r[3]*r[4];[128X[104X
    [4X[28Xr[3]:= r[5]^-1;[128X[104X
    [4X[28X# return values:[128X[104X
    [4X[28X[ r[3] ][128X[104X
    [4X[25Xgap>[125X [27XAsStraightLineProgram( bboxdec );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YRepresentations of Minimal Degree[133X[101X
  
  [33X[0;0YThis   section   deals  with  minimal  degrees  of  permutation  and  matrix
  representations.  We do not provide an algorithm that computes these degrees
  for  an  arbitrary  group,  we  only provide some tools for evaluating known
  databases,    mainly   concerning   [21Xbicyclic   extensions[121X   (see   [CCNPW85,
  Section 6.5])  of simple groups, in order to derive the minimal degrees, see
  Section [14X6.3-4[114X.[133X
  
  [33X[0;0YIn  the  [5XAtlasRep[105X  package,  this  information  can  be used for prescribing
  [21Xminimality conditions[121X in [2XDisplayAtlasInfo[102X ([14X3.5-1[114X), [2XOneAtlasGeneratingSetInfo[102X
  ([14X3.5-5[114X),  and  [2XAllAtlasGeneratingSetInfos[102X ([14X3.5-6[114X). An overview of the stored
  minimal degrees can be shown with [2XBrowseMinimalDegrees[102X ([14X3.6-1[114X).[133X
  
  [1X6.3-1 MinimalRepresentationInfo[101X
  
  [29X[2XMinimalRepresentationInfo[102X( [3Xgrpname[103X, [3Xconditions[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya record with the components [10Xvalue[110X and [10Xsource[110X, or [9Xfail[109X[133X
  
  [33X[0;0YLet  [3Xgrpname[103X be the [5XGAP[105X name of a group [22XG[122X, say. If the information described
  by [3Xconditions[103X about minimal representations of this group can be computed or
  is   stored   then  [2XMinimalRepresentationInfo[102X  returns  a  record  with  the
  components [10Xvalue[110X and [10Xsource[110X, otherwise [9Xfail[109X is returned.[133X
  
  [33X[0;0YThe following values for [3Xconditions[103X are supported.[133X
  
  [30X    [33X[0;6YIf  [3Xconditions[103X  is  [2XNrMovedPoints[102X  ([14XReference:  NrMovedPoints  (for  a
        permutation)[114X)  then  [10Xvalue[110X,  if  known,  is  the  degree  of a minimal
        faithful  (not  necessarily transitive) permutation representation for
        [22XG[122X.[133X
  
  [30X    [33X[0;6YIf  [3Xconditions[103X  consists of [2XCharacteristic[102X ([14XReference: Characteristic[114X)
        and  a  prime  integer  [3Xp[103X  then [10Xvalue[110X, if known, is the dimension of a
        minimal  faithful  (not necessarily irreducible) matrix representation
        in characteristic [3Xp[103X for [22XG[122X.[133X
  
  [30X    [33X[0;6YIf  [3Xconditions[103X  consists of [2XSize[102X ([14XReference: Size[114X) and a prime power [3Xq[103X
        then  [10Xvalue[110X,  if  known,  is  the dimension of a minimal faithful (not
        necessarily  irreducible) matrix representation over the field of size
        [3Xq[103X for [22XG[122X.[133X
  
  [33X[0;0YIn  all  cases,  the value of the component [10Xsource[110X is a list of strings that
  describe  sources  of  the information, which can be the ordinary or modular
  character  table  of [22XG[122X (see [CCNPW85], [JLPW95], [HL89]), the table of marks
  of  [22XG[122X,  or  [Jan05].  For  an overview of minimal degrees of faithful matrix
  representations  for  sporadic  simple groups and their covering groups, see
  also[133X
  
  [33X[0;0Y[7Xhttp://www.math.rwth-aachen.de/~MOC/mindeg/[107X.[133X
  
  [33X[0;0YNote  that  [2XMinimalRepresentationInfo[102X  cannot  provide any information about
  minimal representations over prescribed fields in characteristic zero.[133X
  
  [33X[0;0YInformation  about  groups that occur in the [5XAtlasRep[105X package is precomputed
  in [2XMinimalRepresentationInfoData[102X ([14X6.3-2[114X), so the packages [5XCTblLib[105X and [5XTomLib[105X
  are  not  needed  when [2XMinimalRepresentationInfo[102X is called for these groups.
  (The  only  case  that  is not covered by this list is that one asks for the
  minimal  degree  of  matrix  representations  over  a  prescribed  field  in
  characteristic coprime to the group order.)[133X
  
  [33X[0;0YOne of the following strings can be given as an additional last argument.[133X
  
  [8X[10X"cache"[110X[108X
        [33X[0;6Ymeans  that the function tries to compute (and then store) values that
        are  not  stored  in [2XMinimalRepresentationInfoData[102X ([14X6.3-2[114X), but stored
        values are preferred; this is also the default.[133X
  
  [8X[10X"lookup"[110X[108X
        [33X[0;6Ymeans  that  stored  values  are  returned  but  the function does not
        attempt    to    compute    values    that    are    not   stored   in
        [2XMinimalRepresentationInfoData[102X ([14X6.3-2[114X).[133X
  
  [8X[10X"recompute"[110X[108X
        [33X[0;6Ymeans that the function always tries to compute the desired value, and
        checks the result against stored values.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMinimalRepresentationInfo( "A5", NrMovedPoints );[127X[104X
    [4X[28Xrec( source := [ "computed (alternating group)", "computed (char. table)",[128X[104X
    [4X[28X      "computed (subgroup tables)",[128X[104X
    [4X[28X      "computed (subgroup tables, known repres.)",[128X[104X
    [4X[28X      "computed (table of marks)" ], value := 5 )[128X[104X
    [4X[25Xgap>[125X [27XMinimalRepresentationInfo( "A5", Characteristic, 2 );[127X[104X
    [4X[28Xrec( source := [ "computed (char. table)" ], value := 2 )[128X[104X
    [4X[25Xgap>[125X [27XMinimalRepresentationInfo( "A5", Size, 2 );[127X[104X
    [4X[28Xrec( source := [ "computed (char. table)" ], value := 4 )[128X[104X
  [4X[32X[104X
  
  [1X6.3-2 MinimalRepresentationInfoData[101X
  
  [29X[2XMinimalRepresentationInfoData[102X[32X global variable
  
  [33X[0;0YThis  is  a  record  whose  components  are  [5XGAP[105X  names  of groups for which
  information  about minimal permutation and matrix representations were known
  in  advance  or have been computed in the current [5XGAP[105X session. The value for
  the group [22XG[122X, say, is a record with the following components.[133X
  
  [8X[10XNrMovedPoints[110X[108X
        [33X[0;6Ya  record with the components [10Xvalue[110X (the degree of a smallest faithful
        permutation  representation  of [22XG[122X) and [10Xsource[110X (a string describing the
        source of this information).[133X
  
  [8X[10XCharacteristic[110X[108X
        [33X[0;6Ya  record  whose components are at most [10X0[110X and strings corresponding to
        prime  integers, each bound to a record with the components [10Xvalue[110X (the
        degree  of  a  smallest  faithful  matrix  representation of [22XG[122X in this
        characteristic)  and  [10Xsource[110X  (a  string describing the source of this
        information).[133X
  
  [8X[10XCharacteristicAndSize[110X[108X
        [33X[0;6Ya  record whose components are strings corresponding to prime integers
        [3Xp[103X,  each bound to a record with the components [10Xsizes[110X (a list of powers
        [3Xq[103X  of  [3Xp[103X), [10Xdimensions[110X (the corresponding list of minimal dimensions of
        faithful  matrix representations of [22XG[122X over a field of size [3Xq[103X), [10Xsources[110X
        (the  corresponding  list  of  strings  describing  the source of this
        information),  and [10Xcomplete[110X (a record with the components [10Xval[110X ([9Xtrue[109X if
        the minimal dimension over [13Xany[113X finite field in characteristic [3Xp[103X can be
        derived from the values in the record, and [9Xfalse[109X otherwise) and [10Xsource[110X
        (a string describing the source of this information)).[133X
  
  [33X[0;0YThe values are set by [2XSetMinimalRepresentationInfo[102X ([14X6.3-3[114X).[133X
  
  [1X6.3-3 SetMinimalRepresentationInfo[101X
  
  [29X[2XSetMinimalRepresentationInfo[102X( [3Xgrpname[103X, [3Xop[103X, [3Xvalue[103X, [3Xsource[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X  if  the values were successfully set, [9Xfalse[109X if stored values
            contradict the given ones.[133X
  
  [33X[0;0YThis function sets an entry in [2XMinimalRepresentationInfoData[102X ([14X6.3-2[114X) for the
  group [22XG[122X, say, with [5XGAP[105X name [3Xgrpname[103X.[133X
  
  [33X[0;0YSupported values for [3Xop[103X are[133X
  
  [30X    [33X[0;6Y[10X"NrMovedPoints"[110X  (see  [2XNrMovedPoints[102X  ([14XReference: NrMovedPoints (for a
        permutation)[114X)),  which  means  that  [3Xvalue[103X  is  the  degree of minimal
        faithful  (not  necessarily transitive) permutation representations of
        [22XG[122X,[133X
  
  [30X    [33X[0;6Ya   list   of  length  two  with  first  entry  [10X"Characteristic"[110X  (see
        [2XCharacteristic[102X  ([14XReference:  Characteristic[114X))  and  second  entry [3Xchar[103X
        either  zero  or  a  prime  integer,  which  means  that  [3Xvalue[103X is the
        dimension  of  minimal  faithful  (not necessarily irreducible) matrix
        representations of [22XG[122X in characteristic [3Xchar[103X,[133X
  
  [30X    [33X[0;6Ya  list  of  length  two with first entry [10X"Size"[110X (see [2XSize[102X ([14XReference:
        Size[114X)) and second entry a prime power [3Xq[103X, which means that [3Xvalue[103X is the
        dimension  of  minimal  faithful  (not necessarily irreducible) matrix
        representations of [22XG[122X over the field with [3Xq[103X elements, and[133X
  
  [30X    [33X[0;6Ya  list  of  length  three  with  first  entry  [10X"Characteristic"[110X  (see
        [2XCharacteristic[102X  ([14XReference:  Characteristic[114X)),  second  entry  a prime
        integer [3Xp[103X, and third entry the string [10X"complete"[110X, which means that the
        information  stored for characteristic [3Xp[103X is complete in the sense that
        for any given power [22Xq[122X of [3Xp[103X, the minimal faithful degree over the field
        with [22Xq[122X elements equals that for the largest stored field size of which
        [22Xq[122X is a power.[133X
  
  [33X[0;0YIn each case, [3Xsource[103X is a string describing the source of the data; [13Xcomputed[113X
  values are detected from the prefix [10X"comp"[110X of [3Xsource[103X.[133X
  
  [33X[0;0YIf the intended value is already stored and differs from [3Xvalue[103X then an error
  message is printed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetMinimalRepresentationInfo( "A5", "NrMovedPoints", 5,[127X[104X
    [4X[25X>[125X [27X     "computed (alternating group)" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSetMinimalRepresentationInfo( "A5", [ "Characteristic", 0 ], 3,[127X[104X
    [4X[25X>[125X [27X     "computed (char. table)" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSetMinimalRepresentationInfo( "A5", [ "Characteristic", 2 ], 2,[127X[104X
    [4X[25X>[125X [27X     "computed (char. table)" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSetMinimalRepresentationInfo( "A5", [ "Size", 2 ], 4,[127X[104X
    [4X[25X>[125X [27X     "computed (char. table)" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSetMinimalRepresentationInfo( "A5", [ "Size", 4 ], 2,[127X[104X
    [4X[25X>[125X [27X     "computed (char. table)" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSetMinimalRepresentationInfo( "A5", [ "Characteristic", 3 ], 3,[127X[104X
    [4X[25X>[125X [27X     "computed (char. table)" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X6.3-4 [33X[0;0YCriteria Used to Compute Minimality Information[133X[101X
  
  [33X[0;0YThe information about the minimal degree of a faithful [13Xmatrix representation[113X
  of  [22XG[122X  in  a  given  characteristic  or  over  a  given  field  in  positive
  characteristic  is derived from the relevant (ordinary or modular) character
  table  of  [22XG[122X, except in a few cases where this table itself is not known but
  enough information about the degrees is available in [HL89] and [Jan05].[133X
  
  [33X[0;0YThe  following  criteria  are  used  for  deriving  the  minimal degree of a
  faithful  [13Xpermutation  representation[113X  of  [22XG[122X from the information in the [5XGAP[105X
  libraries of character tables and of tables of marks.[133X
  
  [30X    [33X[0;6YIf the name of [22XG[122X has the form [10X"A[110X[22Xn[122X[10X"[110X or [10X"A[110X[22Xn[122X[10X.2"[110X (denoting alternating and
        symmetric  groups,  respectively) then the minimal degree is [22Xn[122X, except
        if [22Xn[122X is smaller than [22X3[122X or [22X2[122X, respectively.[133X
  
  [30X    [33X[0;6YIf  the  name  of  [22XG[122X has the form [10X"L2([110X[22Xq[122X[10X)"[110X (denoting projective special
        linear  groups  in  dimension  two)  then the minimal degree is [22Xq + 1[122X,
        except if [22Xq ∈ { 2, 3, 5, 7, 9, 11 }[122X, see [Hup67, Satz II.8.28].[133X
  
  [30X    [33X[0;6YIf  the  largest  maximal subgroup of [22XG[122X is core-free then the index of
        this  subgroup  is  the  minimal  degree.  (This  is used when the two
        character  tables  in  question  and the class fusion are available in
        [5XGAP[105X's  Character  Table  Library  ([Bre04]);  this  happens  for  many
        character tables of simple groups.)[133X
  
  [30X    [33X[0;6YIf  [22XG[122X  has a unique minimal normal subgroup then each minimal faithful
        permutation representation is transitive.[133X
  
        [33X[0;6YIn  this  case,  the  minimal degree can be computed directly from the
        information  in  the table of marks of [22XG[122X if this is available in [5XGAP[105X's
        Library of Tables of Marks ([NMP11]).[133X
  
        [33X[0;6YSuppose  that  the  largest maximal subgroup of [22XG[122X is not core-free but
        simple  and normal in [22XG[122X, and that the other maximal subgroups of [22XG[122X are
        core-free.  In  this  case,  we take the minimum of the indices of the
        core-free  maximal  subgroups  and of the product of index and minimal
        degree  of  the  normal  maximal  subgroup.  (This  suffices  since no
        core-free  subgroup of the whole group can contain a nontrivial normal
        subgroup of a normal maximal subgroup.)[133X
  
        [33X[0;6YLet  [22XN[122X be the unique minimal normal subgroup of [22XG[122X, and assume that [22XG/N[122X
        is  simple  and has minimal degree [22Xn[122X, say. If there is a subgroup [22XU[122X of
        index [22Xn ⋅ |N|[122X in [22XG[122X that intersects [22XN[122X trivially then the minimal degree
        of [22XG[122X is [22Xn ⋅ |N|[122X. (This is used for the case that [22XN[122X is central in [22XG[122X and
        [22XN × U[122X occurs as a subgroup of [22XG[122X.)[133X
  
  [30X    [33X[0;6YIf  we  know a subgroup of [22XG[122X whose minimal degree is [22Xn[122X, say, and if we
        know either (a class fusion from) a core-free subgroup of index [22Xn[122X in [22XG[122X
        or  a  faithful permutation representation of degree [22Xn[122X for [22XG[122X then [22Xn[122X is
        the  minimal  degree  for  [22XG[122X. (This happens often for tables of almost
        simple groups.)[133X
  
