| Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/ctbllib/tst/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 3.11.2019 mit Größe 320 kB |
|
SSL mferctbl.gap Sprache: unbekannt |
|
|
#############################################################################
## #W mferctbl.gap character tables of mult.-free endom. rings Thomas Breuer #W Juergen Mueller ## #Y Copyright (C) 2002, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany ## ## This file contains the following {\GAP} objects. ## ## `MULTFREEINFO' ## is the global variable that encodes the character tables of the ## endomorphism rings of the faithful multiplicity-free permutation ## modules of the sporadic simple groups and their cyclic and ## bicyclic extensions. ## ## `MultFreeEndoRingCharacterTables' ## is a function that can be used for computing more detailed data ## about the endomorphism rings from the compact information ## stored in `MULTFREEINFO'. ## ############################################################################# ## ## Print the banner if wanted. ## if not ( GAPInfo.CommandLineOptions.q or GAPInfo.CommandLineOptions.b ) then Print( "---------------------------------------------------------------------------\n", "Loading the database of character tables of endomorphism rings of\n", "multiplicity-free permutation modules of the sporadic simple groups and\n", "their cyclic and bicyclic extensions, compiled by T. Breuer and J. Mueller.\n", "---------------------------------------------------------------------------\n" ); fi; ############################################################################# ## #V MULTFREEINFO ## ## `MULTFREEINFO' is an immutable record. ## Its components are the `Identifier' values of the {\GAP} character ## tables of the sporadic simple groups and their automorphism groups. ## The value of the component corresponding to the group $G$, say, ## is a list containing in the first position a string denoting the name of ## $G$ in {\LaTeX} format, and in each of the remaining positions a triple ## `[ <const>, <subgroup>, <ctbl> ]' ## where <const> is a list of positive integers, <subgroup> is a string ## that denotes the name of a subgroup $H$ of $G$, in {\LaTeX} format, ## and <ctbl> is either the matrix of irreducible characters of the ## endomorphism ring of the permutation module $(1_H)^G$ or `fail' ## (indicating that the character table is not yet known). ## The sum of irreducible characters of $G$ at the positions in <const> ## is a multiplicity-free permutation character of $G$ that is induced from ## the trivial character of $H$. ## ## The ordering of the entries corresponds to the ordering in~\cite{BL96}, ## for the scope of the latter paper. ## if IsBound( MULTFREEINFO ) then # Perhaps the file 'multfree.tst' has been processed in this session. # We will need an extended version of the data used in these tests. MakeReadWriteGlobal( "MULTFREEINFO" ); UnbindGlobal( "MULTFREEINFO" ); fi; BindGlobal( "MULTFREEINFO", rec() ); ############################################################################# ## #F CollapsedAdjacencyMatricesFromCharacterTable( <mat> ) ## ## Let $X$ be the character table of an endomorphism algebra. ## The collapsed adjacency matrix of the orbital graph that corresponds to ## the $j$-th column of $X$ is given by $X^{-1} \cdot D_j \cdot X$, where ## $D_j$ is the diagonal matrix whose diagonal is the $j$-th column of $X$. ## BindGlobal( "CollapsedAdjacencyMatricesFromCharacterTable", function( mat ) local tr,itr; if mat=fail then ## only for 2.B return fail; fi; tr:=TransposedMat(mat); itr:=Inverse(tr); return List([1..Length(mat)], j->tr*List([1..Length(mat)],i->mat[i][j]*itr[i])); end ); ############################################################################# ## #F MultFreeEndoRingCharacterTables( <name> ) ## ## For a string <name> that is the name of a sporadic simple group, ## or a cyclic or bicyclic extension of a sporadic simple group, ## `MultFreeEndoRingCharacterTables' returns a list of records that describe ## the character tables of the endomorphism rings of the faithful ## multiplicity-free permutation modules of this group, ## in a format that is similar to the classification shown in~\cite{BL96}. ## ## If <name> is the string `\"all\"' then `MultFreeEndoRingCharacterTables' ## returns the list of records for all sporadic simple groups and their ## cyclic or bicyclic extensions. ## ## Each entry in the result list has the following components. ## \beginitems ## group & ## {\LaTeX} format of <name>, ## ## name & ## <name>, ## ## character & ## the permutation character, ## ## rank & ## the rank of the character, ## ## subgroup & ## a string that is a name (in {\LaTeX} format) of the subgroup ## from whose trivial character the permutation character is induced, ## ## ctbl & ## the matrix of irreducible characters of the endomorphism ring, ## where the rows are labelled by the irreducible constituents of the ## permutation character and the columns are labelled by the orbits of ## the subgroup on its right cosets, which are ordered according to ## their lengths. ## ## mats & ## the collapsed adjacency matrices, and ## ## ATLAS & ## a string that describes (in {\LaTeX} format) the constituents of the ## permutation character, relative to the perfect group involved; ## the format is described in the section~"ref:PermCharInfoRelative" ## in the {\GAP} Reference Manual. ## \enditems ## ## In the one case where the character table is not (yet) known, ## the components `ctbl' and `mats' have the value `fail'. ## DeclareGlobalFunction( "MultFreeEndoRingCharacterTables" ); InstallGlobalFunction( MultFreeEndoRingCharacterTables, function( name ) local alternat, # list of alternative names result, # the result list tbl, # character table with `Identifier' value `name' group, # value of the `group' component of each result entry len, # length of the list stored for `name' chars, # list of the permutation characters for `name' dername, # name of derived subgroup tblsimp, # character table of the derived subgroup of `tbl' info, # list of `ATLAS' values i, # loop over the permutation characters entry, # one entry in the record for `name' pos; # position in `alternative' alternat:= [ [ "Fi24'" ], [ "F3+" ] ]; result:= []; if IsBound( MULTFREEINFO.( name ) ) then if 9 < Length( name ) and name{ [ 1 .. 9 ] } = "Isoclinic" then tbl:= CharacterTableIsoclinic( CharacterTable( name{ [ 11 .. Length( name )-1 ] } ) ); else tbl:= CharacterTable( name ); fi; group:= MULTFREEINFO.( name )[1]; len:= Length( MULTFREEINFO.( name ) ); chars:= List( MULTFREEINFO.( name ){ [ 2 .. len ] }, x -> Sum( Irr( tbl ){ x[1] } ) ); if Number( name, c -> c = '.' ) = 2 then if 9 < Length( name ) and name{ [ 1 .. 9 ] } = "Isoclinic" then dername:= name{ [ 11 .. Length( name )-1 ] }; else dername:= name; fi; i:= Position( dername, '.' ); dername:= dername{ [ 1 .. Position( dername, '.', i )-1 ] }; tblsimp:= CharacterTable( dername ); if 9 < Length( name ) and name{ [ 1 .. 9 ] } = "Isoclinic" then StoreFusion( tblsimp, GetFusionMap( tblsimp, CharacterTable( name{ [ 11 .. Length( name )-1 ] } ) ), tbl ); fi; info:= PermCharInfoRelative( tblsimp, tbl, chars ).ATLAS; elif '.' in name then dername:= name{ [ 1 .. Position( name, '.' )-1 ] }; if Int( dername ) = fail then tblsimp:= CharacterTable( dername ); info:= PermCharInfoRelative( tblsimp, tbl, chars ).ATLAS; else info:= PermCharInfo( tbl, chars ).ATLAS; fi; else info:= PermCharInfo( tbl, chars ).ATLAS; fi; for i in [ 2 .. len ] do entry:= MULTFREEINFO.( name )[i]; Add( result, rec( group := group, name := name, character := chars[ i-1 ], charnmbs := entry[1], rank := Length( entry[1] ), subgroup := entry[2], ATLAS := info[ i-1 ], ctbl := entry[3], mats := CollapsedAdjacencyMatricesFromCharacterTable( entry[3] ) ) ); od; elif name = "all" then for name in RecNames( MULTFREEINFO ) do Append( result, MultFreeEndoRingCharacterTables( name ) ); od; else pos:= Position( alternat[1], name ); if pos = fail then Error( "<name> must be the name of a ", "(cyclic or bicyclic extension of a)\n", "sporadic simple group, or the string \"all\"" ); else return MultFreeEndoRingCharacterTables( alternat[2][ pos ] ); fi; fi; return result; end ); ############################################################################# ## #F LinearCombinationString( <F>, <vectors>, <strings>, <multstring>, <val> ) ## BindGlobal( "LinearCombinationString", function( F, vectors, strings, multstring, val ) local V, B, coeff, str, i; V:= VectorSpace( F, vectors ); B:= Basis( V, vectors ); coeff:= Coefficients( B, val ); if coeff = fail then return fail; elif IsZero( coeff ) then return String( val ); fi; str:= ""; for i in [ 1 .. Length( coeff ) ] do if coeff[i] > 0 then if str <> "" then Append( str, "+" ); fi; if coeff[i] = 1 and strings[i] = "" then Append( str, "1" ); elif coeff[i] <> 1 then Append( str, String( coeff[i] ) ); if strings[i] <> "" then Append( str, multstring ); fi; fi; Append( str, strings[i] ); elif coeff[i] < 0 then if coeff[i] = -1 then Append( str, "-" ); else Append( str, String( coeff[i] ) ); if strings[i] <> "" then Append( str, multstring ); fi; fi; Append( str, strings[i] ); fi; od; return str; end ); ############################################################################# ## #F Cubic( <irrat> ) ## ## The database contains nonquadratic irrationalities, ## which are either cubic or quartic. ## The function `Cubic' deals with these few cases. ## Cubic:=function( irrat ) local bas, str, dsp, atlas, display; if Conductor( irrat ) = 9 and irrat = ComplexConjugate( irrat ) then bas:= [ 1, EY(9), GaloisCyc( EY(9), 2 ) ]; str:= [ "1", "y9", "y9*2" ]; dsp:= [ "1", "EY(9)", "GaloisCyc(EY(9),2)" ]; elif Conductor( irrat ) = 19 and irrat = ComplexConjugate( irrat ) then bas:= [ EC(19), GaloisCyc( EC(19), 2 ), GaloisCyc( EC(19), 4 ) ]; str:= [ "c19", "c19*2", "c19*4" ]; dsp:= [ "EC(19)", "GaloisCyc(EC(19),2)", "GaloisCyc(EC(19),4)" ]; elif Conductor( irrat ) = 33 then bas:= [1, EV(33), GaloisCyc(EV(33),5), GaloisCyc(EV(33),7)]; str:= [ "1", "v33", "v33*5", "v33*7" ]; dsp:= [ "1", "EV(33)", "GaloisCyc(EV(33),5)", "GaloisCyc(EV(33),7)"]; else Error( "this should not occur" ); fi; atlas := LinearCombinationString( Rationals, bas, str, "", irrat ); display := LinearCombinationString( Rationals, bas, dsp, "*", irrat ); return rec( value:= irrat, ATLAS := atlas, display := display ); end; ############################################################################# ## #F CharacterLabels( <tbl> ) #F CharacterLabels( <tbl2>, <tbl>] ) ## BindGlobal( "CharacterLabels", function( arg ) local tbl, tbl2, tblfustbl2, irr, alp, degreeset, irreds, chi, nccl2, irr2, irreds2, irrnam2, rest, i, j, chi2, k, pos, result; if Length( arg ) = 1 and IsCharacterTable( arg[1] ) then tbl:= arg[1]; tbl2:= false; elif Length( arg ) = 2 and ForAll( arg, IsCharacterTable ) then tbl:= arg[2]; tbl2:= arg[1]; tblfustbl2:= GetFusionMap( tbl, tbl2 ); if tblfustbl2 = fail or Size( tbl2 ) <> 2 * Size( tbl ) then Error( "<tbl> must be of index 2 in <tbl2>, with stored fusion" ); fi; else Error( "usage: CharacterLabels( [<tbl2>, ]<tbl> )" ); fi; irr:= Irr( tbl ); alp:= [ "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m", "n", "o", "p", "q", "r", "s", "t", "u", "v", "w", "x", "y", "z" ]; degreeset:= Set( List( irr, DegreeOfCharacter ) ); # `irreds[i]' contains all irreducibles of `tbl' of the `i'-th degree. irreds:= List( degreeset, x -> [] ); for chi in irr do Add( irreds[ Position( degreeset, chi[1] ) ], chi ); od; # Extend the alphabet if necessary. while Length( alp ) < Maximum( List( irreds, Length ) ) do alp:= Concatenation( alp, List( alp, x -> Concatenation( "(", x, "')" ) ) ); od; if tbl2 <> false then # Construct relative names for the irreducibles of `tbl2'. nccl2:= NrConjugacyClasses( tbl2 ); irr2:= Irr( tbl2 ); irreds2:= []; irrnam2:= []; rest:= List( irr2, x -> x{ tblfustbl2 } ); for i in [ 1 .. Length( irreds ) ] do for j in [ 1 .. Length( irreds[i] ) ] do chi2:= irr2{ Filtered( [ 1 .. nccl2 ], k -> rest[k] = irreds[i][j] ) }; if Length( chi2 ) = 2 then # The `j'-th character of the `i'-th degree of `tbl' extends. Append( irreds2, chi2 ); Add( irrnam2, Concatenation( String( chi2[1][1] ), alp[j], "^+" ) ); Add( irrnam2, Concatenation( String( chi2[1][1] ), alp[j], "^-" ) ); else # The `j'-th character of the `i'-th degree of `tbl' fuses # with another character of `tbl', of the same degree. for k in [ 1 .. nccl2 ] do if rest[k][1] = 2 * degreeset[i] and ScalarProduct( tbl, rest[k], irreds[i][j] ) <> 0 then pos:= Position( irreds2, irr2[k] ); if pos = fail then Add( irreds2, irr2[k] ); Add( irrnam2, Concatenation( String( degreeset[i] ), alp[j] ) ); else Append( irrnam2[ pos ], alp[j] ); fi; fi; od; fi; od; od; result:= List( irr2, chi -> irrnam2[ Position( irreds2, chi ) ] ); else result:= List( irr, chi -> Concatenation( String( chi[1] ), alp[ Position( irreds[ Position( degreeset, chi[1] ) ], chi ) ] ) ); fi; return result; end ); ############################################################################# ## #V DisplayStringLabelledMatrixDefaults ## BindGlobal( "DisplayStringLabelledMatrixDefaults", rec( format := "GAP", rowlabels := [], rowalign := [ "t" ], rowsep := [ , "-" ], collabels := [], colalign := [ "l" ], colsep := [ , "|" ], legend := "", header := "" ) ); ############################################################################# ## #F DisplayStringLabelledMatrix( <matrix>, <inforec> ) ## ## <inforec> must be a record; ## the following components are supported ## \beginitems ## `header' ## (default nothing) ## ## `rowlabels' ## list of row labels (default nothing) ## ## `rowalign' ## list of vertical alignments ("top", "bottom", "center"; ## default top alignment) ## the first is for the column labels! ## ## `rowsep' ## list of strings that separate rows ## the first is for the row above the column labels, ## the second is for the row below the column labels! ## default separate column labels from the other lines by a line ## ## `collabels' ## list of column labels (default nothing) ## ## `colalign' ## list of horizontal alignments ("left", "right", "center"; ## default left alignment of label, right alignment of the rest) ## the first is for the row labels! ## ## `colsep' ## list of strings that separate columns ## the first is for the column before the row labels, ## the second is for the column after the row labels! ## default separate row labels from the other columns with a pipe ## ## `legend' ## ## `format' ## one of `\"GAP\"', `\"HTML\"', or `\"LaTeX\"' (default `\"GAP\"'). ## \enditems ## ## improved version, better than the one in `mferctbl'! ## BindGlobal( "DisplayStringLabelledMatrix", function( matrix, inforec ) local header, rowlabels, collabels, legend, format, # options inforec2, name, ncols, colwidth, row, i, len, fmat, max, fcollabels, openmat, openrow, align1, opencol, closecol, closerow, closemat, str, j; # Get the parameters. inforec2:= ShallowCopy( DisplayStringLabelledMatrixDefaults ); for name in RecNames( inforec ) do inforec2.( name ):= inforec.( name ); od; inforec:= inforec2; if inforec.format = "GAP" then # Compute the column widths (used in GAP format only). ncols:= Length( matrix[1] ); if IsEmpty( inforec.collabels ) then colwidth:= ListWithIdenticalEntries( ncols, 0 ); else colwidth:= List( inforec.collabels, Length ); #T may be non-dense list fi; for row in matrix do for i in [ 1 .. ncols ] do len:= Length( row[i] ); if colwidth[i] < len then colwidth[i]:= len; fi; od; od; matrix:= List( matrix, row -> List( [ 1 .. ncols ], i -> String( row[i], colwidth[i] ) ) ); #T alignment! # Inspect the row labels. if not IsEmpty( inforec.rowlabels ) then max:= Maximum( List( inforec.rowlabels, Length ) ) + 1; inforec.rowlabels:= List( inforec.rowlabels, x -> String( x, -max ) ); #T label alignment! else max:= 0; fi; # Set the parameters. if IsBound( inforec.colsep[1] ) then Append( fcollabels, inforec.colsep[1] ); fi; fcollabels:= String( "", max ); if IsBound( inforec.colsep[2] ) then Append( fcollabels, inforec.colsep[2] ); fi; for i in [ 1 .. ncols ] do Append( fcollabels, " " ); if IsBound( inforec.collabels[i] ) then Append( fcollabels, String( inforec.collabels[i], colwidth[i] ) ); else Append( fcollabels, String( "", colwidth[i] ) ); fi; if IsBound( inforec.colsep[ i+2 ] ) then Append( fcollabels, inforec.colsep[ i+2 ] ); Append( fcollabels, " " ); fi; od; Append( fcollabels, "\n" ); openmat:= Concatenation( ListWithIdenticalEntries( max + Sum( colwidth ) + Length( colwidt openrow:= ShallowCopy( inforec.rowlabels ); for i in [ 1 .. Length( matrix ) + 2 ] do if not IsBound( openrow[i] ) then openrow[i]:= ""; fi; if IsBound( inforec.colsep[1] ) then openrow[i]:= Concatenation( inforec.colsep[1], openrow[i] ); fi; if IsBound( inforec.colsep[2] ) then openrow[i]:= Concatenation( openrow[i], inforec.colsep[2] ); fi; od; opencol := List( matrix[1], x -> " " ); closecol := []; for i in [ 1 .. ncols ] do if IsBound( inforec.colsep[ i+2 ] ) then closecol[i]:= inforec.colsep[ i+2 ]; else closecol[i]:= ""; fi; od; closerow := List( matrix, x -> "\n" ); closemat := ""; elif inforec.format = "HTML" then # Translate alignment values from LaTeX to HTML. inforec.colalign:= ShallowCopy( inforec.colalign ); for i in [ 1 .. Length( inforec.colalign ) ] do if IsBound( inforec.colalign[i] ) then if inforec.colalign[i] = "l" then inforec.colalign[i]:= "left"; elif inforec.colalign[i] = "r" then inforec.colalign[i]:= "right"; elif inforec.colalign[i] = "c" then inforec.colalign[i]:= "center"; fi; fi; od; inforec.rowalign:= ShallowCopy( inforec.rowalign ); for i in [ 1 .. Length( inforec.rowalign ) ] do if IsBound( inforec.rowalign[i] ) then if inforec.rowalign[i] = "l" then inforec.rowalign[i]:= "left"; elif inforec.rowalign[i] = "r" then inforec.rowalign[i]:= "right"; elif inforec.rowalign[i] = "c" then inforec.rowalign[i]:= "center"; fi; fi; od; # Set the parameters. ncols:= Length( matrix[1] ); fcollabels:= "\n<table align=\"center\""; if not IsEmpty( inforec.colsep ) then Append( fcollabels, " border=2" ); fi; if not IsEmpty( inforec.collabels ) then Append( fcollabels, ">\n<tr>\n" ); if not IsEmpty( inforec.rowlabels ) then Append( fcollabels, "<td>\n</td>\n" ); fi; for i in [ 1 .. ncols ] do Append( fcollabels, "<td align=\"" ); if IsBound( inforec.colalign[ i+1 ] ) then Append( fcollabels, inforec.colalign[ i+1 ] ); else Append( fcollabels, "right" ); fi; Append( fcollabels, "\">" ); if IsBound( inforec.collabels[i] ) then Append( fcollabels, inforec.collabels[i] ); fi; Append( fcollabels, "</td>\n" ); od; Append( fcollabels, "</tr>\n" ); fi; openmat:= ""; if IsEmpty( inforec.rowlabels ) then openrow:= List( matrix, x -> "<tr>\n" ); else if IsBound( inforec.colalign[1] ) then align1:= inforec.colalign[1]; else align1:= "right"; fi; openrow:= List( inforec.rowlabels, x -> Concatenation( "<tr>\n<td align=\"", align1, "\">", x, "</td>\n" ) ); for i in [ 1 .. Length( matrix ) ] do if not IsBound( openrow[i] ) then openrow[i]:= Concatenation( "<tr>\n<td align=\"", align1, "\"></td>\n" ); fi; od; fi; opencol:= []; for i in [ 1 .. ncols ] do if IsBound( inforec.colalign[ i+1 ] ) then opencol[i]:= Concatenation( "<td align=\"", inforec.colalign[ i+1 ], "\">\n" ); else opencol[i]:= "<td align=\"right\">\n"; fi; od; closecol:= List( matrix[1], x -> "</td>\n" ); closerow:= List( matrix, x -> "</tr>\n" ); closemat:= "</table>\n"; elif inforec.format = "LaTeX" then # Set the parameters. ncols:= Length( matrix[1] ); fcollabels:= "\\begin{tabular}{"; if IsBound( inforec.colsep[1] ) then Append( fcollabels, inforec.colsep[1] ); fi; if IsBound( inforec.colalign[1] ) then Append( fcollabels, inforec.colalign[1] ); elif not IsEmpty( inforec.rowlabels ) then Append( fcollabels, "l" ); fi; for i in [ 1 .. ncols ] do if IsBound( inforec.colsep[ i+1 ] ) then Append( fcollabels, inforec.colsep[ i+1 ] ); fi; if IsBound( inforec.colalign[ i+1 ] ) then Append( fcollabels, inforec.colalign[ i+1 ] ); else Append( fcollabels, "r" ); fi; od; if IsBound( inforec.colsep[ ncols+2 ] ) then Append( fcollabels, inforec.colsep[ ncols+2 ] ); fi; Append( fcollabels, "}" ); if IsBound( inforec.rowsep[1] ) then if inforec.rowsep[1] = "=" then Append( fcollabels, " \\hline\\hline" ); else Append( fcollabels, " \\hline" ); fi; fi; Append( fcollabels, "\n" ); if not IsEmpty( inforec.collabels ) then for i in [ 1 .. ncols ] do if 1 < i or not IsEmpty( inforec.rowlabels ) then Append( fcollabels, " & " ); fi; if IsBound( inforec.collabels[i] ) then Append( fcollabels, inforec.collabels[i] ); fi; Append( fcollabels, "\n" ); od; Append( fcollabels, " \\rule[-7pt]{0pt}{20pt} \\\\" ); if IsBound( inforec.rowsep[2] ) then if inforec.rowsep[2] = "=" then Append( fcollabels, " \\hline\\hline" ); else Append( fcollabels, " \\hline" ); fi; fi; Append( fcollabels, "\n" ); fi; openmat:= ""; openrow:= ShallowCopy( inforec.rowlabels ); for i in [ 1 .. Length( matrix ) ] do if not IsBound( openrow[i] ) then openrow[i]:= ""; else openrow[i]:= Concatenation( "$", openrow[i], "$" ); fi; od; closerow:= []; for i in [ 1 .. Length( matrix ) ] do if IsBound( inforec.rowsep[ i+2 ] ) then if inforec.rowsep[ i+2 ] = "=" then closerow[i]:= " \\\\ \\hline\\hline\n"; else closerow[i]:= " \\\\ \\hline\n"; fi; else closerow[i]:= " \\\\\n"; fi; od; opencol:= List( matrix[1], x -> " & " ); if IsEmpty( inforec.rowlabels ) then opencol[1]:= ""; fi; closecol:= List( matrix[1], x -> "" ); closemat:= "\\rule[-7pt]{0pt}{5pt}\\end{tabular}\n"; else Error( "<inforec>.format must be one of `\"GAP\"', `\"HTML\"', `\"LaTeX\"'" ); fi; # Put the string for the matrix together. str:= ShallowCopy( inforec.header ); Append( str, fcollabels ); Append( str, openmat ); for i in [ 1 .. Length( matrix ) ] do row:= matrix[i]; Append( str, openrow[i] ); for j in [ 1 .. ncols ] do Append( str, opencol[j] ); Append( str, row[j] ); Append( str, closecol[j] ); od; Append( str, closerow[i] ); od; Append( str, closemat ); Append( str, inforec.legend ); # Return the string. return str; end ); ############################################################################# ## #F DisplayMultFreeEndoRingCharacterTable( <record> ) ## BindGlobal( "DisplayMultFreeEndoRingCharacterTable", function( record ) local name, subname, pos, header, mat, triples, legend, row, mrow, entry, l, triple, tbl, simpname, tbl2, charlabels, rowlabels, collabels, str; name:=record.group; if name[1] = '$' then name:= name{ [ 2 .. Length( name ) - 1 ] }; fi; subname:= record.subgroup; if subname[1] = '$' then subname:= subname{ [ 2 .. Length( subname ) - 1 ] }; fi; pos:= PositionSublist( subname, "\\leq" ); if pos <> fail then subname:= Concatenation( subname{ [ 1 .. pos-1 ] }, "< ", subname{ [ pos+5 .. Length( subname ) ] } ); fi; pos:= PositionSublist( subname, "\\rightarrow" ); if pos <> fail then subname:= Concatenation( subname{ [ 1 .. pos-1 ] }, "---> ", subname{ [ pos+12 .. Length( subname ) ] } ); fi; header:= Concatenation( "G = ", name, ", H = ", subname, "\n\n" ); if IsMatrix( record.ctbl ) then mat:= []; triples:= []; legend:= ""; for row in record.ctbl do mrow:= []; for entry in row do l:= Quadratic( entry ); if IsRecord( l ) then Add( mrow, l.ATLAS ); else l:= Cubic( entry ); if not IsRecord( l ) then Error( "entry <l> not expected" ); fi; Add( mrow, l.ATLAS ); fi; if not IsInt( entry ) then AddSet( triples, [ l.ATLAS, l.display ] ); fi; od; Add( mat, mrow ); od; for triple in triples do Append( legend, "\n" ); Append( legend, triple[1] ); Append( legend, " = " ); Append( legend, triple[2] ); od; #T better legend: only smallest multiple of several #T (example: see first table of M12.2, multiples of r3 and r5) tbl:= CharacterTable( record.name ); if '.' in record.name then simpname:= record.name{ [ 1 .. Position( record.name, '.' )-1 ] }; if Int(simpname)=fail then tbl2:= tbl; tbl:= CharacterTable( simpname ); charlabels:= CharacterLabels( tbl2, tbl ); else charlabels:= CharacterLabels( tbl ); fi; else charlabels:= CharacterLabels( tbl ); fi; rowlabels:= charlabels{ record.charnmbs }; collabels:= List( [ 1 .. record.rank ], i -> Concatenation( "O_", String( i ) ) ); str:= DisplayStringLabelledMatrix( mat, rec( header := header, rowlabels := rowlabels, collabels := collabels, legend := legend, format := "GAP" ) ); else str:= Concatenation( header, "(character table not yet known)" ); fi; Print( str, "\n" ); end ); ############################################################################# ## #F MultFreeFromTOM( <name> ) ## ## For a string <name> that is a valid name for a table of marks in the ## {\GAP} library of tables of marks, `MultFreeFromTOM' returns ## the list of pairs $[ \pi, G ]$ where $\pi$ is a multiplicity free ## permutation character of the group for <name>, ## and $G$ is the corresponding permutation group, ## as computed from the table of marks of this group. ## ## If there is no table of marks for <name> or if the class fusion of the ## character table for <name> into the table of marks is not unique then ## `fail' is returned. ## BindGlobal( "MultFreeFromTOM", function( name ) local tom, # the table of marks G, # the underlying group t, # the character table fus, # fusion map from 't' to 'tom' perms, # list of perm. characters multfree, # list of mult. free characters and perm. groups i, # loop over `perms' H, # point stabilizer tr, # right transversal of `H' in `G' P; # permutation action of `G' on the cosets of `H' tom:= TableOfMarks( name ); if tom <> fail then G:= UnderlyingGroup( tom ); t:= CharacterTable( name ); fus:= FusionCharTableTom( t, tom ); if ForAll( fus, IsInt ) then perms:= PermCharsTom( fus, tom ); multfree:= []; for i in [ 1 .. Length( perms ) ] do if ForAll( Irr( t ), chi -> ScalarProduct( t, chi, perms[i] ) <= 1 ) then H:= RepresentativeTom( tom, i ); tr:= RightTransversal( G, H ); P:= Action( G, tr, OnRight ); Add( multfree, [ Character( t, perms[i] ), P ] ); fi; od; else Print( "#E fusion is not unique!\n" ); multfree:= fail; fi; else multfree:= fail; fi; return multfree; end ); ############################################################################# ## #F OrbitAndTransversal( <gens>, <pt> ) ## ## Let <gens> be a list of permutations, and <pt> be a positive integer. ## `OrbitAndTransversal' returns a record that describes the orbit of <pt> ## under the group generated by <gens>, as follows. ## The component `orb' is the orbit itself, ## the component `from' is the list of predecessors, ## the component `use' is the list of positions of generators used ## OrbitAndTransversal := function( gens, pt ) local orb, # the orbit of `pt' sort, # a sorted copy of `orb', for faster lookup from, # list of points use, i, j, img; # Initialize the lists. orb := [ pt ]; sort := [ pt ]; from := [ fail ]; use := [ fail ]; # Compute the orbit. for i in orb do for j in [ 1 .. Length( gens ) ] do img:= i^gens[j]; if not img in sort then Add( orb, img ); AddSet( sort, img ); Add( from, i ); Add( use, j ); fi; od; od; # Return the result. return rec( orb := orb, from := from, use := use, gens := gens ); end; ############################################################################# ## #F RepresentativeFromTransversal( <orbitinfo>, <gens>, <pt> ) ## ## Let <orbitinfo> be a record returned by `OrbitAndTransversal', and <pt> ## be a point in the orbit `<orbitinfo>.orb'. ## `RepresentativeFromTransversal' returns an element in the group generated ## by `<orbitinfo>.gens' that maps `<orbitinfo>.orb[1]' to <pt>. ## RepresentativeFromTransversal := function( record, gens, pt2 ) local orb, from, use, rep, pos; orb := record.orb; from := record.from; use := record.use; rep := (); while pt2 <> orb[1] do pos:= Position( orb, pt2 ); rep:= gens[ use[ pos ] ] * rep; pt2:= from[ pos ]; od; return rep; end; ############################################################################# ## #F CollapsedAdjacencyMatricesInfo( <G>[, <order>][, <S>] ) ## ## Let <G> be a transitive permutations group, ## $\omega$ the smallest point moved by <G>, ## and $S$ the stabilizer of $\omega$ in <G>. ## `CollapsedAdjacencyMatricesInfo' returns a record describing the ## collapsed adjacency matrices of the permutation module. ## The components are ## `mats' ## the collapsed adjacency matrices, ## ## `points' ## a list of representatives of the orbits of $S$ on the set of the ## points moved by <G>, ## ## `representatives' : ## a list of elements in <G> such that the $i$-th entry maps $\omega$ ## to the $i$-th entry in the `points' list, ## ## The order of <G> can be given as the optional argument <order>, ## and the group $S$ can be given as the optional argument <S>. ## CollapsedAdjacencyMatricesInfo := function( arg ) local G,size,stab,points,gens,orbs,record,reps,mats,r,i,j,k,img; # Get and check the arguments. if Length( arg ) = 1 and IsPermGroup( arg[1] ) then G := arg[1]; size := Size( G ); elif Length( arg ) = 2 and IsPermGroup( arg[1] ) and IsPosInt( arg[2] ) then G := arg[1]; size := arg[2]; elif Length( arg ) = 2 and IsPermGroup( arg[1] ) and IsPermGroup( arg[2] ) then G := arg[1]; size := Size( G ); stab := arg[2]; elif Length( arg ) = 3 and IsPermGroup( arg[1] ) and IsPosInt( arg[2] ) and IsPermGroup( arg[3] ) then G := arg[1]; size := arg[2]; stab := arg[3]; fi; points:= MovedPoints( G ); if not IsBound( stab ) then stab:= Stabilizer( G, points[1] ); fi; # Compute the orbits of the point stabilizer, and sort them. gens:= GeneratorsOfGroup( G ); orbs:= List( Orbits( stab, points ), Set ); SortParallel( List( orbs, Length ), orbs ); # Compute representatives ... record:= OrbitAndTransversal( gens, orbs[1][1] ); reps:= List( orbs, x -> RepresentativeFromTransversal( record, gens, x[1] ) ); # Compute the collapsed adjacency matrices. mats:= []; r:= Length( orbs ); for i in [ 1 .. r ] do mats[i]:= []; for j in [ 1 .. r ] do mats[i][j]:= []; od; for k in [1 .. r] do img:= OnTuples( orbs[i], reps[k] ); for j in [ 1 .. r ] do mats[i][j][k]:= Number( Intersection( orbs[j], img ) ); od; od; od; # Return the result. return rec( mats := mats, points := List( orbs, x -> x[1] ), representatives := reps ); end; ############################################################################# ## ## The following component can be used for running over the groups in the ## original order. ## Note that from GAP 4.5 on, `RecNames' does not return the list of ## component names in the same order as the assignments of the components. ## MULTFREEINFO.allnames:= [ "M11", "M12", "J1", "M22", "J2", "M23", "HS", "J3", "M24", "McL", "He", "Ru", "Suz", "ON", "Co3", "Co2", "Fi22", "HN", "Ly", "Th", "Fi23", "Co1", "J4", "F3+", "B", "M", "M12.2", "M22.2", "J2.2", "HS.2", "J3.2", "McL.2", "He.2", "Suz.2", "ON.2", "Fi22.2", "HN.2", "F3+.2", "2.M12", "2.M22", "3.M22", "4.M22", "6.M22", "12.M22", "2.J2", "2.HS", "3.J3", "3.McL", "2.Ru", "2.Suz", "3.Suz", "6.Suz", "3.ON", "2.Fi22", "3.Fi22", "6.Fi22", "2.Co1", "3.F3+", "2.B", "2.M12.2", "Isoclinic(2.M12.2)", "2.M22.2", "Isoclinic(2.M22.2)", "3.M22.2", "4.M22.2", "Isoclinic(4.M22.2)", "6.M22.2", "Isoclinic(6.M22.2)", "12.M22.2", "Isoclinic(12.M22.2)", "2.J2.2", "Isoclinic(2.J2.2)", "2.HS.2", "Isoclinic(2.HS.2)", "3.J3.2", "3.McL.2", "2.Suz.2", "Isoclinic(2.Suz.2)", "3.Suz.2", "6.Suz.2", "Isoclinic(6.Suz.2)", "3.ON.2", "2.Fi22.2", "Isoclinic(2.Fi22.2)", "3.Fi22.2", "6.Fi22.2", "Isoclinic(6.Fi22.2)", "3.F3+.2" ]; MULTFREEINFO.("M11"):= ["$M_{11}$", ## [[1,2],"$A_6.2_3$", [[1,10], [1,-1]]], ## [[1,2,5],"$A_6 \\leq A_6.2_3$", [[1,1,20], [1,1,-2], [1,-1,0]]], ## [[1,5],"$L_2(11)$", [[1,11], [1,-1]]], ## [[1,5,6,7,9,10],"$11:5 \\leq L_2(11)$", [[1,11,11,11,55,55], [1,-1,-1,11,-5,-5], [1,(-5-3*ER(-11))/2,(-5+3*ER(-11))/2,-1,-5,10], [1,(-5+3*ER(-11))/2,(-5-3*ER(-11))/2,-1,-5,10], [1,3,3,-1,-5,-1], [1,-1,-1,-1,7,-5]]], ## [[1,2,8],"$3^2:Q_8.2$", [[1,18,36], [1,7,-8], [1,-2,1]]], ## [[1,2,8,10],"$3^2:8 \\leq 3^2:Q_8.2$", [[1,1,36,72], [1,1,14,-16], [1,1,-4,2], [1,-1,0,0]]], ## [[1,2,5,8],"$A_5.2$", [[1,15,20,30], [1,-7,-2,8], [1,-3,8,-6], [1,2,-2,-1]]]]; MULTFREEINFO.("M12"):= ["$M_{12}$", ## [[1,2],"$M_{11}$", [[1,11], [1,-1]]], ## [[1,3],"$M_{11}$", [[1,11], [1,-1]]], ## [[1,2,3,8,11],"$L_2(11) \\leq M_{11}$", [[1,11,11,55,66], [1,-1,11,-5,-6], [1,11,-1,-5,-6], [1,-1,-1,7,-6], [1,-1,-1,-5,6]]], ## [[1,2,7],"$A_6.2^2$", [[1,20,45], [1,8,-9], [1,-2,1]]], ## [[1,2,3,7,8],"$A_6.2_1 \\leq A_6.2^2$", [[1,1,40,45,45], [1,-1,0,15,-15], [1,1,16,-9,-9], [1,1,-4,1,1], [1,-1,0,-3,3]]], ## [[1,2,7,11],"$A_6.2_2 \\leq A_6.2^2$", [[1,1,40,90], [1,1,16,-18], [1,1,-4,2], [1,-1,0,0]]], ## [[1,3,7],"$A_6.2^2$", [[1,20,45], [1,8,-9], [1,-2,1]]], ## [[1,2,3,7,8],"$A_6.2_1 \\leq A_6.2^2$", [[1,1,40,45,45], [1,1,16,-9,-9], [1,-1,0,15,-15], [1,1,-4,1,1], [1,-1,0,-3,3]]], ## [[1,3,7,11],"$A_6.2_2 \\leq A_6.2^2$", [[1,1,40,90], [1,1,16,-18], [1,1,-4,2], [1,-1,0,0]]], ## [[1,4,5,6,11],"$L_2(11)$", [[1,11,11,55,66], [1,(-5-3*ER(-11))/2,(-5+3*ER(-11))/2,10,-6], [1,(-5+3*ER(-11))/2,(-5-3*ER(-11))/2,10,-6], [1,3,3,-1,-6], [1,-1,-1,-5,6]]], ## [[1,2,7,8,12],"$3^2.2.S_4$", [[1,12,27,72,108], [1,-4,15,-24,12], [1,1,5,6,-13], [1,5,-3,-6,3], [1,-3,-3,2,3]]], ## [[1,2,6,7,8,10,12,13],"$3^2:2.A_4 \\leq 3^2.2.S_4$", [[1,1,24,27,27,72,72,216], [1,1,-8,15,15,-24,-24,24], [1,-1,0,3,-3,16,-16,0], [1,1,2,5,5,6,6,-26], [1,1,10,-3,-3,-6,-6,6], [1,-1,0,-9,9,0,0,0], [1,1,-6,-3,-3,2,2,6], [1,-1,0,3,-3,-6,6,0]]], ## [[1,3,7,8,12],"$3^2.2.S_4$", [[1,12,27,72,108], [1,-4,15,-24,12], [1,1,5,6,-13], [1,5,-3,-6,3], [1,-3,-3,2,3]]], ## [[1,3,6,7,8,9,12,13],"$3^2:2.A_4 \\leq 3^2.2.S_4$", [[1,1,24,27,27,72,72,216], [1,1,-8,15,15,-24,-24,24], [1,-1,0,3,-3,16,-16,0], [1,1,2,5,5,6,6,-26], [1,1,10,-3,-3,-6,-6,6], [1,-1,0,-9,9,0,0,0], [1,1,-6,-3,-3,2,2,6], [1,-1,0,3,-3,-6,6,0]]]]; MULTFREEINFO.("J1"):= ["$J_1$", ## [[1,2,3,4,6],"$L_2(11)$", [[1,11,12,110,132], [1,(-7-ER(5))/2,(-3+3*ER(5))/2,(5+7*ER(5))/2,(3-9*ER(5))/2], [1,(-7+ER(5))/2,(-3-3*ER(5))/2,(5-7*ER(5))/2,(3+9*ER(5))/2], [1,4,-2,5,-8], [1,1,4,-10,4]]], ## [[1,2,3,4,7,8,9,10,11,12,15],"$2^3.7.3$", [[1,8,28,56,56,56,168,168,168,168,168], [1,-2-ER(5),1+3*ER(5),1+4*ER(5),-11-ER(5),(23-7*ER(5))/2,6*ER(5), (-33+3*ER(5))/2,(27+9*ER(5))/2,-6-6*ER(5),(15-15*ER(5))/2], [1,-2+ER(5),1-3*ER(5),1-4*ER(5),-11+ER(5),(23+7*ER(5))/2,-6*ER(5), (-33-3*ER(5))/2,(27-9*ER(5))/2,-6+6*ER(5),(15+15*ER(5))/2], [1,2,11,-4,4,-2,-24,-6,12,18,-12], [1,-2-ER(5),-7,1+4*ER(5),-4+4*ER(5),-4-ER(5),3,3-9*ER(5),3-3*ER(5), 18+3*ER(5),-12+3*ER(5)], [1,-2+ER(5),-7,1-4*ER(5),-4-4*ER(5),-4+ER(5),3,3+9*ER(5),3+3*ER(5), 18-3*ER(5),-12-3*ER(5)], [1,-EC(19)-2*GaloisCyc(EC(19),2)-2*GaloisCyc(EC(19),4), 2*GaloisCyc(EC(19),4),4*EC(19)-GaloisCyc(EC(19),2), -EC(19)+2*GaloisCyc(EC(19),2)-2*GaloisCyc(EC(19),4), 7*EC(19)+7*GaloisCyc(EC(19),2)+6*GaloisCyc(EC(19),4), -6*EC(19)-2*GaloisCyc(EC(19),2)-GaloisCyc(EC(19),4), 3*EC(19)+GaloisCyc(EC(19),2)+2*GaloisCyc(EC(19),4), -EC(19)+2*GaloisCyc(EC(19),2)-5*GaloisCyc(EC(19),4), 5*EC(19)+4*GaloisCyc(EC(19),2)+8*GaloisCyc(EC(19),4), -9*EC(19)-10*GaloisCyc(EC(19),2)-7*GaloisCyc(EC(19),4)], [1,-2*EC(19)-2*GaloisCyc(EC(19),2)-GaloisCyc(EC(19),4), 2*GaloisCyc(EC(19),2),-EC(19)+4*GaloisCyc(EC(19),4), 2*EC(19)-2*GaloisCyc(EC(19),2)-GaloisCyc(EC(19),4), 7*EC(19)+6*GaloisCyc(EC(19),2)+7*GaloisCyc(EC(19),4), -2*EC(19)-GaloisCyc(EC(19),2)-6*GaloisCyc(EC(19),4), EC(19)+2*GaloisCyc(EC(19),2)+3*GaloisCyc(EC(19),4), 2*EC(19)-5*GaloisCyc(EC(19),2)-GaloisCyc(EC(19),4), 4*EC(19)+8*GaloisCyc(EC(19),2)+5*GaloisCyc(EC(19),4), -10*EC(19)-7*GaloisCyc(EC(19),2)-9*GaloisCyc(EC(19),4)], [1,-2*EC(19)-GaloisCyc(EC(19),2)-2*GaloisCyc(EC(19),4),2*EC(19), 4*GaloisCyc(EC(19),2)-GaloisCyc(EC(19),4), -2*EC(19)-GaloisCyc(EC(19),2)+2*GaloisCyc(EC(19),4), 6*EC(19)+7*GaloisCyc(EC(19),2)+7*GaloisCyc(EC(19),4), -EC(19)-6*GaloisCyc(EC(19),2)-2*GaloisCyc(EC(19),4), 2*EC(19)+3*GaloisCyc(EC(19),2)+GaloisCyc(EC(19),4), -5*EC(19)-GaloisCyc(EC(19),2)+2*GaloisCyc(EC(19),4), 8*EC(19)+5*GaloisCyc(EC(19),2)+4*GaloisCyc(EC(19),4), -7*EC(19)-9*GaloisCyc(EC(19),2)-10*GaloisCyc(EC(19),4)], [1,3,1,1,4,9,15,-4,-14,-1,-15], [1,-3,1,1,4,3,-9,14,-8,-7,3]]]]; MULTFREEINFO.("M22"):= ["$M_{22}$", ## [[1,2],"$L_3(4)$", [[1,21], [1,-1]]], ## [[1,2,5],"$2^4:A_6$", [[1,16,60], [1,-6,5], [1,2,-3]]], ## [[1,2,5,7,9],"$2^4:A_5 \\leq 2^4:A_6$", [[1,5,96,120,240], [1,5,-36,10,20], [1,5,12,-6,-12], [1,-1,0,12,-12], [1,-1,0,-8,8]]], ## [[1,2,7],"$A_7$", [[1,70,105], [1,-18,17], [1,2,-3]]], ## [[1,2,7],"$A_7$", [[1,70,105], [1,-18,17], [1,2,-3]]], ## [[1,2,5,7],"$2^4:S_5$", [[1,30,40,160], [1,-3,18,-16], [1,9,-2,-8], [1,-3,-2,4]]], ## [[1,2,5,6,7],"$2^3:L_3(2)$", [[1,7,42,112,168], [1,-4,9,24,-30], [1,4,9,-8,-6], [1,-3,2,-8,8], [1,1,-6,4,0]]], ## [[1,2,5,7,12],"$A_6.2_3$", [[1,30,45,180,360], [1,8,-21,48,-36], [1,16,3,-16,-4], [1,-2,9,8,-16], [1,-2,-3,-4,8]]], ## [[1,2,5,7,8,9],"$L_2(11)$", [[1,55,55,66,165,330], [1,15,-25,-6,45,-30], [1,13,13,-18,-3,-6], [1,-5,7,6,9,-18], [1,-7,-3,-6,1,14], [1,5,-3,6,-11,2]]]]; MULTFREEINFO.("J2"):= ["$J_2$", ## [[1,6,7],"$U_3(3)$", [[1,36,63], [1,6,-7], [1,-4,3]]], ## [[1,7,10,11],"$3.A_6.2_2$", [[1,36,108,135], [1,-4,-12,15], [1,8,-4,-5], [1,-4,8,-5]]], ## [[1,2,3,6,10,12],"$2^{1+4}_{-}:A_5$", [[1,10,32,32,80,160], [1,-5,2-6*ER(5),2+6*ER(5),20,-20], [1,-5,2+6*ER(5),2-6*ER(5),20,-20], [1,5,-8,-8,10,0], [1,3,4,4,-4,-8], [1,-2,-1,-1,-4,7]]], ## [[1,6,7,10,12,13],"$2^{2+4}.(3 \\times S_3)$", [[1,12,32,96,192,192], [1,7,-8,16,-28,12], [1,-3,12,6,-18,2], [1,5,4,-2,10,-18], [1,0,-1,-12,0,12], [1,-3,-4,6,6,-6]]], ## [[1,7,10,11,12,13,18],"$A_4 \\times A_5$", [[1,15,20,24,180,240,360], [1,5,10,-6,0,20,-30], [1,1,6,10,-16,-12,10], [1,-5,0,4,20,0,-20], [1,6,-4,0,9,-12,0], [1,-3,2,-6,0,-12,18], [1,-1,-4,0,-12,16,0]]]]; MULTFREEINFO.("M23"):= ["$M_{23}$", ## [[1,2],"$M_{22}$", [[1,22], [1,-1]]], ## [[1,2,5],"$L_3(4).2_2$", [[1,112,140], [1,-26,25], [1,2,-3]]], ## [[1,2,5],"$2^4:A_7$", [[1,112,140], [1,-26,25], [1,2,-3]]], ## [[1,2,5,9],"$A_8$", [[1,15,210,280], [1,-8,49,-42], [1,4,1,-6], [1,-3,-6,8]]], ## [[1,2,5,16],"$M_{11}$", [[1,165,330,792], [1,-65,100,-36], [1,19,16,-36], [1,-3,-6,8]]]]; MULTFREEINFO.("HS"):= ["$HS$", ## [[1,2,3],"$M_{22}$", [[1,22,77], [1,-8,7], [1,2,-3]]], ## [[1,7],"$U_3(5).2$", [[1,175], [1,-1]]], ## [[1,2,5,7],"$U_3(5) \\leq U_3(5).2$", [[1,1,175,175], [1,-1,35,-35], [1,-1,-5,5], [1,1,-1,-1]]], ## [[1,7],"$U_3(5).2$", [[1,175], [1,-1]]], ## [[1,2,6,7],"$U_3(5) \\leq U_3(5).2$", [[1,1,175,175], [1,-1,35,-35], [1,-1,-5,5], [1,1,-1,-1]]], ## [[1,2,3,7,13],"$L_3(4).2_1$", [[1,42,105,280,672], [1,12,-45,80,-48], [1,22,5,-20,-8], [1,-2,17,16,-32], [1,-2,-3,-4,8]]], ## [[1,3,4,7,9],"$A_8.2$", [[1,28,105,336,630], [1,-12,25,16,-30], [1,8,15,-24,0], [1,6,-5,28,-30], [1,-2,-5,-4,10]]], ## [[1,2,3,4,5,6,7,9,10],"$A_8 \\leq A_8.2$", [[1,1,28,28,105,105,336,336,1260], [1,-1,0,0,-35,35,-112,112,0], [1,1,-12,-12,25,25,16,16,-60], [1,1,8,8,15,15,-24,-24,0], [1,-1,-10,10,15,-15,-12,12,0], [1,-1,10,-10,15,-15,-12,12,0], [1,1,6,6,-5,-5,28,28,-60], [1,1,-2,-2,-5,-5,-4,-4,20], [1,-1,0,0,-5,5,8,-8,0]]], ## [[1,2,3,4,7,9,10,13,18],"$4^3:L_3(2)$", [[1,28,64,112,336,448,896,896,1344], [1,-7,-36,42,-84,168,-224,56,84], [1,13,14,32,6,28,16,-164,54], [1,13,4,22,36,-32,-64,56,-36], [1,-5,20,2,6,52,16,16,-108], [1,3,4,2,-34,-12,16,16,4], [1,-7,-6,12,6,-12,16,-4,-6], [1,5,-10,-8,6,12,16,-4,-18], [1,-2,4,-8,6,-2,-19,-4,24]]], ## [[1,2,3,5,7,10,13,16,22],"$M_{11}$", [[1,55,132,165,495,660,792,1320,1980], [1,5,52,-85,195,-140,72,-280,180], [1,-20,37,40,45,85,-138,-80,30], [1,-19,-12,-21,27,84,72,-24,-108], [1,7,4,37,63,-44,24,40,-132], [1,-4,13,-16,-3,-11,-18,56,-18], [1,12,-11,-8,13,21,-26,0,-2], [1,6,13,4,-23,9,22,-24,-8], [1,-4,-7,4,-3,-11,2,-4,22]]], ## [[1,2,3,6,7,10,13,16,22],"$M_{11}$", [[1,55,132,165,495,660,792,1320,1980], [1,5,52,-85,195,-140,72,-280,180], [1,-20,37,40,45,85,-138,-80,30], [1,-19,-12,-21,27,84,72,-24,-108], [1,7,4,37,63,-44,24,40,-132], [1,-4,13,-16,-3,-11,-18,56,-18], [1,12,-11,-8,13,21,-26,0,-2], [1,6,13,4,-23,9,22,-24,-8], [1,-4,-7,4,-3,-11,2,-4,22]]], ## [[1,3,4,7,9,13,16,17,18],"$4.2^4:S_5$", [[1,30,80,128,480,640,960,1536,1920], [1,15,20,-32,60,80,0,96,-240], [1,15,20,8,60,-20,0,-144,60], [1,-3,14,40,-48,68,36,-48,-60], [1,5,-10,8,0,20,-60,16,20], [1,7,4,0,-28,-32,16,32,0], [1,-5,10,-12,-10,10,-20,-4,30], [1,-5,0,8,20,-20,0,16,-20], [1,0,-10,-7,0,10,30,-24,0]]]]; MULTFREEINFO.("J3"):= ["$J_3$", ## [[1,4,5,6,10,11,12,13],"$L_2(16).2$", [[1,85,120,510,680,1360,1360,2040], [1,13,12,-30,-40,28-36*ER(5),28+36*ER(5),-12], [1,13,12,-30,-40,28+36*ER(5),28-36*ER(5),-12], [1,-17,-18,0,0,68,68,-102], [1,13,-6,6,32,-8,-8,-30], [1,-4-ER(17),-6+2*ER(17),-11-5*ER(17),-2+6*ER(17),-8,-8,38-2*ER(17)], [1,-4+ER(17),-6-2*ER(17),-11+5*ER(17),-2-6*ER(17),-8,-8,38+2*ER(17)], [1,-5,12,24,-4,-8,-8,-12]]]]; MULTFREEINFO.("M24"):= ["$M_{24}$", ## [[1,2],"$M_{23}$", [[1,23], [1,-1]]], ## [[1,2,7],"$M_{22}.2$", [[1,44,231], [1,20,-21], [1,-2,1]]], ## [[1,2,7,9],"$2^4:A_8$", [[1,30,280,448], [1,-15,70,-56], [1,7,4,-12], [1,-3,-6,8]]], ## [[1,7,14],"$M_{12}.2$", [[1,495,792], [1,35,-36], [1,-9,8]]], ## [[1,2,7,14,17],"$M_{12} \\leq M_{12}.2$", [[1,1,495,495,1584], [1,-1,-165,165,0], [1,1,35,35,-72], [1,1,-9,-9,16], [1,-1,3,-3,0]]], ## [[1,7,9,14],"$2^6:3.S_6$", [[1,90,240,1440], [1,21,10,-32], [1,-9,20,-12], [1,-1,-12,12]]], ## [[1,7,9,14,18],"$2^6:3.A_6 \\leq 2^6:3.S_6$", [[1,1,180,480,2880], [1,1,42,20,-64], [1,1,-18,40,-24], [1,1,-2,-24,24], [1,-1,0,0,0]]], ## [[1,2,7,9,17],"$L_3(4).3.2_2$", [[1,63,210,630,1120], [1,39,-30,150,-160], [1,17,3,-37,16], [1,-3,23,3,-24], [1,-3,-9,3,8]]], ## [[1,2,7,8,9,17,18],"$L_3(4).3 \\leq L_3(4).3.2_2$", [[1,1,63,63,420,1260,2240], [1,1,39,39,-60,300,-320], [1,1,17,17,6,-74,32], [1,-1,-21,21,0,0,0], [1,1,-3,-3,46,6,-48], [1,1,-3,-3,-18,6,16], [1,-1,3,-3,0,0,0]]], ## [[1,7,9,14,19],"$2^6:(L_3(2) \\times S_3)$", [[1,42,56,1008,2688], [1,19,10,42,-72], [1,9,-10,-48,48], [1,-3,10,-24,16], [1,-3,-4,18,-12]]], ## [[1,7,8,9,14,17,19,20], "$2^6:(L_3(2) \\times 3) \\leq 2^6:(L_3(2) \\times S_3)$", [[1,1,56,56,84,2016,2688,2688], [1,1,10,10,38,84,-72,-72], [1,-1,-14,14,0,0,-168,168], [1,1,-10,-10,18,-96,48,48], [1,1,10,10,-6,-48,16,16], [1,-1,10,-10,0,0,-24,24], [1,1,-4,-4,-6,36,-12,-12], [1,-1,-4,4,0,0,32,-32]]], ## [[1,7,9,14,17,18,19,20,23,24,26], "$2^6:(7:3 \\times S_3) \\leq 2^6:(L_3(2) \\times S_3)$", [[1,7,168,168,224,224,2688,2688,8064,8064,8064], [1,7,76,76,40,40,-72,-72,-216,-216,336], [1,7,36,36,-40,-40,48,48,144,144,-384], [1,7,-12,-12,40,40,16,16,48,48,-192], [1,-1,12,-12,-8,8,0,0,-288,288,0], [1,-1,-24,24,-32,32,0,0,0,0,0], [1,7,-12,-12,-16,-16,-12,-12,-36,-36,144], [1,-1,12,-12,-8,8,112,-112,48,-48,0], [1,-1,12,-12,-8,8,-48,48,48,-48,0], [1,-1,-4,4,8,-8,60,60,-60,-60,0], [1,-1,-4,4,8,-8,-32,-32,32,32,0]]]]; MULTFREEINFO.("McL"):= ["$McL$", ## [[1,2,4],"$U_4(3)$", [[1,112,162], [1,-28,27], [1,2,-3]]], ## [[1,2,4,9],"$M_{22}$", [[1,330,462,1232], [1,-120,147,-28], [1,30,27,-58], [1,-3,-6,8]]], ## [[1,2,4,9],"$M_{22}$", [[1,330,462,1232], [1,-120,147,-28], [1,30,27,-58], [1,-3,-6,8]]], ## [[1,2,4,9,14],"$U_3(5)$", [[1,252,750,2625,3500], [1,-126,300,-525,350], [1,54,90,-15,-130], [1,-18,12,51,-46], [1,4,-10,-15,20]]], ## [[1,4,12,14,15],"$3^{1+4}:2S_5$", [[1,90,1215,2430,11664], [1,35,225,-45,-216], [1,-1,-3,-69,72], [1,10,-25,30,-16], [1,-10,15,30,-36]]], ## [[1,4,9,14,15,20],"$2.A_8$", [[1,210,2240,5040,6720,8064], [1,45,260,90,-540,144], [1,39,-28,72,60,-144], [1,-5,60,-60,60,-56], [1,-15,-10,90,-30,-36], [1,3,-28,-36,-12,72]]]]; MULTFREEINFO.("He"):= ["$He$", ## [[1,2,3,6,9],"$S_4(4).2$", [[1,136,136,425,1360], [1,-18-14*ER(-7),-18+14*ER(-7),75,-40], [1,-18+14*ER(-7),-18-14*ER(-7),75,-40], [1,10,10,5,-26], [1,-4,-4,-9,16]]], ## [[1,2,3,6,7,8,9],"$S_4(4) \\leq S_4(4).2$", [[1,1,272,272,425,425,2720], [1,1,-36-28*ER(-7),-36+28*ER(-7),75,75,-80], [1,1,-36+28*ER(-7),-36-28*ER(-7),75,75,-80], [1,1,20,20,5,5,-52], [1,-1,0,0,-5*ER(17),5*ER(17),0], [1,-1,0,0,5*ER(17),-5*ER(17),0], [1,1,-8,-8,-9,-9,32]]], ## [[1,2,3,6,9,12,14],"$2^2.L_3(4).S_3$", [[1,105,720,840,840,1344,4480], [1,35,-120,-70-70*ER(-7),-70+70*ER(-7),224,0], [1,35,-120,-70+70*ER(-7),-70-70*ER(-7),224,0], [1,21,6,42,42,0,-112], [1,7,48,-28,-28,0,0], [1,-14,6,7,7,35,-42], [1,0,-15,0,0,-21,35]]], ## [[1,2,3,6,7,8,9,12,14,15],"$2^2.L_3(4).3 \\leq 2^2.L_3(4).S_3$", [[1,1,105,105,1344,1344,1440,1680,1680,8960], [1,1,35,35,224,224,-240,-140-140*ER(-7),-140+140*ER(-7),0], [1,1,35,35,224,224,-240,-140+140*ER(-7),-140-140*ER(-7),0], [1,1,21,21,0,0,12,84,84,-224], [1,-1,-5*ER(17),5*ER(17),-64,64,0,0,0,0], [1,-1,5*ER(17),-5*ER(17),-64,64,0,0,0,0], [1,1,7,7,0,0,96,-56,-56,0], [1,1,-14,-14,35,35,12,14,14,-84], [1,1,0,0,-21,-21,-30,0,0,70], [1,-1,0,0,21,-21,0,0,0,0]]]]; MULTFREEINFO.("Ru"):= ["$Ru$", ## [[1,5,6],"${^2F_4(2)^{\\prime}}.2$", [[1,1755,2304], [1,-65,64], [1,15,-16]]], ## [[1,4,5,6,7],"${^2F_4(2)^{\\prime}} \\leq {^2F_4(2)^{\\prime}}.2$", [[1,1,2304,2304,3510], [1,-1,144,-144,0], [1,1,64,64,-130], [1,1,-16,-16,30], [1,-1,-16,16,0]]], ## [[1,6,8,14,15,16,21,23,25,32],"$(2^2 \\times Sz(8)):3$", [[1,455,3640,5824,29120,29120,58240,87360,87360,116480], [1,-9,392,-208,816,352,-224,-1728,1056,-448], [1,39,-104,64,192,256,-512,192,384,-512], [1,-9,72,192,256,-288,-64,192,-224,-128], [1,23-2*ER(6),16-14*ER(6),-92+24*ER(6),68+28*ER(6), -64+76*ER(6),208+148*ER(6),216-84*ER(6),-192-72*ER(6),-184-104*ER(6)], [1,23+2*ER(6),16+14*ER(6),-92-24*ER(6),68-28*ER(6), -64-76*ER(6),208-148*ER(6),216+84*ER(6),-192+72*ER(6),-184+104*ER(6)], [1,-25,-40,-16,-80,-160,160,0,480,-320], [1,23,40,64,-256,32,64,-192,96,128], [1,-1,-64,-16,112,-64,-32,-288,-96,448], [1,-22,20,-16,-56,167,-116,153,-159,28]]]]; MULTFREEINFO.("Suz"):= ["$Suz$", ## [[1,4,5],"$G_2(4)$", [[1,416,1365], [1,20,-21], [1,-16,15]]], ## [[1,3,4,9,15],"$3.U_4(3):2$", [[1,280,486,8505,13608], [1,80,-54,405,-432], [1,-28,90,189,-252], [1,20,18,-75,36], [1,-8,-10,9,8]]], ## [[1,2,3,4,9,10,11,15],"$3.U_4(3) \\leq 3.U_4(3):2$", [[1,1,486,486,560,8505,8505,27216], [1,-1,162,-162,0,-945,945,0], [1,1,-54,-54,160,405,405,-864], [1,1,90,90,-56,189,189,-504], [1,1,18,18,40,-75,-75,72], [1,-1,18,-18,0,63,-63,0], [1,-1,-18,18,0,-45,45,0], [1,1,-10,-10,-16,9,9,16]]], ## [[1,2,3,9,11,12],"$U_5(2)$", [[1,891,1980,2816,6336,20736], [1,243,-180,512,-576,0], [1,-99,330,176,-264,-144], [1,33,30,8,96,-168], [1,-27,-30,32,24,0], [1,9,6,-40,-48,72]]], ## [[1,2,4,6,9,12,16,17,27],"$2^{1+6}_-.U_4(2)$", [[1,54,360,1728,5120,9216,17280,46080,55296], [1,-27,90,432,800,-1152,-2160,-1440,3456], [1,21,30,276,-160,768,120,-1440,384], [1,3,-60,132,200,48,-60,360,-624], [1,15,48,12,128,-144,276,-96,-240], [1,-9,24,-36,80,144,-108,48,-144], [1,-11,10,48,-80,-64,80,80,-64], [1,9,6,0,-64,0,-144,192,0], [1,0,-12,-18,8,-9,36,-96,90]]], ## [[1,3,4,5,9,11,12,15,17,27,28,30,33],"$2^{4+6}:3A_6$", [[1,60,480,1536,1920,6144,6144,20480,23040,23040,46080,92160,184320], [1,-15,30,336,120,1104,-96,-1120,-2160,-360,2880,-5040,4320], [1,27,84,216,336,-192,864,1472,72,864,-1440,-2880,576], [1,15,-60,-156,300,96,1104,-1120,-180,-720,720,1440,-1440], [1,21,90,-24,48,216,-96,-112,-360,576,-96,744,-1008], [1,-15,30,-96,120,96,-96,320,0,-360,0,0,0], [1,-3,18,24,-96,264,96,-16,360,-144,-288,-72,-144], [1,7,-36,116,76,48,-96,112,172,-96,240,240,-784], [1,15,36,-12,12,-96,-48,-160,252,-144,144,-288,288], [1,6,0,6,-42,-24,24,128,-180,-144,9,144,72], [1,-9,12,36,12,-48,0,-112,-36,0,-144,144,144], [1,-5,0,-16,-20,-24,24,40,40,120,240,-120,-280], [1,3,-24,-24,12,24,-24,-40,0,72,-144,-72,216]]]]; MULTFREEINFO.("ON"):= ["$ON$", ## [[1,2,7,8,11],"$L_3(7).2$", [[1,5586,6384,52136,58653], [1,196,-106,161,-252], [1,-21,111,161,-252], [1,42,48,-280,189], [1,-56,-64,56,63]]], ## [[1,2,7,8,10,11,18],"$L_3(7) \\leq L_3(7).2$", [[1,1,11172,12768,52136,52136,117306], [1,1,392,-212,161,161,-504], [1,1,-42,222,161,161,-504], [1,1,84,96,-280,-280,378], [1,-1,0,0,-343,343,0], [1,1,-112,-128,56,56,126], [1,-1,0,0,152,-152,0]]], ## [[1,2,7,9,11],"$L_3(7).2$", [[1,5586,6384,52136,58653], [1,196,-106,161,-252], [1,-21,111,161,-252], [1,42,48,-280,189], [1,-56,-64,56,63]]], ## [[1,2,7,9,10,11,18],"$L_3(7) \\leq L_3(7).2$", [[1,1,11172,12768,52136,52136,117306], [1,1,392,-212,161,161,-504], [1,1,-42,222,161,161,-504], [1,1,84,96,-280,-280,378], [1,-1,0,0,-343,343,0], [1,1,-112,-128,56,56,126], [1,-1,0,0,152,-152,0]]]]; MULTFREEINFO.("Co3"):= ["$Co_3$", ## [[1,5],"$McL.2$", [[1,275], [1,-1]]], ## [[1,2,4,5],"$McL \\leq McL.2$", [[1,1,275,275], [1,-1,55,-55], [1,-1,-5,5], [1,1,-1,-1]]], ## [[1,2,5,9,15],"$HS$", [[1,352,1100,4125,5600], [1,-176,440,-825,560], [1,76,134,-15,-196], [1,-26,20,75,-70], [1,4,-10,-15,20]]], ## [[1,2,4,5,9,13,15,24],"$M_{23}$", [[1,253,506,1771,7590,8855,14168,15456], [1,55,-286,847,-330,-2695,3080,-672], [1,-85,-86,147,870,105,-280,-672], [1,1,146,343,-330,455,56,-672], [1,10,-61,97,-105,80,-250,228], [1,-22,31,21,15,-120,-82,156], [1,28,11,1,75,-40,-52,-24], [1,-4,-5,-15,-21,24,44,-24]]], ## [[1,5,13,15,20,31],"$3^5:(2 \\times M_{11})$", [[1,495,2673,32076,40095,53460], [1,35,741,2268,-1305,-1740], [1,-55,123,-324,495,-240], [1,35,93,-324,-225,420], [1,35,-15,0,207,-228], [1,-9,-15,44,-57,36]]], ## [[1,2,4,5,9,13,15,20,22,24,28,31],"$3^5:M_{11} \\leq 3^5:(2 \\times M_{11})$", [[1,1,495,495,2673,2673,32076,32076,40095,40095,53460,53460], [1,-1,165,-165,1485,-1485,10692,-10692,4455,-4455,5940,-5940], [1,-1,-135,135,345,-345,-108,108,2655,-2655,-3660,3660], [1,1,35,35,741,741,2268,2268,-1305,-1305,-1740,-1740], [1,-1,15,-15,315,-315,-108,108,-945,945,-360,360], [1,1,-55,-55,123,123,-324,-324,495,495,-240,-240], [1,1,35,35,93,93,-324,-324,-225,-225,420,420], [1,1,35,35,-15,-15,0,0,207,207,-228,-228], [1,-1,33,-33,9,-9,-108,108,135,-135,36,-36], [1,-1,-27,27,21,-21,-108,108,63,-63,228,-228], [1,-1,-3,3,-27,27,108,-108,-81,81,-108,108], [1,1,-9,-9,-15,-15,44,44,-57,-57,36,36]]], ## [[1,5,14,15,20,27,29],"$2.S_6(2)$", [[1,630,1920,8960,30240,48384,80640], [1,147,-288,1232,1260,2016,-4368], [1,-45,120,-40,540,-216,-360], [1,75,0,80,180,-576,240], [1,3,72,116,-252,72,-12], [1,15,0,-100,0,144,-60], [1,-18,-33,32,0,-54,72]]]]; MULTFREEINFO.("Co2"):= ["$Co_2$", ## [[1,4,6],"$U_6(2).2$", [[1,891,1408], [1,63,-64], [1,-9,8]]], ## [[1,2,4,6,7],"$U_6(2) \\leq U_6(2).2$", [[1,1,891,891,2816], [1,-1,297,-297,0], [1,1,63,63,-128], [1,1,-9,-9,16], [1,-1,-3,3,0]]], ## [[1,4,5,6,8,15,17,28,36,39,44],"$U_5(2).2 \\leq U_6(2).2$", [[1,176,495,5346,8448,8910,14256,142560,253440,427680,684288], [1,176,495,378,-384,630,1008,10080,-11520,30240,-31104], [1,-16,15,1026,-768,270,-1296,4320,7680,-4320,-6912], [1,176,495,-54,48,-90,-144,-1440,1440,-4320,3888], [1,-16,15,378,384,-810,432,4320,-3840,-4320,3456], [1,8,-9,378,288,126,504,-1008,576,0,-864], [1,8,-9,-54,-288,414,360,576,-576,-1296,864], [1,-16,15,-54,-48,-90,144,0,480,0,-432], [1,-16,15,26,32,70,-96,-80,-320,80,288], [1,8,-9,-54,96,30,-24,192,192,-144,-288], [1,8,-9,26,-64,-50,-24,-128,-128,176,192]]], ## [[1,2,3,4,5,6,7,8,15,16,17,19,21,28,35,36,39,42,44,48], "$U_5(2) \\leq U_6(2).2$", [[1,1,176,176,495,495,5346,5346,8910,8910,14256,14256,16896,142560, 142560,253440,253440,427680,427680,1368576], [1,-1,176,-176,495,-495,1782,-1782,-2970,2970,4752,-4752,0,-47520,47520, 0,0,142560,-142560,0], [1,-1,-16,16,15,-15,1782,-1782,1350,-1350,-432,432,0,-12960,12960, -30720,30720,-12960,12960,0], [1,1,176,176,495,495,378,378,630,630,1008,1008,-768,10080,10080, -11520,-11520,30240,30240,-62208], [1,1,-16,-16,15,15,1026,1026,270,270,-1296,-1296,-1536,4320,4320, 7680,7680,-4320,-4320,-13824], [1,1,176,176,495,495,-54,-54,-90,-90,-144,-144,96,-1440,-1440,1440,1440, -4320,-4320,7776], [1,-1,176,-176,495,-495,-18,18,30,-30,-48,48,0,480,-480,0,0,-1440,1440,0], [1,1,-16,-16,15,15,378,378,-810,-810,432,432,768,4320,4320,-3840,-3840, -4320,-4320,6912], [1,1,8,8,-9,-9,378,378,126,126,504,504,576,-1008,-1008,576,576,0,0,-1728], [1,-1,-16,16,15,-15,270,-270,-162,162,-432,432,0,-864,864,1536,-1536, -864,864,0], [1,1,8,8,-9,-9,-54,-54,414,414,360,360,-576,576,576,-576,-576, -1296,-1296,1728], [1,-1,-16,16,15,-15,-18,18,350,-350,368,-368,0,-960,960,1280,-1280, -960,960,0], [1,-1,8,-8,-9,9,270,-270,54,-54,216,-216,0,864,-864,0,0,432,-432,0], [1,1,-16,-16,15,15,-54,-54,-90,-90,144,144,-96,0,0,480,480,0,0,-864], [1,-1,8,-8,-9,9,-18,18,-138,138,120,-120,0,-192,192,0,0,-432,432,0], [1,1,-16,-16,15,15,26,26,70,70,-96,-96,64,-80,-80,-320,-320,80,80,576], [1,1,8,8,-9,-9,-54,-54,30,30,-24,-24,192,192,192,192,192,-144,-144,-576], [1,-1,-16,16,15,-15,-18,18,-18,18,0,0,0,144,-144,-192,192,144,-144,0], [1,1,8,8,-9,-9,26,26,-50,-50,-24,-24,-128,-128,-128,-128,-128,176,176,384], [1,-1,8,-8,-9,9,-18,18,54,-54,-72,72,0,0,0,0,0,144,-144,0]]], ## [[1,4,6,14,17],"$2^{10}:M_{22}:2$", [[1,462,2464,21120,22528], [1,-21,532,-960,448], [1,87,64,120,-272], [1,-21,28,120,-128], [1,3,-20,-48,64]]], ## [[1,2,4,6,7,14,17,20],"$2^{10}:M_{22} \\leq 2^{10}:M_{22}:2$", [[1,1,924,2464,2464,22528,22528,42240], [1,-1,0,1232,-1232,-5632,5632,0], [1,1,-42,532,532,448,448,-1920], [1,1,174,64,64,-272,-272,240], [1,-1,0,182,-182,368,-368,0], [1,1,-42,28,28,-128,-128,240], [1,1,6,-20,-20,64,64,-96], [1,-1,0,-10,10,-16,16,0]]], ## [[1,2,4,7,14,18],"$McL$", [[1,275,2025,7128,15400,22275], [1,-165,945,-2376,3080,-1485], [1,91,369,504,-56,-909], [1,-45,105,24,-280,195], [1,19,9,-72,-56,99], [1,-5,-15,24,40,-45]]], ## [[1,4,6,15,17],"$2^{1+8}:S_6(2)$", [[1,1008,1260,14336,40320], [1,-234,225,1088,-1080], [1,108,135,-64,-180], [1,18,-27,80,-72], [1,-18,9,-64,72]]], ## [[1,4,6,14,17,27,33],"$HS.2$", [[1,3850,4125,44352,61600,132000,231000], [1,-14,1365,-2016,7504,-6000,-840], [1,490,285,-2016,-560,2640,-840], [1,-86,285,576,-416,480,-840], [1,178,21,288,-176,-624,312], [1,10,-35,64,160,160,-360], [1,-30,5,-96,-80,-80,280]]], ## [[1,2,4,6,7,14,17,18,20,27,33,38],"$HS \\leq HS.2$", [[1,1,3850,3850,4125,4125,61600,61600,88704,231000,231000,264000], [1,-1,770,-770,2475,-2475,24640,-24640,0,46200,-46200,0], [1,1,-14,-14,1365,1365,7504,7504,-4032,-840,-840,-12000], [1,1,490,490,285,285,-560,-560,-4032,-840,-840,5280], [1,-1,-70,70,675,-675,1120,-1120,0,-4200,4200,0], [1,1,-86,-86,285,285,-416,-416,1152,-840,-840,960], [1,1,178,178,21,21,-176,-176,576,312,312,-1248], [1,-1,-190,190,75,-75,-320,320,0,600,-600,0], [1,-1,122,-122,99,-99,-416,416,0,408,-408,0], [1,1,10,10,-35,-35,160,160,128,-360,-360,320], [1,1,-30,-30,5,5,-80,-80,-192,280,280,-160], [1,-1,2,-2,-21,21,64,-64,0,-72,72,0]]]]; MULTFREEINFO.("Fi22"):= ["$Fi_{22}$", ## [[1,3,7],"$2.U_6(2)$", [[1,693,2816], [1,63,-64], [1,-9,8]]], ## [[1,3,9],"$O_7(3)$", [[1,3159,10920], [1,279,-280], [1,-9,8]]], ## [[1,3,9],"$O_7(3)$", [[1,3159,10920], [1,279,-280], [1,-9,8]]], ## [[1,7,9,13],"$O_8^+(2):S_3$", [[1,1575,22400,37800], [1,171,-64,-108], [1,-9,224,-216], [1,-9,-64,72]]], ## [[1,4,7,8,9,13,15],"$O_8^+(2):3 \\leq O_8^+(2):S_3$", [[1,1,1575,1575,22400,22400,75600], [1,-1,225,-225,800,-800,0], [1,1,171,171,-64,-64,-216], [1,-1,-63,63,224,-224,0], [1,1,-9,-9,224,224,-432], [1,1,-9,-9,-64,-64,144], [1,-1,9,-9,-64,64,0]]], ## [[1,3,7,9,13,14,17],"$O_8^+(2):2 \\leq O_8^+(2):S_3$", [[1,2,1575,3150,22400,44800,113400], [1,-1,315,-315,2240,-2240,0], [1,2,171,342,-64,-128,-324], [1,2,-9,-18,224,448,-648], [1,2,-9,-18,-64,-128,216], [1,-1,-45,45,80,-80,0], [1,-1,27,-27,-64,64,0]]], ## [[1,2,3,5,7,10,11,17],"$2^{10}:M_{22}$", [[1,154,1024,3696,4928,11264,42240,78848], [1,-77,-320,924,1232,-1408,-5280,4928], [1,49,-176,546,-532,1184,-960,-112], [1,-35,160,294,-364,-400,120,224], [1,37,88,186,248,32,120,-712], [1,13,-32,6,-28,-112,120,32], [1,-17,-20,24,32,32,120,-172], [1,1,16,-30,-4,32,-96,80]]], ## [[1,3,5,7,9,10,13,17,25,28],"$2^6:S_6(2)$", [[1,135,1260,2304,8640,10080,45360,143360,241920,241920], [1,-15,210,624,-960,1680,1260,8960,-1680,-10080], [1,-27,126,-288,216,1008,-2268,-1792,6048,-3024], [1,57,246,120,840,96,1368,-1408,120,-1440], [1,3,-60,192,192,312,-576,-256,-960,1152], [1,21,30,-96,-168,240,180,-256,-672,720], [1,27,36,0,0,-144,-432,512,0,0], [1,-15,66,48,-96,-48,-36,-256,48,288], [1,-9,0,-36,72,0,0,224,-252,0], [1,3,-24,12,-24,-12,72,-112,228,-144]]], ## [[1,4,5,9,10,26,31,32,39,45,53],"${^2F_4(2)^{\\prime}}$", [[1,1755,11700,14976,83200,83200,140400,187200,374400,449280,2246400], [1,-405,900,-576,-3200,-3200,-10800,14400,-14400,17280,0], [1,-189,1980,-576,5440,5440,4320,-7200,-14400,13824,-8640], [1,171,612,-864,832,832,1008,3456,3744,-576,-9216], [1,99,540,576,-320,-320,-1440,-1440,2880,2304,-2880], [1,75,-60,192,-128,-128,624,384,-576,384,-768], [1,27,36,-144,-608,256,0,-288,-144,0,864], [1,27,36,-144,256,-608,0,-288,-144,0,864], [1,-21,132,96,64,64,-48,192,-288,-960,768], [1,27,-108,0,256,256,-432,0,0,0,0], [1,-45,-36,0,-32,-32,144,0,288,288,-576]]]]; MULTFREEINFO.("HN"):= ["$HN$", ## [[1,2,3,4,5,8,10,11,12,18,20,23],"$A_{12}$", [[1,462,2520,2520,10395,16632,30800,69300,166320,166320,311850,362880], [1,-198,360*ER(5),-360*ER(5),2475,792,4400,-9900,-7920*ER(5),7920*ER(5), -14850,17280], [1,-198,-360*ER(5),360*ER(5),2475,792,4400,-9900,7920*ER(5),-7920*ER(5), -14850,17280], [1,132,-540,-540,1485,-1188,1100,4950,-5940,-5940,0,6480], [1,12,120,120,495,1332,-2200,300,-1080,-1080,-900,2880], [1,82,240,240,515,-88,400,900,640,640,-1650,-1920], [1,62,-120,-120,155,632,400,-300,80,80,1050,-1920], [1,-48,60*ER(5),-60*ER(5),225,-108,-100,-150,180*ER(5),-180*ER(5),900,-720], [1,-48,-60*ER(5),60*ER(5),225,-108,-100,-150,-180*ER(5),180*ER(5),900,-720], [1,12,30,30,-45,-18,50,-150,-270,-270,450,180], [1,-18,0,0,-45,72,80,180,0,0,-270,0], [1,12,-20,-20,5,-68,-100,-50,180,180,-200,80]]], [[1,2,3,4,5,8,9,10,11,12,18,20,23,24,32,39,40,41,47],"$A_{11} \\leq A_{12}$", [[1,11,2772,2772,16632,20790,30240,30240,83160,99792,103950,362880,369600, 831600,1247400,1995840,1995840,2494800,3991680], [1,11,-1188,-1188,792,4950,-4320*ER(5),4320*ER(5),3960,4752,24750,17280, 52800,-118800,-59400,95040*ER(5),-95040*ER(5),-118800,190080], [1,11,-1188,-1188,792,4950,4320*ER(5),-4320*ER(5),3960,4752,24750,17280, 52800,-118800,-59400,-95040*ER(5),95040*ER(5),-118800,190080], [1,11,792,792,-1188,2970,-6480,-6480,-5940,-7128,14850,6480,13200,59400,0, --> -------------------- --> maximum size reached --> -------------------- [ Verzeichnis aufwärts0.59unsichere Verbindung Übersetzung europäischer Sprachen durch Browser ] |
2026-03-28