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Spracherkennung für: .tst vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen] # This file was created automatically, do not edit!
############################################################################# ## #W ctblpope.tst GAP 4 package CTblLib Thomas Breuer ## ## This file contains the GAP code of examples in the package ## documentation files. ## ## In order to run the tests, one starts GAP from the 'tst' subdirectory ## of the 'pkg/ctbllib' directory, and calls 'Test( "ctblpope.tst" );'. ## gap> LoadPackage( "CTblLib", false ); true gap> save:= SizeScreen();; gap> SizeScreen( [ 72 ] );; gap> START_TEST( "ctblpope.tst" ); ## gap> if IsBound( BrowseData ) then > data:= BrowseData.defaults.dynamic.replayDefaults; > oldinterval:= data.replayInterval; > data.replayInterval:= 1; > fi; ## doc2/ctblpope.xml (76-79) gap> LoadPackage( "ctbllib", "1.2", false ); true ## doc2/ctblpope.xml (91-98) gap> g:= MathieuGroup( 24 );; gap> SetName( g, "m24" ); gap> Size( g ); IsSimple( g ); NrMovedPoints( g ); 244823040 true 24 ## doc2/ctblpope.xml (109-117) gap> ccl:= AttributeValueNotSet( ConjugacyClasses, g );; gap> HasConjugacyClasses( g ); false gap> invariants:= List( ccl, c -> [ Order( Representative( c ) ), > Size( c ), Size( ConjugacyClass( g, Representative( c )^2 ) ) ] );; gap> SortParallel( invariants, ccl ); gap> SetConjugacyClasses( g, ccl ); ## doc2/ctblpope.xml (127-162) gap> NrConjugacyClasses( g ); 26 gap> pi:= NaturalCharacter( g ); Character( CharacterTable( m24 ), [ 24, 8, 0, 6, 0, 0, 4, 0, 4, 2, 0, 3, 3, 2, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1 ] ) gap> IsTransitive( pi ); Transitivity( pi ); true 5 gap> SetIdentifier( CharacterTable( g ), "M24table" ); gap> Display( pi ); M24table 2 10 10 9 3 3 7 7 5 2 3 3 1 1 4 2 . 2 2 1 3 3 1 1 3 2 1 . 1 1 1 1 1 1 . . . 1 1 . 5 1 . 1 1 . . . . 1 . . . . . 1 . . . . 7 1 1 . . 1 . . . . . . 1 1 . . . . . 1 11 1 . . . . . . . . . . . . . . 1 . . . 23 1 . . . . . . . . . . . . . . . . . . 1a 2a 2b 3a 3b 4a 4b 4c 5a 6a 6b 7a 7b 8a 10a 11a 12a 12b 14a Y.1 24 8 . 6 . . 4 . 4 2 . 3 3 2 . 2 . . 1 2 1 . . . . . . 3 . 1 1 1 1 . . 5 . 1 1 . . . . 7 1 . . 1 1 . . 11 . . . . . . . 23 . . . . . 1 1 14b 15a 15b 21a 21b 23a 23b Y.1 1 1 1 . . 1 1 ## doc2/ctblpope.xml (179-192) gap> piop:= pi * pi; Character( CharacterTable( m24 ), [ 576, 64, 0, 36, 0, 0, 16, 0, 16, 4, 0, 9, 9, 4, 0, 4, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1 ] ) gap> IsTransitive( piop ); false gap> piup:= SymmetricParts( UnderlyingCharacterTable(pi), [ pi ], 2 )[1]; Character( CharacterTable( m24 ), [ 300, 44, 12, 21, 0, 4, 12, 0, 10, 5, 0, 6, 6, 4, 2, 3, 0, 1, 2, 2, 1, 1, 0, 0, 1, 1 ] ) gap> IsTransitive( piup ); false ## doc2/ctblpope.xml (203-214) gap> ScalarProduct( piup, TrivialCharacter( g ) ); 2 gap> comb:= Combinations( [ 1 .. 24 ], 2 );; gap> hom:= ActionHomomorphism( g, comb, OnSets );; gap> pihom:= NaturalCharacter( hom ); Character( CharacterTable( m24 ), [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0 ] ) gap> Transitivity( pihom ); 1 ## doc2/ctblpope.xml (227-237) gap> pi2s:= piup - pi; VirtualCharacter( CharacterTable( m24 ), [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0 ] ) gap> pi2s = pihom; true gap> HasIrr( g ); HasIrr( CharacterTable( g ) ); false false ## doc2/ctblpope.xml (253-256) gap> IsPrimitive( g, comb, OnSets ); true ## doc2/ctblpope.xml (274-286) gap> tbl:= CharacterTable( "M24" ); CharacterTable( "M24" ) gap> maxes:= Maxes( tbl ); [ "M23", "M22.2", "2^4:a8", "M12.2", "2^6:3.s6", "L3(4).3.2_2", "2^6:(psl(3,2)xs3)", "L2(23)", "L3(2)" ] gap> s:= CharacterTable( maxes[2] ); CharacterTable( "M22.2" ) gap> TrivialCharacter( s )^tbl; Character( CharacterTable( "M24" ), [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0 ] ) ## doc2/ctblpope.xml (312-357) gap> m11:= CharacterTable( "M11" );; gap> SetName( m11, "m11" ); gap> perms:= PermChars( m11 ); [ Character( m11, [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( m11, [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ), Character( m11, [ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ), Character( m11, [ 22, 6, 4, 2, 2, 0, 0, 0, 0, 0 ] ), Character( m11, [ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ), Character( m11, [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ), Character( m11, [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ), Character( m11, [ 110, 6, 2, 6, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 110, 14, 2, 2, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 132, 12, 6, 0, 2, 0, 0, 0, 0, 0 ] ), Character( m11, [ 144, 0, 0, 0, 4, 0, 0, 0, 1, 1 ] ), Character( m11, [ 165, 13, 3, 1, 0, 1, 1, 1, 0, 0 ] ), Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 330, 2, 6, 2, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 330, 18, 6, 2, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 396, 12, 0, 4, 1, 0, 0, 0, 0, 0 ] ), Character( m11, [ 440, 8, 8, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 440, 24, 8, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 495, 15, 0, 3, 0, 0, 1, 1, 0, 0 ] ), Character( m11, [ 660, 4, 3, 4, 0, 1, 0, 0, 0, 0 ] ), Character( m11, [ 660, 12, 3, 0, 0, 3, 0, 0, 0, 0 ] ), Character( m11, [ 660, 12, 12, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 660, 28, 3, 0, 0, 1, 0, 0, 0, 0 ] ), Character( m11, [ 720, 0, 0, 0, 0, 0, 0, 0, 5, 5 ] ), Character( m11, [ 792, 24, 0, 0, 2, 0, 0, 0, 0, 0 ] ), Character( m11, [ 880, 0, 16, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 990, 6, 0, 2, 0, 0, 2, 2, 0, 0 ] ), Character( m11, [ 990, 6, 0, 6, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 990, 30, 0, 2, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 1320, 8, 6, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 1320, 24, 6, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 1584, 0, 0, 0, 4, 0, 0, 0, 0, 0 ] ), Character( m11, [ 1980, 12, 0, 4, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 1980, 36, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 2640, 0, 12, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 3960, 24, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 7920, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> Length( perms ); 39 ## doc2/ctblpope.xml (374-382) gap> degrees:= DivisorsInt( Size( m11 ) );; gap> perms2:= [];; gap> for d in degrees do > Append( perms2, PermChars( m11, d ) ); > od; gap> Set( perms ) = Set( perms2 ); true ## doc2/ctblpope.xml (391-398) gap> perms3:= [];; gap> for d in degrees do > Append( perms3, PermChars( m11, rec( torso:= [ d ] ) ) ); > od; gap> Set( perms ) = Set( perms3 ); true ## doc2/ctblpope.xml (412-421) gap> newperms:= TestPerm4( m11, perms );; gap> newperms = perms; true gap> newperms:= TestPerm5( m11, perms, m11 mod 11 );; gap> newperms = perms; false gap> Difference( perms, newperms ); [ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (433-444) gap> tom:= TableOfMarks( "M11" ); TableOfMarks( "M11" ) gap> trueperms:= PermCharsTom( m11, tom );; gap> Length( trueperms ); Length( Set( trueperms ) ); 39 36 gap> Difference( perms, trueperms ); [ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), Character( m11, [ 660, 4, 3, 4, 0, 1, 0, 0, 0, 0 ] ), Character( m11, [ 660, 12, 3, 0, 0, 3, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (466-478) gap> u62:= CharacterTable( "U6(2)" );; gap> m22:= CharacterTable( "M22" );; gap> fus:= PossibleClassFusions( m22, u62 ); [ [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 33, 34 ], [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 34, 33 ], [ 1, 3, 7, 11, 14, 15, 22, 24, 24, 27, 33, 34 ], [ 1, 3, 7, 11, 14, 15, 22, 24, 24, 27, 34, 33 ], [ 1, 3, 7, 12, 14, 15, 22, 24, 24, 28, 33, 34 ], [ 1, 3, 7, 12, 14, 15, 22, 24, 24, 28, 34, 33 ] ] gap> RepresentativesFusions( m22, fus, u62 ); [ [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 33, 34 ] ] ## doc2/ctblpope.xml (487-512) gap> cand:= Set( fus, > x -> Induced( m22, u62, [ TrivialCharacter( m22 ) ], x )[1] ); [ Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 0, 48, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( u62, cand ).ATLAS; [ "1a+22a+252a+616a+1155c+1386a+8064a+9240c", "1a+22a+252a+616a+1155b+1386a+8064a+9240b", "1a+22a+252a+616a+1155a+1386a+8064a+9240a" ] gap> aut:= AutomorphismsOfTable( u62 );; Size( aut ); 24 gap> elms:= Filtered( Elements( aut ), x -> Order( x ) = 3 ); [ (10,11,12)(26,27,28)(40,41,42), (10,12,11)(26,28,27)(40,42,41) ] gap> Position( cand, Permuted( cand[1], elms[1] ) ); 3 gap> Position( cand, Permuted( cand[3], elms[1] ) ); 2 ## doc2/ctblpope.xml (534-548) gap> u622:= CharacterTable( "U6(2).2" );; gap> m222:= CharacterTable( "M22.2" );; gap> fus:= PossibleClassFusions( m222, u622 ); [ [ 1, 3, 7, 10, 13, 14, 20, 22, 22, 24, 29, 38, 39, 42, 41, 46, 50, 53, 58, 59, 59 ] ] gap> cand:= Induced( m222, u622, [ TrivialCharacter( m222 ) ], fus[1] ); [ Character( CharacterTable( "U6(2).2" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1080, 72, 0, 48, 8, 0, 0, 0, 18, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( u622, cand ).ATLAS; [ "1a+22a+252a+616a+1155a+1386a+8064a+9240a" ] ## doc2/ctblpope.xml (673-747) gap> o8p3:= CharacterTable("O8+(3)");; gap> Size( o8p3 ) / (2^9*168); 57572775 gap> o8p2:= CharacterTable( "O8+(2)" );; gap> fus:= PossibleClassFusions( o8p2, o8p3 );; gap> Length( fus ); 24 gap> rep:= RepresentativesFusions( o8p2, fus, o8p3 ); [ [ 1, 5, 2, 3, 4, 5, 7, 8, 12, 16, 17, 19, 23, 20, 21, 22, 23, 24, 25, 26, 37, 38, 42, 31, 32, 36, 49, 52, 51, 50, 43, 44, 45, 53, 55, 56, 57, 71, 71, 71, 72, 73, 74, 78, 79, 83, 88, 89, 90, 94, 100, 101, 105 ] ] gap> fus:= rep[1];; gap> Size( o8p2 ) / (2^9*168); 2025 gap> sub:= CharacterTable( "2^6:A8" );; gap> subfus:= GetFusionMap( sub, o8p2 ); [ 1, 3, 2, 2, 4, 5, 6, 13, 3, 6, 12, 13, 14, 7, 21, 24, 11, 30, 29, 31, 13, 17, 15, 16, 14, 17, 36, 37, 18, 41, 24, 44, 48, 28, 33, 32, 34, 35, 35, 51, 51 ] gap> fus:= CompositionMaps( fus, subfus ); [ 1, 2, 5, 5, 3, 4, 5, 23, 2, 5, 19, 23, 20, 7, 37, 31, 17, 50, 51, 43, 23, 23, 21, 22, 20, 23, 56, 57, 24, 72, 31, 78, 89, 52, 45, 44, 53, 55, 55, 100, 100 ] gap> Size( sub ) / (2^9*168); 15 gap> List( Irr( sub ), Degree ); [ 1, 7, 14, 20, 21, 21, 21, 28, 35, 45, 45, 56, 64, 70, 28, 28, 35, 35, 35, 35, 70, 70, 70, 70, 140, 140, 140, 140, 140, 210, 210, 252, 252, 280, 280, 315, 315, 315, 315, 420, 448 ] gap> cand:= PermChars( sub, 15 ); [ Character( CharacterTable( "2^6:A8" ), [ 15, 15, 15, 7, 7, 7, 7, 7, 3, 3, 3, 3, 3, 0, 0, 0, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0 ] ) ] gap> ind:= Induced( sub, o8p3, cand, fus ); [ Character( CharacterTable( "O8+(3)" ), [ 57572775, 59535, 59535, 59535, 3591, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 135, 135, 243, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 27, 27, 0, 0, 0, 0, 27, 27, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> o8p33:= CharacterTable( "O8+(3).3" );; gap> inv:= InverseMap( GetFusionMap( o8p3, o8p33 ) ); [ 1, [ 2, 3, 4 ], 5, 6, [ 7, 8, 9 ], [ 10, 11, 12 ], 13, [ 14, 15, 16 ], 17, 18, 19, [ 20, 21, 22 ], 23, [ 24, 25, 26 ], [ 27, 28, 29 ], 30, [ 31, 32, 33 ], [ 34, 35, 36 ], [ 37, 38, 39 ], [ 40, 41, 42 ], [ 43, 44, 45 ], 46, [ 47, 48, 49 ], 50, [ 51, 52, 53 ], 54, 55, 56, 57, [ 58, 59, 60 ], [ 61, 62, 63 ], 64, [ 65, 66, 67 ], 68, [ 69, 70, 71 ], [ 72, 73, 74 ], [ 75, 76, 77 ], [ 78, 79, 80 ], [ 81, 82, 83 ], 84, 85, [ 86, 87, 88 ], [ 89, 90, 91 ], [ 92, 93, 94 ], 95, 96, [ 97, 98, 99 ], [ 100, 101, 102 ], [ 103, 104, 105 ], [ 106, 107, 108 ], [ 109, 110, 111 ], [ 112, 113, 114 ] ] gap> ext:= CompositionMaps( ind[1], inv ); [ 57572775, 59535, 3591, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 243, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> perms:= PermChars( o8p33, rec( torso:= ext ) ); [ Character( CharacterTable( "O8+(3).3" ), [ 57572775, 59535, 3591, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 243, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3159, 3159, 243, 243, 39, 39, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( o8p33, perms ).ATLAS; [ "1a+780aabb+2457a+2808abc+9450aaabbcc+18200abcdddef+24192a+54600a^{5\ }b+70200aabb+87360ab+139776a^{5}+147420a^{4}b^{4}+163800ab+184275aabc+\ 199017aa+218700a+245700a+291200aef+332800a^{4}b^{5}c^{5}+491400aaabcd+\ 531441a^{5}b^{4}c^{4}+552825a^{4}+568620aabb+698880a^{4}b^{4}+716800aa\ abbccdddeeff+786240aabb+873600aa+998400aa+1257984a^{6}+1397760aa" ] ## doc2/ctblpope.xml (777-783) gap> o73:= CharacterTable( "O7(3)" );; gap> a7:= CharacterTable( "A7" );; gap> fus:= PossibleClassFusions( a7, o73 ); [ [ 1, 3, 6, 10, 15, 16, 24, 33, 33 ], [ 1, 3, 7, 10, 15, 16, 22, 33, 33 ] ] ## doc2/ctblpope.xml (794-810) gap> ind:= List( fus, > x -> Induced( a7, o73, [ TrivialCharacter( a7 ) ], x )[1] );; gap> mat:= MatScalarProducts( o73, Irr( o73 ), ind );; gap> sum:= Sum( mat ); [ 2, 6, 2, 0, 8, 6, 2, 4, 4, 8, 3, 0, 4, 4, 9, 3, 5, 0, 0, 9, 0, 10, 5, 6, 15, 1, 12, 1, 15, 7, 2, 4, 14, 16, 0, 12, 12, 7, 8, 8, 14, 12, 12, 14, 6, 6, 20, 16, 12, 12, 12, 10, 10, 12, 12, 8, 12, 6 ] gap> const:= Filtered( [ 1 .. Length( sum ) ], x -> sum[x] <> 0 ); [ 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58 ] gap> Length( const ); 52 gap> const:= Irr( o73 ){ const };; gap> rat:= RationalizedMat( const );; ## doc2/ctblpope.xml (819-837) gap> names:= ClassNames( o73 ); [ "1a", "2a", "2b", "2c", "3a", "3b", "3c", "3d", "3e", "3f", "3g", "4a", "4b", "4c", "4d", "5a", "6a", "6b", "6c", "6d", "6e", "6f", "6g", "6h", "6i", "6j", "6k", "6l", "6m", "6n", "6o", "6p", "7a", "8a", "8b", "9a", "9b", "9c", "9d", "10a", "10b", "12a", "12b", "12c", "12d", "12e", "12f", "12g", "12h", "13a", "13b", "14a", "15a", "18a", "18b", "18c", "18d", "20a" ] gap> List( fus, x -> names{ x } ); [ [ "1a", "2b", "3b", "3f", "4d", "5a", "6h", "7a", "7a" ], [ "1a", "2b", "3c", "3f", "4d", "5a", "6f", "7a", "7a" ] ] gap> torso:= [ 28431 ];; gap> zeros:= [ 5, 8, 9, 11, 17, 20, 23, 28, 29, 32, 36, 37, 38, > 43, 46, 47, 48, 53, 54, 55, 56, 57, 58 ];; gap> names{ zeros }; [ "3a", "3d", "3e", "3g", "6a", "6d", "6g", "6l", "6m", "6p", "9a", "9b", "9c", "12b", "12e", "12f", "12g", "15a", "18a", "18b", "18c", "18d", "20a" ] ## doc2/ctblpope.xml (854-869) gap> for i in zeros do > torso[i]:= 0; > od; gap> torso; [ 28431,,,, 0,,, 0, 0,, 0,,,,,, 0,,, 0,,, 0,,,,, 0, 0,,, 0,,,, 0, 0, 0,,,,, 0,,, 0, 0, 0,,,,, 0, 0, 0, 0, 0, 0 ] gap> perms:= PermChars( o73, rec( torso:= torso, chars:= rat ) ); [ Character( CharacterTable( "O7(3)" ), [ 28431, 567, 567, 111, 0, 0, 243, 0, 0, 81, 0, 15, 3, 27, 15, 6, 0, 0, 27, 0, 3, 27, 0, 0, 0, 3, 9, 0, 0, 3, 3, 0, 4, 1, 1, 0, 0, 0, 0, 2, 2, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( o73, perms ).ATLAS; [ "1a+78a+168a+182a+260ab+1092a+2457a+2730a+4095b+5460a+11648a" ] ## doc2/ctblpope.xml (895-917) gap> o8p3:= CharacterTable( "O8+(3)" );; gap> o8p2:= CharacterTable( "O8+(2)" );; gap> fus:= PossibleClassFusions( o8p2, o8p3 );; gap> NamesOfFusionSources( o8p2 ); [ "A9", "2^8:O8+(2)", "(D10xD10).2^2", "(3x3^3:S3):2", "(3x3^(1+2)+:2A4).2", "2^(3+3+3).L3(2)", "NRS(O8+(2),2^(3+3+3)_a)", "NRS(O8+(2),2^(3+3+3)_b)", "O8+(2)N2", "O8+(2)M2", "O8+(2)M3", "O8+(2)M5", "O8+(2)M6", "O8+(2)M8", "O8+(2)M9", "(3xU4(2)):2", "O8+(2)M11", "O8+(2)M12", "2^(1+8)_+:(S3xS3xS3)", "3^4:2^3.S4(a)", "(A5xA5):2^2", "O8+(2)M16", "O8+(2)M17", "2^(1+8)+.O8+(2)", "7:6", "(A5xD10).2", "(D10xA5).2", "O8+(2)N5C", "2^6:A8", "2.O8+(2)", "2^2.O8+(2)", "S6(2)" ] gap> sub:= CharacterTable( "2^6:A8" );; gap> subfus:= GetFusionMap( sub, o8p2 ); [ 1, 3, 2, 2, 4, 5, 6, 13, 3, 6, 12, 13, 14, 7, 21, 24, 11, 30, 29, 31, 13, 17, 15, 16, 14, 17, 36, 37, 18, 41, 24, 44, 48, 28, 33, 32, 34, 35, 35, 51, 51 ] gap> fus:= List( fus, x -> CompositionMaps( x, subfus ) );; gap> fus:= Set( fus );; gap> Length( fus ); 24 ## doc2/ctblpope.xml (926-932) gap> ind:= List( fus, x -> Induced( sub, o8p3, > [ TrivialCharacter( sub ) ], x )[1] );; gap> ind:= Set( ind );; gap> Length( ind ); 6 ## doc2/ctblpope.xml (942-957) gap> o8p32:= CharacterTable( "O8+(3).2_1" );; gap> fus:= GetFusionMap( o8p3, o8p32 );; gap> ext:= List( ind, x -> CompositionMaps( x, InverseMap( fus ) ) );; gap> ext:= Filtered( ext, x -> ForAll( x, IsInt ) ); [ [ 3838185, 17577, 8505, 8505, 873, 0, 0, 0, 0, 6561, 0, 0, 729, 0, 9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 9, 27, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0 ], [ 3838185, 17577, 8505, 8505, 873, 0, 6561, 0, 0, 0, 0, 0, 729, 0, 9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 0, 0, 0, 9, 27, 27, 0, 0, 9, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] ## doc2/ctblpope.xml (970-985) gap> perms:= PermChars( o8p32, rec( torso:= ext[1] ) ); [ Character( CharacterTable( "O8+(3).2_1" ), [ 3838185, 17577, 8505, 8505, 873, 0, 0, 0, 0, 6561, 0, 0, 729, 0, 9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 9, 27, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3159, 1575, 567, 63, 87, 15, 0, 0, 45, 0, 81, 9, 27, 0, 0, 3, 3, 3, 3, 5, 5, 0, 0, 0, 4, 0, 0, 27, 0, 9, 0, 0, 15, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( o8p32, perms ).ATLAS; [ "1a+260abc+520ab+819a+2808b+9450aab+18200a+23400ac+29120b+36400aab+4\ 6592abce+49140d+66339a+98280ab+163800a+189540d+232960d+332800ab+368550\ a+419328a+531441ab" ] ## doc2/ctblpope.xml (994-1013) gap> o8p32:= CharacterTable( "O8+(3).2_2" );; gap> fus:= GetFusionMap( o8p3, o8p32 );; gap> ext:= List( ind, x -> CompositionMaps( x, InverseMap( fus ) ) );; gap> ext:= Filtered( ext, x -> ForAll( x, IsInt ) );; gap> perms:= PermChars( o8p32, rec( torso:= ext[1] ) ); [ Character( CharacterTable( "O8+(3).2_2" ), [ 3838185, 17577, 8505, 873, 0, 0, 0, 6561, 0, 0, 0, 0, 729, 0, 9, 105, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 0, 9, 27, 0, 0, 0, 27, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 199017, 2025, 297, 441, 73, 9, 0, 1215, 0, 0, 0, 0, 0, 81, 0, 0, 0, 0, 27, 27, 0, 1, 9, 12, 0, 0, 45, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( o8p32, perms ).ATLAS; [ "1a+260aac+520ab+819a+2808a+9450aaa+18200accee+23400ac+29120a+36400a\ +46592aa+49140c+66339a+93184a+98280ab+163800a+184275ac+189540c+232960c\ +332800aa+419328a+531441aa" ] ## doc2/ctblpope.xml (1026-1049) gap> o8p322:= CharacterTable( "O8+(3).(2^2)_{122}" );; gap> fus:= GetFusionMap( o8p32, o8p322 );; gap> ext:= List( perms, x -> CompositionMaps( x, InverseMap( fus ) ) );; gap> ext:= Filtered( ext, x -> ForAll( x, IsInt ) );; gap> perms:= PermChars( o8p322, rec( torso:= ext[1] ) ); [ Character( CharacterTable( "O8+(3).(2^2)_{122}" ), [ 3838185, 17577, 8505, 873, 0, 0, 0, 6561, 0, 0, 729, 0, 9, 105, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 9, 9, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 3159, 1575, 567, 63, 87, 15, 0, 0, 45, 0, 81, 9, 27, 0, 0, 3, 3, 3, 5, 0, 0, 4, 0, 0, 27, 0, 9, 0, 0, 15, 0, 3, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 199017, 2025, 297, 441, 73, 9, 0, 1215, 0, 0, 0, 0, 81, 0, 0, 0, 27, 27, 0, 1, 9, 12, 0, 0, 45, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 28431, 1647, 135, 63, 87, 39, 0, 0, 243, 27, 0, 0, 81, 63, 0, 0, 0, 9, 0, 3, 3, 6, 2, 0, 0, 0, 9, 0, 0, 3, 3, 3, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ] ) ] gap> PermCharInfo( o8p322, perms ).ATLAS; [ "1a+260ace+819a+1040a+2808c+9450aac+18200a+23400ae+29120c+36400aac+4\ 6592ac+49140g+66339a+93184a+163800b+189540g+196560a+232960g+332800ac+3\ 68550a+419328a+531441ac" ] ## doc2/ctblpope.xml (1079-1083) gap> s444:= CharacterTable( "S4(4).4" );; gap> NamesOfFusionSources( s444 ); [ "(L3(2)xS4(4):2).2", "S4(4)", "S4(4).2" ] ## doc2/ctblpope.xml (1093-1105) gap> perms:= PermChars( s444, > rec( torso:= [ Size( s444 ) / ( 5^2*2^5 ) ] ) ); [ Character( CharacterTable( "S4(4).4" ), [ 4896, 384, 96, 0, 16, 32, 36, 16, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "S4(4).4" ), [ 4896, 192, 32, 0, 0, 8, 6, 1, 0, 2, 0, 0, 36, 0, 12, 0, 0, 0, 1, 0, 6, 6, 2, 2, 0, 0, 0, 0, 1, 1 ] ), Character( CharacterTable( "S4(4).4" ), [ 4896, 240, 64, 0, 8, 8, 36, 16, 0, 0, 0, 0, 0, 12, 8, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (1116-1121) gap> PermCharInfo( s444, perms ).ATLAS; [ "1abcd+50abcd+153abcd+170a^{4}b^{4}+680aabb", "1a+50ac+153a+170aab+256a+680abb+816a+1020a", "1ac+50ac+68a+153abcd+170aabbb+204a+680abb+1020a" ] ## doc2/ctblpope.xml (1147-1150) gap> t:= CharacterTable( "Co1" ); CharacterTable( "Co1" ) ## doc2/ctblpope.xml (1162-1169) gap> s:= CharacterTable( Maxes( t )[5] ); CharacterTable( "2^(1+8)+.O8+(2)" ) gap> ind:= Induced( s, t, [ TrivialCharacter( s ) ] );; gap> PermCharInfo( t, ind ).ATLAS; [ "1a+299a+17250a+27300a+80730a+313950a+644644a+2816856a+5494125a+1243\ 2420a+24794000a" ] ## doc2/ctblpope.xml (1181-1189) gap> centorder:= SizesCentralizers( t )[3];; gap> maxes:= List( Maxes( t ), CharacterTable );; gap> cand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 ); [ CharacterTable( "(A4xG2(4)):2" ) ] gap> u:= cand[1];; gap> index:= Size( u ) / centorder; 3 ## doc2/ctblpope.xml (1210-1236) gap> subperm:= PermChars( u, rec( degree := index, bounds := false ) ); [ Character( CharacterTable( "(A4xG2(4)):2" ), [ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ] gap> subperm = PermChars( u, rec( torso := [ 3 ] ) ); true gap> ind:= Induced( u, t, subperm ); [ Character( CharacterTable( "Co1" ), [ 2065694400, 181440, 119408, 38016, 2779920, 0, 0, 378, 30240, 864, 0, 720, 316, 80, 2520, 30, 0, 6480, 1508, 0, 0, 0, 0, 0, 38, 18, 105, 0, 600, 120, 56, 24, 0, 12, 0, 0, 0, 120, 48, 18, 0, 0, 6, 0, 360, 144, 108, 0, 0, 10, 0, 0, 0, 0, 0, 4, 2, 3, 9, 0, 0, 15, 3, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 12, 8, 0, 6, 0, 0, 3, 0, 1, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0 ] ) ] gap> PermCharInfo( t, ind ).ATLAS; [ "1a+1771a+8855a+27300aa+313950a+345345a+644644aa+871884aaa+1771000a+\ 2055625a+4100096a+7628985a+9669660a+12432420aa+21528000aa+23244375a+24\ 174150aa+24794000a+31574400aa+40370176a+60435375a+85250880aa+100725625\ a+106142400a+150732800a+184184000a+185912496a+207491625a+299710125a+30\ 2176875a" ] ## doc2/ctblpope.xml (1244-1248) gap> centorder:= SizesCentralizers( t )[4];; gap> cand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 ); [ CharacterTable( "Co2" ), CharacterTable( "2^11:M24" ) ] ## doc2/ctblpope.xml (1258-1269) gap> u:= cand[1];; gap> GetFusionMap( u, t ); [ 1, 2, 2, 4, 7, 6, 9, 11, 11, 10, 11, 12, 14, 17, 16, 21, 23, 20, 22, 22, 24, 28, 30, 33, 31, 32, 33, 33, 37, 42, 41, 43, 44, 48, 52, 49, 53, 55, 53, 52, 54, 60, 60, 60, 64, 65, 65, 67, 66, 70, 73, 72, 78, 79, 84, 85, 87, 92, 93, 93 ] gap> centorder; 389283840 gap> SizesCentralizers( u )[4]; 1474560 ## doc2/ctblpope.xml (1277-1307) gap> u:= cand[2]; CharacterTable( "2^11:M24" ) gap> index:= Size( u ) / centorder; 1288 gap> subperm:= PermChars( u, rec( torso := [ index ] ) ); [ Character( CharacterTable( "2^11:M24" ), [ 1288, 1288, 1288, 56, 56, 56, 56, 56, 56, 48, 48, 48, 48, 48, 10, 10, 10, 10, 7, 7, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 0, 0, 0, 0, 2, 2, 2, 2, 3, 3, 3, 1, 1, 2, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> subperm = PermChars( u, rec( degree:= index, bounds := false ) ); true gap> ind:= Induced( u, t, subperm ); [ Character( CharacterTable( "Co1" ), [ 10680579000, 1988280, 196560, 94744, 0, 17010, 0, 945, 7560, 3432, 2280, 1728, 252, 308, 0, 225, 0, 0, 0, 270, 0, 306, 0, 46, 45, 25, 0, 0, 120, 32, 12, 52, 36, 36, 0, 0, 0, 0, 0, 45, 15, 0, 9, 3, 0, 0, 0, 0, 18, 0, 30, 0, 6, 18, 0, 3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( t, ind ).ATLAS; [ "1a+17250aa+27300a+80730aa+644644aaa+871884a+1821600a+2055625aaa+281\ 6856a+5494125a^{4}+12432420aa+16347825aa+23244375a+24174150aa+24667500\ aa+24794000aaa+31574400a+40370176a+55255200a+66602250a^{4}+83720000aa+\ 85250880aaa+91547820aa+106142400a+150732800a+184184000aaa+185912496aaa\ +185955000aaa+207491625aaa+215547904aa+241741500aaa+247235625a+2578576\ 00aa+259008750a+280280000a+302176875a+326956500a+387317700a+402902500a\ +464257024a+469945476b+502078500a+503513010a+504627200a+522161640a" ] ## doc2/ctblpope.xml (1330-1351) gap> t:= CharacterTable( "G2(3)" ); CharacterTable( "G2(3)" ) gap> t:= CharacterTable( "G2(3)" );; gap> n:= Length( RationalizedMat( Irr( t ) ) );; gap> maxmult:= List( [ 1 .. n ], i -> 1 );; gap> perms:= [];; gap> divs:= DivisorsInt( Size( t ) );; gap> for d in divs do > Append( perms, > PermChars( t, rec( bounds := false, > degree := d, > maxmult := maxmult ) ) ); > od; gap> Length( perms ); 42 gap> List( perms, Degree ); [ 1, 351, 351, 364, 364, 378, 378, 546, 546, 546, 546, 546, 702, 702, 728, 728, 1092, 1092, 1092, 1092, 1092, 1092, 1092, 1092, 1456, 1456, 1638, 1638, 2184, 2184, 2457, 2457, 2457, 2457, 3159, 3276, 3276, 3276, 3276, 4368, 6552, 6552 ] ## doc2/ctblpope.xml (1370-1384) gap> tom:= TableOfMarks( "G2(3)" ); TableOfMarks( "G2(3)" ) gap> tbl:= CharacterTable( "G2(3)" ); CharacterTable( "G2(3)" ) gap> permstom:= PermCharsTom( tbl, tom );; gap> Length( permstom ); 433 gap> multfree:= Intersection( perms, permstom );; gap> Length( multfree ); 15 gap> List( multfree, Degree ); [ 1, 351, 351, 364, 364, 378, 378, 702, 702, 728, 728, 1092, 1092, 2184, 2184 ] ## doc2/ctblpope.xml (1404-1409) gap> tbl2:= CharacterTable("O8+(2).2");; gap> s3:= CharacterTable( "Symmetric", 3 );; gap> s:= CharacterTableWreathSymmetric( s3, 4 ); CharacterTable( "Sym(3)wrS4" ) ## doc2/ctblpope.xml (1419-1442) gap> fus:= PossibleClassFusions( s, tbl2 ); [ [ 1, 41, 6, 3, 48, 9, 42, 19, 51, 8, 5, 50, 24, 49, 7, 2, 44, 22, 42, 12, 53, 17, 58, 21, 5, 47, 26, 50, 37, 52, 23, 60, 18, 4, 46, 25, 14, 61, 20, 9, 53, 30, 51, 26, 64, 8, 52, 31, 13, 56, 38 ] ] gap> pi:= Induced( s, tbl2, [ TrivialCharacter( s ) ], fus[1] )[1]; Character( CharacterTable( "O8+(2).2" ), [ 11200, 256, 160, 160, 80, 40, 40, 76, 13, 0, 0, 8, 8, 4, 0, 0, 16, 16, 4, 4, 4, 1, 1, 1, 1, 5, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1120, 96, 0, 16, 0, 16, 8, 10, 4, 6, 7, 12, 3, 0, 0, 2, 0, 4, 0, 1, 1, 0, 0, 1, 0, 0, 0 ] ) gap> PermCharInfo( tbl2, pi ).ATLAS; [ "1a+84a+168a+175a+300a+700c+972a+1400a+3200a+4200b" ] gap> tbl:= CharacterTable( "O8+(2)" ); CharacterTable( "O8+(2)" ) gap> rest:= RestrictedClassFunction( pi, tbl ); Character( CharacterTable( "O8+(2)" ), [ 11200, 256, 160, 160, 160, 80, 40, 40, 40, 76, 13, 0, 0, 8, 8, 8, 4, 0, 0, 0, 16, 16, 16, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 5, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0 ] ) gap> PermCharInfo( tbl, rest ).ATLAS; [ "1a+84abc+175a+300a+700bcd+972a+3200a+4200a" ] ## doc2/ctblpope.xml (1463-1489) gap> tbl:= CharacterTable( "M22" ); CharacterTable( "M22" ) gap> perms:= PermChars( tbl, rec( torso:= [ 56 ] ) ); [ Character( CharacterTable( "M22" ), [ 56, 8, 2, 4, 0, 1, 2, 0, 0, 2, 1, 1 ] ) ] gap> pi:= perms[1];; gap> Norm( pi ); 2 gap> Display( tbl, rec( chars:= perms ) ); M22 2 7 7 2 5 4 . 2 . . 3 . . 3 2 1 2 . . . 1 . . . . . 5 1 . . . . 1 . . . . . . 7 1 . . . . . . 1 1 . . . 11 1 . . . . . . . . . 1 1 1a 2a 3a 4a 4b 5a 6a 7a 7b 8a 11a 11b 2P 1a 1a 3a 2a 2a 5a 3a 7a 7b 4a 11b 11a 3P 1a 2a 1a 4a 4b 5a 2a 7b 7a 8a 11a 11b 5P 1a 2a 3a 4a 4b 1a 6a 7b 7a 8a 11a 11b 7P 1a 2a 3a 4a 4b 5a 6a 1a 1a 8a 11b 11a 11P 1a 2a 3a 4a 4b 5a 6a 7a 7b 8a 1a 1a Y.1 56 8 2 4 . 1 2 . . 2 1 1 ## doc2/ctblpope.xml (1501-1505) gap> perms:= PermChars( tbl, rec( torso:= [ 56 * 55 ] ) );; gap> Length( perms ); 16 ## doc2/ctblpope.xml (1517-1529) gap> OrdersClassRepresentatives( tbl ); [ 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 11, 11 ] gap> perms:= Filtered( perms, x -> x[5] = 0 and x[10] <> 0 ); [ Character( CharacterTable( "M22" ), [ 3080, 56, 2, 12, 0, 0, 2, 0, 0, 2, 0, 0 ] ), Character( CharacterTable( "M22" ), [ 3080, 8, 2, 8, 0, 0, 2, 0, 0, 4, 0, 0 ] ), Character( CharacterTable( "M22" ), [ 3080, 24, 11, 4, 0, 0, 3, 0, 0, 2, 0, 0 ] ), Character( CharacterTable( "M22" ), [ 3080, 24, 20, 4, 0, 0, 0, 0, 0, 2, 0, 0 ] ) ] ## doc2/ctblpope.xml (1539-1569) gap> infoperms:= PermCharInfo( tbl, perms );; gap> Display( tbl, infoperms.display ); M22 2 7 7 2 5 2 3 3 2 1 2 . 1 . 5 1 . . . . . 7 1 . . . . . 11 1 . . . . . 1a 2a 3a 4a 6a 8a 2P 1a 1a 3a 2a 3a 4a 3P 1a 2a 1a 4a 2a 8a 5P 1a 2a 3a 4a 6a 8a 7P 1a 2a 3a 4a 6a 8a 11P 1a 2a 3a 4a 6a 8a I.1 3080 56 2 12 2 2 I.2 1 21 8 54 24 36 I.3 1 3 4 9 12 18 I.4 3080 8 2 8 2 4 I.5 1 3 8 36 24 72 I.6 1 3 4 9 12 18 I.7 3080 24 11 4 3 2 I.8 1 9 44 18 36 36 I.9 1 3 4 9 12 18 I.10 3080 24 20 4 . 2 I.11 1 9 80 18 . 36 I.12 1 3 4 9 12 18 ## doc2/ctblpope.xml (1659-1670) gap> ly:= CharacterTable( "Ly" );; gap> mcl:= CharacterTable( "McL" );; gap> mcl2:= CharacterTable( "McL.2" );; gap> 3mcl2:= CharacterTable( "3.McL.2" );; gap> perms:= PermChars( mcl, rec( degree:= 15400 ) ); [ Character( CharacterTable( "McL" ), [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 0, 2, 1, 1, 1, 0, 0, 3, 0, 0, 1, 1, 1, 1 ] ), Character( CharacterTable( "McL" ), [ 15400, 280, 10, 37, 20, 0, 5, 10, 1, 0, 0, 2, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (1682-1692) gap> ord10:= Filtered( [ 1 .. NrConjugacyClasses( mcl ) ], > i -> OrdersClassRepresentatives( mcl )[i] = 10 ); [ 15 ] gap> List( perms, pi -> pi[ ord10[1] ] ); [ 1, 0 ] gap> pi:= perms[1]; Character( CharacterTable( "McL" ), [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 0, 2, 1, 1, 1, 0, 0, 3, 0, 0, 1, 1, 1, 1 ] ) ## doc2/ctblpope.xml (1707-1722) gap> map:= InverseMap( GetFusionMap( mcl, mcl2 ) ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, [ 10, 11 ], 12, [ 13, 14 ], 15, 16, 17, 18, [ 19, 20 ], [ 21, 22 ], [ 23, 24 ] ] gap> torso:= CompositionMaps( pi, map ); [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 2, 1, 1, 0, 0, 3, 0, 1, 1 ] gap> perms:= PermChars( mcl2, rec( torso:= torso ) ); [ Character( CharacterTable( "McL.2" ), [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 2, 1, 1, 0, 0, 3, 0, 1, 1, 110, 26, 2, 4, 0, 0, 5, 2, 1, 1, 0, 0, 1, 1 ] ) ] gap> pi:= Inflated( perms[1], 3mcl2 ); Character( CharacterTable( "3.McL.2" ), [ 15400, 15400, 56, 56, 91, 91, 10, 12, 12, 25, 25, 0, 0, 11, 11, 2, 2, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 1, 1, 1, 1, 1, 1, 110, 26, 2, 4, 0, 0, 5, 2, 1, 1, 0, 0, 1, 1 ] ) ## doc2/ctblpope.xml (1732-1738) gap> fus:= PossibleClassFusions( 3mcl2, ly );; Length( fus ); 4 gap> g:= AutomorphismsOfTable( ly );; gap> OrbitLengths( g, fus, OnTuples ); [ 4 ] ## doc2/ctblpope.xml (1744-1750) gap> pi:= Induced( 3mcl2, ly, [ pi ], fus[1] )[1]; Character( CharacterTable( "Ly" ), [ 147934325000, 286440, 1416800, 1082, 784, 12500, 0, 672, 42, 24, 0, 40, 0, 2, 20, 0, 0, 0, 64, 10, 0, 50, 2, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ## doc2/ctblpope.xml (1773-1787) gap> orders:= OrdersClassRepresentatives( ly ); [ 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 14, 15, 15, 15, 18, 20, 21, 21, 22, 22, 24, 24, 24, 25, 28, 30, 30, 31, 31, 31, 31, 31, 33, 33, 37, 37, 40, 40, 42, 42, 67, 67, 67 ] gap> torso:= [];; gap> for i in [ 1 .. Length( orders ) ] do > if orders[i] mod 2 = 1 then > torso[i]:= pi[i]/2; > fi; > od; gap> torso; [ 73967162500,, 708400, 541,, 6250, 0,,,, 0,,, 1,,, 0, 0,,,, 25, 1, 0, ,, 0, 0,,,,,, 0,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0,,,,, 0, 0, 0 ] ## doc2/ctblpope.xml (1796-1807) gap> perms:= PermChars( ly, rec( torso:= torso ) );; gap> Length( perms ); 43 gap> perms:= Filtered( perms, cand -> ForAll( [ 1 .. Length( orders ) ], > i -> cand[i] >= pi[i] / 2 ) ); [ Character( CharacterTable( "Ly" ), [ 73967162500, 204820, 708400, 541, 392, 6250, 0, 1456, 61, 25, 0, 22, 10, 1, 10, 0, 0, 0, 32, 5, 0, 25, 1, 0, 1, 2, 0, 0, 0, 0, 4, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (1831-1843) gap> tbl:= CharacterTable( "U3(5).3.2" ); CharacterTable( "U3(5).3.2" ) gap> deg:= Size( tbl ) / ( 2^4*3^3 ); 1750 gap> pi:= PermChars( tbl, rec( torso:= [ deg ] ) ); [ Character( CharacterTable( "U3(5).3.2" ), [ 1750, 70, 13, 2, 0, 0, 1, 0, 0, 0, 10, 7, 10, 4, 2, 0, 0, 0, 0, 0, 0, 30, 10, 3, 0, 0, 1, 0, 0 ] ), Character( CharacterTable( "U3(5).3.2" ), [ 1750, 30, 4, 6, 0, 0, 0, 0, 0, 0, 40, 7, 0, 6, 0, 0, 0, 0, 0, 0, 0, 20, 0, 2, 2, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (1854-1860) gap> ord8:= Filtered( [ 1 .. NrConjugacyClasses( tbl ) ], > i -> OrdersClassRepresentatives( tbl )[i] = 8 ); [ 9, 25 ] gap> List( pi, x -> x{ ord8 } ); [ [ 0, 0 ], [ 0, 2 ] ] ## doc2/ctblpope.xml (1874-1882) gap> subtbl:= CharacterTable( "U3(5)" ); CharacterTable( "U3(5)" ) gap> rest:= RestrictedClassFunctions( pi, subtbl ); [ Character( CharacterTable( "U3(5)" ), [ 1750, 70, 13, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U3(5)" ), [ 1750, 30, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (1893-1899) gap> ord6:= Filtered( [ 1 .. NrConjugacyClasses( subtbl ) ], > i -> OrdersClassRepresentatives( subtbl )[i] = 6 ); [ 9 ] gap> List( rest, x -> x{ ord6 } ); [ [ 1 ], [ 0 ] ] ## doc2/ctblpope.xml (1910-1916) gap> tom:= TableOfMarks( "U3(5)" ); TableOfMarks( "U3(5)" ) gap> perms:= PermCharsTom( subtbl, tom );; gap> Set( Filtered( perms, x -> x[1] = deg ) ) = Set( rest ); true ## doc2/ctblpope.xml (1928-1932) gap> PermCharInfo( tbl, pi ).ATLAS; [ "1a+21a+42a+84aac+105a+125a+126a+250a+252a+288bc", "1a+42a+84ac+105ab+125a+126a+250a+252b+288bc" ] ## doc2/ctblpope.xml (1942-1953) gap> subtbl2:= CharacterTable( "U3(5).2" );; gap> rest2:= RestrictedClassFunctions( pi, subtbl2 );; gap> PermCharInfo( subtbl2, rest2 ).ATLAS; [ "1a+21aab+28aa+56aa+84a+105a+125aab+126aab+288aa", "1a+21ab+28a+56a+84a+105ab+125aab+126a+252a+288aa" ] gap> subtbl3:= CharacterTable( "U3(5).3" );; gap> rest3:= RestrictedClassFunctions( pi, subtbl3 );; gap> PermCharInfo( subtbl3, rest3 ).ATLAS; [ "1a+21abc+84aab+105a+125abc+126abc+144bcef", "1a+21bc+84ab+105aa+125abc+126adg+144bcef" ] ## doc2/ctblpope.xml (1985-1997) gap> t:= CharacterTable( "O8+(2).3.2" );; gap> s:= CharacterTable( "O8+(2)" );; gap> tom:= TableOfMarks( s );; gap> perms:= PermCharsTom( s, tom );; gap> deg:= 3^4*5^2*7; 14175 gap> perms:= Filtered( perms, x -> x[1] = deg );; gap> Length( perms ); 4 gap> Length( Set( perms ) ); 1 ## doc2/ctblpope.xml (2010-2029) gap> fus:= PossibleClassFusions( s, t ); [ [ 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 10, 10, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 19, 20, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 26, 26, 26, 27, 27, 27 ] ] gap> fus:= fus[1];; gap> inv:= InverseMap( fus );; gap> comp:= CompositionMaps( perms[1], inv ); [ 14175, 1215, 375, 79, 0, 0, 27, 27, 99, 15, 7, 0, 0, 0, 0, 9, 3, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0 ] gap> ext:= PermChars( t, rec( torso:= comp ) ); [ Character( CharacterTable( "O8+(2).3.2" ), [ 14175, 1215, 375, 79, 0, 0, 27, 27, 99, 15, 7, 0, 0, 0, 0, 9, 3, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 63, 9, 15, 7, 1, 0, 3, 3, 3, 1, 0, 0, 1, 1, 945, 129, 45, 69, 21, 25, 13, 0, 0, 0, 9, 0, 3, 3, 7, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( t, ext[1] ).ATLAS; [ "1a+50b+100a+252bb+300b+700b+972bb+1400a+1944a+3200b+4032b" ] ## doc2/ctblpope.xml (2220-2230) gap> co1:= CharacterTable( "Co1" );; gap> order:= 2^(2+11+22-25) * 2 * Size( CharacterTable( "M24" ) ); 501397585920 gap> maxes:= List( Maxes( co1 ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^11:M24" ) ] gap> List( filt, t -> Size( t ) / order ); [ 1 ] gap> k:= filt[1];; ## doc2/ctblpope.xml (2251-2262) gap> f:= CharacterTable( "Symmetric", 3 ); CharacterTable( "Sym(3)" ) gap> OrdersClassRepresentatives( f ); [ 1, 2, 3 ] gap> deg3:= PermChars( f, 3 ); [ Character( CharacterTable( "Sym(3)" ), [ 3, 1, 0 ] ) ] gap> deg6:= PermChars( f, 6 ); [ Character( CharacterTable( "Sym(3)" ), [ 6, 0, 0 ] ) ] gap> deg3[1] - 1/3 * deg6[1]; ClassFunction( CharacterTable( "Sym(3)" ), [ 1, 1, 0 ] ) ## doc2/ctblpope.xml (2274-2280) gap> m:= CharacterTable( "M" ); CharacterTable( "M" ) gap> g:= CharacterTable( "MC2B" ); CharacterTable( "2^1+24.Co1" ) gap> pi:= RestrictedClassFunction( TrivialCharacter( k )^co1, g );; ## doc2/ctblpope.xml (2296-2302) gap> zclasses:= ClassPositionsOfCentre( g );; gap> gmodz:= g / zclasses; CharacterTable( "2^1+24.Co1/[ 1, 2 ]" ) gap> invmap:= InverseMap( GetFusionMap( g, gmodz ) );; gap> pibar:= CompositionMaps( pi, invmap );; ## doc2/ctblpope.xml (2319-2332) gap> factorders:= OrdersClassRepresentatives( gmodz );; gap> phibar:= [];; gap> for i in [ 1 .. NrConjugacyClasses( gmodz ) ] do > if factorders[i] mod 2 = 1 then > phibar[i]:= 2 * pibar[i]; > elif pibar[i] = 0 then > phibar[i]:= 0; > fi; > od; gap> cand:= PermChars( gmodz, rec( torso:= phibar ) );; gap> Length( cand ); 1 ## doc2/ctblpope.xml (2341-2351) gap> phi:= RestrictedClassFunction( cand[1], g )^m;; gap> pi:= pi^m;; gap> cand:= ShallowCopy( pi - 1/3 * phi );; gap> morders:= OrdersClassRepresentatives( m );; gap> for i in [ 1 .. Length( morders ) ] do > if morders[i] mod 3 = 0 and phi[ PowerMap( m, 3 )[i] ] <> 0 then > Unbind( cand[i] ); > fi; > od; ## doc2/ctblpope.xml (2422-2425) gap> constit:= Filtered( RationalizedMat( Irr( m ) ), > chi -> ScalarProduct( m, chi, pi ) <> 0 );; ## doc2/ctblpope.xml (2436-2458) gap> cand:= PermChars( m, > rec( torso:= cand, chars:= constit, > lower:= ShallowCopy( pi - 1/3 * phi ), > normalsubgroup:= [ 1 .. NrConjugacyClasses( m ) ], > nonfaithful:= TrivialCharacter( m ) ) ); [ Character( CharacterTable( "M" ), [ 16009115629875684006343550944921875, 7774182899642733721875, 120168544413337875, 4436049512692980, 215448838605, 131873639625, 760550656275, 110042727795, 943894035, 568854195, 1851609375, 0, 4680311220, 405405, 78624756, 14467005, 178605, 248265, 874650, 0, 76995, 591163, 224055, 34955, 29539, 20727, 0, 0, 375375, 15775, 0, 0, 0, 495, 116532, 3645, 62316, 1017, 11268, 357, 1701, 45, 117, 705, 0, 0, 4410, 1498, 0, 3780, 810, 0, 0, 83, 135, 31, 0, 0, 0, 0, 0, 0, 0, 255, 195, 0, 215, 0, 0, 210, 0, 42, 0, 35, 15, 1, 1, 160, 48, 9, 92, 25, 9, 9, 5, 1, 21, 0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 0, 120, 76, 10, 0, 0, 0, 0, 0, 1, 1, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 3, 0, 0, 0, 18, 0, 10, 0, 3, 3, 0, 1, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 12, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (2473-2479) gap> h:= CharacterTable( "2^(2+11+22).(M24xS3)" );; gap> fus:= PossibleClassFusions( h, m );; gap> cand = Set( fus, map -> InducedClassFunctionsByFusionMap( h, m, > [ TrivialCharacter( h ) ], map )[1] ); true ## doc2/ctblpope.xml (2518-2532) gap> co1:= CharacterTable( "Co1" );; gap> order:= 2^(3+6+12+18-25) * 168 * 3 * Factorial( 6 ) / 7; 849346560 gap> maxes:= List( Maxes( co1 ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^(1+8)+.O8+(2)" ), CharacterTable( "2^(4+12).(S3x3S6)" ) ] gap> List( filt, t -> Size( t ) / order ); [ 105, 1 ] gap> o8p2:= CharacterTable( "O8+(2)" );; gap> PermChars( o8p2, rec( torso:= [ 105 ] ) ); [ ] gap> k:= filt[2];; ## doc2/ctblpope.xml (2584-2598) gap> f:= CharacterTable( "L3(2)" ); CharacterTable( "L3(2)" ) gap> OrdersClassRepresentatives( f ); [ 1, 2, 3, 4, 7, 7 ] gap> deg7:= PermChars( f, 7 ); [ Character( CharacterTable( "L3(2)" ), [ 7, 3, 1, 1, 0, 0 ] ) ] gap> deg42:= PermChars( f, 42 ); [ Character( CharacterTable( "L3(2)" ), [ 42, 2, 0, 2, 0, 0 ] ), Character( CharacterTable( "L3(2)" ), [ 42, 6, 0, 0, 0, 0 ] ) ] gap> deg168:= PermChars( f, 168 ); [ Character( CharacterTable( "L3(2)" ), [ 168, 0, 0, 0, 0, 0 ] ) ] gap> deg7[1] - 1/3 * deg42[2] + 1/21 * deg168[1]; ClassFunction( CharacterTable( "L3(2)" ), [ 1, 1, 1, 1, 0, 0 ] ) ## doc2/ctblpope.xml (2616-2622) gap> m:= CharacterTable( "M" ); CharacterTable( "M" ) gap> g:= CharacterTable( "MC2B" ); CharacterTable( "2^1+24.Co1" ) gap> pi:= RestrictedClassFunction( TrivialCharacter( k )^co1, g );; ## doc2/ctblpope.xml (2633-2643) gap> nsg:= ClassPositionsOfNormalSubgroups( k );; gap> nsgsizes:= List( nsg, x -> Sum( SizesConjugacyClasses( k ){ x } ) );; gap> nn:= nsg[ Position( nsgsizes, Size( k ) / 6 ) ];; gap> psi:= 0 * [ 1 .. NrConjugacyClasses( k ) ];; gap> for i in nn do > psi[i]:= 6; > od; gap> psi:= InducedClassFunction( k, psi, co1 );; gap> psi:= RestrictedClassFunction( psi, g );; ## doc2/ctblpope.xml (2658-2664) gap> zclasses:= ClassPositionsOfCentre( g );; gap> gmodz:= g / zclasses; CharacterTable( "2^1+24.Co1/[ 1, 2 ]" ) gap> invmap:= InverseMap( GetFusionMap( g, gmodz ) );; gap> psibar:= CompositionMaps( psi, invmap );; ## doc2/ctblpope.xml (2682-2700) gap> factorders:= OrdersClassRepresentatives( gmodz );; gap> phibar:= [];; gap> upperphibar:= [];; gap> for i in [ 1 .. NrConjugacyClasses( gmodz ) ] do > if factorders[i] mod 2 = 1 then > phibar[i]:= 2 * psibar[i]; > elif psibar[i] = 0 then > phibar[i]:= 0; > fi; > upperphibar[i]:= 2 * psibar[i]; > od; gap> cand:= PermChars( gmodz, rec( torso:= phibar, > upper:= upperphibar, > normalsubgroup:= [ 1 .. NrConjugacyClasses( gmodz ) ], > nonfaithful:= TrivialCharacter( gmodz ) ) );; gap> Length( cand ); 3 ## doc2/ctblpope.xml (2711-2720) gap> nn:= First( ClassPositionsOfNormalSubgroups( gmodz ), > x -> Sum( SizesConjugacyClasses( gmodz ){x} ) = 2^24 ); [ 1 .. 4 ] gap> cont:= PermCharInfo( gmodz, cand ).contained;; gap> cand:= cand{ Filtered( [ 1 .. Length( cand ) ], > i -> Sum( cont[i]{ nn } ) < 2^24 ) };; gap> Length( cand ); 2 ## doc2/ctblpope.xml (2731-2757) gap> poss:= [];; gap> for v in cand do > phibar:= []; > upperphibar:= []; > for i in [ 1 .. NrConjugacyClasses( gmodz ) ] do > if factorders[i] mod 2 = 1 then > phibar[i]:= 2 * v[i]; > elif v[i] = 0 then > phibar[i]:= 0; > fi; > upperphibar[i]:= 2 * v[i]; > od; > Append( poss, PermChars( gmodz, rec( torso:= phibar, > upper:= upperphibar, > normalsubgroup:= [ 1 .. NrConjugacyClasses( gmodz ) ], > nonfaithful:= TrivialCharacter( gmodz ) ) ) ); > od; gap> Length( poss ); 6 gap> cont:= PermCharInfo( gmodz, poss ).contained;; gap> poss:= poss{ Filtered( [ 1 .. Length( poss ) ], > i -> Sum( cont[i]{ nn } ) < 2^23 ) };; gap> Length( poss ); 4 gap> phicand:= RestrictedClassFunctions( poss, g );; ## doc2/ctblpope.xml (2768-2783) gap> phicand:= RestrictedClassFunctions( poss, g );; gap> phicand:= InducedClassFunctions( phicand, m );; gap> psi:= psi^m;; gap> pi:= pi^m;; gap> cand:= List( phicand, > phi -> ShallowCopy( pi - 1/3 * psi + 1/21 * phi ) );; gap> morders:= OrdersClassRepresentatives( m );; gap> for x in cand do > for i in [ 1 .. Length( morders ) ] do > if morders[i] mod 7 = 0 then > Unbind( x[i] ); > fi; > od; > od; ## doc2/ctblpope.xml (2791-2807) gap> cand:= Filtered( cand, x -> ForAll( x, IsInt ) ); [ [ 4050306254358548053604918389065234375, 148844831270071996434375, 2815847622206994375, 14567365753025085, 3447181417680, 659368198125, 3520153823175, 548464353255, 5706077895, 3056566695, 264515625, 0, 19572895485, 6486480, 186109245, 61410960, 758160, 688365,,, 172503, 1264351, 376155, 137935, 99127, 52731, 0, 0, 119625, 3625, 0, 0, 0, 0, 402813, 29160, 185301, 2781, 21069, 1932, 4212, 360, 576, 1125, 0, 0,,,, 2160, 810, 0, 0, 111, 179, 43, 0, 0, 0, 0, 0, 0, 0, 185, 105, 0, 65, 0, 0,,,,, 0, 0, 0, 0, 337, 105, 36, 157, 37, 18, 18, 16, 4, 21, 0, 0, 0, 0, 0,,,,, 0, 0, 60, 40, 10, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0,,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0,,,, 0, 0, 0, 6, 8, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0,,, 0, 0, 0, 0, 0,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0 ] ] ## doc2/ctblpope.xml (2816-2819) gap> constit:= Filtered( RationalizedMat( Irr( m ) ), > chi -> ScalarProduct( m, chi, pi ) <> 0 );; ## doc2/ctblpope.xml (2829-2847) gap> cand:= PermChars( m, rec( torso:= cand[1], chars:= constit ) ); [ Character( CharacterTable( "M" ), [ 4050306254358548053604918389065234375, 148844831270071996434375, 2815847622206994375, 14567365753025085, 3447181417680, 659368198125, 3520153823175, 548464353255, 5706077895, 3056566695, 264515625, 0, 19572895485, 6486480, 186109245, 61410960, 758160, 688365, 58310, 0, 172503, 1264351, 376155, 137935, 99127, 52731, 0, 0, 119625, 3625, 0, 0, 0, 0, 402813, 29160, 185301, 2781, 21069, 1932, 4212, 360, 576, 1125, 0, 0, 1302, 294, 0, 2160, 810, 0, 0, 111, 179, 43, 0, 0, 0, 0, 0, 0, 0, 185, 105, 0, 65, 0, 0, 224, 0, 14, 0, 0, 0, 0, 0, 337, 105, 36, 157, 37, 18, 18, 16, 4, 21, 0, 0, 0, 0, 0, 70, 38, 14, 0, 0, 0, 60, 40, 10, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 24, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 8, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (2865-2869) gap> h:= CharacterTable( "2^[39].(L3(2)x3.S6)" );; gap> cand[1] = TrivialCharacter( h )^m; true ## doc2/ctblpope.xml (2905-2925) gap> co1:= CharacterTable( "Co1" );; gap> order:= 2^35*Size( CharacterTable( "L5(2)" ) )*6 / 2^25 / 31; 1981808640 gap> maxes:= List( Maxes( co1 ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^11:M24" ), CharacterTable( "2^(1+8)+.O8+(2)" ), CharacterTable( "2^(2+12):(A8xS3)" ) ] gap> List( filt, t -> Size( t ) / order ); [ 253, 45, 1 ] gap> m24:= CharacterTable( "M24" );; gap> cand:= PermChars( m24, rec( torso:=[ 253 ] ) ); [ Character( CharacterTable( "M24" ), [ 253, 29, 13, 10, 1, 5, 5, 1, 3, 2, 1, 1, 1, 1, 3, 0, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0 ] ) ] gap> TestPerm5( m24, cand, m24 mod 11 ); [ ] gap> PermChars( CharacterTable( "O8+(2)" ), rec( torso:=[ 45 ] ) ); [ ] gap> k:= filt[3];; ## doc2/ctblpope.xml (2963-2972) gap> subtbl:= CharacterTable( "2^4:A8" );; gap> subtom:= TableOfMarks( subtbl );; gap> perms:= PermCharsTom( subtbl, subtom );; gap> nsg:= ClassPositionsOfNormalSubgroups( subtbl ); [ [ 1 ], [ 1, 2 ], [ 1 .. 25 ] ] gap> above:= Filtered( perms, x -> x[1] = x[2] );; gap> tbl:= CharacterTable( "L5(2)" );; gap> above:= Set( Induced( subtbl, tbl, above ) );; ## doc2/ctblpope.xml (2985-2990) gap> index2:= Filtered( perms, > x -> Sum( PermCharInfo( subtbl, [x] ).contained[1]{ [1,2] } ) = 8 );; gap> index2:= Filtered( index2, x -> not x[1] in [ 630, 840, 1260, 1680 ] );; gap> index2:= Set( Induced( subtbl, tbl, index2 ) );; ## doc2/ctblpope.xml (2999-3023) gap> orders:= OrdersClassRepresentatives( tbl );; gap> goodclasses:= Filtered( [ 1 .. NrConjugacyClasses( tbl ) ], > i -> not orders[i] in [ 21, 31 ] ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ] gap> matrix:= List( Concatenation( above, index2 ), x -> x{ goodclasses } );; gap> sol:= SolutionMat( matrix, > ListWithIdenticalEntries( Length( goodclasses ), 1 ) ); [ 692/651, 57/217, -78/217, -26/217, 0, 74/651, 11/217, 0, 3/217, 151/651, 0, 22/651, 0, 0, 0, -11/217, 0, 0, 0, 0, 0, 0, 0, 0, -115/651, 0, -3/31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -34/93, -11/651, 0, 2/21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> nonzero:= Filtered( [ 1 .. Length( sol ) ], i -> sol[i] <> 0 ); [ 1, 2, 3, 4, 6, 7, 9, 10, 12, 16, 25, 27, 106, 107, 109, 120 ] gap> sol:= sol{ nonzero };; ## doc2/ctblpope.xml (3041-3065) gap> a8degrees:= List( above{ Filtered( nonzero, > x -> x <= Length( above ) ) }, > x -> x[1] ) / 31; [ 1, 8, 15, 28, 56, 56, 70, 105, 120, 168, 336, 336 ] gap> a8tbl:= subtbl / [ 1, 2 ];; gap> invtoa8:= InverseMap( GetFusionMap( subtbl, a8tbl ) );; gap> nsg:= ClassPositionsOfNormalSubgroups( k );; gap> nn:= First( nsg, x -> Sum( SizesConjugacyClasses( k ){ x } ) = 6*2^14 );; gap> a8tbl_other:= k / nn;; gap> g:= CharacterTable( "MC2B" ); CharacterTable( "2^1+24.Co1" ) gap> constit:= [];; gap> for i in [ 1 .. Length( a8degrees ) ] do > cand:= PermChars( a8tbl_other, rec( torso:= [ a8degrees[i] ] ) ); > filt:= Filtered( perms, x -> x^tbl = above[ nonzero[i] ] ); > filt:= List( filt, x -> CompositionMaps( x, invtoa8 ) ); > cand:= Filtered( cand, > x -> ForAny( filt, y -> Collected( x ) = Collected(y) ) ); > Add( constit, List( Induced( Restricted( Induced( > Restricted( cand, k ), co1 ), g ), m ), ValuesOfClassFunction ) ); > od; gap> List( constit, Length ); [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ## doc2/ctblpope.xml (3082-3123) gap> downdegrees:= List( index2{ Filtered( nonzero, > x -> x > Length( above ) ) > - Length( above ) }, > x -> x[1] ) / 31; [ 30, 210, 210, 1920 ] gap> f:= g / ClassPositionsOfCentre( g );; gap> forders:= OrdersClassRepresentatives( f );; gap> inv:= InverseMap( GetFusionMap( g, f ) );; gap> for j in [ 1 .. Length( downdegrees ) ] do > chars:= []; > cand:= PermChars( a8tbl_other, rec( torso:= [ downdegrees[j]/2 ] ) ); > filt:= Filtered( perms, x -> x^tbl = index2[ nonzero[ > j + Length( a8degrees ) ] - Length( above ) ] ); > filt:= Induced( subtbl, a8tbl, filt, > GetFusionMap( subtbl, a8tbl )); > cand:= Filtered( cand, x -> ForAny( filt, > y -> Collected( x ) = Collected( y ) ) ); > cand:= Restricted( Induced( Restricted( cand, k ), co1 ), g ); > for chi in cand do > cchi:= CompositionMaps( chi, inv ); > upper:= []; > pphi:= []; > for i in [ 1 .. NrConjugacyClasses( f ) ] do > if forders[i] mod 2 = 1 then > pphi[i]:= 2 * cchi[i]; > elif cchi[i] = 0 then > pphi[i]:= 0; > fi; > upper[i]:= 2* cchi[i]; > od; > Append( chars, PermChars( f, rec( torso:= ShallowCopy( pphi ), > upper:= upper, > normalsubgroup:= [ 1 .. 4 ], > nonfaithful:= cchi ) ) ); > od; > Add( constit, List( Induced( Restricted( chars, g ), m ), > ValuesOfClassFunction ) ); > od; gap> List( constit, Length ); [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 10, 10, 2 ] ## doc2/ctblpope.xml (3131-3143) gap> cand:= List( Cartesian( constit ), l -> sol * l );; gap> m:= CharacterTable( "M" ); CharacterTable( "M" ) gap> morders:= OrdersClassRepresentatives( m );; gap> for x in cand do > for i in [ 1 .. Length( morders ) ] do > if morders[i] mod 31 = 0 or morders[i] mod 21 = 0 then > Unbind( x[i] ); > fi; > od; > od; ## doc2/ctblpope.xml (3151-3167) gap> cand:= Filtered( cand, x -> ForAll( x, IsInt ) ); [ [ 391965121389536908413379198941796875, 23914487292951376996875, 474163138042468875, 9500455925885925, 646346515815, 334363486275, 954161764875, 147339103275, 1481392395, 1313281515, 0, 8203125, 9827885925, 1216215, 91556325, 9388791, 115911, 587331, 874650, 0, 79515, 581955, 336375, 104371, 62331, 36855, 0, 0, 0, 0, 28125, 525, 1125, 0, 188325, 16767, 88965, 2403, 9477, 1155, 891, 207, 351, 627, 0, 0, 4410, 1498, 0, 0, 0, 30, 150, 91, 151, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125, 0, 5, 5,,,,, 0, 0, 0, 0, 141, 45, 27, 61, 27, 9, 9, 7, 3, 15, 0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 30, 0, 0, 0, 6, 6, 6,,, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,,,, 0, 0, 0, 0, 0, 0,,, 0, 0, 0, 0, 0, 0,, 0, 0, 0 ] ] ## doc2/ctblpope.xml (3176-3194) gap> cand:= PermChars( m, rec( torso:= cand[1] ) ); [ Character( CharacterTable( "M" ), [ 391965121389536908413379198941796875, 23914487292951376996875, 474163138042468875, 9500455925885925, 646346515815, 334363486275, 954161764875, 147339103275, 1481392395, 1313281515, 0, 8203125, 9827885925, 1216215, 91556325, 9388791, 115911, 587331, 874650, 0, 79515, 581955, 336375, 104371, 62331, 36855, 0, 0, 0, 0, 28125, 525, 1125, 0, 188325, 16767, 88965, 2403, 9477, 1155, 891, 207, 351, 627, 0, 0, 4410, 1498, 0, 0, 0, 30, 150, 91, 151, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125, 0, 5, 5, 210, 0, 42, 0, 0, 0, 0, 0, 141, 45, 27, 61, 27, 9, 9, 7, 3, 15, 0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 30, 0, 0, 0, 6, 6, 6, 3, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 18, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (3211-3217) gap> h:= CharacterTable( "2^(5+10+20).(S3xL5(2))" );; gap> fus:= PossibleClassFusions( h, m );; gap> cand = Set( fus, map -> InducedClassFunctionsByFusionMap( h, m, > [ TrivialCharacter( h ) ], map )[1] ); true ## doc2/ctblpope.xml (3248-3258) gap> Hbar:= CharacterTable( "O10+(2)" );; gap> U_Abar:= CharacterTable( "O10+(2)M1" ); CharacterTable( "S8(2)" ) gap> Index( Hbar, U_Abar ); 496 gap> U_Bbar:= CharacterTable( "O10+(2)M2" ); CharacterTable( "2^8:O8+(2)" ) gap> Index( Hbar, U_Bbar ); 527 ## doc2/ctblpope.xml (3276-3290) gap> b:= CharacterTable( "B" ); CharacterTable( "B" ) gap> Horder:= 2^26 * Size( Hbar ); 1577011055923770163200 gap> order:= Horder / ( 2 * 496 ); 1589728887019929600 gap> maxes:= List( Maxes( b ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^(9+16).S8(2)" ) ] gap> List( filt, t -> Size( t ) / order ); [ 1 ] gap> u1:= filt[1]; CharacterTable( "2^(9+16).S8(2)" ) ## doc2/ctblpope.xml (3307-3318) gap> co1:= CharacterTable( "Co1" );; gap> order:= Horder / ( 2^25 * 527 ); 89181388800 gap> maxes:= List( Maxes( co1 ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^(1+8)+.O8+(2)" ) ] gap> List( filt, t -> Size( t ) / order ); [ 1 ] gap> u2:= filt[1]; CharacterTable( "2^(1+8)+.O8+(2)" ) ## doc2/ctblpope.xml (3327-3336) gap> m:= CharacterTable( "M" ); CharacterTable( "M" ) gap> 2b:= CharacterTable( "MC2A" ); CharacterTable( "2.B" ) gap> mm:= CharacterTable( "MC2B" ); CharacterTable( "2^1+24.Co1" ) gap> pi_A:= RestrictedClassFunction( TrivialCharacter( u1 )^b, 2b )^m;; gap> pi_B:= RestrictedClassFunction( TrivialCharacter( u2 )^co1, mm )^m;; ## doc2/ctblpope.xml (3344-3347) gap> torso:= [ Size( m ) / Horder ]; [ 512372707698741056749515292734375 ] ## doc2/ctblpope.xml (3389-3429) gap> morders:= OrdersClassRepresentatives( m );; gap> 2parts:= Union( [ 1 ], Filtered( Set( morders ), > x -> IsPrimePowerInt( x ) and IsEvenInt( x ) ) ); [ 1, 2, 4, 8, 16, 32 ] gap> factorders:= Set( OrdersClassRepresentatives( Hbar ) );; gap> primes_A:= Filtered( PrimeDivisors( Horder ), p -> 496 mod p <> 0 ); [ 3, 5, 7, 17 ] gap> primes_B:= Filtered( PrimeDivisors( Horder ), p -> 527 mod p <> 0 ); [ 2, 3, 5, 7 ] gap> primes:= Union( primes_A, primes_B );; gap> n:= NrConjugacyClasses( m );; gap> for i in [ 1 .. n ] do > if Horder mod morders[i] <> 0 then > torso[i]:= 0; > elif ForAll( factorders, x -> not morders[i] / x in 2parts ) then > torso[i]:= 0; > else > for p in primes do > if morders[i] mod p = 0 then > pprime:= morders[i]; > while pprime mod p = 0 do pprime:= pprime / p; od; > pos:= PowerMap( m, pprime )[i]; > if p in primes_A and pi_A[ pos ] = 0 then > torso[i]:= 0; > elif p in primes_B and pi_B[ pos ] = 0 then > torso[i]:= 0; > fi; > fi; > od; > fi; > od; gap> torso; [ 512372707698741056749515292734375,,,,, 0,,,,,,,,,,,, 0,, 0,,,,,,,,,, ,,,, 0,,,, 0,,,,,, 0, 0, 0,,, 0,,,, 0,,,,,,,,,, 0,,,,,,,, 0, 0, 0, 0, 0, 0, 0,,,,, 0,,,,, 0, 0, 0, 0, 0, 0,,,, 0, 0,,,,, 0,,,,,,, 0, 0, , 0, 0,,,,, 0, 0, 0, 0, 0,,,,, 0,, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, , 0,, 0, 0, 0, 0,, 0, 0, 0, 0, 0,,,,,, 0,,, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ## doc2/ctblpope.xml (3480-3504) gap> cand:= [ [ pi_A ], [ pi_B ] ];; gap> candbar:= [ TrivialCharacter( U_Abar )^Hbar, > TrivialCharacter( U_Bbar )^Hbar ];; gap> AddSubgroupOfS82:= function( subname ) > local psis82; > > psis82:= TrivialCharacter( CharacterTable( subname ) )^U_Abar; > Add( cand, [ Restricted( Restricted( psis82, u1 )^b, 2b )^m ] ); > Add( candbar, psis82 ^ Hbar ); > end;; gap> tt1:= CharacterTable( "O8+(2)" ); CharacterTable( "O8+(2)" ) gap> AddSubgroupOfO8p2:= function( subname ) > local psi, list, char; > > psi:= TrivialCharacter( CharacterTable( subname ) )^tt1; > list:= []; > for char in Orbit( AutomorphismsOfTable( tt1 ), psi, Permuted ) do > AddSet( list, Restricted( Restricted( char, u2 ) ^ co1, mm ) ^ m ); > od; > Add( cand, list ); > Add( candbar, Restricted( psi, U_Bbar ) ^ Hbar ); > end;; ## doc2/ctblpope.xml (3513-3525) gap> AddSubgroupOfS82( "O8+(2).2" ); gap> AddSubgroupOfO8p2( "S6(2)" ); gap> AddSubgroupOfS82( "O8-(2).2" ); gap> AddSubgroupOfS82( "A10.2" ); gap> AddSubgroupOfS82( "S4(4).2" ); gap> AddSubgroupOfS82( "L2(17)" ); gap> AddSubgroupOfO8p2( "A9" ); gap> AddSubgroupOfO8p2( "2^6:A8" ); gap> AddSubgroupOfO8p2( "(3xU4(2)):2" ); gap> AddSubgroupOfO8p2( "(A5xA5):2^2" ); gap> AddSubgroupOfS82( "S8(2)M4" ); ## doc2/ctblpope.xml (3538-3548) gap> a5:= CharacterTable( "A5" );; gap> fus:= PossibleClassFusions( a5, U_Abar )[1];; gap> NrPolyhedralSubgroups( U_Abar, fus[2], fus[3], fus[4] ); rec( number := 548352, type := "A5" ) gap> psis82:= Induced( a5, U_Abar, [ TrivialCharacter( a5 ) ], fus )[1];; gap> Add( cand, [ Restricted( Restricted( psis82, u1 )^b, 2b )^m ] ); gap> Add( candbar, psis82 ^ Hbar ); gap> List( cand, Length ); [ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1 ] ## doc2/ctblpope.xml (3563-3598) gap> Hbarorders:= OrdersClassRepresentatives( Hbar );; gap> TryCondition:= function( cond ) > local pos, sol, lincomb, oldknown, i; > > pos:= Filtered( [ 1 .. Length( Hbarorders ) ], > i -> cond( Hbarorders[i] ) ); > sol:= SolutionMat( candbar{[1..Length(candbar)]}{ pos }, > ListWithIdenticalEntries( Length( pos ), 1 ) ); > if sol = fail then > return "no solution"; > fi; > > pos:= Filtered( [ 1 .. Length( morders) ], i -> cond( morders[i] ) ); > lincomb:= Filtered( Set( Cartesian( cand ), x -> sol * x ), > x -> ForAll( pos, i -> IsPosInt( x[i] ) or x[i] = 0 ) ); > if Length( lincomb ) <> 1 then > return "solution is not unique"; > fi; > > lincomb:= lincomb[1];; > oldknown:= Number( torso ); > for i in pos do > if IsBound( torso[i] ) then > if torso[i] <> lincomb[i] then > Error( "contradiction of new and known value at position ", i ); > fi; > elif not IsInt( lincomb[i] ) or lincomb[i] < 0 then > Error( "new value at position ", i, " is not a nonneg. integer" ); > fi; > torso[i]:= lincomb[i]; > od; > return Concatenation( "now ", String( Number( torso ) ), " values (", > String( Number( torso ) - oldknown ), " new)" ); > end;; ## doc2/ctblpope.xml (3618-3631) gap> TryCondition( x -> x mod 7 = 0 and x mod 3 <> 0 ); "now 99 values (7 new)" gap> TryCondition( x -> x mod 17 = 0 and x mod 3 <> 0 ); "now 102 values (3 new)" gap> TryCondition( x -> x mod 5 = 0 and x mod 3 <> 0 ); "now 119 values (17 new)" gap> TryCondition( x -> 4 mod x = 0 ); "now 125 values (6 new)" gap> TryCondition( x -> 9 mod x = 0 ); "now 129 values (4 new)" gap> TryCondition( x -> x in [ 9, 18, 36 ] ); "now 138 values (9 new)" ## doc2/ctblpope.xml (3640-3644) gap> constit:= Filtered( RationalizedMat( Irr( m ) ), > x -> ScalarProduct( m, x, pi_A ) <> 0 > and ScalarProduct( m, x, pi_B ) <> 0 );; ## doc2/ctblpope.xml (3657-3668) gap> lower:= [];; gap> Hindex:= Size( m ) / Horder; 512372707698741056749515292734375 gap> for i in [ 1 .. NrConjugacyClasses( m ) ] do > lower[i]:= Maximum( pi_A[i] / ( pi_A[1] / Hindex ), > pi_B[i] / ( pi_B[1] / Hindex ) ); > if not IsInt( lower[i] ) then > lower[i]:= Int( lower[i] + 1 ); > fi; > od; ## doc2/ctblpope.xml (3679-3699) gap> cand:= PermChars( m, rec( torso:= torso, chars:= constit, > lower:= lower, normalsubgroup:= [ 1 .. NrConjugacyClasses( m ) ], > nonfaithful:= TrivialCharacter( m ) ) ); [ Character( CharacterTable( "M" ), [ 512372707698741056749515292734375, 405589064025344574375, 29628786742129575, 658201521662685, 215448838605, 0, 121971774375, 28098354375, 335229607, 108472455, 164587500, 4921875, 2487507165, 2567565, 26157789, 6593805, 398925, 0, 437325, 0, 44983, 234399, 90675, 21391, 41111, 12915, 6561, 6561, 177100, 7660, 6875, 315, 275, 0, 113373, 17901, 57213, 0, 4957, 1197, 909, 301, 397, 0, 0, 0, 3885, 525, 0, 2835, 90, 45, 0, 103, 67, 43, 28, 81, 189, 9, 9, 9, 0, 540, 300, 175, 20, 15, 7, 420, 0, 0, 0, 0, 0, 0, 0, 165, 61, 37, 37, 0, 9, 9, 13, 5, 0, 0, 0, 0, 0, 0, 77, 45, 13, 0, 0, 45, 115, 19, 10, 0, 5, 5, 9, 9, 1, 1, 0, 0, 4, 0, 0, 9, 9, 3, 1, 0, 0, 0, 0, 0, 0, 4, 1, 1, 0, 24, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 3, 1, 1, 2, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] ## doc2/ctblpope.xml (3714-3720) gap> h:= CharacterTable( "2^(10+16).O10+(2)" );; gap> fus:= PossibleClassFusions( h, m );; gap> cand = Set( fus, map -> InducedClassFunctionsByFusionMap( h, m, > [ TrivialCharacter( h ) ], map )[1] ); true ## doc2/ctblpope.xml (3745-3750) gap> t:= CharacterTable( "S5" ); CharacterTable( "A5.2" ) gap> pi:= PermChars( t, rec( torso:= [ 15 ] ) ); [ Character( CharacterTable( "A5.2" ), [ 15, 3, 0, 0, 3, 1, 0 ] ) ] ## doc2/ctblpope.xml (3759-3768) gap> m222:= CharacterTable( "M22.2" ); CharacterTable( "M22.2" ) gap> mx:= List( Maxes( m222 ), CharacterTable );; gap> mx:= Filtered( mx, x -> Size( m222 ) / Size( x ) = 231 ); [ CharacterTable( "M22.2M4" ) ] gap> pi:= pi[1]{ GetFusionMap( mx[1], t ) }; [ 15, 15, 3, 3, 3, 0, 0, 3, 3, 1, 1, 0, 15, 15, 3, 3, 3, 0, 0, 3, 3, 1, 1, 0 ] ## doc2/ctblpope.xml (3778-3783) gap> pi:= InducedClassFunction( mx[1], pi, m222 ); ClassFunction( CharacterTable( "M22.2" ), [ 3465, 105, 0, 9, 5, 0, 0, 0, 0, 1, 0, 189, 45, 9, 13, 0, 1, 0, 0, 0, 0 ] ) ## doc2/ctblpope.xml (3793-3805) gap> co2:= CharacterTable( "Co2" ); CharacterTable( "Co2" ) gap> mx:= List( Maxes( co2 ), CharacterTable );; gap> mx:= Filtered( mx, x -> Size( co2 ) / Size( x ) = 46575 ); [ CharacterTable( "2^10:m22:2" ) ] gap> pi:= pi{ GetFusionMap( mx[1], m222 ) }; [ 3465, 3465, 3465, 3465, 105, 105, 105, 105, 105, 105, 105, 105, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 189, 189, 189, 189, 189, 189, 45, 45, 45, 45, 9, 9, 9, 9, 13, 13, 13, 13, 13, 13, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ## doc2/ctblpope.xml (3813-3820) gap> pi:= InducedClassFunction( mx[1], pi, co2 ); ClassFunction( CharacterTable( "Co2" ), [ 161382375, 626535, 162855, 27495, 0, 0, 6615, 3975, 2727, 855, 567, 975, 115, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 51, 19, 27, 35, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ## doc2/ctblpope.xml (3831-3840) gap> b:= CharacterTable( "B" ); CharacterTable( "B" ) gap> mx:= List( Maxes( b ), CharacterTable );; gap> mx:= Filtered( mx, x -> Size( b ) / Size( x ) = 11707448673375 ); [ CharacterTable( "2^(1+22).Co2" ) ] gap> pi:= pi{ GetFusionMap( mx[1], co2 ) };; gap> pi[1]; 161382375 ## doc2/ctblpope.xml (3848-3863) gap> pi:= InducedClassFunction( mx[1], pi, b ); ClassFunction( CharacterTable( "B" ), [ 1889375872099856765625, 2609385408855225, 62316674429625, 207818526825, 268788490425, 0, 0, 13052741625, 7537207545, 128298681, 270580905, 46366425, 74315385, 35633385, 3937689, 201825, 1233225, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 713097, 241425, 320625, 88521, 275265, 57705, 19305, 20089, 9441, 6489, 2577, 1825, 5345, 753, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 273, 417, 105, 97, 185, 33, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ## doc2/ctblpope.xml (3922-3932) gap> mcl:= CharacterTable( "McL" ); CharacterTable( "McL" ) gap> co2:= CharacterTable( "Co2" ); CharacterTable( "Co2" ) gap> ind:= Induced( mcl, co2, [ TrivialCharacter( mcl ) ] )[1]; Character( CharacterTable( "Co2" ), [ 47104, 0, 1024, 0, 16, 160, 0, 0, 0, 0, 64, 0, 0, 4, 24, 16, 0, 0, 0, 16, 0, 8, 0, 0, 0, 0, 0, 8, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ] ) ## doc2/ctblpope.xml (3941-3968) gap> m:= CharacterTable( "BM2" ); CharacterTable( "2^(1+22).Co2" ) gap> infl:= ind{ GetFusionMap( m, co2 ) }; [ 47104, 47104, 47104, 47104, 47104, 47104, 47104, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 16, 16, 160, 160, 160, 160, 160, 160, 160, 160, 160, 160, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, 24, 24, 24, 24, 24, 24, 24, 24, 16, 16, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 ] ## doc2/ctblpope.xml (3976-3993) gap> b:= CharacterTable( "B" ); CharacterTable( "B" ) gap> ind:= Induced( m, b, [ infl ] )[1]; ClassFunction( CharacterTable( "B" ), [ 551467662310656000, 186911262720, 272993634304, 0, 634521600, 194594400, 69984, 8495104, 17465344, 129024, 276480, 2073600, 16384, 798720, 46080, 0, 5120, 138600, 1000, 110880, 252000, 112480, 432, 12960, 0, 1312, 8352, 864, 432, 0, 2520, 0, 2880, 2880, 3072, 2880, 0, 0, 256, 64, 1152, 576, 640, 192, 96, 0, 108, 2520, 744, 0, 104, 120, 40, 30, 160, 16, 1120, 1024, 0, 0, 96, 288, 64, 144, 0, 96, 0, 108, 16, 48, 0, 32, 12, 0, 0, 0, 168, 0, 104, 48, 0, 4, 0, 0, 0, 0, 32, 16, 8, 8, 0, 24, 12, 4, 0, 0, 0, 0, 24, 4, 24, 24, 0, 0, 0, 0, 4, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 8, 0, 16, 8, 4, 0, 0, 0, 0, 0, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ## doc2/ctblpope.xml (4001-4021) gap> 2b:= CharacterTable( "2.B" ); CharacterTable( "2.B" ) gap> infl:= ind{ GetFusionMap( 2b, b ) }; [ 551467662310656000, 551467662310656000, 186911262720, 272993634304, 272993634304, 0, 634521600, 194594400, 194594400, 69984, 69984, 8495104, 17465344, 129024, 276480, 2073600, 2073600, 16384, 798720, 46080, 0, 5120, 138600, 138600, 1000, 1000, 110880, 252000, 112480, 112480, 432, 12960, 0, 1312, 1312, 8352, 864, 864, 432, 0, 2520, 2520, 0, 2880, 2880, 3072, 2880, 0, 0, 256, 64, 1152, 576, 576, 640, 192, 96, 0, 0, 108, 108, 2520, 744, 744, 0, 104, 104, 120, 40, 40, 30, 30, 160, 16, 1120, 1024, 0, 0, 0, 96, 288, 64, 144, 144, 0, 96, 0, 108, 108, 16, 48, 0, 32, 12, 12, 0, 0, 0, 0, 168, 0, 104, 104, 48, 0, 0, 4, 4, 0, 0, 0, 0, 32, 16, 8, 8, 8, 0, 0, 24, 12, 4, 4, 0, 0, 0, 0, 0, 0, 24, 4, 24, 24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 6, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 0, 16, 8, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ## doc2/ctblpope.xml (4042-4078) gap> centre:= ClassPositionsOfCentre( 2b ); [ 1, 2 ] gap> pi:= PermChars( 2b, rec( torso:= [ 2 * infl[1], 0 ], > normalsubgroup:= centre, > nonfaithful:= infl ) ); [ Character( CharacterTable( "2.B" ), [ 1102935324621312000, 0, 186911262720, 541790208000, 4197060608, 0, 634521600, 389188800, 0, 139968, 0, 8495104, 17465344, 129024, 276480, 4026240, 120960, 16384, 798720, 46080, 0, 5120, 277200, 0, 2000, 0, 110880, 252000, 190080, 34880, 432, 12960, 0, 2592, 32, 8352, 1728, 0, 432, 0, 5040, 0, 0, 2880, 2880, 3072, 2880, 0, 0, 256, 64, 1152, 1008, 144, 640, 192, 96, 0, 0, 216, 0, 2520, 960, 528, 0, 200, 8, 120, 80, 0, 60, 0, 160, 16, 1120, 1024, 0, 0, 0, 96, 288, 64, 216, 72, 0, 96, 0, 216, 0, 16, 48, 0, 32, 24, 0, 0, 0, 0, 0, 168, 0, 160, 48, 48, 0, 0, 8, 0, 0, 0, 0, 0, 32, 16, 8, 12, 4, 0, 0, 24, 12, 0, 8, 0, 0, 0, 0, 0, 0, 24, 4, 24, 24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 6, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 0, 16, 8, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> MatScalarProducts( 2b, Irr( 2b ), pi ); [ [ 1, 1, 2, 1, 2, 0, 2, 3, 2, 0, 0, 1, 4, 1, 2, 0, 3, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 3, 1, 5, 0, 4, 3, 2, 0, 0, 3, 2, 0, 6, 4, 0, 1, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 5, 0, 5, 2, 0, 0, 2, 0, 0, 4, 1, 0, 2, 0, 4, 2, 4, 4, 3, 0, 2, 4, 2, 4, 0, 3, 0, 3, 2, 5, 0, 1, 0, 3, 1, 0, 1, 1, 2, 5, 3, 1, 1, 4, 5, 1, 1, 0, 3, 0, 0, 3, 2, 1, 1, 2, 1, 1, 4, 0, 3, 2, 3, 1, 3, 0, 1, 3, 0, 2, 2, 1, 3, 3, 0, 0, 2, 0, 0, 0, 0, 3, 0, 3, 3, 3, 1, 0, 3, 0, 4, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 3, 3, 0, 0, 0, 1, 1, 1, 1, 2, 3, 2, 0, 0, 2, 2, 4, 3, 5, 2, 4, 0, 0, 0, 0, 5, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 7, 0, 0, 1, 7, 7, 0, 0, 0, 1, 6, 4, 5, 0, 0, 3, 0, 0, 0, 0, 0, 4, 1, 1, 3, 8, 3, 2, 2, 5, 0, 1 ] ] ## doc2/ctblpope.xml (4180-4201) gap> names:= AllCharacterTableNames( IsSporadicSimple, true, > IsDuplicateTable, false : OrderedBy:= Size ); [ "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" ] gap> normindices:= rec( Co1:= 1, J4:= 1, F3\+:= 1, B:= 1, M:= 1 );; gap> for name in names do > n:= CharacterTable( Concatenation( name, "N2" ) ); > if n = fail then > Print( name, "\n" ); > else > 2part:= 2^Length( Positions( Factors( Size( n ) ), 2 ) ); > normindices.( name ):= Size( n ) / 2part; > fi; > od; Co1 J4 F3+ B M ## doc2/ctblpope.xml (4212-4226) gap> maxbound:= [];; gap> for name in names do > t:= CharacterTable( name ); > mx:= List( Maxes( t ), CharacterTable ); > odd:= Filtered( mx, s -> ( Size( t ) / Size( s ) ) mod 2 <> 0 ); > primperm:= List( odd, s -> TrivialCharacter( s )^t ); > sum:= normindices.( name ) * Sum( primperm, pi -> pi / pi[1] ); > Add( maxbound, > [ name, Maximum( sum{ [ 2 .. Length( sum ) ] } ) ] ); > od; gap> SortBy( maxbound, x -> - x[2] ); gap> maxbound[1]; [ "J2", 3/5 ] ## doc2/ctblpope.xml (4279-4336) gap> upper_bound:= function( tblS, tblM, p ) > local ppart, ppartposS, ppartposM, n, N_S, f, subname, u; > > ppart:= Product( Filtered( Factors( Size( tblS ) ), x -> x = p ), 1 ); > ppartposS:= Positions( OrdersClassRepresentatives( tblS ), ppart ); > if 0 < Length( ppartposS ) then > # P is cyclic. > if tblM = fail then > return ( SizesCentralizers( tblS )[ ppartposS[1] ] * Phi( ppart ) > / Length( ppartposS ) ) / ppart; > else > ppartposM:= Positions( OrdersClassRepresentatives( tblM ), ppart ); > return ( SizesCentralizers( tblS )[ ppartposS[1] ] * Phi( ppart ) > / Length( ppartposS ) ) / > ( SizesCentralizers( tblM )[ ppartposM[1] ] * Phi( ppart ) > / Length( ppartposM ) ); > fi; > fi; > > # Compute |N_S(P)|. > n:= CharacterTable( Concatenation( Identifier( tblS ), "N", > String( p ) ) ); > if n <> fail then > N_S:= Size( n ); > elif p = 2 then > N_S:= ppart * normindices.( Identifier( tblS ) ); > elif Identifier( tblS ) = "M" and p = 3 then > # The Sylow 3-normalizer is contained in 3^(3+2+6+6):(L3(3)xSD16) > N_S:= ppart * 2^6; > elif Identifier( tblS ) = "F3+" and p = 3 then > N_S:= ppart * 8; > else > Error( "cannot compute |N_S(P)|" ); > fi; > > if tblM = fail then > return N_S / ppart; > elif Sum( SizesConjugacyClasses( tblM ){ > ClassPositionsOfPCore( tblM, p ) } ) = ppart then > # P is normal in M. > return N_S / Size( tblM ); > fi; > > # Inspect known character tables of subgroups of M. > f:= N_S / ppart; > for subname in NamesOfFusionSources( tblM ) do > u:= CharacterTable( subname ); > if ClassPositionsOfKernel( GetFusionMap( u, tblM ) ) = [ 1 ] and > Sum( SizesConjugacyClasses( u ){ > ClassPositionsOfPCore( u, p ) } ) = ppart then > f:= Minimum( f, N_S / Size( u ) ); > fi; > od; > > return f; > end;; ## doc2/ctblpope.xml (4380-4438) gap> badcases_thm:= [];; gap> badcases_strong:= [];; gap> good:= "dummy";; # Avoid a message about an unbound variable ... gap> for name in Filtered( names, x -> x <> "M" ) do > t:= CharacterTable( name ); > orders:= OrdersClassRepresentatives( t ); > n:= NrConjugacyClasses( t ); > mx:= List( Maxes( t ), CharacterTable ); > for p in PrimeDivisors( Size( t ) ) do > good:= Filtered( mx, s -> ( Size( t ) / Size( s ) ) mod p <> 0 ); > primperm:= []; > for s in good do > if GetFusionMap( s, t ) <> fail then > Add( primperm, TrivialCharacter( s )^t ); > else > ind:= Set( PossibleClassFusions( s, t ), > map -> InducedClassFunctionsByFusionMap( s, t, > [ TrivialCharacter( s ) ], map )[1] ); > if Length( ind ) <> 1 then > Error( "permutation character not uniquely determined" ); > fi; > Add( primperm, ind[1] ); > fi; > od; > sum:= Sum( [ 1 .. Length( good ) ], > i -> upper_bound( t, good[i], p ) > * primperm[i] / primperm[i][1] ); > badpos:= Filtered( [ 2 .. Length( sum ) ], i -> sum[i] >= 1 ); > if badpos <> [] then > Add( badcases_strong, [ name, p, ShallowCopy( badpos ) ] ); > for i in ShallowCopy( badpos ) do > q:= SmallestRootInt( orders[i] ); > if IsPrimeInt( q ) then > if ForAny( [ 2 .. n ], > j -> SmallestRootInt( orders[j] ) = q > and not j in badpos ) then > RemoveSet( badpos, i ); > fi; > fi; > od; > if not IsEmpty( badpos ) then > Add( badcases_thm, [ name, p, badpos ] ); > fi; > fi; > od; > od; gap> badcases_thm; [ [ "M23", 3, [ 3 ] ], [ "HS", 3, [ 4, 11 ] ] ] gap> badcases_strong; [ [ "M11", 5, [ 2 ] ], [ "M12", 5, [ 3, 4 ] ], [ "M22", 3, [ 2 ] ], [ "M22", 5, [ 2 ] ], [ "J2", 3, [ 2 ] ], [ "M23", 3, [ 2, 3 ] ], [ "M23", 5, [ 2 ] ], [ "M23", 7, [ 2 ] ], [ "HS", 3, [ 2, 3, 4, 5, 6, 7, 9, 11 ] ], [ "HS", 5, [ 2, 3, 5 ] ], [ "M24", 5, [ 2, 4 ] ], [ "M24", 7, [ 2, 4 ] ], [ "He", 5, [ 2 ] ], [ "Co2", 3, [ 2, 3 ] ], [ "Fi22", 5, [ 2 ] ], [ "Fi22", 7, [ 2 ] ], [ "Fi23", 5, [ 2, 3, 5 ] ], [ "Fi23", 7, [ 2 ] ], [ "B", 7, [ 2 ] ] ] ## doc2/ctblpope.xml (4461-4496) gap> prove_generation:= function( name, p ) > local S, prg, P, reps, good, x, g, U; > > prg:= AtlasProgram( name, "classes" ); > if prg = fail then > return fail; > fi; > > S:= AtlasGroup( name ); > P:= SylowSubgroup( S, p ); > reps:= ResultOfStraightLineProgram( prg.program, GeneratorsOfGroup( S ) ); > good:= []; > for x in Filtered( reps, x -> Order( x ) <> 1 ) do > repeat > g:= Random( S ); > U:= ClosureGroup( P, x^g ); > until Size( U ) = Size( S ); > Add( good, x^g ); > od; > > return [ P, good ]; > end;; gap> for entry in badcases_strong do > res:= prove_generation( entry[1], entry[2] ); > if res = fail then > Print( "no classes script for ", entry, "\n" ); > fi; > od; no classes script for [ "He", 5, [ 2 ] ] no classes script for [ "Fi22", 5, [ 2 ] ] no classes script for [ "Fi22", 7, [ 2 ] ] no classes script for [ "Fi23", 5, [ 2, 3, 5 ] ] no classes script for [ "Fi23", 7, [ 2 ] ] no classes script for [ "B", 7, [ 2 ] ] ## doc2/ctblpope.xml (4511-4522) gap> S:= AtlasGroup( "He" );; gap> repeat > x:= Random( S ); > until Order( x ) = 10; gap> x:= x^5;; gap> P5:= SylowSubgroup( S, 5 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S ); ## doc2/ctblpope.xml (4531-4547) gap> S:= AtlasGroup( "Fi22" );; gap> repeat > x:= Random( S ); > until Order( x ) = 30; gap> x:= x^15;; gap> P5:= SylowSubgroup( S, 5 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S ); gap> P7:= SylowSubgroup( S, 7 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P7, x^g ); > until Size( U ) = Size( S ); ## doc2/ctblpope.xml (4556-4572) gap> S:= AtlasGroup( "Fi23" );; gap> repeat > x:= Random( S ); > until Order( x ) = 42; gap> x:= x^21;; gap> P5:= SylowSubgroup( S, 5 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S ); gap> P7:= SylowSubgroup( S, 7 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P7, x^g ); > until Size( U ) = Size( S ); ## doc2/ctblpope.xml (4581-4590) gap> repeat > x:= Random( S ); > until Order( x ) = 60; gap> x:= x^30;; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S ); ## doc2/ctblpope.xml (4599-4608) gap> repeat > x:= Random( S ); > until Order( x ) = 60; gap> x:= x^20;; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S ); ## doc2/ctblpope.xml (4624-4635) gap> t:= CharacterTable( "B" );; gap> 7pos:= Positions( OrdersClassRepresentatives( t ), 7 ); [ 31 ] gap> 47pos:= Positions( OrdersClassRepresentatives( t ), 47 ); [ 172, 173 ] gap> ClassMultiplicationCoefficient( t, 2, 7pos[1], 47pos[1] ); 7332 gap> Filtered( Maxes( t ), > x -> Size( CharacterTable( x ) ) mod 47 = 0 ); [ "47:23" ] ## doc2/ctblpope.xml (4668-4699) gap> dir:= DirectoriesPackageLibrary( "ctbllib", "data" );; gap> filename:= Filename( dir, "prim_perm_M.json" );; gap> Monster_prim_data:= EvalString( StringFile( filename ) )[2];; gap> Length( Monster_prim_data ); 46 gap> t:= CharacterTable( "M" );; gap> monstermaxindices:= [];; gap> monstermaxtables:= [];; gap> for entry in Monster_prim_data do > if Length( entry ) = 1 then > s:= CharacterTable( entry[1] ); > Add( monstermaxtables, s ); > Add( monstermaxindices, Size( t ) / Size( s ) ); > else > Add( monstermaxtables, fail ); > Add( monstermaxindices, entry[2][1] ); > fi; > od; gap> odd_prim:= [];; gap> for i in [ 1 .. Length( Monster_prim_data ) ] do > if monstermaxindices[i] mod 2 <> 0 then > if monstermaxtables[i] <> fail then > Add( odd_prim, TrivialCharacter( monstermaxtables[i] )^t ); > else > Add( odd_prim, Monster_prim_data[i][2] ); > fi; > fi; > od; gap> Length( odd_prim ); 5 ## doc2/ctblpope.xml (4707-4713) gap> sum:= normindices.M * Sum( odd_prim, pi -> pi / pi[1] );; gap> max:= Maximum( sum{ [ 2 .. Length( sum ) ] } ); 12784979/103007903752128375 gap> max < 10^-9; true ## doc2/ctblpope.xml (4745-4780) gap> t:= CharacterTable( "M" );; gap> orders:= OrdersClassRepresentatives( t );; gap> for p in Difference( PrimeDivisors( Size( t ) ), [ 19 ] ) do > goodpos:= Filtered( [ 1 .. Length( Monster_prim_data ) ], > i -> monstermaxindices[i] mod p <> 0 ); > sum:= ListWithIdenticalEntries( NrConjugacyClasses( t ), 0 ); > for i in goodpos do > if Length( Monster_prim_data[i] ) = 2 then > # We know the permutation character but not the subgroup table. > sum:= sum + upper_bound( t, fail, p ) > * Monster_prim_data[i][2] / monstermaxindices[i]; > else > s:= monstermaxtables[i]; > if GetFusionMap( s, t ) <> fail then > # We can compute the permutation character. > sum:= sum + upper_bound( t, s, p ) > * TrivialCharacter( s )^t / monstermaxindices[i]; > else > # We get only candidates for the permutation character. > cand:= Set( PossibleClassFusions( s, t ), > map -> InducedClassFunctionsByFusionMap( s, t, > [ TrivialCharacter( s ) ], map )[1] ); > # For each class, take the maximum of the possible values. > sum:= sum + upper_bound( t, s, p ) > * List( TransposedMat( cand ), Maximum ) > / monstermaxindices[i]; > fi; > fi; > od; > badpos:= Filtered( [ 2 .. Length( sum ) ], i -> sum[i] >= 1 ); > if badpos <> [] then > Error( "check open cases in ", badpos, "\n" ); > fi; > od; ## doc2/ctblpope.xml (4795-4808) gap> pos19:= Positions( OrdersClassRepresentatives( t ), 19 ); [ 63 ] gap> pos71:= Positions( OrdersClassRepresentatives( t ), 71 ); [ 169, 170 ] gap> ClassMultiplicationCoefficient( t, pos19[1], pos19[1], pos71[1] ); 621743152370566020417806353602387433415016198936 gap> ForAny( monstermaxindices, > x -> ( Size( t ) / x ) mod ( 19 * 71 ) = 0 ); false gap> ForAny( [ "L2(13)", "U3(4)", "U3(8)", "Sz(8)" ], > x -> Size( CharacterTable( x ) ) mod 71 = 0 ); false ## gap> if IsBound( BrowseData ) then > data:= BrowseData.defaults.dynamic.replayDefaults; > data.replayInterval:= oldinterval; > fi; ## gap> STOP_TEST( "ctblpope.tst" ); gap> SizeScreen( save );; ############################################################################# ## #E [zur Elbe Produktseite wechseln0.79QuellennavigatorsAnalyse erneut starten2026-06-17] |
2026-07-27