pgr12:=function()
local d,g,sg,p;
d:=FreeGroup("a","b","c");
g:=d/[d.1^2,d.2^2,d.3^2,d.1*d.2*d.3*d.1*d.2*d.1*d.3*d.2,d.1*d.2*d.3*d.1*d.3*d.2*d.1*d.3];
sg:=Subgroup(g,[g.1]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G12:=pgr12(); gap> Size(G12); 48 gap> N12:=NormalSubgroups(G12);; gap> NTSize(N12); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 8 Groesse des 4. Normalteilers: 24 Groesse des 5. Normalteilers: 48 gap> Centre(G12); Group([ ( 1,22)( 2,16)( 3,20)( 4, 8)( 5,21)( 6,13)( 7,19)( 9,17)(10,24)(11,15) (12,18)(14,23) ]) gap> Size(Centre(G12)); 2 gap> IsSolvable(G12); true gap> CompositionSeries(G12); [ <permutation group of size 48 with 5 generators>, <permutation group of size 24 with 4 generators>, Group([ ( 1,11,22,15)( 2,23,16,14)( 3,10,20,24)( 4, 7, 8,19)( 5, 6,21,13) ( 9,12,17,18), ( 1,17,22, 9)( 2, 5,16,21)( 3, 4,20, 8)( 6,23,13,14) ( 7,10,19,24)(11,12,15,18), ( 1,22)( 2,16)( 3,20)( 4, 8)( 5,21)( 6,13) ( 7,19)( 9,17)(10,24)(11,15)(12,18)(14,23) ]), Group([ ( 1,17,22, 9)( 2, 5,16,21)( 3, 4,20, 8)( 6,23,13,14)( 7,10,19,24) (11,12,15,18), ( 1,22)( 2,16)( 3,20)( 4, 8)( 5,21)( 6,13)( 7,19) ( 9,17)(10,24)(11,15)(12,18)(14,23) ]), Group([ ( 1,22)( 2,16)( 3,20)( 4, 8)( 5,21)( 6,13)( 7,19)( 9,17)(10,24) (11,15)(12,18)(14,23) ]), Group(()) ]
gap> ct12:=CharacterTable(G12); CharacterTable( Group( [ ( 2, 4)( 3, 6)( 5,10)( 7,14)( 8,16)( 9,18)(11,15)(12,17)(13,20)(19,23) (21,24), ( 1, 2)( 3, 7)( 4, 8)( 5,11)( 6,12)( 9,14)(13,18)(15,21)(16,22) (17,23)(19,20), ( 1, 3)( 2, 5)( 4, 9)( 6,13)( 7,15)( 8,17)(10,12)(11,19) (16,21)(18,24)(20,22) ]) ) gap> Display(ct12); CT1 2 4 2 1 3 3 1 3 4 3 1 . 1 . . 1 . 1 1a 2a 6a 8a 8b 3a 4a 2b 2P 1a 1a 3a 4a 4a 3a 2b 1a 3P 1a 2a 2b 8a 8b 1a 4a 2b 5P 1a 2a 6a 8b 8a 3a 4a 2b 7P 1a 2a 6a 8b 8a 3a 4a 2b X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 1 1 X.3 2 . -1 . . -1 2 2 X.4 2 . 1 A -A -1 . -2 X.5 2 . 1 -A A -1 . -2 X.6 3 -1 . 1 1 . -1 3 X.7 3 1 . -1 -1 . -1 3 X.8 4 . -1 . . 1 . -4 A = -E(8)-E(8)^3 = -ER(-2) = -i2 gap> quit;