pgr24:=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.1*d.2*d.1*d.2*d.1*d.2,d.2*d.3*d.2*d.3*d.2*d.3*d.2*d.3,d.1*d.3*d.1*d.3*d.1*d.3,d.3*d.2*d.3*d.1*d.3*d.2*d.3*d.1*d.3*d.2*d.3*d.1];
sg:=Subgroup(g,[g.1,g.2]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G24:=pgr24(); gap> Size(G24); 336 gap> N24:=NormalSubgroups(G24);; gap> NTSize(N24); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 168 Groesse des 4. Normalteilers: 336 gap> Centre(G24); Group([ ( 1,42)( 2,38)( 3,39)( 4,32)( 5,35)( 6,34)( 7,25)( 8,28)( 9,40)(10,27) (11,33)(12,18)(13,19)(14,37)(15,21)(16,26)(17,36)(20,31)(22,30)(23,41) (24,29) ]) gap> Size(Centre(G24)); 2 gap> IsSolvable(G24); false gap> CompositionSeries(G24); [ <permutation group of size 336 with 4 generators>, Group([ ( 1,42)( 2,38)( 3,39)( 4,32)( 5,35)( 6,34)( 7,25)( 8,28)( 9,40) (10,27)(11,33)(12,18)(13,19)(14,37)(15,21)(16,26)(17,36)(20,31)(22,30) (23,41)(24,29) ]), Group(()) ]
gap> ct24:=CharacterTable(G24); CharacterTable( <permutation group of size 336 with 3 generators> ) gap> Display(ct24); CT1 2 4 4 3 4 1 1 1 1 3 1 1 4 3 1 . . . 1 . . 1 . . . 1 7 1 . . . . 1 1 . . 1 1 1 1a 2a 4a 2b 3a 14a 14b 6a 4b 7a 7b 2c 2P 1a 1a 2b 1a 3a 7a 7b 3a 2b 7a 7b 1a 3P 1a 2a 4a 2b 1a 14b 14a 2c 4b 7b 7a 2c 5P 1a 2a 4a 2b 3a 14b 14a 6a 4b 7b 7a 2c 7P 1a 2a 4a 2b 3a 2c 2c 6a 4b 1a 1a 2c 11P 1a 2a 4a 2b 3a 14a 14b 6a 4b 7a 7b 2c 13P 1a 2a 4a 2b 3a 14b 14a 6a 4b 7b 7a 2c X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 X.3 3 -1 1 -1 . A /A . 1 A /A 3 X.4 3 1 1 -1 . -A -/A . -1 A /A -3 X.5 3 -1 1 -1 . /A A . 1 /A A 3 X.6 3 1 1 -1 . -/A -A . -1 /A A -3 X.7 6 2 . 2 . -1 -1 . . -1 -1 6 X.8 6 -2 . 2 . 1 1 . . -1 -1 -6 X.9 7 -1 -1 -1 1 . . 1 -1 . . 7 X.10 7 1 -1 -1 1 . . -1 1 . . -7 X.11 8 . . . -1 1 1 -1 . 1 1 8 X.12 8 . . . -1 -1 -1 1 . 1 1 -8 A = E(7)^3+E(7)^5+E(7)^6 = (-1-ER(-7))/2 = -1-b7 gap> quit;