pgr13:=function()
local d,g,sg,p;
d:=FreeGroup("a","b","c");
g:=d/[d.1^2,d.2^2,d.3^2,d.3*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.2*d.3*d.2*d.1*d.3];
sg:=Subgroup(g,[g.1]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G13:=pgr13(); gap> Size(G13); 96 gap> N13:=NormalSubgroups(G13);; gap> NTSize(N13); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 4 Groesse des 4. Normalteilers: 8 Groesse des 5. Normalteilers: 16 Groesse des 6. Normalteilers: 24 Groesse des 7. Normalteilers: 48 Groesse des 8. Normalteilers: 48 Groesse des 9. Normalteilers: 48 Groesse des 10. Normalteilers: 96 gap> Centre(G13); Group([ ( 1,40,45,43)( 2,33,38,32)( 3,29,34,37)( 4,21,28,25)( 5,44,27,18) ( 6,17,35,26)( 7,23,22,48)( 8,24,16,13)( 9,10,39,36)(11,42,15,31) (12,30,46,14)(19,20,41,47) ]) gap> Size(Centre(G13)); 4 gap> IsSolvable(G13); true gap> CompositionSeries(G13); [ <permutation group of size 96 with 6 generators>, <permutation group of size 48 with 5 generators>, <permutation group of size 16 with 4 generators>, <permutation group of size 8 with 3 generators>, <permutation group of size 4 with 2 generators>, Group([ ( 1,45)( 2,38)( 3,34)( 4,28)( 5,27)( 6,35)( 7,22)( 8,16)( 9,39) (10,36)(11,15)(12,46)(13,24)(14,30)(17,26)(18,44)(19,41)(20,47)(21,25) (23,48)(29,37)(31,42)(32,33)(40,43) ]), Group(()) ]
gap> ct13:=CharacterTable(G13); CharacterTable( <permutation group of size 96 with 3 generators> ) gap> Display(ct13); CT1 2 5 4 3 4 2 2 2 4 3 4 4 2 4 5 5 5 3 1 . . . 1 1 1 . . . . 1 . 1 1 1 1a 2a 2b 8a 6a 12a 12b 4a 4b 8b 8c 3a 8d 4c 4d 2c 2P 1a 1a 1a 4a 3a 6a 6a 2c 2c 4a 4a 3a 4a 2c 2c 1a 3P 1a 2a 2b 8d 2c 4c 4d 4a 4b 8b 8c 1a 8a 4d 4c 2c 5P 1a 2a 2b 8d 6a 12a 12b 4a 4b 8c 8b 3a 8a 4c 4d 2c 7P 1a 2a 2b 8a 6a 12b 12a 4a 4b 8c 8b 3a 8d 4d 4c 2c 11P 1a 2a 2b 8d 6a 12b 12a 4a 4b 8b 8c 3a 8a 4d 4c 2c X.1 1 1 1 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 -1 -1 -1 1 X.3 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 X.4 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 X.5 2 2 . . -1 -1 -1 2 . . . -1 . 2 2 2 X.6 2 -2 . . -1 1 1 2 . . . -1 . -2 -2 2 X.7 2 . . A 1 B -B . . C -C -1 -A D -D -2 X.8 2 . . A 1 -B B . . -C C -1 -A -D D -2 X.9 2 . . -A 1 B -B . . -C C -1 A D -D -2 X.10 2 . . -A 1 -B B . . C -C -1 A -D D -2 X.11 3 -1 -1 1 . . . -1 -1 1 1 . 1 3 3 3 X.12 3 -1 1 -1 . . . -1 1 -1 -1 . -1 3 3 3 X.13 3 1 -1 -1 . . . -1 1 1 1 . -1 -3 -3 3 X.14 3 1 1 1 . . . -1 -1 -1 -1 . 1 -3 -3 3 X.15 4 . . . -1 -B B . . . . 1 . E -E -4 X.16 4 . . . -1 B -B . . . . 1 . -E E -4 A = -E(8)+E(8)^3 = -ER(2) = -r2 B = -E(4) = -ER(-1) = -i C = E(8)+E(8)^3 = ER(-2) = i2 D = -2*E(4) = -2*ER(-1) = -2i E = -4*E(4) = -4*ER(-1) = -4i gap> quit;