pgr28:=function()
local v,g,sg,p;
v:=FreeGroup("a","b","c","d");
g:=v/[v.1^2,v.2^2,v.3^2,v.4^2,v.1*v.2*v.1*v.2*v.1*v.2,v.2*v.3*v.2*v.3*v.2*v.3*v.2*v.3,v.3*v.4*v.3*v.4*v.3*v.4,v.1*v.3*v.1*v.3,v.1*v.4*v.1*v.4,v.2*v.4*v.2*v.4];
sg:=Subgroup(g,[g.1,g.2,g.3]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G28:=pgr28(); gap> Size(G28); 1152 gap> N28:=NormalSubgroups(G28);; gap> NTSize(N28); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 32 Groesse des 4. Normalteilers: 96 Groesse des 5. Normalteilers: 96 Groesse des 6. Normalteilers: 192 Groesse des 7. Normalteilers: 192 Groesse des 8. Normalteilers: 288 Groesse des 9. Normalteilers: 576 Groesse des 10. Normalteilers: 576 Groesse des 11. Normalteilers: 576 Groesse des 12. Normalteilers: 1152 gap> Centre(G28); Group([ ( 1,24)( 2,23)( 3,22)( 4,21)( 5,20)( 6,19)( 7,17)( 8,18)( 9,15)(10,16) (11,13)(12,14) ]) gap> Size(Centre(G28)); 2 gap> IsSolvable(G28); true gap> CompositionSeries(G28); [ <permutation group of size 1152 with 9 generators>, <permutation group of size 576 with 8 generators>, <permutation group of size 192 with 7 generators>, <permutation group of size 96 with 6 generators>, <permutation group of size 32 with 5 generators>, <permutation group of size 16 with 4 generators>, Group([ ( 2, 7)( 3, 5)( 4,11)( 6, 9)( 8,18)(12,14)(13,21)(15,19)(17,23) (20,22), ( 2, 6)( 3, 4)( 5,11)( 7, 9)(10,16)(12,14)(13,20)(15,17) (19,23)(21,22), ( 1,24)( 2,23)( 3,22)( 4,21)( 5,20)( 6,19)( 7,17) ( 8,18)( 9,15)(10,16)(11,13)(12,14) ]), Group([ ( 2, 6)( 3, 4)( 5,11)( 7, 9)(10,16)(12,14)(13,20)(15,17)(19,23) (21,22), ( 1,24)( 2,23)( 3,22)( 4,21)( 5,20)( 6,19)( 7,17)( 8,18) ( 9,15)(10,16)(11,13)(12,14) ]), Group([ ( 1,24)( 2,23)( 3,22)( 4,21)( 5,20)( 6,19)( 7,17)( 8,18)( 9,15) (10,16)(11,13)(12,14) ]), Group(()) ]
gap> ct28:=CharacterTable(G28); CharacterTable( <permutation group of size 1152 with 4 generators> ) gap> Display(ct28); CT1 2 7 7 6 5 2 2 5 5 4 2 2 2 3 3 2 2 5 5 4 2 2 4 5 3 2 2 . 1 2 2 1 1 . 2 2 1 2 2 1 1 1 1 . 1 1 . . 1a 2a 2b 4a 3a 6a 2c 2d 4b 3b 6b 12a 6c 3c 6d 6e 2e 2f 4c 6f 6g 2g 4d 2P 1a 1a 1a 2a 3a 3a 1a 1a 2b 3b 3b 6c 3c 3c 3b 3b 1a 1a 2b 3a 3a 1a 2b 3P 1a 2a 2b 4a 1a 2a 2c 2d 4b 1a 2a 4a 2a 1a 2c 2d 2e 2f 4c 2f 2e 2g 4d 5P 1a 2a 2b 4a 3a 6a 2c 2d 4b 3b 6b 12a 6c 3c 6d 6e 2e 2f 4c 6f 6g 2g 4d 7P 1a 2a 2b 4a 3a 6a 2c 2d 4b 3b 6b 12a 6c 3c 6d 6e 2e 2f 4c 6f 6g 2g 4d 11P 1a 2a 2b 4a 3a 6a 2c 2d 4b 3b 6b 12a 6c 3c 6d 6e 2e 2f 4c 6f 6g 2g 4d X.1 1 1 1 1 1 1 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 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 -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 -1 -1 -1 -1 -1 1 1 X.5 2 2 2 2 -1 -1 . . . 2 2 -1 -1 -1 . . -2 -2 -2 1 1 . . X.6 2 2 2 2 -1 -1 . . . 2 2 -1 -1 -1 . . 2 2 2 -1 -1 . . X.7 2 2 2 2 2 2 -2 -2 -2 -1 -1 -1 -1 -1 1 1 . . . . . . . X.8 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 . . . . . . . X.9 4 -4 . . 1 -1 -2 2 . 1 -1 . 2 -2 1 -1 -2 2 . -1 1 . 2 X.10 4 -4 . . 1 -1 -2 2 . 1 -1 . 2 -2 1 -1 2 -2 . 1 -1 . -2 X.11 4 -4 . . 1 -1 2 -2 . 1 -1 . 2 -2 -1 1 -2 2 . -1 1 . -2 X.12 4 -4 . . 1 -1 2 -2 . 1 -1 . 2 -2 -1 1 2 -2 . 1 -1 . 2 X.13 4 4 4 4 -2 -2 . . . -2 -2 1 1 1 . . . . . . . . . X.14 6 6 -2 2 . . . . . . . -1 3 3 . . . . . . . -2 2 X.15 6 6 -2 2 . . . . . . . -1 3 3 . . . . . . . 2 -2 X.16 8 -8 . . -1 1 . . . 2 -2 . -2 2 . . -4 4 . 1 -1 . . X.17 8 -8 . . -1 1 . . . 2 -2 . -2 2 . . 4 -4 . -1 1 . . X.18 8 -8 . . 2 -2 -4 4 . -1 1 . -2 2 -1 1 . . . . . . . X.19 8 -8 . . 2 -2 4 -4 . -1 1 . -2 2 1 -1 . . . . . . . X.20 9 9 1 -3 . . -3 -3 1 . . . . . . . -3 -3 1 . . 1 1 X.21 9 9 1 -3 . . -3 -3 1 . . . . . . . 3 3 -1 . . -1 -1 X.22 9 9 1 -3 . . 3 3 -1 . . . . . . . -3 -3 1 . . -1 -1 X.23 9 9 1 -3 . . 3 3 -1 . . . . . . . 3 3 -1 . . 1 1 X.24 12 12 -4 4 . . . . . . . 1 -3 -3 . . . . . . . . . X.25 16 -16 . . -2 2 . . . -2 2 . 2 -2 . . . . . . . . . 2 5 3 3 . . 4e 8a 2P 2b 4a 3P 4e 8a 5P 4e 8a 7P 4e 8a 11P 4e 8a X.1 1 1 X.2 -1 -1 X.3 -1 -1 X.4 1 1 X.5 . . X.6 . . X.7 . . X.8 . . X.9 -2 . X.10 2 . X.11 2 . X.12 -2 . X.13 . . X.14 2 . X.15 -2 . X.16 . . X.17 . . X.18 . . X.19 . . X.20 1 -1 X.21 -1 1 X.22 -1 1 X.23 1 -1 X.24 . . X.25 . . gap> quit;