pgr8:=function()
local z,g,sg,p;
z:=FreeGroup("a","b");
g:=z/[z.1^4,z.2^4,z.1*z.2*z.1*z.2^3*z.1^3*z.2^3];
sg:=Subgroup(g,[g.1]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G8:=pgr8(); gap> Size(G8); 96 gap> N8:=NormalSubgroups(G8);; gap> NTSize(N8); 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: 96 gap> Centre(G8); Group([ ( 1,23,20, 9)( 2,16,22,13)( 3,21,24,18)( 4,14,10, 6)( 5, 7,19,15) ( 8,11,17,12) ]) gap> Size(Centre(G8)); 4 gap> IsSolvable(G8); true gap> CompositionSeries(G8); [ <permutation group of size 96 with 6 generators>, <permutation group of size 48 with 5 generators>, <permutation group of size 24 with 4 generators>, Group([ ( 1, 4,20,10)( 2,13,22,16)( 3,17,24, 8)( 5, 7,19,15)( 6,23,14, 9) (11,21,12,18), ( 1,14,20, 6)( 2,19,22, 5)( 3,21,24,18)( 4,23,10, 9) ( 7,16,15,13)( 8,12,17,11), ( 1,20)( 2,22)( 3,24)( 4,10)( 5,19)( 6,14) ( 7,15)( 8,17)( 9,23)(11,12)(13,16)(18,21) ]), Group([ ( 1,14,20, 6)( 2,19,22, 5)( 3,21,24,18)( 4,23,10, 9)( 7,16,15,13) ( 8,12,17,11), ( 1,20)( 2,22)( 3,24)( 4,10)( 5,19)( 6,14)( 7,15) ( 8,17)( 9,23)(11,12)(13,16)(18,21) ]), Group([ ( 1,20)( 2,22)( 3,24)( 4,10)( 5,19)( 6,14)( 7,15)( 8,17)( 9,23) (11,12)(13,16)(18,21) ]), Group(()) ]
gap> ct8:=CharacterTable(G8); CharacterTable( Group( [ ( 2, 3, 5, 8)( 4, 6,10,14)( 7,11,16,21)(12,13,18,15)(17,22,24,19), ( 1, 2, 4, 7)( 5, 9,13, 6)( 8,12,17,11)(10,15,20,22)(14,19,23,16) ]) ) gap> Display(ct8); CT1 2 5 4 4 4 2 3 2 4 3 2 5 4 2 4 5 5 3 1 . . . 1 . 1 . . 1 1 . 1 . 1 1 1a 4a 2a 4b 12a 8a 6a 4c 8b 12b 4d 4e 3a 4f 2b 4g 2P 1a 2a 1a 2a 6a 4d 3a 2b 4g 6a 2b 2a 3a 2a 1a 2b 3P 1a 4b 2a 4a 4d 8b 2b 4c 8a 4g 4g 4f 1a 4e 2b 4d 5P 1a 4a 2a 4b 12a 8a 6a 4c 8b 12b 4d 4e 3a 4f 2b 4g 7P 1a 4b 2a 4a 12b 8b 6a 4c 8a 12a 4g 4f 3a 4e 2b 4d 11P 1a 4b 2a 4a 12b 8b 6a 4c 8a 12a 4g 4f 3a 4e 2b 4d 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 A -1 -A -1 -A 1 1 A -1 -1 -A 1 A 1 -1 X.4 1 -A -1 A -1 A 1 1 -A -1 -1 A 1 -A 1 -1 X.5 2 . 2 . -1 . -1 2 . -1 2 . -1 . 2 2 X.6 2 . -2 . 1 . -1 2 . 1 -2 . -1 . 2 -2 X.7 2 B . /B A . 1 . . -A C -/B -1 -B -2 -C X.8 2 -B . -/B A . 1 . . -A C /B -1 B -2 -C X.9 2 -/B . -B -A . 1 . . A -C B -1 /B -2 C X.10 2 /B . B -A . 1 . . A -C -B -1 -/B -2 C X.11 3 1 -1 1 . -1 . -1 -1 . 3 1 . 1 3 3 X.12 3 -1 -1 -1 . 1 . -1 1 . 3 -1 . -1 3 3 X.13 3 A 1 -A . A . -1 -A . -3 -A . A 3 -3 X.14 3 -A 1 A . -A . -1 A . -3 A . -A 3 -3 X.15 4 . . . A . -1 . . -A D . 1 . -4 -D X.16 4 . . . -A . -1 . . A -D . 1 . -4 D A = E(4) = ER(-1) = i B = 1+E(4) = 1+ER(-1) = 1+i C = 2*E(4) = 2*ER(-1) = 2i D = -4*E(4) = -4*ER(-1) = -4i gap> quit;