pgr14:=function()
local z,g,sg,p;
z:=FreeGroup("a","b");
g:=z/[z.1^3,z.2^2,z.1*z.2*z.1*z.2*z.1*z.2*z.1*z.2*z.1^2*z.2*z.1^2*z.2*z.1^2*z.2*z.1^2*z.2];
sg:=Subgroup(g,[g.1]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G14:=pgr14(); gap> Size(G14); 144 gap> N14:=NormalSubgroups(G14);; gap> NTSize(N14); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 3 Groesse des 4. Normalteilers: 6 Groesse des 5. Normalteilers: 8 Groesse des 6. Normalteilers: 24 Groesse des 7. Normalteilers: 24 Groesse des 8. Normalteilers: 48 Groesse des 9. Normalteilers: 72 Groesse des 10. Normalteilers: 144 gap> Centre(G14); Group( [ ( 1,32,48,47,45,19)( 2,26,46,44,42,14)( 3,15,40,39,36,18)( 4,20,43,41,37,10) ( 5,11,34,33,30,25)( 6,27,38,35,31, 7)( 8,29,28,24,22, 9)(12,23,21,17,16, 13) ]) gap> Size(Centre(G14)); 6 gap> IsSolvable(G14); true gap> CompositionSeries(G14); [ <permutation group of size 144 with 6 generators>, <permutation group of size 72 with 5 generators>, <permutation group of size 24 with 4 generators>, <permutation group of size 8 with 3 generators>, <permutation group of size 4 with 2 generators>, Group([ ( 1,47)( 2,44)( 3,39)( 4,41)( 5,33)( 6,35)( 7,38)( 8,24)( 9,28) (10,43)(11,30)(12,17)(13,21)(14,46)(15,36)(16,23)(18,40)(19,48)(20,37) (22,29)(25,34)(26,42)(27,31)(32,45) ]), Group(()) ]
gap> ct14:=CharacterTable(G14); CharacterTable( <permutation group of size 144 with 2 generators>, ) gap> Display(ct14); CT1 2 4 1 1 2 3 3 3 1 3 3 2 2 3 4 1 3 1 1 4 3 2 2 2 1 1 1 1 2 1 1 1 1 1 2 2 1 2 2 2 1a 3a 3b 2a 24a 24b 12a 6a 12b 8a 6b 6c 8b 6d 6e 4a 3c 6f 6g 2P 1a 3b 3a 1a 12a 12b 6d 3c 6g 4a 3d 3e 4a 3d 3a 2b 3c 3b 3e 3P 1a 1a 1a 2a 8a 8b 4a 2b 4a 8a 2a 2a 8b 2b 2b 4a 1a 2b 2b 5P 1a 3b 3a 2a 24b 24a 12b 6a 12a 8b 6c 6b 8a 6g 6f 4a 3c 6e 6d 7P 1a 3a 3b 2a 24c 24d 12a 6a 12b 8b 6b 6c 8a 6d 6e 4a 3c 6f 6g 11P 1a 3b 3a 2a 24d 24c 12b 6a 12a 8a 6c 6b 8b 6g 6f 4a 3c 6e 6d 13P 1a 3a 3b 2a 24c 24d 12a 6a 12b 8b 6b 6c 8a 6d 6e 4a 3c 6f 6g 17P 1a 3b 3a 2a 24d 24c 12b 6a 12a 8a 6c 6b 8b 6g 6f 4a 3c 6e 6d 19P 1a 3a 3b 2a 24a 24b 12a 6a 12b 8a 6b 6c 8b 6d 6e 4a 3c 6f 6g 23P 1a 3b 3a 2a 24b 24a 12b 6a 12a 8b 6c 6b 8a 6g 6f 4a 3c 6e 6d X.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 X.3 1 A /A 1 A /A /A 1 A 1 A /A 1 A /A 1 1 A /A X.4 1 A /A -1 -A -/A /A 1 A -1 -A -/A -1 A /A 1 1 A /A X.5 1 /A A 1 /A A A 1 /A 1 /A A 1 /A A 1 1 /A A X.6 1 /A A -1 -/A -A A 1 /A -1 -/A -A -1 /A A 1 1 /A A X.7 2 -1 -1 . . . 2 -1 2 . . . . 2 -1 2 -1 -1 2 X.8 2 -/A -A . B /B . 1 . C . . -C -D A . -1 /A -/D X.9 2 -/A -A . -B -/B . 1 . -C . . C -D A . -1 /A -/D X.10 2 -1 -1 . C -C . 1 . C . . -C -2 1 . -1 1 -2 X.11 2 -1 -1 . -C C . 1 . -C . . C -2 1 . -1 1 -2 X.12 2 -A -/A . -/B -B . 1 . C . . -C -/D /A . -1 A -D X.13 2 -A -/A . /B B . 1 . -C . . C -/D /A . -1 A -D X.14 2 -A -/A . . . D -1 /D . . . . /D -/A 2 -1 -A D X.15 2 -/A -A . . . /D -1 D . . . . D -A 2 -1 -/A /D X.16 3 . . 1 -1 -1 -1 . -1 -1 1 1 -1 3 . -1 . . 3 X.17 3 . . -1 1 1 -1 . -1 1 -1 -1 1 3 . -1 . . 3 X.18 3 . . 1 -/A -A -A . -/A -1 /A A -1 E . -1 . . /E X.19 3 . . -1 /A A -A . -/A 1 -/A -A 1 E . -1 . . /E X.20 3 . . 1 -A -/A -/A . -A -1 A /A -1 /E . -1 . . E X.21 3 . . -1 A /A -/A . -A 1 -A -/A 1 /E . -1 . . E X.22 4 1 1 . . . . -1 . . . . . -4 -1 . 1 -1 -4 X.23 4 A /A . . . . -1 . . . . . F -/A . 1 -A /F X.24 4 /A A . . . . -1 . . . . . /F -A . 1 -/A F 2 3 3 4 4 4 3 1 1 2 2 2 24c 24d 3d 2b 3e 2P 12a 12b 3e 1a 3d 3P 8b 8a 1a 2b 1a 5P 24d 24c 3e 2b 3d 7P 24a 24b 3d 2b 3e 11P 24b 24a 3e 2b 3d 13P 24a 24b 3d 2b 3e 17P 24b 24a 3e 2b 3d 19P 24c 24d 3d 2b 3e 23P 24d 24c 3e 2b 3d X.1 1 1 1 1 1 X.2 -1 -1 1 1 1 X.3 A /A /A 1 A X.4 -A -/A /A 1 A X.5 /A A A 1 /A X.6 -/A -A A 1 /A X.7 . . 2 2 2 X.8 -B -/B /D -2 D X.9 B /B /D -2 D X.10 -C C 2 -2 2 X.11 C -C 2 -2 2 X.12 /B B D -2 /D X.13 -/B -B D -2 /D X.14 . . D 2 /D X.15 . . /D 2 D X.16 -1 -1 3 3 3 X.17 1 1 3 3 3 X.18 -/A -A /E 3 E X.19 /A A /E 3 E X.20 -A -/A E 3 /E X.21 A /A E 3 /E X.22 . . 4 -4 4 X.23 . . -/F -4 -F X.24 . . -F -4 -/F A = E(3) = (-1+ER(-3))/2 = b3 B = -E(24)-E(24)^19 C = -E(8)-E(8)^3 = -ER(-2) = -i2 D = 2*E(3)^2 = -1-ER(-3) = -1-i3 E = 3*E(3)^2 = (-3-3*ER(-3))/2 = -3-3b3 F = -4*E(3) = 2-2*ER(-3) = -4b3 gap> quit;