pgr20:=function()
local z,g,sg,p;
z:=FreeGroup("a","b");
g:=z/[z.1^3,z.2^3,z.1*z.2*z.1*z.2*z.1*z.2^2*z.1^2*z.2^2*z.1^2*z.2^2];
sg:=Subgroup(g,[g.1]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G20:=pgr20(); gap> Size(G20); 360 gap> N20:=NormalSubgroups(G20);; gap> NTSize(N20); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 3 Groesse des 4. Normalteilers: 6 Groesse des 5. Normalteilers: 120 Groesse des 6. Normalteilers: 360 gap> Centre(G20); <permutation group of size 6 with 1 generators> gap> Size(Centre(G20)); 6 gap> IsSolvable(G20); false gap> CompositionSeries(G20); [ <permutation group of size 360 with 7 generators>, <permutation group of size 6 with 2 generators>, <permutation group of size 3 with 1 generators>, Group(()) ]
gap> ct20:=CharacterTable(G20); CharacterTable( <permutation group of size 360 with 2 generators> ) gap> Display(ct20); CT1 2 3 1 1 1 1 1 1 2 1 1 2 1 1 1 1 2 1 1 3 2 2 2 1 1 1 1 1 1 2 1 1 2 1 1 1 1 2 5 1 . . 1 1 1 1 . 1 . . 1 . 1 1 . 1 . 1a 3a 3b 30a 10a 30b 15a 12a 5a 6a 12b 15b 6b 15c 10b 4a 15d 6c 2P 1a 3b 3a 15a 5a 15c 15b 6d 5b 3c 6e 15a 3a 15d 5b 2a 15c 3b 3P 1a 1a 1a 10a 10b 10a 5a 4a 5b 2a 4a 5b 2a 5a 10a 4a 5b 2a 5P 1a 3b 3a 6d 2a 6e 3d 12b 1a 6a 12a 3e 6c 3e 2a 4a 3d 6b 7P 1a 3a 3b 30c 10b 30d 15d 12a 5b 6a 12b 15c 6b 15b 10a 4a 15a 6c 11P 1a 3b 3a 30b 10a 30a 15c 12b 5a 6a 12a 15d 6c 15a 10b 4a 15b 6b 13P 1a 3a 3b 30c 10b 30d 15d 12a 5b 6a 12b 15c 6b 15b 10a 4a 15a 6c 17P 1a 3b 3a 30d 10b 30c 15b 12b 5b 6a 12a 15a 6c 15d 10a 4a 15c 6b 19P 1a 3a 3b 30a 10a 30b 15a 12a 5a 6a 12b 15b 6b 15c 10b 4a 15d 6c 23P 1a 3b 3a 30d 10b 30c 15b 12b 5b 6a 12a 15a 6c 15d 10a 4a 15c 6b 29P 1a 3b 3a 30b 10a 30a 15c 12b 5a 6a 12a 15d 6c 15a 10b 4a 15b 6b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /A /A 1 A A /A 1 1 A /A /A /A 1 1 A A X.3 1 /A A A 1 /A /A A 1 1 /A A A A 1 1 /A /A X.4 2 -A -/A B *C /B -/D . -C 1 . -B /A -D C . -/B A X.5 2 -/A -A /B *C B -D . -C 1 . -/B A -/D C . -B /A X.6 2 -1 -1 C *C C -*C . -C 1 . -C 1 -*C C . -C 1 X.7 2 -A -/A D C /D -/B . -*C 1 . -D /A -B *C . -/D A X.8 2 -/A -A /D C D -B . -*C 1 . -/D A -/B *C . -D /A X.9 2 -1 -1 *C C *C -C . -*C 1 . -*C 1 -C *C . -*C 1 X.10 3 . . *C C *C C -1 *C . -1 *C . C *C -1 *C . X.11 3 . . D C /D /B -/A *C . -A D . B *C -1 /D . X.12 3 . . /D C D B -A *C . -/A /D . /B *C -1 D . X.13 3 . . C *C C *C -1 C . -1 C . *C C -1 C . X.14 3 . . B *C /B /D -/A C . -A B . D C -1 /B . X.15 3 . . /B *C B D -A C . -/A /B . /D C -1 B . X.16 4 1 1 -1 -1 -1 -1 . -1 1 . -1 1 -1 -1 . -1 1 X.17 4 1 1 1 1 1 -1 . -1 -1 . -1 -1 -1 1 . -1 -1 X.18 4 A /A -/A -1 -A -A . -1 1 . -/A /A -/A -1 . -A A X.19 4 /A A -A -1 -/A -/A . -1 1 . -A A -A -1 . -/A /A X.20 4 A /A /A 1 A -A . -1 -1 . -/A -/A -/A 1 . -A -A X.21 4 /A A A 1 /A -/A . -1 -1 . -A -A -A 1 . -/A -/A X.22 5 -1 -1 . . . . 1 . -1 1 . -1 . . 1 . -1 X.23 5 -A -/A . . . . /A . -1 A . -/A . . 1 . -A X.24 5 -/A -A . . . . A . -1 /A . -A . . 1 . -/A X.25 6 . . -1 -1 -1 1 . 1 . . 1 . 1 -1 . 1 . X.26 6 . . -/A -1 -A A . 1 . . /A . /A -1 . A . X.27 6 . . -A -1 -/A /A . 1 . . A . A -1 . /A . 2 3 1 1 1 1 3 3 3 3 3 2 1 1 2 1 2 2 2 2 5 1 1 1 . 1 1 1 1 1 6d 30c 5b 3c 30d 3d 6e 2a 3e 2P 3d 15d 5a 3c 15b 3e 3e 1a 3d 3P 2a 10b 5a 1a 10b 1a 2a 2a 1a 5P 6e 6d 1a 3c 6e 3e 6d 2a 3d 7P 6d 30a 5a 3c 30b 3d 6e 2a 3e 11P 6e 30d 5b 3c 30c 3e 6d 2a 3d 13P 6d 30a 5a 3c 30b 3d 6e 2a 3e 17P 6e 30b 5a 3c 30a 3e 6d 2a 3d 19P 6d 30c 5b 3c 30d 3d 6e 2a 3e 23P 6e 30b 5a 3c 30a 3e 6d 2a 3d 29P 6e 30d 5b 3c 30c 3e 6d 2a 3d X.1 1 1 1 1 1 1 1 1 1 X.2 A /A 1 1 A /A /A 1 A X.3 /A A 1 1 /A A A 1 /A X.4 E D -*C -1 /D -/E /E -2 -E X.5 /E /D -*C -1 D -E E -2 -/E X.6 -2 *C -*C -1 *C 2 -2 -2 2 X.7 E B -C -1 /B -/E /E -2 -E X.8 /E /B -C -1 B -E E -2 -/E X.9 -2 C -C -1 C 2 -2 -2 2 X.10 3 C C . C 3 3 3 3 X.11 F B C . /B /F /F 3 F X.12 /F /B C . B F F 3 /F X.13 3 *C *C . *C 3 3 3 3 X.14 F D *C . /D /F /F 3 F X.15 /F /D *C . D F F 3 /F X.16 4 -1 -1 1 -1 4 4 4 4 X.17 -4 1 -1 1 1 4 -4 -4 4 X.18 G -/A -1 1 -A /G /G 4 G X.19 /G -A -1 1 -/A G G 4 /G X.20 -G /A -1 1 A /G -/G -4 G X.21 -/G A -1 1 /A G -G -4 /G X.22 5 . . -1 . 5 5 5 5 X.23 H . . -1 . /H /H 5 H X.24 /H . . -1 . H H 5 /H X.25 -6 -1 1 . -1 6 -6 -6 6 X.26 I -/A 1 . -A -/I /I -6 -I X.27 /I -A 1 . -/A -I I -6 -/I A = E(3) = (-1+ER(-3))/2 = b3 B = -E(15)^7-E(15)^13 C = -E(5)-E(5)^4 = (1-ER(5))/2 = -b5 D = -E(15)-E(15)^4 E = -2*E(3) = 1-ER(-3) = 1-i3 F = 3*E(3) = (-3+3*ER(-3))/2 = 3b3 G = 4*E(3) = -2+2*ER(-3) = 4b3 H = 5*E(3) = (-5+5*ER(-3))/2 = 5b3 I = -6*E(3) = 3-3*ER(-3) = -6b3 gap> quit;