pgr30:=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.1*v.2*v.1*v.2,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;
G30:=pgr30(); gap> Size(G30); 14400 gap> N30:=NormalSubgroups(G30);; gap> NTSize(N30); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 7200 Groesse des 4. Normalteilers: 14400 gap> Centre(G30); <permutation group of size 2 with 1 generators> gap> Size(Centre(G30)); 2 gap> IsSolvable(G30); false gap> CompositionSeries(G30); [ <permutation group of size 14400 with 5 generators>, <permutation group of size 7200 with 5 generators>, <permutation group of size 120 with 3 generators>, <permutation group of size 2 with 1 generators>, Group(()) ]
gap> ct30:=CharacterTable(G30); CharacterTable( <permutation group of size 14400 with 4 generators> ) gap> Display(ct30); CT1 2 6 6 2 2 2 2 3 3 3 3 1 1 2 2 2 2 1 1 3 2 2 . . . . 1 1 1 1 . . . . . . 1 1 5 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1a 2a 5a 10a 5b 10b 10c 5c 5d 10d 10e 5e 10f 10g 10h 10i 15a 30a 2P 1a 1a 5b 5b 5a 5a 5d 5d 5c 5c 5e 5e 5a 5a 5b 5b 15b 15b 3P 1a 2a 5b 10b 5a 10a 10d 5d 5c 10c 10e 5e 10h 10i 10f 10g 5c 10c 5P 1a 2a 1a 2a 1a 2a 2a 1a 1a 2a 2a 1a 2c 2d 2c 2d 3b 6b 7P 1a 2a 5b 10b 5a 10a 10d 5d 5c 10c 10e 5e 10h 10i 10f 10g 15b 30b 11P 1a 2a 5a 10a 5b 10b 10c 5c 5d 10d 10e 5e 10f 10g 10h 10i 15a 30a 13P 1a 2a 5b 10b 5a 10a 10d 5d 5c 10c 10e 5e 10h 10i 10f 10g 15b 30b 17P 1a 2a 5b 10b 5a 10a 10d 5d 5c 10c 10e 5e 10h 10i 10f 10g 15b 30b 19P 1a 2a 5a 10a 5b 10b 10c 5c 5d 10d 10e 5e 10f 10g 10h 10i 15a 30a 23P 1a 2a 5b 10b 5a 10a 10d 5d 5c 10c 10e 5e 10h 10i 10f 10g 15b 30b 29P 1a 2a 5a 10a 5b 10b 10c 5c 5d 10d 10e 5e 10f 10g 10h 10i 15a 30a X.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 X.3 4 -4 A -A *A -*A *B -*B -B B 1 -1 I -I *I -*I *I -*I X.4 4 -4 A -A *A -*A *B -*B -B B 1 -1 -I I -*I *I *I -*I X.5 4 -4 *A -*A A -A B -B -*B *B 1 -1 *I -*I I -I I -I X.6 4 -4 *A -*A A -A B -B -*B *B 1 -1 -*I *I -I I I -I X.7 6 6 B B *B *B C C *C *C 1 1 . . . . I I X.8 6 6 *B *B B B *C *C C C 1 1 . . . . *I *I X.9 8 8 -2 -2 -2 -2 3 3 3 3 -2 -2 . . . . . . X.10 8 -8 -2 2 -2 2 2 -2 -2 2 -3 3 . . . . 1 -1 X.11 9 9 A A *A *A D D *D *D -1 -1 I I *I *I . . X.12 9 9 A A *A *A D D *D *D -1 -1 -I -I -*I -*I . . X.13 9 9 *A *A A A *D *D D D -1 -1 *I *I I I . . X.14 9 9 *A *A A A *D *D D D -1 -1 -*I -*I -I -I . . X.15 10 10 . . . . 5 5 5 5 . . . . . . -1 -1 X.16 16 16 1 1 1 1 -4 -4 -4 -4 1 1 -1 -1 -1 -1 -1 -1 X.17 16 16 1 1 1 1 -4 -4 -4 -4 1 1 1 1 1 1 -1 -1 X.18 16 -16 1 -1 1 -1 4 -4 -4 4 -1 1 -1 1 -1 1 -1 1 X.19 16 -16 1 -1 1 -1 4 -4 -4 4 -1 1 1 -1 1 -1 -1 1 X.20 16 -16 *B -*B B -B E -E -*E *E -1 1 . . . . I -I X.21 16 -16 B -B *B -*B *E -*E -E E -1 1 . . . . *I -*I X.22 18 18 -2 -2 -2 -2 3 3 3 3 3 3 . . . . . . X.23 24 24 -B -B -*B -*B F F *F *F -1 -1 . . . . I I X.24 24 24 -*B -*B -B -B *F *F F F -1 -1 . . . . *I *I X.25 24 -24 -B B -*B *B G -G -*G *G 1 -1 . . . . -1 1 X.26 24 -24 -*B *B -B B *G -*G -G G 1 -1 . . . . -1 1 X.27 25 25 . . . . . . . . . . . . . . . . X.28 25 25 . . . . . . . . . . . . . . . . X.29 30 30 . . . . H H *H *H . . . . . . -I -I X.30 30 30 . . . . *H *H H H . . . . . . -*I -*I X.31 36 -36 1 -1 1 -1 -6 6 6 -6 -1 1 -1 1 -1 1 . . X.32 36 -36 1 -1 1 -1 -6 6 6 -6 -1 1 1 -1 1 -1 . . X.33 40 40 . . . . -5 -5 -5 -5 . . . . . . 1 1 X.34 48 -48 -2 2 -2 2 2 -2 -2 2 2 -2 . . . . 1 -1 2 1 1 2 2 2 2 3 3 2 2 2 5 3 4 4 4 3 1 1 . . 2 2 2 2 1 1 1 . . 1 1 1 5 1 1 1 1 . . 1 1 . . . . . 1 1 1 15b 30b 20a 20b 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 2P 15a 15a 10c 10d 3a 3a 3b 3b 3a 3a 6b 1a 2b 2a 1a 1a 3P 5d 10d 20b 20a 2a 1a 2a 1a 2d 2c 4b 2b 4a 4b 2c 2d 5P 3b 6b 4b 4b 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 7P 15a 30a 20b 20a 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 11P 15b 30b 20a 20b 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 13P 15a 30a 20b 20a 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 17P 15a 30a 20b 20a 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 19P 15b 30b 20a 20b 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 23P 15a 30a 20b 20a 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 29P 15b 30b 20a 20b 6a 3a 6b 3b 6c 6d 12a 2b 4a 4b 2c 2d 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 I -I . . -1 1 2 -2 -1 1 . . . . -2 2 X.4 I -I . . -1 1 2 -2 1 -1 . . . . 2 -2 X.5 *I -*I . . -1 1 2 -2 -1 1 . . . . -2 2 X.6 *I -*I . . -1 1 2 -2 1 -1 . . . . 2 -2 X.7 *I *I -*I -I . . 3 3 . . -1 -2 . 2 . . X.8 I I -I -*I . . 3 3 . . -1 -2 . 2 . . X.9 . . -1 -1 2 2 5 5 . . 1 . . 4 . . X.10 1 -1 . . -2 2 4 -4 . . . . . . . . X.11 . . -*I -I . . . . . . . 1 -1 -3 3 3 X.12 . . -*I -I . . . . . . . 1 1 -3 -3 -3 X.13 . . -I -*I . . . . . . . 1 -1 -3 3 3 X.14 . . -I -*I . . . . . . . 1 1 -3 -3 -3 X.15 -1 -1 1 1 -2 -2 4 4 . . . 2 . 6 . . X.16 -1 -1 . . 1 1 4 4 1 1 . . . . 4 4 X.17 -1 -1 . . 1 1 4 4 -1 -1 . . . . -4 -4 X.18 -1 1 . . -1 1 -4 4 -1 1 . . . . 4 -4 X.19 -1 1 . . -1 1 -4 4 1 -1 . . . . -4 4 X.20 *I -*I . . 2 -2 2 -2 . . . . . . . . X.21 I -I . . 2 -2 2 -2 . . . . . . . . X.22 . . -1 -1 . . . . . . . 2 . -6 . . X.23 *I *I 1 1 . . 3 3 . . -1 . . -4 . . X.24 I I 1 1 . . 3 3 . . -1 . . -4 . . X.25 -1 1 . . . . 6 -6 . . . . . . . . X.26 -1 1 . . . . 6 -6 . . . . . . . . X.27 . . . . 1 1 -5 -5 -1 -1 -1 1 1 5 5 5 X.28 . . . . 1 1 -5 -5 1 1 -1 1 -1 5 -5 -5 X.29 -*I -*I I *I . . -3 -3 . . 1 -2 . -2 . . X.30 -I -I *I I . . -3 -3 . . 1 -2 . -2 . . X.31 . . . . . . . . . . . . . . -6 6 X.32 . . . . . . . . . . . . . . 6 -6 X.33 1 1 -1 -1 -2 -2 1 1 . . 1 . . 4 . . X.34 1 -1 . . . . -6 6 . . . . . . . . A = -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 = (3-ER(5))/2 = 1-b5 B = -2*E(5)^2-2*E(5)^3 = 1+ER(5) = 1+r5 C = -3*E(5)-4*E(5)^2-4*E(5)^3-3*E(5)^4 = (7+ER(5))/2 = 4+b5 D = -3*E(5)-3*E(5)^4 = (3-3*ER(5))/2 = -3b5 E = -6*E(5)-2*E(5)^2-2*E(5)^3-6*E(5)^4 = 4-2*ER(5) = 4-2r5 F = 3*E(5)-E(5)^2-E(5)^3+3*E(5)^4 = -1+2*ER(5) = 1+4b5 G = 2*E(5)-4*E(5)^2-4*E(5)^3+2*E(5)^4 = 1+3*ER(5) = 1+3r5 H = -5*E(5)^2-5*E(5)^3 = (5+5*ER(5))/2 = 5+5b5 I = -E(5)-E(5)^4 = (1-ER(5))/2 = -b5 gap> quit;