pgr15:=function()
local d,g,sg,p;
d:=FreeGroup("a","b","c");
g:=d/[d.1^2,d.2^2,d.3^3,d.1*d.2*d.3*d.1*d.3^2*d.2,d.3*d.1*d.2*d.3*d.2*d.3^2*d.2*d.3^2*d.2*d.1];
sg:=Subgroup(g,[g.3]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G15:=pgr15(); gap> Size(G15); 288 gap> N15:=NormalSubgroups(G15);; gap> NTSize(N15); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 3 Groesse des 4. Normalteilers: 4 Groesse des 5. Normalteilers: 6 Groesse des 6. Normalteilers: 8 Groesse des 7. Normalteilers: 12 Groesse des 8. Normalteilers: 16 Groesse des 9. Normalteilers: 24 Groesse des 10. Normalteilers: 24 Groesse des 11. Normalteilers: 48 Groesse des 12. Normalteilers: 48 Groesse des 13. Normalteilers: 48 Groesse des 14. Normalteilers: 48 Groesse des 15. Normalteilers: 72 Groesse des 16. Normalteilers: 96 Groesse des 17. Normalteilers: 144 Groesse des 18. Normalteilers: 144 Groesse des 19. Normalteilers: 144 Groesse des 20. Normalteilers: 288 gap> Centre(G15); <permutation group of size 12 with 1 generators> gap> Size(Centre(G15)); 12 gap> IsSolvable(G15); true gap> CompositionSeries(G15); [ <permutation group of size 288 with 7 generators>, <permutation group of size 96 with 6 generators>, <permutation group of size 48 with 5 generators>, <permutation group of size 16 with 4 generators>, <permutation group of size 8 with 3 generators>, <permutation group of size 4 with 2 generators>, <permutation group of size 2 with 1 generators>, Group(()) ]
gap> ct15:=CharacterTable(G15); CharacterTable( <permutation group of size 288 with 3 generators> ) gap> Display(ct15); CT1 2 5 2 2 4 2 2 3 4 4 4 4 3 4 3 3 3 2 4 3 3 2 2 2 1 2 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1a 3a 3b 2a 12a 12b 2b 24a 24b 8a 24c 12c 24d 12d 6a 6b 6c 4a 4b 2P 1a 3b 3a 1a 6c 6e 1a 12i 12e 4a 12e 6i 12i 6h 3d 3e 3a 2c 2c 3P 1a 1a 1a 2a 4c 4d 2b 8c 8b 8d 8a 4b 8a 4b 2b 2b 2c 4a 4b 5P 1a 3b 3a 2a 12k 12j 2b 24b 24a 8d 24e 12d 24g 12c 6b 6a 6e 4a 4b 7P 1a 3a 3b 2a 12j 12k 2b 24h 24f 8a 24c 12c 24d 12d 6a 6b 6c 4a 4b 11P 1a 3b 3a 2a 12b 12a 2b 24f 24h 8d 24e 12d 24g 12c 6b 6a 6e 4a 4b 13P 1a 3a 3b 2a 12a 12b 2b 24h 24f 8d 24g 12c 24e 12d 6a 6b 6c 4a 4b 17P 1a 3b 3a 2a 12k 12j 2b 24f 24h 8a 24d 12d 24c 12c 6b 6a 6e 4a 4b 19P 1a 3a 3b 2a 12j 12k 2b 24a 24b 8d 24g 12c 24e 12d 6a 6b 6c 4a 4b 23P 1a 3b 3a 2a 12b 12a 2b 24b 24a 8a 24d 12d 24c 12c 6b 6a 6e 4a 4b 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 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 X.5 1 A /A 1 A /A 1 /A A 1 A /A /A A A /A /A 1 1 X.6 1 A /A -1 -A -/A 1 /A A -1 -A -/A -/A -A A /A /A 1 -1 X.7 1 A /A 1 A /A -1 -/A -A -1 -A -/A -/A -A -A -/A /A 1 -1 X.8 1 A /A -1 -A -/A -1 -/A -A 1 A /A /A A -A -/A /A 1 1 X.9 1 /A A 1 /A A 1 A /A 1 /A A A /A /A A A 1 1 X.10 1 /A A -1 -/A -A 1 A /A -1 -/A -A -A -/A /A A A 1 -1 X.11 1 /A A 1 /A A -1 -A -/A -1 -/A -A -A -/A -/A -A A 1 -1 X.12 1 /A A -1 -/A -A -1 -A -/A 1 /A A A /A -/A -A A 1 1 X.13 2 -1 -1 -2 1 1 . . . . . . . . . . -1 2 . X.14 2 -1 -1 2 -1 -1 . . . . . . . . . . -1 2 . X.15 2 -A -/A -2 A /A . . . . . . . . . . -/A 2 . X.16 2 -A -/A 2 -A -/A . . . . . . . . . . -/A 2 . X.17 2 -/A -A -2 /A A . . . . . . . . . . -A 2 . X.18 2 -/A -A 2 -/A -A . . . . . . . . . . -A 2 . X.19 2 -1 -1 . B -B . D -D F -F . -F . . . 1 . . X.20 2 -1 -1 . B -B . -D D -F F . F . . . 1 . . X.21 2 -1 -1 . -B B . D -D -F F . F . . . 1 . . X.22 2 -1 -1 . -B B . -D D F -F . -F . . . 1 . . X.23 2 -A -/A . C /C . E /E F G . /G . . . /A . . X.24 2 -A -/A . C /C . -E -/E -F -G . -/G . . . /A . . X.25 2 -A -/A . -C -/C . E /E -F -G . -/G . . . /A . . X.26 2 -A -/A . -C -/C . -E -/E F G . /G . . . /A . . X.27 2 -/A -A . -/C -C . -/E -E F /G . G . . . A . . X.28 2 -/A -A . -/C -C . /E E -F -/G . -G . . . A . . X.29 2 -/A -A . /C C . -/E -E -F -/G . -G . . . A . . X.30 2 -/A -A . /C C . /E E F /G . G . . . A . . X.31 3 . . -1 . . -1 1 1 1 1 -1 1 -1 -1 -1 . -1 -1 X.32 3 . . -1 . . 1 -1 -1 -1 -1 1 -1 1 1 1 . -1 1 X.33 3 . . 1 . . -1 1 1 -1 -1 1 -1 1 -1 -1 . -1 1 X.34 3 . . 1 . . 1 -1 -1 1 1 -1 1 -1 1 1 . -1 -1 X.35 3 . . -1 . . -1 /A A 1 A -/A /A -A -A -/A . -1 -1 X.36 3 . . -1 . . -1 A /A 1 /A -A A -/A -/A -A . -1 -1 X.37 3 . . -1 . . 1 -A -/A -1 -/A A -A /A /A A . -1 1 X.38 3 . . -1 . . 1 -/A -A -1 -A /A -/A A A /A . -1 1 X.39 3 . . 1 . . -1 /A A -1 -A /A -/A A -A -/A . -1 1 X.40 3 . . 1 . . -1 A /A -1 -/A A -A /A -/A -A . -1 1 X.41 3 . . 1 . . 1 -A -/A 1 /A -A A -/A /A A . -1 -1 X.42 3 . . 1 . . 1 -/A -A 1 A -/A /A -A A /A . -1 -1 X.43 4 1 1 . B -B . . . . . . . . . . -1 . . X.44 4 1 1 . -B B . . . . . . . . . . -1 . . X.45 4 /A A . -/C -C . . . . . . . . . . -A . . X.46 4 /A A . /C C . . . . . . . . . . -A . . X.47 4 A /A . C /C . . . . . . . . . . -/A . . X.48 4 A /A . -C -/C . . . . . . . . . . -/A . . 2 2 4 5 2 4 4 2 2 4 4 2 2 4 4 2 4 5 4 5 3 2 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 6d 12e 12f 12g 8b 8c 6e 12h 6f 12i 12j 12k 24e 8d 3c 6g 4c 24f 6h 2P 3c 6i 6i 6d 4a 4a 3b 6d 3e 6h 6c 6e 12i 4a 3c 3d 2c 12e 3e 3P 2c 4a 4c 4c 8b 8c 2c 4d 2a 4a 4d 4c 8d 8a 1a 2a 4d 8c 2c 5P 6d 12i 12m 12g 8c 8b 6c 12h 6g 12e 12b 12a 24c 8a 3c 6f 4c 24h 6i 7P 6d 12e 12n 12h 8c 8b 6e 12g 6f 12i 12a 12b 24e 8d 3c 6g 4d 24b 6h 11P 6d 12i 12l 12h 8b 8c 6c 12g 6g 12e 12k 12j 24c 8a 3c 6f 4d 24a 6i 13P 6d 12e 12f 12g 8c 8b 6e 12h 6f 12i 12j 12k 24d 8a 3c 6g 4c 24b 6h 17P 6d 12i 12m 12g 8b 8c 6c 12h 6g 12e 12b 12a 24g 8d 3c 6f 4c 24a 6i 19P 6d 12e 12n 12h 8b 8c 6e 12g 6f 12i 12a 12b 24d 8a 3c 6g 4d 24f 6h 23P 6d 12i 12l 12h 8c 8b 6c 12g 6g 12e 12k 12j 24g 8d 3c 6f 4d 24h 6i 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 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 X.5 1 /A /A 1 1 1 A 1 /A A A /A /A 1 1 A 1 A /A X.6 1 /A -/A -1 1 1 A -1 -/A A -A -/A -/A -1 1 -A -1 A /A X.7 1 /A /A 1 -1 -1 A 1 /A A A /A -/A -1 1 A 1 -A /A X.8 1 /A -/A -1 -1 -1 A -1 -/A A -A -/A /A 1 1 -A -1 -A /A X.9 1 A A 1 1 1 /A 1 A /A /A A A 1 1 /A 1 /A A X.10 1 A -A -1 1 1 /A -1 -A /A -/A -A -A -1 1 -/A -1 /A A X.11 1 A A 1 -1 -1 /A 1 A /A /A A -A -1 1 /A 1 -/A A X.12 1 A -A -1 -1 -1 /A -1 -A /A -/A -A A 1 1 -/A -1 -/A A X.13 -1 2 -2 1 . . -1 1 -2 2 1 1 . . -1 -2 -2 . 2 X.14 -1 2 2 -1 . . -1 -1 2 2 -1 -1 . . -1 2 2 . 2 X.15 -1 H -H 1 . . -A 1 -H /H A /A . . -1 -/H -2 . H X.16 -1 H H -1 . . -A -1 H /H -A -/A . . -1 /H 2 . H X.17 -1 /H -/H 1 . . -/A 1 -/H H /A A . . -1 -H -2 . /H X.18 -1 /H /H -1 . . -/A -1 /H H -/A -A . . -1 H 2 . /H X.19 1 . I B -D D 1 -B . . -B B F -F -1 . -I D -2 X.20 1 . I B D -D 1 -B . . -B B -F F -1 . -I -D -2 X.21 1 . -I -B -D D 1 B . . B -B -F F -1 . I D -2 X.22 1 . -I -B D -D 1 B . . B -B F -F -1 . I -D -2 X.23 1 . J B -D D A -B . . -C -/C -/G -F -1 . -I -/E -H X.24 1 . J B D -D A -B . . -C -/C /G F -1 . -I /E -H X.25 1 . -J -B -D D A B . . C /C /G F -1 . I -/E -H X.26 1 . -J -B D -D A B . . C /C -/G -F -1 . I /E -H X.27 1 . -/J B -D D /A -B . . /C C -G -F -1 . -I E -/H X.28 1 . -/J B D -D /A -B . . /C C G F -1 . -I -E -/H X.29 1 . /J -B -D D /A B . . -/C -C G F -1 . I E -/H X.30 1 . /J -B D -D /A B . . -/C -C -G -F -1 . I -E -/H X.31 . -1 3 . 1 1 . . -1 -1 . . 1 1 . -1 3 1 3 X.32 . -1 3 . -1 -1 . . -1 -1 . . -1 -1 . -1 3 -1 3 X.33 . -1 -3 . 1 1 . . 1 -1 . . -1 -1 . 1 -3 1 3 X.34 . -1 -3 . -1 -1 . . 1 -1 . . 1 1 . 1 -3 -1 3 X.35 . -/A K . 1 1 . . -/A -A . . /A 1 . -A 3 A K X.36 . -A /K . 1 1 . . -A -/A . . A 1 . -/A 3 /A /K X.37 . -A /K . -1 -1 . . -A -/A . . -A -1 . -/A 3 -/A /K X.38 . -/A K . -1 -1 . . -/A -A . . -/A -1 . -A 3 -A K X.39 . -/A -K . 1 1 . . /A -A . . -/A -1 . A -3 A K X.40 . -A -/K . 1 1 . . A -/A . . -A -1 . /A -3 /A /K X.41 . -A -/K . -1 -1 . . A -/A . . A 1 . /A -3 -/A /K X.42 . -/A -K . -1 -1 . . /A -A . . /A 1 . A -3 -A K X.43 -1 . L B . . -1 -B . . -B B . . 1 . -L . -4 X.44 -1 . -L -B . . -1 B . . B -B . . 1 . L . -4 X.45 -1 . M B . . -/A -B . . /C C . . 1 . -L . N X.46 -1 . -M -B . . -/A B . . -/C -C . . 1 . L . N X.47 -1 . -/M B . . -A -B . . -C -/C . . 1 . -L . /N X.48 -1 . /M -B . . -A B . . C /C . . 1 . L . /N 2 5 4 5 5 4 5 5 5 5 5 3 2 1 2 2 1 2 2 2 2 2 12l 24g 6i 4d 24h 2c 3d 12m 12n 3e 2P 6h 12e 3d 2c 12i 1a 3e 6h 6i 3d 3P 4d 8d 2c 4c 8b 2c 1a 4c 4d 1a 5P 12n 24d 6h 4d 24f 2c 3e 12f 12l 3d 7P 12m 24g 6i 4c 24a 2c 3d 12l 12f 3e 11P 12f 24d 6h 4c 24b 2c 3e 12n 12m 3d 13P 12l 24c 6i 4d 24a 2c 3d 12m 12n 3e 17P 12n 24e 6h 4d 24b 2c 3e 12f 12l 3d 19P 12m 24c 6i 4c 24h 2c 3d 12l 12f 3e 23P 12f 24e 6h 4c 24f 2c 3e 12n 12m 3d X.1 1 1 1 1 1 1 1 1 1 1 X.2 -1 -1 1 -1 1 1 1 -1 -1 1 X.3 1 -1 1 1 -1 1 1 1 1 1 X.4 -1 1 1 -1 -1 1 1 -1 -1 1 X.5 A A A 1 /A 1 /A A /A A X.6 -A -A A -1 /A 1 /A -A -/A A X.7 A -A A 1 -/A 1 /A A /A A X.8 -A A A -1 -/A 1 /A -A -/A A X.9 /A /A /A 1 A 1 A /A A /A X.10 -/A -/A /A -1 A 1 A -/A -A /A X.11 /A -/A /A 1 -A 1 A /A A /A X.12 -/A /A /A -1 -A 1 A -/A -A /A X.13 -2 . 2 -2 . 2 2 -2 -2 2 X.14 2 . 2 2 . 2 2 2 2 2 X.15 -/H . /H -2 . 2 H -/H -H /H X.16 /H . /H 2 . 2 H /H H /H X.17 -H . H -2 . 2 /H -H -/H H X.18 H . H 2 . 2 /H H /H H X.19 -I F -2 I -D -2 2 I -I 2 X.20 -I -F -2 I D -2 2 I -I 2 X.21 I -F -2 -I -D -2 2 -I I 2 X.22 I F -2 -I D -2 2 -I I 2 X.23 /J -G -/H I -E -2 H -/J -J /H X.24 /J G -/H I E -2 H -/J -J /H X.25 -/J G -/H -I -E -2 H /J J /H X.26 -/J -G -/H -I E -2 H /J J /H X.27 -J -/G -H I /E -2 /H J /J H X.28 -J /G -H I -/E -2 /H J /J H X.29 J /G -H -I /E -2 /H -J -/J H X.30 J -/G -H -I -/E -2 /H -J -/J H X.31 3 1 3 3 1 3 3 3 3 3 X.32 3 -1 3 3 -1 3 3 3 3 3 X.33 -3 -1 3 -3 1 3 3 -3 -3 3 X.34 -3 1 3 -3 -1 3 3 -3 -3 3 X.35 /K A /K 3 /A 3 K /K K /K X.36 K /A K 3 A 3 /K K /K K X.37 K -/A K 3 -A 3 /K K /K K X.38 /K -A /K 3 -/A 3 K /K K /K X.39 -/K -A /K -3 /A 3 K -/K -K /K X.40 -K -/A K -3 A 3 /K -K -/K K X.41 -K /A K -3 -A 3 /K -K -/K K X.42 -/K A /K -3 -/A 3 K -/K -K /K X.43 -L . -4 L . -4 4 L -L 4 X.44 L . -4 -L . -4 4 -L L 4 X.45 /M . /N L . -4 -N -/M -M -/N X.46 -/M . /N -L . -4 -N /M M -/N X.47 -M . N L . -4 -/N M /M -N X.48 M . N -L . -4 -/N -M -/M -N A = E(3) = (-1+ER(-3))/2 = b3 B = -E(4) = -ER(-1) = -i C = -E(12)^7 D = -E(8)-E(8)^3 = -ER(-2) = -i2 E = -E(24)-E(24)^19 F = -E(8)+E(8)^3 = -ER(2) = -r2 G = E(24)^11-E(24)^17 H = 2*E(3)^2 = -1-ER(-3) = -1-i3 I = 2*E(4) = 2*ER(-1) = 2i J = 2*E(12)^11 K = 3*E(3)^2 = (-3-3*ER(-3))/2 = -3-3b3 L = -4*E(4) = -4*ER(-1) = -4i M = -4*E(12)^7 N = -4*E(3) = 2-2*ER(-3) = -4b3 gap> quit;