pgr33:=function()
local f,g,sg,s;
f:=FreeGroup("a","b","c","d","e");
g:=f/[f.1^2,f.2^2,f.3^2,f.4^2,f.5^2,f.1*f.2*f.1*f.2*f.1*f.2,f.1*f.4*f.1*f.4,f.1*f.3*f.1*f.3,f.1*f.5*f.1*f.5,f.2*f.4*f.2*f.4*f.2*f.4,f.2*f.3*f.2*f.3*f.2*f.3,f.2*f.5*f.2*f.5,f.3*f.4*f.3*f.4*f.3*f.4,f.3*f.5*f.3*f.5*f.3*f.5,f.4*f.5*f.4*f.5,f.4*f.3*f.4*f.2*f.4*f.3*f.4*f.2*f.4*f.3*f.4*f.2];
sg:=Subgroup(g,[g.2*g.3,g.4*g.5]);
s:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return s;
end;
G33:=pgr33(); gap> Size(G33); 51840 gap> N33:=NormalSubgroups(G33);; gap> NTSize(N33); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 25920 Groesse des 4. Normalteilers: 51840 gap> Centre(G33); <permutation group of size 2 with 1 generators> gap> Size(Centre(G33)); 2 gap> IsSolvable(G33); false gap> CompositionSeries(G33); [ <permutation group of size 51840 with 5 generators>, <permutation group of size 25920 with 4 generators>, Group(()) ]
gap> ct33:=CharacterTable(G33); CharacterTable( <permutation group of size 51840 with 5 generators> ) gap> Display(ct33); CT1 2 7 7 4 4 6 6 3 3 3 3 3 3 7 7 4 4 4 4 4 4 4 3 4 4 . . 1 1 2 2 2 2 3 3 2 2 2 2 2 2 4 4 4 5 1 1 . . . . . . . . . . . . . . . . . . . 1a 2a 4a 4b 2b 2c 6a 6b 6c 6d 3a 6e 2d 2e 6f 6g 6h 6i 3b 6j 3c 2P 1a 1a 2b 2b 1a 1a 3a 3a 3a 3a 3a 3a 1a 1a 3b 3b 3c 3c 3c 3c 3b 3P 1a 2a 4a 4b 2b 2c 2d 2e 2d 2e 1a 2a 2d 2e 2d 2e 2d 2e 1a 2a 1a 5P 1a 2a 4a 4b 2b 2c 6c 6d 6a 6b 3a 6e 2d 2e 6h 6i 6f 6g 3c 6k 3b 7P 1a 2a 4a 4b 2b 2c 6a 6b 6c 6d 3a 6e 2d 2e 6f 6g 6h 6i 3b 6j 3c 11P 1a 2a 4a 4b 2b 2c 6c 6d 6a 6b 3a 6e 2d 2e 6h 6i 6f 6g 3c 6k 3b 13P 1a 2a 4a 4b 2b 2c 6a 6b 6c 6d 3a 6e 2d 2e 6f 6g 6h 6i 3b 6j 3c 17P 1a 2a 4a 4b 2b 2c 6c 6d 6a 6b 3a 6e 2d 2e 6h 6i 6f 6g 3c 6k 3b X.1 1 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 -1 1 X.3 5 5 -1 -1 1 1 A A -A -A -1 -1 -3 -3 C C /C /C /D /D D X.4 5 -5 1 -1 1 -1 -A A A -A -1 1 3 -3 -C C -/C /C /D -/D D X.5 5 5 -1 -1 1 1 -A -A A A -1 -1 -3 -3 /C /C C C D D /D X.6 5 -5 1 -1 1 -1 A -A -A A -1 1 3 -3 -/C /C -C C D -D /D X.7 6 6 . . 2 2 1 1 1 1 3 3 -2 -2 1 1 1 1 -3 -3 -3 X.8 6 -6 . . 2 -2 -1 1 -1 1 3 -3 2 -2 -1 1 -1 1 -3 3 -3 X.9 10 10 . . -2 -2 -1 -1 -1 -1 1 1 2 2 D D /D /D F F /F X.10 10 -10 . . -2 2 1 -1 1 -1 1 -1 -2 2 -D D -/D /D F -F /F X.11 10 10 . . -2 -2 -1 -1 -1 -1 1 1 2 2 /D /D D D /F /F F X.12 10 -10 . . -2 2 1 -1 1 -1 1 -1 -2 2 -/D /D -D D /F -/F F X.13 15 15 -1 -1 -1 -1 -1 -1 -1 -1 3 3 -1 -1 2 2 2 2 6 6 6 X.14 15 -15 1 -1 -1 1 1 -1 1 -1 3 -3 1 -1 -2 2 -2 2 6 -6 6 X.15 15 15 1 1 3 3 -2 -2 -2 -2 . . 7 7 1 1 1 1 -3 -3 -3 X.16 15 -15 -1 1 3 -3 2 -2 2 -2 . . -7 7 -1 1 -1 1 -3 3 -3 X.17 20 20 . . 4 4 1 1 1 1 5 5 4 4 -2 -2 -2 -2 2 2 2 X.18 20 -20 . . 4 -4 -1 1 -1 1 5 -5 -4 4 2 -2 2 -2 2 -2 2 X.19 24 24 . . . . 2 2 2 2 . . 8 8 2 2 2 2 6 6 6 X.20 24 -24 . . . . -2 2 -2 2 . . -8 8 -2 2 -2 2 6 -6 6 X.21 30 30 . . 2 2 -1 -1 -1 -1 3 3 -10 -10 -1 -1 -1 -1 3 3 3 X.22 30 -30 . . 2 -2 1 -1 1 -1 3 -3 10 -10 1 -1 1 -1 3 -3 3 X.23 30 30 . . 2 2 A A -A -A -3 -3 6 6 C C /C /C G G /G X.24 30 -30 . . 2 -2 -A A A -A -3 3 -6 6 -C C -/C /C G -G /G X.25 30 30 . . 2 2 -A -A A A -3 -3 6 6 /C /C C C /G /G G X.26 30 -30 . . 2 -2 A -A -A A -3 3 -6 6 -/C /C -C C /G -/G G X.27 40 40 . . . . B B /B /B -2 -2 -8 -8 B B /B /B H H /H X.28 40 -40 . . . . -B B -/B /B -2 2 8 -8 -B B -/B /B H -H /H X.29 40 40 . . . . /B /B B B -2 -2 -8 -8 /B /B B B /H /H H X.30 40 -40 . . . . -/B /B -B B -2 2 8 -8 -/B /B -B B /H -/H H X.31 45 45 1 1 -3 -3 . . . . . . -3 -3 E E /E /E I I /I X.32 45 -45 -1 1 -3 3 . . . . . . 3 -3 -E E -/E /E I -I /I X.33 45 45 1 1 -3 -3 . . . . . . -3 -3 /E /E E E /I /I I X.34 45 -45 -1 1 -3 3 . . . . . . 3 -3 -/E /E -E E /I -/I I X.35 60 60 . . 4 4 -1 -1 -1 -1 -3 -3 -4 -4 2 2 2 2 6 6 6 X.36 60 -60 . . 4 -4 1 -1 1 -1 -3 3 4 -4 -2 2 -2 2 6 -6 6 X.37 64 64 . . . . . . . . 4 4 . . . . . . -8 -8 -8 X.38 64 -64 . . . . . . . . 4 -4 . . . . . . -8 8 -8 X.39 81 81 -1 -1 -3 -3 . . . . . . 9 9 . . . . . . . X.40 81 -81 1 -1 -3 3 . . . . . . -9 9 . . . . . . . 2 4 3 3 3 3 5 5 1 1 1 1 3 3 1 1 2 2 2 2 3 4 1 1 1 1 1 1 2 2 2 2 1 1 . . 2 2 3 3 5 . . . . . . . . . . . . . 1 1 . . . . 6k 12a 12b 12c 12d 4c 4d 9a 18a 9b 18b 6l 6m 5a 10a 6n 6o 3d 6p 2P 3b 6g 6g 6i 6i 2e 2e 9b 9b 9a 9a 3a 3a 5a 5a 3d 3d 3d 3d 3P 2a 4d 4c 4d 4c 4c 4d 3b 6j 3c 6k 2c 2b 5a 10a 2e 2d 1a 2a 5P 6j 12c 12d 12a 12b 4c 4d 9b 18b 9a 18a 6l 6m 1a 2a 6n 6o 3d 6p 7P 6k 12a 12b 12c 12d 4c 4d 9a 18a 9b 18b 6l 6m 5a 10a 6n 6o 3d 6p 11P 6j 12c 12d 12a 12b 4c 4d 9b 18b 9a 18a 6l 6m 5a 10a 6n 6o 3d 6p 13P 6k 12a 12b 12c 12d 4c 4d 9a 18a 9b 18b 6l 6m 5a 10a 6n 6o 3d 6p 17P 6j 12c 12d 12a 12b 4c 4d 9b 18b 9a 18a 6l 6m 5a 10a 6n 6o 3d 6p 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 D J J /J /J 1 1 -J -J -/J -/J 1 1 . . . . 2 2 X.4 -D J -J /J -/J -1 1 -J J -/J /J -1 1 . . . . 2 -2 X.5 /D /J /J J J 1 1 -/J -/J -J -J 1 1 . . . . 2 2 X.6 -/D /J -/J J -J -1 1 -/J /J -J J -1 1 . . . . 2 -2 X.7 -3 -1 -1 -1 -1 2 2 . . . . -1 -1 1 1 -2 -2 . . X.8 3 -1 1 -1 1 -2 2 . . . . 1 -1 1 -1 -2 2 . . X.9 /F -J -J -/J -/J 2 2 /J /J J J 1 1 . . -1 -1 1 1 X.10 -/F -J J -/J /J -2 2 /J -/J J -J -1 1 . . -1 1 1 -1 X.11 F -/J -/J -J -J 2 2 J J /J /J 1 1 . . -1 -1 1 1 X.12 -F -/J /J -J J -2 2 J -J /J -/J -1 1 . . -1 1 1 -1 X.13 6 . . . . 3 3 . . . . -1 -1 . . 2 2 . . X.14 -6 . . . . -3 3 . . . . 1 -1 . . 2 -2 . . X.15 -3 -1 -1 -1 -1 -1 -1 . . . . . . . . 1 1 3 3 X.16 3 -1 1 -1 1 1 -1 . . . . . . . . 1 -1 3 -3 X.17 2 . . . . . . -1 -1 -1 -1 1 1 . . 1 1 -1 -1 X.18 -2 . . . . . . -1 1 -1 1 -1 1 . . 1 -1 -1 1 X.19 6 . . . . . . . . . . . . -1 -1 -1 -1 3 3 X.20 -6 . . . . . . . . . . . . -1 1 -1 1 3 -3 X.21 3 1 1 1 1 -2 -2 . . . . -1 -1 . . -1 -1 3 3 X.22 -3 1 -1 1 -1 2 -2 . . . . 1 -1 . . -1 1 3 -3 X.23 /G -J -J -/J -/J 2 2 . . . . -1 -1 . . . . . . X.24 -/G -J J -/J /J -2 2 . . . . 1 -1 . . . . . . X.25 G -/J -/J -J -J 2 2 . . . . -1 -1 . . . . . . X.26 -G -/J /J -J J -2 2 . . . . 1 -1 . . . . . . X.27 /H . . . . . . J J /J /J . . . . 1 1 1 1 X.28 -/H . . . . . . J -J /J -/J . . . . 1 -1 1 -1 X.29 H . . . . . . /J /J J J . . . . 1 1 1 1 X.30 -H . . . . . . /J -/J J -J . . . . 1 -1 1 -1 X.31 /I J J /J /J 1 1 . . . . . . . . . . . . X.32 -/I J -J /J -/J -1 1 . . . . . . . . . . . . X.33 I /J /J J J 1 1 . . . . . . . . . . . . X.34 -I /J -/J J -J -1 1 . . . . . . . . . . . . X.35 6 . . . . . . . . . . 1 1 . . -1 -1 -3 -3 X.36 -6 . . . . . . . . . . -1 1 . . -1 1 -3 3 X.37 -8 . . . . . . 1 1 1 1 . . -1 -1 . . -2 -2 X.38 8 . . . . . . 1 -1 1 -1 . . -1 1 . . -2 2 X.39 . . . . . -3 -3 . . . . . . 1 1 . . . . X.40 . . . . . 3 -3 . . . . . . 1 -1 . . . . A = -E(3)+E(3)^2 = -ER(-3) = -i3 B = -2*E(3) = 1-ER(-3) = 1-i3 C = 2*E(3)+E(3)^2 = (-3+ER(-3))/2 = -1+b3 D = -2*E(3)+E(3)^2 = (1-3*ER(-3))/2 = -1-3b3 E = 3*E(3)^2 = (-3-3*ER(-3))/2 = -3-3b3 F = 5*E(3)+2*E(3)^2 = (-7+3*ER(-3))/2 = -2+3b3 G = -3*E(3)+6*E(3)^2 = (-3-9*ER(-3))/2 = -6-9b3 H = 8*E(3)+2*E(3)^2 = -5+3*ER(-3) = -5+3i3 I = -9*E(3) = (9-9*ER(-3))/2 = -9b3 J = E(3) = (-1+ER(-3))/2 = b3 gap> quit;