pgr27:=function()
local d,g,sg,p;
d:=FreeGroup("a","b","c");
g:=d/[d.1^2,d.2^2,d.3^2,d.1*d.2*d.1*d.2*d.1*d.2*d.1*d.2,d.2*d.3*d.2*d.3*d.2*d.3*d.2*d.3*d.2*d.3,d.1*d.3*d.1*d.3*d.1*d.3,d.3*d.2*d.3*d.1*d.3*d.2*d.3*d.1*d.3*d.2*d.3*d.1];
sg:=Subgroup(g,[g.2,g.3]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G27:=pgr27(); gap> Size(G27); 2160 gap> N27:=NormalSubgroups(G27);; gap> NTSize(N27); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 3 Groesse des 4. Normalteilers: 6 Groesse des 5. Normalteilers: 1080 Groesse des 6. Normalteilers: 2160 gap> Centre(G27); <permutation group of size 6 with 1 generators> gap> Size(Centre(G27)); 6 gap> IsSolvable(G27); false gap> CompositionSeries(G27); [ <permutation group of size 2160 with 8 generators>, <permutation group of size 6 with 2 generators>, <permutation group of size 3 with 1 generators>, Group(()) ]
gap> ct27:=CharacterTable(G27); CharacterTable( <permutation group of size 2160 with 3 generators> ) gap> Display(ct27); CT1 2 4 4 4 4 4 4 1 1 4 4 4 4 4 4 1 1 1 1 1 3 3 3 3 3 3 3 2 2 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 . . . . . . . . 1 1 1 1 1 1a 3a 3b 6a 2a 6b 3c 6c 2b 6d 6e 6f 2c 6g 30a 10a 30b 15a 15b 2P 1a 3b 3a 3a 1a 3b 3c 3c 1a 3b 3a 3a 1a 3b 15c 5b 15d 15d 15c 3P 1a 1a 1a 2a 2a 2a 1a 2a 2b 2b 2b 2c 2c 2c 10b 10b 10b 5b 5b 5P 1a 3b 3a 6b 2a 6a 3c 6c 2b 6e 6d 6g 2c 6f 6b 2a 6a 3b 3a 7P 1a 3a 3b 6a 2a 6b 3c 6c 2b 6d 6e 6f 2c 6g 30c 10b 30d 15c 15d 11P 1a 3b 3a 6b 2a 6a 3c 6c 2b 6e 6d 6g 2c 6f 30b 10a 30a 15b 15a 13P 1a 3a 3b 6a 2a 6b 3c 6c 2b 6d 6e 6f 2c 6g 30c 10b 30d 15c 15d 17P 1a 3b 3a 6b 2a 6a 3c 6c 2b 6e 6d 6g 2c 6f 30d 10b 30c 15d 15c 19P 1a 3a 3b 6a 2a 6b 3c 6c 2b 6d 6e 6f 2c 6g 30a 10a 30b 15a 15b 23P 1a 3b 3a 6b 2a 6a 3c 6c 2b 6e 6d 6g 2c 6f 30d 10b 30c 15d 15c 29P 1a 3b 3a 6b 2a 6a 3c 6c 2b 6e 6d 6g 2c 6f 30b 10a 30a 15b 15a 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 3 A /A -/A -3 -A . . 1 E /E -/E -1 -E G I /G -/G -G X.4 3 A /A -/A -3 -A . . 1 E /E -/E -1 -E H *I /H -/H -H X.5 3 A /A /A 3 A . . -1 -E -/E -/E -1 -E -H -*I -/H -/H -H X.6 3 A /A /A 3 A . . -1 -E -/E -/E -1 -E -G -I -/G -/G -G X.7 3 /A A -A -3 -/A . . 1 /E E -E -1 -/E /H *I H -H -/H X.8 3 /A A -A -3 -/A . . 1 /E E -E -1 -/E /G I G -G -/G X.9 3 /A A A 3 /A . . -1 -/E -E -E -1 -/E -/G -I -G -G -/G X.10 3 /A A A 3 /A . . -1 -/E -E -E -1 -/E -/H -*I -H -H -/H X.11 5 5 5 -5 -5 -5 -1 1 -1 -1 -1 1 1 1 . . . . . X.12 5 5 5 -5 -5 -5 2 -2 -1 -1 -1 1 1 1 . . . . . X.13 5 5 5 5 5 5 -1 -1 1 1 1 1 1 1 . . . . . X.14 5 5 5 5 5 5 2 2 1 1 1 1 1 1 . . . . . X.15 6 B /B -/B -6 -B . . -2 F /F -/F 2 -F -/E -1 -E E /E X.16 6 B /B /B 6 B . . 2 -F -/F -/F 2 -F /E 1 E E /E X.17 6 /B B -B -6 -/B . . -2 /F F -F 2 -/F -E -1 -/E /E E X.18 6 /B B B 6 /B . . 2 -/F -F -F 2 -/F E 1 /E /E E X.19 8 8 8 -8 -8 -8 -1 1 . . . . . . I I I -I -I X.20 8 8 8 -8 -8 -8 -1 1 . . . . . . *I *I *I -*I -*I X.21 8 8 8 8 8 8 -1 -1 . . . . . . -*I -*I -*I -*I -*I X.22 8 8 8 8 8 8 -1 -1 . . . . . . -I -I -I -I -I X.23 9 9 9 9 9 9 . . 1 1 1 1 1 1 -1 -1 -1 -1 -1 X.24 9 9 9 -9 -9 -9 . . -1 -1 -1 1 1 1 1 1 1 -1 -1 X.25 9 C /C -/C -9 -C . . -1 -E -/E /E 1 E /E 1 E -E -/E X.26 9 C /C /C 9 C . . 1 E /E /E 1 E -/E -1 -E -E -/E X.27 9 /C C -C -9 -/C . . -1 -/E -E E 1 /E E 1 /E -/E -E X.28 9 /C C C 9 /C . . 1 /E E E 1 /E -E -1 -/E -/E -E X.29 10 10 10 10 10 10 1 1 -2 -2 -2 -2 -2 -2 . . . . . X.30 10 10 10 -10 -10 -10 1 -1 2 2 2 -2 -2 -2 . . . . . X.31 15 D /D -/D -15 -D . . 1 E /E -/E -1 -E . . . . . X.32 15 D /D /D 15 D . . -1 -E -/E -/E -1 -E . . . . . X.33 15 /D D -D -15 -/D . . 1 /E E -E -1 -/E . . . . . X.34 15 /D D D 15 /D . . -1 -/E -E -E -1 -/E . . . . . 2 1 1 1 1 1 1 1 3 3 3 3 3 3 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 5 1 1 1 1 1 1 1 . . . . . . . . 5a 5b 15c 15d 30c 10b 30d 4a 12a 12b 12c 4b 12d 3d 6h 2P 5b 5a 15b 15a 15a 5a 15b 2c 6f 6g 6g 2c 6f 3d 3d 3P 5b 5a 5a 5a 10a 10a 10a 4a 4a 4a 4b 4b 4b 1a 2a 5P 1a 1a 3b 3a 6b 2a 6a 4a 12b 12a 12d 4b 12c 3d 6h 7P 5b 5a 15a 15b 30a 10a 30b 4a 12a 12b 12c 4b 12d 3d 6h 11P 5a 5b 15d 15c 30d 10b 30c 4a 12b 12a 12d 4b 12c 3d 6h 13P 5b 5a 15a 15b 30a 10a 30b 4a 12a 12b 12c 4b 12d 3d 6h 17P 5b 5a 15b 15a 30b 10a 30a 4a 12b 12a 12d 4b 12c 3d 6h 19P 5a 5b 15c 15d 30c 10b 30d 4a 12a 12b 12c 4b 12d 3d 6h 23P 5b 5a 15b 15a 30b 10a 30a 4a 12b 12a 12d 4b 12c 3d 6h 29P 5a 5b 15d 15c 30d 10b 30c 4a 12b 12a 12d 4b 12c 3d 6h X.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 X.3 -I -*I -/H -H H *I /H -1 -E -/E /E 1 E . . X.4 -*I -I -/G -G G I /G -1 -E -/E /E 1 E . . X.5 -*I -I -/G -G -G -I -/G 1 E /E /E 1 E . . X.6 -I -*I -/H -H -H -*I -/H 1 E /E /E 1 E . . X.7 -*I -I -G -/G /G I G -1 -/E -E E 1 /E . . X.8 -I -*I -H -/H /H *I H -1 -/E -E E 1 /E . . X.9 -I -*I -H -/H -/H -*I -H 1 /E E E 1 /E . . X.10 -*I -I -G -/G -/G -I -G 1 /E E E 1 /E . . X.11 . . . . . . . 1 1 1 -1 -1 -1 2 -2 X.12 . . . . . . . 1 1 1 -1 -1 -1 -1 1 X.13 . . . . . . . -1 -1 -1 -1 -1 -1 2 2 X.14 . . . . . . . -1 -1 -1 -1 -1 -1 -1 -1 X.15 1 1 E /E -/E -1 -E . . . . . . . . X.16 1 1 E /E /E 1 E . . . . . . . . X.17 1 1 /E E -E -1 -/E . . . . . . . . X.18 1 1 /E E E 1 /E . . . . . . . . X.19 -I -*I -*I -*I *I *I *I . . . . . . -1 1 X.20 -*I -I -I -I I I I . . . . . . -1 1 X.21 -*I -I -I -I -I -I -I . . . . . . -1 -1 X.22 -I -*I -*I -*I -*I -*I -*I . . . . . . -1 -1 X.23 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 . . X.24 -1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 1 . . X.25 -1 -1 -E -/E /E 1 E -1 -E -/E /E 1 E . . X.26 -1 -1 -E -/E -/E -1 -E 1 E /E /E 1 E . . X.27 -1 -1 -/E -E E 1 /E -1 -/E -E E 1 /E . . X.28 -1 -1 -/E -E -E -1 -/E 1 /E E E 1 /E . . X.29 . . . . . . . . . . . . . 1 1 X.30 . . . . . . . . . . . . . 1 -1 X.31 . . . . . . . 1 E /E -/E -1 -E . . X.32 . . . . . . . -1 -E -/E -/E -1 -E . . X.33 . . . . . . . 1 /E E -E -1 -/E . . X.34 . . . . . . . -1 -/E -E -E -1 -/E . . A = 3*E(3)^2 = (-3-3*ER(-3))/2 = -3-3b3 B = 6*E(3)^2 = -3-3*ER(-3) = -3-3i3 C = 9*E(3)^2 = (-9-9*ER(-3))/2 = -9-9b3 D = 15*E(3)^2 = (-15-15*ER(-3))/2 = -15-15b3 E = E(3)^2 = (-1-ER(-3))/2 = -1-b3 F = -2*E(3)^2 = 1+ER(-3) = 1+i3 G = E(15)^11+E(15)^14 H = E(15)^2+E(15)^8 I = E(5)^2+E(5)^3 = (-1-ER(5))/2 = -1-b5 gap> quit;