pgr7:=function()
local d,g,sg,p;
d:=FreeGroup("a","b","c");
g:=d/[d.1^2,d.2^3,d.3^3,d.1*d.2*d.3*d.1*d.3^2*d.2^2,d.1*d.2*d.3*d.2^2*d.1*d.3^2];
sg:=Subgroup(g,[g.3]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G7:=pgr7(); gap> Size(G7); 144 gap> N7:=NormalSubgroups(G7);; gap> NTSize(N7); 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: 24 Groesse des 12. Normalteilers: 24 Groesse des 13. Normalteilers: 48 Groesse des 14. Normalteilers: 48 Groesse des 15. Normalteilers: 48 Groesse des 16. Normalteilers: 48 Groesse des 17. Normalteilers: 72 Groesse des 18. Normalteilers: 144 gap> Centre(G7); Group( [ ( 1,15,40,43,42,47,46,45,34,33,17, 4)( 2, 7,25,44,29,36,48,35,38,20,19, 8) ( 3,12,31,14,39,37,21,41,26,23,10, 9)( 5,16, 6,24,22,28,27,32,11,30,13,18 ) ]) gap> Size(Centre(G7)); 12 gap> IsSolvable(G7); true gap> CompositionSeries(G7); [ <permutation group of size 144 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>, Group([ ( 1,46)( 2,48)( 3,21)( 4,47)( 5,27)( 6,11)( 7,35)( 8,36)( 9,37) (10,39)(12,41)(13,22)(14,23)(15,45)(16,32)(17,42)(18,28)(19,29)(20,44) (24,30)(25,38)(26,31)(33,43)(34,40) ]), Group(()) ]
gap> ct7:=CharacterTable(G7); CharacterTable( <permutation group of size 144 with 3 generators> ) gap> Display(ct7); CT1 2 4 2 2 3 2 2 2 3 2 2 4 2 3 2 2 3 2 2 3 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 1 2 2 1a 3a 3b 2a 12a 12b 3c 12c 6a 12d 12e 12f 6b 12g 3d 12h 6c 12i 2P 1a 3b 3a 1a 6e 6i 3d 6g 3f 6f 6g 6c 3h 6c 3c 6j 3e 6h 3P 1a 1a 1a 2a 4b 4c 1a 4a 2b 4b 4b 4b 2a 4c 1a 4a 2b 4c 5P 1a 3b 3a 2a 12o 12p 3d 12h 6c 12n 12q 12j 6d 12m 3c 12c 6a 12k 7P 1a 3a 3b 2a 12p 12o 3c 12c 6a 12k 12r 12g 6b 12f 3d 12h 6c 12n 11P 1a 3b 3a 2a 12b 12a 3d 12h 6c 12i 12l 12m 6d 12j 3c 12c 6a 12d 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 1 1 1 1 1 1 A A A A A A A A /A /A /A /A X.4 1 1 1 -1 -1 -1 A A A -A -A -A -A -A /A /A /A -/A X.5 1 1 1 1 1 1 /A /A /A /A /A /A /A /A A A A A X.6 1 1 1 -1 -1 -1 /A /A /A -/A -/A -/A -/A -/A A A A -A X.7 1 A /A 1 A /A 1 A /A 1 A /A A /A 1 /A A 1 X.8 1 A /A -1 -A -/A 1 A /A -1 -A -/A -A -/A 1 /A A -1 X.9 1 A /A 1 A /A A /A 1 A /A 1 /A 1 /A A 1 /A X.10 1 A /A -1 -A -/A A /A 1 -A -/A -1 -/A -1 /A A 1 -/A X.11 1 A /A 1 A /A /A 1 A /A 1 A 1 A A 1 /A A X.12 1 A /A -1 -A -/A /A 1 A -/A -1 -A -1 -A A 1 /A -A X.13 1 /A A 1 /A A 1 /A A 1 /A A /A A 1 A /A 1 X.14 1 /A A -1 -/A -A 1 /A A -1 -/A -A -/A -A 1 A /A -1 X.15 1 /A A 1 /A A A 1 /A A 1 /A 1 /A /A 1 A /A X.16 1 /A A -1 -/A -A A 1 /A -A -1 -/A -1 -/A /A 1 A -/A X.17 1 /A A 1 /A A /A A 1 /A A 1 A 1 A /A 1 A X.18 1 /A A -1 -/A -A /A A 1 -/A -A -1 -A -1 A /A 1 -A X.19 2 -/A -A . B /B -1 . A C D -/B . /B -1 . /A -C X.20 2 -/A -A . -B -/B -1 . A -C -D /B . -/B -1 . /A C X.21 2 -/A -A . B /B -A . /A -/B E B . -B -/A . A -B X.22 2 -/A -A . -B -/B -A . /A /B -E -B . B -/A . A B X.23 2 -/A -A . B /B -/A . 1 B -/D C . -C -A . 1 /B X.24 2 -/A -A . -B -/B -/A . 1 -B /D -C . C -A . 1 -/B X.25 2 -1 -1 . C -C -1 . 1 C E C . -C -1 . 1 -C X.26 2 -1 -1 . -C C -1 . 1 -C -E -C . C -1 . 1 C X.27 2 -1 -1 . C -C -A . A -/B -/D -/B . /B -/A . /A -B X.28 2 -1 -1 . -C C -A . A /B /D /B . -/B -/A . /A B X.29 2 -1 -1 . C -C -/A . /A B D B . -B -A . A /B X.30 2 -1 -1 . -C C -/A . /A -B -D -B . B -A . A -/B X.31 2 -A -/A . -/B -B -1 . /A C -/D B . -B -1 . A -C X.32 2 -A -/A . /B B -1 . /A -C /D -B . B -1 . A C X.33 2 -A -/A . -/B -B -A . 1 -/B D C . -C -/A . 1 -B X.34 2 -A -/A . /B B -A . 1 /B -D -C . C -/A . 1 B X.35 2 -A -/A . -/B -B -/A . A B E -/B . /B -A . /A /B X.36 2 -A -/A . /B B -/A . A -B -E /B . -/B -A . /A -/B X.37 3 . . -1 . . . -1 . . 3 . -1 . . -1 . . X.38 3 . . 1 . . . -1 . . -3 . 1 . . -1 . . X.39 3 . . -1 . . . -A . . F . -A . . -/A . . X.40 3 . . 1 . . . -A . . -F . A . . -/A . . X.41 3 . . -1 . . . -/A . . /F . -/A . . -A . . X.42 3 . . 1 . . . -/A . . -/F . /A . . -A . . 2 2 3 3 2 2 4 2 2 4 2 2 2 2 2 2 2 4 4 4 3 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 12j 6d 4a 6e 12k 12l 12m 6f 6g 3e 6h 12n 6i 3f 12o 12p 4b 3g 6j 2P 6a 3g 2b 3a 6f 6j 6a 3c 3g 3f 3d 6h 3b 3e 6i 6e 2b 3h 3h 3P 4b 2a 4a 2b 4c 4c 4c 2b 2b 1a 2b 4b 2b 1a 4b 4c 4c 1a 2b 5P 12f 6b 4a 6i 12i 12r 12g 6h 6j 3f 6f 12d 6e 3e 12a 12b 4b 3h 6g 7P 12m 6d 4a 6e 12d 12q 12j 6f 6g 3e 6h 12i 6i 3f 12b 12a 4c 3g 6j 11P 12g 6b 4a 6i 12n 12e 12f 6h 6j 3f 6f 12k 6e 3e 12p 12o 4c 3h 6g 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 /A /A 1 1 A /A /A /A /A A A /A 1 /A 1 1 1 A A X.4 -/A -/A 1 1 -A -/A -/A /A /A A A -/A 1 /A -1 -1 -1 A A X.5 A A 1 1 /A A A A A /A /A A 1 A 1 1 1 /A /A X.6 -A -A 1 1 -/A -A -A A A /A /A -A 1 A -1 -1 -1 /A /A X.7 A /A 1 /A 1 /A A 1 /A /A 1 1 A A /A A 1 A A X.8 -A -/A 1 /A -1 -/A -A 1 /A /A 1 -1 A A -/A -A -1 A A X.9 1 A 1 /A A A 1 /A A 1 A /A A 1 /A A 1 /A /A X.10 -1 -A 1 /A -A -A -1 /A A 1 A -/A A 1 -/A -A -1 /A /A X.11 /A 1 1 /A /A 1 /A A 1 A /A A A /A /A A 1 1 1 X.12 -/A -1 1 /A -/A -1 -/A A 1 A /A -A A /A -/A -A -1 1 1 X.13 /A A 1 A 1 A /A 1 A A 1 1 /A /A A /A 1 /A /A X.14 -/A -A 1 A -1 -A -/A 1 A A 1 -1 /A /A -A -/A -1 /A /A X.15 A 1 1 A A 1 A /A 1 /A A /A /A A A /A 1 1 1 X.16 -A -1 1 A -A -1 -A /A 1 /A A -/A /A A -A -/A -1 1 1 X.17 1 /A 1 A /A /A 1 A /A 1 /A A /A 1 A /A 1 A A X.18 -1 -/A 1 A -/A -/A -1 A /A 1 /A -A /A 1 -A -/A -1 A A X.19 B . . A -C /D -B 1 G -A 1 C /A -/A -/B -B -E -/G /G X.20 -B . . A C -/D B 1 G -A 1 -C /A -/A /B B E -/G /G X.21 -/B . . A /B -E /B /A -2 -/A A B /A -A -/B -B -E 2 -2 X.22 /B . . A -/B E -/B /A -2 -/A A -B /A -A /B B E 2 -2 X.23 C . . A -B -D -C A /G -1 /A -/B /A -1 -/B -B -E -G G X.24 -C . . A B D C A /G -1 /A /B /A -1 /B B E -G G X.25 C . . 1 -C -E -C 1 -2 -1 1 C 1 -1 C -C -E 2 -2 X.26 -C . . 1 C E C 1 -2 -1 1 -C 1 -1 -C C E 2 -2 X.27 B . . 1 /B -D -B /A /G -A A B 1 -/A C -C -E -G G X.28 -B . . 1 -/B D B /A /G -A A -B 1 -/A -C C E -G G X.29 -/B . . 1 -B /D /B A G -/A /A -/B 1 -A C -C -E -/G /G X.30 /B . . 1 B -/D -/B A G -/A /A /B 1 -A -C C E -/G /G X.31 -/B . . /A -C -D /B 1 /G -/A 1 C A -A B /B -E -G G X.32 /B . . /A C D -/B 1 /G -/A 1 -C A -A -B -/B E -G G X.33 C . . /A /B /D -C /A G -1 A B A -1 B /B -E -/G /G X.34 -C . . /A -/B -/D C /A G -1 A -B A -1 -B -/B E -/G /G X.35 B . . /A -B -E -B A -2 -A /A -/B A -/A B /B -E 2 -2 X.36 -B . . /A B E B A -2 -A /A /B A -/A -B -/B E 2 -2 X.37 . -1 -1 . . 3 . . 3 . . . . . . . 3 3 3 X.38 . 1 -1 . . -3 . . 3 . . . . . . . -3 3 3 X.39 . -/A -1 . . /F . . /F . . . . . . . 3 F F X.40 . /A -1 . . -/F . . /F . . . . . . . -3 F F X.41 . -A -1 . . F . . F . . . . . . . 3 /F /F X.42 . A -1 . . -F . . F . . . . . . . -3 /F /F 2 4 4 4 4 4 3 2 2 2 2 2 3h 4c 12q 2b 12r 2P 3g 2b 6j 1a 6g 3P 1a 4b 4b 2b 4c 5P 3g 4c 12e 2b 12l 7P 3h 4b 12l 2b 12e 11P 3g 4b 12r 2b 12q X.1 1 1 1 1 1 X.2 1 -1 -1 1 -1 X.3 /A 1 /A 1 A X.4 /A -1 -/A 1 -A X.5 A 1 A 1 /A X.6 A -1 -A 1 -/A X.7 /A 1 /A 1 A X.8 /A -1 -/A 1 -A X.9 A 1 A 1 /A X.10 A -1 -A 1 -/A X.11 1 1 1 1 1 X.12 1 -1 -1 1 -1 X.13 A 1 A 1 /A X.14 A -1 -A 1 -/A X.15 1 1 1 1 1 X.16 1 -1 -1 1 -1 X.17 /A 1 /A 1 A X.18 /A -1 -/A 1 -A X.19 -G E -/D -2 -D X.20 -G -E /D -2 D X.21 2 E E -2 -E X.22 2 -E -E -2 E X.23 -/G E D -2 /D X.24 -/G -E -D -2 -/D X.25 2 E E -2 -E X.26 2 -E -E -2 E X.27 -/G E D -2 /D X.28 -/G -E -D -2 -/D X.29 -G E -/D -2 -D X.30 -G -E /D -2 D X.31 -/G E D -2 /D X.32 -/G -E -D -2 -/D X.33 -G E -/D -2 -D X.34 -G -E /D -2 D X.35 2 E E -2 -E X.36 2 -E -E -2 E X.37 3 3 3 3 3 X.38 3 -3 -3 3 -3 X.39 /F 3 /F 3 F X.40 /F -3 -/F 3 -F X.41 F 3 F 3 /F X.42 F -3 -F 3 -/F A = E(3) = (-1+ER(-3))/2 = b3 B = -E(12)^11 C = -E(4) = -ER(-1) = -i D = 2*E(12)^11 E = 2*E(4) = 2*ER(-1) = 2i F = 3*E(3) = (-3+3*ER(-3))/2 = 3b3 G = -2*E(3) = 1-ER(-3) = 1-i3 gap> quit;