pgr9:=function()
local z,g,sg,p;
z:=FreeGroup("a","b");
g:=z/[z.1^4,z.2^2,z.1*z.2*z.1*z.2*z.1*z.2*z.1^3*z.2*z.1^3*z.2*z.1^3*z.2];
sg:=Subgroup(g,[g.1]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G9:=pgr9(); gap> Size(G9); 192 gap> N9:=NormalSubgroups(G9);; gap> NTSize(N9); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 4 Groesse des 4. Normalteilers: 8 Groesse des 5. Normalteilers: 8 Groesse des 6. Normalteilers: 16 Groesse des 7. Normalteilers: 24 Groesse des 8. Normalteilers: 32 Groesse des 9. Normalteilers: 48 Groesse des 10. Normalteilers: 48 Groesse des 11. Normalteilers: 48 Groesse des 12. Normalteilers: 96 Groesse des 13. Normalteilers: 96 Groesse des 14. Normalteilers: 96 Groesse des 15. Normalteilers: 192 gap> Centre(G9); Group( [ ( 1,31,44,48,41,46,30,12)( 2,24,38,47,33,43,23, 8)( 3, 9,25,39,40,34,29,11) ( 4,13,26,42,21,36,15,16)( 5, 6,18,32,45,27,37,17)( 7,19,20,35,14,28,10,22 ) ]) gap> Size(Centre(G9)); 8 gap> IsSolvable(G9); true gap> CompositionSeries(G9); [ <permutation group of size 192 with 7 generators>, <permutation group of size 96 with 6 generators>, <permutation group of size 48 with 5 generators>, <permutation group of size 24 with 4 generators>, <permutation group of size 8 with 3 generators>, <permutation group of size 4 with 2 generators>, Group([ ( 1,41)( 2,33)( 3,40)( 4,21)( 5,45)( 6,27)( 7,14)( 8,47)( 9,34) (10,20)(11,39)(12,48)(13,36)(15,26)(16,42)(17,32)(18,37)(19,28)(22,35) (23,38)(24,43)(25,29)(30,44)(31,46) ]), Group(()) ]
gap> ct9:=CharacterTable(G9); CharacterTable( <permutation group of size 192 with 2 generators> ) gap> Display(ct9); CT1 2 6 5 5 5 4 3 5 3 3 4 3 5 4 3 6 5 5 4 3 5 3 1 . . . . 1 . 1 1 . 1 . . 1 1 . . . 1 . 1a 4a 2a 4b 2b 24a 8a 24b 12a 8b 6a 4c 8c 12b 8d 8e 8f 4d 24c 8g 2P 1a 2a 1a 2a 1a 12a 4c 12b 6a 4e 3a 2c 4h 6a 4e 4c 4e 2c 12a 4h 3P 1a 4b 2a 4a 2b 8d 8h 8j 4e 8c 2c 4c 8b 4h 8k 8e 8g 4d 8l 8f 5P 1a 4a 2a 4b 2b 24c 8h 24d 12a 8b 6a 4c 8c 12b 8l 8i 8f 4d 24a 8g 7P 1a 4b 2a 4a 2b 24b 8a 24a 12b 8c 6a 4c 8b 12a 8j 8i 8g 4d 24d 8f 11P 1a 4b 2a 4a 2b 24d 8h 24c 12b 8c 6a 4c 8b 12a 8k 8e 8g 4d 24b 8f 13P 1a 4a 2a 4b 2b 24c 8h 24d 12a 8b 6a 4c 8c 12b 8l 8i 8f 4d 24a 8g 17P 1a 4a 2a 4b 2b 24a 8a 24b 12a 8b 6a 4c 8c 12b 8d 8e 8f 4d 24c 8g 19P 1a 4b 2a 4a 2b 24d 8h 24c 12b 8c 6a 4c 8b 12a 8k 8e 8g 4d 24b 8f 23P 1a 4b 2a 4a 2b 24b 8a 24a 12b 8c 6a 4c 8b 12a 8j 8i 8g 4d 24d 8f X.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 X.3 1 -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 1 X.5 1 A -1 -A 1 A -1 -A -1 -A 1 1 A -1 -A 1 A -1 A -A X.6 1 A -1 -A -1 -A 1 A -1 -A 1 1 A -1 A -1 -A 1 -A A X.7 1 -A -1 A 1 -A -1 A -1 A 1 1 -A -1 A 1 -A -1 -A A X.8 1 -A -1 A -1 A 1 -A -1 A 1 1 -A -1 -A -1 A 1 A -A X.9 2 . 2 . . 1 . 1 -1 . -1 2 . -1 -2 . -2 . 1 -2 X.10 2 . 2 . . -1 . -1 -1 . -1 2 . -1 2 . 2 . -1 2 X.11 2 . -2 . . A . -A 1 . -1 2 . 1 E . -E . A E X.12 2 . -2 . . -A . A 1 . -1 2 . 1 -E . E . -A -E X.13 2 B . /B . C D /C A . 1 . . -A F I . . -C . X.14 2 B . /B . -C -D -/C A . 1 . . -A -F -I . . C . X.15 2 /B . B . -/C -D -C -A . 1 . . A -/F I . . /C . X.16 2 /B . B . /C D C -A . 1 . . A /F -I . . -/C . X.17 2 -/B . -B . -/C D -C -A . 1 . . A -/F -I . . /C . X.18 2 -/B . -B . /C -D C -A . 1 . . A /F I . . -/C . X.19 2 -B . -/B . C -D /C A . 1 . . -A F -I . . -C . X.20 2 -B . -/B . -C D -/C A . 1 . . -A -F I . . C . X.21 3 -1 -1 -1 -1 . 1 . . 1 . -1 1 . -3 1 1 -1 . 1 X.22 3 -1 -1 -1 1 . -1 . . 1 . -1 1 . 3 -1 -1 1 . -1 X.23 3 1 -1 1 -1 . 1 . . -1 . -1 -1 . 3 1 -1 -1 . -1 X.24 3 1 -1 1 1 . -1 . . -1 . -1 -1 . -3 -1 1 1 . 1 X.25 3 -A 1 A -1 . -1 . . -A . -1 A . G 1 A 1 . -A X.26 3 -A 1 A 1 . 1 . . -A . -1 A . -G -1 -A -1 . A X.27 3 A 1 -A -1 . -1 . . A . -1 -A . -G 1 -A 1 . A X.28 3 A 1 -A 1 . 1 . . A . -1 -A . G -1 A -1 . -A X.29 4 . . . . -/C . -C A . -1 . . -A H . . . /C . X.30 4 . . . . C . /C -A . -1 . . A -/H . . . -C . X.31 4 . . . . -C . -/C -A . -1 . . A /H . . . C . X.32 4 . . . . /C . C A . -1 . . -A -H . . . -/C . 2 3 5 5 6 5 6 3 5 6 6 6 6 3 1 . . 1 . 1 1 . 1 1 1 1 24d 8h 8i 4e 4f 8j 3a 4g 2c 4h 8k 8l 2P 12b 4c 4c 2c 2a 4h 3a 2a 1a 2c 4h 4e 3P 8k 8a 8i 4h 4g 8l 1a 4f 2c 4e 8d 8j 5P 24b 8a 8e 4e 4f 8k 3a 4g 2c 4h 8j 8d 7P 24c 8h 8e 4h 4g 8d 3a 4f 2c 4e 8l 8k 11P 24a 8a 8i 4h 4g 8l 3a 4f 2c 4e 8d 8j 13P 24b 8a 8e 4e 4f 8k 3a 4g 2c 4h 8j 8d 17P 24d 8h 8i 4e 4f 8j 3a 4g 2c 4h 8k 8l 19P 24a 8a 8i 4h 4g 8l 3a 4f 2c 4e 8d 8j 23P 24c 8h 8e 4h 4g 8d 3a 4f 2c 4e 8l 8k X.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 X.3 -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 X.5 -A -1 1 -1 -A A 1 A 1 -1 A -A X.6 A 1 -1 -1 -A -A 1 A 1 -1 -A A X.7 A -1 1 -1 A -A 1 -A 1 -1 -A A X.8 -A 1 -1 -1 A A 1 -A 1 -1 A -A X.9 1 . . 2 . -2 -1 . 2 2 -2 -2 X.10 -1 . . 2 . 2 -1 . 2 2 2 2 X.11 -A . . -2 . -E -1 . 2 -2 -E E X.12 A . . -2 . E -1 . 2 -2 E -E X.13 -/C -D -I E -/B /F -1 -B -2 -E -/F -F X.14 /C D I E -/B -/F -1 -B -2 -E /F F X.15 C D -I -E -B -F -1 -/B -2 E F /F X.16 -C -D I -E -B F -1 -/B -2 E -F -/F X.17 C -D I -E B -F -1 /B -2 E F /F X.18 -C D -I -E B F -1 /B -2 E -F -/F X.19 -/C D I E /B /F -1 B -2 -E -/F -F X.20 /C -D -I E /B -/F -1 B -2 -E /F F X.21 . 1 1 3 -1 -3 . -1 3 3 -3 -3 X.22 . -1 -1 3 -1 3 . -1 3 3 3 3 X.23 . 1 1 3 1 3 . 1 3 3 3 3 X.24 . -1 -1 3 1 -3 . 1 3 3 -3 -3 X.25 . -1 1 -3 A -G . -A 3 -3 -G G X.26 . 1 -1 -3 A G . -A 3 -3 G -G X.27 . -1 1 -3 -A G . A 3 -3 G -G X.28 . 1 -1 -3 -A -G . A 3 -3 -G G X.29 C . . J . /H 1 . -4 -J -/H -H X.30 -/C . . -J . -H 1 . -4 J H /H X.31 /C . . -J . H 1 . -4 J -H -/H X.32 -C . . J . -/H 1 . -4 -J /H H A = E(4) = ER(-1) = i B = -1-E(4) = -1-ER(-1) = -1-i C = -E(8)^3 D = -E(8)+E(8)^3 = -ER(2) = -r2 E = 2*E(4) = 2*ER(-1) = 2i F = 2*E(8) G = 3*E(4) = 3*ER(-1) = 3i H = -4*E(8)^3 I = -E(8)-E(8)^3 = -ER(-2) = -i2 J = -4*E(4) = -4*ER(-1) = -4i gap> quit;