pgr29:=function()
local v,g,sg,p;
v:=FreeGroup("a","b","c","d");
g:=v/[v.1^2,v.2^2,v.3^2,v.4^2,v.1*v.2*v.1*v.2*v.1*v.2,v.1*v.3*v.1*v.3,v.1*v.4*v.1*v.4,v.2*v.4*v.2*v.4*v.2*v.4,v.3*v.4*v.3*v.4*v.3*v.4,v.2*v.3*v.2*v.3*v.2*v.3*v.2*v.3,v.4*v.2*v.3*v.4*v.2*v.3*v.4*v.3*v.2*v.4*v.3*v.2];
sg:=Subgroup(g,[g.1,g.2,g.3]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G29:=pgr29(); gap> Size(G29); 7680 gap> N29:=NormalSubgroups(G29);; gap> NTSize(N29); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 4 Groesse des 4. Normalteilers: 64 Groesse des 5. Normalteilers: 3840 Groesse des 6. Normalteilers: 7680 gap> Centre(G29); <permutation group of size 4 with 2 generators> gap> Size(Centre(G29)); 4 gap> IsSolvable(G29); false gap> CompositionSeries(G29); [ <permutation group of size 7680 with 10 generators>, <permutation group of size 3840 with 9 generators>, <permutation group of size 64 with 7 generators>, <permutation group of size 32 with 4 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> ct29:=CharacterTable(G29); CharacterTable( <permutation group of size 7680 with 4 generators> ) gap> Display(ct29); CT1 2 9 9 8 8 9 9 6 6 5 6 6 5 3 3 3 3 6 7 7 4 7 7 3 1 1 . . 1 1 1 1 . 1 1 . 1 1 1 1 . . . . . . 5 1 1 . . 1 1 . . . . . . . . . . . . . . . . 1a 2a 2b 4a 4b 4c 2c 2d 4d 4e 4f 4g 3a 6a 12a 12b 2e 4h 4i 8a 4j 4k 2P 1a 1a 1a 2a 2a 2a 1a 1a 2b 2a 2a 2b 3a 3a 6a 6a 1a 2b 2b 4a 2b 2b 3P 1a 2a 2b 4a 4c 4b 2c 2d 4d 4f 4e 4g 1a 2a 4b 4c 2e 4h 4i 8a 4k 4j 5P 1a 2a 2b 4a 4b 4c 2c 2d 4d 4e 4f 4g 3a 6a 12a 12b 2e 4h 4i 8a 4j 4k 7P 1a 2a 2b 4a 4c 4b 2c 2d 4d 4f 4e 4g 3a 6a 12b 12a 2e 4h 4i 8a 4k 4j 11P 1a 2a 2b 4a 4c 4b 2c 2d 4d 4f 4e 4g 3a 6a 12b 12a 2e 4h 4i 8a 4k 4j 13P 1a 2a 2b 4a 4b 4c 2c 2d 4d 4e 4f 4g 3a 6a 12a 12b 2e 4h 4i 8a 4j 4k 17P 1a 2a 2b 4a 4b 4c 2c 2d 4d 4e 4f 4g 3a 6a 12a 12b 2e 4h 4i 8a 4j 4k 19P 1a 2a 2b 4a 4c 4b 2c 2d 4d 4f 4e 4g 3a 6a 12b 12a 2e 4h 4i 8a 4k 4j X.1 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 1 X.3 4 4 4 4 4 4 2 2 2 2 2 2 1 1 1 1 . . . . . . X.4 4 4 4 4 4 4 -2 -2 -2 -2 -2 -2 1 1 1 1 . . . . . . X.5 4 -4 . . A -A -2 2 . E -E . 1 -1 F -F . 2 -2 . -E E X.6 4 -4 . . A -A 2 -2 . -E E . 1 -1 F -F . 2 -2 . -E E X.7 4 -4 . . -A A -2 2 . -E E . 1 -1 -F F . 2 -2 . E -E X.8 4 -4 . . -A A 2 -2 . E -E . 1 -1 -F F . 2 -2 . E -E X.9 5 5 5 5 5 5 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 X.10 5 5 5 5 5 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 X.11 6 6 6 6 6 6 . . . . . . . . . . -2 -2 -2 -2 -2 -2 X.12 6 6 -2 2 -6 -6 . . . . . . . . . . 2 -2 -2 . 2 2 X.13 6 6 -2 2 -6 -6 . . . . . . . . . . 2 -2 -2 . 2 2 X.14 6 6 -2 2 -6 -6 . . . . . . . . . . -2 2 2 . -2 -2 X.15 6 6 -2 2 -6 -6 . . . . . . . . . . -2 2 2 . -2 -2 X.16 10 10 2 -2 -10 -10 -4 -4 . 4 4 . 1 1 -1 -1 2 2 2 . -2 -2 X.17 10 10 2 -2 -10 -10 4 4 . -4 -4 . 1 1 -1 -1 2 2 2 . -2 -2 X.18 10 10 2 -2 -10 -10 -2 -2 2 2 2 -2 1 1 -1 -1 -2 -2 -2 . 2 2 X.19 10 10 2 -2 -10 -10 2 2 -2 -2 -2 2 1 1 -1 -1 -2 -2 -2 . 2 2 X.20 15 15 -1 -1 15 15 -3 -3 1 -3 -3 1 . . . . -1 3 3 -1 3 3 X.21 15 15 -1 -1 15 15 -3 -3 1 -3 -3 1 . . . . 3 -1 -1 -1 -1 -1 X.22 15 15 -1 -1 15 15 3 3 -1 3 3 -1 . . . . -1 3 3 -1 3 3 X.23 15 15 -1 -1 15 15 3 3 -1 3 3 -1 . . . . 3 -1 -1 -1 -1 -1 X.24 16 -16 . . B -B 4 -4 . -A A . 1 -1 F -F . . . . . . X.25 16 -16 . . B -B -4 4 . A -A . 1 -1 F -F . . . . . . X.26 16 -16 . . -B B 4 -4 . A -A . 1 -1 -F F . . . . . . X.27 16 -16 . . -B B -4 4 . -A A . 1 -1 -F F . . . . . . X.28 20 20 4 -4 -20 -20 2 2 2 -2 -2 -2 -1 -1 1 1 . . . . . . X.29 20 20 4 -4 -20 -20 -2 -2 -2 2 2 2 -1 -1 1 1 . . . . . . X.30 20 -20 . . C -C 2 -2 . -E E . -1 1 -F F . 2 -2 . -E E X.31 20 -20 . . C -C -2 2 . E -E . -1 1 -F F . 2 -2 . -E E X.32 20 -20 . . -C C 2 -2 . E -E . -1 1 F -F . 2 -2 . E -E X.33 20 -20 . . -C C -2 2 . -E E . -1 1 F -F . 2 -2 . E -E X.34 24 24 -8 8 -24 -24 . . . . . . . . . . . . . . . . X.35 24 -24 . . D -D . . . . . . . . . . . -4 4 . A -A X.36 24 -24 . . -D D . . . . . . . . . . . -4 4 . -A A X.37 30 30 -2 -2 30 30 . . . . . . . . . . -2 -2 -2 2 -2 -2 2 6 3 3 3 3 4 4 5 5 5 5 2 2 2 2 3 . 1 1 1 1 . . . . . . . . . . 5 . . . . . . . . . . . 1 1 1 1 4l 6b 6c 12c 12d 4m 4n 8b 8c 8d 8e 20a 20b 10a 5a 2P 2a 3a 3a 6a 6a 2e 2e 4k 4k 4j 4j 10a 10a 5a 5a 3P 4l 2d 2c 4e 4f 4m 4n 8d 8e 8b 8c 20b 20a 10a 5a 5P 4l 6b 6c 12c 12d 4m 4n 8b 8c 8d 8e 4c 4b 2a 1a 7P 4l 6b 6c 12d 12c 4m 4n 8d 8e 8b 8c 20b 20a 10a 5a 11P 4l 6b 6c 12d 12c 4m 4n 8d 8e 8b 8c 20b 20a 10a 5a 13P 4l 6b 6c 12c 12d 4m 4n 8b 8c 8d 8e 20a 20b 10a 5a 17P 4l 6b 6c 12c 12d 4m 4n 8b 8c 8d 8e 20a 20b 10a 5a 19P 4l 6b 6c 12d 12c 4m 4n 8d 8e 8b 8c 20b 20a 10a 5a 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 . -1 -1 -1 -1 . . . . . . -1 -1 -1 -1 X.4 . 1 1 1 1 . . . . . . -1 -1 -1 -1 X.5 . -1 1 -F F . . G -G /G -/G -F F 1 -1 X.6 . 1 -1 F -F . . -G G -/G /G -F F 1 -1 X.7 . -1 1 F -F . . /G -/G G -G F -F 1 -1 X.8 . 1 -1 -F F . . -/G /G -G G F -F 1 -1 X.9 1 1 1 1 1 -1 -1 -1 -1 -1 -1 . . . . X.10 1 -1 -1 -1 -1 1 1 1 1 1 1 . . . . X.11 -2 . . . . . . . . . . 1 1 1 1 X.12 -2 . . . . -2 2 . . . . -1 -1 1 1 X.13 -2 . . . . 2 -2 . . . . -1 -1 1 1 X.14 2 . . . . . . E E -E -E -1 -1 1 1 X.15 2 . . . . . . -E -E E E -1 -1 1 1 X.16 -2 -1 -1 1 1 . . . . . . . . . . X.17 -2 1 1 -1 -1 . . . . . . . . . . X.18 2 1 1 -1 -1 . . . . . . . . . . X.19 2 -1 -1 1 1 . . . . . . . . . . X.20 -1 . . . . 1 1 -1 -1 -1 -1 . . . . X.21 3 . . . . -1 -1 1 1 1 1 . . . . X.22 -1 . . . . -1 -1 1 1 1 1 . . . . X.23 3 . . . . 1 1 -1 -1 -1 -1 . . . . X.24 . -1 1 -F F . . . . . . F -F -1 1 X.25 . 1 -1 F -F . . . . . . F -F -1 1 X.26 . -1 1 F -F . . . . . . -F F -1 1 X.27 . 1 -1 -F F . . . . . . -F F -1 1 X.28 . -1 -1 1 1 . . . . . . . . . . X.29 . 1 1 -1 -1 . . . . . . . . . . X.30 . 1 -1 F -F . . G -G /G -/G . . . . X.31 . -1 1 -F F . . -G G -/G /G . . . . X.32 . 1 -1 -F F . . /G -/G G -G . . . . X.33 . -1 1 F -F . . -/G /G -G G . . . . X.34 . . . . . . . . . . . 1 1 -1 -1 X.35 . . . . . . . . . . . -F F 1 -1 X.36 . . . . . . . . . . . F -F 1 -1 X.37 -2 . . . . . . . . . . . . . . A = -4*E(4) = -4*ER(-1) = -4i B = -16*E(4) = -16*ER(-1) = -16i C = -20*E(4) = -20*ER(-1) = -20i D = -24*E(4) = -24*ER(-1) = -24i E = -2*E(4) = -2*ER(-1) = -2i F = E(4) = ER(-1) = i G = -1-E(4) = -1-ER(-1) = -1-i gap> quit;