pgr5:=function()
local z,g,sg,p;
z:=FreeGroup("a","b");
g:=z/[z.1^3,z.2^3,z.1*z.2*z.1*z.2*z.1^2*z.2^2*z.1^2*z.2^2];
sg:=Subgroup(g,[g.1]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G5:=pgr5(); gap> Size(G5); 72 gap> N5:=NormalSubgroups(G5);; gap> NTSize(N5); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 3 Groesse des 4. Normalteilers: 6 Groesse des 5. Normalteilers: 8 Groesse des 6. Normalteilers: 24 Groesse des 7. Normalteilers: 24 Groesse des 8. Normalteilers: 24 Groesse des 9. Normalteilers: 24 Groesse des 10. Normalteilers: 72 gap> Centre(G5); Group( [ ( 1,20,24,22,19, 6)( 2,10,18,21,12, 9)( 3, 4,15,23,17,14)( 5, 7,13,16, 8,11 ) ]) gap> Size(Centre(G5)); 6 gap> IsSolvable(G5); true gap> CompositionSeries(G5); [ <permutation group of size 72 with 5 generators>, <permutation group of size 24 with 4 generators>, Group([ ( 1,13,22,11)( 2, 3,21,23)( 4,12,17,10)( 5,20,16,19)( 6, 7,24, 8) ( 9,14,18,15), ( 1,15,22,14)( 2,16,21, 5)( 3,20,23,19)( 4,24,17, 6) ( 7,10, 8,12)( 9,13,18,11), ( 1,22)( 2,21)( 3,23)( 4,17)( 5,16)( 6,24) ( 7, 8)( 9,18)(10,12)(11,13)(14,15)(19,20) ]), Group([ ( 1,15,22,14)( 2,16,21, 5)( 3,20,23,19)( 4,24,17, 6)( 7,10, 8,12) ( 9,13,18,11), ( 1,22)( 2,21)( 3,23)( 4,17)( 5,16)( 6,24)( 7, 8) ( 9,18)(10,12)(11,13)(14,15)(19,20) ]), Group([ ( 1,22)( 2,21)( 3,23)( 4,17)( 5,16)( 6,24)( 7, 8)( 9,18)(10,12) (11,13)(14,15)(19,20) ]), Group(()) ]
gap> ct5:=CharacterTable(G5);
CharacterTable( Group(
[ ( 2, 3, 5)( 4, 7,10)( 8,12,17)( 9,14,11)(13,18,15)(16,21,23),
( 1, 2, 4)( 3, 6, 9)( 5, 8,13)( 7,11,16)(10,15,20)(12,14,19)(17,22,21)
(18,23,24) ]) )
gap> Display(ct5);
CT1
2 3 1 1 1 2 1 1 1 2 3 1 1 1 2 1 1 1 3 3 3
3 2 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2
1a 3a 3b 3c 12a 6a 3d 6b 12b 6c 6d 3e 6e 4a 6f 3f 6g 3g 6h 2a
2P 1a 3b 3a 3d 6c 3e 3c 3f 6h 3g 3c 3f 3a 2a 3b 3e 3d 3h 3h 1a
3P 1a 1a 1a 1a 4a 2a 1a 2a 4a 2a 2a 1a 2a 4a 2a 1a 2a 1a 2a 2a
5P 1a 3b 3a 3d 12b 6b 3c 6a 12a 6h 6g 3f 6f 4a 6e 3e 6d 3h 6c 2a
7P 1a 3a 3b 3c 12a 6a 3d 6b 12b 6c 6d 3e 6e 4a 6f 3f 6g 3g 6h 2a
11P 1a 3b 3a 3d 12b 6b 3c 6a 12a 6h 6g 3f 6f 4a 6e 3e 6d 3h 6c 2a
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 A /A 1 A /A 1 A /A /A 1 A /A 1 A /A 1 A A 1
X.3 1 /A A 1 /A A 1 /A A A 1 /A A 1 /A A 1 /A /A 1
X.4 1 1 1 A A A /A /A /A /A /A /A 1 1 1 A A A A 1
X.5 1 A /A A /A 1 /A 1 A A /A 1 /A 1 A 1 A /A /A 1
X.6 1 /A A A 1 /A /A A 1 1 /A A A 1 /A /A A 1 1 1
X.7 1 1 1 /A /A /A A A A A A A 1 1 1 /A /A /A /A 1
X.8 1 A /A /A 1 A A /A 1 1 A /A /A 1 A A /A 1 1 1
X.9 1 /A A /A A 1 A 1 /A /A A 1 A 1 /A 1 /A A A 1
X.10 2 -1 -1 -1 . 1 -1 1 . -2 1 -1 1 . 1 -1 1 2 -2 -2
X.11 2 -1 -1 -/A . /A -A A . B A -A 1 . 1 -/A /A -/B /B -2
X.12 2 -A -/A -/A . A -A /A . -2 A -/A /A . A -A /A 2 -2 -2
X.13 2 -/A -A -/A . 1 -A 1 . /B A -1 A . /A -1 /A -B B -2
X.14 2 -A -/A -1 . /A -1 A . /B 1 -A /A . A -/A 1 -B B -2
X.15 2 -/A -A -1 . A -1 /A . B 1 -/A A . /A -A 1 -/B /B -2
X.16 2 -1 -1 -A . A -/A /A . /B /A -/A 1 . 1 -A A -B B -2
X.17 2 -A -/A -A . 1 -/A 1 . B /A -1 /A . A -1 A -/B /B -2
X.18 2 -/A -A -A . /A -/A A . -2 /A -A A . /A -/A A 2 -2 -2
X.19 3 . . . -1 . . . -1 3 . . . -1 . . . 3 3 3
X.20 3 . . . -A . . . -/A C . . . -1 . . . /C /C 3
X.21 3 . . . -/A . . . -A /C . . . -1 . . . C C 3
2 3
3 2
3h
2P 3g
3P 1a
5P 3g
7P 3h
11P 3g
X.1 1
X.2 /A
X.3 A
X.4 /A
X.5 A
X.6 1
X.7 A
X.8 1
X.9 /A
X.10 2
X.11 -B
X.12 2
X.13 -/B
X.14 -/B
X.15 -B
X.16 -/B
X.17 -B
X.18 2
X.19 3
X.20 C
X.21 /C
A = E(3)
= (-1+ER(-3))/2 = b3
B = -2*E(3)
= 1-ER(-3) = 1-i3
C = 3*E(3)^2
= (-3-3*ER(-3))/2 = -3-3b3
gap> quit;