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## G = C2×S3×He3order 324 = 22·34

### Direct product of C2, S3 and He3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×S3×He3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — S3×He3 — C2×S3×He3
 Lower central C3 — C32 — C2×S3×He3
 Upper central C1 — C6 — C2×He3

Generators and relations for C2×S3×He3
G = < a,b,c,d,e,f | a2=b3=c2=d3=e3=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=de-1, ef=fe >

Subgroups: 436 in 144 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3×S3, C3×C6, C3×C6, C3×C6, He3, He3, C33, S3×C6, S3×C6, C62, C2×He3, C2×He3, S3×C32, C32×C6, C3×He3, C22×He3, S3×C3×C6, S3×He3, C6×He3, C2×S3×He3
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, C3×S3, C3×C6, He3, S3×C6, C62, C2×He3, S3×C32, C22×He3, S3×C3×C6, S3×He3, C2×S3×He3

Smallest permutation representation of C2×S3×He3
On 36 points
Generators in S36
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 20)(2 19)(3 21)(4 23)(5 22)(6 24)(7 26)(8 25)(9 27)(10 29)(11 28)(12 30)(13 32)(14 31)(15 33)(16 35)(17 34)(18 36)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 21 20)(22 24 23)(25 27 26)(28 30 29)(31 33 32)(34 36 35)
(1 4 9)(2 5 7)(3 6 8)(10 13 18)(11 14 16)(12 15 17)(19 22 26)(20 23 27)(21 24 25)(28 31 35)(29 32 36)(30 33 34)

G:=sub<Sym(36)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,20)(2,19)(3,21)(4,23)(5,22)(6,24)(7,26)(8,25)(9,27)(10,29)(11,28)(12,30)(13,32)(14,31)(15,33)(16,35)(17,34)(18,36), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35), (1,4,9)(2,5,7)(3,6,8)(10,13,18)(11,14,16)(12,15,17)(19,22,26)(20,23,27)(21,24,25)(28,31,35)(29,32,36)(30,33,34)>;

G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,20)(2,19)(3,21)(4,23)(5,22)(6,24)(7,26)(8,25)(9,27)(10,29)(11,28)(12,30)(13,32)(14,31)(15,33)(16,35)(17,34)(18,36), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35), (1,4,9)(2,5,7)(3,6,8)(10,13,18)(11,14,16)(12,15,17)(19,22,26)(20,23,27)(21,24,25)(28,31,35)(29,32,36)(30,33,34) );

G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,20),(2,19),(3,21),(4,23),(5,22),(6,24),(7,26),(8,25),(9,27),(10,29),(11,28),(12,30),(13,32),(14,31),(15,33),(16,35),(17,34),(18,36)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,21,20),(22,24,23),(25,27,26),(28,30,29),(31,33,32),(34,36,35)], [(1,4,9),(2,5,7),(3,6,8),(10,13,18),(11,14,16),(12,15,17),(19,22,26),(20,23,27),(21,24,25),(28,31,35),(29,32,36),(30,33,34)]])

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F ··· 3M 3N ··· 3U 6A 6B 6C 6D 6E 6F ··· 6Q 6R ··· 6Y 6Z ··· 6AO order 1 2 2 2 3 3 3 3 3 3 ··· 3 3 ··· 3 6 6 6 6 6 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 3 3 1 1 2 2 2 3 ··· 3 6 ··· 6 1 1 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 He3 C2×He3 C2×He3 S3×He3 C2×S3×He3 kernel C2×S3×He3 S3×He3 C6×He3 S3×C3×C6 S3×C32 C32×C6 C2×He3 He3 C3×C6 C32 D6 S3 C6 C2 C1 # reps 1 2 1 8 16 8 1 1 8 8 2 4 2 2 2

Matrix representation of C2×S3×He3 in GL5(𝔽7)

 6 0 0 0 0 0 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 6 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 6 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 2 0 0 0 0 0 2 0 0 0 0 0 1 5 6 0 0 0 0 6 0 0 0 1 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 6 0 0 3 1 6 0 0 1 0 6

G:=sub<GL(5,GF(7))| [6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,1,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,5,0,1,0,0,6,6,6],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,0,3,1,0,0,0,1,0,0,0,6,6,6] >;

C2×S3×He3 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm He}_3
% in TeX

G:=Group("C2xS3xHe3");
// GroupNames label

G:=SmallGroup(324,139);
// by ID

G=gap.SmallGroup(324,139);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,303,7781]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^2=d^3=e^3=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d*e^-1,e*f=f*e>;
// generators/relations

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