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## G = C2×He3⋊4S3order 324 = 22·34

### Direct product of C2 and He3⋊4S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×He3⋊4S3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — He3⋊4S3 — C2×He3⋊4S3
 Lower central C33 — C2×He3⋊4S3
 Upper central C1 — C2

Generators and relations for C2×He34S3
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 988 in 160 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, He3, He3, C33, C33, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, C2×He3, C3×C3⋊S3, C33⋊C2, C32×C6, C32×C6, C3×He3, C2×C32⋊C6, C6×C3⋊S3, C2×C33⋊C2, He34S3, C6×He3, C2×He34S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, C3⋊S3, S3×C6, C2×C3⋊S3, C32⋊C6, C3×C3⋊S3, C2×C32⋊C6, C6×C3⋊S3, He34S3, C2×He34S3

Smallest permutation representation of C2×He34S3
On 54 points
Generators in S54
(1 10)(2 9)(3 11)(4 14)(5 13)(6 15)(7 18)(8 17)(12 16)(19 47)(20 48)(21 46)(22 54)(23 52)(24 53)(25 49)(26 50)(27 51)(28 44)(29 45)(30 43)(31 39)(32 37)(33 38)(34 42)(35 40)(36 41)
(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)(37 38 39)(40 41 42)(43 44 45)(46 47 48)(49 50 51)(52 53 54)
(1 16 3)(2 18 17)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(19 21 20)(22 24 23)(25 26 27)(28 30 29)(31 32 33)(34 35 36)(37 38 39)(40 41 42)(43 45 44)(46 48 47)(49 50 51)(52 54 53)
(1 54 35)(2 48 33)(3 52 34)(4 45 26)(5 44 27)(6 43 25)(7 19 39)(8 21 37)(9 20 38)(10 22 40)(11 23 42)(12 24 41)(13 28 51)(14 29 50)(15 30 49)(16 53 36)(17 46 32)(18 47 31)
(1 17 14)(2 13 16)(3 18 15)(4 10 8)(5 12 9)(6 11 7)(19 43 23)(20 44 24)(21 45 22)(25 42 39)(26 40 37)(27 41 38)(28 53 48)(29 54 46)(30 52 47)(31 49 34)(32 50 35)(33 51 36)
(1 2)(3 18)(4 5)(7 11)(8 12)(9 10)(13 14)(16 17)(19 23)(20 22)(21 24)(26 27)(28 29)(31 34)(32 36)(33 35)(37 41)(38 40)(39 42)(44 45)(46 53)(47 52)(48 54)(50 51)

G:=sub<Sym(54)| (1,10)(2,9)(3,11)(4,14)(5,13)(6,15)(7,18)(8,17)(12,16)(19,47)(20,48)(21,46)(22,54)(23,52)(24,53)(25,49)(26,50)(27,51)(28,44)(29,45)(30,43)(31,39)(32,37)(33,38)(34,42)(35,40)(36,41), (19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,44,45)(46,47,48)(49,50,51)(52,53,54), (1,16,3)(2,18,17)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,45,44)(46,48,47)(49,50,51)(52,54,53), (1,54,35)(2,48,33)(3,52,34)(4,45,26)(5,44,27)(6,43,25)(7,19,39)(8,21,37)(9,20,38)(10,22,40)(11,23,42)(12,24,41)(13,28,51)(14,29,50)(15,30,49)(16,53,36)(17,46,32)(18,47,31), (1,17,14)(2,13,16)(3,18,15)(4,10,8)(5,12,9)(6,11,7)(19,43,23)(20,44,24)(21,45,22)(25,42,39)(26,40,37)(27,41,38)(28,53,48)(29,54,46)(30,52,47)(31,49,34)(32,50,35)(33,51,36), (1,2)(3,18)(4,5)(7,11)(8,12)(9,10)(13,14)(16,17)(19,23)(20,22)(21,24)(26,27)(28,29)(31,34)(32,36)(33,35)(37,41)(38,40)(39,42)(44,45)(46,53)(47,52)(48,54)(50,51)>;

G:=Group( (1,10)(2,9)(3,11)(4,14)(5,13)(6,15)(7,18)(8,17)(12,16)(19,47)(20,48)(21,46)(22,54)(23,52)(24,53)(25,49)(26,50)(27,51)(28,44)(29,45)(30,43)(31,39)(32,37)(33,38)(34,42)(35,40)(36,41), (19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,44,45)(46,47,48)(49,50,51)(52,53,54), (1,16,3)(2,18,17)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,45,44)(46,48,47)(49,50,51)(52,54,53), (1,54,35)(2,48,33)(3,52,34)(4,45,26)(5,44,27)(6,43,25)(7,19,39)(8,21,37)(9,20,38)(10,22,40)(11,23,42)(12,24,41)(13,28,51)(14,29,50)(15,30,49)(16,53,36)(17,46,32)(18,47,31), (1,17,14)(2,13,16)(3,18,15)(4,10,8)(5,12,9)(6,11,7)(19,43,23)(20,44,24)(21,45,22)(25,42,39)(26,40,37)(27,41,38)(28,53,48)(29,54,46)(30,52,47)(31,49,34)(32,50,35)(33,51,36), (1,2)(3,18)(4,5)(7,11)(8,12)(9,10)(13,14)(16,17)(19,23)(20,22)(21,24)(26,27)(28,29)(31,34)(32,36)(33,35)(37,41)(38,40)(39,42)(44,45)(46,53)(47,52)(48,54)(50,51) );

G=PermutationGroup([[(1,10),(2,9),(3,11),(4,14),(5,13),(6,15),(7,18),(8,17),(12,16),(19,47),(20,48),(21,46),(22,54),(23,52),(24,53),(25,49),(26,50),(27,51),(28,44),(29,45),(30,43),(31,39),(32,37),(33,38),(34,42),(35,40),(36,41)], [(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36),(37,38,39),(40,41,42),(43,44,45),(46,47,48),(49,50,51),(52,53,54)], [(1,16,3),(2,18,17),(4,5,6),(7,8,9),(10,12,11),(13,15,14),(19,21,20),(22,24,23),(25,26,27),(28,30,29),(31,32,33),(34,35,36),(37,38,39),(40,41,42),(43,45,44),(46,48,47),(49,50,51),(52,54,53)], [(1,54,35),(2,48,33),(3,52,34),(4,45,26),(5,44,27),(6,43,25),(7,19,39),(8,21,37),(9,20,38),(10,22,40),(11,23,42),(12,24,41),(13,28,51),(14,29,50),(15,30,49),(16,53,36),(17,46,32),(18,47,31)], [(1,17,14),(2,13,16),(3,18,15),(4,10,8),(5,12,9),(6,11,7),(19,43,23),(20,44,24),(21,45,22),(25,42,39),(26,40,37),(27,41,38),(28,53,48),(29,54,46),(30,52,47),(31,49,34),(32,50,35),(33,51,36)], [(1,2),(3,18),(4,5),(7,11),(8,12),(9,10),(13,14),(16,17),(19,23),(20,22),(21,24),(26,27),(28,29),(31,34),(32,36),(33,35),(37,41),(38,40),(39,42),(44,45),(46,53),(47,52),(48,54),(50,51)]])

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G ··· 3Q 6A 6B 6C 6D 6E 6F 6G ··· 6Q 6R 6S 6T 6U order 1 2 2 2 3 3 3 3 3 3 3 ··· 3 6 6 6 6 6 6 6 ··· 6 6 6 6 6 size 1 1 27 27 2 2 2 2 3 3 6 ··· 6 2 2 2 2 3 3 6 ··· 6 27 27 27 27

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 6 6 type + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 S3 D6 D6 C3×S3 S3×C6 C32⋊C6 C2×C32⋊C6 kernel C2×He3⋊4S3 He3⋊4S3 C6×He3 C2×C33⋊C2 C33⋊C2 C32×C6 C2×He3 C32×C6 He3 C33 C3×C6 C32 C6 C3 # reps 1 2 1 2 4 2 3 1 3 1 8 8 3 3

Matrix representation of C2×He34S3 in GL8(ℤ)

 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1] >;

C2×He34S3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_4S_3
% in TeX

G:=Group("C2xHe3:4S3");
// GroupNames label

G:=SmallGroup(324,144);
// by ID

G=gap.SmallGroup(324,144);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,735,2164,7781]);
// Polycyclic

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

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