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G = C2×He3⋊4S3  order 324 = 22·34

Direct product of C2 and He3⋊4S3

direct product, metabelian, supersoluble, monomial

Aliases: C2×He3⋊4S3, He3⋊9D6, C33⋊10D6, C6⋊(C32⋊C6), (C6×He3)⋊3C2, (C2×He3)⋊4S3, C32⋊5(S3×C6), C33⋊5(C2×C6), (C32×C6)⋊4C6, (C32×C6)⋊4S3, C33⋊C2⋊3C6, (C3×He3)⋊8C22, (C3×C6)⋊3(C3×S3), C6.6(C3×C3⋊S3), C3.2(C6×C3⋊S3), (C3×C6)⋊1(C3⋊S3), C32⋊2(C2×C3⋊S3), C3⋊2(C2×C32⋊C6), (C2×C33⋊C2)⋊2C3, SmallGroup(324,144)

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×He3⋊4S3
 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, He3⋊4S3, C6×He3, C2×He3⋊4S3
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, He3⋊4S3, C2×He3⋊4S3

Smallest permutation representation of C2×He3⋊4S3
►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 2A2B2C3A3B3C3D3E3F3G···3Q6A6B6C6D6E6F6G···6Q6R6S6T6U
order12223333333···36666666···66666
size1127272222336···62222336···627272727

42 irreducible representations

dim11111122222266
type+++++++++
imageC1C2C2C3C6C6S3S3D6D6C3×S3S3×C6C32⋊C6C2×C32⋊C6
kernelC2×He3⋊4S3He3⋊4S3C6×He3C2×C33⋊C2C33⋊C2C32×C6C2×He3C32×C6He3C33C3×C6C32C6C3
# reps12124231318833

Matrix representation of C2×He3⋊4S3 ►in GL8(ℤ)

-10000000
0-1000000
00100000
00010000
00001000
00000100
00000010
00000001
,
0-1000000
1-1000000
00-110000
00-100000
00000-100
00001-100
00000010
00000001
,
10000000
01000000
000-10000
001-10000
00000-100
00001-100
0000000-1
0000001-1
,
10000000
01000000
00000010
00000001
00100000
00010000
00001000
00000100
,
0-1000000
1-1000000
000-10000
001-10000
00000-100
00001-100
0000000-1
0000001-1
,
01000000
10000000
00-100000
00-110000
0000-1000
0000-1100
000000-10
000000-11

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×He3⋊4S3 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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