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## G = S3×C33⋊C2order 324 = 22·34

### Direct product of S3 and C33⋊C2

Aliases: S3×C33⋊C2, C3315D6, C344C22, C328S32, (S3×C32)⋊4S3, (S3×C33)⋊3C2, C34⋊C21C2, C31(S3×C3⋊S3), (C3×S3)⋊(C3⋊S3), C324(C2×C3⋊S3), C31(C2×C33⋊C2), (C3×C33⋊C2)⋊2C2, SmallGroup(324,168)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — S3×C33⋊C2
 Chief series C1 — C3 — C32 — C33 — C34 — S3×C33 — S3×C33⋊C2
 Lower central C34 — S3×C33⋊C2
 Upper central C1

Generators and relations for S3×C33⋊C2
G = < a,b,c,d,e,f | a3=b2=c3=d3=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 3880 in 436 conjugacy classes, 88 normal (10 characteristic)
C1, C2, C3, C3, C3, C22, S3, S3, C6, C32, C32, D6, C3×S3, C3×S3, C3⋊S3, C3×C6, C33, C33, C33, S32, C2×C3⋊S3, S3×C32, C3×C3⋊S3, C33⋊C2, C33⋊C2, C32×C6, C34, S3×C3⋊S3, C2×C33⋊C2, S3×C33, C3×C33⋊C2, C34⋊C2, S3×C33⋊C2
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S32, C2×C3⋊S3, C33⋊C2, S3×C3⋊S3, C2×C33⋊C2, S3×C33⋊C2

Smallest permutation representation of S3×C33⋊C2
On 54 points
Generators in S54
(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)(37 38 39)(40 41 42)(43 44 45)(46 47 48)(49 50 51)(52 53 54)
(1 28)(2 30)(3 29)(4 34)(5 36)(6 35)(7 31)(8 33)(9 32)(10 46)(11 48)(12 47)(13 52)(14 54)(15 53)(16 49)(17 51)(18 50)(19 37)(20 39)(21 38)(22 43)(23 45)(24 44)(25 40)(26 42)(27 41)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)
(1 8 6)(2 9 4)(3 7 5)(10 17 15)(11 18 13)(12 16 14)(19 26 24)(20 27 22)(21 25 23)(28 33 35)(29 31 36)(30 32 34)(37 42 44)(38 40 45)(39 41 43)(46 51 53)(47 49 54)(48 50 52)
(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 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3N 3O ··· 3AA 6A ··· 6M 6N order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 size 1 3 27 81 2 ··· 2 4 ··· 4 6 ··· 6 54

45 irreducible representations

 dim 1 1 1 1 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 S3 S3 D6 S32 kernel S3×C33⋊C2 S3×C33 C3×C33⋊C2 C34⋊C2 S3×C32 C33⋊C2 C33 C32 # reps 1 1 1 1 13 1 14 13

Matrix representation of S3×C33⋊C2 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 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 -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 1 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 -1 -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 0 1
,
 -1 -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 0 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 1 0 0 0 0 0 0 0 0 1
,
 -1 -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 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
,
 -1 0 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 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 0 -1

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

S3×C33⋊C2 in GAP, Magma, Sage, TeX

S_3\times C_3^3\rtimes C_2
% in TeX

G:=Group("S3xC3^3:C2");
// GroupNames label

G:=SmallGroup(324,168);
// by ID

G=gap.SmallGroup(324,168);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,297,1090,7781]);
// Polycyclic

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

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