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## G = C3×C33⋊C2order 162 = 2·34

### Direct product of C3 and C33⋊C2

Aliases: C3×C33⋊C2, C343C2, C339C6, C338S3, C326(C3×S3), C323(C3⋊S3), C3⋊(C3×C3⋊S3), SmallGroup(162,53)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C33⋊C2
G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 580 in 190 conjugacy classes, 58 normal (6 characteristic)
C1, C2, C3, C3 [×13], C3 [×13], S3 [×13], C6, C32 [×26], C32 [×52], C3×S3 [×13], C3⋊S3 [×13], C33, C33 [×13], C33 [×13], C3×C3⋊S3 [×13], C33⋊C2, C34, C3×C33⋊C2
Quotients: C1, C2, C3, S3 [×13], C6, C3×S3 [×13], C3⋊S3 [×13], C3×C3⋊S3 [×13], C33⋊C2, C3×C33⋊C2

Smallest permutation representation of C3×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 12 40)(2 10 41)(3 11 42)(4 7 34)(5 8 35)(6 9 36)(13 25 19)(14 26 20)(15 27 21)(16 30 33)(17 28 31)(18 29 32)(22 45 39)(23 43 37)(24 44 38)(46 49 52)(47 50 53)(48 51 54)
(1 25 35)(2 26 36)(3 27 34)(4 11 21)(5 12 19)(6 10 20)(7 42 15)(8 40 13)(9 41 14)(16 53 23)(17 54 24)(18 52 22)(28 48 44)(29 46 45)(30 47 43)(31 51 38)(32 49 39)(33 50 37)
(1 15 6)(2 13 4)(3 14 5)(7 10 25)(8 11 26)(9 12 27)(16 51 45)(17 49 43)(18 50 44)(19 34 41)(20 35 42)(21 36 40)(22 33 48)(23 31 46)(24 32 47)(28 52 37)(29 53 38)(30 54 39)
(1 22)(2 23)(3 24)(4 31)(5 32)(6 33)(7 28)(8 29)(9 30)(10 37)(11 38)(12 39)(13 46)(14 47)(15 48)(16 36)(17 34)(18 35)(19 49)(20 50)(21 51)(25 52)(26 53)(27 54)(40 45)(41 43)(42 44)

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

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

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

C3×C33⋊C2 is a maximal subgroup of
C3×S3×C3⋊S3  C3317D6  C331C18  C332D9  C33⋊C18  C33⋊D9  C336D9  C347S3  C3410C6  C34.11S3  C3413S3
C3×C33⋊C2 is a maximal quotient of
C3410C6  C34.11S3  C9○He33S3  C9○He34S3

45 conjugacy classes

 class 1 2 3A 3B 3C ··· 3AO 6A 6B order 1 2 3 3 3 ··· 3 6 6 size 1 27 1 1 2 ··· 2 27 27

45 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 S3 C3×S3 kernel C3×C33⋊C2 C34 C33⋊C2 C33 C33 C32 # reps 1 1 2 2 13 26

Matrix representation of C3×C33⋊C2 in GL6(𝔽7)

 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 4 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 2 0 0 0 0 0 0 4
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

C3×C33⋊C2 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC3^3:C2");
// GroupNames label

G:=SmallGroup(162,53);
// by ID

G=gap.SmallGroup(162,53);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,182,723,2704]);
// Polycyclic

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

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