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## G = C32×C33⋊C2order 486 = 2·35

### Direct product of C32 and C33⋊C2

Aliases: C32×C33⋊C2, C353C2, C3415S3, C3416C6, C3315(C3×C6), C3319(C3×S3), C3311(C3⋊S3), C326(S3×C32), C3⋊(C32×C3⋊S3), C325(C3×C3⋊S3), SmallGroup(486,258)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C32×C33⋊C2
 Chief series C1 — C3 — C32 — C33 — C34 — C35 — C32×C33⋊C2
 Lower central C33 — C32×C33⋊C2
 Upper central C1 — C32

Generators and relations for C32×C33⋊C2
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, 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: 3768 in 1584 conjugacy classes, 174 normal (6 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, C3×C6, C33, C33, C33, S3×C32, C3×C3⋊S3, C33⋊C2, C34, C34, C32×C3⋊S3, C3×C33⋊C2, C35, C32×C33⋊C2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C3⋊S3, C3×C33⋊C2, C32×C33⋊C2

Smallest permutation representation of C32×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 29 39)(2 30 37)(3 28 38)(4 13 26)(5 14 27)(6 15 25)(7 17 24)(8 18 22)(9 16 23)(10 20 53)(11 21 54)(12 19 52)(31 41 48)(32 42 46)(33 40 47)(34 44 51)(35 45 49)(36 43 50)
(1 22 13)(2 23 14)(3 24 15)(4 39 18)(5 37 16)(6 38 17)(7 25 28)(8 26 29)(9 27 30)(10 33 36)(11 31 34)(12 32 35)(19 42 45)(20 40 43)(21 41 44)(46 49 52)(47 50 53)(48 51 54)
(1 29 39)(2 30 37)(3 28 38)(4 13 26)(5 14 27)(6 15 25)(7 17 24)(8 18 22)(9 16 23)(10 53 20)(11 54 21)(12 52 19)(31 48 41)(32 46 42)(33 47 40)(34 51 44)(35 49 45)(36 50 43)
(1 15 23)(2 13 24)(3 14 22)(4 17 37)(5 18 38)(6 16 39)(7 30 26)(8 28 27)(9 29 25)(10 34 32)(11 35 33)(12 36 31)(19 43 41)(20 44 42)(21 45 40)(46 53 51)(47 54 49)(48 52 50)
(1 52)(2 53)(3 54)(4 42)(5 40)(6 41)(7 34)(8 35)(9 36)(10 30)(11 28)(12 29)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 39)(20 37)(21 38)(22 49)(23 50)(24 51)(25 31)(26 32)(27 33)

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

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

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

135 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3DU 6A ··· 6H order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 27 1 ··· 1 2 ··· 2 27 ··· 27

135 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 S3 C3×S3 kernel C32×C33⋊C2 C35 C3×C33⋊C2 C34 C34 C33 # reps 1 1 8 8 13 104

Matrix representation of C32×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 4 0 0 0 0 0 0 4
,
 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 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 4 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 2
,
 2 0 0 0 0 0 0 4 0 0 0 0 0 0 2 4 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 4 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 2
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 6 0 0 0 0 0 3 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

C32×C33⋊C2 in GAP, Magma, Sage, TeX

C_3^2\times C_3^3\rtimes C_2
% in TeX

G:=Group("C3^2xC3^3:C2");
// GroupNames label

G:=SmallGroup(486,258);
// by ID

G=gap.SmallGroup(486,258);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=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,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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