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## G = C33⋊17D6order 324 = 22·34

### 5th semidirect product of C33 and D6 acting via D6/C3=C22

Aliases: C3317D6, C346C22, C326S32, C33⋊C24S3, C3⋊(C324D6), C33(S3×C3⋊S3), (C3×C3⋊S3)⋊5S3, C3⋊S32(C3⋊S3), C325(C2×C3⋊S3), (C32×C3⋊S3)⋊5C2, (C3×C33⋊C2)⋊3C2, SmallGroup(324,170)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — C33⋊17D6
 Chief series C1 — C3 — C32 — C33 — C34 — C3×C33⋊C2 — C33⋊17D6
 Lower central C34 — C33⋊17D6
 Upper central C1

Generators and relations for C3317D6
G = < a,b,c,d,e | a3=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1432 in 224 conjugacy classes, 35 normal (7 characteristic)
C1, C2 [×3], C3 [×6], C3 [×13], C22, S3 [×21], C6 [×6], C32 [×2], C32 [×8], C32 [×40], D6 [×6], C3×S3 [×30], C3⋊S3, C3⋊S3 [×18], C3×C6, C33 [×6], C33 [×13], S32 [×9], C2×C3⋊S3, S3×C32 [×3], C3×C3⋊S3 [×4], C3×C3⋊S3 [×18], C33⋊C2 [×2], C34, S3×C3⋊S3 [×2], C324D6 [×4], C32×C3⋊S3, C3×C33⋊C2 [×2], C3317D6
Quotients: C1, C2 [×3], C22, S3 [×6], D6 [×6], C3⋊S3, S32 [×9], C2×C3⋊S3, S3×C3⋊S3 [×2], C324D6 [×4], C3317D6

Smallest permutation representation of C3317D6
On 36 points
Generators in S36
```(1 23 35)(2 24 36)(3 19 31)(4 20 32)(5 21 33)(6 22 34)(7 18 28)(8 13 29)(9 14 30)(10 15 25)(11 16 26)(12 17 27)
(1 5 3)(2 4 6)(7 9 11)(8 12 10)(13 17 15)(14 16 18)(19 23 21)(20 22 24)(25 29 27)(26 28 30)(31 35 33)(32 34 36)
(1 33 19)(2 20 34)(3 35 21)(4 22 36)(5 31 23)(6 24 32)(7 26 14)(8 15 27)(9 28 16)(10 17 29)(11 30 18)(12 13 25)
(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)
(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 21)(8 20)(9 19)(10 24)(11 23)(12 22)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)```

`G:=sub<Sym(36)| (1,23,35)(2,24,36)(3,19,31)(4,20,32)(5,21,33)(6,22,34)(7,18,28)(8,13,29)(9,14,30)(10,15,25)(11,16,26)(12,17,27), (1,5,3)(2,4,6)(7,9,11)(8,12,10)(13,17,15)(14,16,18)(19,23,21)(20,22,24)(25,29,27)(26,28,30)(31,35,33)(32,34,36), (1,33,19)(2,20,34)(3,35,21)(4,22,36)(5,31,23)(6,24,32)(7,26,14)(8,15,27)(9,28,16)(10,17,29)(11,30,18)(12,13,25), (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), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,21)(8,20)(9,19)(10,24)(11,23)(12,22)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)>;`

`G:=Group( (1,23,35)(2,24,36)(3,19,31)(4,20,32)(5,21,33)(6,22,34)(7,18,28)(8,13,29)(9,14,30)(10,15,25)(11,16,26)(12,17,27), (1,5,3)(2,4,6)(7,9,11)(8,12,10)(13,17,15)(14,16,18)(19,23,21)(20,22,24)(25,29,27)(26,28,30)(31,35,33)(32,34,36), (1,33,19)(2,20,34)(3,35,21)(4,22,36)(5,31,23)(6,24,32)(7,26,14)(8,15,27)(9,28,16)(10,17,29)(11,30,18)(12,13,25), (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), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,21)(8,20)(9,19)(10,24)(11,23)(12,22)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31) );`

`G=PermutationGroup([(1,23,35),(2,24,36),(3,19,31),(4,20,32),(5,21,33),(6,22,34),(7,18,28),(8,13,29),(9,14,30),(10,15,25),(11,16,26),(12,17,27)], [(1,5,3),(2,4,6),(7,9,11),(8,12,10),(13,17,15),(14,16,18),(19,23,21),(20,22,24),(25,29,27),(26,28,30),(31,35,33),(32,34,36)], [(1,33,19),(2,20,34),(3,35,21),(4,22,36),(5,31,23),(6,24,32),(7,26,14),(8,15,27),(9,28,16),(10,17,29),(11,30,18),(12,13,25)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,21),(8,20),(9,19),(10,24),(11,23),(12,22),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31)])`

33 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3F 3G ··· 3W 6A 6B 6C 6D 6E 6F order 1 2 2 2 3 ··· 3 3 ··· 3 6 6 6 6 6 6 size 1 9 27 27 2 ··· 2 4 ··· 4 18 18 18 18 54 54

33 irreducible representations

 dim 1 1 1 2 2 2 4 4 type + + + + + + + image C1 C2 C2 S3 S3 D6 S32 C32⋊4D6 kernel C33⋊17D6 C32×C3⋊S3 C3×C33⋊C2 C3×C3⋊S3 C33⋊C2 C33 C32 C3 # reps 1 1 2 4 2 6 9 8

Matrix representation of C3317D6 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 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 -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 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 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 0 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 1 0 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 -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 -1 0 0 0 0 0 0 0 -1

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

C3317D6 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_{17}D_6`
`% in TeX`

`G:=Group("C3^3:17D6");`
`// GroupNames label`

`G:=SmallGroup(324,170);`
`// by ID`

`G=gap.SmallGroup(324,170);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,80,579,297,1090,7781]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;`
`// generators/relations`

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