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## G = C32⋊5D18order 324 = 22·34

### 2nd semidirect product of C32 and D18 acting via D18/C9=C22

Aliases: C325D18, C33.10D6, C92S32, C9⋊S35S3, C3⋊S32D9, C33(S3×D9), (C3×C9)⋊14D6, C32.9S32, (C32×C9)⋊5C22, C3.1(C324D6), (C9×C3⋊S3)⋊3C2, (C3×C9⋊S3)⋊3C2, (C3×C3⋊S3).5S3, SmallGroup(324,123)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — C32⋊5D18
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C9×C3⋊S3 — C32⋊5D18
 Lower central C32×C9 — C32⋊5D18
 Upper central C1

Generators and relations for C325D18
G = < a,b,c,d | a3=b3=c18=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 628 in 86 conjugacy classes, 20 normal (10 characteristic)
C1, C2 [×3], C3, C3 [×2], C3 [×4], C22, S3 [×9], C6 [×3], C9, C9 [×3], C32, C32 [×2], C32 [×4], D6 [×3], D9 [×4], C18, C3×S3 [×9], C3⋊S3, C3⋊S3 [×2], C3×C9 [×2], C3×C9 [×4], C33, D18, S32 [×3], C3×D9 [×4], S3×C9 [×3], C9⋊S3 [×2], C3×C3⋊S3, C3×C3⋊S3 [×2], C32×C9, S3×D9 [×2], C324D6, C3×C9⋊S3 [×2], C9×C3⋊S3, C325D18
Quotients: C1, C2 [×3], C22, S3 [×3], D6 [×3], D9, D18, S32 [×3], S3×D9 [×2], C324D6, C325D18

Smallest permutation representation of C325D18
On 36 points
Generators in S36
```(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 31 25)(20 26 32)(21 33 27)(22 28 34)(23 35 29)(24 30 36)
(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 25 31)(20 32 26)(21 27 33)(22 34 28)(23 29 35)(24 36 30)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D ··· 3H 6A 6B 6C 9A 9B 9C 9D ··· 9O 18A 18B 18C order 1 2 2 2 3 3 3 3 ··· 3 6 6 6 9 9 9 9 ··· 9 18 18 18 size 1 9 27 27 2 2 2 4 ··· 4 18 54 54 2 2 2 4 ··· 4 18 18 18

33 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 D9 D18 S32 S32 S3×D9 C32⋊4D6 C32⋊5D18 kernel C32⋊5D18 C3×C9⋊S3 C9×C3⋊S3 C9⋊S3 C3×C3⋊S3 C3×C9 C33 C3⋊S3 C32 C9 C32 C3 C3 C1 # reps 1 2 1 2 1 2 1 3 3 1 2 6 2 6

Matrix representation of C325D18 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 0 0 0 0 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 18 0 0 0 0 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 14 0 0 0 0 0 0 15
,
 0 18 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 0 11 0 0 0 0 7 0

`G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,14,0,0,0,0,0,0,15],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,7,0,0,0,0,11,0] >;`

C325D18 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_5D_{18}`
`% in TeX`

`G:=Group("C3^2:5D18");`
`// GroupNames label`

`G:=SmallGroup(324,123);`
`// by ID`

`G=gap.SmallGroup(324,123);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,80,1593,453,2164,3899]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^18=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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