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## G = C32⋊2D12order 216 = 23·33

### The semidirect product of C32 and D12 acting via D12/C3=D4

Aliases: C333D4, C322D12, C32⋊C4⋊S3, C31S3≀C2, C3⋊S3.2D6, C324D62C2, (C3×C32⋊C4)⋊1C2, (C3×C3⋊S3).5C22, SmallGroup(216,159)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — C32⋊2D12
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C32⋊4D6 — C32⋊2D12
 Lower central C33 — C3×C3⋊S3 — C32⋊2D12
 Upper central C1

Generators and relations for C322D12
G = < a,b,c,d | a3=b3=c12=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

Subgroups: 436 in 60 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C3, C3, C4, C22, S3, C6, D4, C32, C32, C12, D6, C3×S3, C3⋊S3, C3⋊S3, D12, C33, C32⋊C4, S32, C3×C3⋊S3, C3×C3⋊S3, S3≀C2, C3×C32⋊C4, C324D6, C322D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S3≀C2, C322D12

Character table of C322D12

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 6A 6B 6C 12A 12B size 1 9 18 18 2 4 4 8 8 18 18 36 36 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ5 2 2 0 0 -1 2 2 -1 -1 2 -1 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 0 -1 2 2 -1 -1 -2 -1 0 0 1 1 orthogonal lifted from D6 ρ7 2 -2 0 0 2 2 2 2 2 0 -2 0 0 0 0 orthogonal lifted from D4 ρ8 2 -2 0 0 -1 2 2 -1 -1 0 1 0 0 -√3 √3 orthogonal lifted from D12 ρ9 2 -2 0 0 -1 2 2 -1 -1 0 1 0 0 √3 -√3 orthogonal lifted from D12 ρ10 4 0 -2 0 4 -2 1 -2 1 0 0 0 1 0 0 orthogonal lifted from S3≀C2 ρ11 4 0 0 2 4 1 -2 1 -2 0 0 -1 0 0 0 orthogonal lifted from S3≀C2 ρ12 4 0 2 0 4 -2 1 -2 1 0 0 0 -1 0 0 orthogonal lifted from S3≀C2 ρ13 4 0 0 -2 4 1 -2 1 -2 0 0 1 0 0 0 orthogonal lifted from S3≀C2 ρ14 8 0 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 orthogonal faithful ρ15 8 0 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C322D12
On 12 points - transitive group 12T118
Generators in S12
```(2 6 10)(4 12 8)
(1 5 9)(3 11 7)
(1 2 3 4 5 6 7 8 9 10 11 12)
(1 3)(4 12)(5 11)(6 10)(7 9)```

`G:=sub<Sym(12)| (2,6,10)(4,12,8), (1,5,9)(3,11,7), (1,2,3,4,5,6,7,8,9,10,11,12), (1,3)(4,12)(5,11)(6,10)(7,9)>;`

`G:=Group( (2,6,10)(4,12,8), (1,5,9)(3,11,7), (1,2,3,4,5,6,7,8,9,10,11,12), (1,3)(4,12)(5,11)(6,10)(7,9) );`

`G=PermutationGroup([[(2,6,10),(4,12,8)], [(1,5,9),(3,11,7)], [(1,2,3,4,5,6,7,8,9,10,11,12)], [(1,3),(4,12),(5,11),(6,10),(7,9)]])`

`G:=TransitiveGroup(12,118);`

On 18 points - transitive group 18T104
Generators in S18
```(1 10 16)(2 11 17)(3 18 12)(4 7 13)(5 14 8)(6 15 9)
(1 10 16)(2 17 11)(3 18 12)(4 13 7)(5 14 8)(6 9 15)
(1 2 3 4 5 6)(7 8 9 10 11 12 13 14 15 16 17 18)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)```

`G:=sub<Sym(18)| (1,10,16)(2,11,17)(3,18,12)(4,7,13)(5,14,8)(6,15,9), (1,10,16)(2,17,11)(3,18,12)(4,13,7)(5,14,8)(6,9,15), (1,2,3,4,5,6)(7,8,9,10,11,12,13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)>;`

`G:=Group( (1,10,16)(2,11,17)(3,18,12)(4,7,13)(5,14,8)(6,15,9), (1,10,16)(2,17,11)(3,18,12)(4,13,7)(5,14,8)(6,9,15), (1,2,3,4,5,6)(7,8,9,10,11,12,13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16) );`

`G=PermutationGroup([[(1,10,16),(2,11,17),(3,18,12),(4,7,13),(5,14,8),(6,15,9)], [(1,10,16),(2,17,11),(3,18,12),(4,13,7),(5,14,8),(6,9,15)], [(1,2,3,4,5,6),(7,8,9,10,11,12,13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,18),(14,17),(15,16)]])`

`G:=TransitiveGroup(18,104);`

On 24 points - transitive group 24T558
Generators in S24
```(1 5 9)(3 11 7)(14 22 18)(16 20 24)
(2 10 6)(4 8 12)(13 21 17)(15 19 23)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 16)(2 15)(3 14)(4 13)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)```

`G:=sub<Sym(24)| (1,5,9)(3,11,7)(14,22,18)(16,20,24), (2,10,6)(4,8,12)(13,21,17)(15,19,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)>;`

`G:=Group( (1,5,9)(3,11,7)(14,22,18)(16,20,24), (2,10,6)(4,8,12)(13,21,17)(15,19,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17) );`

`G=PermutationGroup([[(1,5,9),(3,11,7),(14,22,18),(16,20,24)], [(2,10,6),(4,8,12),(13,21,17),(15,19,23)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,16),(2,15),(3,14),(4,13),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17)]])`

`G:=TransitiveGroup(24,558);`

On 27 points - transitive group 27T79
Generators in S27
```(1 4 10)(2 8 14)(3 12 6)(5 26 23)(7 25 16)(9 18 27)(11 17 20)(13 22 19)(15 21 24)
(1 7 13)(2 11 5)(3 15 9)(4 25 22)(6 24 27)(8 17 26)(10 16 19)(12 21 18)(14 20 23)
(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)
(1 3)(4 6)(7 15)(8 14)(9 13)(10 12)(16 21)(17 20)(18 19)(22 27)(23 26)(24 25)```

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

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

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

`G:=TransitiveGroup(27,79);`

C322D12 is a maximal subgroup of   C333SD16  F9⋊S3  S3×S3≀C2
C322D12 is a maximal quotient of   (C3×C6).8D12  (C3×C6).9D12  C322D24  C338SD16  C333Q16

Polynomial with Galois group C322D12 over ℚ
actionf(x)Disc(f)
12T118x12-2x9-2x6+3x3-1-315·136

Matrix representation of C322D12 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1
,
 7 10 0 0 0 0 3 10 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 12 0 0 0 0 0 0 0 12 0
,
 7 10 0 0 0 0 3 6 0 0 0 0 0 0 0 0 0 12 0 0 0 12 0 0 0 0 0 0 12 0 0 0 12 0 0 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1],[7,3,0,0,0,0,10,10,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0],[7,3,0,0,0,0,10,6,0,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0,0,0] >;`

C322D12 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_2D_{12}`
`% in TeX`

`G:=Group("C3^2:2D12");`
`// GroupNames label`

`G:=SmallGroup(216,159);`
`// by ID`

`G=gap.SmallGroup(216,159);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,3,-3,73,31,579,585,111,244,130,376,5189]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^12=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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