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## G = C3⋊D12order 72 = 23·32

### The semidirect product of C3 and D12 acting via D12/D6=C2

Aliases: D62S3, C32D12, Dic3⋊S3, C6.4D6, C323D4, C2.4S32, (S3×C6)⋊2C2, C31(C3⋊D4), (C3×Dic3)⋊1C2, (C3×C6).4C22, (C2×C3⋊S3)⋊1C2, SmallGroup(72,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3⋊D12
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C3⋊D12
 Lower central C32 — C3×C6 — C3⋊D12
 Upper central C1 — C2

Generators and relations for C3⋊D12
G = < a,b,c | a3=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C3⋊D12

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 12A 12B size 1 1 6 18 2 2 4 6 2 2 4 6 6 6 6 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 2 0 2 -1 -1 0 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ6 2 2 -2 0 2 -1 -1 0 -1 2 -1 1 1 0 0 orthogonal lifted from D6 ρ7 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ8 2 2 0 0 -1 2 -1 2 2 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ9 2 2 0 0 -1 2 -1 -2 2 -1 -1 0 0 1 1 orthogonal lifted from D6 ρ10 2 -2 0 0 -1 2 -1 0 -2 1 1 0 0 √3 -√3 orthogonal lifted from D12 ρ11 2 -2 0 0 -1 2 -1 0 -2 1 1 0 0 -√3 √3 orthogonal lifted from D12 ρ12 2 -2 0 0 2 -1 -1 0 1 -2 1 -√-3 √-3 0 0 complex lifted from C3⋊D4 ρ13 2 -2 0 0 2 -1 -1 0 1 -2 1 √-3 -√-3 0 0 complex lifted from C3⋊D4 ρ14 4 4 0 0 -2 -2 1 0 -2 -2 1 0 0 0 0 orthogonal lifted from S32 ρ15 4 -4 0 0 -2 -2 1 0 2 2 -1 0 0 0 0 orthogonal faithful

Permutation representations of C3⋊D12
On 12 points - transitive group 12T38
Generators in S12
```(1 9 5)(2 6 10)(3 11 7)(4 8 12)
(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)| (1,9,5)(2,6,10)(3,11,7)(4,8,12), (1,2,3,4,5,6,7,8,9,10,11,12), (1,3)(4,12)(5,11)(6,10)(7,9)>;`

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

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

`G:=TransitiveGroup(12,38);`

On 24 points - transitive group 24T74
Generators in S24
```(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 21 17)(14 18 22)(15 23 19)(16 20 24)
(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 14)(2 13)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)```

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

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

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

`G:=TransitiveGroup(24,74);`

Polynomial with Galois group C3⋊D12 over ℚ
actionf(x)Disc(f)
12T38x12-36x10+486x8-3084x6+9585x4-13608x2+6534235·339·116·296

Matrix representation of C3⋊D12 in GL4(ℤ) generated by

 0 -1 0 0 1 -1 0 0 0 0 -1 1 0 0 -1 0
,
 0 0 0 -1 0 0 1 -1 0 1 0 0 -1 1 0 0
,
 -1 1 0 0 0 1 0 0 0 0 1 -1 0 0 0 -1
`G:=sub<GL(4,Integers())| [0,1,0,0,-1,-1,0,0,0,0,-1,-1,0,0,1,0],[0,0,0,-1,0,0,1,1,0,1,0,0,-1,-1,0,0],[-1,0,0,0,1,1,0,0,0,0,1,0,0,0,-1,-1] >;`

C3⋊D12 in GAP, Magma, Sage, TeX

`C_3\rtimes D_{12}`
`% in TeX`

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

`G:=SmallGroup(72,23);`
`// by ID`

`G=gap.SmallGroup(72,23);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-3,61,26,168,1204]);`
`// Polycyclic`

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

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