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## G = D12⋊S3order 144 = 24·32

### 3rd semidirect product of D12 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — D12⋊S3
 Lower central C32 — C3×C6 — D12⋊S3
 Upper central C1 — C2 — C4

Generators and relations for D12⋊S3
G = < a,b,c,d | a12=b2=c3=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 288 in 88 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×3], C2×C6 [×2], C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, Dic6, C4×S3 [×5], D12, D12 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, C3×Q8, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, D42S3, Q83S3, S3×Dic3 [×2], C3⋊D12 [×2], C3×Dic6, C3×D12, C4×C3⋊S3, D12⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, D42S3, Q83S3, C2×S32, D12⋊S3

Character table of D12⋊S3

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F size 1 1 6 6 18 2 2 4 2 6 6 9 9 2 2 4 12 12 4 4 4 4 12 12 ρ1 1 1 1 1 1 1 1 1 1 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 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 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 1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ6 1 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 linear of order 2 ρ7 1 1 1 -1 -1 1 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ8 1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ9 2 2 0 0 0 -1 2 -1 -2 -2 2 0 0 -1 2 -1 0 0 1 -2 1 1 -1 1 orthogonal lifted from D6 ρ10 2 2 0 0 0 -1 2 -1 2 -2 -2 0 0 -1 2 -1 0 0 -1 2 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 2 -1 -1 2 0 0 0 0 2 -1 -1 -1 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ12 2 2 -2 -2 0 2 -1 -1 2 0 0 0 0 2 -1 -1 1 1 2 -1 -1 -1 0 0 orthogonal lifted from D6 ρ13 2 2 -2 2 0 2 -1 -1 -2 0 0 0 0 2 -1 -1 1 -1 -2 1 1 1 0 0 orthogonal lifted from D6 ρ14 2 2 0 0 0 -1 2 -1 2 2 2 0 0 -1 2 -1 0 0 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 0 0 0 -1 2 -1 -2 2 -2 0 0 -1 2 -1 0 0 1 -2 1 1 1 -1 orthogonal lifted from D6 ρ16 2 2 2 -2 0 2 -1 -1 -2 0 0 0 0 2 -1 -1 -1 1 -2 1 1 1 0 0 orthogonal lifted from D6 ρ17 2 -2 0 0 0 2 2 2 0 0 0 -2i 2i -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 0 0 0 2 2 2 0 0 0 2i -2i -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 4 4 0 0 0 -2 -2 1 -4 0 0 0 0 -2 -2 1 0 0 2 2 -1 -1 0 0 orthogonal lifted from C2×S32 ρ20 4 -4 0 0 0 -2 4 -2 0 0 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ21 4 4 0 0 0 -2 -2 1 4 0 0 0 0 -2 -2 1 0 0 -2 -2 1 1 0 0 orthogonal lifted from S32 ρ22 4 -4 0 0 0 4 -2 -2 0 0 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ23 4 -4 0 0 0 -2 -2 1 0 0 0 0 0 2 2 -1 0 0 0 0 3i -3i 0 0 complex faithful ρ24 4 -4 0 0 0 -2 -2 1 0 0 0 0 0 2 2 -1 0 0 0 0 -3i 3i 0 0 complex faithful

Permutation representations of D12⋊S3
On 24 points - transitive group 24T227
Generators in S24
```(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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(2 6)(3 11)(5 9)(8 12)(13 19)(14 24)(15 17)(16 22)(18 20)(21 23)```

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

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

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

`G:=TransitiveGroup(24,227);`

Matrix representation of D12⋊S3 in GL4(𝔽5) generated by

 0 3 4 1 1 0 0 4 2 4 4 4 3 4 0 1
,
 4 4 1 2 0 1 3 1 4 2 4 1 3 4 0 1
,
 3 4 3 0 0 0 1 4 4 3 3 3 4 4 4 2
,
 1 0 2 0 0 0 4 1 0 0 4 0 0 1 4 0
`G:=sub<GL(4,GF(5))| [0,1,2,3,3,0,4,4,4,0,4,0,1,4,4,1],[4,0,4,3,4,1,2,4,1,3,4,0,2,1,1,1],[3,0,4,4,4,0,3,4,3,1,3,4,0,4,3,2],[1,0,0,0,0,0,0,1,2,4,4,4,0,1,0,0] >;`

D12⋊S3 in GAP, Magma, Sage, TeX

`D_{12}\rtimes S_3`
`% in TeX`

`G:=Group("D12:S3");`
`// GroupNames label`

`G:=SmallGroup(144,139);`
`// by ID`

`G=gap.SmallGroup(144,139);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,50,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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