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## G = D12⋊5D6order 288 = 25·32

### 5th semidirect product of D12 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12⋊5D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12⋊5D6
 Lower central C32 — C3×C6 — C3×C12 — D12⋊5D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D125D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 866 in 163 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×9], C6 [×2], C6 [×6], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3 [×4], C12 [×2], C12 [×2], D6 [×16], C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×4], D12, D12 [×4], C2×Dic3, C3⋊D4 [×5], C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×S3 [×4], C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3 [×3], C62, C8⋊S3 [×2], C24⋊C2, D24, D4⋊S3, D4⋊S3, D4.S3, Q82S3, C3×D8, C3×SD16, S3×D4 [×3], D42S3, Q83S3, C3×C3⋊C8 [×2], S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C22×C3⋊S3, D8⋊S3, Q83D6, C12.31D6, C3⋊D24, C325SD16, C3×D4⋊S3, C3×D4.S3, D12⋊S3, D4×C3⋊S3, D125D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8⋊C22, S32, S3×D4 [×2], C2×S32, D8⋊S3, Q83D6, Dic3⋊D6, D125D6

Character table of D125D6

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 4 12 18 36 2 2 4 2 12 18 2 2 4 8 8 8 8 24 12 12 4 4 8 24 12 12 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 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 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 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 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 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 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 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 1 1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 0 0 0 2 -1 -1 2 2 0 2 -1 -1 2 -1 -1 -1 0 2 0 2 -1 -1 -1 0 -1 -1 0 orthogonal lifted from S3 ρ10 2 2 2 -2 0 0 -1 2 -1 2 0 0 -1 2 -1 -1 -1 -1 2 1 0 -2 -1 2 -1 0 1 0 0 1 orthogonal lifted from D6 ρ11 2 2 -2 0 0 0 2 -1 -1 2 -2 0 2 -1 -1 -2 1 1 1 0 2 0 2 -1 -1 1 0 -1 -1 0 orthogonal lifted from D6 ρ12 2 2 -2 -2 0 0 -1 2 -1 2 0 0 -1 2 -1 1 1 1 -2 1 0 2 -1 2 -1 0 -1 0 0 -1 orthogonal lifted from D6 ρ13 2 2 0 0 -2 0 2 2 2 -2 0 2 2 2 2 0 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 0 2 -1 -1 2 2 0 2 -1 -1 -2 1 1 1 0 -2 0 2 -1 -1 -1 0 1 1 0 orthogonal lifted from D6 ρ15 2 2 2 2 0 0 -1 2 -1 2 0 0 -1 2 -1 -1 -1 -1 2 -1 0 2 -1 2 -1 0 -1 0 0 -1 orthogonal lifted from S3 ρ16 2 2 0 0 2 0 2 2 2 -2 0 -2 2 2 2 0 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 2 0 0 -1 2 -1 2 0 0 -1 2 -1 1 1 1 -2 -1 0 -2 -1 2 -1 0 1 0 0 1 orthogonal lifted from D6 ρ18 2 2 2 0 0 0 2 -1 -1 2 -2 0 2 -1 -1 2 -1 -1 -1 0 -2 0 2 -1 -1 1 0 1 1 0 orthogonal lifted from D6 ρ19 4 4 0 0 0 0 -2 -2 1 -4 0 0 -2 -2 1 0 3 -3 0 0 0 0 2 2 -1 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ20 4 4 0 0 0 0 -2 -2 1 -4 0 0 -2 -2 1 0 -3 3 0 0 0 0 2 2 -1 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ21 4 4 0 0 0 0 -2 4 -2 -4 0 0 -2 4 -2 0 0 0 0 0 0 0 2 -4 2 0 0 0 0 0 orthogonal lifted from S3×D4 ρ22 4 -4 0 0 0 0 4 4 4 0 0 0 -4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ23 4 4 0 0 0 0 4 -2 -2 -4 0 0 4 -2 -2 0 0 0 0 0 0 0 -4 2 2 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 0 0 0 -2 -2 1 4 0 0 -2 -2 1 2 -1 -1 2 0 0 0 -2 -2 1 0 0 0 0 0 orthogonal lifted from C2×S32 ρ25 4 4 4 0 0 0 -2 -2 1 4 0 0 -2 -2 1 -2 1 1 -2 0 0 0 -2 -2 1 0 0 0 0 0 orthogonal lifted from S32 ρ26 4 -4 0 0 0 0 4 -2 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√6 √6 0 orthogonal lifted from Q8⋊3D6 ρ27 4 -4 0 0 0 0 4 -2 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 √6 -√6 0 orthogonal lifted from Q8⋊3D6 ρ28 4 -4 0 0 0 0 -2 4 -2 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 -√-6 0 0 √-6 complex lifted from D8⋊S3 ρ29 4 -4 0 0 0 0 -2 4 -2 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 √-6 0 0 -√-6 complex lifted from D8⋊S3 ρ30 8 -8 0 0 0 0 -4 -4 2 0 0 0 4 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D125D6
On 24 points - transitive group 24T669
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 4 6 8 10 12)(3 11 7)(13 14 21 22 17 18)(15 16 23 24 19 20)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)```

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

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

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

`G:=TransitiveGroup(24,669);`

Matrix representation of D125D6 in GL8(ℤ)

 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 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 1 0 0 0 0 0 0 0 1 -1 1 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 1 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 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0
,
 -1 1 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 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0

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

D125D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_5D_6`
`% in TeX`

`G:=Group("D12:5D6");`
`// GroupNames label`

`G:=SmallGroup(288,585);`
`// by ID`

`G=gap.SmallGroup(288,585);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,675,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;`
`// generators/relations`

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