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## G = D12⋊13D6order 288 = 25·32

### 7th semidirect product of D12 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊13D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×C3⋊D4 — D12⋊13D6
 Lower central C32 — C3×C6 — D12⋊13D6
 Upper central C1 — C2 — D4

Generators and relations for D1213D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=a7, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >

Subgroups: 1506 in 359 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, 2+ 1+4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, Q83S3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3⋊D12, C322Q8, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C2×S32, S3×C2×C6, C22×C3⋊S3, D46D6, D4○D12, D12⋊S3, D6.D6, D6.6D6, S3×D12, D6.3D6, C2×C3⋊D12, S3×C3⋊D4, Dic3⋊D6, C3×S3×D4, C3×D42S3, D4×C3⋊S3, D1213D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D46D6, D4○D12, C22×S32, D1213D6

Permutation representations of D1213D6
On 24 points - transitive group 24T607
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(1 5 9)(2 12 10 8 6 4)(3 7 11)(13 15 17 19 21 23)(14 22 18)(16 24 20)
(1 9)(3 7)(4 12)(6 10)(13 23)(14 16)(15 21)(17 19)(18 24)(20 22)```

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

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

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

`G:=TransitiveGroup(24,607);`

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C ··· 6G 6H 6I 6J 6K 6L 6M 6N 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 2 2 6 6 6 6 18 18 18 2 2 4 2 6 6 6 6 18 2 2 4 ··· 4 6 6 8 8 12 12 12 4 4 6 6 8 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 D6 D6 2+ 1+4 S32 C2×S32 C2×S32 D4⋊6D6 D4○D12 D12⋊13D6 kernel D12⋊13D6 D12⋊S3 D6.D6 D6.6D6 S3×D12 D6.3D6 C2×C3⋊D12 S3×C3⋊D4 Dic3⋊D6 C3×S3×D4 C3×D4⋊2S3 D4×C3⋊S3 S3×D4 D4⋊2S3 Dic6 C4×S3 D12 C2×Dic3 C3⋊D4 C3×D4 C22×S3 C32 D4 C4 C22 C3 C3 C1 # reps 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 2 1 2 4 2 2 1 1 1 2 2 2 1

Matrix representation of D1213D6 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 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 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 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 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 -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

`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],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0],[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,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-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,-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] >;`

D1213D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{13}D_6`
`% in TeX`

`G:=Group("D12:13D6");`
`// GroupNames label`

`G:=SmallGroup(288,962);`
`// by ID`

`G=gap.SmallGroup(288,962);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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