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## G = D12⋊23D6order 288 = 25·32

### 7th semidirect product of D12 and D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊23D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — D12⋊23D6
 Lower central C32 — C3×C6 — D12⋊23D6
 Upper central C1 — C4 — C2×C4

Generators and relations for D1223D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >

Subgroups: 1298 in 355 conjugacy classes, 110 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C2×C4○D4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C4○D12, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C3×C4○D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C2×S32, C22×C3⋊S3, S3×C4○D4, D12⋊S3, Dic3.D6, D6.D6, C4×S32, D6⋊D6, D6.4D6, Dic3⋊D6, C3×C4○D12, C2×C4×C3⋊S3, D1223D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, S3×C23, C2×S32, S3×C4○D4, C22×S32, D1223D6

Permutation representations of D1223D6
On 24 points - transitive group 24T610
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)
(1 5)(2 10)(4 8)(7 11)(13 15)(14 20)(16 18)(17 23)(19 21)(22 24)```

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

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

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

`G:=TransitiveGroup(24,610);`

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C ··· 6G 6H 6I 6J 6K 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N order 1 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 2 6 6 6 6 9 9 18 2 2 4 1 1 2 6 6 6 6 9 9 18 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 D6 C4○D4 S32 C2×S32 C2×S32 S3×C4○D4 D12⋊23D6 kernel D12⋊23D6 D12⋊S3 Dic3.D6 D6.D6 C4×S32 D6⋊D6 D6.4D6 Dic3⋊D6 C3×C4○D12 C2×C4×C3⋊S3 C4○D12 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C3⋊S3 C2×C4 C4 C22 C3 C1 # reps 1 2 1 2 2 1 2 2 2 1 2 2 4 2 4 2 4 1 2 1 4 4

Matrix representation of D1223D6 in GL4(𝔽5) generated by

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

D1223D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{23}D_6`
`% in TeX`

`G:=Group("D12:23D6");`
`// GroupNames label`

`G:=SmallGroup(288,954);`
`// by ID`

`G=gap.SmallGroup(288,954);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,346,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,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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