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## G = D12⋊27D6order 288 = 25·32

### 3rd semidirect product of D12 and D6 acting through Inn(D12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊27D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×D12 — D12⋊27D6
 Lower central C32 — C3×C6 — D12⋊27D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D1227D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a6b, dbd=a4b, dcd=c-1 >

Subgroups: 1602 in 359 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×9], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×14], S3 [×16], C6 [×2], C6 [×9], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], C32, Dic3 [×4], C12 [×4], C12 [×6], D6 [×4], D6 [×28], C2×C6 [×2], C2×C6 [×5], C2×D4 [×9], C4○D4 [×6], C3×S3 [×4], C3⋊S3 [×4], C3×C6, C3×C6, Dic6 [×2], C4×S3 [×4], C4×S3 [×8], D12 [×2], D12 [×20], C3⋊D4 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×14], 2+ 1+4, C3×Dic3 [×4], C3×C12 [×2], S32 [×4], S3×C6 [×4], C2×C3⋊S3 [×4], C2×C3⋊S3 [×2], C62, C2×D12 [×7], C4○D12 [×2], C4○D12 [×4], S3×D4 [×12], Q83S3 [×4], C3×C4○D4 [×2], C6.D6 [×4], C3⋊D12 [×8], C3×Dic6 [×2], S3×C12 [×4], C3×D12 [×2], C3×C3⋊D4 [×4], C12⋊S3 [×4], C6×C12, C2×S32 [×4], C22×C3⋊S3 [×2], D4○D12 [×2], D6.6D6 [×4], S3×D12 [×4], Dic3⋊D6 [×4], C3×C4○D12 [×2], C2×C12⋊S3, D1227D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2+ 1+4, S32, S3×C23 [×2], C2×S32 [×3], D4○D12 [×2], C22×S32, D1227D6

Permutation representations of D1227D6
On 24 points - transitive group 24T611
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 4)(6 12)(7 11)(8 10)(13 15)(16 24)(17 23)(18 22)(19 21)```

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

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

`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),(2,4),(6,12),(7,11),(8,10),(13,15),(16,24),(17,23),(18,22),(19,21)])`

`G:=TransitiveGroup(24,611);`

45 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 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 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 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 2 6 6 6 6 18 18 18 18 2 2 4 2 2 6 6 6 6 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 12 12 12

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 D6 2+ 1+4 S32 C2×S32 C2×S32 D4○D12 D12⋊27D6 kernel D12⋊27D6 D6.6D6 S3×D12 Dic3⋊D6 C3×C4○D12 C2×C12⋊S3 C4○D12 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C32 C2×C4 C4 C22 C3 C1 # reps 1 4 4 4 2 1 2 2 4 2 4 2 1 1 2 1 4 4

Matrix representation of D1227D6 in GL4(𝔽13) generated by

 7 3 0 0 10 10 0 0 9 8 10 3 8 9 10 7
,
 4 9 12 11 4 9 11 12 9 8 10 3 8 9 10 3
,
 0 12 0 0 1 12 0 0 0 3 0 12 9 7 1 1
,
 10 10 0 0 7 3 0 0 3 6 7 10 5 4 3 6
`G:=sub<GL(4,GF(13))| [7,10,9,8,3,10,8,9,0,0,10,10,0,0,3,7],[4,4,9,8,9,9,8,9,12,11,10,10,11,12,3,3],[0,1,0,9,12,12,3,7,0,0,0,1,0,0,12,1],[10,7,3,5,10,3,6,4,0,0,7,3,0,0,10,6] >;`

D1227D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{27}D_6`
`% in TeX`

`G:=Group("D12:27D6");`
`// GroupNames label`

`G:=SmallGroup(288,956);`
`// by ID`

`G=gap.SmallGroup(288,956);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,100,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^6*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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