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G = C12.70D12order 288 = 25·32

1st non-split extension by C12 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.70D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C3×C4.Dic3 — C12.70D12
 Lower central C32 — C3×C6 — C62 — C12.70D12
 Upper central C1 — C2 — C2×C4

Generators and relations for C12.70D12
G = < a,b,c | a12=c2=1, b12=a6, bab-1=cac=a-1, cbc=a9b11 >

Subgroups: 706 in 119 conjugacy classes, 32 normal (10 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×8], C6 [×2], C6 [×5], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×16], C2×C6 [×2], C2×C6, M4(2) [×2], C2×D4, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], D12 [×8], C2×C12 [×2], C2×C12, C22×S3 [×6], C4.D4, C3×C12 [×2], C2×C3⋊S3 [×4], C62, C4.Dic3 [×2], C3×M4(2) [×2], C2×D12 [×3], C3×C3⋊C8 [×2], C12⋊S3 [×2], C6×C12, C22×C3⋊S3 [×2], C12.46D4 [×2], C3×C4.Dic3 [×2], C2×C12⋊S3, C12.70D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C4.D4, S32, D6⋊C4 [×2], C6.D6, C3⋊D12 [×2], C12.46D4 [×2], C6.D12, C12.70D12

Permutation representations of C12.70D12
On 24 points - transitive group 24T617
Generators in S24
```(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)
(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 20)(3 13)(4 18)(5 11)(6 16)(7 9)(8 14)(10 12)(17 23)(19 21)(22 24)```

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

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

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

`G:=TransitiveGroup(24,617);`

39 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 6A 6B 6C ··· 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 24A ··· 24H order 1 2 2 2 2 3 3 3 4 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 36 36 2 2 4 2 2 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 ··· 12

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C4 S3 D4 D6 D12 C3⋊D4 C4×S3 C4.D4 S32 C3⋊D12 C6.D6 C12.46D4 C12.70D12 kernel C12.70D12 C3×C4.Dic3 C2×C12⋊S3 C22×C3⋊S3 C4.Dic3 C3×C12 C2×C12 C12 C12 C2×C6 C32 C2×C4 C4 C22 C3 C1 # reps 1 2 1 4 2 2 2 4 4 4 1 1 2 1 4 4

Matrix representation of C12.70D12 in GL4(𝔽73) generated by

 14 7 0 0 66 7 0 0 0 0 14 7 0 0 66 7
,
 0 0 59 66 0 0 7 14 72 72 0 0 0 1 0 0
,
 72 72 0 0 0 1 0 0 0 0 0 72 0 0 72 0
`G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,14,66,0,0,7,7],[0,0,72,0,0,0,72,1,59,7,0,0,66,14,0,0],[72,0,0,0,72,1,0,0,0,0,0,72,0,0,72,0] >;`

C12.70D12 in GAP, Magma, Sage, TeX

`C_{12}._{70}D_{12}`
`% in TeX`

`G:=Group("C12.70D12");`
`// GroupNames label`

`G:=SmallGroup(288,207);`
`// by ID`

`G=gap.SmallGroup(288,207);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,219,100,675,346,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=c^2=1,b^12=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^9*b^11>;`
`// generators/relations`

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