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G = C12.29D6order 144 = 24·32

3rd non-split extension by C12 of D6 acting via D6/S3=C2

Aliases: C12.29D6, C3⋊C86S3, C3⋊S32C8, C31(S3×C8), C4.14S32, C6.1(C4×S3), C324(C2×C8), C3⋊Dic3.2C4, (C3×C12).28C22, C2.1(C6.D6), (C3×C3⋊C8)⋊6C2, (C2×C3⋊S3).2C4, (C4×C3⋊S3).3C2, (C3×C6).9(C2×C4), SmallGroup(144,53)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C12.29D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C3⋊C8 — C12.29D6
 Lower central C32 — C12.29D6
 Upper central C1 — C4

Generators and relations for C12.29D6
G = < a,b,c | a12=1, b6=a3, c2=a6, bab-1=cac-1=a5, cbc-1=a6b5 >

Permutation representations of C12.29D6
On 24 points - transitive group 24T237
Generators in S24
```(1 11 21 7 17 3 13 23 9 19 5 15)(2 4 6 8 10 12 14 16 18 20 22 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 13 3)(2 8 14 20)(4 18 16 6)(5 11 17 23)(7 21 19 9)(10 24 22 12)```

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

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

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

`G:=TransitiveGroup(24,237);`

C12.29D6 is a maximal subgroup of
S32⋊C8  C3⋊S3.2D8  C3⋊S3.2Q16  C32⋊C4⋊C8  S32×C8  C24⋊D6  C24.64D6  C3⋊C8.22D6  C3⋊C820D6  D12⋊D6  Dic6⋊D6  D12.8D6  D12.9D6  Dic6.9D6  D12.14D6  C36.38D6  C32⋊C6⋊C8  C12.69S32  C12.93S32
C12.29D6 is a maximal quotient of
C24.60D6  C24.62D6  C6.(S3×C8)  C12.78D12  C12.15Dic6  C36.38D6  C12.89S32  C12.69S32  C12.93S32

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 8A ··· 8H 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 8 ··· 8 12 12 12 12 12 12 24 ··· 24 size 1 1 9 9 2 2 4 1 1 9 9 2 2 4 3 ··· 3 2 2 2 2 4 4 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 C8 S3 D6 C4×S3 S3×C8 S32 C6.D6 C12.29D6 kernel C12.29D6 C3×C3⋊C8 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3⋊S3 C3⋊C8 C12 C6 C3 C4 C2 C1 # reps 1 2 1 2 2 8 2 2 4 8 1 1 2

Matrix representation of C12.29D6 in GL4(𝔽5) generated by

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

C12.29D6 in GAP, Magma, Sage, TeX

`C_{12}._{29}D_6`
`% in TeX`

`G:=Group("C12.29D6");`
`// GroupNames label`

`G:=SmallGroup(144,53);`
`// by ID`

`G=gap.SmallGroup(144,53);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,31,50,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^6=a^3,c^2=a^6,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^6*b^5>;`
`// generators/relations`

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