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## G = C12.D6order 144 = 24·32

### 24th non-split extension by C12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.D6
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C12.D6
 Lower central C32 — C3×C6 — C12.D6
 Upper central C1 — C2 — D4

Generators and relations for C12.D6
G = < a,b,c | a12=b6=1, c2=a6, bab-1=a7, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 346 in 120 conjugacy classes, 47 normal (13 characteristic)
C1, C2, C2 [×3], C3 [×4], C4, C4 [×3], C22 [×2], C22, S3 [×4], C6 [×4], C6 [×8], C2×C4 [×3], D4, D4 [×2], Q8, C32, Dic3 [×12], C12 [×4], D6 [×4], C2×C6 [×8], C4○D4, C3⋊S3, C3×C6, C3×C6 [×2], Dic6 [×4], C4×S3 [×4], C2×Dic3 [×8], C3⋊D4 [×8], C3×D4 [×4], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3, C62 [×2], D42S3 [×4], C324Q8, C4×C3⋊S3, C2×C3⋊Dic3 [×2], C327D4 [×2], D4×C32, C12.D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C4○D4, C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], D42S3 [×4], C22×C3⋊S3, C12.D6

Character table of C12.D6

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 12A 12B 12C 12D size 1 1 2 2 18 2 2 2 2 2 9 9 18 18 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ6 1 1 1 -1 -1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 2 2 -2 -2 0 -1 -1 2 -1 2 0 0 0 0 2 -1 -1 -1 1 1 1 1 -2 1 -2 1 2 -1 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 -2 0 -1 2 -1 -1 -2 0 0 0 0 -1 -1 -1 2 -1 2 1 -2 1 1 -1 -1 1 1 1 -2 orthogonal lifted from D6 ρ11 2 2 -2 2 0 -1 2 -1 -1 -2 0 0 0 0 -1 -1 -1 2 1 -2 -1 2 -1 -1 1 1 1 1 1 -2 orthogonal lifted from D6 ρ12 2 2 2 2 0 -1 -1 -1 2 2 0 0 0 0 -1 -1 2 -1 -1 -1 -1 -1 -1 2 -1 2 -1 2 -1 -1 orthogonal lifted from S3 ρ13 2 2 -2 2 0 2 -1 -1 -1 -2 0 0 0 0 -1 2 -1 -1 -2 1 2 -1 -1 -1 1 1 1 1 -2 1 orthogonal lifted from D6 ρ14 2 2 -2 -2 0 -1 -1 -1 2 2 0 0 0 0 -1 -1 2 -1 1 1 1 1 1 -2 1 -2 -1 2 -1 -1 orthogonal lifted from D6 ρ15 2 2 2 -2 0 2 -1 -1 -1 -2 0 0 0 0 -1 2 -1 -1 2 -1 -2 1 1 1 -1 -1 1 1 -2 1 orthogonal lifted from D6 ρ16 2 2 -2 2 0 -1 -1 2 -1 -2 0 0 0 0 2 -1 -1 -1 1 1 -1 -1 2 -1 -2 1 -2 1 1 1 orthogonal lifted from D6 ρ17 2 2 2 2 0 -1 -1 2 -1 2 0 0 0 0 2 -1 -1 -1 -1 -1 -1 -1 2 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ18 2 2 -2 -2 0 2 -1 -1 -1 2 0 0 0 0 -1 2 -1 -1 -2 1 -2 1 1 1 1 1 -1 -1 2 -1 orthogonal lifted from D6 ρ19 2 2 -2 2 0 -1 -1 -1 2 -2 0 0 0 0 -1 -1 2 -1 1 1 -1 -1 -1 2 1 -2 1 -2 1 1 orthogonal lifted from D6 ρ20 2 2 2 2 0 2 -1 -1 -1 2 0 0 0 0 -1 2 -1 -1 2 -1 2 -1 -1 -1 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ21 2 2 2 -2 0 -1 -1 -1 2 -2 0 0 0 0 -1 -1 2 -1 -1 -1 1 1 1 -2 -1 2 1 -2 1 1 orthogonal lifted from D6 ρ22 2 2 2 2 0 -1 2 -1 -1 2 0 0 0 0 -1 -1 -1 2 -1 2 -1 2 -1 -1 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ23 2 2 2 -2 0 -1 -1 2 -1 -2 0 0 0 0 2 -1 -1 -1 -1 -1 1 1 -2 1 2 -1 -2 1 1 1 orthogonal lifted from D6 ρ24 2 2 -2 -2 0 -1 2 -1 -1 2 0 0 0 0 -1 -1 -1 2 1 -2 1 -2 1 1 1 1 -1 -1 -1 2 orthogonal lifted from D6 ρ25 2 -2 0 0 0 2 2 2 2 0 -2i 2i 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 0 0 0 2 2 2 2 0 2i -2i 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 4 -4 0 0 0 -2 -2 4 -2 0 0 0 0 0 -4 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ28 4 -4 0 0 0 4 -2 -2 -2 0 0 0 0 0 2 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ29 4 -4 0 0 0 -2 4 -2 -2 0 0 0 0 0 2 2 2 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ30 4 -4 0 0 0 -2 -2 -2 4 0 0 0 0 0 2 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2

Smallest permutation representation of C12.D6
On 72 points
Generators in S72
```(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 29 54 66 23 40)(2 36 55 61 24 47)(3 31 56 68 13 42)(4 26 57 63 14 37)(5 33 58 70 15 44)(6 28 59 65 16 39)(7 35 60 72 17 46)(8 30 49 67 18 41)(9 25 50 62 19 48)(10 32 51 69 20 43)(11 27 52 64 21 38)(12 34 53 71 22 45)
(1 40 7 46)(2 39 8 45)(3 38 9 44)(4 37 10 43)(5 48 11 42)(6 47 12 41)(13 27 19 33)(14 26 20 32)(15 25 21 31)(16 36 22 30)(17 35 23 29)(18 34 24 28)(49 71 55 65)(50 70 56 64)(51 69 57 63)(52 68 58 62)(53 67 59 61)(54 66 60 72)```

`G:=sub<Sym(72)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,29,54,66,23,40)(2,36,55,61,24,47)(3,31,56,68,13,42)(4,26,57,63,14,37)(5,33,58,70,15,44)(6,28,59,65,16,39)(7,35,60,72,17,46)(8,30,49,67,18,41)(9,25,50,62,19,48)(10,32,51,69,20,43)(11,27,52,64,21,38)(12,34,53,71,22,45), (1,40,7,46)(2,39,8,45)(3,38,9,44)(4,37,10,43)(5,48,11,42)(6,47,12,41)(13,27,19,33)(14,26,20,32)(15,25,21,31)(16,36,22,30)(17,35,23,29)(18,34,24,28)(49,71,55,65)(50,70,56,64)(51,69,57,63)(52,68,58,62)(53,67,59,61)(54,66,60,72)>;`

`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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,29,54,66,23,40)(2,36,55,61,24,47)(3,31,56,68,13,42)(4,26,57,63,14,37)(5,33,58,70,15,44)(6,28,59,65,16,39)(7,35,60,72,17,46)(8,30,49,67,18,41)(9,25,50,62,19,48)(10,32,51,69,20,43)(11,27,52,64,21,38)(12,34,53,71,22,45), (1,40,7,46)(2,39,8,45)(3,38,9,44)(4,37,10,43)(5,48,11,42)(6,47,12,41)(13,27,19,33)(14,26,20,32)(15,25,21,31)(16,36,22,30)(17,35,23,29)(18,34,24,28)(49,71,55,65)(50,70,56,64)(51,69,57,63)(52,68,58,62)(53,67,59,61)(54,66,60,72) );`

`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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,29,54,66,23,40),(2,36,55,61,24,47),(3,31,56,68,13,42),(4,26,57,63,14,37),(5,33,58,70,15,44),(6,28,59,65,16,39),(7,35,60,72,17,46),(8,30,49,67,18,41),(9,25,50,62,19,48),(10,32,51,69,20,43),(11,27,52,64,21,38),(12,34,53,71,22,45)], [(1,40,7,46),(2,39,8,45),(3,38,9,44),(4,37,10,43),(5,48,11,42),(6,47,12,41),(13,27,19,33),(14,26,20,32),(15,25,21,31),(16,36,22,30),(17,35,23,29),(18,34,24,28),(49,71,55,65),(50,70,56,64),(51,69,57,63),(52,68,58,62),(53,67,59,61),(54,66,60,72)])`

Matrix representation of C12.D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;`

C12.D6 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(144,173);`
`// by ID`

`G=gap.SmallGroup(144,173);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,964,3461]);`
`// Polycyclic`

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

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