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## G = D12.15D6order 288 = 25·32

### 15th non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.15D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12.15D6
 Lower central C32 — C3×C6 — C3×C12 — D12.15D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.15D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a9c5 >

Subgroups: 578 in 139 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×5], C6 [×2], C6 [×2], C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], C32, Dic3 [×8], C12 [×2], C12 [×6], D6 [×4], C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×7], C4×S3 [×8], D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8 [×2], C3×Q8 [×2], C8.C22, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3 [×2], C24⋊C2, Dic12, D4.S3, Q82S3, C3⋊Q16, C3⋊Q16, C3×SD16, C3×Q16, D42S3, S3×Q8 [×3], Q83S3, C3×C3⋊C8 [×2], S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, D4.D6, Q16⋊S3, C12.31D6, D12.S3, C323Q16, C3×Q82S3, C3×C3⋊Q16, D12⋊S3, Q8×C3⋊S3, D12.15D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8.C22, S32, S3×D4 [×2], C2×S32, D4.D6, Q16⋊S3, Dic3⋊D6, D12.15D6

Character table of D12.15D6

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B 24C 24D size 1 1 12 18 2 2 4 2 4 12 18 36 2 2 4 24 12 12 4 4 8 8 8 8 8 24 12 12 12 12 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 0 2 -1 -1 2 2 0 0 0 -1 2 -1 1 0 -2 2 -1 -1 2 -1 -1 -1 0 1 1 0 0 orthogonal lifted from D6 ρ10 2 2 0 0 -1 2 -1 2 2 2 0 0 2 -1 -1 0 2 0 -1 2 -1 -1 2 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 2 0 0 -1 2 -1 2 2 -2 0 0 2 -1 -1 0 -2 0 -1 2 -1 -1 2 -1 -1 1 0 0 1 1 orthogonal lifted from D6 ρ12 2 2 0 2 2 2 2 -2 0 0 -2 0 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 2 -1 -1 2 -2 0 0 0 -1 2 -1 1 0 2 2 -1 -1 -2 1 1 1 0 -1 -1 0 0 orthogonal lifted from D6 ρ14 2 2 0 -2 2 2 2 -2 0 0 2 0 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 2 -1 -1 2 -2 0 0 0 -1 2 -1 -1 0 -2 2 -1 -1 -2 1 1 1 0 1 1 0 0 orthogonal lifted from D6 ρ16 2 2 0 0 -1 2 -1 2 -2 2 0 0 2 -1 -1 0 -2 0 -1 2 -1 1 -2 1 1 -1 0 0 1 1 orthogonal lifted from D6 ρ17 2 2 0 0 -1 2 -1 2 -2 -2 0 0 2 -1 -1 0 2 0 -1 2 -1 1 -2 1 1 1 0 0 -1 -1 orthogonal lifted from D6 ρ18 2 2 2 0 2 -1 -1 2 2 0 0 0 -1 2 -1 -1 0 2 2 -1 -1 2 -1 -1 -1 0 -1 -1 0 0 orthogonal lifted from S3 ρ19 4 4 0 0 -2 -2 1 -4 0 0 0 0 -2 -2 1 0 0 0 2 2 -1 0 0 -3 3 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ20 4 4 0 0 -2 -2 1 4 -4 0 0 0 -2 -2 1 0 0 0 -2 -2 1 2 2 -1 -1 0 0 0 0 0 orthogonal lifted from C2×S32 ρ21 4 4 0 0 -2 -2 1 -4 0 0 0 0 -2 -2 1 0 0 0 2 2 -1 0 0 3 -3 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ22 4 4 0 0 -2 4 -2 -4 0 0 0 0 4 -2 -2 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ23 4 4 0 0 -2 -2 1 4 4 0 0 0 -2 -2 1 0 0 0 -2 -2 1 -2 -2 1 1 0 0 0 0 0 orthogonal lifted from S32 ρ24 4 4 0 0 4 -2 -2 -4 0 0 0 0 -2 4 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 4 4 4 0 0 0 0 0 -4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ26 4 -4 0 0 4 -2 -2 0 0 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 √6 -√6 0 0 symplectic lifted from D4.D6, Schur index 2 ρ27 4 -4 0 0 4 -2 -2 0 0 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 -√6 √6 0 0 symplectic lifted from D4.D6, Schur index 2 ρ28 4 -4 0 0 -2 4 -2 0 0 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -√-6 √-6 complex lifted from Q16⋊S3 ρ29 4 -4 0 0 -2 4 -2 0 0 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 √-6 -√-6 complex lifted from Q16⋊S3 ρ30 8 -8 0 0 -4 -4 2 0 0 0 0 0 4 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D12.15D6
On 48 points
Generators in S48
```(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)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 39)(14 38)(15 37)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)
(1 23 3 13 5 15 7 17 9 19 11 21)(2 18 4 20 6 22 8 24 10 14 12 16)(25 39 35 37 33 47 31 45 29 43 27 41)(26 46 36 44 34 42 32 40 30 38 28 48)
(1 44 10 41 7 38 4 47)(2 45 11 42 8 39 5 48)(3 46 12 43 9 40 6 37)(13 29 22 26 19 35 16 32)(14 30 23 27 20 36 17 33)(15 31 24 28 21 25 18 34)```

`G:=sub<Sym(48)| (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,39)(14,38)(15,37)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,23,3,13,5,15,7,17,9,19,11,21)(2,18,4,20,6,22,8,24,10,14,12,16)(25,39,35,37,33,47,31,45,29,43,27,41)(26,46,36,44,34,42,32,40,30,38,28,48), (1,44,10,41,7,38,4,47)(2,45,11,42,8,39,5,48)(3,46,12,43,9,40,6,37)(13,29,22,26,19,35,16,32)(14,30,23,27,20,36,17,33)(15,31,24,28,21,25,18,34)>;`

`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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,39)(14,38)(15,37)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,23,3,13,5,15,7,17,9,19,11,21)(2,18,4,20,6,22,8,24,10,14,12,16)(25,39,35,37,33,47,31,45,29,43,27,41)(26,46,36,44,34,42,32,40,30,38,28,48), (1,44,10,41,7,38,4,47)(2,45,11,42,8,39,5,48)(3,46,12,43,9,40,6,37)(13,29,22,26,19,35,16,32)(14,30,23,27,20,36,17,33)(15,31,24,28,21,25,18,34) );`

`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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,39),(14,38),(15,37),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40)], [(1,23,3,13,5,15,7,17,9,19,11,21),(2,18,4,20,6,22,8,24,10,14,12,16),(25,39,35,37,33,47,31,45,29,43,27,41),(26,46,36,44,34,42,32,40,30,38,28,48)], [(1,44,10,41,7,38,4,47),(2,45,11,42,8,39,5,48),(3,46,12,43,9,40,6,37),(13,29,22,26,19,35,16,32),(14,30,23,27,20,36,17,33),(15,31,24,28,21,25,18,34)])`

Matrix representation of D12.15D6 in GL8(𝔽73)

 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 50 0 72 72 0 0 0 0 0 23 1 0 0 0 0 0 0 0 0 0 0 1 0 71 0 0 0 0 72 72 2 2 0 0 0 0 0 1 0 72 0 0 0 0 72 72 1 1
,
 23 23 1 2 0 0 0 0 23 23 2 1 0 0 0 0 12 13 50 50 0 0 0 0 13 12 50 50 0 0 0 0 0 0 0 0 0 0 63 68 0 0 0 0 0 0 5 10 0 0 0 0 68 34 0 0 0 0 0 0 39 5 0 0
,
 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 23 0 1 0 0 0 0 50 0 72 72 0 0 0 0 0 0 0 0 46 0 54 0 0 0 0 0 0 46 0 54 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 27
,
 50 50 72 71 0 0 0 0 0 0 1 72 0 0 0 0 30 31 0 23 0 0 0 0 30 30 0 23 0 0 0 0 0 0 0 0 62 51 11 22 0 0 0 0 22 11 51 62 0 0 0 0 31 62 0 0 0 0 0 0 11 42 0 0

`G:=sub<GL(8,GF(73))| [0,1,50,0,0,0,0,0,72,72,0,23,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,72,0,0,0,0,1,72,1,72,0,0,0,0,0,2,0,1,0,0,0,0,71,2,72,1],[23,23,12,13,0,0,0,0,23,23,13,12,0,0,0,0,1,2,50,50,0,0,0,0,2,1,50,50,0,0,0,0,0,0,0,0,0,0,68,39,0,0,0,0,0,0,34,5,0,0,0,0,63,5,0,0,0,0,0,0,68,10,0,0],[0,1,0,50,0,0,0,0,72,72,23,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,54,0,27,0,0,0,0,0,0,54,0,27],[50,0,30,30,0,0,0,0,50,0,31,30,0,0,0,0,72,1,0,0,0,0,0,0,71,72,23,23,0,0,0,0,0,0,0,0,62,22,31,11,0,0,0,0,51,11,62,42,0,0,0,0,11,51,0,0,0,0,0,0,22,62,0,0] >;`

D12.15D6 in GAP, Magma, Sage, TeX

`D_{12}._{15}D_6`
`% in TeX`

`G:=Group("D12.15D6");`
`// GroupNames label`

`G:=SmallGroup(288,599);`
`// by ID`

`G=gap.SmallGroup(288,599);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,100,675,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^5>;`
`// generators/relations`

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