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## G = D12.8D6order 288 = 25·32

### 8th non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.8D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12.8D6
 Lower central C32 — C3×C6 — C3×C12 — D12.8D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D12.8D6
G = < a,b,c,d | a12=b2=c6=1, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c-1 >

Subgroups: 602 in 144 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×4], C6 [×2], C6 [×6], C8 [×2], C2×C4 [×3], D4, D4 [×3], Q8 [×2], C32, Dic3 [×8], C12 [×2], C12 [×2], D6 [×4], C2×C6 [×5], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×3], C4×S3 [×4], D12, D12, C2×Dic3 [×5], C3⋊D4 [×5], C3×D4 [×2], C3×D4 [×2], C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, S3×C8 [×2], C24⋊C2, Dic12, D4⋊S3, D4.S3, D4.S3, C3⋊Q16, C3×D8, C3×SD16, D42S3 [×4], Q83S3, C3×C3⋊C8 [×2], S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, D83S3, Q8.7D6, C12.29D6, D12.S3, C323Q16, C3×D4⋊S3, C3×D4.S3, D12⋊S3, C12.D6, D12.8D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C4○D8, S32, S3×D4 [×2], C2×S32, D83S3, Q8.7D6, Dic3⋊D6, D12.8D6

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

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 4 12 18 2 2 4 2 9 9 12 36 2 2 4 8 8 8 8 24 6 6 6 6 4 4 8 24 12 12 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 C4○D8 S32 S3×D4 C2×S32 D8⋊3S3 Q8.7D6 Dic3⋊D6 D12.8D6 kernel D12.8D6 C12.29D6 D12.S3 C32⋊3Q16 C3×D4⋊S3 C3×D4.S3 D12⋊S3 C12.D6 D4⋊S3 D4.S3 C3⋊Dic3 C2×C3⋊S3 C3⋊C8 Dic6 D12 C3×D4 C32 D4 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 4 1 2 1 2 2 2 1

Matrix representation of D12.8D6 in GL6(𝔽73)

 27 0 0 0 0 0 36 46 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 15 14 0 0 0 0 57 58 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 4 67 0 0 0 0 39 69 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 63 0 0 0 0 0 3 22 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 72

`G:=sub<GL(6,GF(73))| [27,36,0,0,0,0,0,46,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,57,0,0,0,0,14,58,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[4,39,0,0,0,0,67,69,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[63,3,0,0,0,0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;`

D12.8D6 in GAP, Magma, Sage, TeX

`D_{12}._8D_6`
`% in TeX`

`G:=Group("D12.8D6");`
`// GroupNames label`

`G:=SmallGroup(288,584);`
`// by ID`

`G=gap.SmallGroup(288,584);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,675,346,185,80,1356,9414]);`
`// Polycyclic`

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

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