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## G = D12⋊15D6order 288 = 25·32

### 9th semidirect product of D12 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊15D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — D12⋊15D6
 Lower central C32 — C3×C6 — D12⋊15D6
 Upper central C1 — C2 — Q8

Generators and relations for D1215D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=a5, ad=da, cbc-1=a10b, dbd=a6b, dcd=c-1 >

Subgroups: 1250 in 347 conjugacy classes, 110 normal (10 characteristic)
C1, C2, C2 [×8], C3 [×2], C3, C4 [×3], C4 [×5], C22 [×13], S3 [×12], C6 [×2], C6 [×7], C2×C4 [×16], D4 [×12], Q8, Q8 [×3], C23 [×3], C32, Dic3 [×2], Dic3 [×9], C12 [×6], C12 [×5], D6 [×6], D6 [×15], C2×C6 [×6], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×6], C3⋊S3 [×2], C3×C6, Dic6 [×9], C4×S3 [×6], C4×S3 [×17], D12 [×6], C2×Dic3 [×6], C3⋊D4 [×12], C2×C12 [×6], C3×D4 [×6], C3×Q8 [×2], C3×Q8, C22×S3 [×6], C2×C4○D4, C3×Dic3 [×2], C3⋊Dic3 [×3], C3×C12 [×3], S32 [×6], S3×C6 [×6], C2×C3⋊S3, S3×C2×C4 [×6], C4○D12 [×6], S3×D4 [×6], D42S3 [×6], S3×Q8 [×3], Q83S3 [×2], C3×C4○D4 [×2], S3×Dic3 [×6], C6.D6, D6⋊S3 [×6], S3×C12 [×6], C3×D12 [×6], C324Q8 [×3], C4×C3⋊S3 [×3], Q8×C32, C2×S32 [×3], S3×C4○D4 [×2], D125S3 [×6], C4×S32 [×3], D6⋊D6 [×3], C3×Q83S3 [×2], Q8×C3⋊S3, D1215D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, S3×C23 [×2], C2×S32 [×3], S3×C4○D4 [×2], C22×S32, D1215D6

Smallest permutation representation of D1215D6
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 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 48)(21 47)(22 46)(23 45)(24 44)
(1 38 5 46 9 42)(2 43 6 39 10 47)(3 48 7 44 11 40)(4 41 8 37 12 45)(13 36 17 32 21 28)(14 29 18 25 22 33)(15 34 19 30 23 26)(16 27 20 35 24 31)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)```

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

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

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

45 conjugacy classes

 class 1 2A 2B ··· 2G 2H 2I 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D ··· 6I 12A ··· 12F 12G 12H 12I 12J 12K 12L 12M order 1 2 2 ··· 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 ··· 6 12 ··· 12 12 12 12 12 12 12 12 size 1 1 6 ··· 6 9 9 2 2 4 2 2 2 3 3 3 3 18 18 18 2 2 4 12 ··· 12 4 ··· 4 6 6 6 6 8 8 8

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 C4○D4 S32 C2×S32 S3×C4○D4 D12⋊15D6 kernel D12⋊15D6 D12⋊5S3 C4×S32 D6⋊D6 C3×Q8⋊3S3 Q8×C3⋊S3 Q8⋊3S3 C4×S3 D12 C3×Q8 C3⋊S3 Q8 C4 C3 C1 # reps 1 6 3 3 2 1 2 6 6 2 4 1 3 4 1

Matrix representation of D1215D6 in GL6(𝔽13)

 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 11 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 1 0 0 0 0 0 0 1 2 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 3 0 0 0 0 5 5 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 10 0 0 0 0 8 8 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,5,0,0,0,0,3,5,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,8,0,0,0,0,10,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D1215D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{15}D_6`
`% in TeX`

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

`G:=SmallGroup(288,967);`
`// by ID`

`G=gap.SmallGroup(288,967);`
`# by ID`

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

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=a^10*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

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