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## G = D12⋊16D6order 288 = 25·32

### 10th semidirect product of D12 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊16D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×D12 — D12⋊16D6
 Lower central C32 — C3×C6 — D12⋊16D6
 Upper central C1 — C2 — Q8

Generators and relations for D1216D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a4b, dcd=c-1 >

Subgroups: 1530 in 352 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×9], C3 [×2], C3, C4 [×3], C4 [×3], C22 [×15], S3 [×15], C6 [×2], C6 [×7], C2×C4 [×9], D4 [×18], Q8, Q8, C23 [×6], C32, Dic3 [×2], Dic3 [×3], C12 [×6], C12 [×5], D6 [×6], D6 [×21], C2×C6 [×6], C2×D4 [×9], C4○D4 [×6], C3×S3 [×6], C3⋊S3 [×3], C3×C6, Dic6 [×2], C4×S3 [×6], C4×S3 [×9], D12 [×6], D12 [×15], C3⋊D4 [×12], C2×C12 [×6], C3×D4 [×6], C3×Q8 [×2], C3×Q8, C22×S3 [×12], 2+ 1+4, C3×Dic3 [×2], C3⋊Dic3, C3×C12 [×3], S32 [×6], S3×C6 [×6], C2×C3⋊S3 [×3], C2×D12 [×6], C4○D12 [×6], S3×D4 [×12], Q83S3 [×2], Q83S3 [×3], C3×C4○D4 [×2], D6⋊S3 [×3], C3⋊D12 [×6], C322Q8, S3×C12 [×6], C3×D12 [×6], C4×C3⋊S3 [×3], C12⋊S3 [×3], Q8×C32, C2×S32 [×6], D4○D12 [×2], D6.D6 [×3], S3×D12 [×6], D6⋊D6 [×3], C3×Q83S3 [×2], C12.26D6, D1216D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2+ 1+4, S32, S3×C23 [×2], C2×S32 [×3], D4○D12 [×2], C22×S32, D1216D6

Smallest permutation representation of D1216D6
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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)
(1 24 5 20 9 16)(2 17 6 13 10 21)(3 22 7 18 11 14)(4 15 8 23 12 19)(25 47 29 43 33 39)(26 40 30 48 34 44)(27 45 31 41 35 37)(28 38 32 46 36 42)
(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 2+ 1+4 S32 C2×S32 D4○D12 D12⋊16D6 kernel D12⋊16D6 D6.D6 S3×D12 D6⋊D6 C3×Q8⋊3S3 C12.26D6 Q8⋊3S3 C4×S3 D12 C3×Q8 C32 Q8 C4 C3 C1 # reps 1 3 6 3 2 1 2 6 6 2 1 1 3 4 1

Matrix representation of D1216D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 3 0 0 0 0 10 6 0 0 0 0 0 0 10 10 0 0 0 0 3 7
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 7 0 0 0 0 10 10 0 0 0 0 0 0 3 7 0 0 0 0 10 10
,
 0 12 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12 0 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 7 3 0 0 0 0 10 6 0 0 0 0 0 0 6 10 0 0 0 0 3 7

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,10,0,0,0,0,3,6,0,0,0,0,0,0,10,3,0,0,0,0,10,7],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,10,0,0,0,0,7,10,0,0,0,0,0,0,3,10,0,0,0,0,7,10],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,10,0,0,0,0,3,6,0,0,0,0,0,0,6,3,0,0,0,0,10,7] >;`

D1216D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{16}D_6`
`% in TeX`

`G:=Group("D12:16D6");`
`// GroupNames label`

`G:=SmallGroup(288,968);`
`// by ID`

`G=gap.SmallGroup(288,968);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,100,675,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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