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## G = D6⋊5D12order 288 = 25·32

### 2nd semidirect product of D6 and D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D6⋊5D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C22×S32 — D6⋊5D12
 Lower central C32 — C62 — D6⋊5D12
 Upper central C1 — C22 — C2×C4

Generators and relations for D65D12
G = < a,b,c,d | a6=b2=c12=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 1538 in 287 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×23], S3 [×14], C6 [×6], C6 [×7], C2×C4, C2×C4 [×2], D4 [×6], C23 [×10], C32, Dic3 [×2], C12 [×6], D6 [×4], D6 [×42], C2×C6 [×2], C2×C6 [×9], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3 [×4], C3⋊S3 [×3], C3×C6, C3×C6 [×2], D12 [×12], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C22×S3 [×2], C22×S3 [×18], C22×C6 [×2], C22≀C2, C3×Dic3 [×2], C3×C12, S32 [×8], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×5], C62, D6⋊C4 [×2], D6⋊C4 [×2], C3×C22⋊C4 [×2], C2×D12 [×5], C2×C3⋊D4 [×2], S3×C23 [×2], C3⋊D12 [×4], C6×Dic3 [×2], C12⋊S3 [×2], C6×C12, C2×S32 [×6], S3×C2×C6 [×2], C22×C3⋊S3 [×2], D6⋊D4 [×2], C6.D12, C3×D6⋊C4 [×2], C2×C3⋊D12 [×2], C2×C12⋊S3, C22×S32, D65D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], D12 [×4], C22×S3 [×2], C22≀C2, S32, C2×D12 [×2], S3×D4 [×4], C2×S32, D6⋊D4 [×2], S3×D12 [×2], Dic3⋊D6, D65D12

Smallest permutation representation of D65D12
On 48 points
Generators in S48
```(1 17 9 13 5 21)(2 18 10 14 6 22)(3 19 11 15 7 23)(4 20 12 16 8 24)(25 40 29 44 33 48)(26 41 30 45 34 37)(27 42 31 46 35 38)(28 43 32 47 36 39)
(1 41)(2 35)(3 43)(4 25)(5 45)(6 27)(7 47)(8 29)(9 37)(10 31)(11 39)(12 33)(13 34)(14 42)(15 36)(16 44)(17 26)(18 46)(19 28)(20 48)(21 30)(22 38)(23 32)(24 40)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 48)(34 47)(35 46)(36 45)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3A 3B 3C 4A 4B 4C 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A ··· 12H 12I 12J 12K 12L order 1 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 6 6 18 18 36 2 2 4 4 12 12 2 ··· 2 4 4 4 12 12 12 12 4 ··· 4 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D12 S32 S3×D4 C2×S32 S3×D12 Dic3⋊D6 kernel D6⋊5D12 C6.D12 C3×D6⋊C4 C2×C3⋊D12 C2×C12⋊S3 C22×S32 D6⋊C4 S3×C6 C2×C3⋊S3 C2×Dic3 C2×C12 C22×S3 D6 C2×C4 C6 C22 C2 C2 # reps 1 1 2 2 1 1 2 4 2 2 2 2 8 1 4 1 4 2

Matrix representation of D65D12 in GL8(ℤ)

 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 -2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 -2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1

`G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1] >;`

D65D12 in GAP, Magma, Sage, TeX

`D_6\rtimes_5D_{12}`
`% in TeX`

`G:=Group("D6:5D12");`
`// GroupNames label`

`G:=SmallGroup(288,571);`
`// by ID`

`G=gap.SmallGroup(288,571);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,142,1356,9414]);`
`// Polycyclic`

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

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