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## G = D12⋊D9order 432 = 24·33

### 3rd semidirect product of D12 and D9 acting via D9/C9=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D12⋊D9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — S3×Dic9 — D12⋊D9
 Lower central C3×C9 — C3×C18 — D12⋊D9
 Upper central C1 — C2 — C4

Generators and relations for D12⋊D9
G = < a,b,c,d | a6=b2=1, c18=d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c17 >

Subgroups: 832 in 136 conjugacy classes, 41 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×C9, Dic9, Dic9, C36, C36, D18, C2×C18, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, D42S3, Q83S3, S3×C9, C9⋊S3, C3×C18, Dic18, C4×D9, C2×Dic9, C9⋊D4, D4×C9, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C3×Dic9, C9⋊Dic3, C3×C36, S3×C18, C2×C9⋊S3, D42D9, D12⋊S3, S3×Dic9, C9⋊D12, C3×Dic18, C9×D12, C4×C9⋊S3, D12⋊D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, D42S3, Q83S3, C22×D9, C2×S32, S3×D9, D42D9, D12⋊S3, C2×S3×D9, D12⋊D9

Smallest permutation representation of D12⋊D9
On 72 points
Generators in S72
```(1 7 13 19 25 31)(2 8 14 20 26 32)(3 9 15 21 27 33)(4 10 16 22 28 34)(5 11 17 23 29 35)(6 12 18 24 30 36)(37 67 61 55 49 43)(38 68 62 56 50 44)(39 69 63 57 51 45)(40 70 64 58 52 46)(41 71 65 59 53 47)(42 72 66 60 54 48)
(1 65)(2 48)(3 67)(4 50)(5 69)(6 52)(7 71)(8 54)(9 37)(10 56)(11 39)(12 58)(13 41)(14 60)(15 43)(16 62)(17 45)(18 64)(19 47)(20 66)(21 49)(22 68)(23 51)(24 70)(25 53)(26 72)(27 55)(28 38)(29 57)(30 40)(31 59)(32 42)(33 61)(34 44)(35 63)(36 46)
(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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 49 19 67)(2 48 20 66)(3 47 21 65)(4 46 22 64)(5 45 23 63)(6 44 24 62)(7 43 25 61)(8 42 26 60)(9 41 27 59)(10 40 28 58)(11 39 29 57)(12 38 30 56)(13 37 31 55)(14 72 32 54)(15 71 33 53)(16 70 34 52)(17 69 35 51)(18 68 36 50)```

`G:=sub<Sym(72)| (1,7,13,19,25,31)(2,8,14,20,26,32)(3,9,15,21,27,33)(4,10,16,22,28,34)(5,11,17,23,29,35)(6,12,18,24,30,36)(37,67,61,55,49,43)(38,68,62,56,50,44)(39,69,63,57,51,45)(40,70,64,58,52,46)(41,71,65,59,53,47)(42,72,66,60,54,48), (1,65)(2,48)(3,67)(4,50)(5,69)(6,52)(7,71)(8,54)(9,37)(10,56)(11,39)(12,58)(13,41)(14,60)(15,43)(16,62)(17,45)(18,64)(19,47)(20,66)(21,49)(22,68)(23,51)(24,70)(25,53)(26,72)(27,55)(28,38)(29,57)(30,40)(31,59)(32,42)(33,61)(34,44)(35,63)(36,46), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,49,19,67)(2,48,20,66)(3,47,21,65)(4,46,22,64)(5,45,23,63)(6,44,24,62)(7,43,25,61)(8,42,26,60)(9,41,27,59)(10,40,28,58)(11,39,29,57)(12,38,30,56)(13,37,31,55)(14,72,32,54)(15,71,33,53)(16,70,34,52)(17,69,35,51)(18,68,36,50)>;`

`G:=Group( (1,7,13,19,25,31)(2,8,14,20,26,32)(3,9,15,21,27,33)(4,10,16,22,28,34)(5,11,17,23,29,35)(6,12,18,24,30,36)(37,67,61,55,49,43)(38,68,62,56,50,44)(39,69,63,57,51,45)(40,70,64,58,52,46)(41,71,65,59,53,47)(42,72,66,60,54,48), (1,65)(2,48)(3,67)(4,50)(5,69)(6,52)(7,71)(8,54)(9,37)(10,56)(11,39)(12,58)(13,41)(14,60)(15,43)(16,62)(17,45)(18,64)(19,47)(20,66)(21,49)(22,68)(23,51)(24,70)(25,53)(26,72)(27,55)(28,38)(29,57)(30,40)(31,59)(32,42)(33,61)(34,44)(35,63)(36,46), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,49,19,67)(2,48,20,66)(3,47,21,65)(4,46,22,64)(5,45,23,63)(6,44,24,62)(7,43,25,61)(8,42,26,60)(9,41,27,59)(10,40,28,58)(11,39,29,57)(12,38,30,56)(13,37,31,55)(14,72,32,54)(15,71,33,53)(16,70,34,52)(17,69,35,51)(18,68,36,50) );`

`G=PermutationGroup([[(1,7,13,19,25,31),(2,8,14,20,26,32),(3,9,15,21,27,33),(4,10,16,22,28,34),(5,11,17,23,29,35),(6,12,18,24,30,36),(37,67,61,55,49,43),(38,68,62,56,50,44),(39,69,63,57,51,45),(40,70,64,58,52,46),(41,71,65,59,53,47),(42,72,66,60,54,48)], [(1,65),(2,48),(3,67),(4,50),(5,69),(6,52),(7,71),(8,54),(9,37),(10,56),(11,39),(12,58),(13,41),(14,60),(15,43),(16,62),(17,45),(18,64),(19,47),(20,66),(21,49),(22,68),(23,51),(24,70),(25,53),(26,72),(27,55),(28,38),(29,57),(30,40),(31,59),(32,42),(33,61),(34,44),(35,63),(36,46)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,49,19,67),(2,48,20,66),(3,47,21,65),(4,46,22,64),(5,45,23,63),(6,44,24,62),(7,43,25,61),(8,42,26,60),(9,41,27,59),(10,40,28,58),(11,39,29,57),(12,38,30,56),(13,37,31,55),(14,72,32,54),(15,71,33,53),(16,70,34,52),(17,69,35,51),(18,68,36,50)]])`

51 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 18A 18B 18C 18D 18E 18F 18G ··· 18L 36A ··· 36I order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 18 18 18 18 18 18 18 ··· 18 36 ··· 36 size 1 1 6 6 54 2 2 4 2 18 18 27 27 2 2 4 12 12 2 2 2 4 4 4 4 4 4 4 36 36 2 2 2 4 4 4 12 ··· 12 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + - + image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 C4○D4 D9 D18 D18 S32 Q8⋊3S3 D4⋊2S3 C2×S32 S3×D9 D4⋊2D9 D12⋊S3 C2×S3×D9 D12⋊D9 kernel D12⋊D9 S3×Dic9 C9⋊D12 C3×Dic18 C9×D12 C4×C9⋊S3 Dic18 C3×D12 Dic9 C36 C3×C12 S3×C6 C3×C9 D12 C12 D6 C12 C9 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 2 1 1 2 2 3 3 6 1 1 1 1 3 3 2 3 6

Matrix representation of D12⋊D9 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 1 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 6 0 0 0 0 31 0 0 0 0 0 0 0 1 0 0 0 0 0 1 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 31 0 0 0 0 0 0 6 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 11 17 0 0 0 0 20 31
,
 0 31 0 0 0 0 31 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 11 17 0 0 0 0 6 26

`G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,31,0,0,0,0,6,0,0,0,0,0,0,0,1,1,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[31,0,0,0,0,0,0,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,11,20,0,0,0,0,17,31],[0,31,0,0,0,0,31,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,11,6,0,0,0,0,17,26] >;`

D12⋊D9 in GAP, Magma, Sage, TeX

`D_{12}\rtimes D_9`
`% in TeX`

`G:=Group("D12:D9");`
`// GroupNames label`

`G:=SmallGroup(432,286);`
`// by ID`

`G=gap.SmallGroup(432,286);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,3091,662,4037,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=1,c^18=d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^17>;`
`// generators/relations`

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