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## G = D6⋊D9order 216 = 23·33

### 1st semidirect product of D6 and D9 acting via D9/C9=C2

Aliases: D61D9, D182S3, C6.6D18, C18.6D6, C6.6S32, (C3×C9)⋊2D4, (C6×D9)⋊2C2, C2.6(S3×D9), (S3×C18)⋊3C2, (S3×C6).2S3, C92(C3⋊D4), C32(C9⋊D4), C9⋊Dic33C2, (C3×C6).27D6, (C3×C18).6C22, C3.2(D6⋊S3), C32.2(C3⋊D4), SmallGroup(216,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D6⋊D9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — D6⋊D9
 Lower central C3×C9 — C3×C18 — D6⋊D9
 Upper central C1 — C2

Generators and relations for D6⋊D9
G = < a,b,c,d | a6=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >

Smallest permutation representation of D6⋊D9
On 72 points
Generators in S72
```(1 11 4 14 7 17)(2 12 5 15 8 18)(3 13 6 16 9 10)(19 31 25 28 22 34)(20 32 26 29 23 35)(21 33 27 30 24 36)(37 49 43 46 40 52)(38 50 44 47 41 53)(39 51 45 48 42 54)(55 70 58 64 61 67)(56 71 59 65 62 68)(57 72 60 66 63 69)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 46)(10 43)(11 44)(12 45)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 64)(26 65)(27 66)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 55)(35 56)(36 57)
(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 21)(2 20)(3 19)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 72)(45 71)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 63)(54 62)```

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

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

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

D6⋊D9 is a maximal subgroup of   D125D9  D6.D18  D365S3  C36⋊D6  D18.4D6  S3×C9⋊D4  D9×C3⋊D4
D6⋊D9 is a maximal quotient of   D36⋊S3  D12.D9  Dic6⋊D9  C12.D18  C18.Dic6  D18⋊Dic3  D6⋊Dic9

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 S3 S3 D4 D6 D6 D9 C3⋊D4 C3⋊D4 D18 C9⋊D4 S32 D6⋊S3 S3×D9 D6⋊D9 kernel D6⋊D9 C9⋊Dic3 C6×D9 S3×C18 D18 S3×C6 C3×C9 C18 C3×C6 D6 C9 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 3 2 2 3 6 1 1 3 3

Matrix representation of D6⋊D9 in GL4(𝔽37) generated by

 1 36 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 1 36 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 17 26 0 0 11 6
,
 7 23 0 0 14 30 0 0 0 0 26 31 0 0 20 11
`G:=sub<GL(4,GF(37))| [1,1,0,0,36,0,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,17,11,0,0,26,6],[7,14,0,0,23,30,0,0,0,0,26,20,0,0,31,11] >;`

D6⋊D9 in GAP, Magma, Sage, TeX

`D_6\rtimes D_9`
`% in TeX`

`G:=Group("D6:D9");`
`// GroupNames label`

`G:=SmallGroup(216,31);`
`// by ID`

`G=gap.SmallGroup(216,31);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,1065,453,1444,2603]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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