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## G = D9⋊3- 1+2order 486 = 2·35

### The semidirect product of D9 and 3- 1+2 acting via 3- 1+2/C32=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — D9⋊3- 1+2
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C9⋊3- 1+2 — D9⋊3- 1+2
 Lower central C9 — C3×C9 — D9⋊3- 1+2
 Upper central C1 — C3 — C32

Generators and relations for D9⋊3- 1+2
G = < a,b,c,d | a9=b2=c9=d3=1, bab=a-1, cac-1=a4, ad=da, cbc-1=a3b, bd=db, dcd-1=c4 >

Subgroups: 238 in 67 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C3×D9, C3×D9, S3×C9, C2×3- 1+2, S3×C32, C32⋊C9, C9⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C9⋊C18, C32×D9, S3×3- 1+2, C9⋊3- 1+2, D9⋊3- 1+2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, 3- 1+2, C9⋊C6, C2×3- 1+2, S3×C32, C3×C9⋊C6, S3×3- 1+2, D9⋊3- 1+2

Smallest permutation representation of D9⋊3- 1+2
On 54 points
Generators in S54
```(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)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 54)(27 53)
(1 13 24 7 10 21 4 16 27)(2 11 19 8 17 25 5 14 22)(3 18 23 9 15 20 6 12 26)(28 41 48 31 44 51 34 38 54)(29 39 52 32 42 46 35 45 49)(30 37 47 33 40 50 36 43 53)
(1 26 17)(2 27 18)(3 19 10)(4 20 11)(5 21 12)(6 22 13)(7 23 14)(8 24 15)(9 25 16)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)```

`G:=sub<Sym(54)| (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,54)(27,53), (1,13,24,7,10,21,4,16,27)(2,11,19,8,17,25,5,14,22)(3,18,23,9,15,20,6,12,26)(28,41,48,31,44,51,34,38,54)(29,39,52,32,42,46,35,45,49)(30,37,47,33,40,50,36,43,53), (1,26,17)(2,27,18)(3,19,10)(4,20,11)(5,21,12)(6,22,13)(7,23,14)(8,24,15)(9,25,16)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45)>;`

`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)(46,47,48,49,50,51,52,53,54), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,54)(27,53), (1,13,24,7,10,21,4,16,27)(2,11,19,8,17,25,5,14,22)(3,18,23,9,15,20,6,12,26)(28,41,48,31,44,51,34,38,54)(29,39,52,32,42,46,35,45,49)(30,37,47,33,40,50,36,43,53), (1,26,17)(2,27,18)(3,19,10)(4,20,11)(5,21,12)(6,22,13)(7,23,14)(8,24,15)(9,25,16)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45) );`

`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),(46,47,48,49,50,51,52,53,54)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,45),(18,44),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,54),(27,53)], [(1,13,24,7,10,21,4,16,27),(2,11,19,8,17,25,5,14,22),(3,18,23,9,15,20,6,12,26),(28,41,48,31,44,51,34,38,54),(29,39,52,32,42,46,35,45,49),(30,37,47,33,40,50,36,43,53)], [(1,26,17),(2,27,18),(3,19,10),(4,20,11),(5,21,12),(6,22,13),(7,23,14),(8,24,15),(9,25,16),(28,46,37),(29,47,38),(30,48,39),(31,49,40),(32,50,41),(33,51,42),(34,52,43),(35,53,44),(36,54,45)]])`

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 9A ··· 9I 9J ··· 9O 9P ··· 9U 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 3 6 6 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 1 2 2 2 3 3 6 6 9 9 27 27 6 ··· 6 9 ··· 9 18 ··· 18 27 ··· 27

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 6 6 6 type + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 3- 1+2 C2×3- 1+2 C9⋊C6 C3×C9⋊C6 S3×3- 1+2 D9⋊3- 1+2 kernel D9⋊3- 1+2 C9⋊3- 1+2 C9⋊C18 C32×D9 C9⋊C9 C32×C9 C3×3- 1+2 C3×C9 C33 D9 C9 C32 C3 C3 C1 # reps 1 1 6 2 6 2 1 6 2 2 2 1 2 2 6

Matrix representation of D9⋊3- 1+2 in GL6(𝔽19)

 0 7 0 0 0 0 0 0 7 0 0 0 11 0 0 0 0 0 0 0 0 0 0 7 0 0 0 11 0 0 0 0 0 0 11 0
,
 0 0 0 0 0 7 0 0 0 11 0 0 0 0 0 0 11 0 0 7 0 0 0 0 0 0 7 0 0 0 11 0 0 0 0 0
,
 0 4 0 0 0 0 0 0 9 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 9 0 0 0 4 0 0
,
 0 16 0 0 0 0 0 0 16 0 0 0 17 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 17 0 0

`G:=sub<GL(6,GF(19))| [0,0,11,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,7,0,0],[0,0,0,0,0,11,0,0,0,7,0,0,0,0,0,0,7,0,0,11,0,0,0,0,0,0,11,0,0,0,7,0,0,0,0,0],[0,0,4,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,9,0],[0,0,17,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,17,0,0,0,16,0,0,0,0,0,0,16,0] >;`

D9⋊3- 1+2 in GAP, Magma, Sage, TeX

`D_9\rtimes 3_-^{1+2}`
`% in TeX`

`G:=Group("D9:ES-(3,1)");`
`// GroupNames label`

`G:=SmallGroup(486,108);`
`// by ID`

`G=gap.SmallGroup(486,108);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,224,8104,3250,208,11669]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^9=b^2=c^9=d^3=1,b*a*b=a^-1,c*a*c^-1=a^4,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^4>;`
`// generators/relations`

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