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## G = C9⋊S3⋊3C9order 486 = 2·35

### 3rd semidirect product of C9⋊S3 and C9 acting via C9/C3=C3

Aliases: C9⋊S33C9, (C3×C9)⋊3C18, C32⋊C9.6S3, (C32×C9).5C6, C32.8(S3×C9), C33.51(C3×S3), C3.4(C32⋊C18), C32.20He31C2, C32.38(C32⋊C6), C3.4(He3.2S3), (C3×C9⋊S3).3C3, SmallGroup(486,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C9⋊S3⋊3C9
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C32.20He3 — C9⋊S3⋊3C9
 Lower central C3×C9 — C9⋊S3⋊3C9
 Upper central C1 — C3

Generators and relations for C9⋊S33C9
G = < a,b,c,d | a9=b3=c2=d9=1, ab=ba, cac=a-1, dad-1=a7b, cbc=b-1, dbd-1=a6b, cd=dc >

Smallest permutation representation of C9⋊S33C9
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)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(1 31)(2 30)(3 29)(4 28)(5 36)(6 35)(7 34)(8 33)(9 32)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 45)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 54)
(1 23 17 4 26 11 7 20 14)(2 27 18 5 21 12 8 24 15)(3 22 10 6 25 13 9 19 16)(28 46 43 34 52 40 31 49 37)(29 50 44 35 47 41 32 53 38)(30 54 45 36 51 42 33 48 39)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 6 6 6 6 type + + + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 C32⋊C6 C32⋊C18 He3.2S3 C9⋊S3⋊3C9 kernel C9⋊S3⋊3C9 C32.20He3 C3×C9⋊S3 C32×C9 C9⋊S3 C3×C9 C32⋊C9 C33 C32 C32 C3 C3 C1 # reps 1 1 2 2 6 6 1 2 6 1 2 3 6

Matrix representation of C9⋊S33C9 in GL6(𝔽19)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 6 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 7 0 0

`G:=sub<GL(6,GF(19))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,6,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,0,7,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C9⋊S33C9 in GAP, Magma, Sage, TeX

`C_9\rtimes S_3\rtimes_3C_9`
`% in TeX`

`G:=Group("C9:S3:3C9");`
`// GroupNames label`

`G:=SmallGroup(486,22);`
`// by ID`

`G=gap.SmallGroup(486,22);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,4755,873,453,3244,3250,11669]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^9=b^3=c^2=d^9=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a^7*b,c*b*c=b^-1,d*b*d^-1=a^6*b,c*d=d*c>;`
`// generators/relations`

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