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## G = C92⋊9C6order 486 = 2·35

### 9th semidirect product of C92 and C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C92 — C92⋊9C6
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C9×3- 1+2 — C92⋊9C6
 Lower central C92 — C92⋊9C6
 Upper central C1

Generators and relations for C929C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a-1, cbc-1=b2 >

Subgroups: 692 in 91 conjugacy classes, 29 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×D9, C9⋊C6, C9⋊S3, C3×C3⋊S3, C92, C92, C32⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C32⋊D9, C9⋊D9, C3×C9⋊S3, C33.S3, C9×3- 1+2, C929C6
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊C6, C9⋊S3, C3×C3⋊S3, C3×C9⋊S3, C33.S3, He3.4S3, C929C6

Smallest permutation representation of C929C6
On 81 points
Generators in S81
```(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)(73 74 75 76 77 78 79 80 81)
(1 42 36 20 67 11 74 57 50)(2 43 28 21 68 12 75 58 51)(3 44 29 22 69 13 76 59 52)(4 45 30 23 70 14 77 60 53)(5 37 31 24 71 15 78 61 54)(6 38 32 25 72 16 79 62 46)(7 39 33 26 64 17 80 63 47)(8 40 34 27 65 18 81 55 48)(9 41 35 19 66 10 73 56 49)
(2 9)(3 8)(4 7)(5 6)(10 58 49 68 35 43)(11 57 50 67 36 42)(12 56 51 66 28 41)(13 55 52 65 29 40)(14 63 53 64 30 39)(15 62 54 72 31 38)(16 61 46 71 32 37)(17 60 47 70 33 45)(18 59 48 69 34 44)(19 75)(20 74)(21 73)(22 81)(23 80)(24 79)(25 78)(26 77)(27 76)```

`G:=sub<Sym(81)| (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)(73,74,75,76,77,78,79,80,81), (1,42,36,20,67,11,74,57,50)(2,43,28,21,68,12,75,58,51)(3,44,29,22,69,13,76,59,52)(4,45,30,23,70,14,77,60,53)(5,37,31,24,71,15,78,61,54)(6,38,32,25,72,16,79,62,46)(7,39,33,26,64,17,80,63,47)(8,40,34,27,65,18,81,55,48)(9,41,35,19,66,10,73,56,49), (2,9)(3,8)(4,7)(5,6)(10,58,49,68,35,43)(11,57,50,67,36,42)(12,56,51,66,28,41)(13,55,52,65,29,40)(14,63,53,64,30,39)(15,62,54,72,31,38)(16,61,46,71,32,37)(17,60,47,70,33,45)(18,59,48,69,34,44)(19,75)(20,74)(21,73)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76)>;`

`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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,42,36,20,67,11,74,57,50)(2,43,28,21,68,12,75,58,51)(3,44,29,22,69,13,76,59,52)(4,45,30,23,70,14,77,60,53)(5,37,31,24,71,15,78,61,54)(6,38,32,25,72,16,79,62,46)(7,39,33,26,64,17,80,63,47)(8,40,34,27,65,18,81,55,48)(9,41,35,19,66,10,73,56,49), (2,9)(3,8)(4,7)(5,6)(10,58,49,68,35,43)(11,57,50,67,36,42)(12,56,51,66,28,41)(13,55,52,65,29,40)(14,63,53,64,30,39)(15,62,54,72,31,38)(16,61,46,71,32,37)(17,60,47,70,33,45)(18,59,48,69,34,44)(19,75)(20,74)(21,73)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76) );`

`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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81)], [(1,42,36,20,67,11,74,57,50),(2,43,28,21,68,12,75,58,51),(3,44,29,22,69,13,76,59,52),(4,45,30,23,70,14,77,60,53),(5,37,31,24,71,15,78,61,54),(6,38,32,25,72,16,79,62,46),(7,39,33,26,64,17,80,63,47),(8,40,34,27,65,18,81,55,48),(9,41,35,19,66,10,73,56,49)], [(2,9),(3,8),(4,7),(5,6),(10,58,49,68,35,43),(11,57,50,67,36,42),(12,56,51,66,28,41),(13,55,52,65,29,40),(14,63,53,64,30,39),(15,62,54,72,31,38),(16,61,46,71,32,37),(17,60,47,70,33,45),(18,59,48,69,34,44),(19,75),(20,74),(21,73),(22,81),(23,80),(24,79),(25,78),(26,77),(27,76)]])`

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A ··· 9I 9J ··· 9AP order 1 2 3 3 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 size 1 81 2 2 2 2 3 3 6 6 81 81 2 ··· 2 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 6 6 type + + + + + + + + image C1 C2 C3 C6 S3 S3 S3 C3×S3 D9 C3×D9 C9⋊C6 He3.4S3 kernel C92⋊9C6 C9×3- 1+2 C9⋊D9 C92 C32⋊C9 C32×C9 C3×3- 1+2 C3×C9 3- 1+2 C9 C9 C3 # reps 1 1 2 2 2 1 1 8 9 18 3 6

Matrix representation of C929C6 in GL8(𝔽19)

 14 17 0 0 0 0 0 0 2 12 0 0 0 0 0 0 0 0 2 5 0 0 0 0 0 0 14 7 0 0 0 0 0 0 5 14 7 5 0 0 0 0 14 0 14 2 0 0 0 0 0 5 0 0 7 5 0 0 5 14 0 0 14 2
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 17 1 18 17 0 0 0 0 0 0 1 18 0 0 0 0 1 0 1 1 0 1 0 0 2 17 1 1 18 18 0 0 18 0 18 0 0 0 0 0 0 0 18 0 0 0
,
 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 18 0 0 0 18 0 0 0 18 0 0 18 0 0 0 18 1 18 0 0 0 0 0 1 0 1 1 0 0

`G:=sub<GL(8,GF(19))| [14,2,0,0,0,0,0,0,17,12,0,0,0,0,0,0,0,0,2,14,5,14,0,5,0,0,5,7,14,0,5,14,0,0,0,0,7,14,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,5,2],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,0,1,2,18,0,0,0,1,0,0,17,0,0,0,0,18,1,1,1,18,18,0,0,17,18,1,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0],[0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,1,0,0,18,1,0,0,0,18,18,18,1,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0] >;`

C929C6 in GAP, Magma, Sage, TeX

`C_9^2\rtimes_9C_6`
`% in TeX`

`G:=Group("C9^2:9C6");`
`// GroupNames label`

`G:=SmallGroup(486,144);`
`// by ID`

`G=gap.SmallGroup(486,144);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,338,4755,2169,453,3244,11669]);`
`// Polycyclic`

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

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