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## G = C9⋊S3⋊C9order 486 = 2·35

### 1st semidirect product of C9⋊S3 and C9 acting via C9/C3=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C9⋊S3⋊C9
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C3.C92 — C9⋊S3⋊C9
 Lower central C3×C9 — C9⋊S3⋊C9
 Upper central C1 — C3

Generators and relations for C9⋊S3⋊C9
G = < a,b,c,d | a9=b3=c2=d9=1, dad-1=ab=ba, cac=a-1, cbc=b-1, bd=db, cd=dc >

Subgroups: 336 in 69 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C32×C9, C32×C9, C3×C9⋊S3, C9×C3⋊S3, C3.C92, C9⋊S3⋊C9
Quotients: C1, C2, C3, S3, C6, C9, D9, C18, C3×S3, C3×D9, S3×C9, C32⋊C6, C9⋊C6, C9×D9, C32⋊C18, C32⋊D9, C9⋊C18, C9⋊S3⋊C9

Smallest permutation representation of C9⋊S3⋊C9
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 38 31)(2 39 32)(3 40 33)(4 41 34)(5 42 35)(6 43 36)(7 44 28)(8 45 29)(9 37 30)(10 25 52)(11 26 53)(12 27 54)(13 19 46)(14 20 47)(15 21 48)(16 22 49)(17 23 50)(18 24 51)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 54)(45 53)
(1 3 42 7 9 39 4 6 45)(2 34 36 8 31 33 5 28 30)(10 50 15 13 53 18 16 47 12)(11 24 22 14 27 25 17 21 19)(20 54 52 23 48 46 26 51 49)(29 38 40 35 44 37 32 41 43)```

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

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

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

63 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 9A ··· 9F 9G ··· 9AM 18A ··· 18F order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 ··· 2 27 27 3 ··· 3 6 ··· 6 27 ··· 27

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 6 6 6 6 type + + + + + + image C1 C2 C3 C6 C9 C18 S3 D9 C3×S3 C3×D9 S3×C9 C9×D9 C32⋊C6 C9⋊C6 C32⋊C18 C9⋊C18 kernel C9⋊S3⋊C9 C3.C92 C3×C9⋊S3 C32×C9 C9⋊S3 C3×C9 C32×C9 C3×C9 C33 C32 C32 C3 C32 C32 C3 C3 # reps 1 1 2 2 6 6 1 3 2 6 6 18 1 2 2 4

Matrix representation of C9⋊S3⋊C9 in GL8(𝔽19)

 5 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 5 10 0 0 0 0 0 0 0 14 1 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 4 0 13 0 0 0 0 0 0 0 3 0 0 0 0 0 0 1 15
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11
,
 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 15 0 0 0 0 0 8 0 2 0 0 0 0 0 16 1 10 0 0 0 0 0 0 0 0 9 0 15 0 0 0 0 0 8 0 2 0 0 0 0 0 16 1 10

`G:=sub<GL(8,GF(19))| [5,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,10,14,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,3,15],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,8,16,0,0,0,0,0,0,0,1,0,0,0,0,0,15,2,10,0,0,0,0,0,0,0,0,9,8,16,0,0,0,0,0,0,0,1,0,0,0,0,0,15,2,10] >;`

C9⋊S3⋊C9 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(486,3);`
`// by ID`

`G=gap.SmallGroup(486,3);`
`# by ID`

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

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

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