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## G = C92⋊3S3order 486 = 2·35

### 3rd semidirect product of C92 and S3 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C92⋊3S3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C92⋊3C3 — C92⋊3S3
 Lower central C3×C9 — C92⋊3S3
 Upper central C1 — C3

Generators and relations for C923S3
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, cac-1=ab3, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 300 in 69 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C92, C92, C32⋊C9, C32⋊C9, C9⋊C9, C9⋊C9, C32×C9, C9×D9, C32⋊C18, C9⋊C18, C3×C9⋊S3, C923C3, C923S3
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, C3×C3⋊S3, C9×C3⋊S3, He3.4S3, C923S3

Smallest permutation representation of C923S3
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 41 35 4 44 29 7 38 32)(2 42 36 5 45 30 8 39 33)(3 43 28 6 37 31 9 40 34)(10 47 23 16 53 20 13 50 26)(11 48 24 17 54 21 14 51 27)(12 49 25 18 46 22 15 52 19)
(2 8 5)(3 6 9)(10 13 16)(11 17 14)(20 23 26)(21 27 24)(28 31 34)(30 36 33)(37 40 43)(39 45 42)(47 50 53)(48 54 51)
(1 18)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(19 44)(20 45)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 51)(29 52)(30 53)(31 54)(32 46)(33 47)(34 48)(35 49)(36 50)```

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

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

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

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 6 6 type + + + + + + image C1 C2 C3 C6 C9 C18 S3 S3 S3 C3×S3 C3×S3 S3×C9 S3×C9 He3.4S3 C92⋊3S3 kernel C92⋊3S3 C92⋊3C3 C3×C9⋊S3 C32×C9 C9⋊S3 C3×C9 C92 C32⋊C9 C9⋊C9 C3×C9 C33 C9 C32 C3 C1 # reps 1 1 2 2 6 6 1 1 2 6 2 18 6 3 6

Matrix representation of C923S3 in GL6(𝔽19)

 11 6 0 0 0 0 0 8 1 0 0 0 8 12 0 0 0 0 0 0 0 11 6 0 0 0 0 0 8 1 0 0 0 8 12 0
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 12 11 0 0 0 0 7 0 7 0 0 0 0 0 0 1 0 0 0 0 0 11 7 0 0 0 0 18 0 11
,
 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

`G:=sub<GL(6,GF(19))| [11,0,8,0,0,0,6,8,12,0,0,0,0,1,0,0,0,0,0,0,0,11,0,8,0,0,0,6,8,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,12,7,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,11,18,0,0,0,0,7,0,0,0,0,0,0,11],[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] >;`

C923S3 in GAP, Magma, Sage, TeX

`C_9^2\rtimes_3S_3`
`% in TeX`

`G:=Group("C9^2:3S3");`
`// GroupNames label`

`G:=SmallGroup(486,139);`
`// by ID`

`G=gap.SmallGroup(486,139);`
`# by ID`

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

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

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