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G = C92.S3order 486 = 2·35

2nd non-split extension by C92 of S3 acting faithfully

Aliases: C92.2S3, C92.C3⋊C2, C3.He3.1C6, C32.2(C32⋊C6), C3.6(He3.2C6), 3- 1+2.S3.1C3, (C3×C9).18(C3×S3), SmallGroup(486,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3.He3 — C92.S3
 Chief series C1 — C3 — C32 — C3×C9 — C3.He3 — C92.C3 — C92.S3
 Lower central C3.He3 — C92.S3
 Upper central C1

Generators and relations for C92.S3
G = < a,b,c,d | a9=b9=d2=1, c3=b6, ab=ba, cac-1=ab-1, ad=da, cbc-1=a3b7, dbd=a3b-1, dcd=b3c2 >

Permutation representations of C92.S3
On 27 points - transitive group 27T184
Generators in S27
```(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 8 5 2 9 6 3 7 4)(10 12 14 16 18 11 13 15 17)(19 20 21 22 23 24 25 26 27)
(1 14 23 3 17 20 2 11 26)(4 15 22 6 18 19 5 12 25)(7 16 21 9 10 27 8 13 24)
(2 3)(4 8)(5 7)(6 9)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 19)(17 20)(18 21)```

`G:=sub<Sym(27)| (10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,8,5,2,9,6,3,7,4)(10,12,14,16,18,11,13,15,17)(19,20,21,22,23,24,25,26,27), (1,14,23,3,17,20,2,11,26)(4,15,22,6,18,19,5,12,25)(7,16,21,9,10,27,8,13,24), (2,3)(4,8)(5,7)(6,9)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21)>;`

`G:=Group( (10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,8,5,2,9,6,3,7,4)(10,12,14,16,18,11,13,15,17)(19,20,21,22,23,24,25,26,27), (1,14,23,3,17,20,2,11,26)(4,15,22,6,18,19,5,12,25)(7,16,21,9,10,27,8,13,24), (2,3)(4,8)(5,7)(6,9)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21) );`

`G=PermutationGroup([[(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,8,5,2,9,6,3,7,4),(10,12,14,16,18,11,13,15,17),(19,20,21,22,23,24,25,26,27)], [(1,14,23,3,17,20,2,11,26),(4,15,22,6,18,19,5,12,25),(7,16,21,9,10,27,8,13,24)], [(2,3),(4,8),(5,7),(6,9),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,19),(17,20),(18,21)]])`

`G:=TransitiveGroup(27,184);`

31 conjugacy classes

 class 1 2 3A 3B 3C 6A 6B 9A ··· 9F 9G ··· 9O 9P 9Q 9R 18A ··· 18F order 1 2 3 3 3 6 6 9 ··· 9 9 ··· 9 9 9 9 18 ··· 18 size 1 27 2 3 3 27 27 3 ··· 3 6 ··· 6 54 54 54 27 ··· 27

31 irreducible representations

 dim 1 1 1 1 2 2 3 6 6 6 type + + + + + image C1 C2 C3 C6 S3 C3×S3 He3.2C6 C32⋊C6 C92.S3 C92.S3 kernel C92.S3 C92.C3 3- 1+2.S3 C3.He3 C92 C3×C9 C3 C32 C1 C1 # reps 1 1 2 2 1 2 12 1 3 6

Matrix representation of C92.S3 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 9 9 7 5 0 0 14 14 14 2 0 0 14 14 0 0 2 14 9 9 0 0 5 7
,
 7 14 0 0 0 0 5 2 0 0 0 0 0 14 5 7 0 0 7 2 12 17 0 0 5 0 0 0 2 14 0 14 0 0 5 7
,
 1 1 0 0 18 17 0 0 0 0 1 18 0 0 0 0 0 18 1 0 0 0 0 18 0 0 1 0 0 18 0 0 0 1 0 18
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(19))| [1,0,9,14,14,9,0,1,9,14,14,9,0,0,7,14,0,0,0,0,5,2,0,0,0,0,0,0,2,5,0,0,0,0,14,7],[7,5,0,7,5,0,14,2,14,2,0,14,0,0,5,12,0,0,0,0,7,17,0,0,0,0,0,0,2,5,0,0,0,0,14,7],[1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,1,0,0,0,0,17,18,18,18,18,18],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;`

C92.S3 in GAP, Magma, Sage, TeX

`C_9^2.S_3`
`% in TeX`

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

`G:=SmallGroup(486,38);`
`// by ID`

`G=gap.SmallGroup(486,38);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,3134,224,986,6051,951,453,1096,11669]);`
`// Polycyclic`

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

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