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## G = C92⋊2C6order 486 = 2·35

### 2nd semidirect product of C92 and C6 acting faithfully

Aliases: C922C6, C9⋊D92C3, C922C32C2, He3⋊C3.2S3, C32.15(C32⋊C6), C3.3(He3.2S3), (C3×C9).28(C3×S3), SmallGroup(486,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C92 — C92⋊2C6
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C92⋊2C3 — C92⋊2C6
 Lower central C92 — C92⋊2C6
 Upper central C1

Generators and relations for C922C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a-1b-1, cbc-1=a3b2 >

81C2
3C3
27C3
54C3
27S3
81C6
81S3
3C9
3C9
3C9
3C9
9C32
18C32
27D9
27D9
27D9
27D9
27C3×S3
3He3
6He3

Character table of C922C6

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L size 1 81 2 6 27 27 54 54 81 81 6 6 6 6 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 -1 1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ7 2 0 2 2 2 2 -1 -1 0 0 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 complex lifted from C3×S3 ρ9 2 0 2 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 complex lifted from C3×S3 ρ10 6 0 6 6 0 0 0 0 0 0 0 0 0 0 0 -3 -3 -3 0 0 0 0 orthogonal lifted from C32⋊C6 ρ11 6 0 6 -3 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 0 0 0 2ζ98-ζ94+ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 orthogonal lifted from He3.2S3 ρ12 6 0 -3 0 0 0 0 0 0 0 2ζ95+2ζ94+2 -ζ95-ζ94-1 -ζ98-ζ9-1 2ζ98+2ζ9+2 -ζ98-ζ9-1 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ97-ζ92-1 2ζ97+2ζ92+2 -ζ97-ζ92-1 -ζ95-ζ94-1 orthogonal faithful ρ13 6 0 -3 0 0 0 0 0 0 0 2ζ98+2ζ9+2 -ζ98-ζ9-1 -ζ97-ζ92-1 2ζ97+2ζ92+2 -ζ97-ζ92-1 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 -ζ95-ζ94-1 2ζ95+2ζ94+2 -ζ95-ζ94-1 -ζ98-ζ9-1 orthogonal faithful ρ14 6 0 -3 0 0 0 0 0 0 0 -ζ95-ζ94-1 -ζ95-ζ94-1 2ζ98+2ζ9+2 -ζ98-ζ9-1 -ζ98-ζ9-1 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 2ζ97+2ζ92+2 -ζ97-ζ92-1 -ζ97-ζ92-1 2ζ95+2ζ94+2 orthogonal faithful ρ15 6 0 6 -3 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 -ζ98+2ζ97+ζ94+ζ92 -ζ98+2ζ97+ζ94+ζ92 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 orthogonal lifted from He3.2S3 ρ16 6 0 -3 0 0 0 0 0 0 0 -ζ97-ζ92-1 -ζ97-ζ92-1 2ζ95+2ζ94+2 -ζ95-ζ94-1 -ζ95-ζ94-1 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98+2ζ9+2 -ζ98-ζ9-1 -ζ98-ζ9-1 2ζ97+2ζ92+2 orthogonal faithful ρ17 6 0 -3 0 0 0 0 0 0 0 -ζ98-ζ9-1 2ζ98+2ζ9+2 -ζ97-ζ92-1 -ζ97-ζ92-1 2ζ97+2ζ92+2 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 -ζ95-ζ94-1 -ζ95-ζ94-1 2ζ95+2ζ94+2 -ζ98-ζ9-1 orthogonal faithful ρ18 6 0 6 -3 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 -ζ98+2ζ97+ζ94+ζ92 -ζ98+2ζ97+ζ94+ζ92 0 0 0 ζ95+2ζ94-ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 orthogonal lifted from He3.2S3 ρ19 6 0 -3 0 0 0 0 0 0 0 -ζ95-ζ94-1 2ζ95+2ζ94+2 -ζ98-ζ9-1 -ζ98-ζ9-1 2ζ98+2ζ9+2 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ97-ζ92-1 -ζ97-ζ92-1 2ζ97+2ζ92+2 -ζ95-ζ94-1 orthogonal faithful ρ20 6 0 -3 0 0 0 0 0 0 0 -ζ97-ζ92-1 2ζ97+2ζ92+2 -ζ95-ζ94-1 -ζ95-ζ94-1 2ζ95+2ζ94+2 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 -ζ98-ζ9-1 -ζ98-ζ9-1 2ζ98+2ζ9+2 -ζ97-ζ92-1 orthogonal faithful ρ21 6 0 -3 0 0 0 0 0 0 0 -ζ98-ζ9-1 -ζ98-ζ9-1 2ζ97+2ζ92+2 -ζ97-ζ92-1 -ζ97-ζ92-1 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ95+2ζ94+2 -ζ95-ζ94-1 -ζ95-ζ94-1 2ζ98+2ζ9+2 orthogonal faithful ρ22 6 0 -3 0 0 0 0 0 0 0 2ζ97+2ζ92+2 -ζ97-ζ92-1 -ζ95-ζ94-1 2ζ95+2ζ94+2 -ζ95-ζ94-1 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 -ζ98-ζ9-1 2ζ98+2ζ9+2 -ζ98-ζ9-1 -ζ97-ζ92-1 orthogonal faithful

Permutation representations of C922C6
On 27 points - transitive group 27T158
Generators in S27
```(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 7 6 2 8 4 3 9 5)(10 17 15 13 11 18 16 14 12)(19 27 26 25 24 23 22 21 20)
(1 10 27)(2 16 24 3 13 21)(4 14 22 8 15 23)(5 11 19 7 18 26)(6 17 25 9 12 20)```

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

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

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

`G:=TransitiveGroup(27,158);`

Matrix representation of C922C6 in GL6(𝔽19)

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

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

C922C6 in GAP, Magma, Sage, TeX

`C_9^2\rtimes_2C_6`
`% in TeX`

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

`G:=SmallGroup(486,37);`
`// by ID`

`G=gap.SmallGroup(486,37);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,338,4755,2817,453,3244,3250,11669]);`
`// Polycyclic`

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

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