Copied to
clipboard

## G = C92⋊7C6order 486 = 2·35

### 7th semidirect product of C92 and C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C92⋊7C6
 Chief series C1 — C3 — C9 — C3×C9 — C92 — C92⋊7C3 — C92⋊7C6
 Lower central C9 — C3×C9 — C92⋊7C6
 Upper central C1 — C3 — C9

Generators and relations for C927C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a4, cbc-1=b2 >

Subgroups: 238 in 66 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C3×D9, S3×C9, C9⋊C6, C2×3- 1+2, S3×C32, C92, C32⋊C9, C9⋊C9, C9⋊C9, C3×3- 1+2, C9×D9, C9⋊C18, C3×C9⋊C6, S3×3- 1+2, C927C3, C927C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, 3- 1+2, C9⋊C6, C2×3- 1+2, S3×C32, C3×C9⋊C6, S3×3- 1+2, C927C6

Smallest permutation representation of C927C6
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 44 31 7 41 28 4 38 34)(2 45 32 8 42 29 5 39 35)(3 37 33 9 43 30 6 40 36)(10 23 52 13 26 46 16 20 49)(11 24 53 14 27 47 17 21 50)(12 25 54 15 19 48 18 22 51)
(1 24 33 49 39 18)(2 22 28 50 37 13)(3 20 32 51 44 17)(4 27 36 52 42 12)(5 25 31 53 40 16)(6 23 35 54 38 11)(7 21 30 46 45 15)(8 19 34 47 43 10)(9 26 29 48 41 14)```

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

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

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

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 6A 6B 6C 6D 9A 9B 9C ··· 9M 9N 9O 9P 9Q 9R ··· 9W 18A ··· 18F order 1 2 3 3 3 3 3 3 3 6 6 6 6 9 9 9 ··· 9 9 9 9 9 9 ··· 9 18 ··· 18 size 1 9 1 1 2 2 2 9 9 9 9 27 27 3 3 6 ··· 6 9 9 9 9 18 ··· 18 27 ··· 27

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 6 6 6 6 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 S3 C3×S3 C3×S3 3- 1+2 C2×3- 1+2 C9⋊C6 C3×C9⋊C6 S3×3- 1+2 C92⋊7C6 kernel C92⋊7C6 C92⋊7C3 C9×D9 C9⋊C18 C3×C9⋊C6 C92 C9⋊C9 C3×3- 1+2 C3×3- 1+2 C3×C9 C33 D9 C9 C9 C3 C3 C1 # reps 1 1 2 4 2 2 4 2 1 6 2 2 2 1 2 2 6

Matrix representation of C927C6 in GL6(𝔽19)

 0 5 0 0 0 0 0 0 5 0 0 0 17 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 17 0 0
,
 0 7 0 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 11 0 0 0 0 0 0 11 0
,
 0 0 0 0 0 16 0 0 0 16 0 0 0 0 0 0 17 0 0 0 16 0 0 0 16 0 0 0 0 0 0 17 0 0 0 0

`G:=sub<GL(6,GF(19))| [0,0,17,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,17,0,0,0,5,0,0,0,0,0,0,5,0],[0,0,1,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,0,0,16,0,0,0,0,0,0,17,0,0,0,16,0,0,0,16,0,0,0,0,0,0,17,0,0,0,16,0,0,0,0,0] >;`

C927C6 in GAP, Magma, Sage, TeX

`C_9^2\rtimes_7C_6`
`% in TeX`

`G:=Group("C9^2:7C6");`
`// GroupNames label`

`G:=SmallGroup(486,109);`
`// by ID`

`G=gap.SmallGroup(486,109);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,68,8104,3250,208,11669]);`
`// Polycyclic`

`G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^4,c*b*c^-1=b^2>;`
`// generators/relations`

׿
×
𝔽