Copied to
clipboard

## G = C32⋊C9.S3order 486 = 2·35

### 1st non-split extension by C32⋊C9 of S3 acting faithfully

Aliases: C32⋊C9.1S3, C32⋊C9.1C6, C33.2(C3×S3), C322D91C3, C32.24He32C2, C32.26(C32⋊C6), C3.5(He3.2S3), C3.1(He3.2C6), SmallGroup(486,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — C32⋊C9.S3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32.24He3 — C32⋊C9.S3
 Lower central C32⋊C9 — C32⋊C9.S3
 Upper central C1 — C3

Generators and relations for C32⋊C9.S3
G = < a,b,c,d | a3=b3=c9=1, d6=b, ab=ba, cac-1=ab-1, dad-1=a-1bc3, bc=cb, bd=db, dcd-1=a-1c5 >

27C2
2C3
9C3
18C3
18C3
18C3
9S3
27C6
27S3
3C32
3C32
6C32
6C32
6C32
6C32
9C9
9C9
18C32
18C32
18C32
9D9
27C18
27C3×S3
6He3
6He3
6He3
6C33

Permutation representations of C32⋊C9.S3
On 18 points - transitive group 18T170
Generators in S18
```(1 13 7)(4 16 10)(5 11 17)(6 12 18)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)
(1 17 9 7 5 15 13 11 3)(2 4 6 14 16 18 8 10 12)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)```

`G:=sub<Sym(18)| (1,13,7)(4,16,10)(5,11,17)(6,12,18), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18), (1,17,9,7,5,15,13,11,3)(2,4,6,14,16,18,8,10,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)>;`

`G:=Group( (1,13,7)(4,16,10)(5,11,17)(6,12,18), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18), (1,17,9,7,5,15,13,11,3)(2,4,6,14,16,18,8,10,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18) );`

`G=PermutationGroup([(1,13,7),(4,16,10),(5,11,17),(6,12,18)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18)], [(1,17,9,7,5,15,13,11,3),(2,4,6,14,16,18,8,10,12)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)])`

`G:=TransitiveGroup(18,170);`

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3L 6A 6B 9A ··· 9F 9G 9H 9I 18A ··· 18F order 1 2 3 3 3 3 3 3 ··· 3 6 6 9 ··· 9 9 9 9 18 ··· 18 size 1 27 1 1 2 2 2 18 ··· 18 27 27 9 ··· 9 18 18 18 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 He3.2S3 C32⋊C9.S3 kernel C32⋊C9.S3 C32.24He3 C32⋊2D9 C32⋊C9 C32⋊C9 C33 C3 C32 C3 C1 # reps 1 1 2 2 1 2 12 1 3 6

Matrix representation of C32⋊C9.S3 in GL6(𝔽19)

 7 0 0 0 0 0 7 1 0 0 0 0 8 0 11 0 0 0 0 0 0 7 0 0 11 0 0 0 11 0 12 0 0 0 0 1
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 7 13 0 0 0 0 0 12 7 0 0 0 7 12 0 0 0 0 12 7 0 0 0 1 11 7 0 11 0 0 11 7 0 0 11 0
,
 8 0 0 0 15 0 8 0 0 0 8 7 8 0 0 7 8 0 0 11 0 0 11 0 11 0 7 0 11 0 11 0 0 0 11 0

`G:=sub<GL(6,GF(19))| [7,7,8,0,11,12,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[7,0,7,12,11,11,13,12,12,7,7,7,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[8,8,8,0,11,11,0,0,0,11,0,0,0,0,0,0,7,0,0,0,7,0,0,0,15,8,8,11,11,11,0,7,0,0,0,0] >;`

C32⋊C9.S3 in GAP, Magma, Sage, TeX

`C_3^2\rtimes C_9.S_3`
`% in TeX`

`G:=Group("C3^2:C9.S3");`
`// GroupNames label`

`G:=SmallGroup(486,5);`
`// by ID`

`G=gap.SmallGroup(486,5);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,224,338,4755,873,735,3244]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^9=1,d^6=b,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b*c^3,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^5>;`
`// generators/relations`

Export

׿
×
𝔽