Copied to
clipboard

## G = C92⋊2S3order 486 = 2·35

### 2nd semidirect product of C92 and S3 acting faithfully

Aliases: C922S3, He3⋊C3⋊S3, C922C33C2, C3.6(He3⋊S3), C32.1(He3⋊C2), (C3×C9).1(C3⋊S3), SmallGroup(486,61)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C92⋊2C3 — C92⋊2S3
 Chief series C1 — C3 — C32 — C3×C9 — He3⋊C3 — C92⋊2C3 — C92⋊2S3
 Lower central C92⋊2C3 — C92⋊2S3
 Upper central C1 — C3

Generators and relations for C922S3
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, cac-1=a7b-1, dad=a-1b-1, cbc-1=a3b, bd=db, dcd=c-1 >

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

Permutation representations of C922S3
On 27 points - transitive group 27T179
Generators in S27
```(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 9 2 5 8 6 4 3 7)(10 17 15 13 11 18 16 14 12)(19 27 26 25 24 23 22 21 20)
(1 27 15)(2 25 17)(3 20 13)(4 21 12)(5 24 18)(6 22 11)(7 19 14)(8 23 10)(9 26 16)
(1 27)(2 25)(3 20)(4 21)(5 24)(6 22)(7 19)(8 23)(9 26)```

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

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

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

`G:=TransitiveGroup(27,179);`

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A ··· 9F 9G ··· 9O 18A ··· 18F order 1 2 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 6 54 54 54 27 27 3 ··· 3 6 ··· 6 27 ··· 27

31 irreducible representations

 dim 1 1 2 2 3 3 6 6 type + + + + + image C1 C2 S3 S3 He3⋊C2 C92⋊2S3 He3⋊S3 C92⋊2S3 kernel C92⋊2S3 C92⋊2C3 C92 He3⋊C3 C32 C1 C3 C1 # reps 1 1 1 3 4 12 3 6

Matrix representation of C922S3 in GL3(𝔽19) generated by

 6 0 0 0 1 0 0 0 16
,
 9 0 0 0 6 0 0 0 6
,
 0 0 1 1 0 0 0 1 0
,
 1 0 0 0 0 1 0 1 0
`G:=sub<GL(3,GF(19))| [6,0,0,0,1,0,0,0,16],[9,0,0,0,6,0,0,0,6],[0,1,0,0,0,1,1,0,0],[1,0,0,0,0,1,0,1,0] >;`

C922S3 in GAP, Magma, Sage, TeX

`C_9^2\rtimes_2S_3`
`% in TeX`

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

`G:=SmallGroup(486,61);`
`// by ID`

`G=gap.SmallGroup(486,61);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,224,873,1167,453,8104,3250,1906]);`
`// 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^7*b^-1,d*a*d=a^-1*b^-1,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽