Copied to
clipboard

## G = C33.S3order 162 = 2·34

### 4th non-split extension by C33 of S3 acting faithfully

Aliases: C33.4S3, 3- 1+23S3, C9⋊(C3×S3), C3⋊(C9⋊C6), C9⋊S34C3, (C3×C9)⋊5C6, C32.5(C3⋊S3), C32.18(C3×S3), (C3×3- 1+2)⋊2C2, C3.3(C3×C3⋊S3), SmallGroup(162,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C33.S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — C33.S3
 Lower central C3×C9 — C33.S3
 Upper central C1

Generators and relations for C33.S3
G = < a,b,c,d,e | a3=b3=c3=e2=1, d3=c, ab=ba, ac=ca, dad-1=ac-1, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=c-1d2 >

Character table of C33.S3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I size 1 27 2 2 2 2 3 3 6 6 27 27 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 trivial ρ2 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 1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 ζ3 1 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 linear of order 6 ρ4 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 linear of order 3 ρ5 1 -1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 ζ32 1 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 linear of order 6 ρ6 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 linear of order 3 ρ7 2 0 2 2 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 -1 -1 -1 2 2 2 -1 -1 0 0 -1 -1 -1 -1 2 -1 2 -1 2 orthogonal lifted from S3 ρ9 2 0 -1 -1 -1 2 2 2 -1 -1 0 0 -1 -1 2 2 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ10 2 0 -1 -1 -1 2 2 2 -1 -1 0 0 2 2 -1 -1 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ11 2 0 2 2 2 2 -1-√-3 -1+√-3 -1-√-3 -1+√-3 0 0 ζ65 -1 -1 ζ6 ζ6 ζ6 -1 ζ65 ζ65 complex lifted from C3×S3 ρ12 2 0 2 2 2 2 -1+√-3 -1-√-3 -1+√-3 -1-√-3 0 0 ζ6 -1 -1 ζ65 ζ65 ζ65 -1 ζ6 ζ6 complex lifted from C3×S3 ρ13 2 0 -1 -1 -1 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 ζ6 -1 2 -1+√-3 ζ65 ζ65 -1 -1-√-3 ζ6 complex lifted from C3×S3 ρ14 2 0 -1 -1 -1 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 -1+√-3 2 -1 ζ6 ζ6 -1-√-3 -1 ζ65 ζ65 complex lifted from C3×S3 ρ15 2 0 -1 -1 -1 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 ζ65 -1 -1 ζ6 -1-√-3 ζ6 2 ζ65 -1+√-3 complex lifted from C3×S3 ρ16 2 0 -1 -1 -1 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 ζ65 -1 2 -1-√-3 ζ6 ζ6 -1 -1+√-3 ζ65 complex lifted from C3×S3 ρ17 2 0 -1 -1 -1 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 -1-√-3 2 -1 ζ65 ζ65 -1+√-3 -1 ζ6 ζ6 complex lifted from C3×S3 ρ18 2 0 -1 -1 -1 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 ζ6 -1 -1 ζ65 -1+√-3 ζ65 2 ζ6 -1-√-3 complex lifted from C3×S3 ρ19 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ20 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ21 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6

Permutation representations of C33.S3
On 27 points - transitive group 27T58
Generators in S27
```(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)
(1 14 26)(2 15 27)(3 16 19)(4 17 20)(5 18 21)(6 10 22)(7 11 23)(8 12 24)(9 13 25)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)
(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)
(2 9)(3 8)(4 7)(5 6)(10 21)(11 20)(12 19)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)```

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

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

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

`G:=TransitiveGroup(27,58);`

Matrix representation of C33.S3 in GL8(𝔽19)

 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 18 18
,
 17 3 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 1 18 18 0 0 0 0 18 0 0 0 0 1 0 0 0 1 0 0 18 18
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 1 18 18 0 0 0 0 18 0 0 0 0 1 0 0 0 1 0 0 18 18
,
 1 16 0 0 0 0 0 0 1 17 0 0 0 0 0 0 0 0 1 1 0 0 18 17 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 18 0 0 0 0 1 0 0 18 0 0 0 0 0 1 0 18
,
 2 16 0 0 0 0 0 0 1 17 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

`G:=sub<GL(8,GF(19))| [7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[17,18,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,18,18,18,0,18,0,0,0,1,0,0,1,0,1,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,18,0,18,0,0,0,1,0,0,1,0,1,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,1,0,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,18,1,0,0,0,0,0,0,17,18,18,18,18,18],[2,1,0,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;`

C33.S3 in GAP, Magma, Sage, TeX

`C_3^3.S_3`
`% in TeX`

`G:=Group("C3^3.S3");`
`// GroupNames label`

`G:=SmallGroup(162,42);`
`// by ID`

`G=gap.SmallGroup(162,42);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-3,-3,992,457,282,723,2704]);`
`// Polycyclic`

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

Export

׿
×
𝔽