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## G = C33⋊C6order 162 = 2·34

### 1st semidirect product of C33 and C6 acting faithfully

Aliases: He31S3, C331C6, C3≀C33C2, C33⋊C21C3, C32.6(C3×S3), C3.2(C32⋊C6), SmallGroup(162,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊C6
 Chief series C1 — C3 — C32 — C33 — C3≀C3 — C33⋊C6
 Lower central C33 — C33⋊C6
 Upper central C1

Generators and relations for C33⋊C6
G = < a,b,c,d | a3=b3=c3=d6=1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

27C2
3C3
3C3
3C3
3C3
9C3
9S3
27S3
27S3
27S3
27S3
27C6
3C32
3C32
3C32
3C32
3C32
6C9

Character table of C33⋊C6

 class 1 2 3A 3B 3C 3D 3E 3F 3G 6A 6B 9A 9B size 1 27 2 6 6 6 6 9 9 27 27 18 18 ρ1 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 linear of order 2 ρ3 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ5 1 -1 1 1 1 1 1 ζ3 ζ32 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ6 1 -1 1 1 1 1 1 ζ32 ζ3 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ7 2 0 2 -1 -1 -1 2 2 2 0 0 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 -1 -1 -1 2 -1+√-3 -1-√-3 0 0 ζ6 ζ65 complex lifted from C3×S3 ρ9 2 0 2 -1 -1 -1 2 -1-√-3 -1+√-3 0 0 ζ65 ζ6 complex lifted from C3×S3 ρ10 6 0 -3 0 3 -3 0 0 0 0 0 0 0 orthogonal faithful ρ11 6 0 -3 -3 0 3 0 0 0 0 0 0 0 orthogonal faithful ρ12 6 0 6 0 0 0 -3 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ13 6 0 -3 3 -3 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C33⋊C6
On 9 points - transitive group 9T22
Generators in S9
```(1 4 7)(2 8 5)
(2 5 8)(3 6 9)
(1 7 4)(2 5 8)(3 9 6)
(1 2 3)(4 5 6 7 8 9)```

`G:=sub<Sym(9)| (1,4,7)(2,8,5), (2,5,8)(3,6,9), (1,7,4)(2,5,8)(3,9,6), (1,2,3)(4,5,6,7,8,9)>;`

`G:=Group( (1,4,7)(2,8,5), (2,5,8)(3,6,9), (1,7,4)(2,5,8)(3,9,6), (1,2,3)(4,5,6,7,8,9) );`

`G=PermutationGroup([[(1,4,7),(2,8,5)], [(2,5,8),(3,6,9)], [(1,7,4),(2,5,8),(3,9,6)], [(1,2,3),(4,5,6,7,8,9)]])`

`G:=TransitiveGroup(9,22);`

On 18 points - transitive group 18T85
Generators in S18
```(1 8 14)(2 15 9)(4 17 11)(5 12 18)
(2 9 15)(3 10 16)(5 18 12)(6 13 7)
(1 14 8)(2 9 15)(3 16 10)(4 11 17)(5 18 12)(6 7 13)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,85);`

On 27 points - transitive group 27T53
Generators in S27
```(1 13 10)(3 5 8)(6 20 24)(9 27 17)(12 19 26)(15 23 16)
(1 10 13)(2 7 4)(3 16 19)(5 15 26)(6 24 20)(8 23 12)(9 17 27)(11 25 18)(14 21 22)
(1 27 24)(2 25 22)(3 23 26)(4 11 21)(5 16 12)(6 13 17)(7 18 14)(8 15 19)(9 20 10)
(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)```

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

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

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

`G:=TransitiveGroup(27,53);`

On 27 points - transitive group 27T62
Generators in S27
```(1 22 25)(2 17 12)(3 15 20)(4 21 27)(5 19 23)(6 13 14)(7 24 18)(8 26 16)(9 11 10)
(1 7 4)(2 5 6)(3 9 8)(10 16 20)(11 26 15)(12 23 14)(13 17 19)(18 27 25)(21 22 24)
(1 3 2)(4 8 6)(5 7 9)(10 23 18)(11 19 24)(12 25 20)(13 21 26)(14 27 16)(15 17 22)
(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)```

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

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

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

`G:=TransitiveGroup(27,62);`

On 27 points - transitive group 27T63
Generators in S27
```(1 8 5)(2 6 9)(3 19 16)(4 14 23)(7 26 11)(10 22 21)(12 24 17)(13 18 25)(15 20 27)
(2 22 25)(3 23 26)(4 11 19)(6 21 13)(7 16 14)(9 10 18)
(1 24 27)(2 22 25)(3 26 23)(4 19 11)(5 12 20)(6 21 13)(7 14 16)(8 17 15)(9 10 18)
(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)```

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

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

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

`G:=TransitiveGroup(27,63);`

C33⋊C6 is a maximal subgroup of   He3⋊D6  (C3×He3)⋊C6  C345C6  C3≀C3.S3
C33⋊C6 is a maximal quotient of   C33⋊C12  C3.C3≀S3  C32⋊C9⋊S3  (C3×He3).C6  C331C18  He3⋊D9  C345C6

Polynomial with Galois group C33⋊C6 over ℚ
actionf(x)Disc(f)
9T22x9-30x7-20x6+279x5+372x4-659x3-1566x2-1044x-232224·39·77·292

Matrix representation of C33⋊C6 in GL6(ℤ)

 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0
,
 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1
,
 0 0 0 0 1 0 0 0 0 0 -1 -1 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 0 0

`G:=sub<GL(6,Integers())| [-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0] >;`

C33⋊C6 in GAP, Magma, Sage, TeX

`C_3^3\rtimes C_6`
`% in TeX`

`G:=Group("C3^3:C6");`
`// GroupNames label`

`G:=SmallGroup(162,11);`
`// by ID`

`G=gap.SmallGroup(162,11);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-3,-3,182,187,723,728,2704]);`
`// Polycyclic`

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

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