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## G = C33⋊A4order 324 = 22·34

### The semidirect product of C33 and A4 acting faithfully

Aliases: C33⋊A4, C324D6⋊C3, SmallGroup(324,160)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C32⋊4D6 — C33⋊A4
 Chief series C1 — C33 — C32⋊4D6 — C33⋊A4
 Lower central C32⋊4D6 — C33⋊A4
 Upper central C1

Generators and relations for C33⋊A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, ac=ca, dad=a-1, ae=ea, faf-1=c, bc=cb, dbd=ebe=b-1, fbf-1=a, cd=dc, ece=c-1, fcf-1=b, fdf-1=de=ed, fef-1=d >

27C2
3C3
4C3
6C3
36C3
27C22
9S3
9S3
18S3
27C6
3C32
4C32
6C32
12C9
12C32
12C9
27D6
27A4
18C3×S3
4He3
9S32

Character table of C33⋊A4

 class 1 2 3A 3B 3C 3D 3E 3F 6 9A 9B 9C 9D size 1 27 4 4 6 12 36 36 54 36 36 36 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 ζ3 ζ32 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ3 1 1 1 1 1 1 ζ32 ζ3 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 3 -1 3 3 3 3 0 0 -1 0 0 0 0 orthogonal lifted from A4 ρ5 4 0 -1-3√-3/2 -1+3√-3/2 -2 1 ζ3 ζ32 0 ζ3 1 ζ32 1 complex faithful ρ6 4 0 -1+3√-3/2 -1-3√-3/2 -2 1 ζ32 ζ3 0 ζ32 1 ζ3 1 complex faithful ρ7 4 0 -1+3√-3/2 -1-3√-3/2 -2 1 ζ3 ζ32 0 1 ζ3 1 ζ32 complex faithful ρ8 4 0 -1-3√-3/2 -1+3√-3/2 -2 1 1 1 0 ζ32 ζ3 ζ3 ζ32 complex faithful ρ9 4 0 -1+3√-3/2 -1-3√-3/2 -2 1 1 1 0 ζ3 ζ32 ζ32 ζ3 complex faithful ρ10 4 0 -1-3√-3/2 -1+3√-3/2 -2 1 ζ32 ζ3 0 1 ζ32 1 ζ3 complex faithful ρ11 6 2 -3 -3 3 0 0 0 -1 0 0 0 0 orthogonal faithful ρ12 6 -2 -3 -3 3 0 0 0 1 0 0 0 0 orthogonal faithful ρ13 12 0 3 3 0 -3 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C33⋊A4
On 9 points - transitive group 9T25
Generators in S9
```(7 8 9)
(4 6 5)
(1 3 2)
(5 6)(8 9)
(2 3)(5 6)
(1 7 4)(2 9 5)(3 8 6)```

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

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

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

`G:=TransitiveGroup(9,25);`

On 12 points - transitive group 12T132
Generators in S12
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)
(1 4)(2 6)(3 5)(7 11)(8 10)(9 12)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)
(1 3 2)(4 7 12)(5 9 10)(6 8 11)```

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

`G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,2,3)(4,5,6)(7,9,8)(10,12,11), (1,2,3)(4,6,5)(7,9,8)(10,11,12), (1,4)(2,6)(3,5)(7,11)(8,10)(9,12), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12), (1,3,2)(4,7,12)(5,9,10)(6,8,11) );`

`G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,2,3),(4,5,6),(7,9,8),(10,12,11)], [(1,2,3),(4,6,5),(7,9,8),(10,11,12)], [(1,4),(2,6),(3,5),(7,11),(8,10),(9,12)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12)], [(1,3,2),(4,7,12),(5,9,10),(6,8,11)]])`

`G:=TransitiveGroup(12,132);`

On 12 points - transitive group 12T133
Generators in S12
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)
(1 4)(2 6)(3 5)(7 11)(8 10)(9 12)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)
(4 8 10)(5 7 11)(6 9 12)```

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

`G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,2,3)(4,5,6)(7,9,8)(10,12,11), (1,2,3)(4,6,5)(7,9,8)(10,11,12), (1,4)(2,6)(3,5)(7,11)(8,10)(9,12), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12), (4,8,10)(5,7,11)(6,9,12) );`

`G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,2,3),(4,5,6),(7,9,8),(10,12,11)], [(1,2,3),(4,6,5),(7,9,8),(10,11,12)], [(1,4),(2,6),(3,5),(7,11),(8,10),(9,12)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12)], [(4,8,10),(5,7,11),(6,9,12)]])`

`G:=TransitiveGroup(12,133);`

On 18 points - transitive group 18T141
Generators in S18
```(13 14 15)(16 17 18)
(1 4 5)(2 9 3)
(6 7 8)(10 11 12)
(1 3)(2 5)(4 9)(6 10)(7 11)(8 12)(14 15)(17 18)
(1 5)(2 3)(6 12)(7 11)(8 10)(13 16)(14 17)(15 18)
(1 6 15)(2 10 18)(3 12 17)(4 7 13)(5 8 14)(9 11 16)```

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

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

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

`G:=TransitiveGroup(18,141);`

On 18 points - transitive group 18T142
Generators in S18
```(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 3 2)(4 5 6)(7 8 9)(10 12 11)
(1 3 2)(4 6 5)(13 14 15)(16 18 17)
(1 4)(2 5)(3 6)(8 9)(11 12)(13 18)(14 17)(15 16)
(2 3)(5 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 14 7)(2 13 9)(3 15 8)(4 17 10)(5 16 11)(6 18 12)```

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

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

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

`G:=TransitiveGroup(18,142);`

On 18 points - transitive group 18T143
Generators in S18
```(13 14 15)(16 17 18)
(1 2 6)(3 4 5)
(7 9 8)(10 11 12)
(1 4)(2 3)(5 6)(13 17)(14 16)(15 18)
(1 4)(2 3)(5 6)(7 10)(8 11)(9 12)
(1 11 17)(2 12 18)(3 9 15)(4 8 13)(5 7 14)(6 10 16)```

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

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

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

`G:=TransitiveGroup(18,143);`

On 27 points - transitive group 27T130
Generators in S27
```(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 7 4)(2 8 5)(3 9 6)(10 15 19)(11 13 20)(12 14 21)
(1 3 2)(4 6 5)(7 9 8)(16 27 23)(17 25 24)(18 26 22)
(4 7)(5 8)(6 9)(11 12)(13 21)(14 20)(15 19)(16 17)(23 24)(25 27)
(2 3)(4 7)(5 9)(6 8)(13 20)(14 21)(15 19)(22 26)(23 27)(24 25)
(1 18 10)(2 17 19)(3 16 15)(4 22 12)(5 24 21)(6 23 14)(7 26 11)(8 25 20)(9 27 13)```

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

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

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

`G:=TransitiveGroup(27,130);`

On 27 points: primitive - transitive group 27T131
Generators in S27
```(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)
(1 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)
(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)
(2 3)(4 13)(5 15)(6 14)(7 9)(10 20)(11 19)(12 21)(16 23)(17 22)(18 24)(25 27)
(4 15)(5 13)(6 14)(7 25)(8 26)(9 27)(10 23)(11 24)(12 22)(16 20)(17 21)(18 19)
(2 6 26)(3 14 8)(4 21 27)(5 12 9)(7 13 17)(10 23 20)(11 18 24)(15 22 25)```

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

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

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

`G:=TransitiveGroup(27,131);`

Polynomial with Galois group C33⋊A4 over ℚ
actionf(x)Disc(f)
9T25x9+3x8-20x7-63x6+87x5+355x4+147x3-313x2-317x-8376·832·1812·42432
12T132x12-3x11-423x10+1162x9+63891x8-150879x7-4461428x6+9070833x5+148490544x4-264799046x3-2074843638x2+2972896206x+6882575909320·58·172·592·612·1638·4912
12T133x12-3x11-103x10+262x9+2631x8-8429x7-18568x6+92583x5-43306x4-228636x3+380622x2-213274x+35969312·58·292·1636·2272·2915119412075556472

Matrix representation of C33⋊A4 in GL4(𝔽7) generated by

 1 4 5 4 6 3 4 0 3 3 6 1 0 0 0 2
,
 3 1 4 5 1 3 3 5 0 0 4 0 0 0 0 2
,
 6 2 5 1 0 4 0 5 4 4 0 6 0 0 0 2
,
 2 4 5 4 3 6 1 5 1 6 3 6 2 2 5 3
,
 1 5 2 6 6 6 5 6 3 4 2 2 1 1 3 5
,
 4 5 2 0 0 0 6 2 0 2 3 5 0 0 0 2
`G:=sub<GL(4,GF(7))| [1,6,3,0,4,3,3,0,5,4,6,0,4,0,1,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[6,0,4,0,2,4,4,0,5,0,0,0,1,5,6,2],[2,3,1,2,4,6,6,2,5,1,3,5,4,5,6,3],[1,6,3,1,5,6,4,1,2,5,2,3,6,6,2,5],[4,0,0,0,5,0,2,0,2,6,3,0,0,2,5,2] >;`

C33⋊A4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes A_4`
`% in TeX`

`G:=Group("C3^3:A4");`
`// GroupNames label`

`G:=SmallGroup(324,160);`
`// by ID`

`G=gap.SmallGroup(324,160);`
`# by ID`

`G:=PCGroup([6,-3,-2,2,-3,3,3,109,56,867,111,3244,730,376,437,2603]);`
`// Polycyclic`

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

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