non-abelian, soluble, monomial
Aliases: C33⋊A4, C32⋊4D6⋊C3, SmallGroup(324,160)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊4D6 — C33⋊A4 |
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 |
►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
| action | f(x) | Disc(f) |
|---|---|---|
| 9T25 | x9+3x8-20x7-63x6+87x5+355x4+147x3-313x2-317x-83 | 76·832·1812·42432 |
| 12T132 | x12-3x11-423x10+1162x9+63891x8-150879x7-4461428x6+9070833x5+148490544x4-264799046x3-2074843638x2+2972896206x+6882575909 | 320·58·172·592·612·1638·4912 |
| 12T133 | x12-3x11-103x10+262x9+2631x8-8429x7-18568x6+92583x5-43306x4-228636x3+380622x2-213274x+35969 | 312·58·292·1636·2272·2915119412075556472 |
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
Export
Subgroup lattice of C33⋊A4 in TeX
Character table of C33⋊A4 in TeX