non-abelian, soluble, monomial
Aliases: C33⋊2D4, S32⋊S3, C3⋊2S3≀C2, C3⋊S3.1D6, C33⋊C4⋊2C2, C32⋊4D6⋊1C2, C32⋊3(C3⋊D4), (C3×S32)⋊2C2, (C3×C3⋊S3).4C22, SmallGroup(216,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=b-1, eae=b, bc=cb, dbd-1=ebe=a, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 384 in 60 conjugacy classes, 11 normal (all characteristic)
C1, C2, C3, C3, C4, C22, S3, C6, D4, C32, C32, Dic3, D6, C2×C6, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3⋊D4, C33, C32⋊C4, S32, S32, S3×C6, S3×C32, C3×C3⋊S3, C3×C3⋊S3, S3≀C2, C33⋊C4, C3×S32, C32⋊4D6, C33⋊D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S3≀C2, C33⋊D4
Character table of C33⋊D4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | |
| size | 1 | 6 | 9 | 18 | 2 | 4 | 4 | 4 | 4 | 8 | 54 | 6 | 6 | 12 | 12 | 12 | 18 | 36 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 0 | -2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | -2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | 0 | 1 | 1 | 1 | 1 | -2 | -1 | 0 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | 0 | -√-3 | √-3 | √-3 | -√-3 | 0 | 1 | 0 | complex lifted from C3⋊D4 |
| ρ9 | 2 | 0 | -2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | 0 | √-3 | -√-3 | -√-3 | √-3 | 0 | 1 | 0 | complex lifted from C3⋊D4 |
| ρ10 | 4 | 2 | 0 | 0 | 4 | 1 | 1 | 1 | -2 | -2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ11 | 4 | -2 | 0 | 0 | 4 | 1 | 1 | 1 | -2 | -2 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ12 | 4 | 0 | 0 | -2 | 4 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from S3≀C2 |
| ρ13 | 4 | 0 | 0 | 2 | 4 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3≀C2 |
| ρ14 | 4 | 2 | 0 | 0 | -2 | 1 | -1-3√-3/2 | -1+3√-3/2 | -2 | 1 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | complex faithful |
| ρ15 | 4 | -2 | 0 | 0 | -2 | 1 | -1+3√-3/2 | -1-3√-3/2 | -2 | 1 | 0 | 1+√-3 | 1-√-3 | ζ3 | ζ32 | 1 | 0 | 0 | complex faithful |
| ρ16 | 4 | 2 | 0 | 0 | -2 | 1 | -1+3√-3/2 | -1-3√-3/2 | -2 | 1 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | complex faithful |
| ρ17 | 4 | -2 | 0 | 0 | -2 | 1 | -1-3√-3/2 | -1+3√-3/2 | -2 | 1 | 0 | 1-√-3 | 1+√-3 | ζ32 | ζ3 | 1 | 0 | 0 | complex faithful |
| ρ18 | 8 | 0 | 0 | 0 | -4 | -4 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 12 points - transitive group 12T116
Generators in S12
(2 10 5)(4 7 12)
(1 8 9)(3 11 6)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)
G:=sub<Sym(12)| (2,10,5)(4,7,12), (1,8,9)(3,11,6), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)>;
G:=Group( (2,10,5)(4,7,12), (1,8,9)(3,11,6), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11) );
G=PermutationGroup([[(2,10,5),(4,7,12)], [(1,8,9),(3,11,6)], [(1,9,8),(2,5,10),(3,11,6),(4,7,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]])
G:=TransitiveGroup(12,116);
►On 12 points - transitive group 12T120
Generators in S12
(1 9 8)(2 10 5)(3 6 11)(4 7 12)
(1 8 9)(2 10 5)(3 11 6)(4 7 12)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 12)(6 11)(7 10)(8 9)
G:=sub<Sym(12)| (1,9,8)(2,10,5)(3,6,11)(4,7,12), (1,8,9)(2,10,5)(3,11,6)(4,7,12), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,12)(6,11)(7,10)(8,9)>;
G:=Group( (1,9,8)(2,10,5)(3,6,11)(4,7,12), (1,8,9)(2,10,5)(3,11,6)(4,7,12), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,12)(6,11)(7,10)(8,9) );
G=PermutationGroup([[(1,9,8),(2,10,5),(3,6,11),(4,7,12)], [(1,8,9),(2,10,5),(3,11,6),(4,7,12)], [(1,9,8),(2,5,10),(3,11,6),(4,7,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(2,4),(5,12),(6,11),(7,10),(8,9)]])
G:=TransitiveGroup(12,120);
►On 18 points - transitive group 18T103
Generators in S18
(1 18 16)(4 8 10)(6 13 11)
(2 15 17)(3 9 7)(5 14 12)
(1 4 6)(2 5 3)(7 17 12)(8 13 18)(9 15 14)(10 11 16)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 4)(5 6)(7 10)(8 9)(11 12)(13 14)(15 18)(16 17)
G:=sub<Sym(18)| (1,18,16)(4,8,10)(6,13,11), (2,15,17)(3,9,7)(5,14,12), (1,4,6)(2,5,3)(7,17,12)(8,13,18)(9,15,14)(10,11,16), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)>;
G:=Group( (1,18,16)(4,8,10)(6,13,11), (2,15,17)(3,9,7)(5,14,12), (1,4,6)(2,5,3)(7,17,12)(8,13,18)(9,15,14)(10,11,16), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17) );
G=PermutationGroup([[(1,18,16),(4,8,10),(6,13,11)], [(2,15,17),(3,9,7),(5,14,12)], [(1,4,6),(2,5,3),(7,17,12),(8,13,18),(9,15,14),(10,11,16)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14),(15,18),(16,17)]])
G:=TransitiveGroup(18,103);
►On 18 points - transitive group 18T106
Generators in S18
(1 13 11)(2 14 12)(3 16 18)(4 15 17)(5 7 9)(6 10 8)
(1 11 13)(2 14 12)(3 16 18)(4 17 15)(5 7 9)(6 8 10)
(1 6 4)(2 3 5)(7 14 16)(8 17 11)(9 12 18)(10 15 13)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 4)(2 3)(8 10)(11 15)(12 18)(13 17)(14 16)
G:=sub<Sym(18)| (1,13,11)(2,14,12)(3,16,18)(4,15,17)(5,7,9)(6,10,8), (1,11,13)(2,14,12)(3,16,18)(4,17,15)(5,7,9)(6,8,10), (1,6,4)(2,3,5)(7,14,16)(8,17,11)(9,12,18)(10,15,13), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,4)(2,3)(8,10)(11,15)(12,18)(13,17)(14,16)>;
G:=Group( (1,13,11)(2,14,12)(3,16,18)(4,15,17)(5,7,9)(6,10,8), (1,11,13)(2,14,12)(3,16,18)(4,17,15)(5,7,9)(6,8,10), (1,6,4)(2,3,5)(7,14,16)(8,17,11)(9,12,18)(10,15,13), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,4)(2,3)(8,10)(11,15)(12,18)(13,17)(14,16) );
G=PermutationGroup([[(1,13,11),(2,14,12),(3,16,18),(4,15,17),(5,7,9),(6,10,8)], [(1,11,13),(2,14,12),(3,16,18),(4,17,15),(5,7,9),(6,8,10)], [(1,6,4),(2,3,5),(7,14,16),(8,17,11),(9,12,18),(10,15,13)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(1,4),(2,3),(8,10),(11,15),(12,18),(13,17),(14,16)]])
G:=TransitiveGroup(18,106);
►On 24 points - transitive group 24T556
Generators in S24
(1 20 21)(3 23 18)(5 14 10)(7 12 16)
(2 17 22)(4 24 19)(6 15 11)(8 9 13)
(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
G:=sub<Sym(24)| (1,20,21)(3,23,18)(5,14,10)(7,12,16), (2,17,22)(4,24,19)(6,15,11)(8,9,13), (1,21,20)(2,17,22)(3,23,18)(4,19,24)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;
G:=Group( (1,20,21)(3,23,18)(5,14,10)(7,12,16), (2,17,22)(4,24,19)(6,15,11)(8,9,13), (1,21,20)(2,17,22)(3,23,18)(4,19,24)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );
G=PermutationGroup([[(1,20,21),(3,23,18),(5,14,10),(7,12,16)], [(2,17,22),(4,24,19),(6,15,11),(8,9,13)], [(1,21,20),(2,17,22),(3,23,18),(4,19,24),(5,14,10),(6,11,15),(7,16,12),(8,9,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)]])
G:=TransitiveGroup(24,556);
►On 24 points - transitive group 24T560
Generators in S24
(1 20 21)(2 22 17)(3 23 18)(4 19 24)(5 14 10)(6 15 11)(7 12 16)(8 9 13)
(1 20 21)(2 17 22)(3 23 18)(4 24 19)(5 10 14)(6 15 11)(7 16 12)(8 9 13)
(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
G:=sub<Sym(24)| (1,20,21)(2,22,17)(3,23,18)(4,19,24)(5,14,10)(6,15,11)(7,12,16)(8,9,13), (1,20,21)(2,17,22)(3,23,18)(4,24,19)(5,10,14)(6,15,11)(7,16,12)(8,9,13), (1,21,20)(2,17,22)(3,23,18)(4,19,24)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;
G:=Group( (1,20,21)(2,22,17)(3,23,18)(4,19,24)(5,14,10)(6,15,11)(7,12,16)(8,9,13), (1,20,21)(2,17,22)(3,23,18)(4,24,19)(5,10,14)(6,15,11)(7,16,12)(8,9,13), (1,21,20)(2,17,22)(3,23,18)(4,19,24)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );
G=PermutationGroup([[(1,20,21),(2,22,17),(3,23,18),(4,19,24),(5,14,10),(6,15,11),(7,12,16),(8,9,13)], [(1,20,21),(2,17,22),(3,23,18),(4,24,19),(5,10,14),(6,15,11),(7,16,12),(8,9,13)], [(1,21,20),(2,17,22),(3,23,18),(4,19,24),(5,14,10),(6,11,15),(7,16,12),(8,9,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)]])
G:=TransitiveGroup(24,560);
►On 27 points - transitive group 27T81
Generators in S27
(1 5 7)(2 23 21)(3 10 8)(4 13 12)(6 14 15)(9 18 25)(11 27 16)(17 22 26)(19 24 20)
(1 6 4)(2 11 9)(3 20 22)(5 14 13)(7 15 12)(8 24 17)(10 19 26)(16 25 21)(18 23 27)
(1 3 2)(4 22 9)(5 10 23)(6 20 11)(7 8 21)(12 17 25)(13 26 18)(14 19 27)(15 24 16)
(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 3)(4 7)(5 6)(8 9)(10 11)(13 15)(16 26)(17 25)(18 24)(19 27)(20 23)(21 22)
G:=sub<Sym(27)| (1,5,7)(2,23,21)(3,10,8)(4,13,12)(6,14,15)(9,18,25)(11,27,16)(17,22,26)(19,24,20), (1,6,4)(2,11,9)(3,20,22)(5,14,13)(7,15,12)(8,24,17)(10,19,26)(16,25,21)(18,23,27), (1,3,2)(4,22,9)(5,10,23)(6,20,11)(7,8,21)(12,17,25)(13,26,18)(14,19,27)(15,24,16), (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), (2,3)(4,7)(5,6)(8,9)(10,11)(13,15)(16,26)(17,25)(18,24)(19,27)(20,23)(21,22)>;
G:=Group( (1,5,7)(2,23,21)(3,10,8)(4,13,12)(6,14,15)(9,18,25)(11,27,16)(17,22,26)(19,24,20), (1,6,4)(2,11,9)(3,20,22)(5,14,13)(7,15,12)(8,24,17)(10,19,26)(16,25,21)(18,23,27), (1,3,2)(4,22,9)(5,10,23)(6,20,11)(7,8,21)(12,17,25)(13,26,18)(14,19,27)(15,24,16), (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), (2,3)(4,7)(5,6)(8,9)(10,11)(13,15)(16,26)(17,25)(18,24)(19,27)(20,23)(21,22) );
G=PermutationGroup([[(1,5,7),(2,23,21),(3,10,8),(4,13,12),(6,14,15),(9,18,25),(11,27,16),(17,22,26),(19,24,20)], [(1,6,4),(2,11,9),(3,20,22),(5,14,13),(7,15,12),(8,24,17),(10,19,26),(16,25,21),(18,23,27)], [(1,3,2),(4,22,9),(5,10,23),(6,20,11),(7,8,21),(12,17,25),(13,26,18),(14,19,27),(15,24,16)], [(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)], [(2,3),(4,7),(5,6),(8,9),(10,11),(13,15),(16,26),(17,25),(18,24),(19,27),(20,23),(21,22)]])
G:=TransitiveGroup(27,81);
C33⋊D4 is a maximal subgroup of
S3×S3≀C2
C33⋊D4 is a maximal quotient of C3⋊S3.2D12 S32⋊Dic3 C33⋊C4⋊C4 C33⋊D8 C33⋊6SD16 C33⋊7SD16 C33⋊Q16
| action | f(x) | Disc(f) |
|---|---|---|
| 12T116 | x12-3x9+3x6-3x3+3 | 323·73 |
| 12T120 | x12-2x9-12x6-8x3-2 | -220·321·76 |
Matrix representation of C33⋊D4 ►in GL4(𝔽7) generated by
| 3 | 2 | 4 | 3 |
| 4 | 5 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 1 |
| 6 | 2 | 1 | 1 |
| 2 | 6 | 6 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 |
| 3 | 1 | 4 | 5 |
| 1 | 3 | 3 | 5 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 5 | 1 | 5 |
| 1 | 5 | 2 | 1 |
| 6 | 1 | 4 | 6 |
| 1 | 1 | 3 | 5 |
| 6 | 3 | 3 | 0 |
| 4 | 4 | 6 | 2 |
| 3 | 4 | 2 | 5 |
| 6 | 6 | 4 | 2 |
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[0,1,6,1,5,5,1,1,1,2,4,3,5,1,6,5],[6,4,3,6,3,4,4,6,3,6,2,4,0,2,5,2] >;
C33⋊D4 in GAP, Magma, Sage, TeX
C_3^3\rtimes D_4
% in TeX
G:=Group("C3^3:D4"); // GroupNames label
G:=SmallGroup(216,158);
// by ID
G=gap.SmallGroup(216,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,3,-3,73,579,201,111,244,376,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=b^-1,e*a*e=b,b*c=c*b,d*b*d^-1=e*b*e=a,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
Export