direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C6, C6⋊C6, C32⋊2C22, C3⋊(C2×C6), (C3×C6)⋊1C2, SmallGroup(36,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C6 |
Generators and relations for S3×C6
G = < a,b,c | a6=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of S3×C6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | |
| 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 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ8 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ9 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ11 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ12 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | linear of order 6 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 1+√-3 | 1-√-3 | ζ3 | 1 | ζ32 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ17 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 1-√-3 | 1+√-3 | ζ32 | 1 | ζ3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
►On 12 points - transitive group 12T18
Generators in S12
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)
G:=sub<Sym(12)| (1,2,3,4,5,6)(7,8,9,10,11,12), (1,3,5)(2,4,6)(7,11,9)(8,12,10), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12), (1,3,5)(2,4,6)(7,11,9)(8,12,10), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10)]])
G:=TransitiveGroup(12,18);
►On 18 points - transitive group 18T6
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 8 14)(2 9 15)(3 10 16)(4 11 17)(5 12 18)(6 7 13)
(1 4)(2 5)(3 6)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)
G:=sub<Sym(18)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,8,14)(2,9,15)(3,10,16)(4,11,17)(5,12,18)(6,7,13), (1,4)(2,5)(3,6)(7,16)(8,17)(9,18)(10,13)(11,14)(12,15)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,8,14)(2,9,15)(3,10,16)(4,11,17)(5,12,18)(6,7,13), (1,4)(2,5)(3,6)(7,16)(8,17)(9,18)(10,13)(11,14)(12,15) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,8,14),(2,9,15),(3,10,16),(4,11,17),(5,12,18),(6,7,13)], [(1,4),(2,5),(3,6),(7,16),(8,17),(9,18),(10,13),(11,14),(12,15)]])
G:=TransitiveGroup(18,6);
S3×C6 is a maximal subgroup of
D6⋊S3 C3⋊D12
| action | f(x) | Disc(f) |
|---|---|---|
| 12T18 | x12-2x11+6x9+x8-8x7+3x6+2x4-6x3+7x2-4x+1 | 212·38·56·5092 |
Matrix representation of S3×C6 ►in GL2(𝔽7) generated by
| 5 | 0 |
| 0 | 5 |
| 4 | 0 |
| 2 | 2 |
| 6 | 1 |
| 0 | 1 |
G:=sub<GL(2,GF(7))| [5,0,0,5],[4,2,0,2],[6,0,1,1] >;
S3×C6 in GAP, Magma, Sage, TeX
S_3\times C_6
% in TeX
G:=Group("S3xC6"); // GroupNames label
G:=SmallGroup(36,12);
// by ID
G=gap.SmallGroup(36,12);
# by ID
G:=PCGroup([4,-2,-2,-3,-3,387]);
// Polycyclic
G:=Group<a,b,c|a^6=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of S3×C6 in TeX
Character table of S3×C6 in TeX