direct product, abelian, monomial, 3-elementary
Aliases: C3×C6, SmallGroup(18,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C6 |
| C1 — C3×C6 |
| C1 — C3×C6 |
Generators and relations for C3×C6
G = < a,b | a3=b6=1, ab=ba >
Character table of C3×C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 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 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ6 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | -1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ65 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | linear of order 3 |
| ρ8 | 1 | -1 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ65 | linear of order 6 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | linear of order 6 |
| ρ11 | 1 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ12 | 1 | -1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | -1 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ13 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | linear of order 3 |
| ρ14 | 1 | -1 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ6 | linear of order 6 |
| ρ15 | 1 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ16 | 1 | -1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | -1 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ17 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ18 | 1 | -1 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | linear of order 6 |
►Regular action on 18 points - transitive group 18T2
Generators in S18
(1 12 15)(2 7 16)(3 8 17)(4 9 18)(5 10 13)(6 11 14)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
G:=sub<Sym(18)| (1,12,15)(2,7,16)(3,8,17)(4,9,18)(5,10,13)(6,11,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
G:=Group( (1,12,15)(2,7,16)(3,8,17)(4,9,18)(5,10,13)(6,11,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
G=PermutationGroup([[(1,12,15),(2,7,16),(3,8,17),(4,9,18),(5,10,13),(6,11,14)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)]])
G:=TransitiveGroup(18,2);
C3×C6 is a maximal subgroup of
C3⋊Dic3
Matrix representation of C3×C6 ►in GL2(𝔽7) generated by
| 4 | 0 |
| 0 | 4 |
| 6 | 0 |
| 0 | 3 |
G:=sub<GL(2,GF(7))| [4,0,0,4],[6,0,0,3] >;
C3×C6 in GAP, Magma, Sage, TeX
C_3\times C_6
% in TeX
G:=Group("C3xC6"); // GroupNames label
G:=SmallGroup(18,5);
// by ID
G=gap.SmallGroup(18,5);
# by ID
G:=PCGroup([3,-2,-3,-3]);
// Polycyclic
G:=Group<a,b|a^3=b^6=1,a*b=b*a>;
// generators/relations
Export
Subgroup lattice of C3×C6 in TeX
Character table of C3×C6 in TeX