direct product, abelian, monomial, 2-elementary
Aliases: C2×C6, SmallGroup(12,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C6 |
| C1 — C2×C6 |
| C1 — C2×C6 |
Generators and relations for C2×C6
G = < a,b | a2=b6=1, ab=ba >
Character table of C2×C6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ8 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ3 | ζ6 | ζ65 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ12 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | linear of order 6 |
►Regular action on 12 points - transitive group 12T2
Generators in S12
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)
G:=sub<Sym(12)| (1,11)(2,12)(3,7)(4,8)(5,9)(6,10), (1,2,3,4,5,6)(7,8,9,10,11,12)>;
G:=Group( (1,11)(2,12)(3,7)(4,8)(5,9)(6,10), (1,2,3,4,5,6)(7,8,9,10,11,12) );
G=PermutationGroup([[(1,11),(2,12),(3,7),(4,8),(5,9),(6,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)]])
G:=TransitiveGroup(12,2);
C2×C6 is a maximal subgroup of
C3⋊D4 C3.A4
| action | f(x) | Disc(f) |
|---|---|---|
| 12T2 | x12-x6+1 | 212·318 |
Matrix representation of C2×C6 ►in GL3(𝔽7) generated by
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 2 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(7))| [1,0,0,0,6,0,0,0,6],[2,0,0,0,6,0,0,0,1] >;
C2×C6 in GAP, Magma, Sage, TeX
C_2\times C_6
% in TeX
G:=Group("C2xC6"); // GroupNames label
G:=SmallGroup(12,5);
// by ID
G=gap.SmallGroup(12,5);
# by ID
G:=PCGroup([3,-2,-2,-3]);
// Polycyclic
G:=Group<a,b|a^2=b^6=1,a*b=b*a>;
// generators/relations
Export
Subgroup lattice of C2×C6 in TeX
Character table of C2×C6 in TeX