Copied to
clipboard

## G = C2×C6order 12 = 22·3

### Abelian group of type [2,6]

Aliases: C2×C6, SmallGroup(12,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6
 Chief series C1 — C3 — C6 — C2×C6
 Lower central C1 — C2×C6
 Upper central 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

Permutation representations of C2×C6
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);

Polynomial with Galois group C2×C6 over ℚ
actionf(x)Disc(f)
12T2x12-x6+1212·318

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

׿
×
𝔽