Copied to
clipboard

G = C6order 6 = 2·3

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C6, also denoted Z6, rotations of a regular hexagon, SmallGroup(6,2)

Series: Derived Chief Lower central Upper central

C1 — C6
C1C3 — C6
C1 — C6
C1 — C6

Generators and relations for C6
 G = < a | a6=1 >


Character table of C6

 class 123A3B6A6B
 size 111111
ρ1111111    trivial
ρ21-111-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ32    linear of order 3
ρ41-1ζ32ζ3ζ65ζ6    linear of order 6 faithful
ρ511ζ3ζ32ζ32ζ3    linear of order 3
ρ61-1ζ3ζ32ζ6ζ65    linear of order 6 faithful

Permutation representations of C6
Regular action on 6 points - transitive group 6T1
Generators in S6
(1 2 3 4 5 6)

G:=sub<Sym(6)| (1,2,3,4,5,6)>;

G:=Group( (1,2,3,4,5,6) );

G=PermutationGroup([(1,2,3,4,5,6)])

G:=TransitiveGroup(6,1);

Polynomial with Galois group C6 over ℚ
actionf(x)Disc(f)
6T1x6+x3+1-39

Matrix representation of C6 in GL1(𝔽7) generated by

3
G:=sub<GL(1,GF(7))| [3] >;

C6 in GAP, Magma, Sage, TeX

C_6
% in TeX

G:=Group("C6");
// GroupNames label

G:=SmallGroup(6,2);
// by ID

G=gap.SmallGroup(6,2);
# by ID

G:=PCGroup([2,-2,-3]);
// Polycyclic

G:=Group<a|a^6=1>;
// generators/relations

׿
×
𝔽