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G = C6  order 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

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

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

C1
C2
C3
C6

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);
 

C6 is a maximal subgroup of
 Dic3  SL2(𝔽3)  C52⋊C6
 Cp⋊C6, p=1 mod 3: F7  C13⋊C6  C19⋊C6  C31⋊C6  C37⋊C6  C43⋊C6  C61⋊C6  C67⋊C6 ...
C6 is a maximal quotient of
 C52⋊C6
 Cp⋊C6, p=1 mod 3: F7  C13⋊C6  C19⋊C6  C31⋊C6  C37⋊C6  C43⋊C6  C61⋊C6  C67⋊C6 ...

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
 

Export

Subgroup lattice of C6 in TeX
Character table of C6 in TeX

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