Copied to
clipboard

## G = C3order 3

### Cyclic group

Aliases: C3, A3, also denoted Z3, Alt3, rotations of a regular triangle, SmallGroup(3,1)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C3
G = < a | a3=1 >

Character table of C3

 class 1 3A 3B size 1 1 1 ρ1 1 1 1 trivial ρ2 1 ζ3 ζ32 linear of order 3 faithful ρ3 1 ζ32 ζ3 linear of order 3 faithful

Permutation representations of C3
Regular action on 3 points - transitive group 3T1
Generators in S3
`(1 2 3)`

`G:=sub<Sym(3)| (1,2,3)>;`

`G:=Group( (1,2,3) );`

`G=PermutationGroup([[(1,2,3)]])`

`G:=TransitiveGroup(3,1);`

C3 is a maximal subgroup of
S3  C9  A4  C52⋊C3  C112⋊C3
Cp⋊C3, p=1 mod 3: C7⋊C3  C13⋊C3  C19⋊C3  C31⋊C3  C37⋊C3  C43⋊C3  C61⋊C3  C67⋊C3 ...
C3 is a maximal quotient of
C9  A4  C52⋊C3  C112⋊C3
Cp⋊C3, p=1 mod 3: C7⋊C3  C13⋊C3  C19⋊C3  C31⋊C3  C37⋊C3  C43⋊C3  C61⋊C3  C67⋊C3 ...

Polynomial with Galois group C3 over ℚ
actionf(x)Disc(f)
3T1x3-3x+134

Matrix representation of C3 in GL1(𝔽7) generated by

 4
`G:=sub<GL(1,GF(7))| [4] >;`

C3 in GAP, Magma, Sage, TeX

`C_3`
`% in TeX`

`G:=Group("C3");`
`// GroupNames label`

`G:=SmallGroup(3,1);`
`// by ID`

`G=gap.SmallGroup(3,1);`
`# by ID`

`G:=PCGroup([1,-3]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a|a^3=1>;`
`// generators/relations`

Export

׿
×
𝔽