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## G = C5order 5

### Cyclic group

Aliases: C5, also denoted Z5, rotations of a regular pentagon, SmallGroup(5,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C5
 Chief series C1 — C5
 Lower central C1 — C5
 Upper central C1 — C5
 Jennings C1 — C5

Generators and relations for C5
G = < a | a5=1 >

Character table of C5

 class 1 5A 5B 5C 5D size 1 1 1 1 1 ρ1 1 1 1 1 1 trivial ρ2 1 ζ5 ζ52 ζ53 ζ54 linear of order 5 faithful ρ3 1 ζ52 ζ54 ζ5 ζ53 linear of order 5 faithful ρ4 1 ζ53 ζ5 ζ54 ζ52 linear of order 5 faithful ρ5 1 ζ54 ζ53 ζ52 ζ5 linear of order 5 faithful

Permutation representations of C5
Regular action on 5 points - transitive group 5T1
Generators in S5
`(1 2 3 4 5)`

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

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

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

`G:=TransitiveGroup(5,1);`

C5 is a maximal subgroup of
D5  C25  C24⋊C5  C34⋊C5
Cp⋊C5, p=1 mod 5: C11⋊C5  C31⋊C5  C41⋊C5  C61⋊C5  C71⋊C5 ...
C5 is a maximal quotient of
C25  C24⋊C5  C34⋊C5
Cp⋊C5, p=1 mod 5: C11⋊C5  C31⋊C5  C41⋊C5  C61⋊C5  C71⋊C5 ...

Polynomial with Galois group C5 over ℚ
actionf(x)Disc(f)
5T1x5-10x3-5x2+10x-158·72

Matrix representation of C5 in GL1(𝔽11) generated by

 4
`G:=sub<GL(1,GF(11))| [4] >;`

C5 in GAP, Magma, Sage, TeX

`C_5`
`% in TeX`

`G:=Group("C5");`
`// GroupNames label`

`G:=SmallGroup(5,1);`
`// by ID`

`G=gap.SmallGroup(5,1);`
`# by ID`

`G:=PCGroup([1,-5]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a|a^5=1>;`
`// generators/relations`

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