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G = C5order 5

Cyclic group

p-group, cyclic, elementary abelian, simple, monomial

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

Series: Derived Chief Lower central Upper central Jennings

C1 — C5
C1 — C5
C1 — C5
C1 — C5
C1 — C5

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


Character table of C5

 class 15A5B5C5D
 size 11111
ρ111111    trivial
ρ21ζ5ζ52ζ53ζ54    linear of order 5 faithful
ρ31ζ52ζ54ζ5ζ53    linear of order 5 faithful
ρ41ζ53ζ5ζ54ζ52    linear of order 5 faithful
ρ51ζ54ζ53ζ52ζ5    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);

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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