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G = C₅  order 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

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

Character table of C₅

(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
 C_p⋊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
 C_p⋊C₅, p=1 mod 5: C₁₁⋊C₅  C₃₁⋊C₅  C₄₁⋊C₅  C₆₁⋊C₅  C₇₁⋊C₅ ...

Matrix representation of C₅ ►in GL₁(𝔽₁₁) generated by

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

Export

Subgroup lattice of C₅ in TeX
Character table of C₅ in TeX