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G = C₃  order 3

Cyclic group

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

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

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

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

Character table of C₃

(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
 C_p⋊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
 C_p⋊C₃, p=1 mod 3: C₇⋊C₃  C₁₃⋊C₃  C₁₉⋊C₃  C₃₁⋊C₃  C₃₇⋊C₃  C₄₃⋊C₃  C₆₁⋊C₃  C₆₇⋊C₃ ...

Matrix representation of C₃ ►in GL₁(𝔽₇) generated by

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

Subgroup lattice of C₃ in TeX
Character table of C₃ in TeX