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G = C₄  order 4 = 2²

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C₄, also denoted Z₄, rotations of a square, SmallGroup(4,1)

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

Generators and relations for C₄
 G = < a | a⁴=1 >

Character table of C₄

(1 2 3 4)

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

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

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

G:=TransitiveGroup(4,1);

C₄ is a maximal subgroup of
 C₈  D₄  C₃²⋊C₄  C₇²⋊C₄  C₁₁²⋊C₄
 Dic_p: Q₈  Dic₃  Dic₅  Dic₇  Dic₁₁  Dic₁₃  Dic₁₇  Dic₁₉ ...
 C_p⋊C₄, p=1 mod 4: F₅  C₁₃⋊C₄  C₁₇⋊C₄  C₂₉⋊C₄  C₃₇⋊C₄  C₄₁⋊C₄  C₅₃⋊C₄  C₆₁⋊C₄ ...
C₄ is a maximal quotient of
 C₈  C₃²⋊C₄  C₇²⋊C₄  A₅⋊C₄  C₁₁²⋊C₄
 Dic_p: Dic₃  Dic₅  Dic₇  Dic₁₁  Dic₁₃  Dic₁₇  Dic₁₉  Dic₂₃ ...
 C_p⋊C₄, p=1 mod 4: F₅  C₁₃⋊C₄  C₁₇⋊C₄  C₂₉⋊C₄  C₃₇⋊C₄  C₄₁⋊C₄  C₅₃⋊C₄  C₆₁⋊C₄ ...

Matrix representation of C₄ ►in GL₁(𝔽₅) generated by

G:=sub<GL(1,GF(5))| [2] >;

C₄ in GAP, Magma, Sage, TeX

C_4

% in TeX

G:=Group("C4");

// GroupNames label

G:=SmallGroup(4,1);

// by ID

G=gap.SmallGroup(4,1);

# by ID

G:=PCGroup([2,-2,-2,4]:ExponentLimit:=1);

// Polycyclic

G:=Group<a|a^4=1>;

// generators/relations

Export

Subgroup lattice of C₄ in TeX
Character table of C₄ in TeX