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G = C₂²  order 4 = 2²

Elementary abelian group of type [2,2]

direct product, p-group, elementary abelian, monomial, rational

Aliases: C₂², symmetries of a (non-square) rectangle, Klein 4-group V₄, SmallGroup(4,2)

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

Generators and relations for C₂²
 G = < a,b | a²=b²=1, ab=ba >

Character table of C₂²

(1 2)(3 4)

(1 4)(2 3)

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

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

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

G:=TransitiveGroup(4,2);

C₂² is a maximal subgroup of   D₄  A₄
C₂² is a maximal quotient of   D₄  Q₈

Matrix representation of C₂² ►in GL₂(ℤ) generated by

G:=sub<GL(2,Integers())| [-1,0,0,1],[1,0,0,-1] >;

C₂² in GAP, Magma, Sage, TeX

C_2^2

% in TeX

G:=Group("C2^2");

// GroupNames label

G:=SmallGroup(4,2);

// by ID

G=gap.SmallGroup(4,2);

# by ID

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

// Polycyclic

G:=Group<a,b|a^2=b^2=1,a*b=b*a>;

// generators/relations

Export

Subgroup lattice of C₂² in TeX
Character table of C₂² in TeX