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G = C22order 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

C1 — C22
C1C2 — C22
C1 — C22
C1 — C22
C1 — C22

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


Character table of C22

 class 12A2B2C
 size 1111
ρ11111    trivial
ρ211-1-1    linear of order 2
ρ31-11-1    linear of order 2
ρ41-1-11    linear of order 2

Permutation representations of C22
Regular action on 4 points - transitive group 4T2
Generators in S4
(1 2)(3 4)
(1 3)(2 4)

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

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

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

G:=TransitiveGroup(4,2);

Polynomial with Galois group C22 over ℚ
actionf(x)Disc(f)
4T2x4+128

Matrix representation of C22 in GL2(ℤ) generated by

-10
01
,
10
0-1
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

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