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## G = C22order 4 = 22

### Elementary abelian group of type [2,2]

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22
 Chief series C1 — C2 — C22
 Lower central C1 — C22
 Upper central C1 — C22
 Jennings C1 — C22

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

Character table of C22

 class 1 2A 2B 2C size 1 1 1 1 ρ1 1 1 1 1 trivial ρ2 1 1 -1 -1 linear of order 2 ρ3 1 -1 1 -1 linear of order 2 ρ4 1 -1 -1 1 linear of order 2

Permutation representations of C22
Regular action on 4 points - transitive group 4T2
Generators in S4
```(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

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

Matrix representation of C22 in GL2(ℤ) generated by

 -1 0 0 1
,
 1 0 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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