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## G = C4order 4 = 22

### Cyclic group

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4
 Chief series C1 — C2 — C4
 Lower central C1 — C4
 Upper central C1 — C4
 Jennings C1 — C2 — C4

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

Character table of C4

 class 1 2 4A 4B size 1 1 1 1 ρ1 1 1 1 1 trivial ρ2 1 1 -1 -1 linear of order 2 ρ3 1 -1 -i i linear of order 4 faithful ρ4 1 -1 i -i linear of order 4 faithful

Permutation representations of C4
Regular action on 4 points - transitive group 4T1
Generators in S4
`(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
Dicp: Q8  Dic3  Dic5  Dic7  Dic11  Dic13  Dic17  Dic19 ...
Cp⋊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
Dicp: Dic3  Dic5  Dic7  Dic11  Dic13  Dic17  Dic19  Dic23 ...
Cp⋊C4, p=1 mod 4: F5  C13⋊C4  C17⋊C4  C29⋊C4  C37⋊C4  C41⋊C4  C53⋊C4  C61⋊C4 ...

Polynomial with Galois group C4 over ℚ
actionf(x)Disc(f)
4T1x4-5x2+524·53

Matrix representation of C4 in GL1(𝔽5) generated by

 2
`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`

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