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G = C2order 2

Cyclic group

Aliases: C2, also denoted Z2, SmallGroup(2,1)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2
G = < a | a2=1 >

Character table of C2

 class 1 2 size 1 1 ρ1 1 1 trivial ρ2 1 -1 linear of order 2 faithful

Permutation representations of C2
Regular action on 2 points - transitive group 2T1
Generators in S2
`(1 2)`

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

`G:=Group( (1,2) );`

`G=PermutationGroup([[(1,2)]])`

`G:=TransitiveGroup(2,1);`

C2 is a maximal subgroup of
C4
Dp: S3  D5  D7  D11  D13  D17  D19  D23 ...
C2 is a maximal quotient of
C4  S5  PGL2(𝔽7)
Dp: S3  D5  D7  D11  D13  D17  D19  D23 ...

Polynomial with Galois group C2 over ℚ
actionf(x)Disc(f)
2T1x2+1-22

Matrix representation of C2 in GL1(ℤ) generated by

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

C2 in GAP, Magma, Sage, TeX

`C_2`
`% in TeX`

`G:=Group("C2");`
`// GroupNames label`

`G:=SmallGroup(2,1);`
`// by ID`

`G=gap.SmallGroup(2,1);`
`# by ID`

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

`G:=Group<a|a^2=1>;`
`// generators/relations`

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