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## G = C8order 8 = 23

### Cyclic group

Aliases: C8, also denoted Z8, SmallGroup(8,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8
 Chief series C1 — C2 — C4 — C8
 Lower central C1 — C8
 Upper central C1 — C8
 Jennings C1 — C2 — C2 — C4 — C8

Generators and relations for C8
G = < a | a8=1 >

Character table of C8

 class 1 2 4A 4B 8A 8B 8C 8D size 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 i -i ζ87 ζ85 ζ83 ζ8 linear of order 8 faithful ρ4 1 1 -1 -1 -i i -i i linear of order 4 ρ5 1 -1 -i i ζ85 ζ87 ζ8 ζ83 linear of order 8 faithful ρ6 1 -1 i -i ζ83 ζ8 ζ87 ζ85 linear of order 8 faithful ρ7 1 1 -1 -1 i -i i -i linear of order 4 ρ8 1 -1 -i i ζ8 ζ83 ζ85 ζ87 linear of order 8 faithful

Permutation representations of C8
Regular action on 8 points - transitive group 8T1
Generators in S8
`(1 2 3 4 5 6 7 8)`

`G:=sub<Sym(8)| (1,2,3,4,5,6,7,8)>;`

`G:=Group( (1,2,3,4,5,6,7,8) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8)])`

`G:=TransitiveGroup(8,1);`

C8 is a maximal subgroup of
C16  D8  SD16  Q16  C322C8  F9  C17⋊C8  C52⋊C8  C41⋊C8  C722C8  C72⋊C8
C2p.C4: M4(2)  C3⋊C8  C52C8  C5⋊C8  C7⋊C8  C11⋊C8  C132C8  C13⋊C8 ...
C8 is a maximal quotient of
C16  C322C8  F9  C52⋊C8  C722C8  C72⋊C8  A5⋊C8
Cp⋊C8: C3⋊C8  C52C8  C5⋊C8  C7⋊C8  C11⋊C8  C132C8  C13⋊C8  C173C8 ...

Polynomial with Galois group C8 over ℚ
actionf(x)Disc(f)
8T1x8-8x6+20x4-16x2+2231

Matrix representation of C8 in GL1(𝔽17) generated by

 2
`G:=sub<GL(1,GF(17))| [2] >;`

C8 in GAP, Magma, Sage, TeX

`C_8`
`% in TeX`

`G:=Group("C8");`
`// GroupNames label`

`G:=SmallGroup(8,1);`
`// by ID`

`G=gap.SmallGroup(8,1);`
`# by ID`

`G:=PCGroup([3,-2,-2,-2,6,16]);`
`// Polycyclic`

`G:=Group<a|a^8=1>;`
`// generators/relations`

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