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## G = C9order 9 = 32

### Cyclic group

Aliases: C9, also denoted Z9, SmallGroup(9,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C9
 Chief series C1 — C3 — C9
 Lower central C1 — C9
 Upper central C1 — C9
 Jennings C1 — C3 — C3 — C9

Generators and relations for C9
G = < a | a9=1 >

Character table of C9

 class 1 3A 3B 9A 9B 9C 9D 9E 9F size 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ32 ζ3 ζ97 ζ95 ζ9 ζ94 ζ98 ζ92 linear of order 9 faithful ρ3 1 ζ3 ζ32 ζ95 ζ9 ζ92 ζ98 ζ97 ζ94 linear of order 9 faithful ρ4 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 ζ32 ζ3 ζ9 ζ92 ζ94 ζ97 ζ95 ζ98 linear of order 9 faithful ρ6 1 ζ3 ζ32 ζ98 ζ97 ζ95 ζ92 ζ94 ζ9 linear of order 9 faithful ρ7 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ8 1 ζ32 ζ3 ζ94 ζ98 ζ97 ζ9 ζ92 ζ95 linear of order 9 faithful ρ9 1 ζ3 ζ32 ζ92 ζ94 ζ98 ζ95 ζ9 ζ97 linear of order 9 faithful

Permutation representations of C9
Regular action on 9 points - transitive group 9T1
Generators in S9
`(1 2 3 4 5 6 7 8 9)`

`G:=sub<Sym(9)| (1,2,3,4,5,6,7,8,9)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9)]])`

`G:=TransitiveGroup(9,1);`

C9 is a maximal subgroup of
D9  C27  3- 1+2  C3.A4  C52⋊C9
Cp⋊C9, p=1 mod 3: C7⋊C9  C13⋊C9  C192C9  C19⋊C9  C31⋊C9  C372C9  C37⋊C9  C43⋊C9 ...
C9 is a maximal quotient of
C27  C3.A4  C52⋊C9
Cp⋊C9, p=1 mod 3: C7⋊C9  C13⋊C9  C192C9  C19⋊C9  C31⋊C9  C372C9  C37⋊C9  C43⋊C9 ...

Polynomial with Galois group C9 over ℚ
actionf(x)Disc(f)
9T1x9-x8-8x7+7x6+21x5-15x4-20x3+10x2+5x-1198

Matrix representation of C9 in GL1(𝔽19) generated by

 4
`G:=sub<GL(1,GF(19))| [4] >;`

C9 in GAP, Magma, Sage, TeX

`C_9`
`% in TeX`

`G:=Group("C9");`
`// GroupNames label`

`G:=SmallGroup(9,1);`
`// by ID`

`G=gap.SmallGroup(9,1);`
`# by ID`

`G:=PCGroup([2,-3,-3,6]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a|a^9=1>;`
`// generators/relations`

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