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## G = C10order 10 = 2·5

### Cyclic group

Aliases: C10, also denoted Z10, SmallGroup(10,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10
 Chief series C1 — C5 — C10
 Lower central C1 — C10
 Upper central C1 — C10

Generators and relations for C10
G = < a | a10=1 >

Character table of C10

 class 1 2 5A 5B 5C 5D 10A 10B 10C 10D size 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ52 ζ53 ζ5 linear of order 5 ρ4 1 -1 ζ52 ζ53 ζ54 ζ5 -ζ54 -ζ52 -ζ53 -ζ5 linear of order 10 faithful ρ5 1 1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ54 ζ5 ζ52 linear of order 5 ρ6 1 -1 ζ54 ζ5 ζ53 ζ52 -ζ53 -ζ54 -ζ5 -ζ52 linear of order 10 faithful ρ7 1 1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ5 ζ54 ζ53 linear of order 5 ρ8 1 -1 ζ5 ζ54 ζ52 ζ53 -ζ52 -ζ5 -ζ54 -ζ53 linear of order 10 faithful ρ9 1 1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ53 ζ52 ζ54 linear of order 5 ρ10 1 -1 ζ53 ζ52 ζ5 ζ54 -ζ5 -ζ53 -ζ52 -ζ54 linear of order 10 faithful

Permutation representations of C10
Regular action on 10 points - transitive group 10T1
Generators in S10
`(1 2 3 4 5 6 7 8 9 10)`

`G:=sub<Sym(10)| (1,2,3,4,5,6,7,8,9,10)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10)]])`

`G:=TransitiveGroup(10,1);`

C10 is a maximal subgroup of   Dic5  F11  2- 1+4⋊C5  C31⋊C10  C41⋊C10
C10 is a maximal quotient of   F11  C31⋊C10  C41⋊C10

Polynomial with Galois group C10 over ℚ
actionf(x)Disc(f)
10T1x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1-119

Matrix representation of C10 in GL1(𝔽11) generated by

 6
`G:=sub<GL(1,GF(11))| [6] >;`

C10 in GAP, Magma, Sage, TeX

`C_{10}`
`% in TeX`

`G:=Group("C10");`
`// GroupNames label`

`G:=SmallGroup(10,2);`
`// by ID`

`G=gap.SmallGroup(10,2);`
`# by ID`

`G:=PCGroup([2,-2,-5]);`
`// Polycyclic`

`G:=Group<a|a^10=1>;`
`// generators/relations`

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