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## G = C20order 20 = 22·5

### Cyclic group

Aliases: C20, also denoted Z20, SmallGroup(20,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20
 Chief series C1 — C2 — C10 — C20
 Lower central C1 — C20
 Upper central C1 — C20

Generators and relations for C20
G = < a | a20=1 >

Character table of C20

 class 1 2 4A 4B 5A 5B 5C 5D 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 -i i 1 1 1 1 -1 -1 -1 -1 i i -i i -i i -i -i linear of order 4 ρ4 1 -1 i -i 1 1 1 1 -1 -1 -1 -1 -i -i i -i i -i i i linear of order 4 ρ5 1 1 1 1 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 ζ54 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 ζ5 linear of order 5 ρ6 1 -1 -i i ζ52 ζ53 ζ54 ζ5 -ζ52 -ζ54 -ζ53 -ζ5 ζ4ζ54 ζ4ζ5 ζ43ζ52 ζ4ζ52 ζ43ζ53 ζ4ζ53 ζ43ζ54 ζ43ζ5 linear of order 20 faithful ρ7 1 1 -1 -1 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 -ζ54 -ζ5 -ζ52 -ζ52 -ζ53 -ζ53 -ζ54 -ζ5 linear of order 10 ρ8 1 -1 i -i ζ52 ζ53 ζ54 ζ5 -ζ52 -ζ54 -ζ53 -ζ5 ζ43ζ54 ζ43ζ5 ζ4ζ52 ζ43ζ52 ζ4ζ53 ζ43ζ53 ζ4ζ54 ζ4ζ5 linear of order 20 faithful ρ9 1 1 1 1 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 ζ53 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 ζ52 linear of order 5 ρ10 1 -1 -i i ζ54 ζ5 ζ53 ζ52 -ζ54 -ζ53 -ζ5 -ζ52 ζ4ζ53 ζ4ζ52 ζ43ζ54 ζ4ζ54 ζ43ζ5 ζ4ζ5 ζ43ζ53 ζ43ζ52 linear of order 20 faithful ρ11 1 1 -1 -1 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 -ζ53 -ζ52 -ζ54 -ζ54 -ζ5 -ζ5 -ζ53 -ζ52 linear of order 10 ρ12 1 -1 i -i ζ54 ζ5 ζ53 ζ52 -ζ54 -ζ53 -ζ5 -ζ52 ζ43ζ53 ζ43ζ52 ζ4ζ54 ζ43ζ54 ζ4ζ5 ζ43ζ5 ζ4ζ53 ζ4ζ52 linear of order 20 faithful ρ13 1 1 1 1 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 ζ52 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 ζ53 linear of order 5 ρ14 1 -1 -i i ζ5 ζ54 ζ52 ζ53 -ζ5 -ζ52 -ζ54 -ζ53 ζ4ζ52 ζ4ζ53 ζ43ζ5 ζ4ζ5 ζ43ζ54 ζ4ζ54 ζ43ζ52 ζ43ζ53 linear of order 20 faithful ρ15 1 1 -1 -1 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 -ζ52 -ζ53 -ζ5 -ζ5 -ζ54 -ζ54 -ζ52 -ζ53 linear of order 10 ρ16 1 -1 i -i ζ5 ζ54 ζ52 ζ53 -ζ5 -ζ52 -ζ54 -ζ53 ζ43ζ52 ζ43ζ53 ζ4ζ5 ζ43ζ5 ζ4ζ54 ζ43ζ54 ζ4ζ52 ζ4ζ53 linear of order 20 faithful ρ17 1 1 1 1 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 ζ5 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 ζ54 linear of order 5 ρ18 1 -1 -i i ζ53 ζ52 ζ5 ζ54 -ζ53 -ζ5 -ζ52 -ζ54 ζ4ζ5 ζ4ζ54 ζ43ζ53 ζ4ζ53 ζ43ζ52 ζ4ζ52 ζ43ζ5 ζ43ζ54 linear of order 20 faithful ρ19 1 1 -1 -1 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 -ζ5 -ζ54 -ζ53 -ζ53 -ζ52 -ζ52 -ζ5 -ζ54 linear of order 10 ρ20 1 -1 i -i ζ53 ζ52 ζ5 ζ54 -ζ53 -ζ5 -ζ52 -ζ54 ζ43ζ5 ζ43ζ54 ζ4ζ53 ζ43ζ53 ζ4ζ52 ζ43ζ52 ζ4ζ5 ζ4ζ54 linear of order 20 faithful

Permutation representations of C20
Regular action on 20 points - transitive group 20T1
Generators in S20
`(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)`

`G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)]])`

`G:=TransitiveGroup(20,1);`

C20 is a maximal subgroup of   C52C8  Dic10  D20  C11⋊C20  2- 1+4.C10
C20 is a maximal quotient of   C11⋊C20

Polynomial with Galois group C20 over ℚ
actionf(x)Disc(f)
20T1x20+x15+x10+x5+1535

Matrix representation of C20 in GL1(𝔽41) generated by

 21
`G:=sub<GL(1,GF(41))| [21] >;`

C20 in GAP, Magma, Sage, TeX

`C_{20}`
`% in TeX`

`G:=Group("C20");`
`// GroupNames label`

`G:=SmallGroup(20,2);`
`// by ID`

`G=gap.SmallGroup(20,2);`
`# by ID`

`G:=PCGroup([3,-2,-5,-2,30]);`
`// Polycyclic`

`G:=Group<a|a^20=1>;`
`// generators/relations`

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