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## G = C59order 59

### Cyclic group

Aliases: C59, also denoted Z59, SmallGroup(59,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C59
 Chief series C1 — C59
 Lower central C1 — C59
 Upper central C1 — C59
 Jennings C1 — C59

Generators and relations for C59
G = < a | a59=1 >

Smallest permutation representation of C59
Regular action on 59 points
Generators in S59
`(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59)`

`G:=sub<Sym(59)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59)])`

C59 is a maximal subgroup of   D59

59 conjugacy classes

 class 1 59A ··· 59BF order 1 59 ··· 59 size 1 1 ··· 1

59 irreducible representations

 dim 1 1 type + image C1 C59 kernel C59 C1 # reps 1 58

Matrix representation of C59 in GL1(𝔽709) generated by

 283
`G:=sub<GL(1,GF(709))| [283] >;`

C59 in GAP, Magma, Sage, TeX

`C_{59}`
`% in TeX`

`G:=Group("C59");`
`// GroupNames label`

`G:=SmallGroup(59,1);`
`// by ID`

`G=gap.SmallGroup(59,1);`
`# by ID`

`G:=PCGroup([1,-59]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a|a^59=1>;`
`// generators/relations`

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