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## G = C65order 65 = 5·13

### Cyclic group

Aliases: C65, also denoted Z65, SmallGroup(65,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C65
 Chief series C1 — C13 — C65
 Lower central C1 — C65
 Upper central C1 — C65

Generators and relations for C65
G = < a | a65=1 >

Smallest permutation representation of C65
Regular action on 65 points
Generators in S65
`(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 60 61 62 63 64 65)`

`G:=sub<Sym(65)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65)]])`

C65 is a maximal subgroup of   D65

65 conjugacy classes

 class 1 5A 5B 5C 5D 13A ··· 13L 65A ··· 65AV order 1 5 5 5 5 13 ··· 13 65 ··· 65 size 1 1 1 1 1 1 ··· 1 1 ··· 1

65 irreducible representations

 dim 1 1 1 1 type + image C1 C5 C13 C65 kernel C65 C13 C5 C1 # reps 1 4 12 48

Matrix representation of C65 in GL1(𝔽131) generated by

 15
`G:=sub<GL(1,GF(131))| [15] >;`

C65 in GAP, Magma, Sage, TeX

`C_{65}`
`% in TeX`

`G:=Group("C65");`
`// GroupNames label`

`G:=SmallGroup(65,1);`
`// by ID`

`G=gap.SmallGroup(65,1);`
`# by ID`

`G:=PCGroup([2,-5,-13]);`
`// Polycyclic`

`G:=Group<a|a^65=1>;`
`// generators/relations`

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