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## G = C69order 69 = 3·23

### Cyclic group

Aliases: C69, also denoted Z69, SmallGroup(69,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C69
 Chief series C1 — C23 — C69
 Lower central C1 — C69
 Upper central C1 — C69

Generators and relations for C69
G = < a | a69=1 >

Smallest permutation representation of C69
Regular action on 69 points
Generators in S69
`(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 66 67 68 69)`

`G:=sub<Sym(69)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)]])`

C69 is a maximal subgroup of   D69

69 conjugacy classes

 class 1 3A 3B 23A ··· 23V 69A ··· 69AR order 1 3 3 23 ··· 23 69 ··· 69 size 1 1 1 1 ··· 1 1 ··· 1

69 irreducible representations

 dim 1 1 1 1 type + image C1 C3 C23 C69 kernel C69 C23 C3 C1 # reps 1 2 22 44

Matrix representation of C69 in GL1(𝔽139) generated by

 37
`G:=sub<GL(1,GF(139))| [37] >;`

C69 in GAP, Magma, Sage, TeX

`C_{69}`
`% in TeX`

`G:=Group("C69");`
`// GroupNames label`

`G:=SmallGroup(69,1);`
`// by ID`

`G=gap.SmallGroup(69,1);`
`# by ID`

`G:=PCGroup([2,-3,-23]);`
`// Polycyclic`

`G:=Group<a|a^69=1>;`
`// generators/relations`

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