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## G = C39order 39 = 3·13

### Cyclic group

Aliases: C39, also denoted Z39, SmallGroup(39,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39
 Chief series C1 — C13 — C39
 Lower central C1 — C39
 Upper central C1 — C39

Generators and relations for C39
G = < a | a39=1 >

Smallest permutation representation of C39
Regular action on 39 points
Generators in S39
`(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)`

`G:=sub<Sym(39)| (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)>;`

`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) );`

`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)])`

C39 is a maximal subgroup of   D39  C13⋊C9

39 conjugacy classes

 class 1 3A 3B 13A ··· 13L 39A ··· 39X order 1 3 3 13 ··· 13 39 ··· 39 size 1 1 1 1 ··· 1 1 ··· 1

39 irreducible representations

 dim 1 1 1 1 type + image C1 C3 C13 C39 kernel C39 C13 C3 C1 # reps 1 2 12 24

Matrix representation of C39 in GL1(𝔽79) generated by

 25
`G:=sub<GL(1,GF(79))| [25] >;`

C39 in GAP, Magma, Sage, TeX

`C_{39}`
`% in TeX`

`G:=Group("C39");`
`// GroupNames label`

`G:=SmallGroup(39,2);`
`// by ID`

`G=gap.SmallGroup(39,2);`
`# by ID`

`G:=PCGroup([2,-3,-13]);`
`// Polycyclic`

`G:=Group<a|a^39=1>;`
`// generators/relations`

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