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## G = C51order 51 = 3·17

### Cyclic group

Aliases: C51, also denoted Z51, SmallGroup(51,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C51
 Chief series C1 — C17 — C51
 Lower central C1 — C51
 Upper central C1 — C51

Generators and relations for C51
G = < a | a51=1 >

Smallest permutation representation of C51
Regular action on 51 points
Generators in S51
`(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)`

`G:=sub<Sym(51)| (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)>;`

`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) );`

`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)])`

C51 is a maximal subgroup of   D51

51 conjugacy classes

 class 1 3A 3B 17A ··· 17P 51A ··· 51AF order 1 3 3 17 ··· 17 51 ··· 51 size 1 1 1 1 ··· 1 1 ··· 1

51 irreducible representations

 dim 1 1 1 1 type + image C1 C3 C17 C51 kernel C51 C17 C3 C1 # reps 1 2 16 32

Matrix representation of C51 in GL1(𝔽103) generated by

 38
`G:=sub<GL(1,GF(103))| [38] >;`

C51 in GAP, Magma, Sage, TeX

`C_{51}`
`% in TeX`

`G:=Group("C51");`
`// GroupNames label`

`G:=SmallGroup(51,1);`
`// by ID`

`G=gap.SmallGroup(51,1);`
`# by ID`

`G:=PCGroup([2,-3,-17]);`
`// Polycyclic`

`G:=Group<a|a^51=1>;`
`// generators/relations`

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