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## G = C52order 52 = 22·13

### Cyclic group

Aliases: C52, also denoted Z52, SmallGroup(52,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52
 Chief series C1 — C2 — C26 — C52
 Lower central C1 — C52
 Upper central C1 — C52

Generators and relations for C52
G = < a | a52=1 >

Smallest permutation representation of C52
Regular action on 52 points
Generators in S52
`(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)`

`G:=sub<Sym(52)| (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)>;`

`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) );`

`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)]])`

C52 is a maximal subgroup of   C132C8  Dic26  D52

52 conjugacy classes

 class 1 2 4A 4B 13A ··· 13L 26A ··· 26L 52A ··· 52X order 1 2 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

52 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C4 C13 C26 C52 kernel C52 C26 C13 C4 C2 C1 # reps 1 1 2 12 12 24

Matrix representation of C52 in GL1(𝔽53) generated by

 14
`G:=sub<GL(1,GF(53))| [14] >;`

C52 in GAP, Magma, Sage, TeX

`C_{52}`
`% in TeX`

`G:=Group("C52");`
`// GroupNames label`

`G:=SmallGroup(52,2);`
`// by ID`

`G=gap.SmallGroup(52,2);`
`# by ID`

`G:=PCGroup([3,-2,-13,-2,78]);`
`// Polycyclic`

`G:=Group<a|a^52=1>;`
`// generators/relations`

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