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## G = C31order 31

### Cyclic group

Aliases: C31, also denoted Z31, SmallGroup(31,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C31
 Chief series C1 — C31
 Lower central C1 — C31
 Upper central C1 — C31
 Jennings C1 — C31

Generators and relations for C31
G = < a | a31=1 >

Permutation representations of C31
Regular action on 31 points - transitive group 31T1
Generators in S31
`(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)`

`G:=sub<Sym(31)| (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)>;`

`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) );`

`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)]])`

`G:=TransitiveGroup(31,1);`

C31 is a maximal subgroup of   D31  C31⋊C3  C31⋊C5

31 conjugacy classes

 class 1 31A ··· 31AD order 1 31 ··· 31 size 1 1 ··· 1

31 irreducible representations

 dim 1 1 type + image C1 C31 kernel C31 C1 # reps 1 30

Matrix representation of C31 in GL1(𝔽311) generated by

 113
`G:=sub<GL(1,GF(311))| [113] >;`

C31 in GAP, Magma, Sage, TeX

`C_{31}`
`% in TeX`

`G:=Group("C31");`
`// GroupNames label`

`G:=SmallGroup(31,1);`
`// by ID`

`G=gap.SmallGroup(31,1);`
`# by ID`

`G:=PCGroup([1,-31]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a|a^31=1>;`
`// generators/relations`

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