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## G = C25order 25 = 52

### Cyclic group

Aliases: C25, also denoted Z25, SmallGroup(25,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C25
 Chief series C1 — C5 — C25
 Lower central C1 — C25
 Upper central C1 — C25
 Jennings C1 — C5 — C5 — C5 — C5 — C25

Generators and relations for C25
G = < a | a25=1 >

Character table of C25

 class 1 5A 5B 5C 5D 25A 25B 25C 25D 25E 25F 25G 25H 25I 25J 25K 25L 25M 25N 25O 25P 25Q 25R 25S 25T size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ255 ζ2510 ζ2515 ζ2520 ζ254 ζ2517 ζ2513 ζ259 ζ25 ζ256 ζ2511 ζ2516 ζ2522 ζ252 ζ257 ζ2512 ζ2518 ζ2523 ζ253 ζ258 ζ2514 ζ2519 ζ2524 ζ2521 linear of order 25 faithful ρ3 1 ζ2510 ζ2520 ζ255 ζ2515 ζ258 ζ259 ζ25 ζ2518 ζ252 ζ2512 ζ2522 ζ257 ζ2519 ζ254 ζ2514 ζ2524 ζ2511 ζ2521 ζ256 ζ2516 ζ253 ζ2513 ζ2523 ζ2517 linear of order 25 faithful ρ4 1 ζ2515 ζ255 ζ2520 ζ2510 ζ2512 ζ25 ζ2514 ζ252 ζ253 ζ2518 ζ258 ζ2523 ζ2516 ζ256 ζ2521 ζ2511 ζ254 ζ2519 ζ259 ζ2524 ζ2517 ζ257 ζ2522 ζ2513 linear of order 25 faithful ρ5 1 ζ2520 ζ2515 ζ2510 ζ255 ζ2516 ζ2518 ζ252 ζ2511 ζ254 ζ2524 ζ2519 ζ2514 ζ2513 ζ258 ζ253 ζ2523 ζ2522 ζ2517 ζ2512 ζ257 ζ256 ζ25 ζ2521 ζ259 linear of order 25 faithful ρ6 1 1 1 1 1 ζ54 ζ52 ζ53 ζ54 ζ5 ζ5 ζ5 ζ5 ζ52 ζ52 ζ52 ζ52 ζ53 ζ53 ζ53 ζ53 ζ54 ζ54 ζ54 ζ5 linear of order 5 ρ7 1 ζ255 ζ2510 ζ2515 ζ2520 ζ2524 ζ252 ζ253 ζ254 ζ256 ζ2511 ζ2516 ζ2521 ζ257 ζ2512 ζ2517 ζ2522 ζ258 ζ2513 ζ2518 ζ2523 ζ259 ζ2514 ζ2519 ζ25 linear of order 25 faithful ρ8 1 ζ2510 ζ2520 ζ255 ζ2515 ζ253 ζ2519 ζ2516 ζ2513 ζ257 ζ2517 ζ252 ζ2512 ζ254 ζ2514 ζ2524 ζ259 ζ25 ζ2511 ζ2521 ζ256 ζ2523 ζ258 ζ2518 ζ2522 linear of order 25 faithful ρ9 1 ζ2515 ζ255 ζ2520 ζ2510 ζ257 ζ2511 ζ254 ζ2522 ζ258 ζ2523 ζ2513 ζ253 ζ25 ζ2516 ζ256 ζ2521 ζ2519 ζ259 ζ2524 ζ2514 ζ2512 ζ252 ζ2517 ζ2518 linear of order 25 faithful ρ10 1 ζ2520 ζ2515 ζ2510 ζ255 ζ2511 ζ253 ζ2517 ζ256 ζ259 ζ254 ζ2524 ζ2519 ζ2523 ζ2518 ζ2513 ζ258 ζ2512 ζ257 ζ252 ζ2522 ζ25 ζ2521 ζ2516 ζ2514 linear of order 25 faithful ρ11 1 1 1 1 1 ζ53 ζ54 ζ5 ζ53 ζ52 ζ52 ζ52 ζ52 ζ54 ζ54 ζ54 ζ54 ζ5 ζ5 ζ5 ζ5 ζ53 ζ53 ζ53 ζ52 linear of order 5 ρ12 1 ζ255 ζ2510 ζ2515 ζ2520 ζ2519 ζ2512 ζ2518 ζ2524 ζ2511 ζ2516 ζ2521 ζ25 ζ2517 ζ2522 ζ252 ζ257 ζ2523 ζ253 ζ258 ζ2513 ζ254 ζ259 ζ2514 ζ256 linear of order 25 faithful ρ13 1 ζ2510 ζ2520 ζ255 ζ2515 ζ2523 ζ254 ζ256 ζ258 ζ2512 ζ2522 ζ257 ζ2517 ζ2514 ζ2524 ζ259 ζ2519 ζ2516 ζ25 ζ2511 ζ2521 ζ2518 ζ253 ζ2513 ζ252 linear of order 25 faithful ρ14 1 ζ2515 ζ255 ζ2520 ζ2510 ζ252 ζ2521 ζ2519 ζ2517 ζ2513 ζ253 ζ2518 ζ258 ζ2511 ζ25 ζ2516 ζ256 ζ259 ζ2524 ζ2514 ζ254 ζ257 ζ2522 ζ2512 ζ2523 linear of order 25 faithful ρ15 1 ζ2520 ζ2515 ζ2510 ζ255 ζ256 ζ2513 ζ257 ζ25 ζ2514 ζ259 ζ254 ζ2524 ζ258 ζ253 ζ2523 ζ2518 ζ252 ζ2522 ζ2517 ζ2512 ζ2521 ζ2516 ζ2511 ζ2519 linear of order 25 faithful ρ16 1 1 1 1 1 ζ52 ζ5 ζ54 ζ52 ζ53 ζ53 ζ53 ζ53 ζ5 ζ5 ζ5 ζ5 ζ54 ζ54 ζ54 ζ54 ζ52 ζ52 ζ52 ζ53 linear of order 5 ρ17 1 ζ255 ζ2510 ζ2515 ζ2520 ζ2514 ζ2522 ζ258 ζ2519 ζ2516 ζ2521 ζ25 ζ256 ζ252 ζ257 ζ2512 ζ2517 ζ2513 ζ2518 ζ2523 ζ253 ζ2524 ζ254 ζ259 ζ2511 linear of order 25 faithful ρ18 1 ζ2510 ζ2520 ζ255 ζ2515 ζ2518 ζ2514 ζ2521 ζ253 ζ2517 ζ252 ζ2512 ζ2522 ζ2524 ζ259 ζ2519 ζ254 ζ256 ζ2516 ζ25 ζ2511 ζ2513 ζ2523 ζ258 ζ257 linear of order 25 faithful ρ19 1 ζ2515 ζ255 ζ2520 ζ2510 ζ2522 ζ256 ζ259 ζ2512 ζ2518 ζ258 ζ2523 ζ2513 ζ2521 ζ2511 ζ25 ζ2516 ζ2524 ζ2514 ζ254 ζ2519 ζ252 ζ2517 ζ257 ζ253 linear of order 25 faithful ρ20 1 ζ2520 ζ2515 ζ2510 ζ255 ζ25 ζ2523 ζ2522 ζ2521 ζ2519 ζ2514 ζ259 ζ254 ζ2518 ζ2513 ζ258 ζ253 ζ2517 ζ2512 ζ257 ζ252 ζ2516 ζ2511 ζ256 ζ2524 linear of order 25 faithful ρ21 1 1 1 1 1 ζ5 ζ53 ζ52 ζ5 ζ54 ζ54 ζ54 ζ54 ζ53 ζ53 ζ53 ζ53 ζ52 ζ52 ζ52 ζ52 ζ5 ζ5 ζ5 ζ54 linear of order 5 ρ22 1 ζ255 ζ2510 ζ2515 ζ2520 ζ259 ζ257 ζ2523 ζ2514 ζ2521 ζ25 ζ256 ζ2511 ζ2512 ζ2517 ζ2522 ζ252 ζ253 ζ258 ζ2513 ζ2518 ζ2519 ζ2524 ζ254 ζ2516 linear of order 25 faithful ρ23 1 ζ2510 ζ2520 ζ255 ζ2515 ζ2513 ζ2524 ζ2511 ζ2523 ζ2522 ζ257 ζ2517 ζ252 ζ259 ζ2519 ζ254 ζ2514 ζ2521 ζ256 ζ2516 ζ25 ζ258 ζ2518 ζ253 ζ2512 linear of order 25 faithful ρ24 1 ζ2515 ζ255 ζ2520 ζ2510 ζ2517 ζ2516 ζ2524 ζ257 ζ2523 ζ2513 ζ253 ζ2518 ζ256 ζ2521 ζ2511 ζ25 ζ2514 ζ254 ζ2519 ζ259 ζ2522 ζ2512 ζ252 ζ258 linear of order 25 faithful ρ25 1 ζ2520 ζ2515 ζ2510 ζ255 ζ2521 ζ258 ζ2512 ζ2516 ζ2524 ζ2519 ζ2514 ζ259 ζ253 ζ2523 ζ2518 ζ2513 ζ257 ζ252 ζ2522 ζ2517 ζ2511 ζ256 ζ25 ζ254 linear of order 25 faithful

Permutation representations of C25
Regular action on 25 points - transitive group 25T1
Generators in S25
`(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)`

`G:=sub<Sym(25)| (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)>;`

`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) );`

`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)]])`

`G:=TransitiveGroup(25,1);`

C25 is a maximal subgroup of   D25  C125  5- 1+2  C11⋊C25  C24⋊C25
C25 is a maximal quotient of   C125  C11⋊C25  C24⋊C25

Matrix representation of C25 in GL1(𝔽101) generated by

 80
`G:=sub<GL(1,GF(101))| [80] >;`

C25 in GAP, Magma, Sage, TeX

`C_{25}`
`% in TeX`

`G:=Group("C25");`
`// GroupNames label`

`G:=SmallGroup(25,1);`
`// by ID`

`G=gap.SmallGroup(25,1);`
`# by ID`

`G:=PCGroup([2,-5,-5,10]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a|a^25=1>;`
`// generators/relations`

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