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## G = C125order 125 = 53

### Cyclic group

Aliases: C125, also denoted Z125, SmallGroup(125,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C125
 Chief series C1 — C5 — C25 — C125
 Lower central C1 — C125
 Upper central C1 — C125
 Jennings C1 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C5 — C25 — C25 — C25 — C25 — C125

Generators and relations for C125
G = < a | a125=1 >

Smallest permutation representation of C125
Regular action on 125 points
Generators in S125
`(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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125)`

`G:=sub<Sym(125)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125)])`

C125 is a maximal subgroup of   D125

125 conjugacy classes

 class 1 5A 5B 5C 5D 25A ··· 25T 125A ··· 125CV order 1 5 5 5 5 25 ··· 25 125 ··· 125 size 1 1 1 1 1 1 ··· 1 1 ··· 1

125 irreducible representations

 dim 1 1 1 1 type + image C1 C5 C25 C125 kernel C125 C25 C5 C1 # reps 1 4 20 100

Matrix representation of C125 in GL1(𝔽251) generated by

 173
`G:=sub<GL(1,GF(251))| [173] >;`

C125 in GAP, Magma, Sage, TeX

`C_{125}`
`% in TeX`

`G:=Group("C125");`
`// GroupNames label`

`G:=SmallGroup(125,1);`
`// by ID`

`G=gap.SmallGroup(125,1);`
`# by ID`

`G:=PCGroup([3,-5,-5,-5,15,34]);`
`// Polycyclic`

`G:=Group<a|a^125=1>;`
`// generators/relations`

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