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## G = C100order 100 = 22·52

### Cyclic group

Aliases: C100, also denoted Z100, SmallGroup(100,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C100
 Chief series C1 — C5 — C10 — C50 — C100
 Lower central C1 — C100
 Upper central C1 — C100

Generators and relations for C100
G = < a | a100=1 >

Smallest permutation representation of C100
Regular action on 100 points
Generators in S100
`(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)`

`G:=sub<Sym(100)| (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)>;`

`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) );`

`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)])`

C100 is a maximal subgroup of   C252C8  Dic50  D100

100 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 10A 10B 10C 10D 20A ··· 20H 25A ··· 25T 50A ··· 50T 100A ··· 100AN order 1 2 4 4 5 5 5 5 10 10 10 10 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 1 1 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C4 C5 C10 C20 C25 C50 C100 kernel C100 C50 C25 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 4 8 20 20 40

Matrix representation of C100 in GL1(𝔽101) generated by

 15
`G:=sub<GL(1,GF(101))| [15] >;`

C100 in GAP, Magma, Sage, TeX

`C_{100}`
`% in TeX`

`G:=Group("C100");`
`// GroupNames label`

`G:=SmallGroup(100,2);`
`// by ID`

`G=gap.SmallGroup(100,2);`
`# by ID`

`G:=PCGroup([4,-2,-5,-2,-5,40,85]);`
`// Polycyclic`

`G:=Group<a|a^100=1>;`
`// generators/relations`

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