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## G = C101order 101

### Cyclic group

Aliases: C101, also denoted Z101, SmallGroup(101,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C101
 Chief series C1 — C101
 Lower central C1 — C101
 Upper central C1 — C101
 Jennings C1 — C101

Generators and relations for C101
G = < a | a101=1 >

Smallest permutation representation of C101
Regular action on 101 points
Generators in S101
`(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)`

`G:=sub<Sym(101)| (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)>;`

`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) );`

`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)])`

C101 is a maximal subgroup of   D101

101 conjugacy classes

 class 1 101A ··· 101CV order 1 101 ··· 101 size 1 1 ··· 1

101 irreducible representations

 dim 1 1 type + image C1 C101 kernel C101 C1 # reps 1 100

Matrix representation of C101 in GL1(𝔽607) generated by

 26
`G:=sub<GL(1,GF(607))| [26] >;`

C101 in GAP, Magma, Sage, TeX

`C_{101}`
`% in TeX`

`G:=Group("C101");`
`// GroupNames label`

`G:=SmallGroup(101,1);`
`// by ID`

`G=gap.SmallGroup(101,1);`
`# by ID`

`G:=PCGroup([1,-101]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a|a^101=1>;`
`// generators/relations`

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