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## G = C105order 105 = 3·5·7

### Cyclic group

Aliases: C105, also denoted Z105, SmallGroup(105,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C105
 Chief series C1 — C7 — C35 — C105
 Lower central C1 — C105
 Upper central C1 — C105

Generators and relations for C105
G = < a | a105=1 >

Smallest permutation representation of C105
Regular action on 105 points
Generators in S105
`(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)`

`G:=sub<Sym(105)| (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)>;`

`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) );`

`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)]])`

C105 is a maximal subgroup of   D105

105 conjugacy classes

 class 1 3A 3B 5A 5B 5C 5D 7A ··· 7F 15A ··· 15H 21A ··· 21L 35A ··· 35X 105A ··· 105AV order 1 3 3 5 5 5 5 7 ··· 7 15 ··· 15 21 ··· 21 35 ··· 35 105 ··· 105 size 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + image C1 C3 C5 C7 C15 C21 C35 C105 kernel C105 C35 C21 C15 C7 C5 C3 C1 # reps 1 2 4 6 8 12 24 48

Matrix representation of C105 in GL1(𝔽211) generated by

 136
`G:=sub<GL(1,GF(211))| [136] >;`

C105 in GAP, Magma, Sage, TeX

`C_{105}`
`% in TeX`

`G:=Group("C105");`
`// GroupNames label`

`G:=SmallGroup(105,2);`
`// by ID`

`G=gap.SmallGroup(105,2);`
`# by ID`

`G:=PCGroup([3,-3,-5,-7]);`
`// Polycyclic`

`G:=Group<a|a^105=1>;`
`// generators/relations`

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