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## G = C119order 119 = 7·17

### Cyclic group

Aliases: C119, also denoted Z119, SmallGroup(119,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C119
 Chief series C1 — C17 — C119
 Lower central C1 — C119
 Upper central C1 — C119

Generators and relations for C119
G = < a | a119=1 >

Smallest permutation representation of C119
Regular action on 119 points
Generators in S119
`(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)`

`G:=sub<Sym(119)| (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)>;`

`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) );`

`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)])`

C119 is a maximal subgroup of   D119

119 conjugacy classes

 class 1 7A ··· 7F 17A ··· 17P 119A ··· 119CR order 1 7 ··· 7 17 ··· 17 119 ··· 119 size 1 1 ··· 1 1 ··· 1 1 ··· 1

119 irreducible representations

 dim 1 1 1 1 type + image C1 C7 C17 C119 kernel C119 C17 C7 C1 # reps 1 6 16 96

Matrix representation of C119 in GL2(𝔽239) generated by

 71 0 0 98
`G:=sub<GL(2,GF(239))| [71,0,0,98] >;`

C119 in GAP, Magma, Sage, TeX

`C_{119}`
`% in TeX`

`G:=Group("C119");`
`// GroupNames label`

`G:=SmallGroup(119,1);`
`// by ID`

`G=gap.SmallGroup(119,1);`
`# by ID`

`G:=PCGroup([2,-7,-17]);`
`// Polycyclic`

`G:=Group<a|a^119=1>;`
`// generators/relations`

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