Copied to
clipboard

## G = C117order 117 = 32·13

### Cyclic group

Aliases: C117, also denoted Z117, SmallGroup(117,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C117
 Chief series C1 — C3 — C39 — C117
 Lower central C1 — C117
 Upper central C1 — C117

Generators and relations for C117
G = < a | a117=1 >

Smallest permutation representation of C117
Regular action on 117 points
Generators in S117
`(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)`

`G:=sub<Sym(117)| (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)>;`

`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) );`

`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)]])`

C117 is a maximal subgroup of   D117  C13⋊C27  C117⋊C3  C1173C3

117 conjugacy classes

 class 1 3A 3B 9A ··· 9F 13A ··· 13L 39A ··· 39X 117A ··· 117BT order 1 3 3 9 ··· 9 13 ··· 13 39 ··· 39 117 ··· 117 size 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

117 irreducible representations

 dim 1 1 1 1 1 1 type + image C1 C3 C9 C13 C39 C117 kernel C117 C39 C13 C9 C3 C1 # reps 1 2 6 12 24 72

Matrix representation of C117 in GL1(𝔽937) generated by

 648
`G:=sub<GL(1,GF(937))| [648] >;`

C117 in GAP, Magma, Sage, TeX

`C_{117}`
`% in TeX`

`G:=Group("C117");`
`// GroupNames label`

`G:=SmallGroup(117,2);`
`// by ID`

`G=gap.SmallGroup(117,2);`
`# by ID`

`G:=PCGroup([3,-3,-13,-3,117]);`
`// Polycyclic`

`G:=Group<a|a^117=1>;`
`// generators/relations`

Export

׿
×
𝔽