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G = C117order 117 = 32·13

Cyclic group

direct product, cyclic, abelian, monomial

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

Series: Derived Chief Lower central Upper central

C1 — C117
C1C3C39 — C117
C1 — C117
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 3A3B9A···9F13A···13L39A···39X117A···117BT
order1339···913···1339···39117···117
size1111···11···11···11···1

117 irreducible representations

dim111111
type+
imageC1C3C9C13C39C117
kernelC117C39C13C9C3C1
# reps126122472

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

Subgroup lattice of C117 in TeX

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