direct product, cyclic, abelian, monomial
Aliases: C117, also denoted Z117, SmallGroup(117,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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 C117⋊3C3
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