metacyclic, supersoluble, monomial, 3-hyperelementary
Aliases: C117⋊2C3, C39.2C32, C13⋊13- 1+2, C13⋊C9⋊1C3, C9⋊1(C13⋊C3), C3.3(C3×C13⋊C3), (C3×C13⋊C3).1C3, SmallGroup(351,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C117⋊C3
G = < a,b | a117=b3=1, bab-1=a61 >
Smallest permutation representation of C117⋊C3
►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)
(2 95 62)(3 72 6)(4 49 67)(5 26 11)(7 97 16)(8 74 77)(9 51 21)(10 28 82)(12 99 87)(13 76 31)(14 53 92)(15 30 36)(17 101 41)(18 78 102)(19 55 46)(20 32 107)(22 103 112)(23 80 56)(24 57 117)(25 34 61)(27 105 66)(29 59 71)(33 84 81)(35 38 86)(37 109 91)(39 63 96)(42 111 45)(43 88 106)(44 65 50)(47 113 116)(48 90 60)(52 115 70)(54 69 75)(58 94 85)(64 73 100)(68 98 110)(83 104 89)(93 108 114)
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), (2,95,62)(3,72,6)(4,49,67)(5,26,11)(7,97,16)(8,74,77)(9,51,21)(10,28,82)(12,99,87)(13,76,31)(14,53,92)(15,30,36)(17,101,41)(18,78,102)(19,55,46)(20,32,107)(22,103,112)(23,80,56)(24,57,117)(25,34,61)(27,105,66)(29,59,71)(33,84,81)(35,38,86)(37,109,91)(39,63,96)(42,111,45)(43,88,106)(44,65,50)(47,113,116)(48,90,60)(52,115,70)(54,69,75)(58,94,85)(64,73,100)(68,98,110)(83,104,89)(93,108,114)>;
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), (2,95,62)(3,72,6)(4,49,67)(5,26,11)(7,97,16)(8,74,77)(9,51,21)(10,28,82)(12,99,87)(13,76,31)(14,53,92)(15,30,36)(17,101,41)(18,78,102)(19,55,46)(20,32,107)(22,103,112)(23,80,56)(24,57,117)(25,34,61)(27,105,66)(29,59,71)(33,84,81)(35,38,86)(37,109,91)(39,63,96)(42,111,45)(43,88,106)(44,65,50)(47,113,116)(48,90,60)(52,115,70)(54,69,75)(58,94,85)(64,73,100)(68,98,110)(83,104,89)(93,108,114) );
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)], [(2,95,62),(3,72,6),(4,49,67),(5,26,11),(7,97,16),(8,74,77),(9,51,21),(10,28,82),(12,99,87),(13,76,31),(14,53,92),(15,30,36),(17,101,41),(18,78,102),(19,55,46),(20,32,107),(22,103,112),(23,80,56),(24,57,117),(25,34,61),(27,105,66),(29,59,71),(33,84,81),(35,38,86),(37,109,91),(39,63,96),(42,111,45),(43,88,106),(44,65,50),(47,113,116),(48,90,60),(52,115,70),(54,69,75),(58,94,85),(64,73,100),(68,98,110),(83,104,89),(93,108,114)]])
47 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 9A | 9B | 9C | 9D | 9E | 9F | 13A | 13B | 13C | 13D | 39A | ··· | 39H | 117A | ··· | 117X |
| order | 1 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 13 | 13 | 13 | 13 | 39 | ··· | 39 | 117 | ··· | 117 |
| size | 1 | 1 | 1 | 39 | 39 | 3 | 3 | 39 | 39 | 39 | 39 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C3 | 3- 1+2 | C13⋊C3 | C3×C13⋊C3 | C117⋊C3 |
| kernel | C117⋊C3 | C13⋊C9 | C117 | C3×C13⋊C3 | C13 | C9 | C3 | C1 |
| # reps | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 24 |
Matrix representation of C117⋊C3 ►in GL3(𝔽937) generated by
| 192 | 541 | 651 |
| 167 | 72 | 142 |
| 541 | 651 | 838 |
| 322 | 322 | 921 |
| 0 | 921 | 742 |
| 0 | 127 | 631 |
G:=sub<GL(3,GF(937))| [192,167,541,541,72,651,651,142,838],[322,0,0,322,921,127,921,742,631] >;
C117⋊C3 in GAP, Magma, Sage, TeX
C_{117}\rtimes C_3 % in TeX
G:=Group("C117:C3"); // GroupNames label
G:=SmallGroup(351,4);
// by ID
G=gap.SmallGroup(351,4);
# by ID
G:=PCGroup([4,-3,-3,-3,-13,97,29,1299]);
// Polycyclic
G:=Group<a,b|a^117=b^3=1,b*a*b^-1=a^61>;
// generators/relations
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