metacyclic, supersoluble, monomial, 3-hyperelementary
Aliases: C117⋊3C3, C39.3C32, C13⋊23- 1+2, C13⋊C9⋊2C3, C9⋊2(C13⋊C3), C3.4(C3×C13⋊C3), (C3×C13⋊C3).2C3, SmallGroup(351,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C117⋊3C3
G = < a,b | a117=b3=1, bab-1=a22 >
Smallest permutation representation of C117⋊3C3
►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 17 23)(3 33 45)(4 49 67)(5 65 89)(6 81 111)(7 97 16)(8 113 38)(9 12 60)(10 28 82)(11 44 104)(13 76 31)(14 92 53)(15 108 75)(18 39 24)(19 55 46)(20 71 68)(21 87 90)(22 103 112)(25 34 61)(26 50 83)(27 66 105)(29 98 32)(30 114 54)(35 77 47)(36 93 69)(37 109 91)(41 56 62)(42 72 84)(43 88 106)(48 51 99)(52 115 70)(57 78 63)(58 94 85)(59 110 107)(64 73 100)(74 116 86)(80 95 101)(96 117 102)
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), (2,17,23)(3,33,45)(4,49,67)(5,65,89)(6,81,111)(7,97,16)(8,113,38)(9,12,60)(10,28,82)(11,44,104)(13,76,31)(14,92,53)(15,108,75)(18,39,24)(19,55,46)(20,71,68)(21,87,90)(22,103,112)(25,34,61)(26,50,83)(27,66,105)(29,98,32)(30,114,54)(35,77,47)(36,93,69)(37,109,91)(41,56,62)(42,72,84)(43,88,106)(48,51,99)(52,115,70)(57,78,63)(58,94,85)(59,110,107)(64,73,100)(74,116,86)(80,95,101)(96,117,102)>;
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), (2,17,23)(3,33,45)(4,49,67)(5,65,89)(6,81,111)(7,97,16)(8,113,38)(9,12,60)(10,28,82)(11,44,104)(13,76,31)(14,92,53)(15,108,75)(18,39,24)(19,55,46)(20,71,68)(21,87,90)(22,103,112)(25,34,61)(26,50,83)(27,66,105)(29,98,32)(30,114,54)(35,77,47)(36,93,69)(37,109,91)(41,56,62)(42,72,84)(43,88,106)(48,51,99)(52,115,70)(57,78,63)(58,94,85)(59,110,107)(64,73,100)(74,116,86)(80,95,101)(96,117,102) );
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)], [(2,17,23),(3,33,45),(4,49,67),(5,65,89),(6,81,111),(7,97,16),(8,113,38),(9,12,60),(10,28,82),(11,44,104),(13,76,31),(14,92,53),(15,108,75),(18,39,24),(19,55,46),(20,71,68),(21,87,90),(22,103,112),(25,34,61),(26,50,83),(27,66,105),(29,98,32),(30,114,54),(35,77,47),(36,93,69),(37,109,91),(41,56,62),(42,72,84),(43,88,106),(48,51,99),(52,115,70),(57,78,63),(58,94,85),(59,110,107),(64,73,100),(74,116,86),(80,95,101),(96,117,102)]])
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⋊3C3 |
| kernel | C117⋊3C3 | C13⋊C9 | C117 | C3×C13⋊C3 | C13 | C9 | C3 | C1 |
| # reps | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 24 |
Matrix representation of C117⋊3C3 ►in GL3(𝔽937) generated by
| 646 | 549 | 548 |
| 85 | 678 | 505 |
| 38 | 852 | 250 |
| 761 | 547 | 75 |
| 153 | 311 | 870 |
| 143 | 778 | 802 |
G:=sub<GL(3,GF(937))| [646,85,38,549,678,852,548,505,250],[761,153,143,547,311,778,75,870,802] >;
C117⋊3C3 in GAP, Magma, Sage, TeX
C_{117}\rtimes_3C_3 % in TeX
G:=Group("C117:3C3"); // GroupNames label
G:=SmallGroup(351,5);
// by ID
G=gap.SmallGroup(351,5);
# by ID
G:=PCGroup([4,-3,-3,-3,-13,97,53,1299]);
// Polycyclic
G:=Group<a,b|a^117=b^3=1,b*a*b^-1=a^22>;
// generators/relations
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