metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D137, C137⋊C2, sometimes denoted D274 or Dih137 or Dih274, SmallGroup(274,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C137 — D137 |
Generators and relations for D137
G = < a,b | a137=b2=1, bab=a-1 >
Smallest permutation representation of D137
►On 137 points: primitive
Generators in S137
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137)
(1 137)(2 136)(3 135)(4 134)(5 133)(6 132)(7 131)(8 130)(9 129)(10 128)(11 127)(12 126)(13 125)(14 124)(15 123)(16 122)(17 121)(18 120)(19 119)(20 118)(21 117)(22 116)(23 115)(24 114)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)
G:=sub<Sym(137)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137), (1,137)(2,136)(3,135)(4,134)(5,133)(6,132)(7,131)(8,130)(9,129)(10,128)(11,127)(12,126)(13,125)(14,124)(15,123)(16,122)(17,121)(18,120)(19,119)(20,118)(21,117)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137), (1,137)(2,136)(3,135)(4,134)(5,133)(6,132)(7,131)(8,130)(9,129)(10,128)(11,127)(12,126)(13,125)(14,124)(15,123)(16,122)(17,121)(18,120)(19,119)(20,118)(21,117)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137)], [(1,137),(2,136),(3,135),(4,134),(5,133),(6,132),(7,131),(8,130),(9,129),(10,128),(11,127),(12,126),(13,125),(14,124),(15,123),(16,122),(17,121),(18,120),(19,119),(20,118),(21,117),(22,116),(23,115),(24,114),(25,113),(26,112),(27,111),(28,110),(29,109),(30,108),(31,107),(32,106),(33,105),(34,104),(35,103),(36,102),(37,101),(38,100),(39,99),(40,98),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70)]])
70 conjugacy classes
| class | 1 | 2 | 137A | ··· | 137BP |
| order | 1 | 2 | 137 | ··· | 137 |
| size | 1 | 137 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D137 |
| kernel | D137 | C137 | C1 |
| # reps | 1 | 1 | 68 |
Matrix representation of D137 ►in GL2(𝔽823) generated by
| 521 | 822 |
| 1 | 0 |
| 521 | 822 |
| 673 | 302 |
G:=sub<GL(2,GF(823))| [521,1,822,0],[521,673,822,302] >;
D137 in GAP, Magma, Sage, TeX
D_{137} % in TeX
G:=Group("D137"); // GroupNames label
G:=SmallGroup(274,1);
// by ID
G=gap.SmallGroup(274,1);
# by ID
G:=PCGroup([2,-2,-137,1089]);
// Polycyclic
G:=Group<a,b|a^137=b^2=1,b*a*b=a^-1>;
// generators/relations
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