direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D158, C2×D79, C158⋊C2, C79⋊C22, sometimes denoted D316 or Dih158 or Dih316, SmallGroup(316,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C79 — D158 |
Generators and relations for D158
G = < a,b | a158=b2=1, bab=a-1 >
Smallest permutation representation of D158
►On 158 points
Generators in S158
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158)
(1 158)(2 157)(3 156)(4 155)(5 154)(6 153)(7 152)(8 151)(9 150)(10 149)(11 148)(12 147)(13 146)(14 145)(15 144)(16 143)(17 142)(18 141)(19 140)(20 139)(21 138)(22 137)(23 136)(24 135)(25 134)(26 133)(27 132)(28 131)(29 130)(30 129)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 120)(40 119)(41 118)(42 117)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 80)
G:=sub<Sym(158)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158), (1,158)(2,157)(3,156)(4,155)(5,154)(6,153)(7,152)(8,151)(9,150)(10,149)(11,148)(12,147)(13,146)(14,145)(15,144)(16,143)(17,142)(18,141)(19,140)(20,139)(21,138)(22,137)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158), (1,158)(2,157)(3,156)(4,155)(5,154)(6,153)(7,152)(8,151)(9,150)(10,149)(11,148)(12,147)(13,146)(14,145)(15,144)(16,143)(17,142)(18,141)(19,140)(20,139)(21,138)(22,137)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158)], [(1,158),(2,157),(3,156),(4,155),(5,154),(6,153),(7,152),(8,151),(9,150),(10,149),(11,148),(12,147),(13,146),(14,145),(15,144),(16,143),(17,142),(18,141),(19,140),(20,139),(21,138),(22,137),(23,136),(24,135),(25,134),(26,133),(27,132),(28,131),(29,130),(30,129),(31,128),(32,127),(33,126),(34,125),(35,124),(36,123),(37,122),(38,121),(39,120),(40,119),(41,118),(42,117),(43,116),(44,115),(45,114),(46,113),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,106),(54,105),(55,104),(56,103),(57,102),(58,101),(59,100),(60,99),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(76,83),(77,82),(78,81),(79,80)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 79A | ··· | 79AM | 158A | ··· | 158AM |
| order | 1 | 2 | 2 | 2 | 79 | ··· | 79 | 158 | ··· | 158 |
| size | 1 | 1 | 79 | 79 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D79 | D158 |
| kernel | D158 | D79 | C158 | C2 | C1 |
| # reps | 1 | 2 | 1 | 39 | 39 |
Matrix representation of D158 ►in GL3(𝔽317) generated by
| 316 | 0 | 0 |
| 0 | 213 | 202 |
| 0 | 269 | 206 |
| 1 | 0 | 0 |
| 0 | 73 | 259 |
| 0 | 70 | 244 |
G:=sub<GL(3,GF(317))| [316,0,0,0,213,269,0,202,206],[1,0,0,0,73,70,0,259,244] >;
D158 in GAP, Magma, Sage, TeX
D_{158} % in TeX
G:=Group("D158"); // GroupNames label
G:=SmallGroup(316,3);
// by ID
G=gap.SmallGroup(316,3);
# by ID
G:=PCGroup([3,-2,-2,-79,2810]);
// Polycyclic
G:=Group<a,b|a^158=b^2=1,b*a*b=a^-1>;
// generators/relations
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