metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D172, C4⋊D43, C43⋊1D4, C172⋊1C2, D86⋊1C2, C2.4D86, C86.3C22, sometimes denoted D344 or Dih172 or Dih344, SmallGroup(344,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D172
G = < a,b | a172=b2=1, bab=a-1 >
Smallest permutation representation of D172
►On 172 points
Generators in S172
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172)
(1 172)(2 171)(3 170)(4 169)(5 168)(6 167)(7 166)(8 165)(9 164)(10 163)(11 162)(12 161)(13 160)(14 159)(15 158)(16 157)(17 156)(18 155)(19 154)(20 153)(21 152)(22 151)(23 150)(24 149)(25 148)(26 147)(27 146)(28 145)(29 144)(30 143)(31 142)(32 141)(33 140)(34 139)(35 138)(36 137)(37 136)(38 135)(39 134)(40 133)(41 132)(42 131)(43 130)(44 129)(45 128)(46 127)(47 126)(48 125)(49 124)(50 123)(51 122)(52 121)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)(61 112)(62 111)(63 110)(64 109)(65 108)(66 107)(67 106)(68 105)(69 104)(70 103)(71 102)(72 101)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)
G:=sub<Sym(172)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,172)(2,171)(3,170)(4,169)(5,168)(6,167)(7,166)(8,165)(9,164)(10,163)(11,162)(12,161)(13,160)(14,159)(15,158)(16,157)(17,156)(18,155)(19,154)(20,153)(21,152)(22,151)(23,150)(24,149)(25,148)(26,147)(27,146)(28,145)(29,144)(30,143)(31,142)(32,141)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,172)(2,171)(3,170)(4,169)(5,168)(6,167)(7,166)(8,165)(9,164)(10,163)(11,162)(12,161)(13,160)(14,159)(15,158)(16,157)(17,156)(18,155)(19,154)(20,153)(21,152)(22,151)(23,150)(24,149)(25,148)(26,147)(27,146)(28,145)(29,144)(30,143)(31,142)(32,141)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172)], [(1,172),(2,171),(3,170),(4,169),(5,168),(6,167),(7,166),(8,165),(9,164),(10,163),(11,162),(12,161),(13,160),(14,159),(15,158),(16,157),(17,156),(18,155),(19,154),(20,153),(21,152),(22,151),(23,150),(24,149),(25,148),(26,147),(27,146),(28,145),(29,144),(30,143),(31,142),(32,141),(33,140),(34,139),(35,138),(36,137),(37,136),(38,135),(39,134),(40,133),(41,132),(42,131),(43,130),(44,129),(45,128),(46,127),(47,126),(48,125),(49,124),(50,123),(51,122),(52,121),(53,120),(54,119),(55,118),(56,117),(57,116),(58,115),(59,114),(60,113),(61,112),(62,111),(63,110),(64,109),(65,108),(66,107),(67,106),(68,105),(69,104),(70,103),(71,102),(72,101),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87)]])
89 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 43A | ··· | 43U | 86A | ··· | 86U | 172A | ··· | 172AP |
| order | 1 | 2 | 2 | 2 | 4 | 43 | ··· | 43 | 86 | ··· | 86 | 172 | ··· | 172 |
| size | 1 | 1 | 86 | 86 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
89 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D43 | D86 | D172 |
| kernel | D172 | C172 | D86 | C43 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 21 | 21 | 42 |
Matrix representation of D172 ►in GL2(𝔽173) generated by
| 112 | 50 |
| 163 | 167 |
| 123 | 111 |
| 71 | 50 |
G:=sub<GL(2,GF(173))| [112,163,50,167],[123,71,111,50] >;
D172 in GAP, Magma, Sage, TeX
D_{172} % in TeX
G:=Group("D172"); // GroupNames label
G:=SmallGroup(344,5);
// by ID
G=gap.SmallGroup(344,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-43,49,21,5379]);
// Polycyclic
G:=Group<a,b|a^172=b^2=1,b*a*b=a^-1>;
// generators/relations
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