metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D176, C11⋊1D16, C176⋊1C2, D88⋊1C2, C16⋊1D11, C22.1D8, C2.3D88, C4.1D44, C8.13D22, C44.24D4, C88.14C22, sometimes denoted D352 or Dih176 or Dih352, SmallGroup(352,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D176
G = < a,b | a176=b2=1, bab=a-1 >
Smallest permutation representation of D176
►On 176 points
Generators in S176
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(1 176)(2 175)(3 174)(4 173)(5 172)(6 171)(7 170)(8 169)(9 168)(10 167)(11 166)(12 165)(13 164)(14 163)(15 162)(16 161)(17 160)(18 159)(19 158)(20 157)(21 156)(22 155)(23 154)(24 153)(25 152)(26 151)(27 150)(28 149)(29 148)(30 147)(31 146)(32 145)(33 144)(34 143)(35 142)(36 141)(37 140)(38 139)(39 138)(40 137)(41 136)(42 135)(43 134)(44 133)(45 132)(46 131)(47 130)(48 129)(49 128)(50 127)(51 126)(52 125)(53 124)(54 123)(55 122)(56 121)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 104)(74 103)(75 102)(76 101)(77 100)(78 99)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)
G:=sub<Sym(176)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,176)(2,175)(3,174)(4,173)(5,172)(6,171)(7,170)(8,169)(9,168)(10,167)(11,166)(12,165)(13,164)(14,163)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,150)(28,149)(29,148)(30,147)(31,146)(32,145)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,176)(2,175)(3,174)(4,173)(5,172)(6,171)(7,170)(8,169)(9,168)(10,167)(11,166)(12,165)(13,164)(14,163)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,150)(28,149)(29,148)(30,147)(31,146)(32,145)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(1,176),(2,175),(3,174),(4,173),(5,172),(6,171),(7,170),(8,169),(9,168),(10,167),(11,166),(12,165),(13,164),(14,163),(15,162),(16,161),(17,160),(18,159),(19,158),(20,157),(21,156),(22,155),(23,154),(24,153),(25,152),(26,151),(27,150),(28,149),(29,148),(30,147),(31,146),(32,145),(33,144),(34,143),(35,142),(36,141),(37,140),(38,139),(39,138),(40,137),(41,136),(42,135),(43,134),(44,133),(45,132),(46,131),(47,130),(48,129),(49,128),(50,127),(51,126),(52,125),(53,124),(54,123),(55,122),(56,121),(57,120),(58,119),(59,118),(60,117),(61,116),(62,115),(63,114),(64,113),(65,112),(66,111),(67,110),(68,109),(69,108),(70,107),(71,106),(72,105),(73,104),(74,103),(75,102),(76,101),(77,100),(78,99),(79,98),(80,97),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89)]])
91 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 11A | ··· | 11E | 16A | 16B | 16C | 16D | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T | 176A | ··· | 176AN |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 11 | ··· | 11 | 16 | 16 | 16 | 16 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 88 | 88 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
91 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D8 | D11 | D16 | D22 | D44 | D88 | D176 |
| kernel | D176 | C176 | D88 | C44 | C22 | C16 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 5 | 4 | 5 | 10 | 20 | 40 |
Matrix representation of D176 ►in GL2(𝔽353) generated by
| 341 | 174 |
| 179 | 199 |
| 341 | 174 |
| 269 | 12 |
G:=sub<GL(2,GF(353))| [341,179,174,199],[341,269,174,12] >;
D176 in GAP, Magma, Sage, TeX
D_{176} % in TeX
G:=Group("D176"); // GroupNames label
G:=SmallGroup(352,5);
// by ID
G=gap.SmallGroup(352,5);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,73,79,218,122,579,69,11525]);
// Polycyclic
G:=Group<a,b|a^176=b^2=1,b*a*b=a^-1>;
// generators/relations
Export