metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D177, C59⋊S3, C3⋊D59, C177⋊1C2, sometimes denoted D354 or Dih177 or Dih354, SmallGroup(354,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C177 — D177 |
Generators and relations for D177
G = < a,b | a177=b2=1, bab=a-1 >
Smallest permutation representation of D177
►On 177 points
Generators in S177
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177)
(1 177)(2 176)(3 175)(4 174)(5 173)(6 172)(7 171)(8 170)(9 169)(10 168)(11 167)(12 166)(13 165)(14 164)(15 163)(16 162)(17 161)(18 160)(19 159)(20 158)(21 157)(22 156)(23 155)(24 154)(25 153)(26 152)(27 151)(28 150)(29 149)(30 148)(31 147)(32 146)(33 145)(34 144)(35 143)(36 142)(37 141)(38 140)(39 139)(40 138)(41 137)(42 136)(43 135)(44 134)(45 133)(46 132)(47 131)(48 130)(49 129)(50 128)(51 127)(52 126)(53 125)(54 124)(55 123)(56 122)(57 121)(58 120)(59 119)(60 118)(61 117)(62 116)(63 115)(64 114)(65 113)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)
G:=sub<Sym(177)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177), (1,177)(2,176)(3,175)(4,174)(5,173)(6,172)(7,171)(8,170)(9,169)(10,168)(11,167)(12,166)(13,165)(14,164)(15,163)(16,162)(17,161)(18,160)(19,159)(20,158)(21,157)(22,156)(23,155)(24,154)(25,153)(26,152)(27,151)(28,150)(29,149)(30,148)(31,147)(32,146)(33,145)(34,144)(35,143)(36,142)(37,141)(38,140)(39,139)(40,138)(41,137)(42,136)(43,135)(44,134)(45,133)(46,132)(47,131)(48,130)(49,129)(50,128)(51,127)(52,126)(53,125)(54,124)(55,123)(56,122)(57,121)(58,120)(59,119)(60,118)(61,117)(62,116)(63,115)(64,114)(65,113)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177), (1,177)(2,176)(3,175)(4,174)(5,173)(6,172)(7,171)(8,170)(9,169)(10,168)(11,167)(12,166)(13,165)(14,164)(15,163)(16,162)(17,161)(18,160)(19,159)(20,158)(21,157)(22,156)(23,155)(24,154)(25,153)(26,152)(27,151)(28,150)(29,149)(30,148)(31,147)(32,146)(33,145)(34,144)(35,143)(36,142)(37,141)(38,140)(39,139)(40,138)(41,137)(42,136)(43,135)(44,134)(45,133)(46,132)(47,131)(48,130)(49,129)(50,128)(51,127)(52,126)(53,125)(54,124)(55,123)(56,122)(57,121)(58,120)(59,119)(60,118)(61,117)(62,116)(63,115)(64,114)(65,113)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177)], [(1,177),(2,176),(3,175),(4,174),(5,173),(6,172),(7,171),(8,170),(9,169),(10,168),(11,167),(12,166),(13,165),(14,164),(15,163),(16,162),(17,161),(18,160),(19,159),(20,158),(21,157),(22,156),(23,155),(24,154),(25,153),(26,152),(27,151),(28,150),(29,149),(30,148),(31,147),(32,146),(33,145),(34,144),(35,143),(36,142),(37,141),(38,140),(39,139),(40,138),(41,137),(42,136),(43,135),(44,134),(45,133),(46,132),(47,131),(48,130),(49,129),(50,128),(51,127),(52,126),(53,125),(54,124),(55,123),(56,122),(57,121),(58,120),(59,119),(60,118),(61,117),(62,116),(63,115),(64,114),(65,113),(66,112),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90)]])
90 conjugacy classes
| class | 1 | 2 | 3 | 59A | ··· | 59AC | 177A | ··· | 177BF |
| order | 1 | 2 | 3 | 59 | ··· | 59 | 177 | ··· | 177 |
| size | 1 | 177 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D59 | D177 |
| kernel | D177 | C177 | C59 | C3 | C1 |
| # reps | 1 | 1 | 1 | 29 | 58 |
Matrix representation of D177 ►in GL2(𝔽709) generated by
| 388 | 460 |
| 249 | 308 |
| 388 | 460 |
| 485 | 321 |
G:=sub<GL(2,GF(709))| [388,249,460,308],[388,485,460,321] >;
D177 in GAP, Magma, Sage, TeX
D_{177} % in TeX
G:=Group("D177"); // GroupNames label
G:=SmallGroup(354,3);
// by ID
G=gap.SmallGroup(354,3);
# by ID
G:=PCGroup([3,-2,-3,-59,25,3134]);
// Polycyclic
G:=Group<a,b|a^177=b^2=1,b*a*b=a^-1>;
// generators/relations
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