metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D185, C37⋊D5, C5⋊D37, C185⋊1C2, sometimes denoted D370 or Dih185 or Dih370, SmallGroup(370,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C185 — D185 |
Generators and relations for D185
G = < a,b | a185=b2=1, bab=a-1 >
Smallest permutation representation of D185
►On 185 points
Generators in S185
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185)
(1 185)(2 184)(3 183)(4 182)(5 181)(6 180)(7 179)(8 178)(9 177)(10 176)(11 175)(12 174)(13 173)(14 172)(15 171)(16 170)(17 169)(18 168)(19 167)(20 166)(21 165)(22 164)(23 163)(24 162)(25 161)(26 160)(27 159)(28 158)(29 157)(30 156)(31 155)(32 154)(33 153)(34 152)(35 151)(36 150)(37 149)(38 148)(39 147)(40 146)(41 145)(42 144)(43 143)(44 142)(45 141)(46 140)(47 139)(48 138)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 125)(62 124)(63 123)(64 122)(65 121)(66 120)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)(73 113)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)
G:=sub<Sym(185)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185), (1,185)(2,184)(3,183)(4,182)(5,181)(6,180)(7,179)(8,178)(9,177)(10,176)(11,175)(12,174)(13,173)(14,172)(15,171)(16,170)(17,169)(18,168)(19,167)(20,166)(21,165)(22,164)(23,163)(24,162)(25,161)(26,160)(27,159)(28,158)(29,157)(30,156)(31,155)(32,154)(33,153)(34,152)(35,151)(36,150)(37,149)(38,148)(39,147)(40,146)(41,145)(42,144)(43,143)(44,142)(45,141)(46,140)(47,139)(48,138)(49,137)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,125)(62,124)(63,123)(64,122)(65,121)(66,120)(67,119)(68,118)(69,117)(70,116)(71,115)(72,114)(73,113)(74,112)(75,111)(76,110)(77,109)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185), (1,185)(2,184)(3,183)(4,182)(5,181)(6,180)(7,179)(8,178)(9,177)(10,176)(11,175)(12,174)(13,173)(14,172)(15,171)(16,170)(17,169)(18,168)(19,167)(20,166)(21,165)(22,164)(23,163)(24,162)(25,161)(26,160)(27,159)(28,158)(29,157)(30,156)(31,155)(32,154)(33,153)(34,152)(35,151)(36,150)(37,149)(38,148)(39,147)(40,146)(41,145)(42,144)(43,143)(44,142)(45,141)(46,140)(47,139)(48,138)(49,137)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,125)(62,124)(63,123)(64,122)(65,121)(66,120)(67,119)(68,118)(69,117)(70,116)(71,115)(72,114)(73,113)(74,112)(75,111)(76,110)(77,109)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185)], [(1,185),(2,184),(3,183),(4,182),(5,181),(6,180),(7,179),(8,178),(9,177),(10,176),(11,175),(12,174),(13,173),(14,172),(15,171),(16,170),(17,169),(18,168),(19,167),(20,166),(21,165),(22,164),(23,163),(24,162),(25,161),(26,160),(27,159),(28,158),(29,157),(30,156),(31,155),(32,154),(33,153),(34,152),(35,151),(36,150),(37,149),(38,148),(39,147),(40,146),(41,145),(42,144),(43,143),(44,142),(45,141),(46,140),(47,139),(48,138),(49,137),(50,136),(51,135),(52,134),(53,133),(54,132),(55,131),(56,130),(57,129),(58,128),(59,127),(60,126),(61,125),(62,124),(63,123),(64,122),(65,121),(66,120),(67,119),(68,118),(69,117),(70,116),(71,115),(72,114),(73,113),(74,112),(75,111),(76,110),(77,109),(78,108),(79,107),(80,106),(81,105),(82,104),(83,103),(84,102),(85,101),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94)]])
94 conjugacy classes
| class | 1 | 2 | 5A | 5B | 37A | ··· | 37R | 185A | ··· | 185BT |
| order | 1 | 2 | 5 | 5 | 37 | ··· | 37 | 185 | ··· | 185 |
| size | 1 | 185 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D5 | D37 | D185 |
| kernel | D185 | C185 | C37 | C5 | C1 |
| # reps | 1 | 1 | 2 | 18 | 72 |
Matrix representation of D185 ►in GL2(𝔽1481) generated by
| 618 | 918 |
| 563 | 1040 |
| 618 | 918 |
| 147 | 863 |
G:=sub<GL(2,GF(1481))| [618,563,918,1040],[618,147,918,863] >;
D185 in GAP, Magma, Sage, TeX
D_{185} % in TeX
G:=Group("D185"); // GroupNames label
G:=SmallGroup(370,3);
// by ID
G=gap.SmallGroup(370,3);
# by ID
G:=PCGroup([3,-2,-5,-37,49,3242]);
// Polycyclic
G:=Group<a,b|a^185=b^2=1,b*a*b=a^-1>;
// generators/relations
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