direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D190, C2×D95, C38⋊D5, C10⋊D19, C5⋊2D38, C19⋊2D10, C190⋊1C2, C95⋊2C22, sometimes denoted D380 or Dih190 or Dih380, SmallGroup(380,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C95 — D190 |
Generators and relations for D190
G = < a,b | a190=b2=1, bab=a-1 >
Smallest permutation representation of D190
►On 190 points
Generators in S190
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190)
(1 190)(2 189)(3 188)(4 187)(5 186)(6 185)(7 184)(8 183)(9 182)(10 181)(11 180)(12 179)(13 178)(14 177)(15 176)(16 175)(17 174)(18 173)(19 172)(20 171)(21 170)(22 169)(23 168)(24 167)(25 166)(26 165)(27 164)(28 163)(29 162)(30 161)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(41 150)(42 149)(43 148)(44 147)(45 146)(46 145)(47 144)(48 143)(49 142)(50 141)(51 140)(52 139)(53 138)(54 137)(55 136)(56 135)(57 134)(58 133)(59 132)(60 131)(61 130)(62 129)(63 128)(64 127)(65 126)(66 125)(67 124)(68 123)(69 122)(70 121)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)
G:=sub<Sym(190)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190), (1,190)(2,189)(3,188)(4,187)(5,186)(6,185)(7,184)(8,183)(9,182)(10,181)(11,180)(12,179)(13,178)(14,177)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190), (1,190)(2,189)(3,188)(4,187)(5,186)(6,185)(7,184)(8,183)(9,182)(10,181)(11,180)(12,179)(13,178)(14,177)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190)], [(1,190),(2,189),(3,188),(4,187),(5,186),(6,185),(7,184),(8,183),(9,182),(10,181),(11,180),(12,179),(13,178),(14,177),(15,176),(16,175),(17,174),(18,173),(19,172),(20,171),(21,170),(22,169),(23,168),(24,167),(25,166),(26,165),(27,164),(28,163),(29,162),(30,161),(31,160),(32,159),(33,158),(34,157),(35,156),(36,155),(37,154),(38,153),(39,152),(40,151),(41,150),(42,149),(43,148),(44,147),(45,146),(46,145),(47,144),(48,143),(49,142),(50,141),(51,140),(52,139),(53,138),(54,137),(55,136),(56,135),(57,134),(58,133),(59,132),(60,131),(61,130),(62,129),(63,128),(64,127),(65,126),(66,125),(67,124),(68,123),(69,122),(70,121),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,110),(82,109),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 19A | ··· | 19I | 38A | ··· | 38I | 95A | ··· | 95AJ | 190A | ··· | 190AJ |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 19 | ··· | 19 | 38 | ··· | 38 | 95 | ··· | 95 | 190 | ··· | 190 |
| size | 1 | 1 | 95 | 95 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D5 | D10 | D19 | D38 | D95 | D190 |
| kernel | D190 | D95 | C190 | C38 | C19 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 9 | 9 | 36 | 36 |
Matrix representation of D190 ►in GL4(𝔽191) generated by
| 44 | 79 | 0 | 0 |
| 100 | 45 | 0 | 0 |
| 0 | 0 | 114 | 130 |
| 0 | 0 | 61 | 88 |
| 44 | 79 | 0 | 0 |
| 152 | 147 | 0 | 0 |
| 0 | 0 | 114 | 130 |
| 0 | 0 | 69 | 77 |
G:=sub<GL(4,GF(191))| [44,100,0,0,79,45,0,0,0,0,114,61,0,0,130,88],[44,152,0,0,79,147,0,0,0,0,114,69,0,0,130,77] >;
D190 in GAP, Magma, Sage, TeX
D_{190} % in TeX
G:=Group("D190"); // GroupNames label
G:=SmallGroup(380,10);
// by ID
G=gap.SmallGroup(380,10);
# by ID
G:=PCGroup([4,-2,-2,-5,-19,194,5763]);
// Polycyclic
G:=Group<a,b|a^190=b^2=1,b*a*b=a^-1>;
// generators/relations
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