direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D250, C2×D125, C250⋊C2, C5.D50, C125⋊C22, C25.D10, C50.2D5, C10.2D25, sometimes denoted D500 or Dih250 or Dih500, SmallGroup(500,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C125 — D250 |
Generators and relations for D250
G = < a,b | a250=b2=1, bab=a-1 >
Smallest permutation representation of D250
►On 250 points
Generators in S250
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250)
(1 250)(2 249)(3 248)(4 247)(5 246)(6 245)(7 244)(8 243)(9 242)(10 241)(11 240)(12 239)(13 238)(14 237)(15 236)(16 235)(17 234)(18 233)(19 232)(20 231)(21 230)(22 229)(23 228)(24 227)(25 226)(26 225)(27 224)(28 223)(29 222)(30 221)(31 220)(32 219)(33 218)(34 217)(35 216)(36 215)(37 214)(38 213)(39 212)(40 211)(41 210)(42 209)(43 208)(44 207)(45 206)(46 205)(47 204)(48 203)(49 202)(50 201)(51 200)(52 199)(53 198)(54 197)(55 196)(56 195)(57 194)(58 193)(59 192)(60 191)(61 190)(62 189)(63 188)(64 187)(65 186)(66 185)(67 184)(68 183)(69 182)(70 181)(71 180)(72 179)(73 178)(74 177)(75 176)(76 175)(77 174)(78 173)(79 172)(80 171)(81 170)(82 169)(83 168)(84 167)(85 166)(86 165)(87 164)(88 163)(89 162)(90 161)(91 160)(92 159)(93 158)(94 157)(95 156)(96 155)(97 154)(98 153)(99 152)(100 151)(101 150)(102 149)(103 148)(104 147)(105 146)(106 145)(107 144)(108 143)(109 142)(110 141)(111 140)(112 139)(113 138)(114 137)(115 136)(116 135)(117 134)(118 133)(119 132)(120 131)(121 130)(122 129)(123 128)(124 127)(125 126)
G:=sub<Sym(250)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250), (1,250)(2,249)(3,248)(4,247)(5,246)(6,245)(7,244)(8,243)(9,242)(10,241)(11,240)(12,239)(13,238)(14,237)(15,236)(16,235)(17,234)(18,233)(19,232)(20,231)(21,230)(22,229)(23,228)(24,227)(25,226)(26,225)(27,224)(28,223)(29,222)(30,221)(31,220)(32,219)(33,218)(34,217)(35,216)(36,215)(37,214)(38,213)(39,212)(40,211)(41,210)(42,209)(43,208)(44,207)(45,206)(46,205)(47,204)(48,203)(49,202)(50,201)(51,200)(52,199)(53,198)(54,197)(55,196)(56,195)(57,194)(58,193)(59,192)(60,191)(61,190)(62,189)(63,188)(64,187)(65,186)(66,185)(67,184)(68,183)(69,182)(70,181)(71,180)(72,179)(73,178)(74,177)(75,176)(76,175)(77,174)(78,173)(79,172)(80,171)(81,170)(82,169)(83,168)(84,167)(85,166)(86,165)(87,164)(88,163)(89,162)(90,161)(91,160)(92,159)(93,158)(94,157)(95,156)(96,155)(97,154)(98,153)(99,152)(100,151)(101,150)(102,149)(103,148)(104,147)(105,146)(106,145)(107,144)(108,143)(109,142)(110,141)(111,140)(112,139)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250), (1,250)(2,249)(3,248)(4,247)(5,246)(6,245)(7,244)(8,243)(9,242)(10,241)(11,240)(12,239)(13,238)(14,237)(15,236)(16,235)(17,234)(18,233)(19,232)(20,231)(21,230)(22,229)(23,228)(24,227)(25,226)(26,225)(27,224)(28,223)(29,222)(30,221)(31,220)(32,219)(33,218)(34,217)(35,216)(36,215)(37,214)(38,213)(39,212)(40,211)(41,210)(42,209)(43,208)(44,207)(45,206)(46,205)(47,204)(48,203)(49,202)(50,201)(51,200)(52,199)(53,198)(54,197)(55,196)(56,195)(57,194)(58,193)(59,192)(60,191)(61,190)(62,189)(63,188)(64,187)(65,186)(66,185)(67,184)(68,183)(69,182)(70,181)(71,180)(72,179)(73,178)(74,177)(75,176)(76,175)(77,174)(78,173)(79,172)(80,171)(81,170)(82,169)(83,168)(84,167)(85,166)(86,165)(87,164)(88,163)(89,162)(90,161)(91,160)(92,159)(93,158)(94,157)(95,156)(96,155)(97,154)(98,153)(99,152)(100,151)(101,150)(102,149)(103,148)(104,147)(105,146)(106,145)(107,144)(108,143)(109,142)(110,141)(111,140)(112,139)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250)], [(1,250),(2,249),(3,248),(4,247),(5,246),(6,245),(7,244),(8,243),(9,242),(10,241),(11,240),(12,239),(13,238),(14,237),(15,236),(16,235),(17,234),(18,233),(19,232),(20,231),(21,230),(22,229),(23,228),(24,227),(25,226),(26,225),(27,224),(28,223),(29,222),(30,221),(31,220),(32,219),(33,218),(34,217),(35,216),(36,215),(37,214),(38,213),(39,212),(40,211),(41,210),(42,209),(43,208),(44,207),(45,206),(46,205),(47,204),(48,203),(49,202),(50,201),(51,200),(52,199),(53,198),(54,197),(55,196),(56,195),(57,194),(58,193),(59,192),(60,191),(61,190),(62,189),(63,188),(64,187),(65,186),(66,185),(67,184),(68,183),(69,182),(70,181),(71,180),(72,179),(73,178),(74,177),(75,176),(76,175),(77,174),(78,173),(79,172),(80,171),(81,170),(82,169),(83,168),(84,167),(85,166),(86,165),(87,164),(88,163),(89,162),(90,161),(91,160),(92,159),(93,158),(94,157),(95,156),(96,155),(97,154),(98,153),(99,152),(100,151),(101,150),(102,149),(103,148),(104,147),(105,146),(106,145),(107,144),(108,143),(109,142),(110,141),(111,140),(112,139),(113,138),(114,137),(115,136),(116,135),(117,134),(118,133),(119,132),(120,131),(121,130),(122,129),(123,128),(124,127),(125,126)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 25A | ··· | 25J | 50A | ··· | 50J | 125A | ··· | 125AX | 250A | ··· | 250AX |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 25 | ··· | 25 | 50 | ··· | 50 | 125 | ··· | 125 | 250 | ··· | 250 |
| size | 1 | 1 | 125 | 125 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D5 | D10 | D25 | D50 | D125 | D250 |
| kernel | D250 | D125 | C250 | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 10 | 10 | 50 | 50 |
Matrix representation of D250 ►in GL2(𝔽251) generated by
| 231 | 159 |
| 92 | 9 |
| 180 | 68 |
| 44 | 71 |
G:=sub<GL(2,GF(251))| [231,92,159,9],[180,44,68,71] >;
D250 in GAP, Magma, Sage, TeX
D_{250} % in TeX
G:=Group("D250"); // GroupNames label
G:=SmallGroup(500,4);
// by ID
G=gap.SmallGroup(500,4);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,542,687,3603,418,10004]);
// Polycyclic
G:=Group<a,b|a^250=b^2=1,b*a*b=a^-1>;
// generators/relations
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