metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D239, C239⋊C2, sometimes denoted D478 or Dih239 or Dih478, SmallGroup(478,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C239 — D239 |
Generators and relations for D239
G = < a,b | a239=b2=1, bab=a-1 >
Smallest permutation representation of D239
►On 239 points: primitive
Generators in S239
(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)
(1 239)(2 238)(3 237)(4 236)(5 235)(6 234)(7 233)(8 232)(9 231)(10 230)(11 229)(12 228)(13 227)(14 226)(15 225)(16 224)(17 223)(18 222)(19 221)(20 220)(21 219)(22 218)(23 217)(24 216)(25 215)(26 214)(27 213)(28 212)(29 211)(30 210)(31 209)(32 208)(33 207)(34 206)(35 205)(36 204)(37 203)(38 202)(39 201)(40 200)(41 199)(42 198)(43 197)(44 196)(45 195)(46 194)(47 193)(48 192)(49 191)(50 190)(51 189)(52 188)(53 187)(54 186)(55 185)(56 184)(57 183)(58 182)(59 181)(60 180)(61 179)(62 178)(63 177)(64 176)(65 175)(66 174)(67 173)(68 172)(69 171)(70 170)(71 169)(72 168)(73 167)(74 166)(75 165)(76 164)(77 163)(78 162)(79 161)(80 160)(81 159)(82 158)(83 157)(84 156)(85 155)(86 154)(87 153)(88 152)(89 151)(90 150)(91 149)(92 148)(93 147)(94 146)(95 145)(96 144)(97 143)(98 142)(99 141)(100 140)(101 139)(102 138)(103 137)(104 136)(105 135)(106 134)(107 133)(108 132)(109 131)(110 130)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)
G:=sub<Sym(239)| (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), (1,239)(2,238)(3,237)(4,236)(5,235)(6,234)(7,233)(8,232)(9,231)(10,230)(11,229)(12,228)(13,227)(14,226)(15,225)(16,224)(17,223)(18,222)(19,221)(20,220)(21,219)(22,218)(23,217)(24,216)(25,215)(26,214)(27,213)(28,212)(29,211)(30,210)(31,209)(32,208)(33,207)(34,206)(35,205)(36,204)(37,203)(38,202)(39,201)(40,200)(41,199)(42,198)(43,197)(44,196)(45,195)(46,194)(47,193)(48,192)(49,191)(50,190)(51,189)(52,188)(53,187)(54,186)(55,185)(56,184)(57,183)(58,182)(59,181)(60,180)(61,179)(62,178)(63,177)(64,176)(65,175)(66,174)(67,173)(68,172)(69,171)(70,170)(71,169)(72,168)(73,167)(74,166)(75,165)(76,164)(77,163)(78,162)(79,161)(80,160)(81,159)(82,158)(83,157)(84,156)(85,155)(86,154)(87,153)(88,152)(89,151)(90,150)(91,149)(92,148)(93,147)(94,146)(95,145)(96,144)(97,143)(98,142)(99,141)(100,140)(101,139)(102,138)(103,137)(104,136)(105,135)(106,134)(107,133)(108,132)(109,131)(110,130)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)>;
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), (1,239)(2,238)(3,237)(4,236)(5,235)(6,234)(7,233)(8,232)(9,231)(10,230)(11,229)(12,228)(13,227)(14,226)(15,225)(16,224)(17,223)(18,222)(19,221)(20,220)(21,219)(22,218)(23,217)(24,216)(25,215)(26,214)(27,213)(28,212)(29,211)(30,210)(31,209)(32,208)(33,207)(34,206)(35,205)(36,204)(37,203)(38,202)(39,201)(40,200)(41,199)(42,198)(43,197)(44,196)(45,195)(46,194)(47,193)(48,192)(49,191)(50,190)(51,189)(52,188)(53,187)(54,186)(55,185)(56,184)(57,183)(58,182)(59,181)(60,180)(61,179)(62,178)(63,177)(64,176)(65,175)(66,174)(67,173)(68,172)(69,171)(70,170)(71,169)(72,168)(73,167)(74,166)(75,165)(76,164)(77,163)(78,162)(79,161)(80,160)(81,159)(82,158)(83,157)(84,156)(85,155)(86,154)(87,153)(88,152)(89,151)(90,150)(91,149)(92,148)(93,147)(94,146)(95,145)(96,144)(97,143)(98,142)(99,141)(100,140)(101,139)(102,138)(103,137)(104,136)(105,135)(106,134)(107,133)(108,132)(109,131)(110,130)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121) );
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)], [(1,239),(2,238),(3,237),(4,236),(5,235),(6,234),(7,233),(8,232),(9,231),(10,230),(11,229),(12,228),(13,227),(14,226),(15,225),(16,224),(17,223),(18,222),(19,221),(20,220),(21,219),(22,218),(23,217),(24,216),(25,215),(26,214),(27,213),(28,212),(29,211),(30,210),(31,209),(32,208),(33,207),(34,206),(35,205),(36,204),(37,203),(38,202),(39,201),(40,200),(41,199),(42,198),(43,197),(44,196),(45,195),(46,194),(47,193),(48,192),(49,191),(50,190),(51,189),(52,188),(53,187),(54,186),(55,185),(56,184),(57,183),(58,182),(59,181),(60,180),(61,179),(62,178),(63,177),(64,176),(65,175),(66,174),(67,173),(68,172),(69,171),(70,170),(71,169),(72,168),(73,167),(74,166),(75,165),(76,164),(77,163),(78,162),(79,161),(80,160),(81,159),(82,158),(83,157),(84,156),(85,155),(86,154),(87,153),(88,152),(89,151),(90,150),(91,149),(92,148),(93,147),(94,146),(95,145),(96,144),(97,143),(98,142),(99,141),(100,140),(101,139),(102,138),(103,137),(104,136),(105,135),(106,134),(107,133),(108,132),(109,131),(110,130),(111,129),(112,128),(113,127),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121)]])
121 conjugacy classes
| class | 1 | 2 | 239A | ··· | 239DO |
| order | 1 | 2 | 239 | ··· | 239 |
| size | 1 | 239 | 2 | ··· | 2 |
121 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D239 |
| kernel | D239 | C239 | C1 |
| # reps | 1 | 1 | 119 |
Matrix representation of D239 ►in GL2(𝔽479) generated by
| 47 | 478 |
| 1 | 0 |
| 47 | 478 |
| 292 | 432 |
G:=sub<GL(2,GF(479))| [47,1,478,0],[47,292,478,432] >;
D239 in GAP, Magma, Sage, TeX
D_{239} % in TeX
G:=Group("D239"); // GroupNames label
G:=SmallGroup(478,1);
// by ID
G=gap.SmallGroup(478,1);
# by ID
G:=PCGroup([2,-2,-239,1905]);
// Polycyclic
G:=Group<a,b|a^239=b^2=1,b*a*b=a^-1>;
// generators/relations
Export