metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D243, C243⋊C2, C81.S3, C3.D81, C27.1D9, C9.1D27, sometimes denoted D486 or Dih243 or Dih486, SmallGroup(486,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C243 — D243 |
Generators and relations for D243
G = < a,b | a243=b2=1, bab=a-1 >
Smallest permutation representation of D243
►On 243 points
Generators in S243
(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)
(2 243)(3 242)(4 241)(5 240)(6 239)(7 238)(8 237)(9 236)(10 235)(11 234)(12 233)(13 232)(14 231)(15 230)(16 229)(17 228)(18 227)(19 226)(20 225)(21 224)(22 223)(23 222)(24 221)(25 220)(26 219)(27 218)(28 217)(29 216)(30 215)(31 214)(32 213)(33 212)(34 211)(35 210)(36 209)(37 208)(38 207)(39 206)(40 205)(41 204)(42 203)(43 202)(44 201)(45 200)(46 199)(47 198)(48 197)(49 196)(50 195)(51 194)(52 193)(53 192)(54 191)(55 190)(56 189)(57 188)(58 187)(59 186)(60 185)(61 184)(62 183)(63 182)(64 181)(65 180)(66 179)(67 178)(68 177)(69 176)(70 175)(71 174)(72 173)(73 172)(74 171)(75 170)(76 169)(77 168)(78 167)(79 166)(80 165)(81 164)(82 163)(83 162)(84 161)(85 160)(86 159)(87 158)(88 157)(89 156)(90 155)(91 154)(92 153)(93 152)(94 151)(95 150)(96 149)(97 148)(98 147)(99 146)(100 145)(101 144)(102 143)(103 142)(104 141)(105 140)(106 139)(107 138)(108 137)(109 136)(110 135)(111 134)(112 133)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)
G:=sub<Sym(243)| (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), (2,243)(3,242)(4,241)(5,240)(6,239)(7,238)(8,237)(9,236)(10,235)(11,234)(12,233)(13,232)(14,231)(15,230)(16,229)(17,228)(18,227)(19,226)(20,225)(21,224)(22,223)(23,222)(24,221)(25,220)(26,219)(27,218)(28,217)(29,216)(30,215)(31,214)(32,213)(33,212)(34,211)(35,210)(36,209)(37,208)(38,207)(39,206)(40,205)(41,204)(42,203)(43,202)(44,201)(45,200)(46,199)(47,198)(48,197)(49,196)(50,195)(51,194)(52,193)(53,192)(54,191)(55,190)(56,189)(57,188)(58,187)(59,186)(60,185)(61,184)(62,183)(63,182)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,164)(82,163)(83,162)(84,161)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137)(109,136)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)>;
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), (2,243)(3,242)(4,241)(5,240)(6,239)(7,238)(8,237)(9,236)(10,235)(11,234)(12,233)(13,232)(14,231)(15,230)(16,229)(17,228)(18,227)(19,226)(20,225)(21,224)(22,223)(23,222)(24,221)(25,220)(26,219)(27,218)(28,217)(29,216)(30,215)(31,214)(32,213)(33,212)(34,211)(35,210)(36,209)(37,208)(38,207)(39,206)(40,205)(41,204)(42,203)(43,202)(44,201)(45,200)(46,199)(47,198)(48,197)(49,196)(50,195)(51,194)(52,193)(53,192)(54,191)(55,190)(56,189)(57,188)(58,187)(59,186)(60,185)(61,184)(62,183)(63,182)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,164)(82,163)(83,162)(84,161)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137)(109,136)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123) );
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)], [(2,243),(3,242),(4,241),(5,240),(6,239),(7,238),(8,237),(9,236),(10,235),(11,234),(12,233),(13,232),(14,231),(15,230),(16,229),(17,228),(18,227),(19,226),(20,225),(21,224),(22,223),(23,222),(24,221),(25,220),(26,219),(27,218),(28,217),(29,216),(30,215),(31,214),(32,213),(33,212),(34,211),(35,210),(36,209),(37,208),(38,207),(39,206),(40,205),(41,204),(42,203),(43,202),(44,201),(45,200),(46,199),(47,198),(48,197),(49,196),(50,195),(51,194),(52,193),(53,192),(54,191),(55,190),(56,189),(57,188),(58,187),(59,186),(60,185),(61,184),(62,183),(63,182),(64,181),(65,180),(66,179),(67,178),(68,177),(69,176),(70,175),(71,174),(72,173),(73,172),(74,171),(75,170),(76,169),(77,168),(78,167),(79,166),(80,165),(81,164),(82,163),(83,162),(84,161),(85,160),(86,159),(87,158),(88,157),(89,156),(90,155),(91,154),(92,153),(93,152),(94,151),(95,150),(96,149),(97,148),(98,147),(99,146),(100,145),(101,144),(102,143),(103,142),(104,141),(105,140),(106,139),(107,138),(108,137),(109,136),(110,135),(111,134),(112,133),(113,132),(114,131),(115,130),(116,129),(117,128),(118,127),(119,126),(120,125),(121,124),(122,123)]])
123 conjugacy classes
| class | 1 | 2 | 3 | 9A | 9B | 9C | 27A | ··· | 27I | 81A | ··· | 81AA | 243A | ··· | 243CC |
| order | 1 | 2 | 3 | 9 | 9 | 9 | 27 | ··· | 27 | 81 | ··· | 81 | 243 | ··· | 243 |
| size | 1 | 243 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | D9 | D27 | D81 | D243 |
| kernel | D243 | C243 | C81 | C27 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 3 | 9 | 27 | 81 |
Matrix representation of D243 ►in GL2(𝔽487) generated by
| 80 | 123 |
| 364 | 444 |
| 1 | 0 |
| 486 | 486 |
G:=sub<GL(2,GF(487))| [80,364,123,444],[1,486,0,486] >;
D243 in GAP, Magma, Sage, TeX
D_{243} % in TeX
G:=Group("D243"); // GroupNames label
G:=SmallGroup(486,1);
// by ID
G=gap.SmallGroup(486,1);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,121,187,542,284,2163,381,8104,208,11669]);
// Polycyclic
G:=Group<a,b|a^243=b^2=1,b*a*b=a^-1>;
// generators/relations
Export