metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D247, C19⋊D13, C13⋊D19, C247⋊1C2, sometimes denoted D494 or Dih247 or Dih494, SmallGroup(494,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C247 — D247 |
Generators and relations for D247
G = < a,b | a247=b2=1, bab=a-1 >
Smallest permutation representation of D247
►On 247 points
Generators in S247
(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)
(1 247)(2 246)(3 245)(4 244)(5 243)(6 242)(7 241)(8 240)(9 239)(10 238)(11 237)(12 236)(13 235)(14 234)(15 233)(16 232)(17 231)(18 230)(19 229)(20 228)(21 227)(22 226)(23 225)(24 224)(25 223)(26 222)(27 221)(28 220)(29 219)(30 218)(31 217)(32 216)(33 215)(34 214)(35 213)(36 212)(37 211)(38 210)(39 209)(40 208)(41 207)(42 206)(43 205)(44 204)(45 203)(46 202)(47 201)(48 200)(49 199)(50 198)(51 197)(52 196)(53 195)(54 194)(55 193)(56 192)(57 191)(58 190)(59 189)(60 188)(61 187)(62 186)(63 185)(64 184)(65 183)(66 182)(67 181)(68 180)(69 179)(70 178)(71 177)(72 176)(73 175)(74 174)(75 173)(76 172)(77 171)(78 170)(79 169)(80 168)(81 167)(82 166)(83 165)(84 164)(85 163)(86 162)(87 161)(88 160)(89 159)(90 158)(91 157)(92 156)(93 155)(94 154)(95 153)(96 152)(97 151)(98 150)(99 149)(100 148)(101 147)(102 146)(103 145)(104 144)(105 143)(106 142)(107 141)(108 140)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)
G:=sub<Sym(247)| (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), (1,247)(2,246)(3,245)(4,244)(5,243)(6,242)(7,241)(8,240)(9,239)(10,238)(11,237)(12,236)(13,235)(14,234)(15,233)(16,232)(17,231)(18,230)(19,229)(20,228)(21,227)(22,226)(23,225)(24,224)(25,223)(26,222)(27,221)(28,220)(29,219)(30,218)(31,217)(32,216)(33,215)(34,214)(35,213)(36,212)(37,211)(38,210)(39,209)(40,208)(41,207)(42,206)(43,205)(44,204)(45,203)(46,202)(47,201)(48,200)(49,199)(50,198)(51,197)(52,196)(53,195)(54,194)(55,193)(56,192)(57,191)(58,190)(59,189)(60,188)(61,187)(62,186)(63,185)(64,184)(65,183)(66,182)(67,181)(68,180)(69,179)(70,178)(71,177)(72,176)(73,175)(74,174)(75,173)(76,172)(77,171)(78,170)(79,169)(80,168)(81,167)(82,166)(83,165)(84,164)(85,163)(86,162)(87,161)(88,160)(89,159)(90,158)(91,157)(92,156)(93,155)(94,154)(95,153)(96,152)(97,151)(98,150)(99,149)(100,148)(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)>;
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), (1,247)(2,246)(3,245)(4,244)(5,243)(6,242)(7,241)(8,240)(9,239)(10,238)(11,237)(12,236)(13,235)(14,234)(15,233)(16,232)(17,231)(18,230)(19,229)(20,228)(21,227)(22,226)(23,225)(24,224)(25,223)(26,222)(27,221)(28,220)(29,219)(30,218)(31,217)(32,216)(33,215)(34,214)(35,213)(36,212)(37,211)(38,210)(39,209)(40,208)(41,207)(42,206)(43,205)(44,204)(45,203)(46,202)(47,201)(48,200)(49,199)(50,198)(51,197)(52,196)(53,195)(54,194)(55,193)(56,192)(57,191)(58,190)(59,189)(60,188)(61,187)(62,186)(63,185)(64,184)(65,183)(66,182)(67,181)(68,180)(69,179)(70,178)(71,177)(72,176)(73,175)(74,174)(75,173)(76,172)(77,171)(78,170)(79,169)(80,168)(81,167)(82,166)(83,165)(84,164)(85,163)(86,162)(87,161)(88,160)(89,159)(90,158)(91,157)(92,156)(93,155)(94,154)(95,153)(96,152)(97,151)(98,150)(99,149)(100,148)(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125) );
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)], [(1,247),(2,246),(3,245),(4,244),(5,243),(6,242),(7,241),(8,240),(9,239),(10,238),(11,237),(12,236),(13,235),(14,234),(15,233),(16,232),(17,231),(18,230),(19,229),(20,228),(21,227),(22,226),(23,225),(24,224),(25,223),(26,222),(27,221),(28,220),(29,219),(30,218),(31,217),(32,216),(33,215),(34,214),(35,213),(36,212),(37,211),(38,210),(39,209),(40,208),(41,207),(42,206),(43,205),(44,204),(45,203),(46,202),(47,201),(48,200),(49,199),(50,198),(51,197),(52,196),(53,195),(54,194),(55,193),(56,192),(57,191),(58,190),(59,189),(60,188),(61,187),(62,186),(63,185),(64,184),(65,183),(66,182),(67,181),(68,180),(69,179),(70,178),(71,177),(72,176),(73,175),(74,174),(75,173),(76,172),(77,171),(78,170),(79,169),(80,168),(81,167),(82,166),(83,165),(84,164),(85,163),(86,162),(87,161),(88,160),(89,159),(90,158),(91,157),(92,156),(93,155),(94,154),(95,153),(96,152),(97,151),(98,150),(99,149),(100,148),(101,147),(102,146),(103,145),(104,144),(105,143),(106,142),(107,141),(108,140),(109,139),(110,138),(111,137),(112,136),(113,135),(114,134),(115,133),(116,132),(117,131),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125)]])
125 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13F | 19A | ··· | 19I | 247A | ··· | 247DD |
| order | 1 | 2 | 13 | ··· | 13 | 19 | ··· | 19 | 247 | ··· | 247 |
| size | 1 | 247 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D13 | D19 | D247 |
| kernel | D247 | C247 | C19 | C13 | C1 |
| # reps | 1 | 1 | 6 | 9 | 108 |
Matrix representation of D247 ►in GL2(𝔽1483) generated by
| 1428 | 926 |
| 557 | 356 |
| 1428 | 926 |
| 490 | 55 |
G:=sub<GL(2,GF(1483))| [1428,557,926,356],[1428,490,926,55] >;
D247 in GAP, Magma, Sage, TeX
D_{247} % in TeX
G:=Group("D247"); // GroupNames label
G:=SmallGroup(494,3);
// by ID
G=gap.SmallGroup(494,3);
# by ID
G:=PCGroup([3,-2,-13,-19,145,4214]);
// Polycyclic
G:=Group<a,b|a^247=b^2=1,b*a*b=a^-1>;
// generators/relations
Export