metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D248, C31⋊1D8, C8⋊1D31, C248⋊1C2, C4.9D62, C62.2D4, D124⋊1C2, C2.4D124, C124.9C22, sometimes denoted D496 or Dih248 or Dih496, SmallGroup(496,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D248
G = < a,b | a248=b2=1, bab=a-1 >
Smallest permutation representation of D248
►On 248 points
Generators in S248
(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)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 248)(33 247)(34 246)(35 245)(36 244)(37 243)(38 242)(39 241)(40 240)(41 239)(42 238)(43 237)(44 236)(45 235)(46 234)(47 233)(48 232)(49 231)(50 230)(51 229)(52 228)(53 227)(54 226)(55 225)(56 224)(57 223)(58 222)(59 221)(60 220)(61 219)(62 218)(63 217)(64 216)(65 215)(66 214)(67 213)(68 212)(69 211)(70 210)(71 209)(72 208)(73 207)(74 206)(75 205)(76 204)(77 203)(78 202)(79 201)(80 200)(81 199)(82 198)(83 197)(84 196)(85 195)(86 194)(87 193)(88 192)(89 191)(90 190)(91 189)(92 188)(93 187)(94 186)(95 185)(96 184)(97 183)(98 182)(99 181)(100 180)(101 179)(102 178)(103 177)(104 176)(105 175)(106 174)(107 173)(108 172)(109 171)(110 170)(111 169)(112 168)(113 167)(114 166)(115 165)(116 164)(117 163)(118 162)(119 161)(120 160)(121 159)(122 158)(123 157)(124 156)(125 155)(126 154)(127 153)(128 152)(129 151)(130 150)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)
G:=sub<Sym(248)| (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,248)(33,247)(34,246)(35,245)(36,244)(37,243)(38,242)(39,241)(40,240)(41,239)(42,238)(43,237)(44,236)(45,235)(46,234)(47,233)(48,232)(49,231)(50,230)(51,229)(52,228)(53,227)(54,226)(55,225)(56,224)(57,223)(58,222)(59,221)(60,220)(61,219)(62,218)(63,217)(64,216)(65,215)(66,214)(67,213)(68,212)(69,211)(70,210)(71,209)(72,208)(73,207)(74,206)(75,205)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,197)(84,196)(85,195)(86,194)(87,193)(88,192)(89,191)(90,190)(91,189)(92,188)(93,187)(94,186)(95,185)(96,184)(97,183)(98,182)(99,181)(100,180)(101,179)(102,178)(103,177)(104,176)(105,175)(106,174)(107,173)(108,172)(109,171)(110,170)(111,169)(112,168)(113,167)(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)>;
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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,248)(33,247)(34,246)(35,245)(36,244)(37,243)(38,242)(39,241)(40,240)(41,239)(42,238)(43,237)(44,236)(45,235)(46,234)(47,233)(48,232)(49,231)(50,230)(51,229)(52,228)(53,227)(54,226)(55,225)(56,224)(57,223)(58,222)(59,221)(60,220)(61,219)(62,218)(63,217)(64,216)(65,215)(66,214)(67,213)(68,212)(69,211)(70,210)(71,209)(72,208)(73,207)(74,206)(75,205)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,197)(84,196)(85,195)(86,194)(87,193)(88,192)(89,191)(90,190)(91,189)(92,188)(93,187)(94,186)(95,185)(96,184)(97,183)(98,182)(99,181)(100,180)(101,179)(102,178)(103,177)(104,176)(105,175)(106,174)(107,173)(108,172)(109,171)(110,170)(111,169)(112,168)(113,167)(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141) );
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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(32,248),(33,247),(34,246),(35,245),(36,244),(37,243),(38,242),(39,241),(40,240),(41,239),(42,238),(43,237),(44,236),(45,235),(46,234),(47,233),(48,232),(49,231),(50,230),(51,229),(52,228),(53,227),(54,226),(55,225),(56,224),(57,223),(58,222),(59,221),(60,220),(61,219),(62,218),(63,217),(64,216),(65,215),(66,214),(67,213),(68,212),(69,211),(70,210),(71,209),(72,208),(73,207),(74,206),(75,205),(76,204),(77,203),(78,202),(79,201),(80,200),(81,199),(82,198),(83,197),(84,196),(85,195),(86,194),(87,193),(88,192),(89,191),(90,190),(91,189),(92,188),(93,187),(94,186),(95,185),(96,184),(97,183),(98,182),(99,181),(100,180),(101,179),(102,178),(103,177),(104,176),(105,175),(106,174),(107,173),(108,172),(109,171),(110,170),(111,169),(112,168),(113,167),(114,166),(115,165),(116,164),(117,163),(118,162),(119,161),(120,160),(121,159),(122,158),(123,157),(124,156),(125,155),(126,154),(127,153),(128,152),(129,151),(130,150),(131,149),(132,148),(133,147),(134,146),(135,145),(136,144),(137,143),(138,142),(139,141)]])
127 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD | 248A | ··· | 248BH |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 124 | 124 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
127 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D8 | D31 | D62 | D124 | D248 |
| kernel | D248 | C248 | D124 | C62 | C31 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 15 | 15 | 30 | 60 |
Matrix representation of D248 ►in GL2(𝔽1489) generated by
| 1161 | 1416 |
| 324 | 1148 |
| 1422 | 1239 |
| 959 | 67 |
G:=sub<GL(2,GF(1489))| [1161,324,1416,1148],[1422,959,1239,67] >;
D248 in GAP, Magma, Sage, TeX
D_{248} % in TeX
G:=Group("D248"); // GroupNames label
G:=SmallGroup(496,6);
// by ID
G=gap.SmallGroup(496,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,61,66,182,42,12004]);
// Polycyclic
G:=Group<a,b|a^248=b^2=1,b*a*b=a^-1>;
// generators/relations
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