metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D31, C248⋊2C2, C4.8D62, C62.1D4, C31⋊1SD16, C2.3D124, Dic62⋊1C2, D124.1C2, C124.8C22, SmallGroup(496,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C248⋊C2
G = < a,b | a248=b2=1, bab=a123 >
Smallest permutation representation of C248⋊C2
►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)
(2 124)(3 247)(4 122)(5 245)(6 120)(7 243)(8 118)(9 241)(10 116)(11 239)(12 114)(13 237)(14 112)(15 235)(16 110)(17 233)(18 108)(19 231)(20 106)(21 229)(22 104)(23 227)(24 102)(25 225)(26 100)(27 223)(28 98)(29 221)(30 96)(31 219)(32 94)(33 217)(34 92)(35 215)(36 90)(37 213)(38 88)(39 211)(40 86)(41 209)(42 84)(43 207)(44 82)(45 205)(46 80)(47 203)(48 78)(49 201)(50 76)(51 199)(52 74)(53 197)(54 72)(55 195)(56 70)(57 193)(58 68)(59 191)(60 66)(61 189)(62 64)(63 187)(65 185)(67 183)(69 181)(71 179)(73 177)(75 175)(77 173)(79 171)(81 169)(83 167)(85 165)(87 163)(89 161)(91 159)(93 157)(95 155)(97 153)(99 151)(101 149)(103 147)(105 145)(107 143)(109 141)(111 139)(113 137)(115 135)(117 133)(119 131)(121 129)(123 127)(126 248)(128 246)(130 244)(132 242)(134 240)(136 238)(138 236)(140 234)(142 232)(144 230)(146 228)(148 226)(150 224)(152 222)(154 220)(156 218)(158 216)(160 214)(162 212)(164 210)(166 208)(168 206)(170 204)(172 202)(174 200)(176 198)(178 196)(180 194)(182 192)(184 190)(186 188)
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), (2,124)(3,247)(4,122)(5,245)(6,120)(7,243)(8,118)(9,241)(10,116)(11,239)(12,114)(13,237)(14,112)(15,235)(16,110)(17,233)(18,108)(19,231)(20,106)(21,229)(22,104)(23,227)(24,102)(25,225)(26,100)(27,223)(28,98)(29,221)(30,96)(31,219)(32,94)(33,217)(34,92)(35,215)(36,90)(37,213)(38,88)(39,211)(40,86)(41,209)(42,84)(43,207)(44,82)(45,205)(46,80)(47,203)(48,78)(49,201)(50,76)(51,199)(52,74)(53,197)(54,72)(55,195)(56,70)(57,193)(58,68)(59,191)(60,66)(61,189)(62,64)(63,187)(65,185)(67,183)(69,181)(71,179)(73,177)(75,175)(77,173)(79,171)(81,169)(83,167)(85,165)(87,163)(89,161)(91,159)(93,157)(95,155)(97,153)(99,151)(101,149)(103,147)(105,145)(107,143)(109,141)(111,139)(113,137)(115,135)(117,133)(119,131)(121,129)(123,127)(126,248)(128,246)(130,244)(132,242)(134,240)(136,238)(138,236)(140,234)(142,232)(144,230)(146,228)(148,226)(150,224)(152,222)(154,220)(156,218)(158,216)(160,214)(162,212)(164,210)(166,208)(168,206)(170,204)(172,202)(174,200)(176,198)(178,196)(180,194)(182,192)(184,190)(186,188)>;
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), (2,124)(3,247)(4,122)(5,245)(6,120)(7,243)(8,118)(9,241)(10,116)(11,239)(12,114)(13,237)(14,112)(15,235)(16,110)(17,233)(18,108)(19,231)(20,106)(21,229)(22,104)(23,227)(24,102)(25,225)(26,100)(27,223)(28,98)(29,221)(30,96)(31,219)(32,94)(33,217)(34,92)(35,215)(36,90)(37,213)(38,88)(39,211)(40,86)(41,209)(42,84)(43,207)(44,82)(45,205)(46,80)(47,203)(48,78)(49,201)(50,76)(51,199)(52,74)(53,197)(54,72)(55,195)(56,70)(57,193)(58,68)(59,191)(60,66)(61,189)(62,64)(63,187)(65,185)(67,183)(69,181)(71,179)(73,177)(75,175)(77,173)(79,171)(81,169)(83,167)(85,165)(87,163)(89,161)(91,159)(93,157)(95,155)(97,153)(99,151)(101,149)(103,147)(105,145)(107,143)(109,141)(111,139)(113,137)(115,135)(117,133)(119,131)(121,129)(123,127)(126,248)(128,246)(130,244)(132,242)(134,240)(136,238)(138,236)(140,234)(142,232)(144,230)(146,228)(148,226)(150,224)(152,222)(154,220)(156,218)(158,216)(160,214)(162,212)(164,210)(166,208)(168,206)(170,204)(172,202)(174,200)(176,198)(178,196)(180,194)(182,192)(184,190)(186,188) );
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)], [(2,124),(3,247),(4,122),(5,245),(6,120),(7,243),(8,118),(9,241),(10,116),(11,239),(12,114),(13,237),(14,112),(15,235),(16,110),(17,233),(18,108),(19,231),(20,106),(21,229),(22,104),(23,227),(24,102),(25,225),(26,100),(27,223),(28,98),(29,221),(30,96),(31,219),(32,94),(33,217),(34,92),(35,215),(36,90),(37,213),(38,88),(39,211),(40,86),(41,209),(42,84),(43,207),(44,82),(45,205),(46,80),(47,203),(48,78),(49,201),(50,76),(51,199),(52,74),(53,197),(54,72),(55,195),(56,70),(57,193),(58,68),(59,191),(60,66),(61,189),(62,64),(63,187),(65,185),(67,183),(69,181),(71,179),(73,177),(75,175),(77,173),(79,171),(81,169),(83,167),(85,165),(87,163),(89,161),(91,159),(93,157),(95,155),(97,153),(99,151),(101,149),(103,147),(105,145),(107,143),(109,141),(111,139),(113,137),(115,135),(117,133),(119,131),(121,129),(123,127),(126,248),(128,246),(130,244),(132,242),(134,240),(136,238),(138,236),(140,234),(142,232),(144,230),(146,228),(148,226),(150,224),(152,222),(154,220),(156,218),(158,216),(160,214),(162,212),(164,210),(166,208),(168,206),(170,204),(172,202),(174,200),(176,198),(178,196),(180,194),(182,192),(184,190),(186,188)]])
127 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD | 248A | ··· | 248BH |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 124 | 2 | 124 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
127 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D31 | D62 | D124 | C248⋊C2 |
| kernel | C248⋊C2 | C248 | Dic62 | D124 | C62 | C31 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 15 | 15 | 30 | 60 |
Matrix representation of C248⋊C2 ►in GL2(𝔽1489) generated by
| 245 | 885 |
| 604 | 474 |
| 1 | 1339 |
| 0 | 1488 |
G:=sub<GL(2,GF(1489))| [245,604,885,474],[1,0,1339,1488] >;
C248⋊C2 in GAP, Magma, Sage, TeX
C_{248}\rtimes C_2 % in TeX
G:=Group("C248:C2"); // GroupNames label
G:=SmallGroup(496,5);
// by ID
G=gap.SmallGroup(496,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,61,26,182,42,12004]);
// Polycyclic
G:=Group<a,b|a^248=b^2=1,b*a*b=a^123>;
// generators/relations
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