direct product, abelian, monomial, 2-elementary
Aliases: C2×C248, SmallGroup(496,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C248 |
| C1 — C2×C248 |
| C1 — C2×C248 |
Generators and relations for C2×C248
G = < a,b | a2=b248=1, ab=ba >
Smallest permutation representation of C2×C248
►Regular action on 496 points
Generators in S496
(1 389)(2 390)(3 391)(4 392)(5 393)(6 394)(7 395)(8 396)(9 397)(10 398)(11 399)(12 400)(13 401)(14 402)(15 403)(16 404)(17 405)(18 406)(19 407)(20 408)(21 409)(22 410)(23 411)(24 412)(25 413)(26 414)(27 415)(28 416)(29 417)(30 418)(31 419)(32 420)(33 421)(34 422)(35 423)(36 424)(37 425)(38 426)(39 427)(40 428)(41 429)(42 430)(43 431)(44 432)(45 433)(46 434)(47 435)(48 436)(49 437)(50 438)(51 439)(52 440)(53 441)(54 442)(55 443)(56 444)(57 445)(58 446)(59 447)(60 448)(61 449)(62 450)(63 451)(64 452)(65 453)(66 454)(67 455)(68 456)(69 457)(70 458)(71 459)(72 460)(73 461)(74 462)(75 463)(76 464)(77 465)(78 466)(79 467)(80 468)(81 469)(82 470)(83 471)(84 472)(85 473)(86 474)(87 475)(88 476)(89 477)(90 478)(91 479)(92 480)(93 481)(94 482)(95 483)(96 484)(97 485)(98 486)(99 487)(100 488)(101 489)(102 490)(103 491)(104 492)(105 493)(106 494)(107 495)(108 496)(109 249)(110 250)(111 251)(112 252)(113 253)(114 254)(115 255)(116 256)(117 257)(118 258)(119 259)(120 260)(121 261)(122 262)(123 263)(124 264)(125 265)(126 266)(127 267)(128 268)(129 269)(130 270)(131 271)(132 272)(133 273)(134 274)(135 275)(136 276)(137 277)(138 278)(139 279)(140 280)(141 281)(142 282)(143 283)(144 284)(145 285)(146 286)(147 287)(148 288)(149 289)(150 290)(151 291)(152 292)(153 293)(154 294)(155 295)(156 296)(157 297)(158 298)(159 299)(160 300)(161 301)(162 302)(163 303)(164 304)(165 305)(166 306)(167 307)(168 308)(169 309)(170 310)(171 311)(172 312)(173 313)(174 314)(175 315)(176 316)(177 317)(178 318)(179 319)(180 320)(181 321)(182 322)(183 323)(184 324)(185 325)(186 326)(187 327)(188 328)(189 329)(190 330)(191 331)(192 332)(193 333)(194 334)(195 335)(196 336)(197 337)(198 338)(199 339)(200 340)(201 341)(202 342)(203 343)(204 344)(205 345)(206 346)(207 347)(208 348)(209 349)(210 350)(211 351)(212 352)(213 353)(214 354)(215 355)(216 356)(217 357)(218 358)(219 359)(220 360)(221 361)(222 362)(223 363)(224 364)(225 365)(226 366)(227 367)(228 368)(229 369)(230 370)(231 371)(232 372)(233 373)(234 374)(235 375)(236 376)(237 377)(238 378)(239 379)(240 380)(241 381)(242 382)(243 383)(244 384)(245 385)(246 386)(247 387)(248 388)
(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)(249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496)
G:=sub<Sym(496)| (1,389)(2,390)(3,391)(4,392)(5,393)(6,394)(7,395)(8,396)(9,397)(10,398)(11,399)(12,400)(13,401)(14,402)(15,403)(16,404)(17,405)(18,406)(19,407)(20,408)(21,409)(22,410)(23,411)(24,412)(25,413)(26,414)(27,415)(28,416)(29,417)(30,418)(31,419)(32,420)(33,421)(34,422)(35,423)(36,424)(37,425)(38,426)(39,427)(40,428)(41,429)(42,430)(43,431)(44,432)(45,433)(46,434)(47,435)(48,436)(49,437)(50,438)(51,439)(52,440)(53,441)(54,442)(55,443)(56,444)(57,445)(58,446)(59,447)(60,448)(61,449)(62,450)(63,451)(64,452)(65,453)(66,454)(67,455)(68,456)(69,457)(70,458)(71,459)(72,460)(73,461)(74,462)(75,463)(76,464)(77,465)(78,466)(79,467)(80,468)(81,469)(82,470)(83,471)(84,472)(85,473)(86,474)(87,475)(88,476)(89,477)(90,478)(91,479)(92,480)(93,481)(94,482)(95,483)(96,484)(97,485)(98,486)(99,487)(100,488)(101,489)(102,490)(103,491)(104,492)(105,493)(106,494)(107,495)(108,496)(109,249)(110,250)(111,251)(112,252)(113,253)(114,254)(115,255)(116,256)(117,257)(118,258)(119,259)(120,260)(121,261)(122,262)(123,263)(124,264)(125,265)(126,266)(127,267)(128,268)(129,269)(130,270)(131,271)(132,272)(133,273)(134,274)(135,275)(136,276)(137,277)(138,278)(139,279)(140,280)(141,281)(142,282)(143,283)(144,284)(145,285)(146,286)(147,287)(148,288)(149,289)(150,290)(151,291)(152,292)(153,293)(154,294)(155,295)(156,296)(157,297)(158,298)(159,299)(160,300)(161,301)(162,302)(163,303)(164,304)(165,305)(166,306)(167,307)(168,308)(169,309)(170,310)(171,311)(172,312)(173,313)(174,314)(175,315)(176,316)(177,317)(178,318)(179,319)(180,320)(181,321)(182,322)(183,323)(184,324)(185,325)(186,326)(187,327)(188,328)(189,329)(190,330)(191,331)(192,332)(193,333)(194,334)(195,335)(196,336)(197,337)(198,338)(199,339)(200,340)(201,341)(202,342)(203,343)(204,344)(205,345)(206,346)(207,347)(208,348)(209,349)(210,350)(211,351)(212,352)(213,353)(214,354)(215,355)(216,356)(217,357)(218,358)(219,359)(220,360)(221,361)(222,362)(223,363)(224,364)(225,365)(226,366)(227,367)(228,368)(229,369)(230,370)(231,371)(232,372)(233,373)(234,374)(235,375)(236,376)(237,377)(238,378)(239,379)(240,380)(241,381)(242,382)(243,383)(244,384)(245,385)(246,386)(247,387)(248,388), (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)(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496)>;
G:=Group( (1,389)(2,390)(3,391)(4,392)(5,393)(6,394)(7,395)(8,396)(9,397)(10,398)(11,399)(12,400)(13,401)(14,402)(15,403)(16,404)(17,405)(18,406)(19,407)(20,408)(21,409)(22,410)(23,411)(24,412)(25,413)(26,414)(27,415)(28,416)(29,417)(30,418)(31,419)(32,420)(33,421)(34,422)(35,423)(36,424)(37,425)(38,426)(39,427)(40,428)(41,429)(42,430)(43,431)(44,432)(45,433)(46,434)(47,435)(48,436)(49,437)(50,438)(51,439)(52,440)(53,441)(54,442)(55,443)(56,444)(57,445)(58,446)(59,447)(60,448)(61,449)(62,450)(63,451)(64,452)(65,453)(66,454)(67,455)(68,456)(69,457)(70,458)(71,459)(72,460)(73,461)(74,462)(75,463)(76,464)(77,465)(78,466)(79,467)(80,468)(81,469)(82,470)(83,471)(84,472)(85,473)(86,474)(87,475)(88,476)(89,477)(90,478)(91,479)(92,480)(93,481)(94,482)(95,483)(96,484)(97,485)(98,486)(99,487)(100,488)(101,489)(102,490)(103,491)(104,492)(105,493)(106,494)(107,495)(108,496)(109,249)(110,250)(111,251)(112,252)(113,253)(114,254)(115,255)(116,256)(117,257)(118,258)(119,259)(120,260)(121,261)(122,262)(123,263)(124,264)(125,265)(126,266)(127,267)(128,268)(129,269)(130,270)(131,271)(132,272)(133,273)(134,274)(135,275)(136,276)(137,277)(138,278)(139,279)(140,280)(141,281)(142,282)(143,283)(144,284)(145,285)(146,286)(147,287)(148,288)(149,289)(150,290)(151,291)(152,292)(153,293)(154,294)(155,295)(156,296)(157,297)(158,298)(159,299)(160,300)(161,301)(162,302)(163,303)(164,304)(165,305)(166,306)(167,307)(168,308)(169,309)(170,310)(171,311)(172,312)(173,313)(174,314)(175,315)(176,316)(177,317)(178,318)(179,319)(180,320)(181,321)(182,322)(183,323)(184,324)(185,325)(186,326)(187,327)(188,328)(189,329)(190,330)(191,331)(192,332)(193,333)(194,334)(195,335)(196,336)(197,337)(198,338)(199,339)(200,340)(201,341)(202,342)(203,343)(204,344)(205,345)(206,346)(207,347)(208,348)(209,349)(210,350)(211,351)(212,352)(213,353)(214,354)(215,355)(216,356)(217,357)(218,358)(219,359)(220,360)(221,361)(222,362)(223,363)(224,364)(225,365)(226,366)(227,367)(228,368)(229,369)(230,370)(231,371)(232,372)(233,373)(234,374)(235,375)(236,376)(237,377)(238,378)(239,379)(240,380)(241,381)(242,382)(243,383)(244,384)(245,385)(246,386)(247,387)(248,388), (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)(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496) );
G=PermutationGroup([[(1,389),(2,390),(3,391),(4,392),(5,393),(6,394),(7,395),(8,396),(9,397),(10,398),(11,399),(12,400),(13,401),(14,402),(15,403),(16,404),(17,405),(18,406),(19,407),(20,408),(21,409),(22,410),(23,411),(24,412),(25,413),(26,414),(27,415),(28,416),(29,417),(30,418),(31,419),(32,420),(33,421),(34,422),(35,423),(36,424),(37,425),(38,426),(39,427),(40,428),(41,429),(42,430),(43,431),(44,432),(45,433),(46,434),(47,435),(48,436),(49,437),(50,438),(51,439),(52,440),(53,441),(54,442),(55,443),(56,444),(57,445),(58,446),(59,447),(60,448),(61,449),(62,450),(63,451),(64,452),(65,453),(66,454),(67,455),(68,456),(69,457),(70,458),(71,459),(72,460),(73,461),(74,462),(75,463),(76,464),(77,465),(78,466),(79,467),(80,468),(81,469),(82,470),(83,471),(84,472),(85,473),(86,474),(87,475),(88,476),(89,477),(90,478),(91,479),(92,480),(93,481),(94,482),(95,483),(96,484),(97,485),(98,486),(99,487),(100,488),(101,489),(102,490),(103,491),(104,492),(105,493),(106,494),(107,495),(108,496),(109,249),(110,250),(111,251),(112,252),(113,253),(114,254),(115,255),(116,256),(117,257),(118,258),(119,259),(120,260),(121,261),(122,262),(123,263),(124,264),(125,265),(126,266),(127,267),(128,268),(129,269),(130,270),(131,271),(132,272),(133,273),(134,274),(135,275),(136,276),(137,277),(138,278),(139,279),(140,280),(141,281),(142,282),(143,283),(144,284),(145,285),(146,286),(147,287),(148,288),(149,289),(150,290),(151,291),(152,292),(153,293),(154,294),(155,295),(156,296),(157,297),(158,298),(159,299),(160,300),(161,301),(162,302),(163,303),(164,304),(165,305),(166,306),(167,307),(168,308),(169,309),(170,310),(171,311),(172,312),(173,313),(174,314),(175,315),(176,316),(177,317),(178,318),(179,319),(180,320),(181,321),(182,322),(183,323),(184,324),(185,325),(186,326),(187,327),(188,328),(189,329),(190,330),(191,331),(192,332),(193,333),(194,334),(195,335),(196,336),(197,337),(198,338),(199,339),(200,340),(201,341),(202,342),(203,343),(204,344),(205,345),(206,346),(207,347),(208,348),(209,349),(210,350),(211,351),(212,352),(213,353),(214,354),(215,355),(216,356),(217,357),(218,358),(219,359),(220,360),(221,361),(222,362),(223,363),(224,364),(225,365),(226,366),(227,367),(228,368),(229,369),(230,370),(231,371),(232,372),(233,373),(234,374),(235,375),(236,376),(237,377),(238,378),(239,379),(240,380),(241,381),(242,382),(243,383),(244,384),(245,385),(246,386),(247,387),(248,388)], [(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),(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496)]])
496 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 31A | ··· | 31AD | 62A | ··· | 62CL | 124A | ··· | 124DP | 248A | ··· | 248IF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
496 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C31 | C62 | C62 | C124 | C124 | C248 |
| kernel | C2×C248 | C248 | C2×C124 | C124 | C2×C62 | C62 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 30 | 60 | 30 | 60 | 60 | 240 |
Matrix representation of C2×C248 ►in GL2(𝔽1489) generated by
| 1488 | 0 |
| 0 | 1 |
| 557 | 0 |
| 0 | 433 |
G:=sub<GL(2,GF(1489))| [1488,0,0,1],[557,0,0,433] >;
C2×C248 in GAP, Magma, Sage, TeX
C_2\times C_{248} % in TeX
G:=Group("C2xC248"); // GroupNames label
G:=SmallGroup(496,22);
// by ID
G=gap.SmallGroup(496,22);
# by ID
G:=PCGroup([5,-2,-2,-31,-2,-2,620,58]);
// Polycyclic
G:=Group<a,b|a^2=b^248=1,a*b=b*a>;
// generators/relations
Export