direct product, abelian, monomial, 2-elementary
Aliases: C2×C250, SmallGroup(500,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C250 |
| C1 — C2×C250 |
| C1 — C2×C250 |
Generators and relations for C2×C250
G = < a,b | a2=b250=1, ab=ba >
Smallest permutation representation of C2×C250
►Regular action on 500 points
Generators in S500
(1 438)(2 439)(3 440)(4 441)(5 442)(6 443)(7 444)(8 445)(9 446)(10 447)(11 448)(12 449)(13 450)(14 451)(15 452)(16 453)(17 454)(18 455)(19 456)(20 457)(21 458)(22 459)(23 460)(24 461)(25 462)(26 463)(27 464)(28 465)(29 466)(30 467)(31 468)(32 469)(33 470)(34 471)(35 472)(36 473)(37 474)(38 475)(39 476)(40 477)(41 478)(42 479)(43 480)(44 481)(45 482)(46 483)(47 484)(48 485)(49 486)(50 487)(51 488)(52 489)(53 490)(54 491)(55 492)(56 493)(57 494)(58 495)(59 496)(60 497)(61 498)(62 499)(63 500)(64 251)(65 252)(66 253)(67 254)(68 255)(69 256)(70 257)(71 258)(72 259)(73 260)(74 261)(75 262)(76 263)(77 264)(78 265)(79 266)(80 267)(81 268)(82 269)(83 270)(84 271)(85 272)(86 273)(87 274)(88 275)(89 276)(90 277)(91 278)(92 279)(93 280)(94 281)(95 282)(96 283)(97 284)(98 285)(99 286)(100 287)(101 288)(102 289)(103 290)(104 291)(105 292)(106 293)(107 294)(108 295)(109 296)(110 297)(111 298)(112 299)(113 300)(114 301)(115 302)(116 303)(117 304)(118 305)(119 306)(120 307)(121 308)(122 309)(123 310)(124 311)(125 312)(126 313)(127 314)(128 315)(129 316)(130 317)(131 318)(132 319)(133 320)(134 321)(135 322)(136 323)(137 324)(138 325)(139 326)(140 327)(141 328)(142 329)(143 330)(144 331)(145 332)(146 333)(147 334)(148 335)(149 336)(150 337)(151 338)(152 339)(153 340)(154 341)(155 342)(156 343)(157 344)(158 345)(159 346)(160 347)(161 348)(162 349)(163 350)(164 351)(165 352)(166 353)(167 354)(168 355)(169 356)(170 357)(171 358)(172 359)(173 360)(174 361)(175 362)(176 363)(177 364)(178 365)(179 366)(180 367)(181 368)(182 369)(183 370)(184 371)(185 372)(186 373)(187 374)(188 375)(189 376)(190 377)(191 378)(192 379)(193 380)(194 381)(195 382)(196 383)(197 384)(198 385)(199 386)(200 387)(201 388)(202 389)(203 390)(204 391)(205 392)(206 393)(207 394)(208 395)(209 396)(210 397)(211 398)(212 399)(213 400)(214 401)(215 402)(216 403)(217 404)(218 405)(219 406)(220 407)(221 408)(222 409)(223 410)(224 411)(225 412)(226 413)(227 414)(228 415)(229 416)(230 417)(231 418)(232 419)(233 420)(234 421)(235 422)(236 423)(237 424)(238 425)(239 426)(240 427)(241 428)(242 429)(243 430)(244 431)(245 432)(246 433)(247 434)(248 435)(249 436)(250 437)
(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)(251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500)
G:=sub<Sym(500)| (1,438)(2,439)(3,440)(4,441)(5,442)(6,443)(7,444)(8,445)(9,446)(10,447)(11,448)(12,449)(13,450)(14,451)(15,452)(16,453)(17,454)(18,455)(19,456)(20,457)(21,458)(22,459)(23,460)(24,461)(25,462)(26,463)(27,464)(28,465)(29,466)(30,467)(31,468)(32,469)(33,470)(34,471)(35,472)(36,473)(37,474)(38,475)(39,476)(40,477)(41,478)(42,479)(43,480)(44,481)(45,482)(46,483)(47,484)(48,485)(49,486)(50,487)(51,488)(52,489)(53,490)(54,491)(55,492)(56,493)(57,494)(58,495)(59,496)(60,497)(61,498)(62,499)(63,500)(64,251)(65,252)(66,253)(67,254)(68,255)(69,256)(70,257)(71,258)(72,259)(73,260)(74,261)(75,262)(76,263)(77,264)(78,265)(79,266)(80,267)(81,268)(82,269)(83,270)(84,271)(85,272)(86,273)(87,274)(88,275)(89,276)(90,277)(91,278)(92,279)(93,280)(94,281)(95,282)(96,283)(97,284)(98,285)(99,286)(100,287)(101,288)(102,289)(103,290)(104,291)(105,292)(106,293)(107,294)(108,295)(109,296)(110,297)(111,298)(112,299)(113,300)(114,301)(115,302)(116,303)(117,304)(118,305)(119,306)(120,307)(121,308)(122,309)(123,310)(124,311)(125,312)(126,313)(127,314)(128,315)(129,316)(130,317)(131,318)(132,319)(133,320)(134,321)(135,322)(136,323)(137,324)(138,325)(139,326)(140,327)(141,328)(142,329)(143,330)(144,331)(145,332)(146,333)(147,334)(148,335)(149,336)(150,337)(151,338)(152,339)(153,340)(154,341)(155,342)(156,343)(157,344)(158,345)(159,346)(160,347)(161,348)(162,349)(163,350)(164,351)(165,352)(166,353)(167,354)(168,355)(169,356)(170,357)(171,358)(172,359)(173,360)(174,361)(175,362)(176,363)(177,364)(178,365)(179,366)(180,367)(181,368)(182,369)(183,370)(184,371)(185,372)(186,373)(187,374)(188,375)(189,376)(190,377)(191,378)(192,379)(193,380)(194,381)(195,382)(196,383)(197,384)(198,385)(199,386)(200,387)(201,388)(202,389)(203,390)(204,391)(205,392)(206,393)(207,394)(208,395)(209,396)(210,397)(211,398)(212,399)(213,400)(214,401)(215,402)(216,403)(217,404)(218,405)(219,406)(220,407)(221,408)(222,409)(223,410)(224,411)(225,412)(226,413)(227,414)(228,415)(229,416)(230,417)(231,418)(232,419)(233,420)(234,421)(235,422)(236,423)(237,424)(238,425)(239,426)(240,427)(241,428)(242,429)(243,430)(244,431)(245,432)(246,433)(247,434)(248,435)(249,436)(250,437), (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)(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500)>;
G:=Group( (1,438)(2,439)(3,440)(4,441)(5,442)(6,443)(7,444)(8,445)(9,446)(10,447)(11,448)(12,449)(13,450)(14,451)(15,452)(16,453)(17,454)(18,455)(19,456)(20,457)(21,458)(22,459)(23,460)(24,461)(25,462)(26,463)(27,464)(28,465)(29,466)(30,467)(31,468)(32,469)(33,470)(34,471)(35,472)(36,473)(37,474)(38,475)(39,476)(40,477)(41,478)(42,479)(43,480)(44,481)(45,482)(46,483)(47,484)(48,485)(49,486)(50,487)(51,488)(52,489)(53,490)(54,491)(55,492)(56,493)(57,494)(58,495)(59,496)(60,497)(61,498)(62,499)(63,500)(64,251)(65,252)(66,253)(67,254)(68,255)(69,256)(70,257)(71,258)(72,259)(73,260)(74,261)(75,262)(76,263)(77,264)(78,265)(79,266)(80,267)(81,268)(82,269)(83,270)(84,271)(85,272)(86,273)(87,274)(88,275)(89,276)(90,277)(91,278)(92,279)(93,280)(94,281)(95,282)(96,283)(97,284)(98,285)(99,286)(100,287)(101,288)(102,289)(103,290)(104,291)(105,292)(106,293)(107,294)(108,295)(109,296)(110,297)(111,298)(112,299)(113,300)(114,301)(115,302)(116,303)(117,304)(118,305)(119,306)(120,307)(121,308)(122,309)(123,310)(124,311)(125,312)(126,313)(127,314)(128,315)(129,316)(130,317)(131,318)(132,319)(133,320)(134,321)(135,322)(136,323)(137,324)(138,325)(139,326)(140,327)(141,328)(142,329)(143,330)(144,331)(145,332)(146,333)(147,334)(148,335)(149,336)(150,337)(151,338)(152,339)(153,340)(154,341)(155,342)(156,343)(157,344)(158,345)(159,346)(160,347)(161,348)(162,349)(163,350)(164,351)(165,352)(166,353)(167,354)(168,355)(169,356)(170,357)(171,358)(172,359)(173,360)(174,361)(175,362)(176,363)(177,364)(178,365)(179,366)(180,367)(181,368)(182,369)(183,370)(184,371)(185,372)(186,373)(187,374)(188,375)(189,376)(190,377)(191,378)(192,379)(193,380)(194,381)(195,382)(196,383)(197,384)(198,385)(199,386)(200,387)(201,388)(202,389)(203,390)(204,391)(205,392)(206,393)(207,394)(208,395)(209,396)(210,397)(211,398)(212,399)(213,400)(214,401)(215,402)(216,403)(217,404)(218,405)(219,406)(220,407)(221,408)(222,409)(223,410)(224,411)(225,412)(226,413)(227,414)(228,415)(229,416)(230,417)(231,418)(232,419)(233,420)(234,421)(235,422)(236,423)(237,424)(238,425)(239,426)(240,427)(241,428)(242,429)(243,430)(244,431)(245,432)(246,433)(247,434)(248,435)(249,436)(250,437), (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)(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500) );
G=PermutationGroup([[(1,438),(2,439),(3,440),(4,441),(5,442),(6,443),(7,444),(8,445),(9,446),(10,447),(11,448),(12,449),(13,450),(14,451),(15,452),(16,453),(17,454),(18,455),(19,456),(20,457),(21,458),(22,459),(23,460),(24,461),(25,462),(26,463),(27,464),(28,465),(29,466),(30,467),(31,468),(32,469),(33,470),(34,471),(35,472),(36,473),(37,474),(38,475),(39,476),(40,477),(41,478),(42,479),(43,480),(44,481),(45,482),(46,483),(47,484),(48,485),(49,486),(50,487),(51,488),(52,489),(53,490),(54,491),(55,492),(56,493),(57,494),(58,495),(59,496),(60,497),(61,498),(62,499),(63,500),(64,251),(65,252),(66,253),(67,254),(68,255),(69,256),(70,257),(71,258),(72,259),(73,260),(74,261),(75,262),(76,263),(77,264),(78,265),(79,266),(80,267),(81,268),(82,269),(83,270),(84,271),(85,272),(86,273),(87,274),(88,275),(89,276),(90,277),(91,278),(92,279),(93,280),(94,281),(95,282),(96,283),(97,284),(98,285),(99,286),(100,287),(101,288),(102,289),(103,290),(104,291),(105,292),(106,293),(107,294),(108,295),(109,296),(110,297),(111,298),(112,299),(113,300),(114,301),(115,302),(116,303),(117,304),(118,305),(119,306),(120,307),(121,308),(122,309),(123,310),(124,311),(125,312),(126,313),(127,314),(128,315),(129,316),(130,317),(131,318),(132,319),(133,320),(134,321),(135,322),(136,323),(137,324),(138,325),(139,326),(140,327),(141,328),(142,329),(143,330),(144,331),(145,332),(146,333),(147,334),(148,335),(149,336),(150,337),(151,338),(152,339),(153,340),(154,341),(155,342),(156,343),(157,344),(158,345),(159,346),(160,347),(161,348),(162,349),(163,350),(164,351),(165,352),(166,353),(167,354),(168,355),(169,356),(170,357),(171,358),(172,359),(173,360),(174,361),(175,362),(176,363),(177,364),(178,365),(179,366),(180,367),(181,368),(182,369),(183,370),(184,371),(185,372),(186,373),(187,374),(188,375),(189,376),(190,377),(191,378),(192,379),(193,380),(194,381),(195,382),(196,383),(197,384),(198,385),(199,386),(200,387),(201,388),(202,389),(203,390),(204,391),(205,392),(206,393),(207,394),(208,395),(209,396),(210,397),(211,398),(212,399),(213,400),(214,401),(215,402),(216,403),(217,404),(218,405),(219,406),(220,407),(221,408),(222,409),(223,410),(224,411),(225,412),(226,413),(227,414),(228,415),(229,416),(230,417),(231,418),(232,419),(233,420),(234,421),(235,422),(236,423),(237,424),(238,425),(239,426),(240,427),(241,428),(242,429),(243,430),(244,431),(245,432),(246,433),(247,434),(248,435),(249,436),(250,437)], [(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),(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500)]])
500 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 25A | ··· | 25T | 50A | ··· | 50BH | 125A | ··· | 125CV | 250A | ··· | 250KN |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 25 | ··· | 25 | 50 | ··· | 50 | 125 | ··· | 125 | 250 | ··· | 250 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
500 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C10 | C25 | C50 | C125 | C250 |
| kernel | C2×C250 | C250 | C2×C50 | C50 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 3 | 4 | 12 | 20 | 60 | 100 | 300 |
Matrix representation of C2×C250 ►in GL2(𝔽251) generated by
| 250 | 0 |
| 0 | 250 |
| 80 | 0 |
| 0 | 146 |
G:=sub<GL(2,GF(251))| [250,0,0,250],[80,0,0,146] >;
C2×C250 in GAP, Magma, Sage, TeX
C_2\times C_{250} % in TeX
G:=Group("C2xC250"); // GroupNames label
G:=SmallGroup(500,5);
// by ID
G=gap.SmallGroup(500,5);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,87,118]);
// Polycyclic
G:=Group<a,b|a^2=b^250=1,a*b=b*a>;
// generators/relations
Export