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G = C2×C250order 500 = 22·53

Abelian group of type [2,250]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C250, SmallGroup(500,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C250
C1C5C25C125C250 — 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
(1 444)(2 445)(3 446)(4 447)(5 448)(6 449)(7 450)(8 451)(9 452)(10 453)(11 454)(12 455)(13 456)(14 457)(15 458)(16 459)(17 460)(18 461)(19 462)(20 463)(21 464)(22 465)(23 466)(24 467)(25 468)(26 469)(27 470)(28 471)(29 472)(30 473)(31 474)(32 475)(33 476)(34 477)(35 478)(36 479)(37 480)(38 481)(39 482)(40 483)(41 484)(42 485)(43 486)(44 487)(45 488)(46 489)(47 490)(48 491)(49 492)(50 493)(51 494)(52 495)(53 496)(54 497)(55 498)(56 499)(57 500)(58 251)(59 252)(60 253)(61 254)(62 255)(63 256)(64 257)(65 258)(66 259)(67 260)(68 261)(69 262)(70 263)(71 264)(72 265)(73 266)(74 267)(75 268)(76 269)(77 270)(78 271)(79 272)(80 273)(81 274)(82 275)(83 276)(84 277)(85 278)(86 279)(87 280)(88 281)(89 282)(90 283)(91 284)(92 285)(93 286)(94 287)(95 288)(96 289)(97 290)(98 291)(99 292)(100 293)(101 294)(102 295)(103 296)(104 297)(105 298)(106 299)(107 300)(108 301)(109 302)(110 303)(111 304)(112 305)(113 306)(114 307)(115 308)(116 309)(117 310)(118 311)(119 312)(120 313)(121 314)(122 315)(123 316)(124 317)(125 318)(126 319)(127 320)(128 321)(129 322)(130 323)(131 324)(132 325)(133 326)(134 327)(135 328)(136 329)(137 330)(138 331)(139 332)(140 333)(141 334)(142 335)(143 336)(144 337)(145 338)(146 339)(147 340)(148 341)(149 342)(150 343)(151 344)(152 345)(153 346)(154 347)(155 348)(156 349)(157 350)(158 351)(159 352)(160 353)(161 354)(162 355)(163 356)(164 357)(165 358)(166 359)(167 360)(168 361)(169 362)(170 363)(171 364)(172 365)(173 366)(174 367)(175 368)(176 369)(177 370)(178 371)(179 372)(180 373)(181 374)(182 375)(183 376)(184 377)(185 378)(186 379)(187 380)(188 381)(189 382)(190 383)(191 384)(192 385)(193 386)(194 387)(195 388)(196 389)(197 390)(198 391)(199 392)(200 393)(201 394)(202 395)(203 396)(204 397)(205 398)(206 399)(207 400)(208 401)(209 402)(210 403)(211 404)(212 405)(213 406)(214 407)(215 408)(216 409)(217 410)(218 411)(219 412)(220 413)(221 414)(222 415)(223 416)(224 417)(225 418)(226 419)(227 420)(228 421)(229 422)(230 423)(231 424)(232 425)(233 426)(234 427)(235 428)(236 429)(237 430)(238 431)(239 432)(240 433)(241 434)(242 435)(243 436)(244 437)(245 438)(246 439)(247 440)(248 441)(249 442)(250 443)
(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)| (1,444)(2,445)(3,446)(4,447)(5,448)(6,449)(7,450)(8,451)(9,452)(10,453)(11,454)(12,455)(13,456)(14,457)(15,458)(16,459)(17,460)(18,461)(19,462)(20,463)(21,464)(22,465)(23,466)(24,467)(25,468)(26,469)(27,470)(28,471)(29,472)(30,473)(31,474)(32,475)(33,476)(34,477)(35,478)(36,479)(37,480)(38,481)(39,482)(40,483)(41,484)(42,485)(43,486)(44,487)(45,488)(46,489)(47,490)(48,491)(49,492)(50,493)(51,494)(52,495)(53,496)(54,497)(55,498)(56,499)(57,500)(58,251)(59,252)(60,253)(61,254)(62,255)(63,256)(64,257)(65,258)(66,259)(67,260)(68,261)(69,262)(70,263)(71,264)(72,265)(73,266)(74,267)(75,268)(76,269)(77,270)(78,271)(79,272)(80,273)(81,274)(82,275)(83,276)(84,277)(85,278)(86,279)(87,280)(88,281)(89,282)(90,283)(91,284)(92,285)(93,286)(94,287)(95,288)(96,289)(97,290)(98,291)(99,292)(100,293)(101,294)(102,295)(103,296)(104,297)(105,298)(106,299)(107,300)(108,301)(109,302)(110,303)(111,304)(112,305)(113,306)(114,307)(115,308)(116,309)(117,310)(118,311)(119,312)(120,313)(121,314)(122,315)(123,316)(124,317)(125,318)(126,319)(127,320)(128,321)(129,322)(130,323)(131,324)(132,325)(133,326)(134,327)(135,328)(136,329)(137,330)(138,331)(139,332)(140,333)(141,334)(142,335)(143,336)(144,337)(145,338)(146,339)(147,340)(148,341)(149,342)(150,343)(151,344)(152,345)(153,346)(154,347)(155,348)(156,349)(157,350)(158,351)(159,352)(160,353)(161,354)(162,355)(163,356)(164,357)(165,358)(166,359)(167,360)(168,361)(169,362)(170,363)(171,364)(172,365)(173,366)(174,367)(175,368)(176,369)(177,370)(178,371)(179,372)(180,373)(181,374)(182,375)(183,376)(184,377)(185,378)(186,379)(187,380)(188,381)(189,382)(190,383)(191,384)(192,385)(193,386)(194,387)(195,388)(196,389)(197,390)(198,391)(199,392)(200,393)(201,394)(202,395)(203,396)(204,397)(205,398)(206,399)(207,400)(208,401)(209,402)(210,403)(211,404)(212,405)(213,406)(214,407)(215,408)(216,409)(217,410)(218,411)(219,412)(220,413)(221,414)(222,415)(223,416)(224,417)(225,418)(226,419)(227,420)(228,421)(229,422)(230,423)(231,424)(232,425)(233,426)(234,427)(235,428)(236,429)(237,430)(238,431)(239,432)(240,433)(241,434)(242,435)(243,436)(244,437)(245,438)(246,439)(247,440)(248,441)(249,442)(250,443), (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( (1,444)(2,445)(3,446)(4,447)(5,448)(6,449)(7,450)(8,451)(9,452)(10,453)(11,454)(12,455)(13,456)(14,457)(15,458)(16,459)(17,460)(18,461)(19,462)(20,463)(21,464)(22,465)(23,466)(24,467)(25,468)(26,469)(27,470)(28,471)(29,472)(30,473)(31,474)(32,475)(33,476)(34,477)(35,478)(36,479)(37,480)(38,481)(39,482)(40,483)(41,484)(42,485)(43,486)(44,487)(45,488)(46,489)(47,490)(48,491)(49,492)(50,493)(51,494)(52,495)(53,496)(54,497)(55,498)(56,499)(57,500)(58,251)(59,252)(60,253)(61,254)(62,255)(63,256)(64,257)(65,258)(66,259)(67,260)(68,261)(69,262)(70,263)(71,264)(72,265)(73,266)(74,267)(75,268)(76,269)(77,270)(78,271)(79,272)(80,273)(81,274)(82,275)(83,276)(84,277)(85,278)(86,279)(87,280)(88,281)(89,282)(90,283)(91,284)(92,285)(93,286)(94,287)(95,288)(96,289)(97,290)(98,291)(99,292)(100,293)(101,294)(102,295)(103,296)(104,297)(105,298)(106,299)(107,300)(108,301)(109,302)(110,303)(111,304)(112,305)(113,306)(114,307)(115,308)(116,309)(117,310)(118,311)(119,312)(120,313)(121,314)(122,315)(123,316)(124,317)(125,318)(126,319)(127,320)(128,321)(129,322)(130,323)(131,324)(132,325)(133,326)(134,327)(135,328)(136,329)(137,330)(138,331)(139,332)(140,333)(141,334)(142,335)(143,336)(144,337)(145,338)(146,339)(147,340)(148,341)(149,342)(150,343)(151,344)(152,345)(153,346)(154,347)(155,348)(156,349)(157,350)(158,351)(159,352)(160,353)(161,354)(162,355)(163,356)(164,357)(165,358)(166,359)(167,360)(168,361)(169,362)(170,363)(171,364)(172,365)(173,366)(174,367)(175,368)(176,369)(177,370)(178,371)(179,372)(180,373)(181,374)(182,375)(183,376)(184,377)(185,378)(186,379)(187,380)(188,381)(189,382)(190,383)(191,384)(192,385)(193,386)(194,387)(195,388)(196,389)(197,390)(198,391)(199,392)(200,393)(201,394)(202,395)(203,396)(204,397)(205,398)(206,399)(207,400)(208,401)(209,402)(210,403)(211,404)(212,405)(213,406)(214,407)(215,408)(216,409)(217,410)(218,411)(219,412)(220,413)(221,414)(222,415)(223,416)(224,417)(225,418)(226,419)(227,420)(228,421)(229,422)(230,423)(231,424)(232,425)(233,426)(234,427)(235,428)(236,429)(237,430)(238,431)(239,432)(240,433)(241,434)(242,435)(243,436)(244,437)(245,438)(246,439)(247,440)(248,441)(249,442)(250,443), (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([(1,444),(2,445),(3,446),(4,447),(5,448),(6,449),(7,450),(8,451),(9,452),(10,453),(11,454),(12,455),(13,456),(14,457),(15,458),(16,459),(17,460),(18,461),(19,462),(20,463),(21,464),(22,465),(23,466),(24,467),(25,468),(26,469),(27,470),(28,471),(29,472),(30,473),(31,474),(32,475),(33,476),(34,477),(35,478),(36,479),(37,480),(38,481),(39,482),(40,483),(41,484),(42,485),(43,486),(44,487),(45,488),(46,489),(47,490),(48,491),(49,492),(50,493),(51,494),(52,495),(53,496),(54,497),(55,498),(56,499),(57,500),(58,251),(59,252),(60,253),(61,254),(62,255),(63,256),(64,257),(65,258),(66,259),(67,260),(68,261),(69,262),(70,263),(71,264),(72,265),(73,266),(74,267),(75,268),(76,269),(77,270),(78,271),(79,272),(80,273),(81,274),(82,275),(83,276),(84,277),(85,278),(86,279),(87,280),(88,281),(89,282),(90,283),(91,284),(92,285),(93,286),(94,287),(95,288),(96,289),(97,290),(98,291),(99,292),(100,293),(101,294),(102,295),(103,296),(104,297),(105,298),(106,299),(107,300),(108,301),(109,302),(110,303),(111,304),(112,305),(113,306),(114,307),(115,308),(116,309),(117,310),(118,311),(119,312),(120,313),(121,314),(122,315),(123,316),(124,317),(125,318),(126,319),(127,320),(128,321),(129,322),(130,323),(131,324),(132,325),(133,326),(134,327),(135,328),(136,329),(137,330),(138,331),(139,332),(140,333),(141,334),(142,335),(143,336),(144,337),(145,338),(146,339),(147,340),(148,341),(149,342),(150,343),(151,344),(152,345),(153,346),(154,347),(155,348),(156,349),(157,350),(158,351),(159,352),(160,353),(161,354),(162,355),(163,356),(164,357),(165,358),(166,359),(167,360),(168,361),(169,362),(170,363),(171,364),(172,365),(173,366),(174,367),(175,368),(176,369),(177,370),(178,371),(179,372),(180,373),(181,374),(182,375),(183,376),(184,377),(185,378),(186,379),(187,380),(188,381),(189,382),(190,383),(191,384),(192,385),(193,386),(194,387),(195,388),(196,389),(197,390),(198,391),(199,392),(200,393),(201,394),(202,395),(203,396),(204,397),(205,398),(206,399),(207,400),(208,401),(209,402),(210,403),(211,404),(212,405),(213,406),(214,407),(215,408),(216,409),(217,410),(218,411),(219,412),(220,413),(221,414),(222,415),(223,416),(224,417),(225,418),(226,419),(227,420),(228,421),(229,422),(230,423),(231,424),(232,425),(233,426),(234,427),(235,428),(236,429),(237,430),(238,431),(239,432),(240,433),(241,434),(242,435),(243,436),(244,437),(245,438),(246,439),(247,440),(248,441),(249,442),(250,443)], [(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 2A2B2C5A5B5C5D10A···10L25A···25T50A···50BH125A···125CV250A···250KN
order1222555510···1025···2550···50125···125250···250
size111111111···11···11···11···11···1

500 irreducible representations

dim11111111
type++
imageC1C2C5C10C25C50C125C250
kernelC2×C250C250C2×C50C50C2×C10C10C22C2
# reps134122060100300

Matrix representation of C2×C250 in GL2(𝔽251) generated by

2500
0250
,
800
0146
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

Subgroup lattice of C2×C250 in TeX

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