direct product, abelian, monomial, 2-elementary
Aliases: C2×C242, SmallGroup(484,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C242 |
| C1 — C2×C242 |
| C1 — C2×C242 |
Generators and relations for C2×C242
G = < a,b | a2=b242=1, ab=ba >
Smallest permutation representation of C2×C242
►Regular action on 484 points
Generators in S484
(1 244)(2 245)(3 246)(4 247)(5 248)(6 249)(7 250)(8 251)(9 252)(10 253)(11 254)(12 255)(13 256)(14 257)(15 258)(16 259)(17 260)(18 261)(19 262)(20 263)(21 264)(22 265)(23 266)(24 267)(25 268)(26 269)(27 270)(28 271)(29 272)(30 273)(31 274)(32 275)(33 276)(34 277)(35 278)(36 279)(37 280)(38 281)(39 282)(40 283)(41 284)(42 285)(43 286)(44 287)(45 288)(46 289)(47 290)(48 291)(49 292)(50 293)(51 294)(52 295)(53 296)(54 297)(55 298)(56 299)(57 300)(58 301)(59 302)(60 303)(61 304)(62 305)(63 306)(64 307)(65 308)(66 309)(67 310)(68 311)(69 312)(70 313)(71 314)(72 315)(73 316)(74 317)(75 318)(76 319)(77 320)(78 321)(79 322)(80 323)(81 324)(82 325)(83 326)(84 327)(85 328)(86 329)(87 330)(88 331)(89 332)(90 333)(91 334)(92 335)(93 336)(94 337)(95 338)(96 339)(97 340)(98 341)(99 342)(100 343)(101 344)(102 345)(103 346)(104 347)(105 348)(106 349)(107 350)(108 351)(109 352)(110 353)(111 354)(112 355)(113 356)(114 357)(115 358)(116 359)(117 360)(118 361)(119 362)(120 363)(121 364)(122 365)(123 366)(124 367)(125 368)(126 369)(127 370)(128 371)(129 372)(130 373)(131 374)(132 375)(133 376)(134 377)(135 378)(136 379)(137 380)(138 381)(139 382)(140 383)(141 384)(142 385)(143 386)(144 387)(145 388)(146 389)(147 390)(148 391)(149 392)(150 393)(151 394)(152 395)(153 396)(154 397)(155 398)(156 399)(157 400)(158 401)(159 402)(160 403)(161 404)(162 405)(163 406)(164 407)(165 408)(166 409)(167 410)(168 411)(169 412)(170 413)(171 414)(172 415)(173 416)(174 417)(175 418)(176 419)(177 420)(178 421)(179 422)(180 423)(181 424)(182 425)(183 426)(184 427)(185 428)(186 429)(187 430)(188 431)(189 432)(190 433)(191 434)(192 435)(193 436)(194 437)(195 438)(196 439)(197 440)(198 441)(199 442)(200 443)(201 444)(202 445)(203 446)(204 447)(205 448)(206 449)(207 450)(208 451)(209 452)(210 453)(211 454)(212 455)(213 456)(214 457)(215 458)(216 459)(217 460)(218 461)(219 462)(220 463)(221 464)(222 465)(223 466)(224 467)(225 468)(226 469)(227 470)(228 471)(229 472)(230 473)(231 474)(232 475)(233 476)(234 477)(235 478)(236 479)(237 480)(238 481)(239 482)(240 483)(241 484)(242 243)
(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)
G:=sub<Sym(484)| (1,244)(2,245)(3,246)(4,247)(5,248)(6,249)(7,250)(8,251)(9,252)(10,253)(11,254)(12,255)(13,256)(14,257)(15,258)(16,259)(17,260)(18,261)(19,262)(20,263)(21,264)(22,265)(23,266)(24,267)(25,268)(26,269)(27,270)(28,271)(29,272)(30,273)(31,274)(32,275)(33,276)(34,277)(35,278)(36,279)(37,280)(38,281)(39,282)(40,283)(41,284)(42,285)(43,286)(44,287)(45,288)(46,289)(47,290)(48,291)(49,292)(50,293)(51,294)(52,295)(53,296)(54,297)(55,298)(56,299)(57,300)(58,301)(59,302)(60,303)(61,304)(62,305)(63,306)(64,307)(65,308)(66,309)(67,310)(68,311)(69,312)(70,313)(71,314)(72,315)(73,316)(74,317)(75,318)(76,319)(77,320)(78,321)(79,322)(80,323)(81,324)(82,325)(83,326)(84,327)(85,328)(86,329)(87,330)(88,331)(89,332)(90,333)(91,334)(92,335)(93,336)(94,337)(95,338)(96,339)(97,340)(98,341)(99,342)(100,343)(101,344)(102,345)(103,346)(104,347)(105,348)(106,349)(107,350)(108,351)(109,352)(110,353)(111,354)(112,355)(113,356)(114,357)(115,358)(116,359)(117,360)(118,361)(119,362)(120,363)(121,364)(122,365)(123,366)(124,367)(125,368)(126,369)(127,370)(128,371)(129,372)(130,373)(131,374)(132,375)(133,376)(134,377)(135,378)(136,379)(137,380)(138,381)(139,382)(140,383)(141,384)(142,385)(143,386)(144,387)(145,388)(146,389)(147,390)(148,391)(149,392)(150,393)(151,394)(152,395)(153,396)(154,397)(155,398)(156,399)(157,400)(158,401)(159,402)(160,403)(161,404)(162,405)(163,406)(164,407)(165,408)(166,409)(167,410)(168,411)(169,412)(170,413)(171,414)(172,415)(173,416)(174,417)(175,418)(176,419)(177,420)(178,421)(179,422)(180,423)(181,424)(182,425)(183,426)(184,427)(185,428)(186,429)(187,430)(188,431)(189,432)(190,433)(191,434)(192,435)(193,436)(194,437)(195,438)(196,439)(197,440)(198,441)(199,442)(200,443)(201,444)(202,445)(203,446)(204,447)(205,448)(206,449)(207,450)(208,451)(209,452)(210,453)(211,454)(212,455)(213,456)(214,457)(215,458)(216,459)(217,460)(218,461)(219,462)(220,463)(221,464)(222,465)(223,466)(224,467)(225,468)(226,469)(227,470)(228,471)(229,472)(230,473)(231,474)(232,475)(233,476)(234,477)(235,478)(236,479)(237,480)(238,481)(239,482)(240,483)(241,484)(242,243), (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)>;
G:=Group( (1,244)(2,245)(3,246)(4,247)(5,248)(6,249)(7,250)(8,251)(9,252)(10,253)(11,254)(12,255)(13,256)(14,257)(15,258)(16,259)(17,260)(18,261)(19,262)(20,263)(21,264)(22,265)(23,266)(24,267)(25,268)(26,269)(27,270)(28,271)(29,272)(30,273)(31,274)(32,275)(33,276)(34,277)(35,278)(36,279)(37,280)(38,281)(39,282)(40,283)(41,284)(42,285)(43,286)(44,287)(45,288)(46,289)(47,290)(48,291)(49,292)(50,293)(51,294)(52,295)(53,296)(54,297)(55,298)(56,299)(57,300)(58,301)(59,302)(60,303)(61,304)(62,305)(63,306)(64,307)(65,308)(66,309)(67,310)(68,311)(69,312)(70,313)(71,314)(72,315)(73,316)(74,317)(75,318)(76,319)(77,320)(78,321)(79,322)(80,323)(81,324)(82,325)(83,326)(84,327)(85,328)(86,329)(87,330)(88,331)(89,332)(90,333)(91,334)(92,335)(93,336)(94,337)(95,338)(96,339)(97,340)(98,341)(99,342)(100,343)(101,344)(102,345)(103,346)(104,347)(105,348)(106,349)(107,350)(108,351)(109,352)(110,353)(111,354)(112,355)(113,356)(114,357)(115,358)(116,359)(117,360)(118,361)(119,362)(120,363)(121,364)(122,365)(123,366)(124,367)(125,368)(126,369)(127,370)(128,371)(129,372)(130,373)(131,374)(132,375)(133,376)(134,377)(135,378)(136,379)(137,380)(138,381)(139,382)(140,383)(141,384)(142,385)(143,386)(144,387)(145,388)(146,389)(147,390)(148,391)(149,392)(150,393)(151,394)(152,395)(153,396)(154,397)(155,398)(156,399)(157,400)(158,401)(159,402)(160,403)(161,404)(162,405)(163,406)(164,407)(165,408)(166,409)(167,410)(168,411)(169,412)(170,413)(171,414)(172,415)(173,416)(174,417)(175,418)(176,419)(177,420)(178,421)(179,422)(180,423)(181,424)(182,425)(183,426)(184,427)(185,428)(186,429)(187,430)(188,431)(189,432)(190,433)(191,434)(192,435)(193,436)(194,437)(195,438)(196,439)(197,440)(198,441)(199,442)(200,443)(201,444)(202,445)(203,446)(204,447)(205,448)(206,449)(207,450)(208,451)(209,452)(210,453)(211,454)(212,455)(213,456)(214,457)(215,458)(216,459)(217,460)(218,461)(219,462)(220,463)(221,464)(222,465)(223,466)(224,467)(225,468)(226,469)(227,470)(228,471)(229,472)(230,473)(231,474)(232,475)(233,476)(234,477)(235,478)(236,479)(237,480)(238,481)(239,482)(240,483)(241,484)(242,243), (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) );
G=PermutationGroup([[(1,244),(2,245),(3,246),(4,247),(5,248),(6,249),(7,250),(8,251),(9,252),(10,253),(11,254),(12,255),(13,256),(14,257),(15,258),(16,259),(17,260),(18,261),(19,262),(20,263),(21,264),(22,265),(23,266),(24,267),(25,268),(26,269),(27,270),(28,271),(29,272),(30,273),(31,274),(32,275),(33,276),(34,277),(35,278),(36,279),(37,280),(38,281),(39,282),(40,283),(41,284),(42,285),(43,286),(44,287),(45,288),(46,289),(47,290),(48,291),(49,292),(50,293),(51,294),(52,295),(53,296),(54,297),(55,298),(56,299),(57,300),(58,301),(59,302),(60,303),(61,304),(62,305),(63,306),(64,307),(65,308),(66,309),(67,310),(68,311),(69,312),(70,313),(71,314),(72,315),(73,316),(74,317),(75,318),(76,319),(77,320),(78,321),(79,322),(80,323),(81,324),(82,325),(83,326),(84,327),(85,328),(86,329),(87,330),(88,331),(89,332),(90,333),(91,334),(92,335),(93,336),(94,337),(95,338),(96,339),(97,340),(98,341),(99,342),(100,343),(101,344),(102,345),(103,346),(104,347),(105,348),(106,349),(107,350),(108,351),(109,352),(110,353),(111,354),(112,355),(113,356),(114,357),(115,358),(116,359),(117,360),(118,361),(119,362),(120,363),(121,364),(122,365),(123,366),(124,367),(125,368),(126,369),(127,370),(128,371),(129,372),(130,373),(131,374),(132,375),(133,376),(134,377),(135,378),(136,379),(137,380),(138,381),(139,382),(140,383),(141,384),(142,385),(143,386),(144,387),(145,388),(146,389),(147,390),(148,391),(149,392),(150,393),(151,394),(152,395),(153,396),(154,397),(155,398),(156,399),(157,400),(158,401),(159,402),(160,403),(161,404),(162,405),(163,406),(164,407),(165,408),(166,409),(167,410),(168,411),(169,412),(170,413),(171,414),(172,415),(173,416),(174,417),(175,418),(176,419),(177,420),(178,421),(179,422),(180,423),(181,424),(182,425),(183,426),(184,427),(185,428),(186,429),(187,430),(188,431),(189,432),(190,433),(191,434),(192,435),(193,436),(194,437),(195,438),(196,439),(197,440),(198,441),(199,442),(200,443),(201,444),(202,445),(203,446),(204,447),(205,448),(206,449),(207,450),(208,451),(209,452),(210,453),(211,454),(212,455),(213,456),(214,457),(215,458),(216,459),(217,460),(218,461),(219,462),(220,463),(221,464),(222,465),(223,466),(224,467),(225,468),(226,469),(227,470),(228,471),(229,472),(230,473),(231,474),(232,475),(233,476),(234,477),(235,478),(236,479),(237,480),(238,481),(239,482),(240,483),(241,484),(242,243)], [(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)]])
484 conjugacy classes
| class | 1 | 2A | 2B | 2C | 11A | ··· | 11J | 22A | ··· | 22AD | 121A | ··· | 121DF | 242A | ··· | 242LR |
| order | 1 | 2 | 2 | 2 | 11 | ··· | 11 | 22 | ··· | 22 | 121 | ··· | 121 | 242 | ··· | 242 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
484 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C11 | C22 | C121 | C242 |
| kernel | C2×C242 | C242 | C2×C22 | C22 | C22 | C2 |
| # reps | 1 | 3 | 10 | 30 | 110 | 330 |
Matrix representation of C2×C242 ►in GL2(𝔽727) generated by
| 726 | 0 |
| 0 | 1 |
| 606 | 0 |
| 0 | 108 |
G:=sub<GL(2,GF(727))| [726,0,0,1],[606,0,0,108] >;
C2×C242 in GAP, Magma, Sage, TeX
C_2\times C_{242} % in TeX
G:=Group("C2xC242"); // GroupNames label
G:=SmallGroup(484,4);
// by ID
G=gap.SmallGroup(484,4);
# by ID
G:=PCGroup([4,-2,-2,-11,-11,142]);
// Polycyclic
G:=Group<a,b|a^2=b^242=1,a*b=b*a>;
// generators/relations
Export