direct product, abelian, monomial, 2-elementary
Aliases: C2×C244, SmallGroup(488,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C244 |
| C1 — C2×C244 |
| C1 — C2×C244 |
Generators and relations for C2×C244
G = < a,b | a2=b244=1, ab=ba >
Smallest permutation representation of C2×C244
►Regular action on 488 points
Generators in S488
(1 473)(2 474)(3 475)(4 476)(5 477)(6 478)(7 479)(8 480)(9 481)(10 482)(11 483)(12 484)(13 485)(14 486)(15 487)(16 488)(17 245)(18 246)(19 247)(20 248)(21 249)(22 250)(23 251)(24 252)(25 253)(26 254)(27 255)(28 256)(29 257)(30 258)(31 259)(32 260)(33 261)(34 262)(35 263)(36 264)(37 265)(38 266)(39 267)(40 268)(41 269)(42 270)(43 271)(44 272)(45 273)(46 274)(47 275)(48 276)(49 277)(50 278)(51 279)(52 280)(53 281)(54 282)(55 283)(56 284)(57 285)(58 286)(59 287)(60 288)(61 289)(62 290)(63 291)(64 292)(65 293)(66 294)(67 295)(68 296)(69 297)(70 298)(71 299)(72 300)(73 301)(74 302)(75 303)(76 304)(77 305)(78 306)(79 307)(80 308)(81 309)(82 310)(83 311)(84 312)(85 313)(86 314)(87 315)(88 316)(89 317)(90 318)(91 319)(92 320)(93 321)(94 322)(95 323)(96 324)(97 325)(98 326)(99 327)(100 328)(101 329)(102 330)(103 331)(104 332)(105 333)(106 334)(107 335)(108 336)(109 337)(110 338)(111 339)(112 340)(113 341)(114 342)(115 343)(116 344)(117 345)(118 346)(119 347)(120 348)(121 349)(122 350)(123 351)(124 352)(125 353)(126 354)(127 355)(128 356)(129 357)(130 358)(131 359)(132 360)(133 361)(134 362)(135 363)(136 364)(137 365)(138 366)(139 367)(140 368)(141 369)(142 370)(143 371)(144 372)(145 373)(146 374)(147 375)(148 376)(149 377)(150 378)(151 379)(152 380)(153 381)(154 382)(155 383)(156 384)(157 385)(158 386)(159 387)(160 388)(161 389)(162 390)(163 391)(164 392)(165 393)(166 394)(167 395)(168 396)(169 397)(170 398)(171 399)(172 400)(173 401)(174 402)(175 403)(176 404)(177 405)(178 406)(179 407)(180 408)(181 409)(182 410)(183 411)(184 412)(185 413)(186 414)(187 415)(188 416)(189 417)(190 418)(191 419)(192 420)(193 421)(194 422)(195 423)(196 424)(197 425)(198 426)(199 427)(200 428)(201 429)(202 430)(203 431)(204 432)(205 433)(206 434)(207 435)(208 436)(209 437)(210 438)(211 439)(212 440)(213 441)(214 442)(215 443)(216 444)(217 445)(218 446)(219 447)(220 448)(221 449)(222 450)(223 451)(224 452)(225 453)(226 454)(227 455)(228 456)(229 457)(230 458)(231 459)(232 460)(233 461)(234 462)(235 463)(236 464)(237 465)(238 466)(239 467)(240 468)(241 469)(242 470)(243 471)(244 472)
(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)
G:=sub<Sym(488)| (1,473)(2,474)(3,475)(4,476)(5,477)(6,478)(7,479)(8,480)(9,481)(10,482)(11,483)(12,484)(13,485)(14,486)(15,487)(16,488)(17,245)(18,246)(19,247)(20,248)(21,249)(22,250)(23,251)(24,252)(25,253)(26,254)(27,255)(28,256)(29,257)(30,258)(31,259)(32,260)(33,261)(34,262)(35,263)(36,264)(37,265)(38,266)(39,267)(40,268)(41,269)(42,270)(43,271)(44,272)(45,273)(46,274)(47,275)(48,276)(49,277)(50,278)(51,279)(52,280)(53,281)(54,282)(55,283)(56,284)(57,285)(58,286)(59,287)(60,288)(61,289)(62,290)(63,291)(64,292)(65,293)(66,294)(67,295)(68,296)(69,297)(70,298)(71,299)(72,300)(73,301)(74,302)(75,303)(76,304)(77,305)(78,306)(79,307)(80,308)(81,309)(82,310)(83,311)(84,312)(85,313)(86,314)(87,315)(88,316)(89,317)(90,318)(91,319)(92,320)(93,321)(94,322)(95,323)(96,324)(97,325)(98,326)(99,327)(100,328)(101,329)(102,330)(103,331)(104,332)(105,333)(106,334)(107,335)(108,336)(109,337)(110,338)(111,339)(112,340)(113,341)(114,342)(115,343)(116,344)(117,345)(118,346)(119,347)(120,348)(121,349)(122,350)(123,351)(124,352)(125,353)(126,354)(127,355)(128,356)(129,357)(130,358)(131,359)(132,360)(133,361)(134,362)(135,363)(136,364)(137,365)(138,366)(139,367)(140,368)(141,369)(142,370)(143,371)(144,372)(145,373)(146,374)(147,375)(148,376)(149,377)(150,378)(151,379)(152,380)(153,381)(154,382)(155,383)(156,384)(157,385)(158,386)(159,387)(160,388)(161,389)(162,390)(163,391)(164,392)(165,393)(166,394)(167,395)(168,396)(169,397)(170,398)(171,399)(172,400)(173,401)(174,402)(175,403)(176,404)(177,405)(178,406)(179,407)(180,408)(181,409)(182,410)(183,411)(184,412)(185,413)(186,414)(187,415)(188,416)(189,417)(190,418)(191,419)(192,420)(193,421)(194,422)(195,423)(196,424)(197,425)(198,426)(199,427)(200,428)(201,429)(202,430)(203,431)(204,432)(205,433)(206,434)(207,435)(208,436)(209,437)(210,438)(211,439)(212,440)(213,441)(214,442)(215,443)(216,444)(217,445)(218,446)(219,447)(220,448)(221,449)(222,450)(223,451)(224,452)(225,453)(226,454)(227,455)(228,456)(229,457)(230,458)(231,459)(232,460)(233,461)(234,462)(235,463)(236,464)(237,465)(238,466)(239,467)(240,468)(241,469)(242,470)(243,471)(244,472), (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)>;
G:=Group( (1,473)(2,474)(3,475)(4,476)(5,477)(6,478)(7,479)(8,480)(9,481)(10,482)(11,483)(12,484)(13,485)(14,486)(15,487)(16,488)(17,245)(18,246)(19,247)(20,248)(21,249)(22,250)(23,251)(24,252)(25,253)(26,254)(27,255)(28,256)(29,257)(30,258)(31,259)(32,260)(33,261)(34,262)(35,263)(36,264)(37,265)(38,266)(39,267)(40,268)(41,269)(42,270)(43,271)(44,272)(45,273)(46,274)(47,275)(48,276)(49,277)(50,278)(51,279)(52,280)(53,281)(54,282)(55,283)(56,284)(57,285)(58,286)(59,287)(60,288)(61,289)(62,290)(63,291)(64,292)(65,293)(66,294)(67,295)(68,296)(69,297)(70,298)(71,299)(72,300)(73,301)(74,302)(75,303)(76,304)(77,305)(78,306)(79,307)(80,308)(81,309)(82,310)(83,311)(84,312)(85,313)(86,314)(87,315)(88,316)(89,317)(90,318)(91,319)(92,320)(93,321)(94,322)(95,323)(96,324)(97,325)(98,326)(99,327)(100,328)(101,329)(102,330)(103,331)(104,332)(105,333)(106,334)(107,335)(108,336)(109,337)(110,338)(111,339)(112,340)(113,341)(114,342)(115,343)(116,344)(117,345)(118,346)(119,347)(120,348)(121,349)(122,350)(123,351)(124,352)(125,353)(126,354)(127,355)(128,356)(129,357)(130,358)(131,359)(132,360)(133,361)(134,362)(135,363)(136,364)(137,365)(138,366)(139,367)(140,368)(141,369)(142,370)(143,371)(144,372)(145,373)(146,374)(147,375)(148,376)(149,377)(150,378)(151,379)(152,380)(153,381)(154,382)(155,383)(156,384)(157,385)(158,386)(159,387)(160,388)(161,389)(162,390)(163,391)(164,392)(165,393)(166,394)(167,395)(168,396)(169,397)(170,398)(171,399)(172,400)(173,401)(174,402)(175,403)(176,404)(177,405)(178,406)(179,407)(180,408)(181,409)(182,410)(183,411)(184,412)(185,413)(186,414)(187,415)(188,416)(189,417)(190,418)(191,419)(192,420)(193,421)(194,422)(195,423)(196,424)(197,425)(198,426)(199,427)(200,428)(201,429)(202,430)(203,431)(204,432)(205,433)(206,434)(207,435)(208,436)(209,437)(210,438)(211,439)(212,440)(213,441)(214,442)(215,443)(216,444)(217,445)(218,446)(219,447)(220,448)(221,449)(222,450)(223,451)(224,452)(225,453)(226,454)(227,455)(228,456)(229,457)(230,458)(231,459)(232,460)(233,461)(234,462)(235,463)(236,464)(237,465)(238,466)(239,467)(240,468)(241,469)(242,470)(243,471)(244,472), (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) );
G=PermutationGroup([[(1,473),(2,474),(3,475),(4,476),(5,477),(6,478),(7,479),(8,480),(9,481),(10,482),(11,483),(12,484),(13,485),(14,486),(15,487),(16,488),(17,245),(18,246),(19,247),(20,248),(21,249),(22,250),(23,251),(24,252),(25,253),(26,254),(27,255),(28,256),(29,257),(30,258),(31,259),(32,260),(33,261),(34,262),(35,263),(36,264),(37,265),(38,266),(39,267),(40,268),(41,269),(42,270),(43,271),(44,272),(45,273),(46,274),(47,275),(48,276),(49,277),(50,278),(51,279),(52,280),(53,281),(54,282),(55,283),(56,284),(57,285),(58,286),(59,287),(60,288),(61,289),(62,290),(63,291),(64,292),(65,293),(66,294),(67,295),(68,296),(69,297),(70,298),(71,299),(72,300),(73,301),(74,302),(75,303),(76,304),(77,305),(78,306),(79,307),(80,308),(81,309),(82,310),(83,311),(84,312),(85,313),(86,314),(87,315),(88,316),(89,317),(90,318),(91,319),(92,320),(93,321),(94,322),(95,323),(96,324),(97,325),(98,326),(99,327),(100,328),(101,329),(102,330),(103,331),(104,332),(105,333),(106,334),(107,335),(108,336),(109,337),(110,338),(111,339),(112,340),(113,341),(114,342),(115,343),(116,344),(117,345),(118,346),(119,347),(120,348),(121,349),(122,350),(123,351),(124,352),(125,353),(126,354),(127,355),(128,356),(129,357),(130,358),(131,359),(132,360),(133,361),(134,362),(135,363),(136,364),(137,365),(138,366),(139,367),(140,368),(141,369),(142,370),(143,371),(144,372),(145,373),(146,374),(147,375),(148,376),(149,377),(150,378),(151,379),(152,380),(153,381),(154,382),(155,383),(156,384),(157,385),(158,386),(159,387),(160,388),(161,389),(162,390),(163,391),(164,392),(165,393),(166,394),(167,395),(168,396),(169,397),(170,398),(171,399),(172,400),(173,401),(174,402),(175,403),(176,404),(177,405),(178,406),(179,407),(180,408),(181,409),(182,410),(183,411),(184,412),(185,413),(186,414),(187,415),(188,416),(189,417),(190,418),(191,419),(192,420),(193,421),(194,422),(195,423),(196,424),(197,425),(198,426),(199,427),(200,428),(201,429),(202,430),(203,431),(204,432),(205,433),(206,434),(207,435),(208,436),(209,437),(210,438),(211,439),(212,440),(213,441),(214,442),(215,443),(216,444),(217,445),(218,446),(219,447),(220,448),(221,449),(222,450),(223,451),(224,452),(225,453),(226,454),(227,455),(228,456),(229,457),(230,458),(231,459),(232,460),(233,461),(234,462),(235,463),(236,464),(237,465),(238,466),(239,467),(240,468),(241,469),(242,470),(243,471),(244,472)], [(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)]])
488 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 61A | ··· | 61BH | 122A | ··· | 122FX | 244A | ··· | 244IF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 61 | ··· | 61 | 122 | ··· | 122 | 244 | ··· | 244 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
488 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C61 | C122 | C122 | C244 |
| kernel | C2×C244 | C244 | C2×C122 | C122 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 60 | 120 | 60 | 240 |
Matrix representation of C2×C244 ►in GL2(𝔽733) generated by
| 732 | 0 |
| 0 | 732 |
| 440 | 0 |
| 0 | 432 |
G:=sub<GL(2,GF(733))| [732,0,0,732],[440,0,0,432] >;
C2×C244 in GAP, Magma, Sage, TeX
C_2\times C_{244} % in TeX
G:=Group("C2xC244"); // GroupNames label
G:=SmallGroup(488,9);
// by ID
G=gap.SmallGroup(488,9);
# by ID
G:=PCGroup([4,-2,-2,-61,-2,976]);
// Polycyclic
G:=Group<a,b|a^2=b^244=1,a*b=b*a>;
// generators/relations
Export