direct product, abelian, monomial, 2-elementary
Aliases: C2×C246, SmallGroup(492,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C246 |
| C1 — C2×C246 |
| C1 — C2×C246 |
Generators and relations for C2×C246
G = < a,b | a2=b246=1, ab=ba >
Smallest permutation representation of C2×C246
►Regular action on 492 points
Generators in S492
(1 421)(2 422)(3 423)(4 424)(5 425)(6 426)(7 427)(8 428)(9 429)(10 430)(11 431)(12 432)(13 433)(14 434)(15 435)(16 436)(17 437)(18 438)(19 439)(20 440)(21 441)(22 442)(23 443)(24 444)(25 445)(26 446)(27 447)(28 448)(29 449)(30 450)(31 451)(32 452)(33 453)(34 454)(35 455)(36 456)(37 457)(38 458)(39 459)(40 460)(41 461)(42 462)(43 463)(44 464)(45 465)(46 466)(47 467)(48 468)(49 469)(50 470)(51 471)(52 472)(53 473)(54 474)(55 475)(56 476)(57 477)(58 478)(59 479)(60 480)(61 481)(62 482)(63 483)(64 484)(65 485)(66 486)(67 487)(68 488)(69 489)(70 490)(71 491)(72 492)(73 247)(74 248)(75 249)(76 250)(77 251)(78 252)(79 253)(80 254)(81 255)(82 256)(83 257)(84 258)(85 259)(86 260)(87 261)(88 262)(89 263)(90 264)(91 265)(92 266)(93 267)(94 268)(95 269)(96 270)(97 271)(98 272)(99 273)(100 274)(101 275)(102 276)(103 277)(104 278)(105 279)(106 280)(107 281)(108 282)(109 283)(110 284)(111 285)(112 286)(113 287)(114 288)(115 289)(116 290)(117 291)(118 292)(119 293)(120 294)(121 295)(122 296)(123 297)(124 298)(125 299)(126 300)(127 301)(128 302)(129 303)(130 304)(131 305)(132 306)(133 307)(134 308)(135 309)(136 310)(137 311)(138 312)(139 313)(140 314)(141 315)(142 316)(143 317)(144 318)(145 319)(146 320)(147 321)(148 322)(149 323)(150 324)(151 325)(152 326)(153 327)(154 328)(155 329)(156 330)(157 331)(158 332)(159 333)(160 334)(161 335)(162 336)(163 337)(164 338)(165 339)(166 340)(167 341)(168 342)(169 343)(170 344)(171 345)(172 346)(173 347)(174 348)(175 349)(176 350)(177 351)(178 352)(179 353)(180 354)(181 355)(182 356)(183 357)(184 358)(185 359)(186 360)(187 361)(188 362)(189 363)(190 364)(191 365)(192 366)(193 367)(194 368)(195 369)(196 370)(197 371)(198 372)(199 373)(200 374)(201 375)(202 376)(203 377)(204 378)(205 379)(206 380)(207 381)(208 382)(209 383)(210 384)(211 385)(212 386)(213 387)(214 388)(215 389)(216 390)(217 391)(218 392)(219 393)(220 394)(221 395)(222 396)(223 397)(224 398)(225 399)(226 400)(227 401)(228 402)(229 403)(230 404)(231 405)(232 406)(233 407)(234 408)(235 409)(236 410)(237 411)(238 412)(239 413)(240 414)(241 415)(242 416)(243 417)(244 418)(245 419)(246 420)
(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)
G:=sub<Sym(492)| (1,421)(2,422)(3,423)(4,424)(5,425)(6,426)(7,427)(8,428)(9,429)(10,430)(11,431)(12,432)(13,433)(14,434)(15,435)(16,436)(17,437)(18,438)(19,439)(20,440)(21,441)(22,442)(23,443)(24,444)(25,445)(26,446)(27,447)(28,448)(29,449)(30,450)(31,451)(32,452)(33,453)(34,454)(35,455)(36,456)(37,457)(38,458)(39,459)(40,460)(41,461)(42,462)(43,463)(44,464)(45,465)(46,466)(47,467)(48,468)(49,469)(50,470)(51,471)(52,472)(53,473)(54,474)(55,475)(56,476)(57,477)(58,478)(59,479)(60,480)(61,481)(62,482)(63,483)(64,484)(65,485)(66,486)(67,487)(68,488)(69,489)(70,490)(71,491)(72,492)(73,247)(74,248)(75,249)(76,250)(77,251)(78,252)(79,253)(80,254)(81,255)(82,256)(83,257)(84,258)(85,259)(86,260)(87,261)(88,262)(89,263)(90,264)(91,265)(92,266)(93,267)(94,268)(95,269)(96,270)(97,271)(98,272)(99,273)(100,274)(101,275)(102,276)(103,277)(104,278)(105,279)(106,280)(107,281)(108,282)(109,283)(110,284)(111,285)(112,286)(113,287)(114,288)(115,289)(116,290)(117,291)(118,292)(119,293)(120,294)(121,295)(122,296)(123,297)(124,298)(125,299)(126,300)(127,301)(128,302)(129,303)(130,304)(131,305)(132,306)(133,307)(134,308)(135,309)(136,310)(137,311)(138,312)(139,313)(140,314)(141,315)(142,316)(143,317)(144,318)(145,319)(146,320)(147,321)(148,322)(149,323)(150,324)(151,325)(152,326)(153,327)(154,328)(155,329)(156,330)(157,331)(158,332)(159,333)(160,334)(161,335)(162,336)(163,337)(164,338)(165,339)(166,340)(167,341)(168,342)(169,343)(170,344)(171,345)(172,346)(173,347)(174,348)(175,349)(176,350)(177,351)(178,352)(179,353)(180,354)(181,355)(182,356)(183,357)(184,358)(185,359)(186,360)(187,361)(188,362)(189,363)(190,364)(191,365)(192,366)(193,367)(194,368)(195,369)(196,370)(197,371)(198,372)(199,373)(200,374)(201,375)(202,376)(203,377)(204,378)(205,379)(206,380)(207,381)(208,382)(209,383)(210,384)(211,385)(212,386)(213,387)(214,388)(215,389)(216,390)(217,391)(218,392)(219,393)(220,394)(221,395)(222,396)(223,397)(224,398)(225,399)(226,400)(227,401)(228,402)(229,403)(230,404)(231,405)(232,406)(233,407)(234,408)(235,409)(236,410)(237,411)(238,412)(239,413)(240,414)(241,415)(242,416)(243,417)(244,418)(245,419)(246,420), (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)>;
G:=Group( (1,421)(2,422)(3,423)(4,424)(5,425)(6,426)(7,427)(8,428)(9,429)(10,430)(11,431)(12,432)(13,433)(14,434)(15,435)(16,436)(17,437)(18,438)(19,439)(20,440)(21,441)(22,442)(23,443)(24,444)(25,445)(26,446)(27,447)(28,448)(29,449)(30,450)(31,451)(32,452)(33,453)(34,454)(35,455)(36,456)(37,457)(38,458)(39,459)(40,460)(41,461)(42,462)(43,463)(44,464)(45,465)(46,466)(47,467)(48,468)(49,469)(50,470)(51,471)(52,472)(53,473)(54,474)(55,475)(56,476)(57,477)(58,478)(59,479)(60,480)(61,481)(62,482)(63,483)(64,484)(65,485)(66,486)(67,487)(68,488)(69,489)(70,490)(71,491)(72,492)(73,247)(74,248)(75,249)(76,250)(77,251)(78,252)(79,253)(80,254)(81,255)(82,256)(83,257)(84,258)(85,259)(86,260)(87,261)(88,262)(89,263)(90,264)(91,265)(92,266)(93,267)(94,268)(95,269)(96,270)(97,271)(98,272)(99,273)(100,274)(101,275)(102,276)(103,277)(104,278)(105,279)(106,280)(107,281)(108,282)(109,283)(110,284)(111,285)(112,286)(113,287)(114,288)(115,289)(116,290)(117,291)(118,292)(119,293)(120,294)(121,295)(122,296)(123,297)(124,298)(125,299)(126,300)(127,301)(128,302)(129,303)(130,304)(131,305)(132,306)(133,307)(134,308)(135,309)(136,310)(137,311)(138,312)(139,313)(140,314)(141,315)(142,316)(143,317)(144,318)(145,319)(146,320)(147,321)(148,322)(149,323)(150,324)(151,325)(152,326)(153,327)(154,328)(155,329)(156,330)(157,331)(158,332)(159,333)(160,334)(161,335)(162,336)(163,337)(164,338)(165,339)(166,340)(167,341)(168,342)(169,343)(170,344)(171,345)(172,346)(173,347)(174,348)(175,349)(176,350)(177,351)(178,352)(179,353)(180,354)(181,355)(182,356)(183,357)(184,358)(185,359)(186,360)(187,361)(188,362)(189,363)(190,364)(191,365)(192,366)(193,367)(194,368)(195,369)(196,370)(197,371)(198,372)(199,373)(200,374)(201,375)(202,376)(203,377)(204,378)(205,379)(206,380)(207,381)(208,382)(209,383)(210,384)(211,385)(212,386)(213,387)(214,388)(215,389)(216,390)(217,391)(218,392)(219,393)(220,394)(221,395)(222,396)(223,397)(224,398)(225,399)(226,400)(227,401)(228,402)(229,403)(230,404)(231,405)(232,406)(233,407)(234,408)(235,409)(236,410)(237,411)(238,412)(239,413)(240,414)(241,415)(242,416)(243,417)(244,418)(245,419)(246,420), (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) );
G=PermutationGroup([[(1,421),(2,422),(3,423),(4,424),(5,425),(6,426),(7,427),(8,428),(9,429),(10,430),(11,431),(12,432),(13,433),(14,434),(15,435),(16,436),(17,437),(18,438),(19,439),(20,440),(21,441),(22,442),(23,443),(24,444),(25,445),(26,446),(27,447),(28,448),(29,449),(30,450),(31,451),(32,452),(33,453),(34,454),(35,455),(36,456),(37,457),(38,458),(39,459),(40,460),(41,461),(42,462),(43,463),(44,464),(45,465),(46,466),(47,467),(48,468),(49,469),(50,470),(51,471),(52,472),(53,473),(54,474),(55,475),(56,476),(57,477),(58,478),(59,479),(60,480),(61,481),(62,482),(63,483),(64,484),(65,485),(66,486),(67,487),(68,488),(69,489),(70,490),(71,491),(72,492),(73,247),(74,248),(75,249),(76,250),(77,251),(78,252),(79,253),(80,254),(81,255),(82,256),(83,257),(84,258),(85,259),(86,260),(87,261),(88,262),(89,263),(90,264),(91,265),(92,266),(93,267),(94,268),(95,269),(96,270),(97,271),(98,272),(99,273),(100,274),(101,275),(102,276),(103,277),(104,278),(105,279),(106,280),(107,281),(108,282),(109,283),(110,284),(111,285),(112,286),(113,287),(114,288),(115,289),(116,290),(117,291),(118,292),(119,293),(120,294),(121,295),(122,296),(123,297),(124,298),(125,299),(126,300),(127,301),(128,302),(129,303),(130,304),(131,305),(132,306),(133,307),(134,308),(135,309),(136,310),(137,311),(138,312),(139,313),(140,314),(141,315),(142,316),(143,317),(144,318),(145,319),(146,320),(147,321),(148,322),(149,323),(150,324),(151,325),(152,326),(153,327),(154,328),(155,329),(156,330),(157,331),(158,332),(159,333),(160,334),(161,335),(162,336),(163,337),(164,338),(165,339),(166,340),(167,341),(168,342),(169,343),(170,344),(171,345),(172,346),(173,347),(174,348),(175,349),(176,350),(177,351),(178,352),(179,353),(180,354),(181,355),(182,356),(183,357),(184,358),(185,359),(186,360),(187,361),(188,362),(189,363),(190,364),(191,365),(192,366),(193,367),(194,368),(195,369),(196,370),(197,371),(198,372),(199,373),(200,374),(201,375),(202,376),(203,377),(204,378),(205,379),(206,380),(207,381),(208,382),(209,383),(210,384),(211,385),(212,386),(213,387),(214,388),(215,389),(216,390),(217,391),(218,392),(219,393),(220,394),(221,395),(222,396),(223,397),(224,398),(225,399),(226,400),(227,401),(228,402),(229,403),(230,404),(231,405),(232,406),(233,407),(234,408),(235,409),(236,410),(237,411),(238,412),(239,413),(240,414),(241,415),(242,416),(243,417),(244,418),(245,419),(246,420)], [(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)]])
492 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 41A | ··· | 41AN | 82A | ··· | 82DP | 123A | ··· | 123CB | 246A | ··· | 246IF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 41 | ··· | 41 | 82 | ··· | 82 | 123 | ··· | 123 | 246 | ··· | 246 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
492 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C41 | C82 | C123 | C246 |
| kernel | C2×C246 | C246 | C2×C82 | C82 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 40 | 120 | 80 | 240 |
Matrix representation of C2×C246 ►in GL2(𝔽739) generated by
| 738 | 0 |
| 0 | 738 |
| 237 | 0 |
| 0 | 689 |
G:=sub<GL(2,GF(739))| [738,0,0,738],[237,0,0,689] >;
C2×C246 in GAP, Magma, Sage, TeX
C_2\times C_{246} % in TeX
G:=Group("C2xC246"); // GroupNames label
G:=SmallGroup(492,12);
// by ID
G=gap.SmallGroup(492,12);
# by ID
G:=PCGroup([4,-2,-2,-3,-41]);
// Polycyclic
G:=Group<a,b|a^2=b^246=1,a*b=b*a>;
// generators/relations
Export