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G = C2×C246order 492 = 22·3·41

Abelian group of type [2,246]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C246, SmallGroup(492,12)

Series: Derived Chief Lower central Upper central

C1 — C2×C246
C1C41C123C246 — 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
(1 260)(2 261)(3 262)(4 263)(5 264)(6 265)(7 266)(8 267)(9 268)(10 269)(11 270)(12 271)(13 272)(14 273)(15 274)(16 275)(17 276)(18 277)(19 278)(20 279)(21 280)(22 281)(23 282)(24 283)(25 284)(26 285)(27 286)(28 287)(29 288)(30 289)(31 290)(32 291)(33 292)(34 293)(35 294)(36 295)(37 296)(38 297)(39 298)(40 299)(41 300)(42 301)(43 302)(44 303)(45 304)(46 305)(47 306)(48 307)(49 308)(50 309)(51 310)(52 311)(53 312)(54 313)(55 314)(56 315)(57 316)(58 317)(59 318)(60 319)(61 320)(62 321)(63 322)(64 323)(65 324)(66 325)(67 326)(68 327)(69 328)(70 329)(71 330)(72 331)(73 332)(74 333)(75 334)(76 335)(77 336)(78 337)(79 338)(80 339)(81 340)(82 341)(83 342)(84 343)(85 344)(86 345)(87 346)(88 347)(89 348)(90 349)(91 350)(92 351)(93 352)(94 353)(95 354)(96 355)(97 356)(98 357)(99 358)(100 359)(101 360)(102 361)(103 362)(104 363)(105 364)(106 365)(107 366)(108 367)(109 368)(110 369)(111 370)(112 371)(113 372)(114 373)(115 374)(116 375)(117 376)(118 377)(119 378)(120 379)(121 380)(122 381)(123 382)(124 383)(125 384)(126 385)(127 386)(128 387)(129 388)(130 389)(131 390)(132 391)(133 392)(134 393)(135 394)(136 395)(137 396)(138 397)(139 398)(140 399)(141 400)(142 401)(143 402)(144 403)(145 404)(146 405)(147 406)(148 407)(149 408)(150 409)(151 410)(152 411)(153 412)(154 413)(155 414)(156 415)(157 416)(158 417)(159 418)(160 419)(161 420)(162 421)(163 422)(164 423)(165 424)(166 425)(167 426)(168 427)(169 428)(170 429)(171 430)(172 431)(173 432)(174 433)(175 434)(176 435)(177 436)(178 437)(179 438)(180 439)(181 440)(182 441)(183 442)(184 443)(185 444)(186 445)(187 446)(188 447)(189 448)(190 449)(191 450)(192 451)(193 452)(194 453)(195 454)(196 455)(197 456)(198 457)(199 458)(200 459)(201 460)(202 461)(203 462)(204 463)(205 464)(206 465)(207 466)(208 467)(209 468)(210 469)(211 470)(212 471)(213 472)(214 473)(215 474)(216 475)(217 476)(218 477)(219 478)(220 479)(221 480)(222 481)(223 482)(224 483)(225 484)(226 485)(227 486)(228 487)(229 488)(230 489)(231 490)(232 491)(233 492)(234 247)(235 248)(236 249)(237 250)(238 251)(239 252)(240 253)(241 254)(242 255)(243 256)(244 257)(245 258)(246 259)
(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)| (1,260)(2,261)(3,262)(4,263)(5,264)(6,265)(7,266)(8,267)(9,268)(10,269)(11,270)(12,271)(13,272)(14,273)(15,274)(16,275)(17,276)(18,277)(19,278)(20,279)(21,280)(22,281)(23,282)(24,283)(25,284)(26,285)(27,286)(28,287)(29,288)(30,289)(31,290)(32,291)(33,292)(34,293)(35,294)(36,295)(37,296)(38,297)(39,298)(40,299)(41,300)(42,301)(43,302)(44,303)(45,304)(46,305)(47,306)(48,307)(49,308)(50,309)(51,310)(52,311)(53,312)(54,313)(55,314)(56,315)(57,316)(58,317)(59,318)(60,319)(61,320)(62,321)(63,322)(64,323)(65,324)(66,325)(67,326)(68,327)(69,328)(70,329)(71,330)(72,331)(73,332)(74,333)(75,334)(76,335)(77,336)(78,337)(79,338)(80,339)(81,340)(82,341)(83,342)(84,343)(85,344)(86,345)(87,346)(88,347)(89,348)(90,349)(91,350)(92,351)(93,352)(94,353)(95,354)(96,355)(97,356)(98,357)(99,358)(100,359)(101,360)(102,361)(103,362)(104,363)(105,364)(106,365)(107,366)(108,367)(109,368)(110,369)(111,370)(112,371)(113,372)(114,373)(115,374)(116,375)(117,376)(118,377)(119,378)(120,379)(121,380)(122,381)(123,382)(124,383)(125,384)(126,385)(127,386)(128,387)(129,388)(130,389)(131,390)(132,391)(133,392)(134,393)(135,394)(136,395)(137,396)(138,397)(139,398)(140,399)(141,400)(142,401)(143,402)(144,403)(145,404)(146,405)(147,406)(148,407)(149,408)(150,409)(151,410)(152,411)(153,412)(154,413)(155,414)(156,415)(157,416)(158,417)(159,418)(160,419)(161,420)(162,421)(163,422)(164,423)(165,424)(166,425)(167,426)(168,427)(169,428)(170,429)(171,430)(172,431)(173,432)(174,433)(175,434)(176,435)(177,436)(178,437)(179,438)(180,439)(181,440)(182,441)(183,442)(184,443)(185,444)(186,445)(187,446)(188,447)(189,448)(190,449)(191,450)(192,451)(193,452)(194,453)(195,454)(196,455)(197,456)(198,457)(199,458)(200,459)(201,460)(202,461)(203,462)(204,463)(205,464)(206,465)(207,466)(208,467)(209,468)(210,469)(211,470)(212,471)(213,472)(214,473)(215,474)(216,475)(217,476)(218,477)(219,478)(220,479)(221,480)(222,481)(223,482)(224,483)(225,484)(226,485)(227,486)(228,487)(229,488)(230,489)(231,490)(232,491)(233,492)(234,247)(235,248)(236,249)(237,250)(238,251)(239,252)(240,253)(241,254)(242,255)(243,256)(244,257)(245,258)(246,259), (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( (1,260)(2,261)(3,262)(4,263)(5,264)(6,265)(7,266)(8,267)(9,268)(10,269)(11,270)(12,271)(13,272)(14,273)(15,274)(16,275)(17,276)(18,277)(19,278)(20,279)(21,280)(22,281)(23,282)(24,283)(25,284)(26,285)(27,286)(28,287)(29,288)(30,289)(31,290)(32,291)(33,292)(34,293)(35,294)(36,295)(37,296)(38,297)(39,298)(40,299)(41,300)(42,301)(43,302)(44,303)(45,304)(46,305)(47,306)(48,307)(49,308)(50,309)(51,310)(52,311)(53,312)(54,313)(55,314)(56,315)(57,316)(58,317)(59,318)(60,319)(61,320)(62,321)(63,322)(64,323)(65,324)(66,325)(67,326)(68,327)(69,328)(70,329)(71,330)(72,331)(73,332)(74,333)(75,334)(76,335)(77,336)(78,337)(79,338)(80,339)(81,340)(82,341)(83,342)(84,343)(85,344)(86,345)(87,346)(88,347)(89,348)(90,349)(91,350)(92,351)(93,352)(94,353)(95,354)(96,355)(97,356)(98,357)(99,358)(100,359)(101,360)(102,361)(103,362)(104,363)(105,364)(106,365)(107,366)(108,367)(109,368)(110,369)(111,370)(112,371)(113,372)(114,373)(115,374)(116,375)(117,376)(118,377)(119,378)(120,379)(121,380)(122,381)(123,382)(124,383)(125,384)(126,385)(127,386)(128,387)(129,388)(130,389)(131,390)(132,391)(133,392)(134,393)(135,394)(136,395)(137,396)(138,397)(139,398)(140,399)(141,400)(142,401)(143,402)(144,403)(145,404)(146,405)(147,406)(148,407)(149,408)(150,409)(151,410)(152,411)(153,412)(154,413)(155,414)(156,415)(157,416)(158,417)(159,418)(160,419)(161,420)(162,421)(163,422)(164,423)(165,424)(166,425)(167,426)(168,427)(169,428)(170,429)(171,430)(172,431)(173,432)(174,433)(175,434)(176,435)(177,436)(178,437)(179,438)(180,439)(181,440)(182,441)(183,442)(184,443)(185,444)(186,445)(187,446)(188,447)(189,448)(190,449)(191,450)(192,451)(193,452)(194,453)(195,454)(196,455)(197,456)(198,457)(199,458)(200,459)(201,460)(202,461)(203,462)(204,463)(205,464)(206,465)(207,466)(208,467)(209,468)(210,469)(211,470)(212,471)(213,472)(214,473)(215,474)(216,475)(217,476)(218,477)(219,478)(220,479)(221,480)(222,481)(223,482)(224,483)(225,484)(226,485)(227,486)(228,487)(229,488)(230,489)(231,490)(232,491)(233,492)(234,247)(235,248)(236,249)(237,250)(238,251)(239,252)(240,253)(241,254)(242,255)(243,256)(244,257)(245,258)(246,259), (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([(1,260),(2,261),(3,262),(4,263),(5,264),(6,265),(7,266),(8,267),(9,268),(10,269),(11,270),(12,271),(13,272),(14,273),(15,274),(16,275),(17,276),(18,277),(19,278),(20,279),(21,280),(22,281),(23,282),(24,283),(25,284),(26,285),(27,286),(28,287),(29,288),(30,289),(31,290),(32,291),(33,292),(34,293),(35,294),(36,295),(37,296),(38,297),(39,298),(40,299),(41,300),(42,301),(43,302),(44,303),(45,304),(46,305),(47,306),(48,307),(49,308),(50,309),(51,310),(52,311),(53,312),(54,313),(55,314),(56,315),(57,316),(58,317),(59,318),(60,319),(61,320),(62,321),(63,322),(64,323),(65,324),(66,325),(67,326),(68,327),(69,328),(70,329),(71,330),(72,331),(73,332),(74,333),(75,334),(76,335),(77,336),(78,337),(79,338),(80,339),(81,340),(82,341),(83,342),(84,343),(85,344),(86,345),(87,346),(88,347),(89,348),(90,349),(91,350),(92,351),(93,352),(94,353),(95,354),(96,355),(97,356),(98,357),(99,358),(100,359),(101,360),(102,361),(103,362),(104,363),(105,364),(106,365),(107,366),(108,367),(109,368),(110,369),(111,370),(112,371),(113,372),(114,373),(115,374),(116,375),(117,376),(118,377),(119,378),(120,379),(121,380),(122,381),(123,382),(124,383),(125,384),(126,385),(127,386),(128,387),(129,388),(130,389),(131,390),(132,391),(133,392),(134,393),(135,394),(136,395),(137,396),(138,397),(139,398),(140,399),(141,400),(142,401),(143,402),(144,403),(145,404),(146,405),(147,406),(148,407),(149,408),(150,409),(151,410),(152,411),(153,412),(154,413),(155,414),(156,415),(157,416),(158,417),(159,418),(160,419),(161,420),(162,421),(163,422),(164,423),(165,424),(166,425),(167,426),(168,427),(169,428),(170,429),(171,430),(172,431),(173,432),(174,433),(175,434),(176,435),(177,436),(178,437),(179,438),(180,439),(181,440),(182,441),(183,442),(184,443),(185,444),(186,445),(187,446),(188,447),(189,448),(190,449),(191,450),(192,451),(193,452),(194,453),(195,454),(196,455),(197,456),(198,457),(199,458),(200,459),(201,460),(202,461),(203,462),(204,463),(205,464),(206,465),(207,466),(208,467),(209,468),(210,469),(211,470),(212,471),(213,472),(214,473),(215,474),(216,475),(217,476),(218,477),(219,478),(220,479),(221,480),(222,481),(223,482),(224,483),(225,484),(226,485),(227,486),(228,487),(229,488),(230,489),(231,490),(232,491),(233,492),(234,247),(235,248),(236,249),(237,250),(238,251),(239,252),(240,253),(241,254),(242,255),(243,256),(244,257),(245,258),(246,259)], [(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 2A2B2C3A3B6A···6F41A···41AN82A···82DP123A···123CB246A···246IF
order1222336···641···4182···82123···123246···246
size1111111···11···11···11···11···1

492 irreducible representations

dim11111111
type++
imageC1C2C3C6C41C82C123C246
kernelC2×C246C246C2×C82C82C2×C6C6C22C2
# reps13264012080240

Matrix representation of C2×C246 in GL2(𝔽739) generated by

7380
0738
,
2370
0689
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

Subgroup lattice of C2×C246 in TeX

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