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G = C2×C248order 496 = 24·31

Abelian group of type [2,248]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C248, SmallGroup(496,22)

Series: Derived Chief Lower central Upper central

C1 — C2×C248
C1C2C4C124C248 — C2×C248
C1 — C2×C248
C1 — C2×C248

Generators and relations for C2×C248
 G = < a,b | a2=b248=1, ab=ba >


Smallest permutation representation of C2×C248
Regular action on 496 points
Generators in S496
(1 266)(2 267)(3 268)(4 269)(5 270)(6 271)(7 272)(8 273)(9 274)(10 275)(11 276)(12 277)(13 278)(14 279)(15 280)(16 281)(17 282)(18 283)(19 284)(20 285)(21 286)(22 287)(23 288)(24 289)(25 290)(26 291)(27 292)(28 293)(29 294)(30 295)(31 296)(32 297)(33 298)(34 299)(35 300)(36 301)(37 302)(38 303)(39 304)(40 305)(41 306)(42 307)(43 308)(44 309)(45 310)(46 311)(47 312)(48 313)(49 314)(50 315)(51 316)(52 317)(53 318)(54 319)(55 320)(56 321)(57 322)(58 323)(59 324)(60 325)(61 326)(62 327)(63 328)(64 329)(65 330)(66 331)(67 332)(68 333)(69 334)(70 335)(71 336)(72 337)(73 338)(74 339)(75 340)(76 341)(77 342)(78 343)(79 344)(80 345)(81 346)(82 347)(83 348)(84 349)(85 350)(86 351)(87 352)(88 353)(89 354)(90 355)(91 356)(92 357)(93 358)(94 359)(95 360)(96 361)(97 362)(98 363)(99 364)(100 365)(101 366)(102 367)(103 368)(104 369)(105 370)(106 371)(107 372)(108 373)(109 374)(110 375)(111 376)(112 377)(113 378)(114 379)(115 380)(116 381)(117 382)(118 383)(119 384)(120 385)(121 386)(122 387)(123 388)(124 389)(125 390)(126 391)(127 392)(128 393)(129 394)(130 395)(131 396)(132 397)(133 398)(134 399)(135 400)(136 401)(137 402)(138 403)(139 404)(140 405)(141 406)(142 407)(143 408)(144 409)(145 410)(146 411)(147 412)(148 413)(149 414)(150 415)(151 416)(152 417)(153 418)(154 419)(155 420)(156 421)(157 422)(158 423)(159 424)(160 425)(161 426)(162 427)(163 428)(164 429)(165 430)(166 431)(167 432)(168 433)(169 434)(170 435)(171 436)(172 437)(173 438)(174 439)(175 440)(176 441)(177 442)(178 443)(179 444)(180 445)(181 446)(182 447)(183 448)(184 449)(185 450)(186 451)(187 452)(188 453)(189 454)(190 455)(191 456)(192 457)(193 458)(194 459)(195 460)(196 461)(197 462)(198 463)(199 464)(200 465)(201 466)(202 467)(203 468)(204 469)(205 470)(206 471)(207 472)(208 473)(209 474)(210 475)(211 476)(212 477)(213 478)(214 479)(215 480)(216 481)(217 482)(218 483)(219 484)(220 485)(221 486)(222 487)(223 488)(224 489)(225 490)(226 491)(227 492)(228 493)(229 494)(230 495)(231 496)(232 249)(233 250)(234 251)(235 252)(236 253)(237 254)(238 255)(239 256)(240 257)(241 258)(242 259)(243 260)(244 261)(245 262)(246 263)(247 264)(248 265)
(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)

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

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

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

496 conjugacy classes

class 1 2A2B2C4A4B4C4D8A···8H31A···31AD62A···62CL124A···124DP248A···248IF
order122244448···831···3162···62124···124248···248
size111111111···11···11···11···11···1

496 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C8C31C62C62C124C124C248
kernelC2×C248C248C2×C124C124C2×C62C62C2×C8C8C2×C4C4C22C2
# reps1212283060306060240

Matrix representation of C2×C248 in GL2(𝔽1489) generated by

14880
01
,
5570
0433
G:=sub<GL(2,GF(1489))| [1488,0,0,1],[557,0,0,433] >;

C2×C248 in GAP, Magma, Sage, TeX

C_2\times C_{248}
% in TeX

G:=Group("C2xC248");
// GroupNames label

G:=SmallGroup(496,22);
// by ID

G=gap.SmallGroup(496,22);
# by ID

G:=PCGroup([5,-2,-2,-31,-2,-2,620,58]);
// Polycyclic

G:=Group<a,b|a^2=b^248=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C248 in TeX

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