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G = C2×C244order 488 = 23·61

Abelian group of type [2,244]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C244, SmallGroup(488,9)

Series: Derived Chief Lower central Upper central

C1 — C2×C244
C1C2C122C244 — 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
(1 247)(2 248)(3 249)(4 250)(5 251)(6 252)(7 253)(8 254)(9 255)(10 256)(11 257)(12 258)(13 259)(14 260)(15 261)(16 262)(17 263)(18 264)(19 265)(20 266)(21 267)(22 268)(23 269)(24 270)(25 271)(26 272)(27 273)(28 274)(29 275)(30 276)(31 277)(32 278)(33 279)(34 280)(35 281)(36 282)(37 283)(38 284)(39 285)(40 286)(41 287)(42 288)(43 289)(44 290)(45 291)(46 292)(47 293)(48 294)(49 295)(50 296)(51 297)(52 298)(53 299)(54 300)(55 301)(56 302)(57 303)(58 304)(59 305)(60 306)(61 307)(62 308)(63 309)(64 310)(65 311)(66 312)(67 313)(68 314)(69 315)(70 316)(71 317)(72 318)(73 319)(74 320)(75 321)(76 322)(77 323)(78 324)(79 325)(80 326)(81 327)(82 328)(83 329)(84 330)(85 331)(86 332)(87 333)(88 334)(89 335)(90 336)(91 337)(92 338)(93 339)(94 340)(95 341)(96 342)(97 343)(98 344)(99 345)(100 346)(101 347)(102 348)(103 349)(104 350)(105 351)(106 352)(107 353)(108 354)(109 355)(110 356)(111 357)(112 358)(113 359)(114 360)(115 361)(116 362)(117 363)(118 364)(119 365)(120 366)(121 367)(122 368)(123 369)(124 370)(125 371)(126 372)(127 373)(128 374)(129 375)(130 376)(131 377)(132 378)(133 379)(134 380)(135 381)(136 382)(137 383)(138 384)(139 385)(140 386)(141 387)(142 388)(143 389)(144 390)(145 391)(146 392)(147 393)(148 394)(149 395)(150 396)(151 397)(152 398)(153 399)(154 400)(155 401)(156 402)(157 403)(158 404)(159 405)(160 406)(161 407)(162 408)(163 409)(164 410)(165 411)(166 412)(167 413)(168 414)(169 415)(170 416)(171 417)(172 418)(173 419)(174 420)(175 421)(176 422)(177 423)(178 424)(179 425)(180 426)(181 427)(182 428)(183 429)(184 430)(185 431)(186 432)(187 433)(188 434)(189 435)(190 436)(191 437)(192 438)(193 439)(194 440)(195 441)(196 442)(197 443)(198 444)(199 445)(200 446)(201 447)(202 448)(203 449)(204 450)(205 451)(206 452)(207 453)(208 454)(209 455)(210 456)(211 457)(212 458)(213 459)(214 460)(215 461)(216 462)(217 463)(218 464)(219 465)(220 466)(221 467)(222 468)(223 469)(224 470)(225 471)(226 472)(227 473)(228 474)(229 475)(230 476)(231 477)(232 478)(233 479)(234 480)(235 481)(236 482)(237 483)(238 484)(239 485)(240 486)(241 487)(242 488)(243 245)(244 246)
(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)| (1,247)(2,248)(3,249)(4,250)(5,251)(6,252)(7,253)(8,254)(9,255)(10,256)(11,257)(12,258)(13,259)(14,260)(15,261)(16,262)(17,263)(18,264)(19,265)(20,266)(21,267)(22,268)(23,269)(24,270)(25,271)(26,272)(27,273)(28,274)(29,275)(30,276)(31,277)(32,278)(33,279)(34,280)(35,281)(36,282)(37,283)(38,284)(39,285)(40,286)(41,287)(42,288)(43,289)(44,290)(45,291)(46,292)(47,293)(48,294)(49,295)(50,296)(51,297)(52,298)(53,299)(54,300)(55,301)(56,302)(57,303)(58,304)(59,305)(60,306)(61,307)(62,308)(63,309)(64,310)(65,311)(66,312)(67,313)(68,314)(69,315)(70,316)(71,317)(72,318)(73,319)(74,320)(75,321)(76,322)(77,323)(78,324)(79,325)(80,326)(81,327)(82,328)(83,329)(84,330)(85,331)(86,332)(87,333)(88,334)(89,335)(90,336)(91,337)(92,338)(93,339)(94,340)(95,341)(96,342)(97,343)(98,344)(99,345)(100,346)(101,347)(102,348)(103,349)(104,350)(105,351)(106,352)(107,353)(108,354)(109,355)(110,356)(111,357)(112,358)(113,359)(114,360)(115,361)(116,362)(117,363)(118,364)(119,365)(120,366)(121,367)(122,368)(123,369)(124,370)(125,371)(126,372)(127,373)(128,374)(129,375)(130,376)(131,377)(132,378)(133,379)(134,380)(135,381)(136,382)(137,383)(138,384)(139,385)(140,386)(141,387)(142,388)(143,389)(144,390)(145,391)(146,392)(147,393)(148,394)(149,395)(150,396)(151,397)(152,398)(153,399)(154,400)(155,401)(156,402)(157,403)(158,404)(159,405)(160,406)(161,407)(162,408)(163,409)(164,410)(165,411)(166,412)(167,413)(168,414)(169,415)(170,416)(171,417)(172,418)(173,419)(174,420)(175,421)(176,422)(177,423)(178,424)(179,425)(180,426)(181,427)(182,428)(183,429)(184,430)(185,431)(186,432)(187,433)(188,434)(189,435)(190,436)(191,437)(192,438)(193,439)(194,440)(195,441)(196,442)(197,443)(198,444)(199,445)(200,446)(201,447)(202,448)(203,449)(204,450)(205,451)(206,452)(207,453)(208,454)(209,455)(210,456)(211,457)(212,458)(213,459)(214,460)(215,461)(216,462)(217,463)(218,464)(219,465)(220,466)(221,467)(222,468)(223,469)(224,470)(225,471)(226,472)(227,473)(228,474)(229,475)(230,476)(231,477)(232,478)(233,479)(234,480)(235,481)(236,482)(237,483)(238,484)(239,485)(240,486)(241,487)(242,488)(243,245)(244,246), (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( (1,247)(2,248)(3,249)(4,250)(5,251)(6,252)(7,253)(8,254)(9,255)(10,256)(11,257)(12,258)(13,259)(14,260)(15,261)(16,262)(17,263)(18,264)(19,265)(20,266)(21,267)(22,268)(23,269)(24,270)(25,271)(26,272)(27,273)(28,274)(29,275)(30,276)(31,277)(32,278)(33,279)(34,280)(35,281)(36,282)(37,283)(38,284)(39,285)(40,286)(41,287)(42,288)(43,289)(44,290)(45,291)(46,292)(47,293)(48,294)(49,295)(50,296)(51,297)(52,298)(53,299)(54,300)(55,301)(56,302)(57,303)(58,304)(59,305)(60,306)(61,307)(62,308)(63,309)(64,310)(65,311)(66,312)(67,313)(68,314)(69,315)(70,316)(71,317)(72,318)(73,319)(74,320)(75,321)(76,322)(77,323)(78,324)(79,325)(80,326)(81,327)(82,328)(83,329)(84,330)(85,331)(86,332)(87,333)(88,334)(89,335)(90,336)(91,337)(92,338)(93,339)(94,340)(95,341)(96,342)(97,343)(98,344)(99,345)(100,346)(101,347)(102,348)(103,349)(104,350)(105,351)(106,352)(107,353)(108,354)(109,355)(110,356)(111,357)(112,358)(113,359)(114,360)(115,361)(116,362)(117,363)(118,364)(119,365)(120,366)(121,367)(122,368)(123,369)(124,370)(125,371)(126,372)(127,373)(128,374)(129,375)(130,376)(131,377)(132,378)(133,379)(134,380)(135,381)(136,382)(137,383)(138,384)(139,385)(140,386)(141,387)(142,388)(143,389)(144,390)(145,391)(146,392)(147,393)(148,394)(149,395)(150,396)(151,397)(152,398)(153,399)(154,400)(155,401)(156,402)(157,403)(158,404)(159,405)(160,406)(161,407)(162,408)(163,409)(164,410)(165,411)(166,412)(167,413)(168,414)(169,415)(170,416)(171,417)(172,418)(173,419)(174,420)(175,421)(176,422)(177,423)(178,424)(179,425)(180,426)(181,427)(182,428)(183,429)(184,430)(185,431)(186,432)(187,433)(188,434)(189,435)(190,436)(191,437)(192,438)(193,439)(194,440)(195,441)(196,442)(197,443)(198,444)(199,445)(200,446)(201,447)(202,448)(203,449)(204,450)(205,451)(206,452)(207,453)(208,454)(209,455)(210,456)(211,457)(212,458)(213,459)(214,460)(215,461)(216,462)(217,463)(218,464)(219,465)(220,466)(221,467)(222,468)(223,469)(224,470)(225,471)(226,472)(227,473)(228,474)(229,475)(230,476)(231,477)(232,478)(233,479)(234,480)(235,481)(236,482)(237,483)(238,484)(239,485)(240,486)(241,487)(242,488)(243,245)(244,246), (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([(1,247),(2,248),(3,249),(4,250),(5,251),(6,252),(7,253),(8,254),(9,255),(10,256),(11,257),(12,258),(13,259),(14,260),(15,261),(16,262),(17,263),(18,264),(19,265),(20,266),(21,267),(22,268),(23,269),(24,270),(25,271),(26,272),(27,273),(28,274),(29,275),(30,276),(31,277),(32,278),(33,279),(34,280),(35,281),(36,282),(37,283),(38,284),(39,285),(40,286),(41,287),(42,288),(43,289),(44,290),(45,291),(46,292),(47,293),(48,294),(49,295),(50,296),(51,297),(52,298),(53,299),(54,300),(55,301),(56,302),(57,303),(58,304),(59,305),(60,306),(61,307),(62,308),(63,309),(64,310),(65,311),(66,312),(67,313),(68,314),(69,315),(70,316),(71,317),(72,318),(73,319),(74,320),(75,321),(76,322),(77,323),(78,324),(79,325),(80,326),(81,327),(82,328),(83,329),(84,330),(85,331),(86,332),(87,333),(88,334),(89,335),(90,336),(91,337),(92,338),(93,339),(94,340),(95,341),(96,342),(97,343),(98,344),(99,345),(100,346),(101,347),(102,348),(103,349),(104,350),(105,351),(106,352),(107,353),(108,354),(109,355),(110,356),(111,357),(112,358),(113,359),(114,360),(115,361),(116,362),(117,363),(118,364),(119,365),(120,366),(121,367),(122,368),(123,369),(124,370),(125,371),(126,372),(127,373),(128,374),(129,375),(130,376),(131,377),(132,378),(133,379),(134,380),(135,381),(136,382),(137,383),(138,384),(139,385),(140,386),(141,387),(142,388),(143,389),(144,390),(145,391),(146,392),(147,393),(148,394),(149,395),(150,396),(151,397),(152,398),(153,399),(154,400),(155,401),(156,402),(157,403),(158,404),(159,405),(160,406),(161,407),(162,408),(163,409),(164,410),(165,411),(166,412),(167,413),(168,414),(169,415),(170,416),(171,417),(172,418),(173,419),(174,420),(175,421),(176,422),(177,423),(178,424),(179,425),(180,426),(181,427),(182,428),(183,429),(184,430),(185,431),(186,432),(187,433),(188,434),(189,435),(190,436),(191,437),(192,438),(193,439),(194,440),(195,441),(196,442),(197,443),(198,444),(199,445),(200,446),(201,447),(202,448),(203,449),(204,450),(205,451),(206,452),(207,453),(208,454),(209,455),(210,456),(211,457),(212,458),(213,459),(214,460),(215,461),(216,462),(217,463),(218,464),(219,465),(220,466),(221,467),(222,468),(223,469),(224,470),(225,471),(226,472),(227,473),(228,474),(229,475),(230,476),(231,477),(232,478),(233,479),(234,480),(235,481),(236,482),(237,483),(238,484),(239,485),(240,486),(241,487),(242,488),(243,245),(244,246)], [(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 2A2B2C4A4B4C4D61A···61BH122A···122FX244A···244IF
order1222444461···61122···122244···244
size111111111···11···11···1

488 irreducible representations

dim11111111
type+++
imageC1C2C2C4C61C122C122C244
kernelC2×C244C244C2×C122C122C2×C4C4C22C2
# reps12146012060240

Matrix representation of C2×C244 in GL2(𝔽733) generated by

7320
0732
,
4400
0432
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

Subgroup lattice of C2×C244 in TeX

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