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G = C2×C242order 484 = 22·112

Abelian group of type [2,242]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C242, SmallGroup(484,4)

Series: Derived Chief Lower central Upper central

C1 — C2×C242
C1C11C121C242 — C2×C242
C1 — C2×C242
C1 — C2×C242

Generators and relations for C2×C242
 G = < a,b | a2=b242=1, ab=ba >


Smallest permutation representation of C2×C242
Regular action on 484 points
Generators in S484
(1 255)(2 256)(3 257)(4 258)(5 259)(6 260)(7 261)(8 262)(9 263)(10 264)(11 265)(12 266)(13 267)(14 268)(15 269)(16 270)(17 271)(18 272)(19 273)(20 274)(21 275)(22 276)(23 277)(24 278)(25 279)(26 280)(27 281)(28 282)(29 283)(30 284)(31 285)(32 286)(33 287)(34 288)(35 289)(36 290)(37 291)(38 292)(39 293)(40 294)(41 295)(42 296)(43 297)(44 298)(45 299)(46 300)(47 301)(48 302)(49 303)(50 304)(51 305)(52 306)(53 307)(54 308)(55 309)(56 310)(57 311)(58 312)(59 313)(60 314)(61 315)(62 316)(63 317)(64 318)(65 319)(66 320)(67 321)(68 322)(69 323)(70 324)(71 325)(72 326)(73 327)(74 328)(75 329)(76 330)(77 331)(78 332)(79 333)(80 334)(81 335)(82 336)(83 337)(84 338)(85 339)(86 340)(87 341)(88 342)(89 343)(90 344)(91 345)(92 346)(93 347)(94 348)(95 349)(96 350)(97 351)(98 352)(99 353)(100 354)(101 355)(102 356)(103 357)(104 358)(105 359)(106 360)(107 361)(108 362)(109 363)(110 364)(111 365)(112 366)(113 367)(114 368)(115 369)(116 370)(117 371)(118 372)(119 373)(120 374)(121 375)(122 376)(123 377)(124 378)(125 379)(126 380)(127 381)(128 382)(129 383)(130 384)(131 385)(132 386)(133 387)(134 388)(135 389)(136 390)(137 391)(138 392)(139 393)(140 394)(141 395)(142 396)(143 397)(144 398)(145 399)(146 400)(147 401)(148 402)(149 403)(150 404)(151 405)(152 406)(153 407)(154 408)(155 409)(156 410)(157 411)(158 412)(159 413)(160 414)(161 415)(162 416)(163 417)(164 418)(165 419)(166 420)(167 421)(168 422)(169 423)(170 424)(171 425)(172 426)(173 427)(174 428)(175 429)(176 430)(177 431)(178 432)(179 433)(180 434)(181 435)(182 436)(183 437)(184 438)(185 439)(186 440)(187 441)(188 442)(189 443)(190 444)(191 445)(192 446)(193 447)(194 448)(195 449)(196 450)(197 451)(198 452)(199 453)(200 454)(201 455)(202 456)(203 457)(204 458)(205 459)(206 460)(207 461)(208 462)(209 463)(210 464)(211 465)(212 466)(213 467)(214 468)(215 469)(216 470)(217 471)(218 472)(219 473)(220 474)(221 475)(222 476)(223 477)(224 478)(225 479)(226 480)(227 481)(228 482)(229 483)(230 484)(231 243)(232 244)(233 245)(234 246)(235 247)(236 248)(237 249)(238 250)(239 251)(240 252)(241 253)(242 254)
(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)

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

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

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

484 conjugacy classes

class 1 2A2B2C11A···11J22A···22AD121A···121DF242A···242LR
order122211···1122···22121···121242···242
size11111···11···11···11···1

484 irreducible representations

dim111111
type++
imageC1C2C11C22C121C242
kernelC2×C242C242C2×C22C22C22C2
# reps131030110330

Matrix representation of C2×C242 in GL2(𝔽727) generated by

7260
01
,
6060
0108
G:=sub<GL(2,GF(727))| [726,0,0,1],[606,0,0,108] >;

C2×C242 in GAP, Magma, Sage, TeX

C_2\times C_{242}
% in TeX

G:=Group("C2xC242");
// GroupNames label

G:=SmallGroup(484,4);
// by ID

G=gap.SmallGroup(484,4);
# by ID

G:=PCGroup([4,-2,-2,-11,-11,142]);
// Polycyclic

G:=Group<a,b|a^2=b^242=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C242 in TeX

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