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G = C2×C240order 480 = 25·3·5

Abelian group of type [2,240]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C240, SmallGroup(480,212)

Series: Derived Chief Lower central Upper central

C1 — C2×C240
C1C2C4C8C40C120C240 — C2×C240
C1 — C2×C240
C1 — C2×C240

Generators and relations for C2×C240
 G = < a,b | a2=b240=1, ab=ba >


Smallest permutation representation of C2×C240
Regular action on 480 points
Generators in S480
(1 294)(2 295)(3 296)(4 297)(5 298)(6 299)(7 300)(8 301)(9 302)(10 303)(11 304)(12 305)(13 306)(14 307)(15 308)(16 309)(17 310)(18 311)(19 312)(20 313)(21 314)(22 315)(23 316)(24 317)(25 318)(26 319)(27 320)(28 321)(29 322)(30 323)(31 324)(32 325)(33 326)(34 327)(35 328)(36 329)(37 330)(38 331)(39 332)(40 333)(41 334)(42 335)(43 336)(44 337)(45 338)(46 339)(47 340)(48 341)(49 342)(50 343)(51 344)(52 345)(53 346)(54 347)(55 348)(56 349)(57 350)(58 351)(59 352)(60 353)(61 354)(62 355)(63 356)(64 357)(65 358)(66 359)(67 360)(68 361)(69 362)(70 363)(71 364)(72 365)(73 366)(74 367)(75 368)(76 369)(77 370)(78 371)(79 372)(80 373)(81 374)(82 375)(83 376)(84 377)(85 378)(86 379)(87 380)(88 381)(89 382)(90 383)(91 384)(92 385)(93 386)(94 387)(95 388)(96 389)(97 390)(98 391)(99 392)(100 393)(101 394)(102 395)(103 396)(104 397)(105 398)(106 399)(107 400)(108 401)(109 402)(110 403)(111 404)(112 405)(113 406)(114 407)(115 408)(116 409)(117 410)(118 411)(119 412)(120 413)(121 414)(122 415)(123 416)(124 417)(125 418)(126 419)(127 420)(128 421)(129 422)(130 423)(131 424)(132 425)(133 426)(134 427)(135 428)(136 429)(137 430)(138 431)(139 432)(140 433)(141 434)(142 435)(143 436)(144 437)(145 438)(146 439)(147 440)(148 441)(149 442)(150 443)(151 444)(152 445)(153 446)(154 447)(155 448)(156 449)(157 450)(158 451)(159 452)(160 453)(161 454)(162 455)(163 456)(164 457)(165 458)(166 459)(167 460)(168 461)(169 462)(170 463)(171 464)(172 465)(173 466)(174 467)(175 468)(176 469)(177 470)(178 471)(179 472)(180 473)(181 474)(182 475)(183 476)(184 477)(185 478)(186 479)(187 480)(188 241)(189 242)(190 243)(191 244)(192 245)(193 246)(194 247)(195 248)(196 249)(197 250)(198 251)(199 252)(200 253)(201 254)(202 255)(203 256)(204 257)(205 258)(206 259)(207 260)(208 261)(209 262)(210 263)(211 264)(212 265)(213 266)(214 267)(215 268)(216 269)(217 270)(218 271)(219 272)(220 273)(221 274)(222 275)(223 276)(224 277)(225 278)(226 279)(227 280)(228 281)(229 282)(230 283)(231 284)(232 285)(233 286)(234 287)(235 288)(236 289)(237 290)(238 291)(239 292)(240 293)
(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)

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

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

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

480 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D5A5B5C5D6A···6F8A···8H10A···10L12A···12H15A···15H16A···16P20A···20P24A···24P30A···30X40A···40AF48A···48AF60A···60AF80A···80BL120A···120BL240A···240DX
order122233444455556···68···810···1012···1215···1516···1620···2024···2430···3040···4048···4860···6080···80120···120240···240
size111111111111111···11···11···11···11···11···11···11···11···11···11···11···11···11···11···1

480 irreducible representations

dim11111111111111111111111111111111
type+++
imageC1C2C2C3C4C4C5C6C6C8C8C10C10C12C12C15C16C20C20C24C24C30C30C40C40C48C60C60C80C120C120C240
kernelC2×C240C240C2×C120C2×C80C120C2×C60C2×C48C80C2×C40C60C2×C30C48C2×C24C40C2×C20C2×C16C30C24C2×C12C20C2×C10C16C2×C8C12C2×C6C10C8C2×C4C6C4C22C2
# reps12122244244844481688881681616321616643232128

Matrix representation of C2×C240 in GL2(𝔽241) generated by

2400
01
,
1060
0172
G:=sub<GL(2,GF(241))| [240,0,0,1],[106,0,0,172] >;

C2×C240 in GAP, Magma, Sage, TeX

C_2\times C_{240}
% in TeX

G:=Group("C2xC240");
// GroupNames label

G:=SmallGroup(480,212);
// by ID

G=gap.SmallGroup(480,212);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,420,102,124]);
// Polycyclic

G:=Group<a,b|a^2=b^240=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C240 in TeX

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