direct product, abelian, monomial, 2-elementary
Aliases: C2×C238, SmallGroup(476,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C238 |
| C1 — C2×C238 |
| C1 — C2×C238 |
Generators and relations for C2×C238
G = < a,b | a2=b238=1, ab=ba >
Smallest permutation representation of C2×C238
►Regular action on 476 points
Generators in S476
(1 357)(2 358)(3 359)(4 360)(5 361)(6 362)(7 363)(8 364)(9 365)(10 366)(11 367)(12 368)(13 369)(14 370)(15 371)(16 372)(17 373)(18 374)(19 375)(20 376)(21 377)(22 378)(23 379)(24 380)(25 381)(26 382)(27 383)(28 384)(29 385)(30 386)(31 387)(32 388)(33 389)(34 390)(35 391)(36 392)(37 393)(38 394)(39 395)(40 396)(41 397)(42 398)(43 399)(44 400)(45 401)(46 402)(47 403)(48 404)(49 405)(50 406)(51 407)(52 408)(53 409)(54 410)(55 411)(56 412)(57 413)(58 414)(59 415)(60 416)(61 417)(62 418)(63 419)(64 420)(65 421)(66 422)(67 423)(68 424)(69 425)(70 426)(71 427)(72 428)(73 429)(74 430)(75 431)(76 432)(77 433)(78 434)(79 435)(80 436)(81 437)(82 438)(83 439)(84 440)(85 441)(86 442)(87 443)(88 444)(89 445)(90 446)(91 447)(92 448)(93 449)(94 450)(95 451)(96 452)(97 453)(98 454)(99 455)(100 456)(101 457)(102 458)(103 459)(104 460)(105 461)(106 462)(107 463)(108 464)(109 465)(110 466)(111 467)(112 468)(113 469)(114 470)(115 471)(116 472)(117 473)(118 474)(119 475)(120 476)(121 239)(122 240)(123 241)(124 242)(125 243)(126 244)(127 245)(128 246)(129 247)(130 248)(131 249)(132 250)(133 251)(134 252)(135 253)(136 254)(137 255)(138 256)(139 257)(140 258)(141 259)(142 260)(143 261)(144 262)(145 263)(146 264)(147 265)(148 266)(149 267)(150 268)(151 269)(152 270)(153 271)(154 272)(155 273)(156 274)(157 275)(158 276)(159 277)(160 278)(161 279)(162 280)(163 281)(164 282)(165 283)(166 284)(167 285)(168 286)(169 287)(170 288)(171 289)(172 290)(173 291)(174 292)(175 293)(176 294)(177 295)(178 296)(179 297)(180 298)(181 299)(182 300)(183 301)(184 302)(185 303)(186 304)(187 305)(188 306)(189 307)(190 308)(191 309)(192 310)(193 311)(194 312)(195 313)(196 314)(197 315)(198 316)(199 317)(200 318)(201 319)(202 320)(203 321)(204 322)(205 323)(206 324)(207 325)(208 326)(209 327)(210 328)(211 329)(212 330)(213 331)(214 332)(215 333)(216 334)(217 335)(218 336)(219 337)(220 338)(221 339)(222 340)(223 341)(224 342)(225 343)(226 344)(227 345)(228 346)(229 347)(230 348)(231 349)(232 350)(233 351)(234 352)(235 353)(236 354)(237 355)(238 356)
(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)
G:=sub<Sym(476)| (1,357)(2,358)(3,359)(4,360)(5,361)(6,362)(7,363)(8,364)(9,365)(10,366)(11,367)(12,368)(13,369)(14,370)(15,371)(16,372)(17,373)(18,374)(19,375)(20,376)(21,377)(22,378)(23,379)(24,380)(25,381)(26,382)(27,383)(28,384)(29,385)(30,386)(31,387)(32,388)(33,389)(34,390)(35,391)(36,392)(37,393)(38,394)(39,395)(40,396)(41,397)(42,398)(43,399)(44,400)(45,401)(46,402)(47,403)(48,404)(49,405)(50,406)(51,407)(52,408)(53,409)(54,410)(55,411)(56,412)(57,413)(58,414)(59,415)(60,416)(61,417)(62,418)(63,419)(64,420)(65,421)(66,422)(67,423)(68,424)(69,425)(70,426)(71,427)(72,428)(73,429)(74,430)(75,431)(76,432)(77,433)(78,434)(79,435)(80,436)(81,437)(82,438)(83,439)(84,440)(85,441)(86,442)(87,443)(88,444)(89,445)(90,446)(91,447)(92,448)(93,449)(94,450)(95,451)(96,452)(97,453)(98,454)(99,455)(100,456)(101,457)(102,458)(103,459)(104,460)(105,461)(106,462)(107,463)(108,464)(109,465)(110,466)(111,467)(112,468)(113,469)(114,470)(115,471)(116,472)(117,473)(118,474)(119,475)(120,476)(121,239)(122,240)(123,241)(124,242)(125,243)(126,244)(127,245)(128,246)(129,247)(130,248)(131,249)(132,250)(133,251)(134,252)(135,253)(136,254)(137,255)(138,256)(139,257)(140,258)(141,259)(142,260)(143,261)(144,262)(145,263)(146,264)(147,265)(148,266)(149,267)(150,268)(151,269)(152,270)(153,271)(154,272)(155,273)(156,274)(157,275)(158,276)(159,277)(160,278)(161,279)(162,280)(163,281)(164,282)(165,283)(166,284)(167,285)(168,286)(169,287)(170,288)(171,289)(172,290)(173,291)(174,292)(175,293)(176,294)(177,295)(178,296)(179,297)(180,298)(181,299)(182,300)(183,301)(184,302)(185,303)(186,304)(187,305)(188,306)(189,307)(190,308)(191,309)(192,310)(193,311)(194,312)(195,313)(196,314)(197,315)(198,316)(199,317)(200,318)(201,319)(202,320)(203,321)(204,322)(205,323)(206,324)(207,325)(208,326)(209,327)(210,328)(211,329)(212,330)(213,331)(214,332)(215,333)(216,334)(217,335)(218,336)(219,337)(220,338)(221,339)(222,340)(223,341)(224,342)(225,343)(226,344)(227,345)(228,346)(229,347)(230,348)(231,349)(232,350)(233,351)(234,352)(235,353)(236,354)(237,355)(238,356), (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)>;
G:=Group( (1,357)(2,358)(3,359)(4,360)(5,361)(6,362)(7,363)(8,364)(9,365)(10,366)(11,367)(12,368)(13,369)(14,370)(15,371)(16,372)(17,373)(18,374)(19,375)(20,376)(21,377)(22,378)(23,379)(24,380)(25,381)(26,382)(27,383)(28,384)(29,385)(30,386)(31,387)(32,388)(33,389)(34,390)(35,391)(36,392)(37,393)(38,394)(39,395)(40,396)(41,397)(42,398)(43,399)(44,400)(45,401)(46,402)(47,403)(48,404)(49,405)(50,406)(51,407)(52,408)(53,409)(54,410)(55,411)(56,412)(57,413)(58,414)(59,415)(60,416)(61,417)(62,418)(63,419)(64,420)(65,421)(66,422)(67,423)(68,424)(69,425)(70,426)(71,427)(72,428)(73,429)(74,430)(75,431)(76,432)(77,433)(78,434)(79,435)(80,436)(81,437)(82,438)(83,439)(84,440)(85,441)(86,442)(87,443)(88,444)(89,445)(90,446)(91,447)(92,448)(93,449)(94,450)(95,451)(96,452)(97,453)(98,454)(99,455)(100,456)(101,457)(102,458)(103,459)(104,460)(105,461)(106,462)(107,463)(108,464)(109,465)(110,466)(111,467)(112,468)(113,469)(114,470)(115,471)(116,472)(117,473)(118,474)(119,475)(120,476)(121,239)(122,240)(123,241)(124,242)(125,243)(126,244)(127,245)(128,246)(129,247)(130,248)(131,249)(132,250)(133,251)(134,252)(135,253)(136,254)(137,255)(138,256)(139,257)(140,258)(141,259)(142,260)(143,261)(144,262)(145,263)(146,264)(147,265)(148,266)(149,267)(150,268)(151,269)(152,270)(153,271)(154,272)(155,273)(156,274)(157,275)(158,276)(159,277)(160,278)(161,279)(162,280)(163,281)(164,282)(165,283)(166,284)(167,285)(168,286)(169,287)(170,288)(171,289)(172,290)(173,291)(174,292)(175,293)(176,294)(177,295)(178,296)(179,297)(180,298)(181,299)(182,300)(183,301)(184,302)(185,303)(186,304)(187,305)(188,306)(189,307)(190,308)(191,309)(192,310)(193,311)(194,312)(195,313)(196,314)(197,315)(198,316)(199,317)(200,318)(201,319)(202,320)(203,321)(204,322)(205,323)(206,324)(207,325)(208,326)(209,327)(210,328)(211,329)(212,330)(213,331)(214,332)(215,333)(216,334)(217,335)(218,336)(219,337)(220,338)(221,339)(222,340)(223,341)(224,342)(225,343)(226,344)(227,345)(228,346)(229,347)(230,348)(231,349)(232,350)(233,351)(234,352)(235,353)(236,354)(237,355)(238,356), (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) );
G=PermutationGroup([[(1,357),(2,358),(3,359),(4,360),(5,361),(6,362),(7,363),(8,364),(9,365),(10,366),(11,367),(12,368),(13,369),(14,370),(15,371),(16,372),(17,373),(18,374),(19,375),(20,376),(21,377),(22,378),(23,379),(24,380),(25,381),(26,382),(27,383),(28,384),(29,385),(30,386),(31,387),(32,388),(33,389),(34,390),(35,391),(36,392),(37,393),(38,394),(39,395),(40,396),(41,397),(42,398),(43,399),(44,400),(45,401),(46,402),(47,403),(48,404),(49,405),(50,406),(51,407),(52,408),(53,409),(54,410),(55,411),(56,412),(57,413),(58,414),(59,415),(60,416),(61,417),(62,418),(63,419),(64,420),(65,421),(66,422),(67,423),(68,424),(69,425),(70,426),(71,427),(72,428),(73,429),(74,430),(75,431),(76,432),(77,433),(78,434),(79,435),(80,436),(81,437),(82,438),(83,439),(84,440),(85,441),(86,442),(87,443),(88,444),(89,445),(90,446),(91,447),(92,448),(93,449),(94,450),(95,451),(96,452),(97,453),(98,454),(99,455),(100,456),(101,457),(102,458),(103,459),(104,460),(105,461),(106,462),(107,463),(108,464),(109,465),(110,466),(111,467),(112,468),(113,469),(114,470),(115,471),(116,472),(117,473),(118,474),(119,475),(120,476),(121,239),(122,240),(123,241),(124,242),(125,243),(126,244),(127,245),(128,246),(129,247),(130,248),(131,249),(132,250),(133,251),(134,252),(135,253),(136,254),(137,255),(138,256),(139,257),(140,258),(141,259),(142,260),(143,261),(144,262),(145,263),(146,264),(147,265),(148,266),(149,267),(150,268),(151,269),(152,270),(153,271),(154,272),(155,273),(156,274),(157,275),(158,276),(159,277),(160,278),(161,279),(162,280),(163,281),(164,282),(165,283),(166,284),(167,285),(168,286),(169,287),(170,288),(171,289),(172,290),(173,291),(174,292),(175,293),(176,294),(177,295),(178,296),(179,297),(180,298),(181,299),(182,300),(183,301),(184,302),(185,303),(186,304),(187,305),(188,306),(189,307),(190,308),(191,309),(192,310),(193,311),(194,312),(195,313),(196,314),(197,315),(198,316),(199,317),(200,318),(201,319),(202,320),(203,321),(204,322),(205,323),(206,324),(207,325),(208,326),(209,327),(210,328),(211,329),(212,330),(213,331),(214,332),(215,333),(216,334),(217,335),(218,336),(219,337),(220,338),(221,339),(222,340),(223,341),(224,342),(225,343),(226,344),(227,345),(228,346),(229,347),(230,348),(231,349),(232,350),(233,351),(234,352),(235,353),(236,354),(237,355),(238,356)], [(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)]])
476 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | ··· | 7F | 14A | ··· | 14R | 17A | ··· | 17P | 34A | ··· | 34AV | 119A | ··· | 119CR | 238A | ··· | 238KB |
| order | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 17 | ··· | 17 | 34 | ··· | 34 | 119 | ··· | 119 | 238 | ··· | 238 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
476 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C7 | C14 | C17 | C34 | C119 | C238 |
| kernel | C2×C238 | C238 | C2×C34 | C34 | C2×C14 | C14 | C22 | C2 |
| # reps | 1 | 3 | 6 | 18 | 16 | 48 | 96 | 288 |
Matrix representation of C2×C238 ►in GL2(𝔽239) generated by
| 238 | 0 |
| 0 | 238 |
| 211 | 0 |
| 0 | 179 |
G:=sub<GL(2,GF(239))| [238,0,0,238],[211,0,0,179] >;
C2×C238 in GAP, Magma, Sage, TeX
C_2\times C_{238} % in TeX
G:=Group("C2xC238"); // GroupNames label
G:=SmallGroup(476,11);
// by ID
G=gap.SmallGroup(476,11);
# by ID
G:=PCGroup([4,-2,-2,-7,-17]);
// Polycyclic
G:=Group<a,b|a^2=b^238=1,a*b=b*a>;
// generators/relations
Export