direct product, abelian, monomial, 2-elementary
Aliases: C2×C236, SmallGroup(472,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C236 |
| C1 — C2×C236 |
| C1 — C2×C236 |
Generators and relations for C2×C236
G = < a,b | a2=b236=1, ab=ba >
Smallest permutation representation of C2×C236
►Regular action on 472 points
Generators in S472
(1 321)(2 322)(3 323)(4 324)(5 325)(6 326)(7 327)(8 328)(9 329)(10 330)(11 331)(12 332)(13 333)(14 334)(15 335)(16 336)(17 337)(18 338)(19 339)(20 340)(21 341)(22 342)(23 343)(24 344)(25 345)(26 346)(27 347)(28 348)(29 349)(30 350)(31 351)(32 352)(33 353)(34 354)(35 355)(36 356)(37 357)(38 358)(39 359)(40 360)(41 361)(42 362)(43 363)(44 364)(45 365)(46 366)(47 367)(48 368)(49 369)(50 370)(51 371)(52 372)(53 373)(54 374)(55 375)(56 376)(57 377)(58 378)(59 379)(60 380)(61 381)(62 382)(63 383)(64 384)(65 385)(66 386)(67 387)(68 388)(69 389)(70 390)(71 391)(72 392)(73 393)(74 394)(75 395)(76 396)(77 397)(78 398)(79 399)(80 400)(81 401)(82 402)(83 403)(84 404)(85 405)(86 406)(87 407)(88 408)(89 409)(90 410)(91 411)(92 412)(93 413)(94 414)(95 415)(96 416)(97 417)(98 418)(99 419)(100 420)(101 421)(102 422)(103 423)(104 424)(105 425)(106 426)(107 427)(108 428)(109 429)(110 430)(111 431)(112 432)(113 433)(114 434)(115 435)(116 436)(117 437)(118 438)(119 439)(120 440)(121 441)(122 442)(123 443)(124 444)(125 445)(126 446)(127 447)(128 448)(129 449)(130 450)(131 451)(132 452)(133 453)(134 454)(135 455)(136 456)(137 457)(138 458)(139 459)(140 460)(141 461)(142 462)(143 463)(144 464)(145 465)(146 466)(147 467)(148 468)(149 469)(150 470)(151 471)(152 472)(153 237)(154 238)(155 239)(156 240)(157 241)(158 242)(159 243)(160 244)(161 245)(162 246)(163 247)(164 248)(165 249)(166 250)(167 251)(168 252)(169 253)(170 254)(171 255)(172 256)(173 257)(174 258)(175 259)(176 260)(177 261)(178 262)(179 263)(180 264)(181 265)(182 266)(183 267)(184 268)(185 269)(186 270)(187 271)(188 272)(189 273)(190 274)(191 275)(192 276)(193 277)(194 278)(195 279)(196 280)(197 281)(198 282)(199 283)(200 284)(201 285)(202 286)(203 287)(204 288)(205 289)(206 290)(207 291)(208 292)(209 293)(210 294)(211 295)(212 296)(213 297)(214 298)(215 299)(216 300)(217 301)(218 302)(219 303)(220 304)(221 305)(222 306)(223 307)(224 308)(225 309)(226 310)(227 311)(228 312)(229 313)(230 314)(231 315)(232 316)(233 317)(234 318)(235 319)(236 320)
(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)
G:=sub<Sym(472)| (1,321)(2,322)(3,323)(4,324)(5,325)(6,326)(7,327)(8,328)(9,329)(10,330)(11,331)(12,332)(13,333)(14,334)(15,335)(16,336)(17,337)(18,338)(19,339)(20,340)(21,341)(22,342)(23,343)(24,344)(25,345)(26,346)(27,347)(28,348)(29,349)(30,350)(31,351)(32,352)(33,353)(34,354)(35,355)(36,356)(37,357)(38,358)(39,359)(40,360)(41,361)(42,362)(43,363)(44,364)(45,365)(46,366)(47,367)(48,368)(49,369)(50,370)(51,371)(52,372)(53,373)(54,374)(55,375)(56,376)(57,377)(58,378)(59,379)(60,380)(61,381)(62,382)(63,383)(64,384)(65,385)(66,386)(67,387)(68,388)(69,389)(70,390)(71,391)(72,392)(73,393)(74,394)(75,395)(76,396)(77,397)(78,398)(79,399)(80,400)(81,401)(82,402)(83,403)(84,404)(85,405)(86,406)(87,407)(88,408)(89,409)(90,410)(91,411)(92,412)(93,413)(94,414)(95,415)(96,416)(97,417)(98,418)(99,419)(100,420)(101,421)(102,422)(103,423)(104,424)(105,425)(106,426)(107,427)(108,428)(109,429)(110,430)(111,431)(112,432)(113,433)(114,434)(115,435)(116,436)(117,437)(118,438)(119,439)(120,440)(121,441)(122,442)(123,443)(124,444)(125,445)(126,446)(127,447)(128,448)(129,449)(130,450)(131,451)(132,452)(133,453)(134,454)(135,455)(136,456)(137,457)(138,458)(139,459)(140,460)(141,461)(142,462)(143,463)(144,464)(145,465)(146,466)(147,467)(148,468)(149,469)(150,470)(151,471)(152,472)(153,237)(154,238)(155,239)(156,240)(157,241)(158,242)(159,243)(160,244)(161,245)(162,246)(163,247)(164,248)(165,249)(166,250)(167,251)(168,252)(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)(209,293)(210,294)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)(217,301)(218,302)(219,303)(220,304)(221,305)(222,306)(223,307)(224,308)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)(233,317)(234,318)(235,319)(236,320), (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)>;
G:=Group( (1,321)(2,322)(3,323)(4,324)(5,325)(6,326)(7,327)(8,328)(9,329)(10,330)(11,331)(12,332)(13,333)(14,334)(15,335)(16,336)(17,337)(18,338)(19,339)(20,340)(21,341)(22,342)(23,343)(24,344)(25,345)(26,346)(27,347)(28,348)(29,349)(30,350)(31,351)(32,352)(33,353)(34,354)(35,355)(36,356)(37,357)(38,358)(39,359)(40,360)(41,361)(42,362)(43,363)(44,364)(45,365)(46,366)(47,367)(48,368)(49,369)(50,370)(51,371)(52,372)(53,373)(54,374)(55,375)(56,376)(57,377)(58,378)(59,379)(60,380)(61,381)(62,382)(63,383)(64,384)(65,385)(66,386)(67,387)(68,388)(69,389)(70,390)(71,391)(72,392)(73,393)(74,394)(75,395)(76,396)(77,397)(78,398)(79,399)(80,400)(81,401)(82,402)(83,403)(84,404)(85,405)(86,406)(87,407)(88,408)(89,409)(90,410)(91,411)(92,412)(93,413)(94,414)(95,415)(96,416)(97,417)(98,418)(99,419)(100,420)(101,421)(102,422)(103,423)(104,424)(105,425)(106,426)(107,427)(108,428)(109,429)(110,430)(111,431)(112,432)(113,433)(114,434)(115,435)(116,436)(117,437)(118,438)(119,439)(120,440)(121,441)(122,442)(123,443)(124,444)(125,445)(126,446)(127,447)(128,448)(129,449)(130,450)(131,451)(132,452)(133,453)(134,454)(135,455)(136,456)(137,457)(138,458)(139,459)(140,460)(141,461)(142,462)(143,463)(144,464)(145,465)(146,466)(147,467)(148,468)(149,469)(150,470)(151,471)(152,472)(153,237)(154,238)(155,239)(156,240)(157,241)(158,242)(159,243)(160,244)(161,245)(162,246)(163,247)(164,248)(165,249)(166,250)(167,251)(168,252)(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)(209,293)(210,294)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)(217,301)(218,302)(219,303)(220,304)(221,305)(222,306)(223,307)(224,308)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)(233,317)(234,318)(235,319)(236,320), (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) );
G=PermutationGroup([[(1,321),(2,322),(3,323),(4,324),(5,325),(6,326),(7,327),(8,328),(9,329),(10,330),(11,331),(12,332),(13,333),(14,334),(15,335),(16,336),(17,337),(18,338),(19,339),(20,340),(21,341),(22,342),(23,343),(24,344),(25,345),(26,346),(27,347),(28,348),(29,349),(30,350),(31,351),(32,352),(33,353),(34,354),(35,355),(36,356),(37,357),(38,358),(39,359),(40,360),(41,361),(42,362),(43,363),(44,364),(45,365),(46,366),(47,367),(48,368),(49,369),(50,370),(51,371),(52,372),(53,373),(54,374),(55,375),(56,376),(57,377),(58,378),(59,379),(60,380),(61,381),(62,382),(63,383),(64,384),(65,385),(66,386),(67,387),(68,388),(69,389),(70,390),(71,391),(72,392),(73,393),(74,394),(75,395),(76,396),(77,397),(78,398),(79,399),(80,400),(81,401),(82,402),(83,403),(84,404),(85,405),(86,406),(87,407),(88,408),(89,409),(90,410),(91,411),(92,412),(93,413),(94,414),(95,415),(96,416),(97,417),(98,418),(99,419),(100,420),(101,421),(102,422),(103,423),(104,424),(105,425),(106,426),(107,427),(108,428),(109,429),(110,430),(111,431),(112,432),(113,433),(114,434),(115,435),(116,436),(117,437),(118,438),(119,439),(120,440),(121,441),(122,442),(123,443),(124,444),(125,445),(126,446),(127,447),(128,448),(129,449),(130,450),(131,451),(132,452),(133,453),(134,454),(135,455),(136,456),(137,457),(138,458),(139,459),(140,460),(141,461),(142,462),(143,463),(144,464),(145,465),(146,466),(147,467),(148,468),(149,469),(150,470),(151,471),(152,472),(153,237),(154,238),(155,239),(156,240),(157,241),(158,242),(159,243),(160,244),(161,245),(162,246),(163,247),(164,248),(165,249),(166,250),(167,251),(168,252),(169,253),(170,254),(171,255),(172,256),(173,257),(174,258),(175,259),(176,260),(177,261),(178,262),(179,263),(180,264),(181,265),(182,266),(183,267),(184,268),(185,269),(186,270),(187,271),(188,272),(189,273),(190,274),(191,275),(192,276),(193,277),(194,278),(195,279),(196,280),(197,281),(198,282),(199,283),(200,284),(201,285),(202,286),(203,287),(204,288),(205,289),(206,290),(207,291),(208,292),(209,293),(210,294),(211,295),(212,296),(213,297),(214,298),(215,299),(216,300),(217,301),(218,302),(219,303),(220,304),(221,305),(222,306),(223,307),(224,308),(225,309),(226,310),(227,311),(228,312),(229,313),(230,314),(231,315),(232,316),(233,317),(234,318),(235,319),(236,320)], [(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)]])
472 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 59A | ··· | 59BF | 118A | ··· | 118FR | 236A | ··· | 236HX |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 59 | ··· | 59 | 118 | ··· | 118 | 236 | ··· | 236 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
472 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C59 | C118 | C118 | C236 |
| kernel | C2×C236 | C236 | C2×C118 | C118 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 58 | 116 | 58 | 232 |
Matrix representation of C2×C236 ►in GL2(𝔽709) generated by
| 708 | 0 |
| 0 | 1 |
| 502 | 0 |
| 0 | 237 |
G:=sub<GL(2,GF(709))| [708,0,0,1],[502,0,0,237] >;
C2×C236 in GAP, Magma, Sage, TeX
C_2\times C_{236} % in TeX
G:=Group("C2xC236"); // GroupNames label
G:=SmallGroup(472,8);
// by ID
G=gap.SmallGroup(472,8);
# by ID
G:=PCGroup([4,-2,-2,-59,-2,944]);
// Polycyclic
G:=Group<a,b|a^2=b^236=1,a*b=b*a>;
// generators/relations
Export