direct product, abelian, monomial, 2-elementary
Aliases: C2×C234, SmallGroup(468,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C234 |
| C1 — C2×C234 |
| C1 — C2×C234 |
Generators and relations for C2×C234
G = < a,b | a2=b234=1, ab=ba >
Smallest permutation representation of C2×C234
►Regular action on 468 points
Generators in S468
(1 439)(2 440)(3 441)(4 442)(5 443)(6 444)(7 445)(8 446)(9 447)(10 448)(11 449)(12 450)(13 451)(14 452)(15 453)(16 454)(17 455)(18 456)(19 457)(20 458)(21 459)(22 460)(23 461)(24 462)(25 463)(26 464)(27 465)(28 466)(29 467)(30 468)(31 235)(32 236)(33 237)(34 238)(35 239)(36 240)(37 241)(38 242)(39 243)(40 244)(41 245)(42 246)(43 247)(44 248)(45 249)(46 250)(47 251)(48 252)(49 253)(50 254)(51 255)(52 256)(53 257)(54 258)(55 259)(56 260)(57 261)(58 262)(59 263)(60 264)(61 265)(62 266)(63 267)(64 268)(65 269)(66 270)(67 271)(68 272)(69 273)(70 274)(71 275)(72 276)(73 277)(74 278)(75 279)(76 280)(77 281)(78 282)(79 283)(80 284)(81 285)(82 286)(83 287)(84 288)(85 289)(86 290)(87 291)(88 292)(89 293)(90 294)(91 295)(92 296)(93 297)(94 298)(95 299)(96 300)(97 301)(98 302)(99 303)(100 304)(101 305)(102 306)(103 307)(104 308)(105 309)(106 310)(107 311)(108 312)(109 313)(110 314)(111 315)(112 316)(113 317)(114 318)(115 319)(116 320)(117 321)(118 322)(119 323)(120 324)(121 325)(122 326)(123 327)(124 328)(125 329)(126 330)(127 331)(128 332)(129 333)(130 334)(131 335)(132 336)(133 337)(134 338)(135 339)(136 340)(137 341)(138 342)(139 343)(140 344)(141 345)(142 346)(143 347)(144 348)(145 349)(146 350)(147 351)(148 352)(149 353)(150 354)(151 355)(152 356)(153 357)(154 358)(155 359)(156 360)(157 361)(158 362)(159 363)(160 364)(161 365)(162 366)(163 367)(164 368)(165 369)(166 370)(167 371)(168 372)(169 373)(170 374)(171 375)(172 376)(173 377)(174 378)(175 379)(176 380)(177 381)(178 382)(179 383)(180 384)(181 385)(182 386)(183 387)(184 388)(185 389)(186 390)(187 391)(188 392)(189 393)(190 394)(191 395)(192 396)(193 397)(194 398)(195 399)(196 400)(197 401)(198 402)(199 403)(200 404)(201 405)(202 406)(203 407)(204 408)(205 409)(206 410)(207 411)(208 412)(209 413)(210 414)(211 415)(212 416)(213 417)(214 418)(215 419)(216 420)(217 421)(218 422)(219 423)(220 424)(221 425)(222 426)(223 427)(224 428)(225 429)(226 430)(227 431)(228 432)(229 433)(230 434)(231 435)(232 436)(233 437)(234 438)
(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)
G:=sub<Sym(468)| (1,439)(2,440)(3,441)(4,442)(5,443)(6,444)(7,445)(8,446)(9,447)(10,448)(11,449)(12,450)(13,451)(14,452)(15,453)(16,454)(17,455)(18,456)(19,457)(20,458)(21,459)(22,460)(23,461)(24,462)(25,463)(26,464)(27,465)(28,466)(29,467)(30,468)(31,235)(32,236)(33,237)(34,238)(35,239)(36,240)(37,241)(38,242)(39,243)(40,244)(41,245)(42,246)(43,247)(44,248)(45,249)(46,250)(47,251)(48,252)(49,253)(50,254)(51,255)(52,256)(53,257)(54,258)(55,259)(56,260)(57,261)(58,262)(59,263)(60,264)(61,265)(62,266)(63,267)(64,268)(65,269)(66,270)(67,271)(68,272)(69,273)(70,274)(71,275)(72,276)(73,277)(74,278)(75,279)(76,280)(77,281)(78,282)(79,283)(80,284)(81,285)(82,286)(83,287)(84,288)(85,289)(86,290)(87,291)(88,292)(89,293)(90,294)(91,295)(92,296)(93,297)(94,298)(95,299)(96,300)(97,301)(98,302)(99,303)(100,304)(101,305)(102,306)(103,307)(104,308)(105,309)(106,310)(107,311)(108,312)(109,313)(110,314)(111,315)(112,316)(113,317)(114,318)(115,319)(116,320)(117,321)(118,322)(119,323)(120,324)(121,325)(122,326)(123,327)(124,328)(125,329)(126,330)(127,331)(128,332)(129,333)(130,334)(131,335)(132,336)(133,337)(134,338)(135,339)(136,340)(137,341)(138,342)(139,343)(140,344)(141,345)(142,346)(143,347)(144,348)(145,349)(146,350)(147,351)(148,352)(149,353)(150,354)(151,355)(152,356)(153,357)(154,358)(155,359)(156,360)(157,361)(158,362)(159,363)(160,364)(161,365)(162,366)(163,367)(164,368)(165,369)(166,370)(167,371)(168,372)(169,373)(170,374)(171,375)(172,376)(173,377)(174,378)(175,379)(176,380)(177,381)(178,382)(179,383)(180,384)(181,385)(182,386)(183,387)(184,388)(185,389)(186,390)(187,391)(188,392)(189,393)(190,394)(191,395)(192,396)(193,397)(194,398)(195,399)(196,400)(197,401)(198,402)(199,403)(200,404)(201,405)(202,406)(203,407)(204,408)(205,409)(206,410)(207,411)(208,412)(209,413)(210,414)(211,415)(212,416)(213,417)(214,418)(215,419)(216,420)(217,421)(218,422)(219,423)(220,424)(221,425)(222,426)(223,427)(224,428)(225,429)(226,430)(227,431)(228,432)(229,433)(230,434)(231,435)(232,436)(233,437)(234,438), (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)>;
G:=Group( (1,439)(2,440)(3,441)(4,442)(5,443)(6,444)(7,445)(8,446)(9,447)(10,448)(11,449)(12,450)(13,451)(14,452)(15,453)(16,454)(17,455)(18,456)(19,457)(20,458)(21,459)(22,460)(23,461)(24,462)(25,463)(26,464)(27,465)(28,466)(29,467)(30,468)(31,235)(32,236)(33,237)(34,238)(35,239)(36,240)(37,241)(38,242)(39,243)(40,244)(41,245)(42,246)(43,247)(44,248)(45,249)(46,250)(47,251)(48,252)(49,253)(50,254)(51,255)(52,256)(53,257)(54,258)(55,259)(56,260)(57,261)(58,262)(59,263)(60,264)(61,265)(62,266)(63,267)(64,268)(65,269)(66,270)(67,271)(68,272)(69,273)(70,274)(71,275)(72,276)(73,277)(74,278)(75,279)(76,280)(77,281)(78,282)(79,283)(80,284)(81,285)(82,286)(83,287)(84,288)(85,289)(86,290)(87,291)(88,292)(89,293)(90,294)(91,295)(92,296)(93,297)(94,298)(95,299)(96,300)(97,301)(98,302)(99,303)(100,304)(101,305)(102,306)(103,307)(104,308)(105,309)(106,310)(107,311)(108,312)(109,313)(110,314)(111,315)(112,316)(113,317)(114,318)(115,319)(116,320)(117,321)(118,322)(119,323)(120,324)(121,325)(122,326)(123,327)(124,328)(125,329)(126,330)(127,331)(128,332)(129,333)(130,334)(131,335)(132,336)(133,337)(134,338)(135,339)(136,340)(137,341)(138,342)(139,343)(140,344)(141,345)(142,346)(143,347)(144,348)(145,349)(146,350)(147,351)(148,352)(149,353)(150,354)(151,355)(152,356)(153,357)(154,358)(155,359)(156,360)(157,361)(158,362)(159,363)(160,364)(161,365)(162,366)(163,367)(164,368)(165,369)(166,370)(167,371)(168,372)(169,373)(170,374)(171,375)(172,376)(173,377)(174,378)(175,379)(176,380)(177,381)(178,382)(179,383)(180,384)(181,385)(182,386)(183,387)(184,388)(185,389)(186,390)(187,391)(188,392)(189,393)(190,394)(191,395)(192,396)(193,397)(194,398)(195,399)(196,400)(197,401)(198,402)(199,403)(200,404)(201,405)(202,406)(203,407)(204,408)(205,409)(206,410)(207,411)(208,412)(209,413)(210,414)(211,415)(212,416)(213,417)(214,418)(215,419)(216,420)(217,421)(218,422)(219,423)(220,424)(221,425)(222,426)(223,427)(224,428)(225,429)(226,430)(227,431)(228,432)(229,433)(230,434)(231,435)(232,436)(233,437)(234,438), (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) );
G=PermutationGroup([[(1,439),(2,440),(3,441),(4,442),(5,443),(6,444),(7,445),(8,446),(9,447),(10,448),(11,449),(12,450),(13,451),(14,452),(15,453),(16,454),(17,455),(18,456),(19,457),(20,458),(21,459),(22,460),(23,461),(24,462),(25,463),(26,464),(27,465),(28,466),(29,467),(30,468),(31,235),(32,236),(33,237),(34,238),(35,239),(36,240),(37,241),(38,242),(39,243),(40,244),(41,245),(42,246),(43,247),(44,248),(45,249),(46,250),(47,251),(48,252),(49,253),(50,254),(51,255),(52,256),(53,257),(54,258),(55,259),(56,260),(57,261),(58,262),(59,263),(60,264),(61,265),(62,266),(63,267),(64,268),(65,269),(66,270),(67,271),(68,272),(69,273),(70,274),(71,275),(72,276),(73,277),(74,278),(75,279),(76,280),(77,281),(78,282),(79,283),(80,284),(81,285),(82,286),(83,287),(84,288),(85,289),(86,290),(87,291),(88,292),(89,293),(90,294),(91,295),(92,296),(93,297),(94,298),(95,299),(96,300),(97,301),(98,302),(99,303),(100,304),(101,305),(102,306),(103,307),(104,308),(105,309),(106,310),(107,311),(108,312),(109,313),(110,314),(111,315),(112,316),(113,317),(114,318),(115,319),(116,320),(117,321),(118,322),(119,323),(120,324),(121,325),(122,326),(123,327),(124,328),(125,329),(126,330),(127,331),(128,332),(129,333),(130,334),(131,335),(132,336),(133,337),(134,338),(135,339),(136,340),(137,341),(138,342),(139,343),(140,344),(141,345),(142,346),(143,347),(144,348),(145,349),(146,350),(147,351),(148,352),(149,353),(150,354),(151,355),(152,356),(153,357),(154,358),(155,359),(156,360),(157,361),(158,362),(159,363),(160,364),(161,365),(162,366),(163,367),(164,368),(165,369),(166,370),(167,371),(168,372),(169,373),(170,374),(171,375),(172,376),(173,377),(174,378),(175,379),(176,380),(177,381),(178,382),(179,383),(180,384),(181,385),(182,386),(183,387),(184,388),(185,389),(186,390),(187,391),(188,392),(189,393),(190,394),(191,395),(192,396),(193,397),(194,398),(195,399),(196,400),(197,401),(198,402),(199,403),(200,404),(201,405),(202,406),(203,407),(204,408),(205,409),(206,410),(207,411),(208,412),(209,413),(210,414),(211,415),(212,416),(213,417),(214,418),(215,419),(216,420),(217,421),(218,422),(219,423),(220,424),(221,425),(222,426),(223,427),(224,428),(225,429),(226,430),(227,431),(228,432),(229,433),(230,434),(231,435),(232,436),(233,437),(234,438)], [(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)]])
468 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 9A | ··· | 9F | 13A | ··· | 13L | 18A | ··· | 18R | 26A | ··· | 26AJ | 39A | ··· | 39X | 78A | ··· | 78BT | 117A | ··· | 117BT | 234A | ··· | 234HH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 13 | ··· | 13 | 18 | ··· | 18 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 | 117 | ··· | 117 | 234 | ··· | 234 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
468 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C13 | C18 | C26 | C39 | C78 | C117 | C234 |
| kernel | C2×C234 | C234 | C2×C78 | C78 | C2×C26 | C2×C18 | C26 | C18 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 6 | 12 | 18 | 36 | 24 | 72 | 72 | 216 |
Matrix representation of C2×C234 ►in GL2(𝔽937) generated by
| 936 | 0 |
| 0 | 936 |
| 312 | 0 |
| 0 | 527 |
G:=sub<GL(2,GF(937))| [936,0,0,936],[312,0,0,527] >;
C2×C234 in GAP, Magma, Sage, TeX
C_2\times C_{234} % in TeX
G:=Group("C2xC234"); // GroupNames label
G:=SmallGroup(468,18);
// by ID
G=gap.SmallGroup(468,18);
# by ID
G:=PCGroup([5,-2,-2,-3,-13,-3,597]);
// Polycyclic
G:=Group<a,b|a^2=b^234=1,a*b=b*a>;
// generators/relations
Export