direct product, abelian, monomial, 2-elementary
Aliases: C2×C228, SmallGroup(456,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C228 |
| C1 — C2×C228 |
| C1 — C2×C228 |
Generators and relations for C2×C228
G = < a,b | a2=b228=1, ab=ba >
Smallest permutation representation of C2×C228
►Regular action on 456 points
Generators in S456
(1 302)(2 303)(3 304)(4 305)(5 306)(6 307)(7 308)(8 309)(9 310)(10 311)(11 312)(12 313)(13 314)(14 315)(15 316)(16 317)(17 318)(18 319)(19 320)(20 321)(21 322)(22 323)(23 324)(24 325)(25 326)(26 327)(27 328)(28 329)(29 330)(30 331)(31 332)(32 333)(33 334)(34 335)(35 336)(36 337)(37 338)(38 339)(39 340)(40 341)(41 342)(42 343)(43 344)(44 345)(45 346)(46 347)(47 348)(48 349)(49 350)(50 351)(51 352)(52 353)(53 354)(54 355)(55 356)(56 357)(57 358)(58 359)(59 360)(60 361)(61 362)(62 363)(63 364)(64 365)(65 366)(66 367)(67 368)(68 369)(69 370)(70 371)(71 372)(72 373)(73 374)(74 375)(75 376)(76 377)(77 378)(78 379)(79 380)(80 381)(81 382)(82 383)(83 384)(84 385)(85 386)(86 387)(87 388)(88 389)(89 390)(90 391)(91 392)(92 393)(93 394)(94 395)(95 396)(96 397)(97 398)(98 399)(99 400)(100 401)(101 402)(102 403)(103 404)(104 405)(105 406)(106 407)(107 408)(108 409)(109 410)(110 411)(111 412)(112 413)(113 414)(114 415)(115 416)(116 417)(117 418)(118 419)(119 420)(120 421)(121 422)(122 423)(123 424)(124 425)(125 426)(126 427)(127 428)(128 429)(129 430)(130 431)(131 432)(132 433)(133 434)(134 435)(135 436)(136 437)(137 438)(138 439)(139 440)(140 441)(141 442)(142 443)(143 444)(144 445)(145 446)(146 447)(147 448)(148 449)(149 450)(150 451)(151 452)(152 453)(153 454)(154 455)(155 456)(156 229)(157 230)(158 231)(159 232)(160 233)(161 234)(162 235)(163 236)(164 237)(165 238)(166 239)(167 240)(168 241)(169 242)(170 243)(171 244)(172 245)(173 246)(174 247)(175 248)(176 249)(177 250)(178 251)(179 252)(180 253)(181 254)(182 255)(183 256)(184 257)(185 258)(186 259)(187 260)(188 261)(189 262)(190 263)(191 264)(192 265)(193 266)(194 267)(195 268)(196 269)(197 270)(198 271)(199 272)(200 273)(201 274)(202 275)(203 276)(204 277)(205 278)(206 279)(207 280)(208 281)(209 282)(210 283)(211 284)(212 285)(213 286)(214 287)(215 288)(216 289)(217 290)(218 291)(219 292)(220 293)(221 294)(222 295)(223 296)(224 297)(225 298)(226 299)(227 300)(228 301)
(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)
G:=sub<Sym(456)| (1,302)(2,303)(3,304)(4,305)(5,306)(6,307)(7,308)(8,309)(9,310)(10,311)(11,312)(12,313)(13,314)(14,315)(15,316)(16,317)(17,318)(18,319)(19,320)(20,321)(21,322)(22,323)(23,324)(24,325)(25,326)(26,327)(27,328)(28,329)(29,330)(30,331)(31,332)(32,333)(33,334)(34,335)(35,336)(36,337)(37,338)(38,339)(39,340)(40,341)(41,342)(42,343)(43,344)(44,345)(45,346)(46,347)(47,348)(48,349)(49,350)(50,351)(51,352)(52,353)(53,354)(54,355)(55,356)(56,357)(57,358)(58,359)(59,360)(60,361)(61,362)(62,363)(63,364)(64,365)(65,366)(66,367)(67,368)(68,369)(69,370)(70,371)(71,372)(72,373)(73,374)(74,375)(75,376)(76,377)(77,378)(78,379)(79,380)(80,381)(81,382)(82,383)(83,384)(84,385)(85,386)(86,387)(87,388)(88,389)(89,390)(90,391)(91,392)(92,393)(93,394)(94,395)(95,396)(96,397)(97,398)(98,399)(99,400)(100,401)(101,402)(102,403)(103,404)(104,405)(105,406)(106,407)(107,408)(108,409)(109,410)(110,411)(111,412)(112,413)(113,414)(114,415)(115,416)(116,417)(117,418)(118,419)(119,420)(120,421)(121,422)(122,423)(123,424)(124,425)(125,426)(126,427)(127,428)(128,429)(129,430)(130,431)(131,432)(132,433)(133,434)(134,435)(135,436)(136,437)(137,438)(138,439)(139,440)(140,441)(141,442)(142,443)(143,444)(144,445)(145,446)(146,447)(147,448)(148,449)(149,450)(150,451)(151,452)(152,453)(153,454)(154,455)(155,456)(156,229)(157,230)(158,231)(159,232)(160,233)(161,234)(162,235)(163,236)(164,237)(165,238)(166,239)(167,240)(168,241)(169,242)(170,243)(171,244)(172,245)(173,246)(174,247)(175,248)(176,249)(177,250)(178,251)(179,252)(180,253)(181,254)(182,255)(183,256)(184,257)(185,258)(186,259)(187,260)(188,261)(189,262)(190,263)(191,264)(192,265)(193,266)(194,267)(195,268)(196,269)(197,270)(198,271)(199,272)(200,273)(201,274)(202,275)(203,276)(204,277)(205,278)(206,279)(207,280)(208,281)(209,282)(210,283)(211,284)(212,285)(213,286)(214,287)(215,288)(216,289)(217,290)(218,291)(219,292)(220,293)(221,294)(222,295)(223,296)(224,297)(225,298)(226,299)(227,300)(228,301), (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)>;
G:=Group( (1,302)(2,303)(3,304)(4,305)(5,306)(6,307)(7,308)(8,309)(9,310)(10,311)(11,312)(12,313)(13,314)(14,315)(15,316)(16,317)(17,318)(18,319)(19,320)(20,321)(21,322)(22,323)(23,324)(24,325)(25,326)(26,327)(27,328)(28,329)(29,330)(30,331)(31,332)(32,333)(33,334)(34,335)(35,336)(36,337)(37,338)(38,339)(39,340)(40,341)(41,342)(42,343)(43,344)(44,345)(45,346)(46,347)(47,348)(48,349)(49,350)(50,351)(51,352)(52,353)(53,354)(54,355)(55,356)(56,357)(57,358)(58,359)(59,360)(60,361)(61,362)(62,363)(63,364)(64,365)(65,366)(66,367)(67,368)(68,369)(69,370)(70,371)(71,372)(72,373)(73,374)(74,375)(75,376)(76,377)(77,378)(78,379)(79,380)(80,381)(81,382)(82,383)(83,384)(84,385)(85,386)(86,387)(87,388)(88,389)(89,390)(90,391)(91,392)(92,393)(93,394)(94,395)(95,396)(96,397)(97,398)(98,399)(99,400)(100,401)(101,402)(102,403)(103,404)(104,405)(105,406)(106,407)(107,408)(108,409)(109,410)(110,411)(111,412)(112,413)(113,414)(114,415)(115,416)(116,417)(117,418)(118,419)(119,420)(120,421)(121,422)(122,423)(123,424)(124,425)(125,426)(126,427)(127,428)(128,429)(129,430)(130,431)(131,432)(132,433)(133,434)(134,435)(135,436)(136,437)(137,438)(138,439)(139,440)(140,441)(141,442)(142,443)(143,444)(144,445)(145,446)(146,447)(147,448)(148,449)(149,450)(150,451)(151,452)(152,453)(153,454)(154,455)(155,456)(156,229)(157,230)(158,231)(159,232)(160,233)(161,234)(162,235)(163,236)(164,237)(165,238)(166,239)(167,240)(168,241)(169,242)(170,243)(171,244)(172,245)(173,246)(174,247)(175,248)(176,249)(177,250)(178,251)(179,252)(180,253)(181,254)(182,255)(183,256)(184,257)(185,258)(186,259)(187,260)(188,261)(189,262)(190,263)(191,264)(192,265)(193,266)(194,267)(195,268)(196,269)(197,270)(198,271)(199,272)(200,273)(201,274)(202,275)(203,276)(204,277)(205,278)(206,279)(207,280)(208,281)(209,282)(210,283)(211,284)(212,285)(213,286)(214,287)(215,288)(216,289)(217,290)(218,291)(219,292)(220,293)(221,294)(222,295)(223,296)(224,297)(225,298)(226,299)(227,300)(228,301), (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) );
G=PermutationGroup([[(1,302),(2,303),(3,304),(4,305),(5,306),(6,307),(7,308),(8,309),(9,310),(10,311),(11,312),(12,313),(13,314),(14,315),(15,316),(16,317),(17,318),(18,319),(19,320),(20,321),(21,322),(22,323),(23,324),(24,325),(25,326),(26,327),(27,328),(28,329),(29,330),(30,331),(31,332),(32,333),(33,334),(34,335),(35,336),(36,337),(37,338),(38,339),(39,340),(40,341),(41,342),(42,343),(43,344),(44,345),(45,346),(46,347),(47,348),(48,349),(49,350),(50,351),(51,352),(52,353),(53,354),(54,355),(55,356),(56,357),(57,358),(58,359),(59,360),(60,361),(61,362),(62,363),(63,364),(64,365),(65,366),(66,367),(67,368),(68,369),(69,370),(70,371),(71,372),(72,373),(73,374),(74,375),(75,376),(76,377),(77,378),(78,379),(79,380),(80,381),(81,382),(82,383),(83,384),(84,385),(85,386),(86,387),(87,388),(88,389),(89,390),(90,391),(91,392),(92,393),(93,394),(94,395),(95,396),(96,397),(97,398),(98,399),(99,400),(100,401),(101,402),(102,403),(103,404),(104,405),(105,406),(106,407),(107,408),(108,409),(109,410),(110,411),(111,412),(112,413),(113,414),(114,415),(115,416),(116,417),(117,418),(118,419),(119,420),(120,421),(121,422),(122,423),(123,424),(124,425),(125,426),(126,427),(127,428),(128,429),(129,430),(130,431),(131,432),(132,433),(133,434),(134,435),(135,436),(136,437),(137,438),(138,439),(139,440),(140,441),(141,442),(142,443),(143,444),(144,445),(145,446),(146,447),(147,448),(148,449),(149,450),(150,451),(151,452),(152,453),(153,454),(154,455),(155,456),(156,229),(157,230),(158,231),(159,232),(160,233),(161,234),(162,235),(163,236),(164,237),(165,238),(166,239),(167,240),(168,241),(169,242),(170,243),(171,244),(172,245),(173,246),(174,247),(175,248),(176,249),(177,250),(178,251),(179,252),(180,253),(181,254),(182,255),(183,256),(184,257),(185,258),(186,259),(187,260),(188,261),(189,262),(190,263),(191,264),(192,265),(193,266),(194,267),(195,268),(196,269),(197,270),(198,271),(199,272),(200,273),(201,274),(202,275),(203,276),(204,277),(205,278),(206,279),(207,280),(208,281),(209,282),(210,283),(211,284),(212,285),(213,286),(214,287),(215,288),(216,289),(217,290),(218,291),(219,292),(220,293),(221,294),(222,295),(223,296),(224,297),(225,298),(226,299),(227,300),(228,301)], [(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)]])
456 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 19A | ··· | 19R | 38A | ··· | 38BB | 57A | ··· | 57AJ | 76A | ··· | 76BT | 114A | ··· | 114DD | 228A | ··· | 228EN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| 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 | 1 | ··· | 1 |
456 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | C19 | C38 | C38 | C57 | C76 | C114 | C114 | C228 |
| kernel | C2×C228 | C228 | C2×C114 | C2×C76 | C114 | C76 | C2×C38 | C38 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 18 | 36 | 18 | 36 | 72 | 72 | 36 | 144 |
Matrix representation of C2×C228 ►in GL2(𝔽229) generated by
| 228 | 0 |
| 0 | 1 |
| 3 | 0 |
| 0 | 211 |
G:=sub<GL(2,GF(229))| [228,0,0,1],[3,0,0,211] >;
C2×C228 in GAP, Magma, Sage, TeX
C_2\times C_{228} % in TeX
G:=Group("C2xC228"); // GroupNames label
G:=SmallGroup(456,39);
// by ID
G=gap.SmallGroup(456,39);
# by ID
G:=PCGroup([5,-2,-2,-3,-19,-2,1140]);
// Polycyclic
G:=Group<a,b|a^2=b^228=1,a*b=b*a>;
// generators/relations
Export