direct product, abelian, monomial, 2-elementary
Aliases: C2×C226, SmallGroup(452,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C226 |
| C1 — C2×C226 |
| C1 — C2×C226 |
Generators and relations for C2×C226
G = < a,b | a2=b226=1, ab=ba >
Smallest permutation representation of C2×C226
►Regular action on 452 points
Generators in S452
(1 424)(2 425)(3 426)(4 427)(5 428)(6 429)(7 430)(8 431)(9 432)(10 433)(11 434)(12 435)(13 436)(14 437)(15 438)(16 439)(17 440)(18 441)(19 442)(20 443)(21 444)(22 445)(23 446)(24 447)(25 448)(26 449)(27 450)(28 451)(29 452)(30 227)(31 228)(32 229)(33 230)(34 231)(35 232)(36 233)(37 234)(38 235)(39 236)(40 237)(41 238)(42 239)(43 240)(44 241)(45 242)(46 243)(47 244)(48 245)(49 246)(50 247)(51 248)(52 249)(53 250)(54 251)(55 252)(56 253)(57 254)(58 255)(59 256)(60 257)(61 258)(62 259)(63 260)(64 261)(65 262)(66 263)(67 264)(68 265)(69 266)(70 267)(71 268)(72 269)(73 270)(74 271)(75 272)(76 273)(77 274)(78 275)(79 276)(80 277)(81 278)(82 279)(83 280)(84 281)(85 282)(86 283)(87 284)(88 285)(89 286)(90 287)(91 288)(92 289)(93 290)(94 291)(95 292)(96 293)(97 294)(98 295)(99 296)(100 297)(101 298)(102 299)(103 300)(104 301)(105 302)(106 303)(107 304)(108 305)(109 306)(110 307)(111 308)(112 309)(113 310)(114 311)(115 312)(116 313)(117 314)(118 315)(119 316)(120 317)(121 318)(122 319)(123 320)(124 321)(125 322)(126 323)(127 324)(128 325)(129 326)(130 327)(131 328)(132 329)(133 330)(134 331)(135 332)(136 333)(137 334)(138 335)(139 336)(140 337)(141 338)(142 339)(143 340)(144 341)(145 342)(146 343)(147 344)(148 345)(149 346)(150 347)(151 348)(152 349)(153 350)(154 351)(155 352)(156 353)(157 354)(158 355)(159 356)(160 357)(161 358)(162 359)(163 360)(164 361)(165 362)(166 363)(167 364)(168 365)(169 366)(170 367)(171 368)(172 369)(173 370)(174 371)(175 372)(176 373)(177 374)(178 375)(179 376)(180 377)(181 378)(182 379)(183 380)(184 381)(185 382)(186 383)(187 384)(188 385)(189 386)(190 387)(191 388)(192 389)(193 390)(194 391)(195 392)(196 393)(197 394)(198 395)(199 396)(200 397)(201 398)(202 399)(203 400)(204 401)(205 402)(206 403)(207 404)(208 405)(209 406)(210 407)(211 408)(212 409)(213 410)(214 411)(215 412)(216 413)(217 414)(218 415)(219 416)(220 417)(221 418)(222 419)(223 420)(224 421)(225 422)(226 423)
(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)
G:=sub<Sym(452)| (1,424)(2,425)(3,426)(4,427)(5,428)(6,429)(7,430)(8,431)(9,432)(10,433)(11,434)(12,435)(13,436)(14,437)(15,438)(16,439)(17,440)(18,441)(19,442)(20,443)(21,444)(22,445)(23,446)(24,447)(25,448)(26,449)(27,450)(28,451)(29,452)(30,227)(31,228)(32,229)(33,230)(34,231)(35,232)(36,233)(37,234)(38,235)(39,236)(40,237)(41,238)(42,239)(43,240)(44,241)(45,242)(46,243)(47,244)(48,245)(49,246)(50,247)(51,248)(52,249)(53,250)(54,251)(55,252)(56,253)(57,254)(58,255)(59,256)(60,257)(61,258)(62,259)(63,260)(64,261)(65,262)(66,263)(67,264)(68,265)(69,266)(70,267)(71,268)(72,269)(73,270)(74,271)(75,272)(76,273)(77,274)(78,275)(79,276)(80,277)(81,278)(82,279)(83,280)(84,281)(85,282)(86,283)(87,284)(88,285)(89,286)(90,287)(91,288)(92,289)(93,290)(94,291)(95,292)(96,293)(97,294)(98,295)(99,296)(100,297)(101,298)(102,299)(103,300)(104,301)(105,302)(106,303)(107,304)(108,305)(109,306)(110,307)(111,308)(112,309)(113,310)(114,311)(115,312)(116,313)(117,314)(118,315)(119,316)(120,317)(121,318)(122,319)(123,320)(124,321)(125,322)(126,323)(127,324)(128,325)(129,326)(130,327)(131,328)(132,329)(133,330)(134,331)(135,332)(136,333)(137,334)(138,335)(139,336)(140,337)(141,338)(142,339)(143,340)(144,341)(145,342)(146,343)(147,344)(148,345)(149,346)(150,347)(151,348)(152,349)(153,350)(154,351)(155,352)(156,353)(157,354)(158,355)(159,356)(160,357)(161,358)(162,359)(163,360)(164,361)(165,362)(166,363)(167,364)(168,365)(169,366)(170,367)(171,368)(172,369)(173,370)(174,371)(175,372)(176,373)(177,374)(178,375)(179,376)(180,377)(181,378)(182,379)(183,380)(184,381)(185,382)(186,383)(187,384)(188,385)(189,386)(190,387)(191,388)(192,389)(193,390)(194,391)(195,392)(196,393)(197,394)(198,395)(199,396)(200,397)(201,398)(202,399)(203,400)(204,401)(205,402)(206,403)(207,404)(208,405)(209,406)(210,407)(211,408)(212,409)(213,410)(214,411)(215,412)(216,413)(217,414)(218,415)(219,416)(220,417)(221,418)(222,419)(223,420)(224,421)(225,422)(226,423), (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)>;
G:=Group( (1,424)(2,425)(3,426)(4,427)(5,428)(6,429)(7,430)(8,431)(9,432)(10,433)(11,434)(12,435)(13,436)(14,437)(15,438)(16,439)(17,440)(18,441)(19,442)(20,443)(21,444)(22,445)(23,446)(24,447)(25,448)(26,449)(27,450)(28,451)(29,452)(30,227)(31,228)(32,229)(33,230)(34,231)(35,232)(36,233)(37,234)(38,235)(39,236)(40,237)(41,238)(42,239)(43,240)(44,241)(45,242)(46,243)(47,244)(48,245)(49,246)(50,247)(51,248)(52,249)(53,250)(54,251)(55,252)(56,253)(57,254)(58,255)(59,256)(60,257)(61,258)(62,259)(63,260)(64,261)(65,262)(66,263)(67,264)(68,265)(69,266)(70,267)(71,268)(72,269)(73,270)(74,271)(75,272)(76,273)(77,274)(78,275)(79,276)(80,277)(81,278)(82,279)(83,280)(84,281)(85,282)(86,283)(87,284)(88,285)(89,286)(90,287)(91,288)(92,289)(93,290)(94,291)(95,292)(96,293)(97,294)(98,295)(99,296)(100,297)(101,298)(102,299)(103,300)(104,301)(105,302)(106,303)(107,304)(108,305)(109,306)(110,307)(111,308)(112,309)(113,310)(114,311)(115,312)(116,313)(117,314)(118,315)(119,316)(120,317)(121,318)(122,319)(123,320)(124,321)(125,322)(126,323)(127,324)(128,325)(129,326)(130,327)(131,328)(132,329)(133,330)(134,331)(135,332)(136,333)(137,334)(138,335)(139,336)(140,337)(141,338)(142,339)(143,340)(144,341)(145,342)(146,343)(147,344)(148,345)(149,346)(150,347)(151,348)(152,349)(153,350)(154,351)(155,352)(156,353)(157,354)(158,355)(159,356)(160,357)(161,358)(162,359)(163,360)(164,361)(165,362)(166,363)(167,364)(168,365)(169,366)(170,367)(171,368)(172,369)(173,370)(174,371)(175,372)(176,373)(177,374)(178,375)(179,376)(180,377)(181,378)(182,379)(183,380)(184,381)(185,382)(186,383)(187,384)(188,385)(189,386)(190,387)(191,388)(192,389)(193,390)(194,391)(195,392)(196,393)(197,394)(198,395)(199,396)(200,397)(201,398)(202,399)(203,400)(204,401)(205,402)(206,403)(207,404)(208,405)(209,406)(210,407)(211,408)(212,409)(213,410)(214,411)(215,412)(216,413)(217,414)(218,415)(219,416)(220,417)(221,418)(222,419)(223,420)(224,421)(225,422)(226,423), (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) );
G=PermutationGroup([[(1,424),(2,425),(3,426),(4,427),(5,428),(6,429),(7,430),(8,431),(9,432),(10,433),(11,434),(12,435),(13,436),(14,437),(15,438),(16,439),(17,440),(18,441),(19,442),(20,443),(21,444),(22,445),(23,446),(24,447),(25,448),(26,449),(27,450),(28,451),(29,452),(30,227),(31,228),(32,229),(33,230),(34,231),(35,232),(36,233),(37,234),(38,235),(39,236),(40,237),(41,238),(42,239),(43,240),(44,241),(45,242),(46,243),(47,244),(48,245),(49,246),(50,247),(51,248),(52,249),(53,250),(54,251),(55,252),(56,253),(57,254),(58,255),(59,256),(60,257),(61,258),(62,259),(63,260),(64,261),(65,262),(66,263),(67,264),(68,265),(69,266),(70,267),(71,268),(72,269),(73,270),(74,271),(75,272),(76,273),(77,274),(78,275),(79,276),(80,277),(81,278),(82,279),(83,280),(84,281),(85,282),(86,283),(87,284),(88,285),(89,286),(90,287),(91,288),(92,289),(93,290),(94,291),(95,292),(96,293),(97,294),(98,295),(99,296),(100,297),(101,298),(102,299),(103,300),(104,301),(105,302),(106,303),(107,304),(108,305),(109,306),(110,307),(111,308),(112,309),(113,310),(114,311),(115,312),(116,313),(117,314),(118,315),(119,316),(120,317),(121,318),(122,319),(123,320),(124,321),(125,322),(126,323),(127,324),(128,325),(129,326),(130,327),(131,328),(132,329),(133,330),(134,331),(135,332),(136,333),(137,334),(138,335),(139,336),(140,337),(141,338),(142,339),(143,340),(144,341),(145,342),(146,343),(147,344),(148,345),(149,346),(150,347),(151,348),(152,349),(153,350),(154,351),(155,352),(156,353),(157,354),(158,355),(159,356),(160,357),(161,358),(162,359),(163,360),(164,361),(165,362),(166,363),(167,364),(168,365),(169,366),(170,367),(171,368),(172,369),(173,370),(174,371),(175,372),(176,373),(177,374),(178,375),(179,376),(180,377),(181,378),(182,379),(183,380),(184,381),(185,382),(186,383),(187,384),(188,385),(189,386),(190,387),(191,388),(192,389),(193,390),(194,391),(195,392),(196,393),(197,394),(198,395),(199,396),(200,397),(201,398),(202,399),(203,400),(204,401),(205,402),(206,403),(207,404),(208,405),(209,406),(210,407),(211,408),(212,409),(213,410),(214,411),(215,412),(216,413),(217,414),(218,415),(219,416),(220,417),(221,418),(222,419),(223,420),(224,421),(225,422),(226,423)], [(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)]])
452 conjugacy classes
| class | 1 | 2A | 2B | 2C | 113A | ··· | 113DH | 226A | ··· | 226LX |
| order | 1 | 2 | 2 | 2 | 113 | ··· | 113 | 226 | ··· | 226 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
452 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C113 | C226 |
| kernel | C2×C226 | C226 | C22 | C2 |
| # reps | 1 | 3 | 112 | 336 |
Matrix representation of C2×C226 ►in GL2(𝔽227) generated by
| 1 | 0 |
| 0 | 226 |
| 55 | 0 |
| 0 | 15 |
G:=sub<GL(2,GF(227))| [1,0,0,226],[55,0,0,15] >;
C2×C226 in GAP, Magma, Sage, TeX
C_2\times C_{226} % in TeX
G:=Group("C2xC226"); // GroupNames label
G:=SmallGroup(452,5);
// by ID
G=gap.SmallGroup(452,5);
# by ID
G:=PCGroup([3,-2,-2,-113]);
// Polycyclic
G:=Group<a,b|a^2=b^226=1,a*b=b*a>;
// generators/relations
Export