direct product, abelian, monomial, 2-elementary
Aliases: C2×C224, SmallGroup(448,173)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C224 |
| C1 — C2×C224 |
| C1 — C2×C224 |
Generators and relations for C2×C224
G = < a,b | a2=b224=1, ab=ba >
Smallest permutation representation of C2×C224
►Regular action on 448 points
Generators in S448
(1 267)(2 268)(3 269)(4 270)(5 271)(6 272)(7 273)(8 274)(9 275)(10 276)(11 277)(12 278)(13 279)(14 280)(15 281)(16 282)(17 283)(18 284)(19 285)(20 286)(21 287)(22 288)(23 289)(24 290)(25 291)(26 292)(27 293)(28 294)(29 295)(30 296)(31 297)(32 298)(33 299)(34 300)(35 301)(36 302)(37 303)(38 304)(39 305)(40 306)(41 307)(42 308)(43 309)(44 310)(45 311)(46 312)(47 313)(48 314)(49 315)(50 316)(51 317)(52 318)(53 319)(54 320)(55 321)(56 322)(57 323)(58 324)(59 325)(60 326)(61 327)(62 328)(63 329)(64 330)(65 331)(66 332)(67 333)(68 334)(69 335)(70 336)(71 337)(72 338)(73 339)(74 340)(75 341)(76 342)(77 343)(78 344)(79 345)(80 346)(81 347)(82 348)(83 349)(84 350)(85 351)(86 352)(87 353)(88 354)(89 355)(90 356)(91 357)(92 358)(93 359)(94 360)(95 361)(96 362)(97 363)(98 364)(99 365)(100 366)(101 367)(102 368)(103 369)(104 370)(105 371)(106 372)(107 373)(108 374)(109 375)(110 376)(111 377)(112 378)(113 379)(114 380)(115 381)(116 382)(117 383)(118 384)(119 385)(120 386)(121 387)(122 388)(123 389)(124 390)(125 391)(126 392)(127 393)(128 394)(129 395)(130 396)(131 397)(132 398)(133 399)(134 400)(135 401)(136 402)(137 403)(138 404)(139 405)(140 406)(141 407)(142 408)(143 409)(144 410)(145 411)(146 412)(147 413)(148 414)(149 415)(150 416)(151 417)(152 418)(153 419)(154 420)(155 421)(156 422)(157 423)(158 424)(159 425)(160 426)(161 427)(162 428)(163 429)(164 430)(165 431)(166 432)(167 433)(168 434)(169 435)(170 436)(171 437)(172 438)(173 439)(174 440)(175 441)(176 442)(177 443)(178 444)(179 445)(180 446)(181 447)(182 448)(183 225)(184 226)(185 227)(186 228)(187 229)(188 230)(189 231)(190 232)(191 233)(192 234)(193 235)(194 236)(195 237)(196 238)(197 239)(198 240)(199 241)(200 242)(201 243)(202 244)(203 245)(204 246)(205 247)(206 248)(207 249)(208 250)(209 251)(210 252)(211 253)(212 254)(213 255)(214 256)(215 257)(216 258)(217 259)(218 260)(219 261)(220 262)(221 263)(222 264)(223 265)(224 266)
(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)
G:=sub<Sym(448)| (1,267)(2,268)(3,269)(4,270)(5,271)(6,272)(7,273)(8,274)(9,275)(10,276)(11,277)(12,278)(13,279)(14,280)(15,281)(16,282)(17,283)(18,284)(19,285)(20,286)(21,287)(22,288)(23,289)(24,290)(25,291)(26,292)(27,293)(28,294)(29,295)(30,296)(31,297)(32,298)(33,299)(34,300)(35,301)(36,302)(37,303)(38,304)(39,305)(40,306)(41,307)(42,308)(43,309)(44,310)(45,311)(46,312)(47,313)(48,314)(49,315)(50,316)(51,317)(52,318)(53,319)(54,320)(55,321)(56,322)(57,323)(58,324)(59,325)(60,326)(61,327)(62,328)(63,329)(64,330)(65,331)(66,332)(67,333)(68,334)(69,335)(70,336)(71,337)(72,338)(73,339)(74,340)(75,341)(76,342)(77,343)(78,344)(79,345)(80,346)(81,347)(82,348)(83,349)(84,350)(85,351)(86,352)(87,353)(88,354)(89,355)(90,356)(91,357)(92,358)(93,359)(94,360)(95,361)(96,362)(97,363)(98,364)(99,365)(100,366)(101,367)(102,368)(103,369)(104,370)(105,371)(106,372)(107,373)(108,374)(109,375)(110,376)(111,377)(112,378)(113,379)(114,380)(115,381)(116,382)(117,383)(118,384)(119,385)(120,386)(121,387)(122,388)(123,389)(124,390)(125,391)(126,392)(127,393)(128,394)(129,395)(130,396)(131,397)(132,398)(133,399)(134,400)(135,401)(136,402)(137,403)(138,404)(139,405)(140,406)(141,407)(142,408)(143,409)(144,410)(145,411)(146,412)(147,413)(148,414)(149,415)(150,416)(151,417)(152,418)(153,419)(154,420)(155,421)(156,422)(157,423)(158,424)(159,425)(160,426)(161,427)(162,428)(163,429)(164,430)(165,431)(166,432)(167,433)(168,434)(169,435)(170,436)(171,437)(172,438)(173,439)(174,440)(175,441)(176,442)(177,443)(178,444)(179,445)(180,446)(181,447)(182,448)(183,225)(184,226)(185,227)(186,228)(187,229)(188,230)(189,231)(190,232)(191,233)(192,234)(193,235)(194,236)(195,237)(196,238)(197,239)(198,240)(199,241)(200,242)(201,243)(202,244)(203,245)(204,246)(205,247)(206,248)(207,249)(208,250)(209,251)(210,252)(211,253)(212,254)(213,255)(214,256)(215,257)(216,258)(217,259)(218,260)(219,261)(220,262)(221,263)(222,264)(223,265)(224,266), (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)>;
G:=Group( (1,267)(2,268)(3,269)(4,270)(5,271)(6,272)(7,273)(8,274)(9,275)(10,276)(11,277)(12,278)(13,279)(14,280)(15,281)(16,282)(17,283)(18,284)(19,285)(20,286)(21,287)(22,288)(23,289)(24,290)(25,291)(26,292)(27,293)(28,294)(29,295)(30,296)(31,297)(32,298)(33,299)(34,300)(35,301)(36,302)(37,303)(38,304)(39,305)(40,306)(41,307)(42,308)(43,309)(44,310)(45,311)(46,312)(47,313)(48,314)(49,315)(50,316)(51,317)(52,318)(53,319)(54,320)(55,321)(56,322)(57,323)(58,324)(59,325)(60,326)(61,327)(62,328)(63,329)(64,330)(65,331)(66,332)(67,333)(68,334)(69,335)(70,336)(71,337)(72,338)(73,339)(74,340)(75,341)(76,342)(77,343)(78,344)(79,345)(80,346)(81,347)(82,348)(83,349)(84,350)(85,351)(86,352)(87,353)(88,354)(89,355)(90,356)(91,357)(92,358)(93,359)(94,360)(95,361)(96,362)(97,363)(98,364)(99,365)(100,366)(101,367)(102,368)(103,369)(104,370)(105,371)(106,372)(107,373)(108,374)(109,375)(110,376)(111,377)(112,378)(113,379)(114,380)(115,381)(116,382)(117,383)(118,384)(119,385)(120,386)(121,387)(122,388)(123,389)(124,390)(125,391)(126,392)(127,393)(128,394)(129,395)(130,396)(131,397)(132,398)(133,399)(134,400)(135,401)(136,402)(137,403)(138,404)(139,405)(140,406)(141,407)(142,408)(143,409)(144,410)(145,411)(146,412)(147,413)(148,414)(149,415)(150,416)(151,417)(152,418)(153,419)(154,420)(155,421)(156,422)(157,423)(158,424)(159,425)(160,426)(161,427)(162,428)(163,429)(164,430)(165,431)(166,432)(167,433)(168,434)(169,435)(170,436)(171,437)(172,438)(173,439)(174,440)(175,441)(176,442)(177,443)(178,444)(179,445)(180,446)(181,447)(182,448)(183,225)(184,226)(185,227)(186,228)(187,229)(188,230)(189,231)(190,232)(191,233)(192,234)(193,235)(194,236)(195,237)(196,238)(197,239)(198,240)(199,241)(200,242)(201,243)(202,244)(203,245)(204,246)(205,247)(206,248)(207,249)(208,250)(209,251)(210,252)(211,253)(212,254)(213,255)(214,256)(215,257)(216,258)(217,259)(218,260)(219,261)(220,262)(221,263)(222,264)(223,265)(224,266), (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) );
G=PermutationGroup([[(1,267),(2,268),(3,269),(4,270),(5,271),(6,272),(7,273),(8,274),(9,275),(10,276),(11,277),(12,278),(13,279),(14,280),(15,281),(16,282),(17,283),(18,284),(19,285),(20,286),(21,287),(22,288),(23,289),(24,290),(25,291),(26,292),(27,293),(28,294),(29,295),(30,296),(31,297),(32,298),(33,299),(34,300),(35,301),(36,302),(37,303),(38,304),(39,305),(40,306),(41,307),(42,308),(43,309),(44,310),(45,311),(46,312),(47,313),(48,314),(49,315),(50,316),(51,317),(52,318),(53,319),(54,320),(55,321),(56,322),(57,323),(58,324),(59,325),(60,326),(61,327),(62,328),(63,329),(64,330),(65,331),(66,332),(67,333),(68,334),(69,335),(70,336),(71,337),(72,338),(73,339),(74,340),(75,341),(76,342),(77,343),(78,344),(79,345),(80,346),(81,347),(82,348),(83,349),(84,350),(85,351),(86,352),(87,353),(88,354),(89,355),(90,356),(91,357),(92,358),(93,359),(94,360),(95,361),(96,362),(97,363),(98,364),(99,365),(100,366),(101,367),(102,368),(103,369),(104,370),(105,371),(106,372),(107,373),(108,374),(109,375),(110,376),(111,377),(112,378),(113,379),(114,380),(115,381),(116,382),(117,383),(118,384),(119,385),(120,386),(121,387),(122,388),(123,389),(124,390),(125,391),(126,392),(127,393),(128,394),(129,395),(130,396),(131,397),(132,398),(133,399),(134,400),(135,401),(136,402),(137,403),(138,404),(139,405),(140,406),(141,407),(142,408),(143,409),(144,410),(145,411),(146,412),(147,413),(148,414),(149,415),(150,416),(151,417),(152,418),(153,419),(154,420),(155,421),(156,422),(157,423),(158,424),(159,425),(160,426),(161,427),(162,428),(163,429),(164,430),(165,431),(166,432),(167,433),(168,434),(169,435),(170,436),(171,437),(172,438),(173,439),(174,440),(175,441),(176,442),(177,443),(178,444),(179,445),(180,446),(181,447),(182,448),(183,225),(184,226),(185,227),(186,228),(187,229),(188,230),(189,231),(190,232),(191,233),(192,234),(193,235),(194,236),(195,237),(196,238),(197,239),(198,240),(199,241),(200,242),(201,243),(202,244),(203,245),(204,246),(205,247),(206,248),(207,249),(208,250),(209,251),(210,252),(211,253),(212,254),(213,255),(214,256),(215,257),(216,258),(217,259),(218,260),(219,261),(220,262),(221,263),(222,264),(223,265),(224,266)], [(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)]])
448 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 16A | ··· | 16P | 28A | ··· | 28X | 32A | ··· | 32AF | 56A | ··· | 56AV | 112A | ··· | 112CR | 224A | ··· | 224GJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| 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 |
448 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C8 | C8 | C14 | C14 | C16 | C16 | C28 | C28 | C32 | C56 | C56 | C112 | C112 | C224 |
| kernel | C2×C224 | C224 | C2×C112 | C112 | C2×C56 | C2×C32 | C56 | C2×C28 | C32 | C2×C16 | C28 | C2×C14 | C16 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 4 | 4 | 12 | 6 | 8 | 8 | 12 | 12 | 32 | 24 | 24 | 48 | 48 | 192 |
Matrix representation of C2×C224 ►in GL2(𝔽449) generated by
| 1 | 0 |
| 0 | 448 |
| 205 | 0 |
| 0 | 77 |
G:=sub<GL(2,GF(449))| [1,0,0,448],[205,0,0,77] >;
C2×C224 in GAP, Magma, Sage, TeX
C_2\times C_{224} % in TeX
G:=Group("C2xC224"); // GroupNames label
G:=SmallGroup(448,173);
// by ID
G=gap.SmallGroup(448,173);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,80,102,124]);
// Polycyclic
G:=Group<a,b|a^2=b^224=1,a*b=b*a>;
// generators/relations
Export