direct product, abelian, monomial, 2-elementary
Aliases: C2×C214, SmallGroup(428,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C214 |
| C1 — C2×C214 |
| C1 — C2×C214 |
Generators and relations for C2×C214
G = < a,b | a2=b214=1, ab=ba >
Smallest permutation representation of C2×C214
►Regular action on 428 points
Generators in S428
(1 411)(2 412)(3 413)(4 414)(5 415)(6 416)(7 417)(8 418)(9 419)(10 420)(11 421)(12 422)(13 423)(14 424)(15 425)(16 426)(17 427)(18 428)(19 215)(20 216)(21 217)(22 218)(23 219)(24 220)(25 221)(26 222)(27 223)(28 224)(29 225)(30 226)(31 227)(32 228)(33 229)(34 230)(35 231)(36 232)(37 233)(38 234)(39 235)(40 236)(41 237)(42 238)(43 239)(44 240)(45 241)(46 242)(47 243)(48 244)(49 245)(50 246)(51 247)(52 248)(53 249)(54 250)(55 251)(56 252)(57 253)(58 254)(59 255)(60 256)(61 257)(62 258)(63 259)(64 260)(65 261)(66 262)(67 263)(68 264)(69 265)(70 266)(71 267)(72 268)(73 269)(74 270)(75 271)(76 272)(77 273)(78 274)(79 275)(80 276)(81 277)(82 278)(83 279)(84 280)(85 281)(86 282)(87 283)(88 284)(89 285)(90 286)(91 287)(92 288)(93 289)(94 290)(95 291)(96 292)(97 293)(98 294)(99 295)(100 296)(101 297)(102 298)(103 299)(104 300)(105 301)(106 302)(107 303)(108 304)(109 305)(110 306)(111 307)(112 308)(113 309)(114 310)(115 311)(116 312)(117 313)(118 314)(119 315)(120 316)(121 317)(122 318)(123 319)(124 320)(125 321)(126 322)(127 323)(128 324)(129 325)(130 326)(131 327)(132 328)(133 329)(134 330)(135 331)(136 332)(137 333)(138 334)(139 335)(140 336)(141 337)(142 338)(143 339)(144 340)(145 341)(146 342)(147 343)(148 344)(149 345)(150 346)(151 347)(152 348)(153 349)(154 350)(155 351)(156 352)(157 353)(158 354)(159 355)(160 356)(161 357)(162 358)(163 359)(164 360)(165 361)(166 362)(167 363)(168 364)(169 365)(170 366)(171 367)(172 368)(173 369)(174 370)(175 371)(176 372)(177 373)(178 374)(179 375)(180 376)(181 377)(182 378)(183 379)(184 380)(185 381)(186 382)(187 383)(188 384)(189 385)(190 386)(191 387)(192 388)(193 389)(194 390)(195 391)(196 392)(197 393)(198 394)(199 395)(200 396)(201 397)(202 398)(203 399)(204 400)(205 401)(206 402)(207 403)(208 404)(209 405)(210 406)(211 407)(212 408)(213 409)(214 410)
(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)
G:=sub<Sym(428)| (1,411)(2,412)(3,413)(4,414)(5,415)(6,416)(7,417)(8,418)(9,419)(10,420)(11,421)(12,422)(13,423)(14,424)(15,425)(16,426)(17,427)(18,428)(19,215)(20,216)(21,217)(22,218)(23,219)(24,220)(25,221)(26,222)(27,223)(28,224)(29,225)(30,226)(31,227)(32,228)(33,229)(34,230)(35,231)(36,232)(37,233)(38,234)(39,235)(40,236)(41,237)(42,238)(43,239)(44,240)(45,241)(46,242)(47,243)(48,244)(49,245)(50,246)(51,247)(52,248)(53,249)(54,250)(55,251)(56,252)(57,253)(58,254)(59,255)(60,256)(61,257)(62,258)(63,259)(64,260)(65,261)(66,262)(67,263)(68,264)(69,265)(70,266)(71,267)(72,268)(73,269)(74,270)(75,271)(76,272)(77,273)(78,274)(79,275)(80,276)(81,277)(82,278)(83,279)(84,280)(85,281)(86,282)(87,283)(88,284)(89,285)(90,286)(91,287)(92,288)(93,289)(94,290)(95,291)(96,292)(97,293)(98,294)(99,295)(100,296)(101,297)(102,298)(103,299)(104,300)(105,301)(106,302)(107,303)(108,304)(109,305)(110,306)(111,307)(112,308)(113,309)(114,310)(115,311)(116,312)(117,313)(118,314)(119,315)(120,316)(121,317)(122,318)(123,319)(124,320)(125,321)(126,322)(127,323)(128,324)(129,325)(130,326)(131,327)(132,328)(133,329)(134,330)(135,331)(136,332)(137,333)(138,334)(139,335)(140,336)(141,337)(142,338)(143,339)(144,340)(145,341)(146,342)(147,343)(148,344)(149,345)(150,346)(151,347)(152,348)(153,349)(154,350)(155,351)(156,352)(157,353)(158,354)(159,355)(160,356)(161,357)(162,358)(163,359)(164,360)(165,361)(166,362)(167,363)(168,364)(169,365)(170,366)(171,367)(172,368)(173,369)(174,370)(175,371)(176,372)(177,373)(178,374)(179,375)(180,376)(181,377)(182,378)(183,379)(184,380)(185,381)(186,382)(187,383)(188,384)(189,385)(190,386)(191,387)(192,388)(193,389)(194,390)(195,391)(196,392)(197,393)(198,394)(199,395)(200,396)(201,397)(202,398)(203,399)(204,400)(205,401)(206,402)(207,403)(208,404)(209,405)(210,406)(211,407)(212,408)(213,409)(214,410), (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)>;
G:=Group( (1,411)(2,412)(3,413)(4,414)(5,415)(6,416)(7,417)(8,418)(9,419)(10,420)(11,421)(12,422)(13,423)(14,424)(15,425)(16,426)(17,427)(18,428)(19,215)(20,216)(21,217)(22,218)(23,219)(24,220)(25,221)(26,222)(27,223)(28,224)(29,225)(30,226)(31,227)(32,228)(33,229)(34,230)(35,231)(36,232)(37,233)(38,234)(39,235)(40,236)(41,237)(42,238)(43,239)(44,240)(45,241)(46,242)(47,243)(48,244)(49,245)(50,246)(51,247)(52,248)(53,249)(54,250)(55,251)(56,252)(57,253)(58,254)(59,255)(60,256)(61,257)(62,258)(63,259)(64,260)(65,261)(66,262)(67,263)(68,264)(69,265)(70,266)(71,267)(72,268)(73,269)(74,270)(75,271)(76,272)(77,273)(78,274)(79,275)(80,276)(81,277)(82,278)(83,279)(84,280)(85,281)(86,282)(87,283)(88,284)(89,285)(90,286)(91,287)(92,288)(93,289)(94,290)(95,291)(96,292)(97,293)(98,294)(99,295)(100,296)(101,297)(102,298)(103,299)(104,300)(105,301)(106,302)(107,303)(108,304)(109,305)(110,306)(111,307)(112,308)(113,309)(114,310)(115,311)(116,312)(117,313)(118,314)(119,315)(120,316)(121,317)(122,318)(123,319)(124,320)(125,321)(126,322)(127,323)(128,324)(129,325)(130,326)(131,327)(132,328)(133,329)(134,330)(135,331)(136,332)(137,333)(138,334)(139,335)(140,336)(141,337)(142,338)(143,339)(144,340)(145,341)(146,342)(147,343)(148,344)(149,345)(150,346)(151,347)(152,348)(153,349)(154,350)(155,351)(156,352)(157,353)(158,354)(159,355)(160,356)(161,357)(162,358)(163,359)(164,360)(165,361)(166,362)(167,363)(168,364)(169,365)(170,366)(171,367)(172,368)(173,369)(174,370)(175,371)(176,372)(177,373)(178,374)(179,375)(180,376)(181,377)(182,378)(183,379)(184,380)(185,381)(186,382)(187,383)(188,384)(189,385)(190,386)(191,387)(192,388)(193,389)(194,390)(195,391)(196,392)(197,393)(198,394)(199,395)(200,396)(201,397)(202,398)(203,399)(204,400)(205,401)(206,402)(207,403)(208,404)(209,405)(210,406)(211,407)(212,408)(213,409)(214,410), (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) );
G=PermutationGroup([[(1,411),(2,412),(3,413),(4,414),(5,415),(6,416),(7,417),(8,418),(9,419),(10,420),(11,421),(12,422),(13,423),(14,424),(15,425),(16,426),(17,427),(18,428),(19,215),(20,216),(21,217),(22,218),(23,219),(24,220),(25,221),(26,222),(27,223),(28,224),(29,225),(30,226),(31,227),(32,228),(33,229),(34,230),(35,231),(36,232),(37,233),(38,234),(39,235),(40,236),(41,237),(42,238),(43,239),(44,240),(45,241),(46,242),(47,243),(48,244),(49,245),(50,246),(51,247),(52,248),(53,249),(54,250),(55,251),(56,252),(57,253),(58,254),(59,255),(60,256),(61,257),(62,258),(63,259),(64,260),(65,261),(66,262),(67,263),(68,264),(69,265),(70,266),(71,267),(72,268),(73,269),(74,270),(75,271),(76,272),(77,273),(78,274),(79,275),(80,276),(81,277),(82,278),(83,279),(84,280),(85,281),(86,282),(87,283),(88,284),(89,285),(90,286),(91,287),(92,288),(93,289),(94,290),(95,291),(96,292),(97,293),(98,294),(99,295),(100,296),(101,297),(102,298),(103,299),(104,300),(105,301),(106,302),(107,303),(108,304),(109,305),(110,306),(111,307),(112,308),(113,309),(114,310),(115,311),(116,312),(117,313),(118,314),(119,315),(120,316),(121,317),(122,318),(123,319),(124,320),(125,321),(126,322),(127,323),(128,324),(129,325),(130,326),(131,327),(132,328),(133,329),(134,330),(135,331),(136,332),(137,333),(138,334),(139,335),(140,336),(141,337),(142,338),(143,339),(144,340),(145,341),(146,342),(147,343),(148,344),(149,345),(150,346),(151,347),(152,348),(153,349),(154,350),(155,351),(156,352),(157,353),(158,354),(159,355),(160,356),(161,357),(162,358),(163,359),(164,360),(165,361),(166,362),(167,363),(168,364),(169,365),(170,366),(171,367),(172,368),(173,369),(174,370),(175,371),(176,372),(177,373),(178,374),(179,375),(180,376),(181,377),(182,378),(183,379),(184,380),(185,381),(186,382),(187,383),(188,384),(189,385),(190,386),(191,387),(192,388),(193,389),(194,390),(195,391),(196,392),(197,393),(198,394),(199,395),(200,396),(201,397),(202,398),(203,399),(204,400),(205,401),(206,402),(207,403),(208,404),(209,405),(210,406),(211,407),(212,408),(213,409),(214,410)], [(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)]])
428 conjugacy classes
| class | 1 | 2A | 2B | 2C | 107A | ··· | 107DB | 214A | ··· | 214LF |
| order | 1 | 2 | 2 | 2 | 107 | ··· | 107 | 214 | ··· | 214 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
428 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C107 | C214 |
| kernel | C2×C214 | C214 | C22 | C2 |
| # reps | 1 | 3 | 106 | 318 |
Matrix representation of C2×C214 ►in GL2(𝔽643) generated by
| 642 | 0 |
| 0 | 642 |
| 476 | 0 |
| 0 | 536 |
G:=sub<GL(2,GF(643))| [642,0,0,642],[476,0,0,536] >;
C2×C214 in GAP, Magma, Sage, TeX
C_2\times C_{214} % in TeX
G:=Group("C2xC214"); // GroupNames label
G:=SmallGroup(428,4);
// by ID
G=gap.SmallGroup(428,4);
# by ID
G:=PCGroup([3,-2,-2,-107]);
// Polycyclic
G:=Group<a,b|a^2=b^214=1,a*b=b*a>;
// generators/relations
Export