direct product, abelian, monomial, 2-elementary
Aliases: C2×C212, SmallGroup(424,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C212 |
| C1 — C2×C212 |
| C1 — C2×C212 |
Generators and relations for C2×C212
G = < a,b | a2=b212=1, ab=ba >
Smallest permutation representation of C2×C212
►Regular action on 424 points
Generators in S424
(1 291)(2 292)(3 293)(4 294)(5 295)(6 296)(7 297)(8 298)(9 299)(10 300)(11 301)(12 302)(13 303)(14 304)(15 305)(16 306)(17 307)(18 308)(19 309)(20 310)(21 311)(22 312)(23 313)(24 314)(25 315)(26 316)(27 317)(28 318)(29 319)(30 320)(31 321)(32 322)(33 323)(34 324)(35 325)(36 326)(37 327)(38 328)(39 329)(40 330)(41 331)(42 332)(43 333)(44 334)(45 335)(46 336)(47 337)(48 338)(49 339)(50 340)(51 341)(52 342)(53 343)(54 344)(55 345)(56 346)(57 347)(58 348)(59 349)(60 350)(61 351)(62 352)(63 353)(64 354)(65 355)(66 356)(67 357)(68 358)(69 359)(70 360)(71 361)(72 362)(73 363)(74 364)(75 365)(76 366)(77 367)(78 368)(79 369)(80 370)(81 371)(82 372)(83 373)(84 374)(85 375)(86 376)(87 377)(88 378)(89 379)(90 380)(91 381)(92 382)(93 383)(94 384)(95 385)(96 386)(97 387)(98 388)(99 389)(100 390)(101 391)(102 392)(103 393)(104 394)(105 395)(106 396)(107 397)(108 398)(109 399)(110 400)(111 401)(112 402)(113 403)(114 404)(115 405)(116 406)(117 407)(118 408)(119 409)(120 410)(121 411)(122 412)(123 413)(124 414)(125 415)(126 416)(127 417)(128 418)(129 419)(130 420)(131 421)(132 422)(133 423)(134 424)(135 213)(136 214)(137 215)(138 216)(139 217)(140 218)(141 219)(142 220)(143 221)(144 222)(145 223)(146 224)(147 225)(148 226)(149 227)(150 228)(151 229)(152 230)(153 231)(154 232)(155 233)(156 234)(157 235)(158 236)(159 237)(160 238)(161 239)(162 240)(163 241)(164 242)(165 243)(166 244)(167 245)(168 246)(169 247)(170 248)(171 249)(172 250)(173 251)(174 252)(175 253)(176 254)(177 255)(178 256)(179 257)(180 258)(181 259)(182 260)(183 261)(184 262)(185 263)(186 264)(187 265)(188 266)(189 267)(190 268)(191 269)(192 270)(193 271)(194 272)(195 273)(196 274)(197 275)(198 276)(199 277)(200 278)(201 279)(202 280)(203 281)(204 282)(205 283)(206 284)(207 285)(208 286)(209 287)(210 288)(211 289)(212 290)
(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)
G:=sub<Sym(424)| (1,291)(2,292)(3,293)(4,294)(5,295)(6,296)(7,297)(8,298)(9,299)(10,300)(11,301)(12,302)(13,303)(14,304)(15,305)(16,306)(17,307)(18,308)(19,309)(20,310)(21,311)(22,312)(23,313)(24,314)(25,315)(26,316)(27,317)(28,318)(29,319)(30,320)(31,321)(32,322)(33,323)(34,324)(35,325)(36,326)(37,327)(38,328)(39,329)(40,330)(41,331)(42,332)(43,333)(44,334)(45,335)(46,336)(47,337)(48,338)(49,339)(50,340)(51,341)(52,342)(53,343)(54,344)(55,345)(56,346)(57,347)(58,348)(59,349)(60,350)(61,351)(62,352)(63,353)(64,354)(65,355)(66,356)(67,357)(68,358)(69,359)(70,360)(71,361)(72,362)(73,363)(74,364)(75,365)(76,366)(77,367)(78,368)(79,369)(80,370)(81,371)(82,372)(83,373)(84,374)(85,375)(86,376)(87,377)(88,378)(89,379)(90,380)(91,381)(92,382)(93,383)(94,384)(95,385)(96,386)(97,387)(98,388)(99,389)(100,390)(101,391)(102,392)(103,393)(104,394)(105,395)(106,396)(107,397)(108,398)(109,399)(110,400)(111,401)(112,402)(113,403)(114,404)(115,405)(116,406)(117,407)(118,408)(119,409)(120,410)(121,411)(122,412)(123,413)(124,414)(125,415)(126,416)(127,417)(128,418)(129,419)(130,420)(131,421)(132,422)(133,423)(134,424)(135,213)(136,214)(137,215)(138,216)(139,217)(140,218)(141,219)(142,220)(143,221)(144,222)(145,223)(146,224)(147,225)(148,226)(149,227)(150,228)(151,229)(152,230)(153,231)(154,232)(155,233)(156,234)(157,235)(158,236)(159,237)(160,238)(161,239)(162,240)(163,241)(164,242)(165,243)(166,244)(167,245)(168,246)(169,247)(170,248)(171,249)(172,250)(173,251)(174,252)(175,253)(176,254)(177,255)(178,256)(179,257)(180,258)(181,259)(182,260)(183,261)(184,262)(185,263)(186,264)(187,265)(188,266)(189,267)(190,268)(191,269)(192,270)(193,271)(194,272)(195,273)(196,274)(197,275)(198,276)(199,277)(200,278)(201,279)(202,280)(203,281)(204,282)(205,283)(206,284)(207,285)(208,286)(209,287)(210,288)(211,289)(212,290), (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)>;
G:=Group( (1,291)(2,292)(3,293)(4,294)(5,295)(6,296)(7,297)(8,298)(9,299)(10,300)(11,301)(12,302)(13,303)(14,304)(15,305)(16,306)(17,307)(18,308)(19,309)(20,310)(21,311)(22,312)(23,313)(24,314)(25,315)(26,316)(27,317)(28,318)(29,319)(30,320)(31,321)(32,322)(33,323)(34,324)(35,325)(36,326)(37,327)(38,328)(39,329)(40,330)(41,331)(42,332)(43,333)(44,334)(45,335)(46,336)(47,337)(48,338)(49,339)(50,340)(51,341)(52,342)(53,343)(54,344)(55,345)(56,346)(57,347)(58,348)(59,349)(60,350)(61,351)(62,352)(63,353)(64,354)(65,355)(66,356)(67,357)(68,358)(69,359)(70,360)(71,361)(72,362)(73,363)(74,364)(75,365)(76,366)(77,367)(78,368)(79,369)(80,370)(81,371)(82,372)(83,373)(84,374)(85,375)(86,376)(87,377)(88,378)(89,379)(90,380)(91,381)(92,382)(93,383)(94,384)(95,385)(96,386)(97,387)(98,388)(99,389)(100,390)(101,391)(102,392)(103,393)(104,394)(105,395)(106,396)(107,397)(108,398)(109,399)(110,400)(111,401)(112,402)(113,403)(114,404)(115,405)(116,406)(117,407)(118,408)(119,409)(120,410)(121,411)(122,412)(123,413)(124,414)(125,415)(126,416)(127,417)(128,418)(129,419)(130,420)(131,421)(132,422)(133,423)(134,424)(135,213)(136,214)(137,215)(138,216)(139,217)(140,218)(141,219)(142,220)(143,221)(144,222)(145,223)(146,224)(147,225)(148,226)(149,227)(150,228)(151,229)(152,230)(153,231)(154,232)(155,233)(156,234)(157,235)(158,236)(159,237)(160,238)(161,239)(162,240)(163,241)(164,242)(165,243)(166,244)(167,245)(168,246)(169,247)(170,248)(171,249)(172,250)(173,251)(174,252)(175,253)(176,254)(177,255)(178,256)(179,257)(180,258)(181,259)(182,260)(183,261)(184,262)(185,263)(186,264)(187,265)(188,266)(189,267)(190,268)(191,269)(192,270)(193,271)(194,272)(195,273)(196,274)(197,275)(198,276)(199,277)(200,278)(201,279)(202,280)(203,281)(204,282)(205,283)(206,284)(207,285)(208,286)(209,287)(210,288)(211,289)(212,290), (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) );
G=PermutationGroup([[(1,291),(2,292),(3,293),(4,294),(5,295),(6,296),(7,297),(8,298),(9,299),(10,300),(11,301),(12,302),(13,303),(14,304),(15,305),(16,306),(17,307),(18,308),(19,309),(20,310),(21,311),(22,312),(23,313),(24,314),(25,315),(26,316),(27,317),(28,318),(29,319),(30,320),(31,321),(32,322),(33,323),(34,324),(35,325),(36,326),(37,327),(38,328),(39,329),(40,330),(41,331),(42,332),(43,333),(44,334),(45,335),(46,336),(47,337),(48,338),(49,339),(50,340),(51,341),(52,342),(53,343),(54,344),(55,345),(56,346),(57,347),(58,348),(59,349),(60,350),(61,351),(62,352),(63,353),(64,354),(65,355),(66,356),(67,357),(68,358),(69,359),(70,360),(71,361),(72,362),(73,363),(74,364),(75,365),(76,366),(77,367),(78,368),(79,369),(80,370),(81,371),(82,372),(83,373),(84,374),(85,375),(86,376),(87,377),(88,378),(89,379),(90,380),(91,381),(92,382),(93,383),(94,384),(95,385),(96,386),(97,387),(98,388),(99,389),(100,390),(101,391),(102,392),(103,393),(104,394),(105,395),(106,396),(107,397),(108,398),(109,399),(110,400),(111,401),(112,402),(113,403),(114,404),(115,405),(116,406),(117,407),(118,408),(119,409),(120,410),(121,411),(122,412),(123,413),(124,414),(125,415),(126,416),(127,417),(128,418),(129,419),(130,420),(131,421),(132,422),(133,423),(134,424),(135,213),(136,214),(137,215),(138,216),(139,217),(140,218),(141,219),(142,220),(143,221),(144,222),(145,223),(146,224),(147,225),(148,226),(149,227),(150,228),(151,229),(152,230),(153,231),(154,232),(155,233),(156,234),(157,235),(158,236),(159,237),(160,238),(161,239),(162,240),(163,241),(164,242),(165,243),(166,244),(167,245),(168,246),(169,247),(170,248),(171,249),(172,250),(173,251),(174,252),(175,253),(176,254),(177,255),(178,256),(179,257),(180,258),(181,259),(182,260),(183,261),(184,262),(185,263),(186,264),(187,265),(188,266),(189,267),(190,268),(191,269),(192,270),(193,271),(194,272),(195,273),(196,274),(197,275),(198,276),(199,277),(200,278),(201,279),(202,280),(203,281),(204,282),(205,283),(206,284),(207,285),(208,286),(209,287),(210,288),(211,289),(212,290)], [(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)]])
424 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 53A | ··· | 53AZ | 106A | ··· | 106EZ | 212A | ··· | 212GZ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 53 | ··· | 53 | 106 | ··· | 106 | 212 | ··· | 212 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
424 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C53 | C106 | C106 | C212 |
| kernel | C2×C212 | C212 | C2×C106 | C106 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 52 | 104 | 52 | 208 |
Matrix representation of C2×C212 ►in GL2(𝔽1061) generated by
| 1 | 0 |
| 0 | 1060 |
| 203 | 0 |
| 0 | 658 |
G:=sub<GL(2,GF(1061))| [1,0,0,1060],[203,0,0,658] >;
C2×C212 in GAP, Magma, Sage, TeX
C_2\times C_{212} % in TeX
G:=Group("C2xC212"); // GroupNames label
G:=SmallGroup(424,9);
// by ID
G=gap.SmallGroup(424,9);
# by ID
G:=PCGroup([4,-2,-2,-53,-2,848]);
// Polycyclic
G:=Group<a,b|a^2=b^212=1,a*b=b*a>;
// generators/relations
Export