direct product, abelian, monomial, 2-elementary
Aliases: C2×C210, SmallGroup(420,41)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C210 |
| C1 — C2×C210 |
| C1 — C2×C210 |
Generators and relations for C2×C210
G = < a,b | a2=b210=1, ab=ba >
Smallest permutation representation of C2×C210
►Regular action on 420 points
Generators in S420
(1 276)(2 277)(3 278)(4 279)(5 280)(6 281)(7 282)(8 283)(9 284)(10 285)(11 286)(12 287)(13 288)(14 289)(15 290)(16 291)(17 292)(18 293)(19 294)(20 295)(21 296)(22 297)(23 298)(24 299)(25 300)(26 301)(27 302)(28 303)(29 304)(30 305)(31 306)(32 307)(33 308)(34 309)(35 310)(36 311)(37 312)(38 313)(39 314)(40 315)(41 316)(42 317)(43 318)(44 319)(45 320)(46 321)(47 322)(48 323)(49 324)(50 325)(51 326)(52 327)(53 328)(54 329)(55 330)(56 331)(57 332)(58 333)(59 334)(60 335)(61 336)(62 337)(63 338)(64 339)(65 340)(66 341)(67 342)(68 343)(69 344)(70 345)(71 346)(72 347)(73 348)(74 349)(75 350)(76 351)(77 352)(78 353)(79 354)(80 355)(81 356)(82 357)(83 358)(84 359)(85 360)(86 361)(87 362)(88 363)(89 364)(90 365)(91 366)(92 367)(93 368)(94 369)(95 370)(96 371)(97 372)(98 373)(99 374)(100 375)(101 376)(102 377)(103 378)(104 379)(105 380)(106 381)(107 382)(108 383)(109 384)(110 385)(111 386)(112 387)(113 388)(114 389)(115 390)(116 391)(117 392)(118 393)(119 394)(120 395)(121 396)(122 397)(123 398)(124 399)(125 400)(126 401)(127 402)(128 403)(129 404)(130 405)(131 406)(132 407)(133 408)(134 409)(135 410)(136 411)(137 412)(138 413)(139 414)(140 415)(141 416)(142 417)(143 418)(144 419)(145 420)(146 211)(147 212)(148 213)(149 214)(150 215)(151 216)(152 217)(153 218)(154 219)(155 220)(156 221)(157 222)(158 223)(159 224)(160 225)(161 226)(162 227)(163 228)(164 229)(165 230)(166 231)(167 232)(168 233)(169 234)(170 235)(171 236)(172 237)(173 238)(174 239)(175 240)(176 241)(177 242)(178 243)(179 244)(180 245)(181 246)(182 247)(183 248)(184 249)(185 250)(186 251)(187 252)(188 253)(189 254)(190 255)(191 256)(192 257)(193 258)(194 259)(195 260)(196 261)(197 262)(198 263)(199 264)(200 265)(201 266)(202 267)(203 268)(204 269)(205 270)(206 271)(207 272)(208 273)(209 274)(210 275)
(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)
G:=sub<Sym(420)| (1,276)(2,277)(3,278)(4,279)(5,280)(6,281)(7,282)(8,283)(9,284)(10,285)(11,286)(12,287)(13,288)(14,289)(15,290)(16,291)(17,292)(18,293)(19,294)(20,295)(21,296)(22,297)(23,298)(24,299)(25,300)(26,301)(27,302)(28,303)(29,304)(30,305)(31,306)(32,307)(33,308)(34,309)(35,310)(36,311)(37,312)(38,313)(39,314)(40,315)(41,316)(42,317)(43,318)(44,319)(45,320)(46,321)(47,322)(48,323)(49,324)(50,325)(51,326)(52,327)(53,328)(54,329)(55,330)(56,331)(57,332)(58,333)(59,334)(60,335)(61,336)(62,337)(63,338)(64,339)(65,340)(66,341)(67,342)(68,343)(69,344)(70,345)(71,346)(72,347)(73,348)(74,349)(75,350)(76,351)(77,352)(78,353)(79,354)(80,355)(81,356)(82,357)(83,358)(84,359)(85,360)(86,361)(87,362)(88,363)(89,364)(90,365)(91,366)(92,367)(93,368)(94,369)(95,370)(96,371)(97,372)(98,373)(99,374)(100,375)(101,376)(102,377)(103,378)(104,379)(105,380)(106,381)(107,382)(108,383)(109,384)(110,385)(111,386)(112,387)(113,388)(114,389)(115,390)(116,391)(117,392)(118,393)(119,394)(120,395)(121,396)(122,397)(123,398)(124,399)(125,400)(126,401)(127,402)(128,403)(129,404)(130,405)(131,406)(132,407)(133,408)(134,409)(135,410)(136,411)(137,412)(138,413)(139,414)(140,415)(141,416)(142,417)(143,418)(144,419)(145,420)(146,211)(147,212)(148,213)(149,214)(150,215)(151,216)(152,217)(153,218)(154,219)(155,220)(156,221)(157,222)(158,223)(159,224)(160,225)(161,226)(162,227)(163,228)(164,229)(165,230)(166,231)(167,232)(168,233)(169,234)(170,235)(171,236)(172,237)(173,238)(174,239)(175,240)(176,241)(177,242)(178,243)(179,244)(180,245)(181,246)(182,247)(183,248)(184,249)(185,250)(186,251)(187,252)(188,253)(189,254)(190,255)(191,256)(192,257)(193,258)(194,259)(195,260)(196,261)(197,262)(198,263)(199,264)(200,265)(201,266)(202,267)(203,268)(204,269)(205,270)(206,271)(207,272)(208,273)(209,274)(210,275), (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)>;
G:=Group( (1,276)(2,277)(3,278)(4,279)(5,280)(6,281)(7,282)(8,283)(9,284)(10,285)(11,286)(12,287)(13,288)(14,289)(15,290)(16,291)(17,292)(18,293)(19,294)(20,295)(21,296)(22,297)(23,298)(24,299)(25,300)(26,301)(27,302)(28,303)(29,304)(30,305)(31,306)(32,307)(33,308)(34,309)(35,310)(36,311)(37,312)(38,313)(39,314)(40,315)(41,316)(42,317)(43,318)(44,319)(45,320)(46,321)(47,322)(48,323)(49,324)(50,325)(51,326)(52,327)(53,328)(54,329)(55,330)(56,331)(57,332)(58,333)(59,334)(60,335)(61,336)(62,337)(63,338)(64,339)(65,340)(66,341)(67,342)(68,343)(69,344)(70,345)(71,346)(72,347)(73,348)(74,349)(75,350)(76,351)(77,352)(78,353)(79,354)(80,355)(81,356)(82,357)(83,358)(84,359)(85,360)(86,361)(87,362)(88,363)(89,364)(90,365)(91,366)(92,367)(93,368)(94,369)(95,370)(96,371)(97,372)(98,373)(99,374)(100,375)(101,376)(102,377)(103,378)(104,379)(105,380)(106,381)(107,382)(108,383)(109,384)(110,385)(111,386)(112,387)(113,388)(114,389)(115,390)(116,391)(117,392)(118,393)(119,394)(120,395)(121,396)(122,397)(123,398)(124,399)(125,400)(126,401)(127,402)(128,403)(129,404)(130,405)(131,406)(132,407)(133,408)(134,409)(135,410)(136,411)(137,412)(138,413)(139,414)(140,415)(141,416)(142,417)(143,418)(144,419)(145,420)(146,211)(147,212)(148,213)(149,214)(150,215)(151,216)(152,217)(153,218)(154,219)(155,220)(156,221)(157,222)(158,223)(159,224)(160,225)(161,226)(162,227)(163,228)(164,229)(165,230)(166,231)(167,232)(168,233)(169,234)(170,235)(171,236)(172,237)(173,238)(174,239)(175,240)(176,241)(177,242)(178,243)(179,244)(180,245)(181,246)(182,247)(183,248)(184,249)(185,250)(186,251)(187,252)(188,253)(189,254)(190,255)(191,256)(192,257)(193,258)(194,259)(195,260)(196,261)(197,262)(198,263)(199,264)(200,265)(201,266)(202,267)(203,268)(204,269)(205,270)(206,271)(207,272)(208,273)(209,274)(210,275), (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) );
G=PermutationGroup([[(1,276),(2,277),(3,278),(4,279),(5,280),(6,281),(7,282),(8,283),(9,284),(10,285),(11,286),(12,287),(13,288),(14,289),(15,290),(16,291),(17,292),(18,293),(19,294),(20,295),(21,296),(22,297),(23,298),(24,299),(25,300),(26,301),(27,302),(28,303),(29,304),(30,305),(31,306),(32,307),(33,308),(34,309),(35,310),(36,311),(37,312),(38,313),(39,314),(40,315),(41,316),(42,317),(43,318),(44,319),(45,320),(46,321),(47,322),(48,323),(49,324),(50,325),(51,326),(52,327),(53,328),(54,329),(55,330),(56,331),(57,332),(58,333),(59,334),(60,335),(61,336),(62,337),(63,338),(64,339),(65,340),(66,341),(67,342),(68,343),(69,344),(70,345),(71,346),(72,347),(73,348),(74,349),(75,350),(76,351),(77,352),(78,353),(79,354),(80,355),(81,356),(82,357),(83,358),(84,359),(85,360),(86,361),(87,362),(88,363),(89,364),(90,365),(91,366),(92,367),(93,368),(94,369),(95,370),(96,371),(97,372),(98,373),(99,374),(100,375),(101,376),(102,377),(103,378),(104,379),(105,380),(106,381),(107,382),(108,383),(109,384),(110,385),(111,386),(112,387),(113,388),(114,389),(115,390),(116,391),(117,392),(118,393),(119,394),(120,395),(121,396),(122,397),(123,398),(124,399),(125,400),(126,401),(127,402),(128,403),(129,404),(130,405),(131,406),(132,407),(133,408),(134,409),(135,410),(136,411),(137,412),(138,413),(139,414),(140,415),(141,416),(142,417),(143,418),(144,419),(145,420),(146,211),(147,212),(148,213),(149,214),(150,215),(151,216),(152,217),(153,218),(154,219),(155,220),(156,221),(157,222),(158,223),(159,224),(160,225),(161,226),(162,227),(163,228),(164,229),(165,230),(166,231),(167,232),(168,233),(169,234),(170,235),(171,236),(172,237),(173,238),(174,239),(175,240),(176,241),(177,242),(178,243),(179,244),(180,245),(181,246),(182,247),(183,248),(184,249),(185,250),(186,251),(187,252),(188,253),(189,254),(190,255),(191,256),(192,257),(193,258),(194,259),(195,260),(196,261),(197,262),(198,263),(199,264),(200,265),(201,266),(202,267),(203,268),(204,269),(205,270),(206,271),(207,272),(208,273),(209,274),(210,275)], [(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)]])
420 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 7A | ··· | 7F | 10A | ··· | 10L | 14A | ··· | 14R | 15A | ··· | 15H | 21A | ··· | 21L | 30A | ··· | 30X | 35A | ··· | 35X | 42A | ··· | 42AJ | 70A | ··· | 70BT | 105A | ··· | 105AV | 210A | ··· | 210EN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 7 | ··· | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 15 | ··· | 15 | 21 | ··· | 21 | 30 | ··· | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 70 | ··· | 70 | 105 | ··· | 105 | 210 | ··· | 210 |
| 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 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
420 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C7 | C10 | C14 | C15 | C21 | C30 | C35 | C42 | C70 | C105 | C210 |
| kernel | C2×C210 | C210 | C2×C70 | C2×C42 | C70 | C2×C30 | C42 | C30 | C2×C14 | C2×C10 | C14 | C2×C6 | C10 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 4 | 6 | 6 | 12 | 18 | 8 | 12 | 24 | 24 | 36 | 72 | 48 | 144 |
Matrix representation of C2×C210 ►in GL3(𝔽211) generated by
| 1 | 0 | 0 |
| 0 | 210 | 0 |
| 0 | 0 | 1 |
| 42 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 201 |
G:=sub<GL(3,GF(211))| [1,0,0,0,210,0,0,0,1],[42,0,0,0,8,0,0,0,201] >;
C2×C210 in GAP, Magma, Sage, TeX
C_2\times C_{210} % in TeX
G:=Group("C2xC210"); // GroupNames label
G:=SmallGroup(420,41);
// by ID
G=gap.SmallGroup(420,41);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7]);
// Polycyclic
G:=Group<a,b|a^2=b^210=1,a*b=b*a>;
// generators/relations
Export