direct product, abelian, monomial, 2-elementary
Aliases: C2×C220, SmallGroup(440,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C220 |
| C1 — C2×C220 |
| C1 — C2×C220 |
Generators and relations for C2×C220
G = < a,b | a2=b220=1, ab=ba >
Smallest permutation representation of C2×C220
►Regular action on 440 points
Generators in S440
(1 407)(2 408)(3 409)(4 410)(5 411)(6 412)(7 413)(8 414)(9 415)(10 416)(11 417)(12 418)(13 419)(14 420)(15 421)(16 422)(17 423)(18 424)(19 425)(20 426)(21 427)(22 428)(23 429)(24 430)(25 431)(26 432)(27 433)(28 434)(29 435)(30 436)(31 437)(32 438)(33 439)(34 440)(35 221)(36 222)(37 223)(38 224)(39 225)(40 226)(41 227)(42 228)(43 229)(44 230)(45 231)(46 232)(47 233)(48 234)(49 235)(50 236)(51 237)(52 238)(53 239)(54 240)(55 241)(56 242)(57 243)(58 244)(59 245)(60 246)(61 247)(62 248)(63 249)(64 250)(65 251)(66 252)(67 253)(68 254)(69 255)(70 256)(71 257)(72 258)(73 259)(74 260)(75 261)(76 262)(77 263)(78 264)(79 265)(80 266)(81 267)(82 268)(83 269)(84 270)(85 271)(86 272)(87 273)(88 274)(89 275)(90 276)(91 277)(92 278)(93 279)(94 280)(95 281)(96 282)(97 283)(98 284)(99 285)(100 286)(101 287)(102 288)(103 289)(104 290)(105 291)(106 292)(107 293)(108 294)(109 295)(110 296)(111 297)(112 298)(113 299)(114 300)(115 301)(116 302)(117 303)(118 304)(119 305)(120 306)(121 307)(122 308)(123 309)(124 310)(125 311)(126 312)(127 313)(128 314)(129 315)(130 316)(131 317)(132 318)(133 319)(134 320)(135 321)(136 322)(137 323)(138 324)(139 325)(140 326)(141 327)(142 328)(143 329)(144 330)(145 331)(146 332)(147 333)(148 334)(149 335)(150 336)(151 337)(152 338)(153 339)(154 340)(155 341)(156 342)(157 343)(158 344)(159 345)(160 346)(161 347)(162 348)(163 349)(164 350)(165 351)(166 352)(167 353)(168 354)(169 355)(170 356)(171 357)(172 358)(173 359)(174 360)(175 361)(176 362)(177 363)(178 364)(179 365)(180 366)(181 367)(182 368)(183 369)(184 370)(185 371)(186 372)(187 373)(188 374)(189 375)(190 376)(191 377)(192 378)(193 379)(194 380)(195 381)(196 382)(197 383)(198 384)(199 385)(200 386)(201 387)(202 388)(203 389)(204 390)(205 391)(206 392)(207 393)(208 394)(209 395)(210 396)(211 397)(212 398)(213 399)(214 400)(215 401)(216 402)(217 403)(218 404)(219 405)(220 406)
(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)
G:=sub<Sym(440)| (1,407)(2,408)(3,409)(4,410)(5,411)(6,412)(7,413)(8,414)(9,415)(10,416)(11,417)(12,418)(13,419)(14,420)(15,421)(16,422)(17,423)(18,424)(19,425)(20,426)(21,427)(22,428)(23,429)(24,430)(25,431)(26,432)(27,433)(28,434)(29,435)(30,436)(31,437)(32,438)(33,439)(34,440)(35,221)(36,222)(37,223)(38,224)(39,225)(40,226)(41,227)(42,228)(43,229)(44,230)(45,231)(46,232)(47,233)(48,234)(49,235)(50,236)(51,237)(52,238)(53,239)(54,240)(55,241)(56,242)(57,243)(58,244)(59,245)(60,246)(61,247)(62,248)(63,249)(64,250)(65,251)(66,252)(67,253)(68,254)(69,255)(70,256)(71,257)(72,258)(73,259)(74,260)(75,261)(76,262)(77,263)(78,264)(79,265)(80,266)(81,267)(82,268)(83,269)(84,270)(85,271)(86,272)(87,273)(88,274)(89,275)(90,276)(91,277)(92,278)(93,279)(94,280)(95,281)(96,282)(97,283)(98,284)(99,285)(100,286)(101,287)(102,288)(103,289)(104,290)(105,291)(106,292)(107,293)(108,294)(109,295)(110,296)(111,297)(112,298)(113,299)(114,300)(115,301)(116,302)(117,303)(118,304)(119,305)(120,306)(121,307)(122,308)(123,309)(124,310)(125,311)(126,312)(127,313)(128,314)(129,315)(130,316)(131,317)(132,318)(133,319)(134,320)(135,321)(136,322)(137,323)(138,324)(139,325)(140,326)(141,327)(142,328)(143,329)(144,330)(145,331)(146,332)(147,333)(148,334)(149,335)(150,336)(151,337)(152,338)(153,339)(154,340)(155,341)(156,342)(157,343)(158,344)(159,345)(160,346)(161,347)(162,348)(163,349)(164,350)(165,351)(166,352)(167,353)(168,354)(169,355)(170,356)(171,357)(172,358)(173,359)(174,360)(175,361)(176,362)(177,363)(178,364)(179,365)(180,366)(181,367)(182,368)(183,369)(184,370)(185,371)(186,372)(187,373)(188,374)(189,375)(190,376)(191,377)(192,378)(193,379)(194,380)(195,381)(196,382)(197,383)(198,384)(199,385)(200,386)(201,387)(202,388)(203,389)(204,390)(205,391)(206,392)(207,393)(208,394)(209,395)(210,396)(211,397)(212,398)(213,399)(214,400)(215,401)(216,402)(217,403)(218,404)(219,405)(220,406), (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)>;
G:=Group( (1,407)(2,408)(3,409)(4,410)(5,411)(6,412)(7,413)(8,414)(9,415)(10,416)(11,417)(12,418)(13,419)(14,420)(15,421)(16,422)(17,423)(18,424)(19,425)(20,426)(21,427)(22,428)(23,429)(24,430)(25,431)(26,432)(27,433)(28,434)(29,435)(30,436)(31,437)(32,438)(33,439)(34,440)(35,221)(36,222)(37,223)(38,224)(39,225)(40,226)(41,227)(42,228)(43,229)(44,230)(45,231)(46,232)(47,233)(48,234)(49,235)(50,236)(51,237)(52,238)(53,239)(54,240)(55,241)(56,242)(57,243)(58,244)(59,245)(60,246)(61,247)(62,248)(63,249)(64,250)(65,251)(66,252)(67,253)(68,254)(69,255)(70,256)(71,257)(72,258)(73,259)(74,260)(75,261)(76,262)(77,263)(78,264)(79,265)(80,266)(81,267)(82,268)(83,269)(84,270)(85,271)(86,272)(87,273)(88,274)(89,275)(90,276)(91,277)(92,278)(93,279)(94,280)(95,281)(96,282)(97,283)(98,284)(99,285)(100,286)(101,287)(102,288)(103,289)(104,290)(105,291)(106,292)(107,293)(108,294)(109,295)(110,296)(111,297)(112,298)(113,299)(114,300)(115,301)(116,302)(117,303)(118,304)(119,305)(120,306)(121,307)(122,308)(123,309)(124,310)(125,311)(126,312)(127,313)(128,314)(129,315)(130,316)(131,317)(132,318)(133,319)(134,320)(135,321)(136,322)(137,323)(138,324)(139,325)(140,326)(141,327)(142,328)(143,329)(144,330)(145,331)(146,332)(147,333)(148,334)(149,335)(150,336)(151,337)(152,338)(153,339)(154,340)(155,341)(156,342)(157,343)(158,344)(159,345)(160,346)(161,347)(162,348)(163,349)(164,350)(165,351)(166,352)(167,353)(168,354)(169,355)(170,356)(171,357)(172,358)(173,359)(174,360)(175,361)(176,362)(177,363)(178,364)(179,365)(180,366)(181,367)(182,368)(183,369)(184,370)(185,371)(186,372)(187,373)(188,374)(189,375)(190,376)(191,377)(192,378)(193,379)(194,380)(195,381)(196,382)(197,383)(198,384)(199,385)(200,386)(201,387)(202,388)(203,389)(204,390)(205,391)(206,392)(207,393)(208,394)(209,395)(210,396)(211,397)(212,398)(213,399)(214,400)(215,401)(216,402)(217,403)(218,404)(219,405)(220,406), (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) );
G=PermutationGroup([[(1,407),(2,408),(3,409),(4,410),(5,411),(6,412),(7,413),(8,414),(9,415),(10,416),(11,417),(12,418),(13,419),(14,420),(15,421),(16,422),(17,423),(18,424),(19,425),(20,426),(21,427),(22,428),(23,429),(24,430),(25,431),(26,432),(27,433),(28,434),(29,435),(30,436),(31,437),(32,438),(33,439),(34,440),(35,221),(36,222),(37,223),(38,224),(39,225),(40,226),(41,227),(42,228),(43,229),(44,230),(45,231),(46,232),(47,233),(48,234),(49,235),(50,236),(51,237),(52,238),(53,239),(54,240),(55,241),(56,242),(57,243),(58,244),(59,245),(60,246),(61,247),(62,248),(63,249),(64,250),(65,251),(66,252),(67,253),(68,254),(69,255),(70,256),(71,257),(72,258),(73,259),(74,260),(75,261),(76,262),(77,263),(78,264),(79,265),(80,266),(81,267),(82,268),(83,269),(84,270),(85,271),(86,272),(87,273),(88,274),(89,275),(90,276),(91,277),(92,278),(93,279),(94,280),(95,281),(96,282),(97,283),(98,284),(99,285),(100,286),(101,287),(102,288),(103,289),(104,290),(105,291),(106,292),(107,293),(108,294),(109,295),(110,296),(111,297),(112,298),(113,299),(114,300),(115,301),(116,302),(117,303),(118,304),(119,305),(120,306),(121,307),(122,308),(123,309),(124,310),(125,311),(126,312),(127,313),(128,314),(129,315),(130,316),(131,317),(132,318),(133,319),(134,320),(135,321),(136,322),(137,323),(138,324),(139,325),(140,326),(141,327),(142,328),(143,329),(144,330),(145,331),(146,332),(147,333),(148,334),(149,335),(150,336),(151,337),(152,338),(153,339),(154,340),(155,341),(156,342),(157,343),(158,344),(159,345),(160,346),(161,347),(162,348),(163,349),(164,350),(165,351),(166,352),(167,353),(168,354),(169,355),(170,356),(171,357),(172,358),(173,359),(174,360),(175,361),(176,362),(177,363),(178,364),(179,365),(180,366),(181,367),(182,368),(183,369),(184,370),(185,371),(186,372),(187,373),(188,374),(189,375),(190,376),(191,377),(192,378),(193,379),(194,380),(195,381),(196,382),(197,383),(198,384),(199,385),(200,386),(201,387),(202,388),(203,389),(204,390),(205,391),(206,392),(207,393),(208,394),(209,395),(210,396),(211,397),(212,398),(213,399),(214,400),(215,401),(216,402),(217,403),(218,404),(219,405),(220,406)], [(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)]])
440 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 11A | ··· | 11J | 20A | ··· | 20P | 22A | ··· | 22AD | 44A | ··· | 44AN | 55A | ··· | 55AN | 110A | ··· | 110DP | 220A | ··· | 220FD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 11 | ··· | 11 | 20 | ··· | 20 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| 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 |
440 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C11 | C20 | C22 | C22 | C44 | C55 | C110 | C110 | C220 |
| kernel | C2×C220 | C220 | C2×C110 | C110 | C2×C44 | C44 | C2×C22 | C2×C20 | C22 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 10 | 16 | 20 | 10 | 40 | 40 | 80 | 40 | 160 |
Matrix representation of C2×C220 ►in GL2(𝔽661) generated by
| 1 | 0 |
| 0 | 660 |
| 603 | 0 |
| 0 | 201 |
G:=sub<GL(2,GF(661))| [1,0,0,660],[603,0,0,201] >;
C2×C220 in GAP, Magma, Sage, TeX
C_2\times C_{220} % in TeX
G:=Group("C2xC220"); // GroupNames label
G:=SmallGroup(440,39);
// by ID
G=gap.SmallGroup(440,39);
# by ID
G:=PCGroup([5,-2,-2,-5,-11,-2,1100]);
// Polycyclic
G:=Group<a,b|a^2=b^220=1,a*b=b*a>;
// generators/relations
Export