direct product, abelian, monomial, 2-elementary
Aliases: C2×C222, SmallGroup(444,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C222 |
| C1 — C2×C222 |
| C1 — C2×C222 |
Generators and relations for C2×C222
G = < a,b | a2=b222=1, ab=ba >
Smallest permutation representation of C2×C222
►Regular action on 444 points
Generators in S444
(1 271)(2 272)(3 273)(4 274)(5 275)(6 276)(7 277)(8 278)(9 279)(10 280)(11 281)(12 282)(13 283)(14 284)(15 285)(16 286)(17 287)(18 288)(19 289)(20 290)(21 291)(22 292)(23 293)(24 294)(25 295)(26 296)(27 297)(28 298)(29 299)(30 300)(31 301)(32 302)(33 303)(34 304)(35 305)(36 306)(37 307)(38 308)(39 309)(40 310)(41 311)(42 312)(43 313)(44 314)(45 315)(46 316)(47 317)(48 318)(49 319)(50 320)(51 321)(52 322)(53 323)(54 324)(55 325)(56 326)(57 327)(58 328)(59 329)(60 330)(61 331)(62 332)(63 333)(64 334)(65 335)(66 336)(67 337)(68 338)(69 339)(70 340)(71 341)(72 342)(73 343)(74 344)(75 345)(76 346)(77 347)(78 348)(79 349)(80 350)(81 351)(82 352)(83 353)(84 354)(85 355)(86 356)(87 357)(88 358)(89 359)(90 360)(91 361)(92 362)(93 363)(94 364)(95 365)(96 366)(97 367)(98 368)(99 369)(100 370)(101 371)(102 372)(103 373)(104 374)(105 375)(106 376)(107 377)(108 378)(109 379)(110 380)(111 381)(112 382)(113 383)(114 384)(115 385)(116 386)(117 387)(118 388)(119 389)(120 390)(121 391)(122 392)(123 393)(124 394)(125 395)(126 396)(127 397)(128 398)(129 399)(130 400)(131 401)(132 402)(133 403)(134 404)(135 405)(136 406)(137 407)(138 408)(139 409)(140 410)(141 411)(142 412)(143 413)(144 414)(145 415)(146 416)(147 417)(148 418)(149 419)(150 420)(151 421)(152 422)(153 423)(154 424)(155 425)(156 426)(157 427)(158 428)(159 429)(160 430)(161 431)(162 432)(163 433)(164 434)(165 435)(166 436)(167 437)(168 438)(169 439)(170 440)(171 441)(172 442)(173 443)(174 444)(175 223)(176 224)(177 225)(178 226)(179 227)(180 228)(181 229)(182 230)(183 231)(184 232)(185 233)(186 234)(187 235)(188 236)(189 237)(190 238)(191 239)(192 240)(193 241)(194 242)(195 243)(196 244)(197 245)(198 246)(199 247)(200 248)(201 249)(202 250)(203 251)(204 252)(205 253)(206 254)(207 255)(208 256)(209 257)(210 258)(211 259)(212 260)(213 261)(214 262)(215 263)(216 264)(217 265)(218 266)(219 267)(220 268)(221 269)(222 270)
(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)
G:=sub<Sym(444)| (1,271)(2,272)(3,273)(4,274)(5,275)(6,276)(7,277)(8,278)(9,279)(10,280)(11,281)(12,282)(13,283)(14,284)(15,285)(16,286)(17,287)(18,288)(19,289)(20,290)(21,291)(22,292)(23,293)(24,294)(25,295)(26,296)(27,297)(28,298)(29,299)(30,300)(31,301)(32,302)(33,303)(34,304)(35,305)(36,306)(37,307)(38,308)(39,309)(40,310)(41,311)(42,312)(43,313)(44,314)(45,315)(46,316)(47,317)(48,318)(49,319)(50,320)(51,321)(52,322)(53,323)(54,324)(55,325)(56,326)(57,327)(58,328)(59,329)(60,330)(61,331)(62,332)(63,333)(64,334)(65,335)(66,336)(67,337)(68,338)(69,339)(70,340)(71,341)(72,342)(73,343)(74,344)(75,345)(76,346)(77,347)(78,348)(79,349)(80,350)(81,351)(82,352)(83,353)(84,354)(85,355)(86,356)(87,357)(88,358)(89,359)(90,360)(91,361)(92,362)(93,363)(94,364)(95,365)(96,366)(97,367)(98,368)(99,369)(100,370)(101,371)(102,372)(103,373)(104,374)(105,375)(106,376)(107,377)(108,378)(109,379)(110,380)(111,381)(112,382)(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)(120,390)(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)(128,398)(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)(136,406)(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)(144,414)(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)(161,431)(162,432)(163,433)(164,434)(165,435)(166,436)(167,437)(168,438)(169,439)(170,440)(171,441)(172,442)(173,443)(174,444)(175,223)(176,224)(177,225)(178,226)(179,227)(180,228)(181,229)(182,230)(183,231)(184,232)(185,233)(186,234)(187,235)(188,236)(189,237)(190,238)(191,239)(192,240)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)(200,248)(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)(208,256)(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)(216,264)(217,265)(218,266)(219,267)(220,268)(221,269)(222,270), (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)>;
G:=Group( (1,271)(2,272)(3,273)(4,274)(5,275)(6,276)(7,277)(8,278)(9,279)(10,280)(11,281)(12,282)(13,283)(14,284)(15,285)(16,286)(17,287)(18,288)(19,289)(20,290)(21,291)(22,292)(23,293)(24,294)(25,295)(26,296)(27,297)(28,298)(29,299)(30,300)(31,301)(32,302)(33,303)(34,304)(35,305)(36,306)(37,307)(38,308)(39,309)(40,310)(41,311)(42,312)(43,313)(44,314)(45,315)(46,316)(47,317)(48,318)(49,319)(50,320)(51,321)(52,322)(53,323)(54,324)(55,325)(56,326)(57,327)(58,328)(59,329)(60,330)(61,331)(62,332)(63,333)(64,334)(65,335)(66,336)(67,337)(68,338)(69,339)(70,340)(71,341)(72,342)(73,343)(74,344)(75,345)(76,346)(77,347)(78,348)(79,349)(80,350)(81,351)(82,352)(83,353)(84,354)(85,355)(86,356)(87,357)(88,358)(89,359)(90,360)(91,361)(92,362)(93,363)(94,364)(95,365)(96,366)(97,367)(98,368)(99,369)(100,370)(101,371)(102,372)(103,373)(104,374)(105,375)(106,376)(107,377)(108,378)(109,379)(110,380)(111,381)(112,382)(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)(120,390)(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)(128,398)(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)(136,406)(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)(144,414)(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)(161,431)(162,432)(163,433)(164,434)(165,435)(166,436)(167,437)(168,438)(169,439)(170,440)(171,441)(172,442)(173,443)(174,444)(175,223)(176,224)(177,225)(178,226)(179,227)(180,228)(181,229)(182,230)(183,231)(184,232)(185,233)(186,234)(187,235)(188,236)(189,237)(190,238)(191,239)(192,240)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)(200,248)(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)(208,256)(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)(216,264)(217,265)(218,266)(219,267)(220,268)(221,269)(222,270), (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) );
G=PermutationGroup([[(1,271),(2,272),(3,273),(4,274),(5,275),(6,276),(7,277),(8,278),(9,279),(10,280),(11,281),(12,282),(13,283),(14,284),(15,285),(16,286),(17,287),(18,288),(19,289),(20,290),(21,291),(22,292),(23,293),(24,294),(25,295),(26,296),(27,297),(28,298),(29,299),(30,300),(31,301),(32,302),(33,303),(34,304),(35,305),(36,306),(37,307),(38,308),(39,309),(40,310),(41,311),(42,312),(43,313),(44,314),(45,315),(46,316),(47,317),(48,318),(49,319),(50,320),(51,321),(52,322),(53,323),(54,324),(55,325),(56,326),(57,327),(58,328),(59,329),(60,330),(61,331),(62,332),(63,333),(64,334),(65,335),(66,336),(67,337),(68,338),(69,339),(70,340),(71,341),(72,342),(73,343),(74,344),(75,345),(76,346),(77,347),(78,348),(79,349),(80,350),(81,351),(82,352),(83,353),(84,354),(85,355),(86,356),(87,357),(88,358),(89,359),(90,360),(91,361),(92,362),(93,363),(94,364),(95,365),(96,366),(97,367),(98,368),(99,369),(100,370),(101,371),(102,372),(103,373),(104,374),(105,375),(106,376),(107,377),(108,378),(109,379),(110,380),(111,381),(112,382),(113,383),(114,384),(115,385),(116,386),(117,387),(118,388),(119,389),(120,390),(121,391),(122,392),(123,393),(124,394),(125,395),(126,396),(127,397),(128,398),(129,399),(130,400),(131,401),(132,402),(133,403),(134,404),(135,405),(136,406),(137,407),(138,408),(139,409),(140,410),(141,411),(142,412),(143,413),(144,414),(145,415),(146,416),(147,417),(148,418),(149,419),(150,420),(151,421),(152,422),(153,423),(154,424),(155,425),(156,426),(157,427),(158,428),(159,429),(160,430),(161,431),(162,432),(163,433),(164,434),(165,435),(166,436),(167,437),(168,438),(169,439),(170,440),(171,441),(172,442),(173,443),(174,444),(175,223),(176,224),(177,225),(178,226),(179,227),(180,228),(181,229),(182,230),(183,231),(184,232),(185,233),(186,234),(187,235),(188,236),(189,237),(190,238),(191,239),(192,240),(193,241),(194,242),(195,243),(196,244),(197,245),(198,246),(199,247),(200,248),(201,249),(202,250),(203,251),(204,252),(205,253),(206,254),(207,255),(208,256),(209,257),(210,258),(211,259),(212,260),(213,261),(214,262),(215,263),(216,264),(217,265),(218,266),(219,267),(220,268),(221,269),(222,270)], [(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)]])
444 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 37A | ··· | 37AJ | 74A | ··· | 74DD | 111A | ··· | 111BT | 222A | ··· | 222HH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 37 | ··· | 37 | 74 | ··· | 74 | 111 | ··· | 111 | 222 | ··· | 222 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
444 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C37 | C74 | C111 | C222 |
| kernel | C2×C222 | C222 | C2×C74 | C74 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 36 | 108 | 72 | 216 |
Matrix representation of C2×C222 ►in GL2(𝔽223) generated by
| 1 | 0 |
| 0 | 222 |
| 151 | 0 |
| 0 | 198 |
G:=sub<GL(2,GF(223))| [1,0,0,222],[151,0,0,198] >;
C2×C222 in GAP, Magma, Sage, TeX
C_2\times C_{222} % in TeX
G:=Group("C2xC222"); // GroupNames label
G:=SmallGroup(444,18);
// by ID
G=gap.SmallGroup(444,18);
# by ID
G:=PCGroup([4,-2,-2,-3,-37]);
// Polycyclic
G:=Group<a,b|a^2=b^222=1,a*b=b*a>;
// generators/relations
Export