direct product, abelian, monomial, 3-elementary
Aliases: C3×C162, SmallGroup(486,83)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C162 |
| C1 — C3×C162 |
| C1 — C3×C162 |
Generators and relations for C3×C162
G = < a,b | a3=b162=1, ab=ba >
Smallest permutation representation of C3×C162
►Regular action on 486 points
Generators in S486
(1 358 282)(2 359 283)(3 360 284)(4 361 285)(5 362 286)(6 363 287)(7 364 288)(8 365 289)(9 366 290)(10 367 291)(11 368 292)(12 369 293)(13 370 294)(14 371 295)(15 372 296)(16 373 297)(17 374 298)(18 375 299)(19 376 300)(20 377 301)(21 378 302)(22 379 303)(23 380 304)(24 381 305)(25 382 306)(26 383 307)(27 384 308)(28 385 309)(29 386 310)(30 387 311)(31 388 312)(32 389 313)(33 390 314)(34 391 315)(35 392 316)(36 393 317)(37 394 318)(38 395 319)(39 396 320)(40 397 321)(41 398 322)(42 399 323)(43 400 324)(44 401 163)(45 402 164)(46 403 165)(47 404 166)(48 405 167)(49 406 168)(50 407 169)(51 408 170)(52 409 171)(53 410 172)(54 411 173)(55 412 174)(56 413 175)(57 414 176)(58 415 177)(59 416 178)(60 417 179)(61 418 180)(62 419 181)(63 420 182)(64 421 183)(65 422 184)(66 423 185)(67 424 186)(68 425 187)(69 426 188)(70 427 189)(71 428 190)(72 429 191)(73 430 192)(74 431 193)(75 432 194)(76 433 195)(77 434 196)(78 435 197)(79 436 198)(80 437 199)(81 438 200)(82 439 201)(83 440 202)(84 441 203)(85 442 204)(86 443 205)(87 444 206)(88 445 207)(89 446 208)(90 447 209)(91 448 210)(92 449 211)(93 450 212)(94 451 213)(95 452 214)(96 453 215)(97 454 216)(98 455 217)(99 456 218)(100 457 219)(101 458 220)(102 459 221)(103 460 222)(104 461 223)(105 462 224)(106 463 225)(107 464 226)(108 465 227)(109 466 228)(110 467 229)(111 468 230)(112 469 231)(113 470 232)(114 471 233)(115 472 234)(116 473 235)(117 474 236)(118 475 237)(119 476 238)(120 477 239)(121 478 240)(122 479 241)(123 480 242)(124 481 243)(125 482 244)(126 483 245)(127 484 246)(128 485 247)(129 486 248)(130 325 249)(131 326 250)(132 327 251)(133 328 252)(134 329 253)(135 330 254)(136 331 255)(137 332 256)(138 333 257)(139 334 258)(140 335 259)(141 336 260)(142 337 261)(143 338 262)(144 339 263)(145 340 264)(146 341 265)(147 342 266)(148 343 267)(149 344 268)(150 345 269)(151 346 270)(152 347 271)(153 348 272)(154 349 273)(155 350 274)(156 351 275)(157 352 276)(158 353 277)(159 354 278)(160 355 279)(161 356 280)(162 357 281)
(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)(325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486)
G:=sub<Sym(486)| (1,358,282)(2,359,283)(3,360,284)(4,361,285)(5,362,286)(6,363,287)(7,364,288)(8,365,289)(9,366,290)(10,367,291)(11,368,292)(12,369,293)(13,370,294)(14,371,295)(15,372,296)(16,373,297)(17,374,298)(18,375,299)(19,376,300)(20,377,301)(21,378,302)(22,379,303)(23,380,304)(24,381,305)(25,382,306)(26,383,307)(27,384,308)(28,385,309)(29,386,310)(30,387,311)(31,388,312)(32,389,313)(33,390,314)(34,391,315)(35,392,316)(36,393,317)(37,394,318)(38,395,319)(39,396,320)(40,397,321)(41,398,322)(42,399,323)(43,400,324)(44,401,163)(45,402,164)(46,403,165)(47,404,166)(48,405,167)(49,406,168)(50,407,169)(51,408,170)(52,409,171)(53,410,172)(54,411,173)(55,412,174)(56,413,175)(57,414,176)(58,415,177)(59,416,178)(60,417,179)(61,418,180)(62,419,181)(63,420,182)(64,421,183)(65,422,184)(66,423,185)(67,424,186)(68,425,187)(69,426,188)(70,427,189)(71,428,190)(72,429,191)(73,430,192)(74,431,193)(75,432,194)(76,433,195)(77,434,196)(78,435,197)(79,436,198)(80,437,199)(81,438,200)(82,439,201)(83,440,202)(84,441,203)(85,442,204)(86,443,205)(87,444,206)(88,445,207)(89,446,208)(90,447,209)(91,448,210)(92,449,211)(93,450,212)(94,451,213)(95,452,214)(96,453,215)(97,454,216)(98,455,217)(99,456,218)(100,457,219)(101,458,220)(102,459,221)(103,460,222)(104,461,223)(105,462,224)(106,463,225)(107,464,226)(108,465,227)(109,466,228)(110,467,229)(111,468,230)(112,469,231)(113,470,232)(114,471,233)(115,472,234)(116,473,235)(117,474,236)(118,475,237)(119,476,238)(120,477,239)(121,478,240)(122,479,241)(123,480,242)(124,481,243)(125,482,244)(126,483,245)(127,484,246)(128,485,247)(129,486,248)(130,325,249)(131,326,250)(132,327,251)(133,328,252)(134,329,253)(135,330,254)(136,331,255)(137,332,256)(138,333,257)(139,334,258)(140,335,259)(141,336,260)(142,337,261)(143,338,262)(144,339,263)(145,340,264)(146,341,265)(147,342,266)(148,343,267)(149,344,268)(150,345,269)(151,346,270)(152,347,271)(153,348,272)(154,349,273)(155,350,274)(156,351,275)(157,352,276)(158,353,277)(159,354,278)(160,355,279)(161,356,280)(162,357,281), (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)(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486)>;
G:=Group( (1,358,282)(2,359,283)(3,360,284)(4,361,285)(5,362,286)(6,363,287)(7,364,288)(8,365,289)(9,366,290)(10,367,291)(11,368,292)(12,369,293)(13,370,294)(14,371,295)(15,372,296)(16,373,297)(17,374,298)(18,375,299)(19,376,300)(20,377,301)(21,378,302)(22,379,303)(23,380,304)(24,381,305)(25,382,306)(26,383,307)(27,384,308)(28,385,309)(29,386,310)(30,387,311)(31,388,312)(32,389,313)(33,390,314)(34,391,315)(35,392,316)(36,393,317)(37,394,318)(38,395,319)(39,396,320)(40,397,321)(41,398,322)(42,399,323)(43,400,324)(44,401,163)(45,402,164)(46,403,165)(47,404,166)(48,405,167)(49,406,168)(50,407,169)(51,408,170)(52,409,171)(53,410,172)(54,411,173)(55,412,174)(56,413,175)(57,414,176)(58,415,177)(59,416,178)(60,417,179)(61,418,180)(62,419,181)(63,420,182)(64,421,183)(65,422,184)(66,423,185)(67,424,186)(68,425,187)(69,426,188)(70,427,189)(71,428,190)(72,429,191)(73,430,192)(74,431,193)(75,432,194)(76,433,195)(77,434,196)(78,435,197)(79,436,198)(80,437,199)(81,438,200)(82,439,201)(83,440,202)(84,441,203)(85,442,204)(86,443,205)(87,444,206)(88,445,207)(89,446,208)(90,447,209)(91,448,210)(92,449,211)(93,450,212)(94,451,213)(95,452,214)(96,453,215)(97,454,216)(98,455,217)(99,456,218)(100,457,219)(101,458,220)(102,459,221)(103,460,222)(104,461,223)(105,462,224)(106,463,225)(107,464,226)(108,465,227)(109,466,228)(110,467,229)(111,468,230)(112,469,231)(113,470,232)(114,471,233)(115,472,234)(116,473,235)(117,474,236)(118,475,237)(119,476,238)(120,477,239)(121,478,240)(122,479,241)(123,480,242)(124,481,243)(125,482,244)(126,483,245)(127,484,246)(128,485,247)(129,486,248)(130,325,249)(131,326,250)(132,327,251)(133,328,252)(134,329,253)(135,330,254)(136,331,255)(137,332,256)(138,333,257)(139,334,258)(140,335,259)(141,336,260)(142,337,261)(143,338,262)(144,339,263)(145,340,264)(146,341,265)(147,342,266)(148,343,267)(149,344,268)(150,345,269)(151,346,270)(152,347,271)(153,348,272)(154,349,273)(155,350,274)(156,351,275)(157,352,276)(158,353,277)(159,354,278)(160,355,279)(161,356,280)(162,357,281), (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)(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486) );
G=PermutationGroup([[(1,358,282),(2,359,283),(3,360,284),(4,361,285),(5,362,286),(6,363,287),(7,364,288),(8,365,289),(9,366,290),(10,367,291),(11,368,292),(12,369,293),(13,370,294),(14,371,295),(15,372,296),(16,373,297),(17,374,298),(18,375,299),(19,376,300),(20,377,301),(21,378,302),(22,379,303),(23,380,304),(24,381,305),(25,382,306),(26,383,307),(27,384,308),(28,385,309),(29,386,310),(30,387,311),(31,388,312),(32,389,313),(33,390,314),(34,391,315),(35,392,316),(36,393,317),(37,394,318),(38,395,319),(39,396,320),(40,397,321),(41,398,322),(42,399,323),(43,400,324),(44,401,163),(45,402,164),(46,403,165),(47,404,166),(48,405,167),(49,406,168),(50,407,169),(51,408,170),(52,409,171),(53,410,172),(54,411,173),(55,412,174),(56,413,175),(57,414,176),(58,415,177),(59,416,178),(60,417,179),(61,418,180),(62,419,181),(63,420,182),(64,421,183),(65,422,184),(66,423,185),(67,424,186),(68,425,187),(69,426,188),(70,427,189),(71,428,190),(72,429,191),(73,430,192),(74,431,193),(75,432,194),(76,433,195),(77,434,196),(78,435,197),(79,436,198),(80,437,199),(81,438,200),(82,439,201),(83,440,202),(84,441,203),(85,442,204),(86,443,205),(87,444,206),(88,445,207),(89,446,208),(90,447,209),(91,448,210),(92,449,211),(93,450,212),(94,451,213),(95,452,214),(96,453,215),(97,454,216),(98,455,217),(99,456,218),(100,457,219),(101,458,220),(102,459,221),(103,460,222),(104,461,223),(105,462,224),(106,463,225),(107,464,226),(108,465,227),(109,466,228),(110,467,229),(111,468,230),(112,469,231),(113,470,232),(114,471,233),(115,472,234),(116,473,235),(117,474,236),(118,475,237),(119,476,238),(120,477,239),(121,478,240),(122,479,241),(123,480,242),(124,481,243),(125,482,244),(126,483,245),(127,484,246),(128,485,247),(129,486,248),(130,325,249),(131,326,250),(132,327,251),(133,328,252),(134,329,253),(135,330,254),(136,331,255),(137,332,256),(138,333,257),(139,334,258),(140,335,259),(141,336,260),(142,337,261),(143,338,262),(144,339,263),(145,340,264),(146,341,265),(147,342,266),(148,343,267),(149,344,268),(150,345,269),(151,346,270),(152,347,271),(153,348,272),(154,349,273),(155,350,274),(156,351,275),(157,352,276),(158,353,277),(159,354,278),(160,355,279),(161,356,280),(162,357,281)], [(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),(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486)]])
486 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 9A | ··· | 9R | 18A | ··· | 18R | 27A | ··· | 27BB | 54A | ··· | 54BB | 81A | ··· | 81FF | 162A | ··· | 162FF |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 | 81 | ··· | 81 | 162 | ··· | 162 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
486 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C9 | C18 | C18 | C27 | C27 | C54 | C54 | C81 | C162 |
| kernel | C3×C162 | C3×C81 | C162 | C3×C54 | C81 | C3×C27 | C54 | C3×C18 | C27 | C3×C9 | C18 | C3×C6 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 12 | 6 | 12 | 6 | 36 | 18 | 36 | 18 | 162 | 162 |
Matrix representation of C3×C162 ►in GL2(𝔽163) generated by
| 104 | 0 |
| 0 | 58 |
| 125 | 0 |
| 0 | 42 |
G:=sub<GL(2,GF(163))| [104,0,0,58],[125,0,0,42] >;
C3×C162 in GAP, Magma, Sage, TeX
C_3\times C_{162} % in TeX
G:=Group("C3xC162"); // GroupNames label
G:=SmallGroup(486,83);
// by ID
G=gap.SmallGroup(486,83);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,93,118]);
// Polycyclic
G:=Group<a,b|a^3=b^162=1,a*b=b*a>;
// generators/relations
Export