direct product, abelian, monomial, 3-elementary
Aliases: C3×C159, SmallGroup(477,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C159 |
| C1 — C3×C159 |
| C1 — C3×C159 |
Generators and relations for C3×C159
G = < a,b | a3=b159=1, ab=ba >
Smallest permutation representation of C3×C159
►Regular action on 477 points
Generators in S477
(1 462 243)(2 463 244)(3 464 245)(4 465 246)(5 466 247)(6 467 248)(7 468 249)(8 469 250)(9 470 251)(10 471 252)(11 472 253)(12 473 254)(13 474 255)(14 475 256)(15 476 257)(16 477 258)(17 319 259)(18 320 260)(19 321 261)(20 322 262)(21 323 263)(22 324 264)(23 325 265)(24 326 266)(25 327 267)(26 328 268)(27 329 269)(28 330 270)(29 331 271)(30 332 272)(31 333 273)(32 334 274)(33 335 275)(34 336 276)(35 337 277)(36 338 278)(37 339 279)(38 340 280)(39 341 281)(40 342 282)(41 343 283)(42 344 284)(43 345 285)(44 346 286)(45 347 287)(46 348 288)(47 349 289)(48 350 290)(49 351 291)(50 352 292)(51 353 293)(52 354 294)(53 355 295)(54 356 296)(55 357 297)(56 358 298)(57 359 299)(58 360 300)(59 361 301)(60 362 302)(61 363 303)(62 364 304)(63 365 305)(64 366 306)(65 367 307)(66 368 308)(67 369 309)(68 370 310)(69 371 311)(70 372 312)(71 373 313)(72 374 314)(73 375 315)(74 376 316)(75 377 317)(76 378 318)(77 379 160)(78 380 161)(79 381 162)(80 382 163)(81 383 164)(82 384 165)(83 385 166)(84 386 167)(85 387 168)(86 388 169)(87 389 170)(88 390 171)(89 391 172)(90 392 173)(91 393 174)(92 394 175)(93 395 176)(94 396 177)(95 397 178)(96 398 179)(97 399 180)(98 400 181)(99 401 182)(100 402 183)(101 403 184)(102 404 185)(103 405 186)(104 406 187)(105 407 188)(106 408 189)(107 409 190)(108 410 191)(109 411 192)(110 412 193)(111 413 194)(112 414 195)(113 415 196)(114 416 197)(115 417 198)(116 418 199)(117 419 200)(118 420 201)(119 421 202)(120 422 203)(121 423 204)(122 424 205)(123 425 206)(124 426 207)(125 427 208)(126 428 209)(127 429 210)(128 430 211)(129 431 212)(130 432 213)(131 433 214)(132 434 215)(133 435 216)(134 436 217)(135 437 218)(136 438 219)(137 439 220)(138 440 221)(139 441 222)(140 442 223)(141 443 224)(142 444 225)(143 445 226)(144 446 227)(145 447 228)(146 448 229)(147 449 230)(148 450 231)(149 451 232)(150 452 233)(151 453 234)(152 454 235)(153 455 236)(154 456 237)(155 457 238)(156 458 239)(157 459 240)(158 460 241)(159 461 242)
(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)
G:=sub<Sym(477)| (1,462,243)(2,463,244)(3,464,245)(4,465,246)(5,466,247)(6,467,248)(7,468,249)(8,469,250)(9,470,251)(10,471,252)(11,472,253)(12,473,254)(13,474,255)(14,475,256)(15,476,257)(16,477,258)(17,319,259)(18,320,260)(19,321,261)(20,322,262)(21,323,263)(22,324,264)(23,325,265)(24,326,266)(25,327,267)(26,328,268)(27,329,269)(28,330,270)(29,331,271)(30,332,272)(31,333,273)(32,334,274)(33,335,275)(34,336,276)(35,337,277)(36,338,278)(37,339,279)(38,340,280)(39,341,281)(40,342,282)(41,343,283)(42,344,284)(43,345,285)(44,346,286)(45,347,287)(46,348,288)(47,349,289)(48,350,290)(49,351,291)(50,352,292)(51,353,293)(52,354,294)(53,355,295)(54,356,296)(55,357,297)(56,358,298)(57,359,299)(58,360,300)(59,361,301)(60,362,302)(61,363,303)(62,364,304)(63,365,305)(64,366,306)(65,367,307)(66,368,308)(67,369,309)(68,370,310)(69,371,311)(70,372,312)(71,373,313)(72,374,314)(73,375,315)(74,376,316)(75,377,317)(76,378,318)(77,379,160)(78,380,161)(79,381,162)(80,382,163)(81,383,164)(82,384,165)(83,385,166)(84,386,167)(85,387,168)(86,388,169)(87,389,170)(88,390,171)(89,391,172)(90,392,173)(91,393,174)(92,394,175)(93,395,176)(94,396,177)(95,397,178)(96,398,179)(97,399,180)(98,400,181)(99,401,182)(100,402,183)(101,403,184)(102,404,185)(103,405,186)(104,406,187)(105,407,188)(106,408,189)(107,409,190)(108,410,191)(109,411,192)(110,412,193)(111,413,194)(112,414,195)(113,415,196)(114,416,197)(115,417,198)(116,418,199)(117,419,200)(118,420,201)(119,421,202)(120,422,203)(121,423,204)(122,424,205)(123,425,206)(124,426,207)(125,427,208)(126,428,209)(127,429,210)(128,430,211)(129,431,212)(130,432,213)(131,433,214)(132,434,215)(133,435,216)(134,436,217)(135,437,218)(136,438,219)(137,439,220)(138,440,221)(139,441,222)(140,442,223)(141,443,224)(142,444,225)(143,445,226)(144,446,227)(145,447,228)(146,448,229)(147,449,230)(148,450,231)(149,451,232)(150,452,233)(151,453,234)(152,454,235)(153,455,236)(154,456,237)(155,457,238)(156,458,239)(157,459,240)(158,460,241)(159,461,242), (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)>;
G:=Group( (1,462,243)(2,463,244)(3,464,245)(4,465,246)(5,466,247)(6,467,248)(7,468,249)(8,469,250)(9,470,251)(10,471,252)(11,472,253)(12,473,254)(13,474,255)(14,475,256)(15,476,257)(16,477,258)(17,319,259)(18,320,260)(19,321,261)(20,322,262)(21,323,263)(22,324,264)(23,325,265)(24,326,266)(25,327,267)(26,328,268)(27,329,269)(28,330,270)(29,331,271)(30,332,272)(31,333,273)(32,334,274)(33,335,275)(34,336,276)(35,337,277)(36,338,278)(37,339,279)(38,340,280)(39,341,281)(40,342,282)(41,343,283)(42,344,284)(43,345,285)(44,346,286)(45,347,287)(46,348,288)(47,349,289)(48,350,290)(49,351,291)(50,352,292)(51,353,293)(52,354,294)(53,355,295)(54,356,296)(55,357,297)(56,358,298)(57,359,299)(58,360,300)(59,361,301)(60,362,302)(61,363,303)(62,364,304)(63,365,305)(64,366,306)(65,367,307)(66,368,308)(67,369,309)(68,370,310)(69,371,311)(70,372,312)(71,373,313)(72,374,314)(73,375,315)(74,376,316)(75,377,317)(76,378,318)(77,379,160)(78,380,161)(79,381,162)(80,382,163)(81,383,164)(82,384,165)(83,385,166)(84,386,167)(85,387,168)(86,388,169)(87,389,170)(88,390,171)(89,391,172)(90,392,173)(91,393,174)(92,394,175)(93,395,176)(94,396,177)(95,397,178)(96,398,179)(97,399,180)(98,400,181)(99,401,182)(100,402,183)(101,403,184)(102,404,185)(103,405,186)(104,406,187)(105,407,188)(106,408,189)(107,409,190)(108,410,191)(109,411,192)(110,412,193)(111,413,194)(112,414,195)(113,415,196)(114,416,197)(115,417,198)(116,418,199)(117,419,200)(118,420,201)(119,421,202)(120,422,203)(121,423,204)(122,424,205)(123,425,206)(124,426,207)(125,427,208)(126,428,209)(127,429,210)(128,430,211)(129,431,212)(130,432,213)(131,433,214)(132,434,215)(133,435,216)(134,436,217)(135,437,218)(136,438,219)(137,439,220)(138,440,221)(139,441,222)(140,442,223)(141,443,224)(142,444,225)(143,445,226)(144,446,227)(145,447,228)(146,448,229)(147,449,230)(148,450,231)(149,451,232)(150,452,233)(151,453,234)(152,454,235)(153,455,236)(154,456,237)(155,457,238)(156,458,239)(157,459,240)(158,460,241)(159,461,242), (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) );
G=PermutationGroup([[(1,462,243),(2,463,244),(3,464,245),(4,465,246),(5,466,247),(6,467,248),(7,468,249),(8,469,250),(9,470,251),(10,471,252),(11,472,253),(12,473,254),(13,474,255),(14,475,256),(15,476,257),(16,477,258),(17,319,259),(18,320,260),(19,321,261),(20,322,262),(21,323,263),(22,324,264),(23,325,265),(24,326,266),(25,327,267),(26,328,268),(27,329,269),(28,330,270),(29,331,271),(30,332,272),(31,333,273),(32,334,274),(33,335,275),(34,336,276),(35,337,277),(36,338,278),(37,339,279),(38,340,280),(39,341,281),(40,342,282),(41,343,283),(42,344,284),(43,345,285),(44,346,286),(45,347,287),(46,348,288),(47,349,289),(48,350,290),(49,351,291),(50,352,292),(51,353,293),(52,354,294),(53,355,295),(54,356,296),(55,357,297),(56,358,298),(57,359,299),(58,360,300),(59,361,301),(60,362,302),(61,363,303),(62,364,304),(63,365,305),(64,366,306),(65,367,307),(66,368,308),(67,369,309),(68,370,310),(69,371,311),(70,372,312),(71,373,313),(72,374,314),(73,375,315),(74,376,316),(75,377,317),(76,378,318),(77,379,160),(78,380,161),(79,381,162),(80,382,163),(81,383,164),(82,384,165),(83,385,166),(84,386,167),(85,387,168),(86,388,169),(87,389,170),(88,390,171),(89,391,172),(90,392,173),(91,393,174),(92,394,175),(93,395,176),(94,396,177),(95,397,178),(96,398,179),(97,399,180),(98,400,181),(99,401,182),(100,402,183),(101,403,184),(102,404,185),(103,405,186),(104,406,187),(105,407,188),(106,408,189),(107,409,190),(108,410,191),(109,411,192),(110,412,193),(111,413,194),(112,414,195),(113,415,196),(114,416,197),(115,417,198),(116,418,199),(117,419,200),(118,420,201),(119,421,202),(120,422,203),(121,423,204),(122,424,205),(123,425,206),(124,426,207),(125,427,208),(126,428,209),(127,429,210),(128,430,211),(129,431,212),(130,432,213),(131,433,214),(132,434,215),(133,435,216),(134,436,217),(135,437,218),(136,438,219),(137,439,220),(138,440,221),(139,441,222),(140,442,223),(141,443,224),(142,444,225),(143,445,226),(144,446,227),(145,447,228),(146,448,229),(147,449,230),(148,450,231),(149,451,232),(150,452,233),(151,453,234),(152,454,235),(153,455,236),(154,456,237),(155,457,238),(156,458,239),(157,459,240),(158,460,241),(159,461,242)], [(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)]])
477 conjugacy classes
| class | 1 | 3A | ··· | 3H | 53A | ··· | 53AZ | 159A | ··· | 159OZ |
| order | 1 | 3 | ··· | 3 | 53 | ··· | 53 | 159 | ··· | 159 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
477 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C53 | C159 |
| kernel | C3×C159 | C159 | C32 | C3 |
| # reps | 1 | 8 | 52 | 416 |
Matrix representation of C3×C159 ►in GL2(𝔽3181) generated by
| 440 | 0 |
| 0 | 440 |
| 2745 | 0 |
| 0 | 1527 |
G:=sub<GL(2,GF(3181))| [440,0,0,440],[2745,0,0,1527] >;
C3×C159 in GAP, Magma, Sage, TeX
C_3\times C_{159} % in TeX
G:=Group("C3xC159"); // GroupNames label
G:=SmallGroup(477,2);
// by ID
G=gap.SmallGroup(477,2);
# by ID
G:=PCGroup([3,-3,-3,-53]);
// Polycyclic
G:=Group<a,b|a^3=b^159=1,a*b=b*a>;
// generators/relations
Export