direct product, abelian, monomial, 3-elementary
Aliases: C3×C165, SmallGroup(495,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C165 |
| C1 — C3×C165 |
| C1 — C3×C165 |
Generators and relations for C3×C165
G = < a,b | a3=b165=1, ab=ba >
Smallest permutation representation of C3×C165
►Regular action on 495 points
Generators in S495
(1 448 181)(2 449 182)(3 450 183)(4 451 184)(5 452 185)(6 453 186)(7 454 187)(8 455 188)(9 456 189)(10 457 190)(11 458 191)(12 459 192)(13 460 193)(14 461 194)(15 462 195)(16 463 196)(17 464 197)(18 465 198)(19 466 199)(20 467 200)(21 468 201)(22 469 202)(23 470 203)(24 471 204)(25 472 205)(26 473 206)(27 474 207)(28 475 208)(29 476 209)(30 477 210)(31 478 211)(32 479 212)(33 480 213)(34 481 214)(35 482 215)(36 483 216)(37 484 217)(38 485 218)(39 486 219)(40 487 220)(41 488 221)(42 489 222)(43 490 223)(44 491 224)(45 492 225)(46 493 226)(47 494 227)(48 495 228)(49 331 229)(50 332 230)(51 333 231)(52 334 232)(53 335 233)(54 336 234)(55 337 235)(56 338 236)(57 339 237)(58 340 238)(59 341 239)(60 342 240)(61 343 241)(62 344 242)(63 345 243)(64 346 244)(65 347 245)(66 348 246)(67 349 247)(68 350 248)(69 351 249)(70 352 250)(71 353 251)(72 354 252)(73 355 253)(74 356 254)(75 357 255)(76 358 256)(77 359 257)(78 360 258)(79 361 259)(80 362 260)(81 363 261)(82 364 262)(83 365 263)(84 366 264)(85 367 265)(86 368 266)(87 369 267)(88 370 268)(89 371 269)(90 372 270)(91 373 271)(92 374 272)(93 375 273)(94 376 274)(95 377 275)(96 378 276)(97 379 277)(98 380 278)(99 381 279)(100 382 280)(101 383 281)(102 384 282)(103 385 283)(104 386 284)(105 387 285)(106 388 286)(107 389 287)(108 390 288)(109 391 289)(110 392 290)(111 393 291)(112 394 292)(113 395 293)(114 396 294)(115 397 295)(116 398 296)(117 399 297)(118 400 298)(119 401 299)(120 402 300)(121 403 301)(122 404 302)(123 405 303)(124 406 304)(125 407 305)(126 408 306)(127 409 307)(128 410 308)(129 411 309)(130 412 310)(131 413 311)(132 414 312)(133 415 313)(134 416 314)(135 417 315)(136 418 316)(137 419 317)(138 420 318)(139 421 319)(140 422 320)(141 423 321)(142 424 322)(143 425 323)(144 426 324)(145 427 325)(146 428 326)(147 429 327)(148 430 328)(149 431 329)(150 432 330)(151 433 166)(152 434 167)(153 435 168)(154 436 169)(155 437 170)(156 438 171)(157 439 172)(158 440 173)(159 441 174)(160 442 175)(161 443 176)(162 444 177)(163 445 178)(164 446 179)(165 447 180)
(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)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495)
G:=sub<Sym(495)| (1,448,181)(2,449,182)(3,450,183)(4,451,184)(5,452,185)(6,453,186)(7,454,187)(8,455,188)(9,456,189)(10,457,190)(11,458,191)(12,459,192)(13,460,193)(14,461,194)(15,462,195)(16,463,196)(17,464,197)(18,465,198)(19,466,199)(20,467,200)(21,468,201)(22,469,202)(23,470,203)(24,471,204)(25,472,205)(26,473,206)(27,474,207)(28,475,208)(29,476,209)(30,477,210)(31,478,211)(32,479,212)(33,480,213)(34,481,214)(35,482,215)(36,483,216)(37,484,217)(38,485,218)(39,486,219)(40,487,220)(41,488,221)(42,489,222)(43,490,223)(44,491,224)(45,492,225)(46,493,226)(47,494,227)(48,495,228)(49,331,229)(50,332,230)(51,333,231)(52,334,232)(53,335,233)(54,336,234)(55,337,235)(56,338,236)(57,339,237)(58,340,238)(59,341,239)(60,342,240)(61,343,241)(62,344,242)(63,345,243)(64,346,244)(65,347,245)(66,348,246)(67,349,247)(68,350,248)(69,351,249)(70,352,250)(71,353,251)(72,354,252)(73,355,253)(74,356,254)(75,357,255)(76,358,256)(77,359,257)(78,360,258)(79,361,259)(80,362,260)(81,363,261)(82,364,262)(83,365,263)(84,366,264)(85,367,265)(86,368,266)(87,369,267)(88,370,268)(89,371,269)(90,372,270)(91,373,271)(92,374,272)(93,375,273)(94,376,274)(95,377,275)(96,378,276)(97,379,277)(98,380,278)(99,381,279)(100,382,280)(101,383,281)(102,384,282)(103,385,283)(104,386,284)(105,387,285)(106,388,286)(107,389,287)(108,390,288)(109,391,289)(110,392,290)(111,393,291)(112,394,292)(113,395,293)(114,396,294)(115,397,295)(116,398,296)(117,399,297)(118,400,298)(119,401,299)(120,402,300)(121,403,301)(122,404,302)(123,405,303)(124,406,304)(125,407,305)(126,408,306)(127,409,307)(128,410,308)(129,411,309)(130,412,310)(131,413,311)(132,414,312)(133,415,313)(134,416,314)(135,417,315)(136,418,316)(137,419,317)(138,420,318)(139,421,319)(140,422,320)(141,423,321)(142,424,322)(143,425,323)(144,426,324)(145,427,325)(146,428,326)(147,429,327)(148,430,328)(149,431,329)(150,432,330)(151,433,166)(152,434,167)(153,435,168)(154,436,169)(155,437,170)(156,438,171)(157,439,172)(158,440,173)(159,441,174)(160,442,175)(161,443,176)(162,444,177)(163,445,178)(164,446,179)(165,447,180), (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)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495)>;
G:=Group( (1,448,181)(2,449,182)(3,450,183)(4,451,184)(5,452,185)(6,453,186)(7,454,187)(8,455,188)(9,456,189)(10,457,190)(11,458,191)(12,459,192)(13,460,193)(14,461,194)(15,462,195)(16,463,196)(17,464,197)(18,465,198)(19,466,199)(20,467,200)(21,468,201)(22,469,202)(23,470,203)(24,471,204)(25,472,205)(26,473,206)(27,474,207)(28,475,208)(29,476,209)(30,477,210)(31,478,211)(32,479,212)(33,480,213)(34,481,214)(35,482,215)(36,483,216)(37,484,217)(38,485,218)(39,486,219)(40,487,220)(41,488,221)(42,489,222)(43,490,223)(44,491,224)(45,492,225)(46,493,226)(47,494,227)(48,495,228)(49,331,229)(50,332,230)(51,333,231)(52,334,232)(53,335,233)(54,336,234)(55,337,235)(56,338,236)(57,339,237)(58,340,238)(59,341,239)(60,342,240)(61,343,241)(62,344,242)(63,345,243)(64,346,244)(65,347,245)(66,348,246)(67,349,247)(68,350,248)(69,351,249)(70,352,250)(71,353,251)(72,354,252)(73,355,253)(74,356,254)(75,357,255)(76,358,256)(77,359,257)(78,360,258)(79,361,259)(80,362,260)(81,363,261)(82,364,262)(83,365,263)(84,366,264)(85,367,265)(86,368,266)(87,369,267)(88,370,268)(89,371,269)(90,372,270)(91,373,271)(92,374,272)(93,375,273)(94,376,274)(95,377,275)(96,378,276)(97,379,277)(98,380,278)(99,381,279)(100,382,280)(101,383,281)(102,384,282)(103,385,283)(104,386,284)(105,387,285)(106,388,286)(107,389,287)(108,390,288)(109,391,289)(110,392,290)(111,393,291)(112,394,292)(113,395,293)(114,396,294)(115,397,295)(116,398,296)(117,399,297)(118,400,298)(119,401,299)(120,402,300)(121,403,301)(122,404,302)(123,405,303)(124,406,304)(125,407,305)(126,408,306)(127,409,307)(128,410,308)(129,411,309)(130,412,310)(131,413,311)(132,414,312)(133,415,313)(134,416,314)(135,417,315)(136,418,316)(137,419,317)(138,420,318)(139,421,319)(140,422,320)(141,423,321)(142,424,322)(143,425,323)(144,426,324)(145,427,325)(146,428,326)(147,429,327)(148,430,328)(149,431,329)(150,432,330)(151,433,166)(152,434,167)(153,435,168)(154,436,169)(155,437,170)(156,438,171)(157,439,172)(158,440,173)(159,441,174)(160,442,175)(161,443,176)(162,444,177)(163,445,178)(164,446,179)(165,447,180), (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)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495) );
G=PermutationGroup([[(1,448,181),(2,449,182),(3,450,183),(4,451,184),(5,452,185),(6,453,186),(7,454,187),(8,455,188),(9,456,189),(10,457,190),(11,458,191),(12,459,192),(13,460,193),(14,461,194),(15,462,195),(16,463,196),(17,464,197),(18,465,198),(19,466,199),(20,467,200),(21,468,201),(22,469,202),(23,470,203),(24,471,204),(25,472,205),(26,473,206),(27,474,207),(28,475,208),(29,476,209),(30,477,210),(31,478,211),(32,479,212),(33,480,213),(34,481,214),(35,482,215),(36,483,216),(37,484,217),(38,485,218),(39,486,219),(40,487,220),(41,488,221),(42,489,222),(43,490,223),(44,491,224),(45,492,225),(46,493,226),(47,494,227),(48,495,228),(49,331,229),(50,332,230),(51,333,231),(52,334,232),(53,335,233),(54,336,234),(55,337,235),(56,338,236),(57,339,237),(58,340,238),(59,341,239),(60,342,240),(61,343,241),(62,344,242),(63,345,243),(64,346,244),(65,347,245),(66,348,246),(67,349,247),(68,350,248),(69,351,249),(70,352,250),(71,353,251),(72,354,252),(73,355,253),(74,356,254),(75,357,255),(76,358,256),(77,359,257),(78,360,258),(79,361,259),(80,362,260),(81,363,261),(82,364,262),(83,365,263),(84,366,264),(85,367,265),(86,368,266),(87,369,267),(88,370,268),(89,371,269),(90,372,270),(91,373,271),(92,374,272),(93,375,273),(94,376,274),(95,377,275),(96,378,276),(97,379,277),(98,380,278),(99,381,279),(100,382,280),(101,383,281),(102,384,282),(103,385,283),(104,386,284),(105,387,285),(106,388,286),(107,389,287),(108,390,288),(109,391,289),(110,392,290),(111,393,291),(112,394,292),(113,395,293),(114,396,294),(115,397,295),(116,398,296),(117,399,297),(118,400,298),(119,401,299),(120,402,300),(121,403,301),(122,404,302),(123,405,303),(124,406,304),(125,407,305),(126,408,306),(127,409,307),(128,410,308),(129,411,309),(130,412,310),(131,413,311),(132,414,312),(133,415,313),(134,416,314),(135,417,315),(136,418,316),(137,419,317),(138,420,318),(139,421,319),(140,422,320),(141,423,321),(142,424,322),(143,425,323),(144,426,324),(145,427,325),(146,428,326),(147,429,327),(148,430,328),(149,431,329),(150,432,330),(151,433,166),(152,434,167),(153,435,168),(154,436,169),(155,437,170),(156,438,171),(157,439,172),(158,440,173),(159,441,174),(160,442,175),(161,443,176),(162,444,177),(163,445,178),(164,446,179),(165,447,180)], [(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),(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495)]])
495 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 11A | ··· | 11J | 15A | ··· | 15AF | 33A | ··· | 33CB | 55A | ··· | 55AN | 165A | ··· | 165LH |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 11 | ··· | 11 | 15 | ··· | 15 | 33 | ··· | 33 | 55 | ··· | 55 | 165 | ··· | 165 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
495 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C5 | C11 | C15 | C33 | C55 | C165 |
| kernel | C3×C165 | C165 | C3×C33 | C3×C15 | C33 | C15 | C32 | C3 |
| # reps | 1 | 8 | 4 | 10 | 32 | 80 | 40 | 320 |
Matrix representation of C3×C165 ►in GL2(𝔽331) generated by
| 299 | 0 |
| 0 | 31 |
| 26 | 0 |
| 0 | 313 |
G:=sub<GL(2,GF(331))| [299,0,0,31],[26,0,0,313] >;
C3×C165 in GAP, Magma, Sage, TeX
C_3\times C_{165} % in TeX
G:=Group("C3xC165"); // GroupNames label
G:=SmallGroup(495,4);
// by ID
G=gap.SmallGroup(495,4);
# by ID
G:=PCGroup([4,-3,-3,-5,-11]);
// Polycyclic
G:=Group<a,b|a^3=b^165=1,a*b=b*a>;
// generators/relations
Export