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G = C3×C162order 486 = 2·35

Abelian group of type [3,162]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C162, SmallGroup(486,83)

Series: Derived Chief Lower central Upper central

C1 — C3×C162
C1C3C9C27C3×C27C3×C81 — 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
(1 175 461)(2 176 462)(3 177 463)(4 178 464)(5 179 465)(6 180 466)(7 181 467)(8 182 468)(9 183 469)(10 184 470)(11 185 471)(12 186 472)(13 187 473)(14 188 474)(15 189 475)(16 190 476)(17 191 477)(18 192 478)(19 193 479)(20 194 480)(21 195 481)(22 196 482)(23 197 483)(24 198 484)(25 199 485)(26 200 486)(27 201 325)(28 202 326)(29 203 327)(30 204 328)(31 205 329)(32 206 330)(33 207 331)(34 208 332)(35 209 333)(36 210 334)(37 211 335)(38 212 336)(39 213 337)(40 214 338)(41 215 339)(42 216 340)(43 217 341)(44 218 342)(45 219 343)(46 220 344)(47 221 345)(48 222 346)(49 223 347)(50 224 348)(51 225 349)(52 226 350)(53 227 351)(54 228 352)(55 229 353)(56 230 354)(57 231 355)(58 232 356)(59 233 357)(60 234 358)(61 235 359)(62 236 360)(63 237 361)(64 238 362)(65 239 363)(66 240 364)(67 241 365)(68 242 366)(69 243 367)(70 244 368)(71 245 369)(72 246 370)(73 247 371)(74 248 372)(75 249 373)(76 250 374)(77 251 375)(78 252 376)(79 253 377)(80 254 378)(81 255 379)(82 256 380)(83 257 381)(84 258 382)(85 259 383)(86 260 384)(87 261 385)(88 262 386)(89 263 387)(90 264 388)(91 265 389)(92 266 390)(93 267 391)(94 268 392)(95 269 393)(96 270 394)(97 271 395)(98 272 396)(99 273 397)(100 274 398)(101 275 399)(102 276 400)(103 277 401)(104 278 402)(105 279 403)(106 280 404)(107 281 405)(108 282 406)(109 283 407)(110 284 408)(111 285 409)(112 286 410)(113 287 411)(114 288 412)(115 289 413)(116 290 414)(117 291 415)(118 292 416)(119 293 417)(120 294 418)(121 295 419)(122 296 420)(123 297 421)(124 298 422)(125 299 423)(126 300 424)(127 301 425)(128 302 426)(129 303 427)(130 304 428)(131 305 429)(132 306 430)(133 307 431)(134 308 432)(135 309 433)(136 310 434)(137 311 435)(138 312 436)(139 313 437)(140 314 438)(141 315 439)(142 316 440)(143 317 441)(144 318 442)(145 319 443)(146 320 444)(147 321 445)(148 322 446)(149 323 447)(150 324 448)(151 163 449)(152 164 450)(153 165 451)(154 166 452)(155 167 453)(156 168 454)(157 169 455)(158 170 456)(159 171 457)(160 172 458)(161 173 459)(162 174 460)
(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)| (1,175,461)(2,176,462)(3,177,463)(4,178,464)(5,179,465)(6,180,466)(7,181,467)(8,182,468)(9,183,469)(10,184,470)(11,185,471)(12,186,472)(13,187,473)(14,188,474)(15,189,475)(16,190,476)(17,191,477)(18,192,478)(19,193,479)(20,194,480)(21,195,481)(22,196,482)(23,197,483)(24,198,484)(25,199,485)(26,200,486)(27,201,325)(28,202,326)(29,203,327)(30,204,328)(31,205,329)(32,206,330)(33,207,331)(34,208,332)(35,209,333)(36,210,334)(37,211,335)(38,212,336)(39,213,337)(40,214,338)(41,215,339)(42,216,340)(43,217,341)(44,218,342)(45,219,343)(46,220,344)(47,221,345)(48,222,346)(49,223,347)(50,224,348)(51,225,349)(52,226,350)(53,227,351)(54,228,352)(55,229,353)(56,230,354)(57,231,355)(58,232,356)(59,233,357)(60,234,358)(61,235,359)(62,236,360)(63,237,361)(64,238,362)(65,239,363)(66,240,364)(67,241,365)(68,242,366)(69,243,367)(70,244,368)(71,245,369)(72,246,370)(73,247,371)(74,248,372)(75,249,373)(76,250,374)(77,251,375)(78,252,376)(79,253,377)(80,254,378)(81,255,379)(82,256,380)(83,257,381)(84,258,382)(85,259,383)(86,260,384)(87,261,385)(88,262,386)(89,263,387)(90,264,388)(91,265,389)(92,266,390)(93,267,391)(94,268,392)(95,269,393)(96,270,394)(97,271,395)(98,272,396)(99,273,397)(100,274,398)(101,275,399)(102,276,400)(103,277,401)(104,278,402)(105,279,403)(106,280,404)(107,281,405)(108,282,406)(109,283,407)(110,284,408)(111,285,409)(112,286,410)(113,287,411)(114,288,412)(115,289,413)(116,290,414)(117,291,415)(118,292,416)(119,293,417)(120,294,418)(121,295,419)(122,296,420)(123,297,421)(124,298,422)(125,299,423)(126,300,424)(127,301,425)(128,302,426)(129,303,427)(130,304,428)(131,305,429)(132,306,430)(133,307,431)(134,308,432)(135,309,433)(136,310,434)(137,311,435)(138,312,436)(139,313,437)(140,314,438)(141,315,439)(142,316,440)(143,317,441)(144,318,442)(145,319,443)(146,320,444)(147,321,445)(148,322,446)(149,323,447)(150,324,448)(151,163,449)(152,164,450)(153,165,451)(154,166,452)(155,167,453)(156,168,454)(157,169,455)(158,170,456)(159,171,457)(160,172,458)(161,173,459)(162,174,460), (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( (1,175,461)(2,176,462)(3,177,463)(4,178,464)(5,179,465)(6,180,466)(7,181,467)(8,182,468)(9,183,469)(10,184,470)(11,185,471)(12,186,472)(13,187,473)(14,188,474)(15,189,475)(16,190,476)(17,191,477)(18,192,478)(19,193,479)(20,194,480)(21,195,481)(22,196,482)(23,197,483)(24,198,484)(25,199,485)(26,200,486)(27,201,325)(28,202,326)(29,203,327)(30,204,328)(31,205,329)(32,206,330)(33,207,331)(34,208,332)(35,209,333)(36,210,334)(37,211,335)(38,212,336)(39,213,337)(40,214,338)(41,215,339)(42,216,340)(43,217,341)(44,218,342)(45,219,343)(46,220,344)(47,221,345)(48,222,346)(49,223,347)(50,224,348)(51,225,349)(52,226,350)(53,227,351)(54,228,352)(55,229,353)(56,230,354)(57,231,355)(58,232,356)(59,233,357)(60,234,358)(61,235,359)(62,236,360)(63,237,361)(64,238,362)(65,239,363)(66,240,364)(67,241,365)(68,242,366)(69,243,367)(70,244,368)(71,245,369)(72,246,370)(73,247,371)(74,248,372)(75,249,373)(76,250,374)(77,251,375)(78,252,376)(79,253,377)(80,254,378)(81,255,379)(82,256,380)(83,257,381)(84,258,382)(85,259,383)(86,260,384)(87,261,385)(88,262,386)(89,263,387)(90,264,388)(91,265,389)(92,266,390)(93,267,391)(94,268,392)(95,269,393)(96,270,394)(97,271,395)(98,272,396)(99,273,397)(100,274,398)(101,275,399)(102,276,400)(103,277,401)(104,278,402)(105,279,403)(106,280,404)(107,281,405)(108,282,406)(109,283,407)(110,284,408)(111,285,409)(112,286,410)(113,287,411)(114,288,412)(115,289,413)(116,290,414)(117,291,415)(118,292,416)(119,293,417)(120,294,418)(121,295,419)(122,296,420)(123,297,421)(124,298,422)(125,299,423)(126,300,424)(127,301,425)(128,302,426)(129,303,427)(130,304,428)(131,305,429)(132,306,430)(133,307,431)(134,308,432)(135,309,433)(136,310,434)(137,311,435)(138,312,436)(139,313,437)(140,314,438)(141,315,439)(142,316,440)(143,317,441)(144,318,442)(145,319,443)(146,320,444)(147,321,445)(148,322,446)(149,323,447)(150,324,448)(151,163,449)(152,164,450)(153,165,451)(154,166,452)(155,167,453)(156,168,454)(157,169,455)(158,170,456)(159,171,457)(160,172,458)(161,173,459)(162,174,460), (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([(1,175,461),(2,176,462),(3,177,463),(4,178,464),(5,179,465),(6,180,466),(7,181,467),(8,182,468),(9,183,469),(10,184,470),(11,185,471),(12,186,472),(13,187,473),(14,188,474),(15,189,475),(16,190,476),(17,191,477),(18,192,478),(19,193,479),(20,194,480),(21,195,481),(22,196,482),(23,197,483),(24,198,484),(25,199,485),(26,200,486),(27,201,325),(28,202,326),(29,203,327),(30,204,328),(31,205,329),(32,206,330),(33,207,331),(34,208,332),(35,209,333),(36,210,334),(37,211,335),(38,212,336),(39,213,337),(40,214,338),(41,215,339),(42,216,340),(43,217,341),(44,218,342),(45,219,343),(46,220,344),(47,221,345),(48,222,346),(49,223,347),(50,224,348),(51,225,349),(52,226,350),(53,227,351),(54,228,352),(55,229,353),(56,230,354),(57,231,355),(58,232,356),(59,233,357),(60,234,358),(61,235,359),(62,236,360),(63,237,361),(64,238,362),(65,239,363),(66,240,364),(67,241,365),(68,242,366),(69,243,367),(70,244,368),(71,245,369),(72,246,370),(73,247,371),(74,248,372),(75,249,373),(76,250,374),(77,251,375),(78,252,376),(79,253,377),(80,254,378),(81,255,379),(82,256,380),(83,257,381),(84,258,382),(85,259,383),(86,260,384),(87,261,385),(88,262,386),(89,263,387),(90,264,388),(91,265,389),(92,266,390),(93,267,391),(94,268,392),(95,269,393),(96,270,394),(97,271,395),(98,272,396),(99,273,397),(100,274,398),(101,275,399),(102,276,400),(103,277,401),(104,278,402),(105,279,403),(106,280,404),(107,281,405),(108,282,406),(109,283,407),(110,284,408),(111,285,409),(112,286,410),(113,287,411),(114,288,412),(115,289,413),(116,290,414),(117,291,415),(118,292,416),(119,293,417),(120,294,418),(121,295,419),(122,296,420),(123,297,421),(124,298,422),(125,299,423),(126,300,424),(127,301,425),(128,302,426),(129,303,427),(130,304,428),(131,305,429),(132,306,430),(133,307,431),(134,308,432),(135,309,433),(136,310,434),(137,311,435),(138,312,436),(139,313,437),(140,314,438),(141,315,439),(142,316,440),(143,317,441),(144,318,442),(145,319,443),(146,320,444),(147,321,445),(148,322,446),(149,323,447),(150,324,448),(151,163,449),(152,164,450),(153,165,451),(154,166,452),(155,167,453),(156,168,454),(157,169,455),(158,170,456),(159,171,457),(160,172,458),(161,173,459),(162,174,460)], [(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···3H6A···6H9A···9R18A···18R27A···27BB54A···54BB81A···81FF162A···162FF
order123···36···69···918···1827···2754···5481···81162···162
size111···11···11···11···11···11···11···11···1

486 irreducible representations

dim1111111111111111
type++
imageC1C2C3C3C6C6C9C9C18C18C27C27C54C54C81C162
kernelC3×C162C3×C81C162C3×C54C81C3×C27C54C3×C18C27C3×C9C18C3×C6C9C32C6C3
# reps11626212612636183618162162

Matrix representation of C3×C162 in GL2(𝔽163) generated by

1040
058
,
1250
042
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

Subgroup lattice of C3×C162 in TeX

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