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G = C3×C165order 495 = 32·5·11

Abelian group of type [3,165]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C165, SmallGroup(495,4)

Series: Derived Chief Lower central Upper central

C1 — C3×C165
C1C11C55C165 — 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
(1 473 193)(2 474 194)(3 475 195)(4 476 196)(5 477 197)(6 478 198)(7 479 199)(8 480 200)(9 481 201)(10 482 202)(11 483 203)(12 484 204)(13 485 205)(14 486 206)(15 487 207)(16 488 208)(17 489 209)(18 490 210)(19 491 211)(20 492 212)(21 493 213)(22 494 214)(23 495 215)(24 331 216)(25 332 217)(26 333 218)(27 334 219)(28 335 220)(29 336 221)(30 337 222)(31 338 223)(32 339 224)(33 340 225)(34 341 226)(35 342 227)(36 343 228)(37 344 229)(38 345 230)(39 346 231)(40 347 232)(41 348 233)(42 349 234)(43 350 235)(44 351 236)(45 352 237)(46 353 238)(47 354 239)(48 355 240)(49 356 241)(50 357 242)(51 358 243)(52 359 244)(53 360 245)(54 361 246)(55 362 247)(56 363 248)(57 364 249)(58 365 250)(59 366 251)(60 367 252)(61 368 253)(62 369 254)(63 370 255)(64 371 256)(65 372 257)(66 373 258)(67 374 259)(68 375 260)(69 376 261)(70 377 262)(71 378 263)(72 379 264)(73 380 265)(74 381 266)(75 382 267)(76 383 268)(77 384 269)(78 385 270)(79 386 271)(80 387 272)(81 388 273)(82 389 274)(83 390 275)(84 391 276)(85 392 277)(86 393 278)(87 394 279)(88 395 280)(89 396 281)(90 397 282)(91 398 283)(92 399 284)(93 400 285)(94 401 286)(95 402 287)(96 403 288)(97 404 289)(98 405 290)(99 406 291)(100 407 292)(101 408 293)(102 409 294)(103 410 295)(104 411 296)(105 412 297)(106 413 298)(107 414 299)(108 415 300)(109 416 301)(110 417 302)(111 418 303)(112 419 304)(113 420 305)(114 421 306)(115 422 307)(116 423 308)(117 424 309)(118 425 310)(119 426 311)(120 427 312)(121 428 313)(122 429 314)(123 430 315)(124 431 316)(125 432 317)(126 433 318)(127 434 319)(128 435 320)(129 436 321)(130 437 322)(131 438 323)(132 439 324)(133 440 325)(134 441 326)(135 442 327)(136 443 328)(137 444 329)(138 445 330)(139 446 166)(140 447 167)(141 448 168)(142 449 169)(143 450 170)(144 451 171)(145 452 172)(146 453 173)(147 454 174)(148 455 175)(149 456 176)(150 457 177)(151 458 178)(152 459 179)(153 460 180)(154 461 181)(155 462 182)(156 463 183)(157 464 184)(158 465 185)(159 466 186)(160 467 187)(161 468 188)(162 469 189)(163 470 190)(164 471 191)(165 472 192)
(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)| (1,473,193)(2,474,194)(3,475,195)(4,476,196)(5,477,197)(6,478,198)(7,479,199)(8,480,200)(9,481,201)(10,482,202)(11,483,203)(12,484,204)(13,485,205)(14,486,206)(15,487,207)(16,488,208)(17,489,209)(18,490,210)(19,491,211)(20,492,212)(21,493,213)(22,494,214)(23,495,215)(24,331,216)(25,332,217)(26,333,218)(27,334,219)(28,335,220)(29,336,221)(30,337,222)(31,338,223)(32,339,224)(33,340,225)(34,341,226)(35,342,227)(36,343,228)(37,344,229)(38,345,230)(39,346,231)(40,347,232)(41,348,233)(42,349,234)(43,350,235)(44,351,236)(45,352,237)(46,353,238)(47,354,239)(48,355,240)(49,356,241)(50,357,242)(51,358,243)(52,359,244)(53,360,245)(54,361,246)(55,362,247)(56,363,248)(57,364,249)(58,365,250)(59,366,251)(60,367,252)(61,368,253)(62,369,254)(63,370,255)(64,371,256)(65,372,257)(66,373,258)(67,374,259)(68,375,260)(69,376,261)(70,377,262)(71,378,263)(72,379,264)(73,380,265)(74,381,266)(75,382,267)(76,383,268)(77,384,269)(78,385,270)(79,386,271)(80,387,272)(81,388,273)(82,389,274)(83,390,275)(84,391,276)(85,392,277)(86,393,278)(87,394,279)(88,395,280)(89,396,281)(90,397,282)(91,398,283)(92,399,284)(93,400,285)(94,401,286)(95,402,287)(96,403,288)(97,404,289)(98,405,290)(99,406,291)(100,407,292)(101,408,293)(102,409,294)(103,410,295)(104,411,296)(105,412,297)(106,413,298)(107,414,299)(108,415,300)(109,416,301)(110,417,302)(111,418,303)(112,419,304)(113,420,305)(114,421,306)(115,422,307)(116,423,308)(117,424,309)(118,425,310)(119,426,311)(120,427,312)(121,428,313)(122,429,314)(123,430,315)(124,431,316)(125,432,317)(126,433,318)(127,434,319)(128,435,320)(129,436,321)(130,437,322)(131,438,323)(132,439,324)(133,440,325)(134,441,326)(135,442,327)(136,443,328)(137,444,329)(138,445,330)(139,446,166)(140,447,167)(141,448,168)(142,449,169)(143,450,170)(144,451,171)(145,452,172)(146,453,173)(147,454,174)(148,455,175)(149,456,176)(150,457,177)(151,458,178)(152,459,179)(153,460,180)(154,461,181)(155,462,182)(156,463,183)(157,464,184)(158,465,185)(159,466,186)(160,467,187)(161,468,188)(162,469,189)(163,470,190)(164,471,191)(165,472,192), (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( (1,473,193)(2,474,194)(3,475,195)(4,476,196)(5,477,197)(6,478,198)(7,479,199)(8,480,200)(9,481,201)(10,482,202)(11,483,203)(12,484,204)(13,485,205)(14,486,206)(15,487,207)(16,488,208)(17,489,209)(18,490,210)(19,491,211)(20,492,212)(21,493,213)(22,494,214)(23,495,215)(24,331,216)(25,332,217)(26,333,218)(27,334,219)(28,335,220)(29,336,221)(30,337,222)(31,338,223)(32,339,224)(33,340,225)(34,341,226)(35,342,227)(36,343,228)(37,344,229)(38,345,230)(39,346,231)(40,347,232)(41,348,233)(42,349,234)(43,350,235)(44,351,236)(45,352,237)(46,353,238)(47,354,239)(48,355,240)(49,356,241)(50,357,242)(51,358,243)(52,359,244)(53,360,245)(54,361,246)(55,362,247)(56,363,248)(57,364,249)(58,365,250)(59,366,251)(60,367,252)(61,368,253)(62,369,254)(63,370,255)(64,371,256)(65,372,257)(66,373,258)(67,374,259)(68,375,260)(69,376,261)(70,377,262)(71,378,263)(72,379,264)(73,380,265)(74,381,266)(75,382,267)(76,383,268)(77,384,269)(78,385,270)(79,386,271)(80,387,272)(81,388,273)(82,389,274)(83,390,275)(84,391,276)(85,392,277)(86,393,278)(87,394,279)(88,395,280)(89,396,281)(90,397,282)(91,398,283)(92,399,284)(93,400,285)(94,401,286)(95,402,287)(96,403,288)(97,404,289)(98,405,290)(99,406,291)(100,407,292)(101,408,293)(102,409,294)(103,410,295)(104,411,296)(105,412,297)(106,413,298)(107,414,299)(108,415,300)(109,416,301)(110,417,302)(111,418,303)(112,419,304)(113,420,305)(114,421,306)(115,422,307)(116,423,308)(117,424,309)(118,425,310)(119,426,311)(120,427,312)(121,428,313)(122,429,314)(123,430,315)(124,431,316)(125,432,317)(126,433,318)(127,434,319)(128,435,320)(129,436,321)(130,437,322)(131,438,323)(132,439,324)(133,440,325)(134,441,326)(135,442,327)(136,443,328)(137,444,329)(138,445,330)(139,446,166)(140,447,167)(141,448,168)(142,449,169)(143,450,170)(144,451,171)(145,452,172)(146,453,173)(147,454,174)(148,455,175)(149,456,176)(150,457,177)(151,458,178)(152,459,179)(153,460,180)(154,461,181)(155,462,182)(156,463,183)(157,464,184)(158,465,185)(159,466,186)(160,467,187)(161,468,188)(162,469,189)(163,470,190)(164,471,191)(165,472,192), (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([(1,473,193),(2,474,194),(3,475,195),(4,476,196),(5,477,197),(6,478,198),(7,479,199),(8,480,200),(9,481,201),(10,482,202),(11,483,203),(12,484,204),(13,485,205),(14,486,206),(15,487,207),(16,488,208),(17,489,209),(18,490,210),(19,491,211),(20,492,212),(21,493,213),(22,494,214),(23,495,215),(24,331,216),(25,332,217),(26,333,218),(27,334,219),(28,335,220),(29,336,221),(30,337,222),(31,338,223),(32,339,224),(33,340,225),(34,341,226),(35,342,227),(36,343,228),(37,344,229),(38,345,230),(39,346,231),(40,347,232),(41,348,233),(42,349,234),(43,350,235),(44,351,236),(45,352,237),(46,353,238),(47,354,239),(48,355,240),(49,356,241),(50,357,242),(51,358,243),(52,359,244),(53,360,245),(54,361,246),(55,362,247),(56,363,248),(57,364,249),(58,365,250),(59,366,251),(60,367,252),(61,368,253),(62,369,254),(63,370,255),(64,371,256),(65,372,257),(66,373,258),(67,374,259),(68,375,260),(69,376,261),(70,377,262),(71,378,263),(72,379,264),(73,380,265),(74,381,266),(75,382,267),(76,383,268),(77,384,269),(78,385,270),(79,386,271),(80,387,272),(81,388,273),(82,389,274),(83,390,275),(84,391,276),(85,392,277),(86,393,278),(87,394,279),(88,395,280),(89,396,281),(90,397,282),(91,398,283),(92,399,284),(93,400,285),(94,401,286),(95,402,287),(96,403,288),(97,404,289),(98,405,290),(99,406,291),(100,407,292),(101,408,293),(102,409,294),(103,410,295),(104,411,296),(105,412,297),(106,413,298),(107,414,299),(108,415,300),(109,416,301),(110,417,302),(111,418,303),(112,419,304),(113,420,305),(114,421,306),(115,422,307),(116,423,308),(117,424,309),(118,425,310),(119,426,311),(120,427,312),(121,428,313),(122,429,314),(123,430,315),(124,431,316),(125,432,317),(126,433,318),(127,434,319),(128,435,320),(129,436,321),(130,437,322),(131,438,323),(132,439,324),(133,440,325),(134,441,326),(135,442,327),(136,443,328),(137,444,329),(138,445,330),(139,446,166),(140,447,167),(141,448,168),(142,449,169),(143,450,170),(144,451,171),(145,452,172),(146,453,173),(147,454,174),(148,455,175),(149,456,176),(150,457,177),(151,458,178),(152,459,179),(153,460,180),(154,461,181),(155,462,182),(156,463,183),(157,464,184),(158,465,185),(159,466,186),(160,467,187),(161,468,188),(162,469,189),(163,470,190),(164,471,191),(165,472,192)], [(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···3H5A5B5C5D11A···11J15A···15AF33A···33CB55A···55AN165A···165LH
order13···3555511···1115···1533···3355···55165···165
size11···111111···11···11···11···11···1

495 irreducible representations

dim11111111
type+
imageC1C3C5C11C15C33C55C165
kernelC3×C165C165C3×C33C3×C15C33C15C32C3
# reps18410328040320

Matrix representation of C3×C165 in GL2(𝔽331) generated by

2990
031
,
260
0313
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

Subgroup lattice of C3×C165 in TeX

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