direct product, abelian, monomial, 3-elementary
Aliases: C32×C54, SmallGroup(486,207)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C54 |
| C1 — C32×C54 |
| C1 — C32×C54 |
Generators and relations for C32×C54
G = < a,b,c | a3=b3=c54=1, ab=ba, ac=ca, bc=cb >
Subgroups: 144, all normal (12 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C18, C18, C3×C6, C27, C3×C9, C33, C54, C3×C18, C32×C6, C3×C27, C32×C9, C3×C54, C32×C18, C32×C27, C32×C54
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C27, C3×C9, C33, C54, C3×C18, C32×C6, C3×C27, C32×C9, C3×C54, C32×C18, C32×C27, C32×C54
►Regular action on 486 points
Generators in S486
(1 475 62)(2 476 63)(3 477 64)(4 478 65)(5 479 66)(6 480 67)(7 481 68)(8 482 69)(9 483 70)(10 484 71)(11 485 72)(12 486 73)(13 433 74)(14 434 75)(15 435 76)(16 436 77)(17 437 78)(18 438 79)(19 439 80)(20 440 81)(21 441 82)(22 442 83)(23 443 84)(24 444 85)(25 445 86)(26 446 87)(27 447 88)(28 448 89)(29 449 90)(30 450 91)(31 451 92)(32 452 93)(33 453 94)(34 454 95)(35 455 96)(36 456 97)(37 457 98)(38 458 99)(39 459 100)(40 460 101)(41 461 102)(42 462 103)(43 463 104)(44 464 105)(45 465 106)(46 466 107)(47 467 108)(48 468 55)(49 469 56)(50 470 57)(51 471 58)(52 472 59)(53 473 60)(54 474 61)(109 193 364)(110 194 365)(111 195 366)(112 196 367)(113 197 368)(114 198 369)(115 199 370)(116 200 371)(117 201 372)(118 202 373)(119 203 374)(120 204 375)(121 205 376)(122 206 377)(123 207 378)(124 208 325)(125 209 326)(126 210 327)(127 211 328)(128 212 329)(129 213 330)(130 214 331)(131 215 332)(132 216 333)(133 163 334)(134 164 335)(135 165 336)(136 166 337)(137 167 338)(138 168 339)(139 169 340)(140 170 341)(141 171 342)(142 172 343)(143 173 344)(144 174 345)(145 175 346)(146 176 347)(147 177 348)(148 178 349)(149 179 350)(150 180 351)(151 181 352)(152 182 353)(153 183 354)(154 184 355)(155 185 356)(156 186 357)(157 187 358)(158 188 359)(159 189 360)(160 190 361)(161 191 362)(162 192 363)(217 293 413)(218 294 414)(219 295 415)(220 296 416)(221 297 417)(222 298 418)(223 299 419)(224 300 420)(225 301 421)(226 302 422)(227 303 423)(228 304 424)(229 305 425)(230 306 426)(231 307 427)(232 308 428)(233 309 429)(234 310 430)(235 311 431)(236 312 432)(237 313 379)(238 314 380)(239 315 381)(240 316 382)(241 317 383)(242 318 384)(243 319 385)(244 320 386)(245 321 387)(246 322 388)(247 323 389)(248 324 390)(249 271 391)(250 272 392)(251 273 393)(252 274 394)(253 275 395)(254 276 396)(255 277 397)(256 278 398)(257 279 399)(258 280 400)(259 281 401)(260 282 402)(261 283 403)(262 284 404)(263 285 405)(264 286 406)(265 287 407)(266 288 408)(267 289 409)(268 290 410)(269 291 411)(270 292 412)
(1 158 408)(2 159 409)(3 160 410)(4 161 411)(5 162 412)(6 109 413)(7 110 414)(8 111 415)(9 112 416)(10 113 417)(11 114 418)(12 115 419)(13 116 420)(14 117 421)(15 118 422)(16 119 423)(17 120 424)(18 121 425)(19 122 426)(20 123 427)(21 124 428)(22 125 429)(23 126 430)(24 127 431)(25 128 432)(26 129 379)(27 130 380)(28 131 381)(29 132 382)(30 133 383)(31 134 384)(32 135 385)(33 136 386)(34 137 387)(35 138 388)(36 139 389)(37 140 390)(38 141 391)(39 142 392)(40 143 393)(41 144 394)(42 145 395)(43 146 396)(44 147 397)(45 148 398)(46 149 399)(47 150 400)(48 151 401)(49 152 402)(50 153 403)(51 154 404)(52 155 405)(53 156 406)(54 157 407)(55 352 281)(56 353 282)(57 354 283)(58 355 284)(59 356 285)(60 357 286)(61 358 287)(62 359 288)(63 360 289)(64 361 290)(65 362 291)(66 363 292)(67 364 293)(68 365 294)(69 366 295)(70 367 296)(71 368 297)(72 369 298)(73 370 299)(74 371 300)(75 372 301)(76 373 302)(77 374 303)(78 375 304)(79 376 305)(80 377 306)(81 378 307)(82 325 308)(83 326 309)(84 327 310)(85 328 311)(86 329 312)(87 330 313)(88 331 314)(89 332 315)(90 333 316)(91 334 317)(92 335 318)(93 336 319)(94 337 320)(95 338 321)(96 339 322)(97 340 323)(98 341 324)(99 342 271)(100 343 272)(101 344 273)(102 345 274)(103 346 275)(104 347 276)(105 348 277)(106 349 278)(107 350 279)(108 351 280)(163 241 450)(164 242 451)(165 243 452)(166 244 453)(167 245 454)(168 246 455)(169 247 456)(170 248 457)(171 249 458)(172 250 459)(173 251 460)(174 252 461)(175 253 462)(176 254 463)(177 255 464)(178 256 465)(179 257 466)(180 258 467)(181 259 468)(182 260 469)(183 261 470)(184 262 471)(185 263 472)(186 264 473)(187 265 474)(188 266 475)(189 267 476)(190 268 477)(191 269 478)(192 270 479)(193 217 480)(194 218 481)(195 219 482)(196 220 483)(197 221 484)(198 222 485)(199 223 486)(200 224 433)(201 225 434)(202 226 435)(203 227 436)(204 228 437)(205 229 438)(206 230 439)(207 231 440)(208 232 441)(209 233 442)(210 234 443)(211 235 444)(212 236 445)(213 237 446)(214 238 447)(215 239 448)(216 240 449)
(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)| (1,475,62)(2,476,63)(3,477,64)(4,478,65)(5,479,66)(6,480,67)(7,481,68)(8,482,69)(9,483,70)(10,484,71)(11,485,72)(12,486,73)(13,433,74)(14,434,75)(15,435,76)(16,436,77)(17,437,78)(18,438,79)(19,439,80)(20,440,81)(21,441,82)(22,442,83)(23,443,84)(24,444,85)(25,445,86)(26,446,87)(27,447,88)(28,448,89)(29,449,90)(30,450,91)(31,451,92)(32,452,93)(33,453,94)(34,454,95)(35,455,96)(36,456,97)(37,457,98)(38,458,99)(39,459,100)(40,460,101)(41,461,102)(42,462,103)(43,463,104)(44,464,105)(45,465,106)(46,466,107)(47,467,108)(48,468,55)(49,469,56)(50,470,57)(51,471,58)(52,472,59)(53,473,60)(54,474,61)(109,193,364)(110,194,365)(111,195,366)(112,196,367)(113,197,368)(114,198,369)(115,199,370)(116,200,371)(117,201,372)(118,202,373)(119,203,374)(120,204,375)(121,205,376)(122,206,377)(123,207,378)(124,208,325)(125,209,326)(126,210,327)(127,211,328)(128,212,329)(129,213,330)(130,214,331)(131,215,332)(132,216,333)(133,163,334)(134,164,335)(135,165,336)(136,166,337)(137,167,338)(138,168,339)(139,169,340)(140,170,341)(141,171,342)(142,172,343)(143,173,344)(144,174,345)(145,175,346)(146,176,347)(147,177,348)(148,178,349)(149,179,350)(150,180,351)(151,181,352)(152,182,353)(153,183,354)(154,184,355)(155,185,356)(156,186,357)(157,187,358)(158,188,359)(159,189,360)(160,190,361)(161,191,362)(162,192,363)(217,293,413)(218,294,414)(219,295,415)(220,296,416)(221,297,417)(222,298,418)(223,299,419)(224,300,420)(225,301,421)(226,302,422)(227,303,423)(228,304,424)(229,305,425)(230,306,426)(231,307,427)(232,308,428)(233,309,429)(234,310,430)(235,311,431)(236,312,432)(237,313,379)(238,314,380)(239,315,381)(240,316,382)(241,317,383)(242,318,384)(243,319,385)(244,320,386)(245,321,387)(246,322,388)(247,323,389)(248,324,390)(249,271,391)(250,272,392)(251,273,393)(252,274,394)(253,275,395)(254,276,396)(255,277,397)(256,278,398)(257,279,399)(258,280,400)(259,281,401)(260,282,402)(261,283,403)(262,284,404)(263,285,405)(264,286,406)(265,287,407)(266,288,408)(267,289,409)(268,290,410)(269,291,411)(270,292,412), (1,158,408)(2,159,409)(3,160,410)(4,161,411)(5,162,412)(6,109,413)(7,110,414)(8,111,415)(9,112,416)(10,113,417)(11,114,418)(12,115,419)(13,116,420)(14,117,421)(15,118,422)(16,119,423)(17,120,424)(18,121,425)(19,122,426)(20,123,427)(21,124,428)(22,125,429)(23,126,430)(24,127,431)(25,128,432)(26,129,379)(27,130,380)(28,131,381)(29,132,382)(30,133,383)(31,134,384)(32,135,385)(33,136,386)(34,137,387)(35,138,388)(36,139,389)(37,140,390)(38,141,391)(39,142,392)(40,143,393)(41,144,394)(42,145,395)(43,146,396)(44,147,397)(45,148,398)(46,149,399)(47,150,400)(48,151,401)(49,152,402)(50,153,403)(51,154,404)(52,155,405)(53,156,406)(54,157,407)(55,352,281)(56,353,282)(57,354,283)(58,355,284)(59,356,285)(60,357,286)(61,358,287)(62,359,288)(63,360,289)(64,361,290)(65,362,291)(66,363,292)(67,364,293)(68,365,294)(69,366,295)(70,367,296)(71,368,297)(72,369,298)(73,370,299)(74,371,300)(75,372,301)(76,373,302)(77,374,303)(78,375,304)(79,376,305)(80,377,306)(81,378,307)(82,325,308)(83,326,309)(84,327,310)(85,328,311)(86,329,312)(87,330,313)(88,331,314)(89,332,315)(90,333,316)(91,334,317)(92,335,318)(93,336,319)(94,337,320)(95,338,321)(96,339,322)(97,340,323)(98,341,324)(99,342,271)(100,343,272)(101,344,273)(102,345,274)(103,346,275)(104,347,276)(105,348,277)(106,349,278)(107,350,279)(108,351,280)(163,241,450)(164,242,451)(165,243,452)(166,244,453)(167,245,454)(168,246,455)(169,247,456)(170,248,457)(171,249,458)(172,250,459)(173,251,460)(174,252,461)(175,253,462)(176,254,463)(177,255,464)(178,256,465)(179,257,466)(180,258,467)(181,259,468)(182,260,469)(183,261,470)(184,262,471)(185,263,472)(186,264,473)(187,265,474)(188,266,475)(189,267,476)(190,268,477)(191,269,478)(192,270,479)(193,217,480)(194,218,481)(195,219,482)(196,220,483)(197,221,484)(198,222,485)(199,223,486)(200,224,433)(201,225,434)(202,226,435)(203,227,436)(204,228,437)(205,229,438)(206,230,439)(207,231,440)(208,232,441)(209,233,442)(210,234,443)(211,235,444)(212,236,445)(213,237,446)(214,238,447)(215,239,448)(216,240,449), (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( (1,475,62)(2,476,63)(3,477,64)(4,478,65)(5,479,66)(6,480,67)(7,481,68)(8,482,69)(9,483,70)(10,484,71)(11,485,72)(12,486,73)(13,433,74)(14,434,75)(15,435,76)(16,436,77)(17,437,78)(18,438,79)(19,439,80)(20,440,81)(21,441,82)(22,442,83)(23,443,84)(24,444,85)(25,445,86)(26,446,87)(27,447,88)(28,448,89)(29,449,90)(30,450,91)(31,451,92)(32,452,93)(33,453,94)(34,454,95)(35,455,96)(36,456,97)(37,457,98)(38,458,99)(39,459,100)(40,460,101)(41,461,102)(42,462,103)(43,463,104)(44,464,105)(45,465,106)(46,466,107)(47,467,108)(48,468,55)(49,469,56)(50,470,57)(51,471,58)(52,472,59)(53,473,60)(54,474,61)(109,193,364)(110,194,365)(111,195,366)(112,196,367)(113,197,368)(114,198,369)(115,199,370)(116,200,371)(117,201,372)(118,202,373)(119,203,374)(120,204,375)(121,205,376)(122,206,377)(123,207,378)(124,208,325)(125,209,326)(126,210,327)(127,211,328)(128,212,329)(129,213,330)(130,214,331)(131,215,332)(132,216,333)(133,163,334)(134,164,335)(135,165,336)(136,166,337)(137,167,338)(138,168,339)(139,169,340)(140,170,341)(141,171,342)(142,172,343)(143,173,344)(144,174,345)(145,175,346)(146,176,347)(147,177,348)(148,178,349)(149,179,350)(150,180,351)(151,181,352)(152,182,353)(153,183,354)(154,184,355)(155,185,356)(156,186,357)(157,187,358)(158,188,359)(159,189,360)(160,190,361)(161,191,362)(162,192,363)(217,293,413)(218,294,414)(219,295,415)(220,296,416)(221,297,417)(222,298,418)(223,299,419)(224,300,420)(225,301,421)(226,302,422)(227,303,423)(228,304,424)(229,305,425)(230,306,426)(231,307,427)(232,308,428)(233,309,429)(234,310,430)(235,311,431)(236,312,432)(237,313,379)(238,314,380)(239,315,381)(240,316,382)(241,317,383)(242,318,384)(243,319,385)(244,320,386)(245,321,387)(246,322,388)(247,323,389)(248,324,390)(249,271,391)(250,272,392)(251,273,393)(252,274,394)(253,275,395)(254,276,396)(255,277,397)(256,278,398)(257,279,399)(258,280,400)(259,281,401)(260,282,402)(261,283,403)(262,284,404)(263,285,405)(264,286,406)(265,287,407)(266,288,408)(267,289,409)(268,290,410)(269,291,411)(270,292,412), (1,158,408)(2,159,409)(3,160,410)(4,161,411)(5,162,412)(6,109,413)(7,110,414)(8,111,415)(9,112,416)(10,113,417)(11,114,418)(12,115,419)(13,116,420)(14,117,421)(15,118,422)(16,119,423)(17,120,424)(18,121,425)(19,122,426)(20,123,427)(21,124,428)(22,125,429)(23,126,430)(24,127,431)(25,128,432)(26,129,379)(27,130,380)(28,131,381)(29,132,382)(30,133,383)(31,134,384)(32,135,385)(33,136,386)(34,137,387)(35,138,388)(36,139,389)(37,140,390)(38,141,391)(39,142,392)(40,143,393)(41,144,394)(42,145,395)(43,146,396)(44,147,397)(45,148,398)(46,149,399)(47,150,400)(48,151,401)(49,152,402)(50,153,403)(51,154,404)(52,155,405)(53,156,406)(54,157,407)(55,352,281)(56,353,282)(57,354,283)(58,355,284)(59,356,285)(60,357,286)(61,358,287)(62,359,288)(63,360,289)(64,361,290)(65,362,291)(66,363,292)(67,364,293)(68,365,294)(69,366,295)(70,367,296)(71,368,297)(72,369,298)(73,370,299)(74,371,300)(75,372,301)(76,373,302)(77,374,303)(78,375,304)(79,376,305)(80,377,306)(81,378,307)(82,325,308)(83,326,309)(84,327,310)(85,328,311)(86,329,312)(87,330,313)(88,331,314)(89,332,315)(90,333,316)(91,334,317)(92,335,318)(93,336,319)(94,337,320)(95,338,321)(96,339,322)(97,340,323)(98,341,324)(99,342,271)(100,343,272)(101,344,273)(102,345,274)(103,346,275)(104,347,276)(105,348,277)(106,349,278)(107,350,279)(108,351,280)(163,241,450)(164,242,451)(165,243,452)(166,244,453)(167,245,454)(168,246,455)(169,247,456)(170,248,457)(171,249,458)(172,250,459)(173,251,460)(174,252,461)(175,253,462)(176,254,463)(177,255,464)(178,256,465)(179,257,466)(180,258,467)(181,259,468)(182,260,469)(183,261,470)(184,262,471)(185,263,472)(186,264,473)(187,265,474)(188,266,475)(189,267,476)(190,268,477)(191,269,478)(192,270,479)(193,217,480)(194,218,481)(195,219,482)(196,220,483)(197,221,484)(198,222,485)(199,223,486)(200,224,433)(201,225,434)(202,226,435)(203,227,436)(204,228,437)(205,229,438)(206,230,439)(207,231,440)(208,232,441)(209,233,442)(210,234,443)(211,235,444)(212,236,445)(213,237,446)(214,238,447)(215,239,448)(216,240,449), (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([[(1,475,62),(2,476,63),(3,477,64),(4,478,65),(5,479,66),(6,480,67),(7,481,68),(8,482,69),(9,483,70),(10,484,71),(11,485,72),(12,486,73),(13,433,74),(14,434,75),(15,435,76),(16,436,77),(17,437,78),(18,438,79),(19,439,80),(20,440,81),(21,441,82),(22,442,83),(23,443,84),(24,444,85),(25,445,86),(26,446,87),(27,447,88),(28,448,89),(29,449,90),(30,450,91),(31,451,92),(32,452,93),(33,453,94),(34,454,95),(35,455,96),(36,456,97),(37,457,98),(38,458,99),(39,459,100),(40,460,101),(41,461,102),(42,462,103),(43,463,104),(44,464,105),(45,465,106),(46,466,107),(47,467,108),(48,468,55),(49,469,56),(50,470,57),(51,471,58),(52,472,59),(53,473,60),(54,474,61),(109,193,364),(110,194,365),(111,195,366),(112,196,367),(113,197,368),(114,198,369),(115,199,370),(116,200,371),(117,201,372),(118,202,373),(119,203,374),(120,204,375),(121,205,376),(122,206,377),(123,207,378),(124,208,325),(125,209,326),(126,210,327),(127,211,328),(128,212,329),(129,213,330),(130,214,331),(131,215,332),(132,216,333),(133,163,334),(134,164,335),(135,165,336),(136,166,337),(137,167,338),(138,168,339),(139,169,340),(140,170,341),(141,171,342),(142,172,343),(143,173,344),(144,174,345),(145,175,346),(146,176,347),(147,177,348),(148,178,349),(149,179,350),(150,180,351),(151,181,352),(152,182,353),(153,183,354),(154,184,355),(155,185,356),(156,186,357),(157,187,358),(158,188,359),(159,189,360),(160,190,361),(161,191,362),(162,192,363),(217,293,413),(218,294,414),(219,295,415),(220,296,416),(221,297,417),(222,298,418),(223,299,419),(224,300,420),(225,301,421),(226,302,422),(227,303,423),(228,304,424),(229,305,425),(230,306,426),(231,307,427),(232,308,428),(233,309,429),(234,310,430),(235,311,431),(236,312,432),(237,313,379),(238,314,380),(239,315,381),(240,316,382),(241,317,383),(242,318,384),(243,319,385),(244,320,386),(245,321,387),(246,322,388),(247,323,389),(248,324,390),(249,271,391),(250,272,392),(251,273,393),(252,274,394),(253,275,395),(254,276,396),(255,277,397),(256,278,398),(257,279,399),(258,280,400),(259,281,401),(260,282,402),(261,283,403),(262,284,404),(263,285,405),(264,286,406),(265,287,407),(266,288,408),(267,289,409),(268,290,410),(269,291,411),(270,292,412)], [(1,158,408),(2,159,409),(3,160,410),(4,161,411),(5,162,412),(6,109,413),(7,110,414),(8,111,415),(9,112,416),(10,113,417),(11,114,418),(12,115,419),(13,116,420),(14,117,421),(15,118,422),(16,119,423),(17,120,424),(18,121,425),(19,122,426),(20,123,427),(21,124,428),(22,125,429),(23,126,430),(24,127,431),(25,128,432),(26,129,379),(27,130,380),(28,131,381),(29,132,382),(30,133,383),(31,134,384),(32,135,385),(33,136,386),(34,137,387),(35,138,388),(36,139,389),(37,140,390),(38,141,391),(39,142,392),(40,143,393),(41,144,394),(42,145,395),(43,146,396),(44,147,397),(45,148,398),(46,149,399),(47,150,400),(48,151,401),(49,152,402),(50,153,403),(51,154,404),(52,155,405),(53,156,406),(54,157,407),(55,352,281),(56,353,282),(57,354,283),(58,355,284),(59,356,285),(60,357,286),(61,358,287),(62,359,288),(63,360,289),(64,361,290),(65,362,291),(66,363,292),(67,364,293),(68,365,294),(69,366,295),(70,367,296),(71,368,297),(72,369,298),(73,370,299),(74,371,300),(75,372,301),(76,373,302),(77,374,303),(78,375,304),(79,376,305),(80,377,306),(81,378,307),(82,325,308),(83,326,309),(84,327,310),(85,328,311),(86,329,312),(87,330,313),(88,331,314),(89,332,315),(90,333,316),(91,334,317),(92,335,318),(93,336,319),(94,337,320),(95,338,321),(96,339,322),(97,340,323),(98,341,324),(99,342,271),(100,343,272),(101,344,273),(102,345,274),(103,346,275),(104,347,276),(105,348,277),(106,349,278),(107,350,279),(108,351,280),(163,241,450),(164,242,451),(165,243,452),(166,244,453),(167,245,454),(168,246,455),(169,247,456),(170,248,457),(171,249,458),(172,250,459),(173,251,460),(174,252,461),(175,253,462),(176,254,463),(177,255,464),(178,256,465),(179,257,466),(180,258,467),(181,259,468),(182,260,469),(183,261,470),(184,262,471),(185,263,472),(186,264,473),(187,265,474),(188,266,475),(189,267,476),(190,268,477),(191,269,478),(192,270,479),(193,217,480),(194,218,481),(195,219,482),(196,220,483),(197,221,484),(198,222,485),(199,223,486),(200,224,433),(201,225,434),(202,226,435),(203,227,436),(204,228,437),(205,229,438),(206,230,439),(207,231,440),(208,232,441),(209,233,442),(210,234,443),(211,235,444),(212,236,445),(213,237,446),(214,238,447),(215,239,448),(216,240,449)], [(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 | ··· | 3Z | 6A | ··· | 6Z | 9A | ··· | 9BB | 18A | ··· | 18BB | 27A | ··· | 27FF | 54A | ··· | 54FF |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 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 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C9 | C18 | C18 | C27 | C54 |
| kernel | C32×C54 | C32×C27 | C3×C54 | C32×C18 | C3×C27 | C32×C9 | C3×C18 | C32×C6 | C3×C9 | C33 | C3×C6 | C32 |
| # reps | 1 | 1 | 24 | 2 | 24 | 2 | 48 | 6 | 48 | 6 | 162 | 162 |
Matrix representation of C32×C54 ►in GL3(𝔽109) generated by
| 45 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 63 |
| 9 | 0 | 0 |
| 0 | 31 | 0 |
| 0 | 0 | 82 |
G:=sub<GL(3,GF(109))| [45,0,0,0,63,0,0,0,1],[1,0,0,0,1,0,0,0,63],[9,0,0,0,31,0,0,0,82] >;
C32×C54 in GAP, Magma, Sage, TeX
C_3^2\times C_{54} % in TeX
G:=Group("C3^2xC54"); // GroupNames label
G:=SmallGroup(486,207);
// by ID
G=gap.SmallGroup(486,207);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,118]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^54=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations