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## G = C32×C54order 486 = 2·35

### Abelian group of type [3,3,54]

Aliases: C32×C54, SmallGroup(486,207)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C54
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C32×C27 — C32×C54
 Lower central C1 — C32×C54
 Upper central 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 [×12], C6, C6 [×12], C9, C9 [×8], C32 [×13], C18, C18 [×8], C3×C6 [×13], C27 [×9], C3×C9 [×12], C33, C54 [×9], C3×C18 [×12], C32×C6, C3×C27 [×12], C32×C9, C3×C54 [×12], C32×C18, C32×C27, C32×C54
Quotients: C1, C2, C3 [×13], C6 [×13], C9 [×9], C32 [×13], C18 [×9], C3×C6 [×13], C27 [×9], C3×C9 [×12], C33, C54 [×9], C3×C18 [×12], C32×C6, C3×C27 [×12], C32×C9, C3×C54 [×12], C32×C18, C32×C27, C32×C54

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

׿
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𝔽