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G = C33×C18  order 486 = 2·35

Abelian group of type [3,3,3,18]

direct product, abelian, monomial, 3-elementary

Aliases: C33×C18, SmallGroup(486,250)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33×C18
Chief series
C1 — C3 — C32 — C33 — C34 — C33×C9 — C33×C18
Lower central
C1 — C33×C18
Upper central
C1 — C33×C18

Generators and relations for C33×C18
 G = < a,b,c,d | a3=b3=c3=d18=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 792, all normal (8 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C18, C3×C6, C3×C9, C33, C3×C18, C32×C6, C32×C9, C34, C32×C18, C33×C6, C33×C9, C33×C18
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, C33, C3×C18, C32×C6, C32×C9, C34, C32×C18, C33×C6, C33×C9, C33×C18

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

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

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

(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)| (1,173,306)(2,174,289)(3,175,290)(4,176,291)(5,177,292)(6,178,293)(7,179,294)(8,180,295)(9,163,296)(10,164,297)(11,165,298)(12,166,299)(13,167,300)(14,168,301)(15,169,302)(16,170,303)(17,171,304)(18,172,305)(19,332,250)(20,333,251)(21,334,252)(22,335,235)(23,336,236)(24,337,237)(25,338,238)(26,339,239)(27,340,240)(28,341,241)(29,342,242)(30,325,243)(31,326,244)(32,327,245)(33,328,246)(34,329,247)(35,330,248)(36,331,249)(37,68,384)(38,69,385)(39,70,386)(40,71,387)(41,72,388)(42,55,389)(43,56,390)(44,57,391)(45,58,392)(46,59,393)(47,60,394)(48,61,395)(49,62,396)(50,63,379)(51,64,380)(52,65,381)(53,66,382)(54,67,383)(73,362,140)(74,363,141)(75,364,142)(76,365,143)(77,366,144)(78,367,127)(79,368,128)(80,369,129)(81,370,130)(82,371,131)(83,372,132)(84,373,133)(85,374,134)(86,375,135)(87,376,136)(88,377,137)(89,378,138)(90,361,139)(91,184,152)(92,185,153)(93,186,154)(94,187,155)(95,188,156)(96,189,157)(97,190,158)(98,191,159)(99,192,160)(100,193,161)(101,194,162)(102,195,145)(103,196,146)(104,197,147)(105,198,148)(106,181,149)(107,182,150)(108,183,151)(109,280,416)(110,281,417)(111,282,418)(112,283,419)(113,284,420)(114,285,421)(115,286,422)(116,287,423)(117,288,424)(118,271,425)(119,272,426)(120,273,427)(121,274,428)(122,275,429)(123,276,430)(124,277,431)(125,278,432)(126,279,415)(199,409,343)(200,410,344)(201,411,345)(202,412,346)(203,413,347)(204,414,348)(205,397,349)(206,398,350)(207,399,351)(208,400,352)(209,401,353)(210,402,354)(211,403,355)(212,404,356)(213,405,357)(214,406,358)(215,407,359)(216,408,360)(217,476,439)(218,477,440)(219,478,441)(220,479,442)(221,480,443)(222,481,444)(223,482,445)(224,483,446)(225,484,447)(226,485,448)(227,486,449)(228,469,450)(229,470,433)(230,471,434)(231,472,435)(232,473,436)(233,474,437)(234,475,438)(253,468,323)(254,451,324)(255,452,307)(256,453,308)(257,454,309)(258,455,310)(259,456,311)(260,457,312)(261,458,313)(262,459,314)(263,460,315)(264,461,316)(265,462,317)(266,463,318)(267,464,319)(268,465,320)(269,466,321)(270,467,322), (1,381,264)(2,382,265)(3,383,266)(4,384,267)(5,385,268)(6,386,269)(7,387,270)(8,388,253)(9,389,254)(10,390,255)(11,391,256)(12,392,257)(13,393,258)(14,394,259)(15,395,260)(16,396,261)(17,379,262)(18,380,263)(19,188,444)(20,189,445)(21,190,446)(22,191,447)(23,192,448)(24,193,449)(25,194,450)(26,195,433)(27,196,434)(28,197,435)(29,198,436)(30,181,437)(31,182,438)(32,183,439)(33,184,440)(34,185,441)(35,186,442)(36,187,443)(37,464,176)(38,465,177)(39,466,178)(40,467,179)(41,468,180)(42,451,163)(43,452,164)(44,453,165)(45,454,166)(46,455,167)(47,456,168)(48,457,169)(49,458,170)(50,459,171)(51,460,172)(52,461,173)(53,462,174)(54,463,175)(55,324,296)(56,307,297)(57,308,298)(58,309,299)(59,310,300)(60,311,301)(61,312,302)(62,313,303)(63,314,304)(64,315,305)(65,316,306)(66,317,289)(67,318,290)(68,319,291)(69,320,292)(70,321,293)(71,322,294)(72,323,295)(73,429,405)(74,430,406)(75,431,407)(76,432,408)(77,415,409)(78,416,410)(79,417,411)(80,418,412)(81,419,413)(82,420,414)(83,421,397)(84,422,398)(85,423,399)(86,424,400)(87,425,401)(88,426,402)(89,427,403)(90,428,404)(91,477,246)(92,478,247)(93,479,248)(94,480,249)(95,481,250)(96,482,251)(97,483,252)(98,484,235)(99,485,236)(100,486,237)(101,469,238)(102,470,239)(103,471,240)(104,472,241)(105,473,242)(106,474,243)(107,475,244)(108,476,245)(109,344,367)(110,345,368)(111,346,369)(112,347,370)(113,348,371)(114,349,372)(115,350,373)(116,351,374)(117,352,375)(118,353,376)(119,354,377)(120,355,378)(121,356,361)(122,357,362)(123,358,363)(124,359,364)(125,360,365)(126,343,366)(127,280,200)(128,281,201)(129,282,202)(130,283,203)(131,284,204)(132,285,205)(133,286,206)(134,287,207)(135,288,208)(136,271,209)(137,272,210)(138,273,211)(139,274,212)(140,275,213)(141,276,214)(142,277,215)(143,278,216)(144,279,199)(145,229,339)(146,230,340)(147,231,341)(148,232,342)(149,233,325)(150,234,326)(151,217,327)(152,218,328)(153,219,329)(154,220,330)(155,221,331)(156,222,332)(157,223,333)(158,224,334)(159,225,335)(160,226,336)(161,227,337)(162,228,338), (1,233,206)(2,234,207)(3,217,208)(4,218,209)(5,219,210)(6,220,211)(7,221,212)(8,222,213)(9,223,214)(10,224,215)(11,225,216)(12,226,199)(13,227,200)(14,228,201)(15,229,202)(16,230,203)(17,231,204)(18,232,205)(19,362,72)(20,363,55)(21,364,56)(22,365,57)(23,366,58)(24,367,59)(25,368,60)(26,369,61)(27,370,62)(28,371,63)(29,372,64)(30,373,65)(31,374,66)(32,375,67)(33,376,68)(34,377,69)(35,378,70)(36,361,71)(37,246,87)(38,247,88)(39,248,89)(40,249,90)(41,250,73)(42,251,74)(43,252,75)(44,235,76)(45,236,77)(46,237,78)(47,238,79)(48,239,80)(49,240,81)(50,241,82)(51,242,83)(52,243,84)(53,244,85)(54,245,86)(91,425,464)(92,426,465)(93,427,466)(94,428,467)(95,429,468)(96,430,451)(97,431,452)(98,432,453)(99,415,454)(100,416,455)(101,417,456)(102,418,457)(103,419,458)(104,420,459)(105,421,460)(106,422,461)(107,423,462)(108,424,463)(109,310,193)(110,311,194)(111,312,195)(112,313,196)(113,314,197)(114,315,198)(115,316,181)(116,317,182)(117,318,183)(118,319,184)(119,320,185)(120,321,186)(121,322,187)(122,323,188)(123,324,189)(124,307,190)(125,308,191)(126,309,192)(127,393,337)(128,394,338)(129,395,339)(130,396,340)(131,379,341)(132,380,342)(133,381,325)(134,382,326)(135,383,327)(136,384,328)(137,385,329)(138,386,330)(139,387,331)(140,388,332)(141,389,333)(142,390,334)(143,391,335)(144,392,336)(145,282,260)(146,283,261)(147,284,262)(148,285,263)(149,286,264)(150,287,265)(151,288,266)(152,271,267)(153,272,268)(154,273,269)(155,274,270)(156,275,253)(157,276,254)(158,277,255)(159,278,256)(160,279,257)(161,280,258)(162,281,259)(163,482,406)(164,483,407)(165,484,408)(166,485,409)(167,486,410)(168,469,411)(169,470,412)(170,471,413)(171,472,414)(172,473,397)(173,474,398)(174,475,399)(175,476,400)(176,477,401)(177,478,402)(178,479,403)(179,480,404)(180,481,405)(289,438,351)(290,439,352)(291,440,353)(292,441,354)(293,442,355)(294,443,356)(295,444,357)(296,445,358)(297,446,359)(298,447,360)(299,448,343)(300,449,344)(301,450,345)(302,433,346)(303,434,347)(304,435,348)(305,436,349)(306,437,350), (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( (1,173,306)(2,174,289)(3,175,290)(4,176,291)(5,177,292)(6,178,293)(7,179,294)(8,180,295)(9,163,296)(10,164,297)(11,165,298)(12,166,299)(13,167,300)(14,168,301)(15,169,302)(16,170,303)(17,171,304)(18,172,305)(19,332,250)(20,333,251)(21,334,252)(22,335,235)(23,336,236)(24,337,237)(25,338,238)(26,339,239)(27,340,240)(28,341,241)(29,342,242)(30,325,243)(31,326,244)(32,327,245)(33,328,246)(34,329,247)(35,330,248)(36,331,249)(37,68,384)(38,69,385)(39,70,386)(40,71,387)(41,72,388)(42,55,389)(43,56,390)(44,57,391)(45,58,392)(46,59,393)(47,60,394)(48,61,395)(49,62,396)(50,63,379)(51,64,380)(52,65,381)(53,66,382)(54,67,383)(73,362,140)(74,363,141)(75,364,142)(76,365,143)(77,366,144)(78,367,127)(79,368,128)(80,369,129)(81,370,130)(82,371,131)(83,372,132)(84,373,133)(85,374,134)(86,375,135)(87,376,136)(88,377,137)(89,378,138)(90,361,139)(91,184,152)(92,185,153)(93,186,154)(94,187,155)(95,188,156)(96,189,157)(97,190,158)(98,191,159)(99,192,160)(100,193,161)(101,194,162)(102,195,145)(103,196,146)(104,197,147)(105,198,148)(106,181,149)(107,182,150)(108,183,151)(109,280,416)(110,281,417)(111,282,418)(112,283,419)(113,284,420)(114,285,421)(115,286,422)(116,287,423)(117,288,424)(118,271,425)(119,272,426)(120,273,427)(121,274,428)(122,275,429)(123,276,430)(124,277,431)(125,278,432)(126,279,415)(199,409,343)(200,410,344)(201,411,345)(202,412,346)(203,413,347)(204,414,348)(205,397,349)(206,398,350)(207,399,351)(208,400,352)(209,401,353)(210,402,354)(211,403,355)(212,404,356)(213,405,357)(214,406,358)(215,407,359)(216,408,360)(217,476,439)(218,477,440)(219,478,441)(220,479,442)(221,480,443)(222,481,444)(223,482,445)(224,483,446)(225,484,447)(226,485,448)(227,486,449)(228,469,450)(229,470,433)(230,471,434)(231,472,435)(232,473,436)(233,474,437)(234,475,438)(253,468,323)(254,451,324)(255,452,307)(256,453,308)(257,454,309)(258,455,310)(259,456,311)(260,457,312)(261,458,313)(262,459,314)(263,460,315)(264,461,316)(265,462,317)(266,463,318)(267,464,319)(268,465,320)(269,466,321)(270,467,322), (1,381,264)(2,382,265)(3,383,266)(4,384,267)(5,385,268)(6,386,269)(7,387,270)(8,388,253)(9,389,254)(10,390,255)(11,391,256)(12,392,257)(13,393,258)(14,394,259)(15,395,260)(16,396,261)(17,379,262)(18,380,263)(19,188,444)(20,189,445)(21,190,446)(22,191,447)(23,192,448)(24,193,449)(25,194,450)(26,195,433)(27,196,434)(28,197,435)(29,198,436)(30,181,437)(31,182,438)(32,183,439)(33,184,440)(34,185,441)(35,186,442)(36,187,443)(37,464,176)(38,465,177)(39,466,178)(40,467,179)(41,468,180)(42,451,163)(43,452,164)(44,453,165)(45,454,166)(46,455,167)(47,456,168)(48,457,169)(49,458,170)(50,459,171)(51,460,172)(52,461,173)(53,462,174)(54,463,175)(55,324,296)(56,307,297)(57,308,298)(58,309,299)(59,310,300)(60,311,301)(61,312,302)(62,313,303)(63,314,304)(64,315,305)(65,316,306)(66,317,289)(67,318,290)(68,319,291)(69,320,292)(70,321,293)(71,322,294)(72,323,295)(73,429,405)(74,430,406)(75,431,407)(76,432,408)(77,415,409)(78,416,410)(79,417,411)(80,418,412)(81,419,413)(82,420,414)(83,421,397)(84,422,398)(85,423,399)(86,424,400)(87,425,401)(88,426,402)(89,427,403)(90,428,404)(91,477,246)(92,478,247)(93,479,248)(94,480,249)(95,481,250)(96,482,251)(97,483,252)(98,484,235)(99,485,236)(100,486,237)(101,469,238)(102,470,239)(103,471,240)(104,472,241)(105,473,242)(106,474,243)(107,475,244)(108,476,245)(109,344,367)(110,345,368)(111,346,369)(112,347,370)(113,348,371)(114,349,372)(115,350,373)(116,351,374)(117,352,375)(118,353,376)(119,354,377)(120,355,378)(121,356,361)(122,357,362)(123,358,363)(124,359,364)(125,360,365)(126,343,366)(127,280,200)(128,281,201)(129,282,202)(130,283,203)(131,284,204)(132,285,205)(133,286,206)(134,287,207)(135,288,208)(136,271,209)(137,272,210)(138,273,211)(139,274,212)(140,275,213)(141,276,214)(142,277,215)(143,278,216)(144,279,199)(145,229,339)(146,230,340)(147,231,341)(148,232,342)(149,233,325)(150,234,326)(151,217,327)(152,218,328)(153,219,329)(154,220,330)(155,221,331)(156,222,332)(157,223,333)(158,224,334)(159,225,335)(160,226,336)(161,227,337)(162,228,338), (1,233,206)(2,234,207)(3,217,208)(4,218,209)(5,219,210)(6,220,211)(7,221,212)(8,222,213)(9,223,214)(10,224,215)(11,225,216)(12,226,199)(13,227,200)(14,228,201)(15,229,202)(16,230,203)(17,231,204)(18,232,205)(19,362,72)(20,363,55)(21,364,56)(22,365,57)(23,366,58)(24,367,59)(25,368,60)(26,369,61)(27,370,62)(28,371,63)(29,372,64)(30,373,65)(31,374,66)(32,375,67)(33,376,68)(34,377,69)(35,378,70)(36,361,71)(37,246,87)(38,247,88)(39,248,89)(40,249,90)(41,250,73)(42,251,74)(43,252,75)(44,235,76)(45,236,77)(46,237,78)(47,238,79)(48,239,80)(49,240,81)(50,241,82)(51,242,83)(52,243,84)(53,244,85)(54,245,86)(91,425,464)(92,426,465)(93,427,466)(94,428,467)(95,429,468)(96,430,451)(97,431,452)(98,432,453)(99,415,454)(100,416,455)(101,417,456)(102,418,457)(103,419,458)(104,420,459)(105,421,460)(106,422,461)(107,423,462)(108,424,463)(109,310,193)(110,311,194)(111,312,195)(112,313,196)(113,314,197)(114,315,198)(115,316,181)(116,317,182)(117,318,183)(118,319,184)(119,320,185)(120,321,186)(121,322,187)(122,323,188)(123,324,189)(124,307,190)(125,308,191)(126,309,192)(127,393,337)(128,394,338)(129,395,339)(130,396,340)(131,379,341)(132,380,342)(133,381,325)(134,382,326)(135,383,327)(136,384,328)(137,385,329)(138,386,330)(139,387,331)(140,388,332)(141,389,333)(142,390,334)(143,391,335)(144,392,336)(145,282,260)(146,283,261)(147,284,262)(148,285,263)(149,286,264)(150,287,265)(151,288,266)(152,271,267)(153,272,268)(154,273,269)(155,274,270)(156,275,253)(157,276,254)(158,277,255)(159,278,256)(160,279,257)(161,280,258)(162,281,259)(163,482,406)(164,483,407)(165,484,408)(166,485,409)(167,486,410)(168,469,411)(169,470,412)(170,471,413)(171,472,414)(172,473,397)(173,474,398)(174,475,399)(175,476,400)(176,477,401)(177,478,402)(178,479,403)(179,480,404)(180,481,405)(289,438,351)(290,439,352)(291,440,353)(292,441,354)(293,442,355)(294,443,356)(295,444,357)(296,445,358)(297,446,359)(298,447,360)(299,448,343)(300,449,344)(301,450,345)(302,433,346)(303,434,347)(304,435,348)(305,436,349)(306,437,350), (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([[(1,173,306),(2,174,289),(3,175,290),(4,176,291),(5,177,292),(6,178,293),(7,179,294),(8,180,295),(9,163,296),(10,164,297),(11,165,298),(12,166,299),(13,167,300),(14,168,301),(15,169,302),(16,170,303),(17,171,304),(18,172,305),(19,332,250),(20,333,251),(21,334,252),(22,335,235),(23,336,236),(24,337,237),(25,338,238),(26,339,239),(27,340,240),(28,341,241),(29,342,242),(30,325,243),(31,326,244),(32,327,245),(33,328,246),(34,329,247),(35,330,248),(36,331,249),(37,68,384),(38,69,385),(39,70,386),(40,71,387),(41,72,388),(42,55,389),(43,56,390),(44,57,391),(45,58,392),(46,59,393),(47,60,394),(48,61,395),(49,62,396),(50,63,379),(51,64,380),(52,65,381),(53,66,382),(54,67,383),(73,362,140),(74,363,141),(75,364,142),(76,365,143),(77,366,144),(78,367,127),(79,368,128),(80,369,129),(81,370,130),(82,371,131),(83,372,132),(84,373,133),(85,374,134),(86,375,135),(87,376,136),(88,377,137),(89,378,138),(90,361,139),(91,184,152),(92,185,153),(93,186,154),(94,187,155),(95,188,156),(96,189,157),(97,190,158),(98,191,159),(99,192,160),(100,193,161),(101,194,162),(102,195,145),(103,196,146),(104,197,147),(105,198,148),(106,181,149),(107,182,150),(108,183,151),(109,280,416),(110,281,417),(111,282,418),(112,283,419),(113,284,420),(114,285,421),(115,286,422),(116,287,423),(117,288,424),(118,271,425),(119,272,426),(120,273,427),(121,274,428),(122,275,429),(123,276,430),(124,277,431),(125,278,432),(126,279,415),(199,409,343),(200,410,344),(201,411,345),(202,412,346),(203,413,347),(204,414,348),(205,397,349),(206,398,350),(207,399,351),(208,400,352),(209,401,353),(210,402,354),(211,403,355),(212,404,356),(213,405,357),(214,406,358),(215,407,359),(216,408,360),(217,476,439),(218,477,440),(219,478,441),(220,479,442),(221,480,443),(222,481,444),(223,482,445),(224,483,446),(225,484,447),(226,485,448),(227,486,449),(228,469,450),(229,470,433),(230,471,434),(231,472,435),(232,473,436),(233,474,437),(234,475,438),(253,468,323),(254,451,324),(255,452,307),(256,453,308),(257,454,309),(258,455,310),(259,456,311),(260,457,312),(261,458,313),(262,459,314),(263,460,315),(264,461,316),(265,462,317),(266,463,318),(267,464,319),(268,465,320),(269,466,321),(270,467,322)], [(1,381,264),(2,382,265),(3,383,266),(4,384,267),(5,385,268),(6,386,269),(7,387,270),(8,388,253),(9,389,254),(10,390,255),(11,391,256),(12,392,257),(13,393,258),(14,394,259),(15,395,260),(16,396,261),(17,379,262),(18,380,263),(19,188,444),(20,189,445),(21,190,446),(22,191,447),(23,192,448),(24,193,449),(25,194,450),(26,195,433),(27,196,434),(28,197,435),(29,198,436),(30,181,437),(31,182,438),(32,183,439),(33,184,440),(34,185,441),(35,186,442),(36,187,443),(37,464,176),(38,465,177),(39,466,178),(40,467,179),(41,468,180),(42,451,163),(43,452,164),(44,453,165),(45,454,166),(46,455,167),(47,456,168),(48,457,169),(49,458,170),(50,459,171),(51,460,172),(52,461,173),(53,462,174),(54,463,175),(55,324,296),(56,307,297),(57,308,298),(58,309,299),(59,310,300),(60,311,301),(61,312,302),(62,313,303),(63,314,304),(64,315,305),(65,316,306),(66,317,289),(67,318,290),(68,319,291),(69,320,292),(70,321,293),(71,322,294),(72,323,295),(73,429,405),(74,430,406),(75,431,407),(76,432,408),(77,415,409),(78,416,410),(79,417,411),(80,418,412),(81,419,413),(82,420,414),(83,421,397),(84,422,398),(85,423,399),(86,424,400),(87,425,401),(88,426,402),(89,427,403),(90,428,404),(91,477,246),(92,478,247),(93,479,248),(94,480,249),(95,481,250),(96,482,251),(97,483,252),(98,484,235),(99,485,236),(100,486,237),(101,469,238),(102,470,239),(103,471,240),(104,472,241),(105,473,242),(106,474,243),(107,475,244),(108,476,245),(109,344,367),(110,345,368),(111,346,369),(112,347,370),(113,348,371),(114,349,372),(115,350,373),(116,351,374),(117,352,375),(118,353,376),(119,354,377),(120,355,378),(121,356,361),(122,357,362),(123,358,363),(124,359,364),(125,360,365),(126,343,366),(127,280,200),(128,281,201),(129,282,202),(130,283,203),(131,284,204),(132,285,205),(133,286,206),(134,287,207),(135,288,208),(136,271,209),(137,272,210),(138,273,211),(139,274,212),(140,275,213),(141,276,214),(142,277,215),(143,278,216),(144,279,199),(145,229,339),(146,230,340),(147,231,341),(148,232,342),(149,233,325),(150,234,326),(151,217,327),(152,218,328),(153,219,329),(154,220,330),(155,221,331),(156,222,332),(157,223,333),(158,224,334),(159,225,335),(160,226,336),(161,227,337),(162,228,338)], [(1,233,206),(2,234,207),(3,217,208),(4,218,209),(5,219,210),(6,220,211),(7,221,212),(8,222,213),(9,223,214),(10,224,215),(11,225,216),(12,226,199),(13,227,200),(14,228,201),(15,229,202),(16,230,203),(17,231,204),(18,232,205),(19,362,72),(20,363,55),(21,364,56),(22,365,57),(23,366,58),(24,367,59),(25,368,60),(26,369,61),(27,370,62),(28,371,63),(29,372,64),(30,373,65),(31,374,66),(32,375,67),(33,376,68),(34,377,69),(35,378,70),(36,361,71),(37,246,87),(38,247,88),(39,248,89),(40,249,90),(41,250,73),(42,251,74),(43,252,75),(44,235,76),(45,236,77),(46,237,78),(47,238,79),(48,239,80),(49,240,81),(50,241,82),(51,242,83),(52,243,84),(53,244,85),(54,245,86),(91,425,464),(92,426,465),(93,427,466),(94,428,467),(95,429,468),(96,430,451),(97,431,452),(98,432,453),(99,415,454),(100,416,455),(101,417,456),(102,418,457),(103,419,458),(104,420,459),(105,421,460),(106,422,461),(107,423,462),(108,424,463),(109,310,193),(110,311,194),(111,312,195),(112,313,196),(113,314,197),(114,315,198),(115,316,181),(116,317,182),(117,318,183),(118,319,184),(119,320,185),(120,321,186),(121,322,187),(122,323,188),(123,324,189),(124,307,190),(125,308,191),(126,309,192),(127,393,337),(128,394,338),(129,395,339),(130,396,340),(131,379,341),(132,380,342),(133,381,325),(134,382,326),(135,383,327),(136,384,328),(137,385,329),(138,386,330),(139,387,331),(140,388,332),(141,389,333),(142,390,334),(143,391,335),(144,392,336),(145,282,260),(146,283,261),(147,284,262),(148,285,263),(149,286,264),(150,287,265),(151,288,266),(152,271,267),(153,272,268),(154,273,269),(155,274,270),(156,275,253),(157,276,254),(158,277,255),(159,278,256),(160,279,257),(161,280,258),(162,281,259),(163,482,406),(164,483,407),(165,484,408),(166,485,409),(167,486,410),(168,469,411),(169,470,412),(170,471,413),(171,472,414),(172,473,397),(173,474,398),(174,475,399),(175,476,400),(176,477,401),(177,478,402),(178,479,403),(179,480,404),(180,481,405),(289,438,351),(290,439,352),(291,440,353),(292,441,354),(293,442,355),(294,443,356),(295,444,357),(296,445,358),(297,446,359),(298,447,360),(299,448,343),(300,449,344),(301,450,345),(302,433,346),(303,434,347),(304,435,348),(305,436,349),(306,437,350)], [(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···3CB6A···6CB9A···9FF18A···18FF
order123···36···69···918···18
size111···11···11···11···1

486 irreducible representations

dim11111111
type++
imageC1C2C3C3C6C6C9C18
kernelC33×C18C33×C9C32×C18C33×C6C32×C9C34C32×C6C33
# reps11782782162162

Matrix representation of C33×C18 ►in GL4(𝔽19) generated by

1000
0700
00110
00011
,
7000
0700
0070
00011
,
7000
01100
0070
0001
,
8000
0900
0090
0009
G:=sub<GL(4,GF(19))| [1,0,0,0,0,7,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,11],[7,0,0,0,0,11,0,0,0,0,7,0,0,0,0,1],[8,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9] >;
 

C33×C18 in GAP, Magma, Sage, TeX 

C_3^3\times C_{18}
 
% in TeX
 
G:=Group("C3^3xC18");
 
// GroupNames label
 
G:=SmallGroup(486,250);
 
// by ID
 
G=gap.SmallGroup(486,250);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=d^18=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
 
// generators/relations
 

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