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## G = C3×C9×C18order 486 = 2·35

### Abelian group of type [3,9,18]

Aliases: C3×C9×C18, SmallGroup(486,190)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9×C18
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×C92 — C3×C9×C18
 Lower central C1 — C3×C9×C18
 Upper central C1 — C3×C9×C18

Generators and relations for C3×C9×C18
G = < a,b,c | a3=b9=c18=1, ab=ba, ac=ca, bc=cb >

Subgroups: 252, all normal (8 characteristic)
C1, C2, C3, C6, C9, C32, C32, C18, C3×C6, C3×C6, C3×C9, C33, C3×C18, C32×C6, C92, C32×C9, C9×C18, C32×C18, C3×C92, C3×C9×C18
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, C33, C3×C18, C32×C6, C92, C32×C9, C9×C18, C32×C18, C3×C92, C3×C9×C18

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

486 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C3 C6 C6 C9 C18 kernel C3×C9×C18 C3×C92 C9×C18 C32×C18 C92 C32×C9 C3×C18 C3×C9 # reps 1 1 18 8 18 8 216 216

Matrix representation of C3×C9×C18 in GL3(𝔽19) generated by

 11 0 0 0 1 0 0 0 7
,
 6 0 0 0 9 0 0 0 6
,
 16 0 0 0 13 0 0 0 13
G:=sub<GL(3,GF(19))| [11,0,0,0,1,0,0,0,7],[6,0,0,0,9,0,0,0,6],[16,0,0,0,13,0,0,0,13] >;

C3×C9×C18 in GAP, Magma, Sage, TeX

C_3\times C_9\times C_{18}
% in TeX

G:=Group("C3xC9xC18");
// GroupNames label

G:=SmallGroup(486,190);
// by ID

G=gap.SmallGroup(486,190);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,500]);
// Polycyclic

G:=Group<a,b,c|a^3=b^9=c^18=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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