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## G = C2×C27⋊2C9order 486 = 2·35

### Direct product of C2 and C27⋊2C9

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C27⋊2C9
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C27⋊2C9 — C2×C27⋊2C9
 Lower central C1 — C3 — C2×C27⋊2C9
 Upper central C1 — C3×C18 — C2×C27⋊2C9

Generators and relations for C2×C272C9
G = < a,b,c | a2=b27=c9=1, ab=ba, ac=ca, cbc-1=b10 >

Smallest permutation representation of C2×C272C9
Regular action on 486 points
Generators in S486
(1 413)(2 414)(3 415)(4 416)(5 417)(6 418)(7 419)(8 420)(9 421)(10 422)(11 423)(12 424)(13 425)(14 426)(15 427)(16 428)(17 429)(18 430)(19 431)(20 432)(21 406)(22 407)(23 408)(24 409)(25 410)(26 411)(27 412)(28 343)(29 344)(30 345)(31 346)(32 347)(33 348)(34 349)(35 350)(36 351)(37 325)(38 326)(39 327)(40 328)(41 329)(42 330)(43 331)(44 332)(45 333)(46 334)(47 335)(48 336)(49 337)(50 338)(51 339)(52 340)(53 341)(54 342)(55 152)(56 153)(57 154)(58 155)(59 156)(60 157)(61 158)(62 159)(63 160)(64 161)(65 162)(66 136)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(81 151)(82 450)(83 451)(84 452)(85 453)(86 454)(87 455)(88 456)(89 457)(90 458)(91 459)(92 433)(93 434)(94 435)(95 436)(96 437)(97 438)(98 439)(99 440)(100 441)(101 442)(102 443)(103 444)(104 445)(105 446)(106 447)(107 448)(108 449)(109 284)(110 285)(111 286)(112 287)(113 288)(114 289)(115 290)(116 291)(117 292)(118 293)(119 294)(120 295)(121 296)(122 297)(123 271)(124 272)(125 273)(126 274)(127 275)(128 276)(129 277)(130 278)(131 279)(132 280)(133 281)(134 282)(135 283)(163 269)(164 270)(165 244)(166 245)(167 246)(168 247)(169 248)(170 249)(171 250)(172 251)(173 252)(174 253)(175 254)(176 255)(177 256)(178 257)(179 258)(180 259)(181 260)(182 261)(183 262)(184 263)(185 264)(186 265)(187 266)(188 267)(189 268)(190 355)(191 356)(192 357)(193 358)(194 359)(195 360)(196 361)(197 362)(198 363)(199 364)(200 365)(201 366)(202 367)(203 368)(204 369)(205 370)(206 371)(207 372)(208 373)(209 374)(210 375)(211 376)(212 377)(213 378)(214 352)(215 353)(216 354)(217 470)(218 471)(219 472)(220 473)(221 474)(222 475)(223 476)(224 477)(225 478)(226 479)(227 480)(228 481)(229 482)(230 483)(231 484)(232 485)(233 486)(234 460)(235 461)(236 462)(237 463)(238 464)(239 465)(240 466)(241 467)(242 468)(243 469)(298 389)(299 390)(300 391)(301 392)(302 393)(303 394)(304 395)(305 396)(306 397)(307 398)(308 399)(309 400)(310 401)(311 402)(312 403)(313 404)(314 405)(315 379)(316 380)(317 381)(318 382)(319 383)(320 384)(321 385)(322 386)(323 387)(324 388)
(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)
(1 465 40 127 67 392 456 192 169)(2 484 50 128 59 402 457 211 179)(3 476 33 129 78 385 458 203 189)(4 468 43 130 70 395 459 195 172)(5 460 53 131 62 405 433 214 182)(6 479 36 132 81 388 434 206 165)(7 471 46 133 73 398 435 198 175)(8 463 29 134 65 381 436 190 185)(9 482 39 135 57 391 437 209 168)(10 474 49 109 76 401 438 201 178)(11 466 32 110 68 384 439 193 188)(12 485 42 111 60 394 440 212 171)(13 477 52 112 79 404 441 204 181)(14 469 35 113 71 387 442 196 164)(15 461 45 114 63 397 443 215 174)(16 480 28 115 55 380 444 207 184)(17 472 38 116 74 390 445 199 167)(18 464 48 117 66 400 446 191 177)(19 483 31 118 58 383 447 210 187)(20 475 41 119 77 393 448 202 170)(21 467 51 120 69 403 449 194 180)(22 486 34 121 61 386 450 213 163)(23 478 44 122 80 396 451 205 173)(24 470 54 123 72 379 452 197 183)(25 462 37 124 64 389 453 216 166)(26 481 47 125 56 399 454 208 176)(27 473 30 126 75 382 455 200 186)(82 378 269 407 233 349 296 158 322)(83 370 252 408 225 332 297 150 305)(84 362 262 409 217 342 271 142 315)(85 354 245 410 236 325 272 161 298)(86 373 255 411 228 335 273 153 308)(87 365 265 412 220 345 274 145 318)(88 357 248 413 239 328 275 137 301)(89 376 258 414 231 338 276 156 311)(90 368 268 415 223 348 277 148 321)(91 360 251 416 242 331 278 140 304)(92 352 261 417 234 341 279 159 314)(93 371 244 418 226 351 280 151 324)(94 363 254 419 218 334 281 143 307)(95 355 264 420 237 344 282 162 317)(96 374 247 421 229 327 283 154 300)(97 366 257 422 221 337 284 146 310)(98 358 267 423 240 347 285 138 320)(99 377 250 424 232 330 286 157 303)(100 369 260 425 224 340 287 149 313)(101 361 270 426 243 350 288 141 323)(102 353 253 427 235 333 289 160 306)(103 372 263 428 227 343 290 152 316)(104 364 246 429 219 326 291 144 299)(105 356 256 430 238 336 292 136 309)(106 375 266 431 230 346 293 155 319)(107 367 249 432 222 329 294 147 302)(108 359 259 406 241 339 295 139 312)

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

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

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

198 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 9A ··· 9R 9S ··· 9AJ 18A ··· 18R 18S ··· 18AJ 27A ··· 27BB 54A ··· 54BB order 1 2 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 1 ··· 1 1 ··· 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

198 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C6 C6 C9 C9 C18 C18 C27⋊C3 C2×C27⋊C3 kernel C2×C27⋊2C9 C27⋊2C9 C9×C18 C3×C54 C92 C3×C27 C54 C3×C18 C27 C3×C9 C6 C3 # reps 1 1 2 6 2 6 54 18 54 18 18 18

Matrix representation of C2×C272C9 in GL5(𝔽109)

 108 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 66 0 0 0 0 0 3 2 60 0 0 84 80 17 0 0 2 60 26
,
 1 0 0 0 0 0 38 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 63 0 0

G:=sub<GL(5,GF(109))| [108,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,66,0,0,0,0,0,3,84,2,0,0,2,80,60,0,0,60,17,26],[1,0,0,0,0,0,38,0,0,0,0,0,0,0,63,0,0,1,0,0,0,0,0,1,0] >;

C2×C272C9 in GAP, Magma, Sage, TeX

C_2\times C_{27}\rtimes_2C_9
% in TeX

G:=Group("C2xC27:2C9");
// GroupNames label

G:=SmallGroup(486,71);
// by ID

G=gap.SmallGroup(486,71);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,1520,176,237]);
// Polycyclic

G:=Group<a,b,c|a^2=b^27=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

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