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## G = C2×C9⋊C27order 486 = 2·35

### Direct product of C2 and C9⋊C27

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C9⋊C27
G = < a,b,c | a2=b9=c27=1, ab=ba, ac=ca, cbc-1=b7 >

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

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 3 3 type + + image C1 C2 C3 C3 C6 C6 C9 C18 C27 C54 3- 1+2 C2×3- 1+2 C27⋊C3 C2×C27⋊C3 kernel C2×C9⋊C27 C9⋊C27 C9×C18 C3×C54 C92 C3×C27 C3×C18 C3×C9 C18 C9 C18 C9 C6 C3 # reps 1 1 2 6 2 6 18 18 54 54 6 6 12 12

Matrix representation of C2×C9⋊C27 in GL4(𝔽109) generated by

 108 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 45 0 0 0 0 0 63 0 0 0 0 63 0 1 0 0
,
 49 0 0 0 0 90 6 76 0 76 2 51 0 51 101 17
G:=sub<GL(4,GF(109))| [108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[45,0,0,0,0,0,0,1,0,63,0,0,0,0,63,0],[49,0,0,0,0,90,76,51,0,6,2,101,0,76,51,17] >;

C2×C9⋊C27 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes C_{27}
% in TeX

G:=Group("C2xC9:C27");
// GroupNames label

G:=SmallGroup(486,81);
// by ID

G=gap.SmallGroup(486,81);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,68,118]);
// Polycyclic

G:=Group<a,b,c|a^2=b^9=c^27=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations

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