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G = C2×C3.C92  order 486 = 2·35

Direct product of C2 and C3.C92

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

Aliases: C2×C3.C92, C6.1C92, (C3×C9)⋊8C18, (C3×C18)⋊1C9, C6.1(C9⋊C9), C3.1(C9×C18), (C3×C6).13He3, C6.1(C32⋊C9), C33.56(C3×C6), (C32×C9).13C6, (C32×C18).1C3, C32.18(C3×C18), C32.18(C2×He3), (C32×C6).37C32, (C3×C6).93- 1+2, C32.11(C2×3- 1+2), C3.1(C2×C9⋊C9), (C3×C6).13(C3×C9), C3.1(C2×C32⋊C9), SmallGroup(486,62)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×C3.C92
Chief series
C1 — C3 — C32 — C33 — C32×C9 — C3.C92 — C2×C3.C92
Lower central
C1 — C3 — C2×C3.C92
Upper central
C1 — C32×C6 — C2×C3.C92

Generators and relations for C2×C3.C92
 G = < a,b,c,d | a2=b3=c9=d9=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 234 in 138 conjugacy classes, 90 normal (10 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C18, C3×C6, C3×C6, C3×C9, C3×C9, C33, C3×C18, C3×C18, C32×C6, C32×C9, C32×C18, C3.C92, C2×C3.C92
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, He3, 3- 1+2, C3×C18, C2×He3, C2×3- 1+2, C92, C32⋊C9, C9⋊C9, C9×C18, C2×C32⋊C9, C2×C9⋊C9, C3.C92, C2×C3.C92

Smallest permutation representation of C2×C3.C92
►Regular action on 486 points
Generators in S486
(1 259)(2 260)(3 261)(4 253)(5 254)(6 255)(7 256)(8 257)(9 258)(10 410)(11 411)(12 412)(13 413)(14 414)(15 406)(16 407)(17 408)(18 409)(19 262)(20 263)(21 264)(22 265)(23 266)(24 267)(25 268)(26 269)(27 270)(28 271)(29 272)(30 273)(31 274)(32 275)(33 276)(34 277)(35 278)(36 279)(37 280)(38 281)(39 282)(40 283)(41 284)(42 285)(43 286)(44 287)(45 288)(46 289)(47 290)(48 291)(49 292)(50 293)(51 294)(52 295)(53 296)(54 297)(55 298)(56 299)(57 300)(58 301)(59 302)(60 303)(61 304)(62 305)(63 306)(64 307)(65 308)(66 309)(67 310)(68 311)(69 312)(70 313)(71 314)(72 315)(73 316)(74 317)(75 318)(76 319)(77 320)(78 321)(79 322)(80 323)(81 324)(82 246)(83 247)(84 248)(85 249)(86 250)(87 251)(88 252)(89 244)(90 245)(91 334)(92 335)(93 336)(94 337)(95 338)(96 339)(97 340)(98 341)(99 342)(100 343)(101 344)(102 345)(103 346)(104 347)(105 348)(106 349)(107 350)(108 351)(109 352)(110 353)(111 354)(112 355)(113 356)(114 357)(115 358)(116 359)(117 360)(118 361)(119 362)(120 363)(121 364)(122 365)(123 366)(124 367)(125 368)(126 369)(127 370)(128 371)(129 372)(130 373)(131 374)(132 375)(133 376)(134 377)(135 378)(136 379)(137 380)(138 381)(139 382)(140 383)(141 384)(142 385)(143 386)(144 387)(145 388)(146 389)(147 390)(148 391)(149 392)(150 393)(151 394)(152 395)(153 396)(154 397)(155 398)(156 399)(157 400)(158 401)(159 402)(160 403)(161 404)(162 405)(163 327)(164 328)(165 329)(166 330)(167 331)(168 332)(169 333)(170 325)(171 326)(172 415)(173 416)(174 417)(175 418)(176 419)(177 420)(178 421)(179 422)(180 423)(181 424)(182 425)(183 426)(184 427)(185 428)(186 429)(187 430)(188 431)(189 432)(190 433)(191 434)(192 435)(193 436)(194 437)(195 438)(196 439)(197 440)(198 441)(199 442)(200 443)(201 444)(202 445)(203 446)(204 447)(205 448)(206 449)(207 450)(208 451)(209 452)(210 453)(211 454)(212 455)(213 456)(214 457)(215 458)(216 459)(217 460)(218 461)(219 462)(220 463)(221 464)(222 465)(223 466)(224 467)(225 468)(226 469)(227 470)(228 471)(229 472)(230 473)(231 474)(232 475)(233 476)(234 477)(235 478)(236 479)(237 480)(238 481)(239 482)(240 483)(241 484)(242 485)(243 486)

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

(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)

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

198 conjugacy classes

class 1  2 3A···3Z6A···6Z9A···9BT18A···18BT
order123···36···69···918···18
size111···11···13···33···3

198 irreducible representations

dim1111113333
type++
imageC1C2C3C6C9C18He33- 1+2C2×He3C2×3- 1+2
kernelC2×C3.C92C3.C92C32×C18C32×C9C3×C18C3×C9C3×C6C3×C6C32C32
# reps11887272216216

Matrix representation of C2×C3.C92 ►in GL5(𝔽19)

10000
018000
00100
00010
00001
,
10000
01000
00700
00070
00007
,
40000
01000
006105
005915
0015174
,
170000
04000
00010
00001
00700

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[4,0,0,0,0,0,1,0,0,0,0,0,6,5,15,0,0,10,9,17,0,0,5,15,4],[17,0,0,0,0,0,4,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,1,0] >;
 

C2×C3.C92 in GAP, Magma, Sage, TeX 

C_2\times C_3.C_9^2
 
% in TeX
 
G:=Group("C2xC3.C9^2");
 
// GroupNames label
 
G:=SmallGroup(486,62);
 
// by ID
 
G=gap.SmallGroup(486,62);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,500]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^3=c^9=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
 
// generators/relations
 

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