direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C6×C9⋊C9, C9⋊4(C3×C18), (C3×C18)⋊4C9, C18⋊2(C3×C9), (C3×C9)⋊11C18, C6.2(C32×C9), (C32×C9).18C6, (C3×C6).16C33, (C32×C18).6C3, C33.57(C3×C6), C3.2(C32×C18), (C3×C18).26C32, C32.22(C3×C18), C32.18(C32×C6), (C32×C6).39C32, C3.2(C6×3- 1+2), C6.2(C3×3- 1+2), (C3×C6).113- 1+2, C32.13(C2×3- 1+2), (C3×C6).17(C3×C9), (C3×C9).20(C3×C6), SmallGroup(486,192)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C9⋊C9
G = < a,b,c | a6=b9=c9=1, ab=ba, ac=ca, cbc-1=b7 >
Subgroups: 252 in 180 conjugacy classes, 144 normal (12 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C32, C18, C18, C3×C6, C3×C6, C3×C9, C3×C9, C33, C3×C18, C3×C18, C32×C6, C9⋊C9, C32×C9, C32×C9, C2×C9⋊C9, C32×C18, C32×C18, C3×C9⋊C9, C6×C9⋊C9
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, 3- 1+2, C33, C3×C18, C2×3- 1+2, C32×C6, C9⋊C9, C32×C9, C3×3- 1+2, C2×C9⋊C9, C32×C18, C6×3- 1+2, C3×C9⋊C9, C6×C9⋊C9
►Regular action on 486 points
Generators in S486
(1 251 77 275 33 320)(2 252 78 276 34 321)(3 244 79 277 35 322)(4 245 80 278 36 323)(5 246 81 279 28 324)(6 247 73 271 29 316)(7 248 74 272 30 317)(8 249 75 273 31 318)(9 250 76 274 32 319)(10 301 121 253 25 364)(11 302 122 254 26 365)(12 303 123 255 27 366)(13 304 124 256 19 367)(14 305 125 257 20 368)(15 306 126 258 21 369)(16 298 118 259 22 361)(17 299 119 260 23 362)(18 300 120 261 24 363)(37 343 55 289 100 307)(38 344 56 290 101 308)(39 345 57 291 102 309)(40 346 58 292 103 310)(41 347 59 293 104 311)(42 348 60 294 105 312)(43 349 61 295 106 313)(44 350 62 296 107 314)(45 351 63 297 108 315)(46 234 459 270 477 216)(47 226 451 262 469 208)(48 227 452 263 470 209)(49 228 453 264 471 210)(50 229 454 265 472 211)(51 230 455 266 473 212)(52 231 456 267 474 213)(53 232 457 268 475 214)(54 233 458 269 476 215)(64 177 468 288 420 225)(65 178 460 280 421 217)(66 179 461 281 422 218)(67 180 462 282 423 219)(68 172 463 283 415 220)(69 173 464 284 416 221)(70 174 465 285 417 222)(71 175 466 286 418 223)(72 176 467 287 419 224)(82 403 115 325 160 358)(83 404 116 326 161 359)(84 405 117 327 162 360)(85 397 109 328 154 352)(86 398 110 329 155 353)(87 399 111 330 156 354)(88 400 112 331 157 355)(89 401 113 332 158 356)(90 402 114 333 159 357)(91 382 202 334 139 445)(92 383 203 335 140 446)(93 384 204 336 141 447)(94 385 205 337 142 448)(95 386 206 338 143 449)(96 387 207 339 144 450)(97 379 199 340 136 442)(98 380 200 341 137 443)(99 381 201 342 138 444)(127 424 145 370 181 388)(128 425 146 371 182 389)(129 426 147 372 183 390)(130 427 148 373 184 391)(131 428 149 374 185 392)(132 429 150 375 186 393)(133 430 151 376 187 394)(134 431 152 377 188 395)(135 432 153 378 189 396)(163 484 196 406 241 439)(164 485 197 407 242 440)(165 486 198 408 243 441)(166 478 190 409 235 433)(167 479 191 410 236 434)(168 480 192 411 237 435)(169 481 193 412 238 436)(170 482 194 413 239 437)(171 483 195 414 240 438)
(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)
(1 214 89 27 239 201 100 177 148)(2 209 87 19 243 199 101 172 146)(3 213 85 20 238 206 102 176 153)(4 208 83 21 242 204 103 180 151)(5 212 90 22 237 202 104 175 149)(6 216 88 23 241 200 105 179 147)(7 211 86 24 236 207 106 174 145)(8 215 84 25 240 205 107 178 152)(9 210 82 26 235 203 108 173 150)(10 171 142 44 280 188 75 233 117)(11 166 140 45 284 186 76 228 115)(12 170 138 37 288 184 77 232 113)(13 165 136 38 283 182 78 227 111)(14 169 143 39 287 189 79 231 109)(15 164 141 40 282 187 80 226 116)(16 168 139 41 286 185 81 230 114)(17 163 137 42 281 183 73 234 112)(18 167 144 43 285 181 74 229 110)(28 266 159 118 192 91 59 223 131)(29 270 157 119 196 98 60 218 129)(30 265 155 120 191 96 61 222 127)(31 269 162 121 195 94 62 217 134)(32 264 160 122 190 92 63 221 132)(33 268 158 123 194 99 55 225 130)(34 263 156 124 198 97 56 220 128)(35 267 154 125 193 95 57 224 135)(36 262 161 126 197 93 58 219 133)(46 400 362 439 341 312 461 372 247)(47 404 369 440 336 310 462 376 245)(48 399 367 441 340 308 463 371 252)(49 403 365 433 335 315 464 375 250)(50 398 363 434 339 313 465 370 248)(51 402 361 435 334 311 466 374 246)(52 397 368 436 338 309 467 378 244)(53 401 366 437 342 307 468 373 251)(54 405 364 438 337 314 460 377 249)(64 427 320 475 356 255 413 381 289)(65 431 318 476 360 253 414 385 296)(66 426 316 477 355 260 406 380 294)(67 430 323 469 359 258 407 384 292)(68 425 321 470 354 256 408 379 290)(69 429 319 471 358 254 409 383 297)(70 424 317 472 353 261 410 387 295)(71 428 324 473 357 259 411 382 293)(72 432 322 474 352 257 412 386 291)(271 459 331 299 484 443 348 422 390)(272 454 329 300 479 450 349 417 388)(273 458 327 301 483 448 350 421 395)(274 453 325 302 478 446 351 416 393)(275 457 332 303 482 444 343 420 391)(276 452 330 304 486 442 344 415 389)(277 456 328 305 481 449 345 419 396)(278 451 326 306 485 447 346 423 394)(279 455 333 298 480 445 347 418 392)
G:=sub<Sym(486)| (1,251,77,275,33,320)(2,252,78,276,34,321)(3,244,79,277,35,322)(4,245,80,278,36,323)(5,246,81,279,28,324)(6,247,73,271,29,316)(7,248,74,272,30,317)(8,249,75,273,31,318)(9,250,76,274,32,319)(10,301,121,253,25,364)(11,302,122,254,26,365)(12,303,123,255,27,366)(13,304,124,256,19,367)(14,305,125,257,20,368)(15,306,126,258,21,369)(16,298,118,259,22,361)(17,299,119,260,23,362)(18,300,120,261,24,363)(37,343,55,289,100,307)(38,344,56,290,101,308)(39,345,57,291,102,309)(40,346,58,292,103,310)(41,347,59,293,104,311)(42,348,60,294,105,312)(43,349,61,295,106,313)(44,350,62,296,107,314)(45,351,63,297,108,315)(46,234,459,270,477,216)(47,226,451,262,469,208)(48,227,452,263,470,209)(49,228,453,264,471,210)(50,229,454,265,472,211)(51,230,455,266,473,212)(52,231,456,267,474,213)(53,232,457,268,475,214)(54,233,458,269,476,215)(64,177,468,288,420,225)(65,178,460,280,421,217)(66,179,461,281,422,218)(67,180,462,282,423,219)(68,172,463,283,415,220)(69,173,464,284,416,221)(70,174,465,285,417,222)(71,175,466,286,418,223)(72,176,467,287,419,224)(82,403,115,325,160,358)(83,404,116,326,161,359)(84,405,117,327,162,360)(85,397,109,328,154,352)(86,398,110,329,155,353)(87,399,111,330,156,354)(88,400,112,331,157,355)(89,401,113,332,158,356)(90,402,114,333,159,357)(91,382,202,334,139,445)(92,383,203,335,140,446)(93,384,204,336,141,447)(94,385,205,337,142,448)(95,386,206,338,143,449)(96,387,207,339,144,450)(97,379,199,340,136,442)(98,380,200,341,137,443)(99,381,201,342,138,444)(127,424,145,370,181,388)(128,425,146,371,182,389)(129,426,147,372,183,390)(130,427,148,373,184,391)(131,428,149,374,185,392)(132,429,150,375,186,393)(133,430,151,376,187,394)(134,431,152,377,188,395)(135,432,153,378,189,396)(163,484,196,406,241,439)(164,485,197,407,242,440)(165,486,198,408,243,441)(166,478,190,409,235,433)(167,479,191,410,236,434)(168,480,192,411,237,435)(169,481,193,412,238,436)(170,482,194,413,239,437)(171,483,195,414,240,438), (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), (1,214,89,27,239,201,100,177,148)(2,209,87,19,243,199,101,172,146)(3,213,85,20,238,206,102,176,153)(4,208,83,21,242,204,103,180,151)(5,212,90,22,237,202,104,175,149)(6,216,88,23,241,200,105,179,147)(7,211,86,24,236,207,106,174,145)(8,215,84,25,240,205,107,178,152)(9,210,82,26,235,203,108,173,150)(10,171,142,44,280,188,75,233,117)(11,166,140,45,284,186,76,228,115)(12,170,138,37,288,184,77,232,113)(13,165,136,38,283,182,78,227,111)(14,169,143,39,287,189,79,231,109)(15,164,141,40,282,187,80,226,116)(16,168,139,41,286,185,81,230,114)(17,163,137,42,281,183,73,234,112)(18,167,144,43,285,181,74,229,110)(28,266,159,118,192,91,59,223,131)(29,270,157,119,196,98,60,218,129)(30,265,155,120,191,96,61,222,127)(31,269,162,121,195,94,62,217,134)(32,264,160,122,190,92,63,221,132)(33,268,158,123,194,99,55,225,130)(34,263,156,124,198,97,56,220,128)(35,267,154,125,193,95,57,224,135)(36,262,161,126,197,93,58,219,133)(46,400,362,439,341,312,461,372,247)(47,404,369,440,336,310,462,376,245)(48,399,367,441,340,308,463,371,252)(49,403,365,433,335,315,464,375,250)(50,398,363,434,339,313,465,370,248)(51,402,361,435,334,311,466,374,246)(52,397,368,436,338,309,467,378,244)(53,401,366,437,342,307,468,373,251)(54,405,364,438,337,314,460,377,249)(64,427,320,475,356,255,413,381,289)(65,431,318,476,360,253,414,385,296)(66,426,316,477,355,260,406,380,294)(67,430,323,469,359,258,407,384,292)(68,425,321,470,354,256,408,379,290)(69,429,319,471,358,254,409,383,297)(70,424,317,472,353,261,410,387,295)(71,428,324,473,357,259,411,382,293)(72,432,322,474,352,257,412,386,291)(271,459,331,299,484,443,348,422,390)(272,454,329,300,479,450,349,417,388)(273,458,327,301,483,448,350,421,395)(274,453,325,302,478,446,351,416,393)(275,457,332,303,482,444,343,420,391)(276,452,330,304,486,442,344,415,389)(277,456,328,305,481,449,345,419,396)(278,451,326,306,485,447,346,423,394)(279,455,333,298,480,445,347,418,392)>;
G:=Group( (1,251,77,275,33,320)(2,252,78,276,34,321)(3,244,79,277,35,322)(4,245,80,278,36,323)(5,246,81,279,28,324)(6,247,73,271,29,316)(7,248,74,272,30,317)(8,249,75,273,31,318)(9,250,76,274,32,319)(10,301,121,253,25,364)(11,302,122,254,26,365)(12,303,123,255,27,366)(13,304,124,256,19,367)(14,305,125,257,20,368)(15,306,126,258,21,369)(16,298,118,259,22,361)(17,299,119,260,23,362)(18,300,120,261,24,363)(37,343,55,289,100,307)(38,344,56,290,101,308)(39,345,57,291,102,309)(40,346,58,292,103,310)(41,347,59,293,104,311)(42,348,60,294,105,312)(43,349,61,295,106,313)(44,350,62,296,107,314)(45,351,63,297,108,315)(46,234,459,270,477,216)(47,226,451,262,469,208)(48,227,452,263,470,209)(49,228,453,264,471,210)(50,229,454,265,472,211)(51,230,455,266,473,212)(52,231,456,267,474,213)(53,232,457,268,475,214)(54,233,458,269,476,215)(64,177,468,288,420,225)(65,178,460,280,421,217)(66,179,461,281,422,218)(67,180,462,282,423,219)(68,172,463,283,415,220)(69,173,464,284,416,221)(70,174,465,285,417,222)(71,175,466,286,418,223)(72,176,467,287,419,224)(82,403,115,325,160,358)(83,404,116,326,161,359)(84,405,117,327,162,360)(85,397,109,328,154,352)(86,398,110,329,155,353)(87,399,111,330,156,354)(88,400,112,331,157,355)(89,401,113,332,158,356)(90,402,114,333,159,357)(91,382,202,334,139,445)(92,383,203,335,140,446)(93,384,204,336,141,447)(94,385,205,337,142,448)(95,386,206,338,143,449)(96,387,207,339,144,450)(97,379,199,340,136,442)(98,380,200,341,137,443)(99,381,201,342,138,444)(127,424,145,370,181,388)(128,425,146,371,182,389)(129,426,147,372,183,390)(130,427,148,373,184,391)(131,428,149,374,185,392)(132,429,150,375,186,393)(133,430,151,376,187,394)(134,431,152,377,188,395)(135,432,153,378,189,396)(163,484,196,406,241,439)(164,485,197,407,242,440)(165,486,198,408,243,441)(166,478,190,409,235,433)(167,479,191,410,236,434)(168,480,192,411,237,435)(169,481,193,412,238,436)(170,482,194,413,239,437)(171,483,195,414,240,438), (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), (1,214,89,27,239,201,100,177,148)(2,209,87,19,243,199,101,172,146)(3,213,85,20,238,206,102,176,153)(4,208,83,21,242,204,103,180,151)(5,212,90,22,237,202,104,175,149)(6,216,88,23,241,200,105,179,147)(7,211,86,24,236,207,106,174,145)(8,215,84,25,240,205,107,178,152)(9,210,82,26,235,203,108,173,150)(10,171,142,44,280,188,75,233,117)(11,166,140,45,284,186,76,228,115)(12,170,138,37,288,184,77,232,113)(13,165,136,38,283,182,78,227,111)(14,169,143,39,287,189,79,231,109)(15,164,141,40,282,187,80,226,116)(16,168,139,41,286,185,81,230,114)(17,163,137,42,281,183,73,234,112)(18,167,144,43,285,181,74,229,110)(28,266,159,118,192,91,59,223,131)(29,270,157,119,196,98,60,218,129)(30,265,155,120,191,96,61,222,127)(31,269,162,121,195,94,62,217,134)(32,264,160,122,190,92,63,221,132)(33,268,158,123,194,99,55,225,130)(34,263,156,124,198,97,56,220,128)(35,267,154,125,193,95,57,224,135)(36,262,161,126,197,93,58,219,133)(46,400,362,439,341,312,461,372,247)(47,404,369,440,336,310,462,376,245)(48,399,367,441,340,308,463,371,252)(49,403,365,433,335,315,464,375,250)(50,398,363,434,339,313,465,370,248)(51,402,361,435,334,311,466,374,246)(52,397,368,436,338,309,467,378,244)(53,401,366,437,342,307,468,373,251)(54,405,364,438,337,314,460,377,249)(64,427,320,475,356,255,413,381,289)(65,431,318,476,360,253,414,385,296)(66,426,316,477,355,260,406,380,294)(67,430,323,469,359,258,407,384,292)(68,425,321,470,354,256,408,379,290)(69,429,319,471,358,254,409,383,297)(70,424,317,472,353,261,410,387,295)(71,428,324,473,357,259,411,382,293)(72,432,322,474,352,257,412,386,291)(271,459,331,299,484,443,348,422,390)(272,454,329,300,479,450,349,417,388)(273,458,327,301,483,448,350,421,395)(274,453,325,302,478,446,351,416,393)(275,457,332,303,482,444,343,420,391)(276,452,330,304,486,442,344,415,389)(277,456,328,305,481,449,345,419,396)(278,451,326,306,485,447,346,423,394)(279,455,333,298,480,445,347,418,392) );
G=PermutationGroup([[(1,251,77,275,33,320),(2,252,78,276,34,321),(3,244,79,277,35,322),(4,245,80,278,36,323),(5,246,81,279,28,324),(6,247,73,271,29,316),(7,248,74,272,30,317),(8,249,75,273,31,318),(9,250,76,274,32,319),(10,301,121,253,25,364),(11,302,122,254,26,365),(12,303,123,255,27,366),(13,304,124,256,19,367),(14,305,125,257,20,368),(15,306,126,258,21,369),(16,298,118,259,22,361),(17,299,119,260,23,362),(18,300,120,261,24,363),(37,343,55,289,100,307),(38,344,56,290,101,308),(39,345,57,291,102,309),(40,346,58,292,103,310),(41,347,59,293,104,311),(42,348,60,294,105,312),(43,349,61,295,106,313),(44,350,62,296,107,314),(45,351,63,297,108,315),(46,234,459,270,477,216),(47,226,451,262,469,208),(48,227,452,263,470,209),(49,228,453,264,471,210),(50,229,454,265,472,211),(51,230,455,266,473,212),(52,231,456,267,474,213),(53,232,457,268,475,214),(54,233,458,269,476,215),(64,177,468,288,420,225),(65,178,460,280,421,217),(66,179,461,281,422,218),(67,180,462,282,423,219),(68,172,463,283,415,220),(69,173,464,284,416,221),(70,174,465,285,417,222),(71,175,466,286,418,223),(72,176,467,287,419,224),(82,403,115,325,160,358),(83,404,116,326,161,359),(84,405,117,327,162,360),(85,397,109,328,154,352),(86,398,110,329,155,353),(87,399,111,330,156,354),(88,400,112,331,157,355),(89,401,113,332,158,356),(90,402,114,333,159,357),(91,382,202,334,139,445),(92,383,203,335,140,446),(93,384,204,336,141,447),(94,385,205,337,142,448),(95,386,206,338,143,449),(96,387,207,339,144,450),(97,379,199,340,136,442),(98,380,200,341,137,443),(99,381,201,342,138,444),(127,424,145,370,181,388),(128,425,146,371,182,389),(129,426,147,372,183,390),(130,427,148,373,184,391),(131,428,149,374,185,392),(132,429,150,375,186,393),(133,430,151,376,187,394),(134,431,152,377,188,395),(135,432,153,378,189,396),(163,484,196,406,241,439),(164,485,197,407,242,440),(165,486,198,408,243,441),(166,478,190,409,235,433),(167,479,191,410,236,434),(168,480,192,411,237,435),(169,481,193,412,238,436),(170,482,194,413,239,437),(171,483,195,414,240,438)], [(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)], [(1,214,89,27,239,201,100,177,148),(2,209,87,19,243,199,101,172,146),(3,213,85,20,238,206,102,176,153),(4,208,83,21,242,204,103,180,151),(5,212,90,22,237,202,104,175,149),(6,216,88,23,241,200,105,179,147),(7,211,86,24,236,207,106,174,145),(8,215,84,25,240,205,107,178,152),(9,210,82,26,235,203,108,173,150),(10,171,142,44,280,188,75,233,117),(11,166,140,45,284,186,76,228,115),(12,170,138,37,288,184,77,232,113),(13,165,136,38,283,182,78,227,111),(14,169,143,39,287,189,79,231,109),(15,164,141,40,282,187,80,226,116),(16,168,139,41,286,185,81,230,114),(17,163,137,42,281,183,73,234,112),(18,167,144,43,285,181,74,229,110),(28,266,159,118,192,91,59,223,131),(29,270,157,119,196,98,60,218,129),(30,265,155,120,191,96,61,222,127),(31,269,162,121,195,94,62,217,134),(32,264,160,122,190,92,63,221,132),(33,268,158,123,194,99,55,225,130),(34,263,156,124,198,97,56,220,128),(35,267,154,125,193,95,57,224,135),(36,262,161,126,197,93,58,219,133),(46,400,362,439,341,312,461,372,247),(47,404,369,440,336,310,462,376,245),(48,399,367,441,340,308,463,371,252),(49,403,365,433,335,315,464,375,250),(50,398,363,434,339,313,465,370,248),(51,402,361,435,334,311,466,374,246),(52,397,368,436,338,309,467,378,244),(53,401,366,437,342,307,468,373,251),(54,405,364,438,337,314,460,377,249),(64,427,320,475,356,255,413,381,289),(65,431,318,476,360,253,414,385,296),(66,426,316,477,355,260,406,380,294),(67,430,323,469,359,258,407,384,292),(68,425,321,470,354,256,408,379,290),(69,429,319,471,358,254,409,383,297),(70,424,317,472,353,261,410,387,295),(71,428,324,473,357,259,411,382,293),(72,432,322,474,352,257,412,386,291),(271,459,331,299,484,443,348,422,390),(272,454,329,300,479,450,349,417,388),(273,458,327,301,483,448,350,421,395),(274,453,325,302,478,446,351,416,393),(275,457,332,303,482,444,343,420,391),(276,452,330,304,486,442,344,415,389),(277,456,328,305,481,449,345,419,396),(278,451,326,306,485,447,346,423,394),(279,455,333,298,480,445,347,418,392)]])
198 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 6A | ··· | 6Z | 9A | ··· | 9BT | 18A | ··· | 18BT |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
198 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 | 3- 1+2 | C2×3- 1+2 |
| kernel | C6×C9⋊C9 | C3×C9⋊C9 | C2×C9⋊C9 | C32×C18 | C9⋊C9 | C32×C9 | C3×C18 | C3×C9 | C3×C6 | C32 |
| # reps | 1 | 1 | 18 | 8 | 18 | 8 | 54 | 54 | 18 | 18 |
Matrix representation of C6×C9⋊C9 ►in GL5(𝔽19)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 7 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 8 | 16 | 9 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 2 | 10 | 15 |
| 0 | 0 | 7 | 0 | 9 |
G:=sub<GL(5,GF(19))| [12,0,0,0,0,0,7,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[7,0,0,0,0,0,7,0,0,0,0,0,6,0,8,0,0,0,4,16,0,0,0,0,9],[6,0,0,0,0,0,7,0,0,0,0,0,0,2,7,0,0,7,10,0,0,0,0,15,9] >;
C6×C9⋊C9 in GAP, Magma, Sage, TeX
C_6\times C_9\rtimes C_9
% in TeX
G:=Group("C6xC9:C9"); // GroupNames label
G:=SmallGroup(486,192);
// by ID
G=gap.SmallGroup(486,192);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,176]);
// Polycyclic
G:=Group<a,b,c|a^6=b^9=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations