direct product, abelian, monomial, 3-elementary
Aliases: C9×C54, SmallGroup(486,70)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C9×C54 |
| C1 — C9×C54 |
| C1 — C9×C54 |
Generators and relations for C9×C54
G = < a,b | a9=b54=1, ab=ba >
Subgroups: 72, all normal (12 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C18, C3×C6, C27, C3×C9, C3×C9, C54, C3×C18, C3×C18, C92, C3×C27, C9×C18, C3×C54, C9×C27, C9×C54
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C27, C3×C9, C54, C3×C18, C92, C3×C27, C9×C18, C3×C54, C9×C27, C9×C54
►Regular action on 486 points
Generators in S486
(1 77 351 113 180 397 265 472 320)(2 78 352 114 181 398 266 473 321)(3 79 353 115 182 399 267 474 322)(4 80 354 116 183 400 268 475 323)(5 81 355 117 184 401 269 476 324)(6 82 356 118 185 402 270 477 271)(7 83 357 119 186 403 217 478 272)(8 84 358 120 187 404 218 479 273)(9 85 359 121 188 405 219 480 274)(10 86 360 122 189 406 220 481 275)(11 87 361 123 190 407 221 482 276)(12 88 362 124 191 408 222 483 277)(13 89 363 125 192 409 223 484 278)(14 90 364 126 193 410 224 485 279)(15 91 365 127 194 411 225 486 280)(16 92 366 128 195 412 226 433 281)(17 93 367 129 196 413 227 434 282)(18 94 368 130 197 414 228 435 283)(19 95 369 131 198 415 229 436 284)(20 96 370 132 199 416 230 437 285)(21 97 371 133 200 417 231 438 286)(22 98 372 134 201 418 232 439 287)(23 99 373 135 202 419 233 440 288)(24 100 374 136 203 420 234 441 289)(25 101 375 137 204 421 235 442 290)(26 102 376 138 205 422 236 443 291)(27 103 377 139 206 423 237 444 292)(28 104 378 140 207 424 238 445 293)(29 105 325 141 208 425 239 446 294)(30 106 326 142 209 426 240 447 295)(31 107 327 143 210 427 241 448 296)(32 108 328 144 211 428 242 449 297)(33 55 329 145 212 429 243 450 298)(34 56 330 146 213 430 244 451 299)(35 57 331 147 214 431 245 452 300)(36 58 332 148 215 432 246 453 301)(37 59 333 149 216 379 247 454 302)(38 60 334 150 163 380 248 455 303)(39 61 335 151 164 381 249 456 304)(40 62 336 152 165 382 250 457 305)(41 63 337 153 166 383 251 458 306)(42 64 338 154 167 384 252 459 307)(43 65 339 155 168 385 253 460 308)(44 66 340 156 169 386 254 461 309)(45 67 341 157 170 387 255 462 310)(46 68 342 158 171 388 256 463 311)(47 69 343 159 172 389 257 464 312)(48 70 344 160 173 390 258 465 313)(49 71 345 161 174 391 259 466 314)(50 72 346 162 175 392 260 467 315)(51 73 347 109 176 393 261 468 316)(52 74 348 110 177 394 262 469 317)(53 75 349 111 178 395 263 470 318)(54 76 350 112 179 396 264 471 319)
(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)| (1,77,351,113,180,397,265,472,320)(2,78,352,114,181,398,266,473,321)(3,79,353,115,182,399,267,474,322)(4,80,354,116,183,400,268,475,323)(5,81,355,117,184,401,269,476,324)(6,82,356,118,185,402,270,477,271)(7,83,357,119,186,403,217,478,272)(8,84,358,120,187,404,218,479,273)(9,85,359,121,188,405,219,480,274)(10,86,360,122,189,406,220,481,275)(11,87,361,123,190,407,221,482,276)(12,88,362,124,191,408,222,483,277)(13,89,363,125,192,409,223,484,278)(14,90,364,126,193,410,224,485,279)(15,91,365,127,194,411,225,486,280)(16,92,366,128,195,412,226,433,281)(17,93,367,129,196,413,227,434,282)(18,94,368,130,197,414,228,435,283)(19,95,369,131,198,415,229,436,284)(20,96,370,132,199,416,230,437,285)(21,97,371,133,200,417,231,438,286)(22,98,372,134,201,418,232,439,287)(23,99,373,135,202,419,233,440,288)(24,100,374,136,203,420,234,441,289)(25,101,375,137,204,421,235,442,290)(26,102,376,138,205,422,236,443,291)(27,103,377,139,206,423,237,444,292)(28,104,378,140,207,424,238,445,293)(29,105,325,141,208,425,239,446,294)(30,106,326,142,209,426,240,447,295)(31,107,327,143,210,427,241,448,296)(32,108,328,144,211,428,242,449,297)(33,55,329,145,212,429,243,450,298)(34,56,330,146,213,430,244,451,299)(35,57,331,147,214,431,245,452,300)(36,58,332,148,215,432,246,453,301)(37,59,333,149,216,379,247,454,302)(38,60,334,150,163,380,248,455,303)(39,61,335,151,164,381,249,456,304)(40,62,336,152,165,382,250,457,305)(41,63,337,153,166,383,251,458,306)(42,64,338,154,167,384,252,459,307)(43,65,339,155,168,385,253,460,308)(44,66,340,156,169,386,254,461,309)(45,67,341,157,170,387,255,462,310)(46,68,342,158,171,388,256,463,311)(47,69,343,159,172,389,257,464,312)(48,70,344,160,173,390,258,465,313)(49,71,345,161,174,391,259,466,314)(50,72,346,162,175,392,260,467,315)(51,73,347,109,176,393,261,468,316)(52,74,348,110,177,394,262,469,317)(53,75,349,111,178,395,263,470,318)(54,76,350,112,179,396,264,471,319), (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( (1,77,351,113,180,397,265,472,320)(2,78,352,114,181,398,266,473,321)(3,79,353,115,182,399,267,474,322)(4,80,354,116,183,400,268,475,323)(5,81,355,117,184,401,269,476,324)(6,82,356,118,185,402,270,477,271)(7,83,357,119,186,403,217,478,272)(8,84,358,120,187,404,218,479,273)(9,85,359,121,188,405,219,480,274)(10,86,360,122,189,406,220,481,275)(11,87,361,123,190,407,221,482,276)(12,88,362,124,191,408,222,483,277)(13,89,363,125,192,409,223,484,278)(14,90,364,126,193,410,224,485,279)(15,91,365,127,194,411,225,486,280)(16,92,366,128,195,412,226,433,281)(17,93,367,129,196,413,227,434,282)(18,94,368,130,197,414,228,435,283)(19,95,369,131,198,415,229,436,284)(20,96,370,132,199,416,230,437,285)(21,97,371,133,200,417,231,438,286)(22,98,372,134,201,418,232,439,287)(23,99,373,135,202,419,233,440,288)(24,100,374,136,203,420,234,441,289)(25,101,375,137,204,421,235,442,290)(26,102,376,138,205,422,236,443,291)(27,103,377,139,206,423,237,444,292)(28,104,378,140,207,424,238,445,293)(29,105,325,141,208,425,239,446,294)(30,106,326,142,209,426,240,447,295)(31,107,327,143,210,427,241,448,296)(32,108,328,144,211,428,242,449,297)(33,55,329,145,212,429,243,450,298)(34,56,330,146,213,430,244,451,299)(35,57,331,147,214,431,245,452,300)(36,58,332,148,215,432,246,453,301)(37,59,333,149,216,379,247,454,302)(38,60,334,150,163,380,248,455,303)(39,61,335,151,164,381,249,456,304)(40,62,336,152,165,382,250,457,305)(41,63,337,153,166,383,251,458,306)(42,64,338,154,167,384,252,459,307)(43,65,339,155,168,385,253,460,308)(44,66,340,156,169,386,254,461,309)(45,67,341,157,170,387,255,462,310)(46,68,342,158,171,388,256,463,311)(47,69,343,159,172,389,257,464,312)(48,70,344,160,173,390,258,465,313)(49,71,345,161,174,391,259,466,314)(50,72,346,162,175,392,260,467,315)(51,73,347,109,176,393,261,468,316)(52,74,348,110,177,394,262,469,317)(53,75,349,111,178,395,263,470,318)(54,76,350,112,179,396,264,471,319), (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([[(1,77,351,113,180,397,265,472,320),(2,78,352,114,181,398,266,473,321),(3,79,353,115,182,399,267,474,322),(4,80,354,116,183,400,268,475,323),(5,81,355,117,184,401,269,476,324),(6,82,356,118,185,402,270,477,271),(7,83,357,119,186,403,217,478,272),(8,84,358,120,187,404,218,479,273),(9,85,359,121,188,405,219,480,274),(10,86,360,122,189,406,220,481,275),(11,87,361,123,190,407,221,482,276),(12,88,362,124,191,408,222,483,277),(13,89,363,125,192,409,223,484,278),(14,90,364,126,193,410,224,485,279),(15,91,365,127,194,411,225,486,280),(16,92,366,128,195,412,226,433,281),(17,93,367,129,196,413,227,434,282),(18,94,368,130,197,414,228,435,283),(19,95,369,131,198,415,229,436,284),(20,96,370,132,199,416,230,437,285),(21,97,371,133,200,417,231,438,286),(22,98,372,134,201,418,232,439,287),(23,99,373,135,202,419,233,440,288),(24,100,374,136,203,420,234,441,289),(25,101,375,137,204,421,235,442,290),(26,102,376,138,205,422,236,443,291),(27,103,377,139,206,423,237,444,292),(28,104,378,140,207,424,238,445,293),(29,105,325,141,208,425,239,446,294),(30,106,326,142,209,426,240,447,295),(31,107,327,143,210,427,241,448,296),(32,108,328,144,211,428,242,449,297),(33,55,329,145,212,429,243,450,298),(34,56,330,146,213,430,244,451,299),(35,57,331,147,214,431,245,452,300),(36,58,332,148,215,432,246,453,301),(37,59,333,149,216,379,247,454,302),(38,60,334,150,163,380,248,455,303),(39,61,335,151,164,381,249,456,304),(40,62,336,152,165,382,250,457,305),(41,63,337,153,166,383,251,458,306),(42,64,338,154,167,384,252,459,307),(43,65,339,155,168,385,253,460,308),(44,66,340,156,169,386,254,461,309),(45,67,341,157,170,387,255,462,310),(46,68,342,158,171,388,256,463,311),(47,69,343,159,172,389,257,464,312),(48,70,344,160,173,390,258,465,313),(49,71,345,161,174,391,259,466,314),(50,72,346,162,175,392,260,467,315),(51,73,347,109,176,393,261,468,316),(52,74,348,110,177,394,262,469,317),(53,75,349,111,178,395,263,470,318),(54,76,350,112,179,396,264,471,319)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162),(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270),(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324),(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378),(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432),(433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486)]])
486 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 9A | ··· | 9BT | 18A | ··· | 18BT | 27A | ··· | 27FF | 54A | ··· | 54FF |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
486 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C9 | C18 | C18 | C27 | C54 |
| kernel | C9×C54 | C9×C27 | C9×C18 | C3×C54 | C92 | C3×C27 | C54 | C3×C18 | C27 | C3×C9 | C18 | C9 |
| # reps | 1 | 1 | 2 | 6 | 2 | 6 | 54 | 18 | 54 | 18 | 162 | 162 |
Matrix representation of C9×C54 ►in GL2(𝔽109) generated by
| 16 | 0 |
| 0 | 45 |
| 29 | 0 |
| 0 | 29 |
G:=sub<GL(2,GF(109))| [16,0,0,45],[29,0,0,29] >;
C9×C54 in GAP, Magma, Sage, TeX
C_9\times C_{54} % in TeX
G:=Group("C9xC54"); // GroupNames label
G:=SmallGroup(486,70);
// by ID
G=gap.SmallGroup(486,70);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,176,237]);
// Polycyclic
G:=Group<a,b|a^9=b^54=1,a*b=b*a>;
// generators/relations