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G = C9×C54order 486 = 2·35

Abelian group of type [9,54]

direct product, abelian, monomial, 3-elementary

Aliases: C9×C54, SmallGroup(486,70)

Series: Derived Chief Lower central Upper central

C1 — C9×C54
C1C3C32C3×C9C92C9×C27 — 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 [×3], C6, C6 [×3], C9 [×12], C32, C18 [×12], C3×C6, C27 [×9], C3×C9, C3×C9 [×3], C54 [×9], C3×C18, C3×C18 [×3], C92, C3×C27 [×3], C9×C18, C3×C54 [×3], C9×C27, C9×C54
Quotients: C1, C2, C3 [×4], C6 [×4], C9 [×12], C32, C18 [×12], C3×C6, C27 [×9], C3×C9 [×4], C54 [×9], C3×C18 [×4], C92, C3×C27 [×3], C9×C18, C3×C54 [×3], C9×C27, C9×C54

Smallest permutation representation of C9×C54
Regular action on 486 points
Generators in S486
(1 261 196 118 406 456 73 320 374)(2 262 197 119 407 457 74 321 375)(3 263 198 120 408 458 75 322 376)(4 264 199 121 409 459 76 323 377)(5 265 200 122 410 460 77 324 378)(6 266 201 123 411 461 78 271 325)(7 267 202 124 412 462 79 272 326)(8 268 203 125 413 463 80 273 327)(9 269 204 126 414 464 81 274 328)(10 270 205 127 415 465 82 275 329)(11 217 206 128 416 466 83 276 330)(12 218 207 129 417 467 84 277 331)(13 219 208 130 418 468 85 278 332)(14 220 209 131 419 469 86 279 333)(15 221 210 132 420 470 87 280 334)(16 222 211 133 421 471 88 281 335)(17 223 212 134 422 472 89 282 336)(18 224 213 135 423 473 90 283 337)(19 225 214 136 424 474 91 284 338)(20 226 215 137 425 475 92 285 339)(21 227 216 138 426 476 93 286 340)(22 228 163 139 427 477 94 287 341)(23 229 164 140 428 478 95 288 342)(24 230 165 141 429 479 96 289 343)(25 231 166 142 430 480 97 290 344)(26 232 167 143 431 481 98 291 345)(27 233 168 144 432 482 99 292 346)(28 234 169 145 379 483 100 293 347)(29 235 170 146 380 484 101 294 348)(30 236 171 147 381 485 102 295 349)(31 237 172 148 382 486 103 296 350)(32 238 173 149 383 433 104 297 351)(33 239 174 150 384 434 105 298 352)(34 240 175 151 385 435 106 299 353)(35 241 176 152 386 436 107 300 354)(36 242 177 153 387 437 108 301 355)(37 243 178 154 388 438 55 302 356)(38 244 179 155 389 439 56 303 357)(39 245 180 156 390 440 57 304 358)(40 246 181 157 391 441 58 305 359)(41 247 182 158 392 442 59 306 360)(42 248 183 159 393 443 60 307 361)(43 249 184 160 394 444 61 308 362)(44 250 185 161 395 445 62 309 363)(45 251 186 162 396 446 63 310 364)(46 252 187 109 397 447 64 311 365)(47 253 188 110 398 448 65 312 366)(48 254 189 111 399 449 66 313 367)(49 255 190 112 400 450 67 314 368)(50 256 191 113 401 451 68 315 369)(51 257 192 114 402 452 69 316 370)(52 258 193 115 403 453 70 317 371)(53 259 194 116 404 454 71 318 372)(54 260 195 117 405 455 72 319 373)
(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)| (1,261,196,118,406,456,73,320,374)(2,262,197,119,407,457,74,321,375)(3,263,198,120,408,458,75,322,376)(4,264,199,121,409,459,76,323,377)(5,265,200,122,410,460,77,324,378)(6,266,201,123,411,461,78,271,325)(7,267,202,124,412,462,79,272,326)(8,268,203,125,413,463,80,273,327)(9,269,204,126,414,464,81,274,328)(10,270,205,127,415,465,82,275,329)(11,217,206,128,416,466,83,276,330)(12,218,207,129,417,467,84,277,331)(13,219,208,130,418,468,85,278,332)(14,220,209,131,419,469,86,279,333)(15,221,210,132,420,470,87,280,334)(16,222,211,133,421,471,88,281,335)(17,223,212,134,422,472,89,282,336)(18,224,213,135,423,473,90,283,337)(19,225,214,136,424,474,91,284,338)(20,226,215,137,425,475,92,285,339)(21,227,216,138,426,476,93,286,340)(22,228,163,139,427,477,94,287,341)(23,229,164,140,428,478,95,288,342)(24,230,165,141,429,479,96,289,343)(25,231,166,142,430,480,97,290,344)(26,232,167,143,431,481,98,291,345)(27,233,168,144,432,482,99,292,346)(28,234,169,145,379,483,100,293,347)(29,235,170,146,380,484,101,294,348)(30,236,171,147,381,485,102,295,349)(31,237,172,148,382,486,103,296,350)(32,238,173,149,383,433,104,297,351)(33,239,174,150,384,434,105,298,352)(34,240,175,151,385,435,106,299,353)(35,241,176,152,386,436,107,300,354)(36,242,177,153,387,437,108,301,355)(37,243,178,154,388,438,55,302,356)(38,244,179,155,389,439,56,303,357)(39,245,180,156,390,440,57,304,358)(40,246,181,157,391,441,58,305,359)(41,247,182,158,392,442,59,306,360)(42,248,183,159,393,443,60,307,361)(43,249,184,160,394,444,61,308,362)(44,250,185,161,395,445,62,309,363)(45,251,186,162,396,446,63,310,364)(46,252,187,109,397,447,64,311,365)(47,253,188,110,398,448,65,312,366)(48,254,189,111,399,449,66,313,367)(49,255,190,112,400,450,67,314,368)(50,256,191,113,401,451,68,315,369)(51,257,192,114,402,452,69,316,370)(52,258,193,115,403,453,70,317,371)(53,259,194,116,404,454,71,318,372)(54,260,195,117,405,455,72,319,373), (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( (1,261,196,118,406,456,73,320,374)(2,262,197,119,407,457,74,321,375)(3,263,198,120,408,458,75,322,376)(4,264,199,121,409,459,76,323,377)(5,265,200,122,410,460,77,324,378)(6,266,201,123,411,461,78,271,325)(7,267,202,124,412,462,79,272,326)(8,268,203,125,413,463,80,273,327)(9,269,204,126,414,464,81,274,328)(10,270,205,127,415,465,82,275,329)(11,217,206,128,416,466,83,276,330)(12,218,207,129,417,467,84,277,331)(13,219,208,130,418,468,85,278,332)(14,220,209,131,419,469,86,279,333)(15,221,210,132,420,470,87,280,334)(16,222,211,133,421,471,88,281,335)(17,223,212,134,422,472,89,282,336)(18,224,213,135,423,473,90,283,337)(19,225,214,136,424,474,91,284,338)(20,226,215,137,425,475,92,285,339)(21,227,216,138,426,476,93,286,340)(22,228,163,139,427,477,94,287,341)(23,229,164,140,428,478,95,288,342)(24,230,165,141,429,479,96,289,343)(25,231,166,142,430,480,97,290,344)(26,232,167,143,431,481,98,291,345)(27,233,168,144,432,482,99,292,346)(28,234,169,145,379,483,100,293,347)(29,235,170,146,380,484,101,294,348)(30,236,171,147,381,485,102,295,349)(31,237,172,148,382,486,103,296,350)(32,238,173,149,383,433,104,297,351)(33,239,174,150,384,434,105,298,352)(34,240,175,151,385,435,106,299,353)(35,241,176,152,386,436,107,300,354)(36,242,177,153,387,437,108,301,355)(37,243,178,154,388,438,55,302,356)(38,244,179,155,389,439,56,303,357)(39,245,180,156,390,440,57,304,358)(40,246,181,157,391,441,58,305,359)(41,247,182,158,392,442,59,306,360)(42,248,183,159,393,443,60,307,361)(43,249,184,160,394,444,61,308,362)(44,250,185,161,395,445,62,309,363)(45,251,186,162,396,446,63,310,364)(46,252,187,109,397,447,64,311,365)(47,253,188,110,398,448,65,312,366)(48,254,189,111,399,449,66,313,367)(49,255,190,112,400,450,67,314,368)(50,256,191,113,401,451,68,315,369)(51,257,192,114,402,452,69,316,370)(52,258,193,115,403,453,70,317,371)(53,259,194,116,404,454,71,318,372)(54,260,195,117,405,455,72,319,373), (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([(1,261,196,118,406,456,73,320,374),(2,262,197,119,407,457,74,321,375),(3,263,198,120,408,458,75,322,376),(4,264,199,121,409,459,76,323,377),(5,265,200,122,410,460,77,324,378),(6,266,201,123,411,461,78,271,325),(7,267,202,124,412,462,79,272,326),(8,268,203,125,413,463,80,273,327),(9,269,204,126,414,464,81,274,328),(10,270,205,127,415,465,82,275,329),(11,217,206,128,416,466,83,276,330),(12,218,207,129,417,467,84,277,331),(13,219,208,130,418,468,85,278,332),(14,220,209,131,419,469,86,279,333),(15,221,210,132,420,470,87,280,334),(16,222,211,133,421,471,88,281,335),(17,223,212,134,422,472,89,282,336),(18,224,213,135,423,473,90,283,337),(19,225,214,136,424,474,91,284,338),(20,226,215,137,425,475,92,285,339),(21,227,216,138,426,476,93,286,340),(22,228,163,139,427,477,94,287,341),(23,229,164,140,428,478,95,288,342),(24,230,165,141,429,479,96,289,343),(25,231,166,142,430,480,97,290,344),(26,232,167,143,431,481,98,291,345),(27,233,168,144,432,482,99,292,346),(28,234,169,145,379,483,100,293,347),(29,235,170,146,380,484,101,294,348),(30,236,171,147,381,485,102,295,349),(31,237,172,148,382,486,103,296,350),(32,238,173,149,383,433,104,297,351),(33,239,174,150,384,434,105,298,352),(34,240,175,151,385,435,106,299,353),(35,241,176,152,386,436,107,300,354),(36,242,177,153,387,437,108,301,355),(37,243,178,154,388,438,55,302,356),(38,244,179,155,389,439,56,303,357),(39,245,180,156,390,440,57,304,358),(40,246,181,157,391,441,58,305,359),(41,247,182,158,392,442,59,306,360),(42,248,183,159,393,443,60,307,361),(43,249,184,160,394,444,61,308,362),(44,250,185,161,395,445,62,309,363),(45,251,186,162,396,446,63,310,364),(46,252,187,109,397,447,64,311,365),(47,253,188,110,398,448,65,312,366),(48,254,189,111,399,449,66,313,367),(49,255,190,112,400,450,67,314,368),(50,256,191,113,401,451,68,315,369),(51,257,192,114,402,452,69,316,370),(52,258,193,115,403,453,70,317,371),(53,259,194,116,404,454,71,318,372),(54,260,195,117,405,455,72,319,373)], [(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···3H6A···6H9A···9BT18A···18BT27A···27FF54A···54FF
order123···36···69···918···1827···2754···54
size111···11···11···11···11···11···1

486 irreducible representations

dim111111111111
type++
imageC1C2C3C3C6C6C9C9C18C18C27C54
kernelC9×C54C9×C27C9×C18C3×C54C92C3×C27C54C3×C18C27C3×C9C18C9
# reps11262654185418162162

Matrix representation of C9×C54 in GL2(𝔽109) generated by

160
045
,
290
029
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

׿
×
𝔽