direct product, abelian, monomial, 11-elementary
Aliases: C11×C44, SmallGroup(484,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C11×C44 |
| C1 — C11×C44 |
| C1 — C11×C44 |
Generators and relations for C11×C44
G = < a,b | a11=b44=1, ab=ba >
Smallest permutation representation of C11×C44
►Regular action on 484 points
Generators in S484
(1 456 404 308 51 243 144 343 122 369 208)(2 457 405 265 52 244 145 344 123 370 209)(3 458 406 266 53 245 146 345 124 371 210)(4 459 407 267 54 246 147 346 125 372 211)(5 460 408 268 55 247 148 347 126 373 212)(6 461 409 269 56 248 149 348 127 374 213)(7 462 410 270 57 249 150 349 128 375 214)(8 463 411 271 58 250 151 350 129 376 215)(9 464 412 272 59 251 152 351 130 377 216)(10 465 413 273 60 252 153 352 131 378 217)(11 466 414 274 61 253 154 309 132 379 218)(12 467 415 275 62 254 155 310 89 380 219)(13 468 416 276 63 255 156 311 90 381 220)(14 469 417 277 64 256 157 312 91 382 177)(15 470 418 278 65 257 158 313 92 383 178)(16 471 419 279 66 258 159 314 93 384 179)(17 472 420 280 67 259 160 315 94 385 180)(18 473 421 281 68 260 161 316 95 386 181)(19 474 422 282 69 261 162 317 96 387 182)(20 475 423 283 70 262 163 318 97 388 183)(21 476 424 284 71 263 164 319 98 389 184)(22 477 425 285 72 264 165 320 99 390 185)(23 478 426 286 73 221 166 321 100 391 186)(24 479 427 287 74 222 167 322 101 392 187)(25 480 428 288 75 223 168 323 102 393 188)(26 481 429 289 76 224 169 324 103 394 189)(27 482 430 290 77 225 170 325 104 395 190)(28 483 431 291 78 226 171 326 105 396 191)(29 484 432 292 79 227 172 327 106 353 192)(30 441 433 293 80 228 173 328 107 354 193)(31 442 434 294 81 229 174 329 108 355 194)(32 443 435 295 82 230 175 330 109 356 195)(33 444 436 296 83 231 176 331 110 357 196)(34 445 437 297 84 232 133 332 111 358 197)(35 446 438 298 85 233 134 333 112 359 198)(36 447 439 299 86 234 135 334 113 360 199)(37 448 440 300 87 235 136 335 114 361 200)(38 449 397 301 88 236 137 336 115 362 201)(39 450 398 302 45 237 138 337 116 363 202)(40 451 399 303 46 238 139 338 117 364 203)(41 452 400 304 47 239 140 339 118 365 204)(42 453 401 305 48 240 141 340 119 366 205)(43 454 402 306 49 241 142 341 120 367 206)(44 455 403 307 50 242 143 342 121 368 207)
(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)
G:=sub<Sym(484)| (1,456,404,308,51,243,144,343,122,369,208)(2,457,405,265,52,244,145,344,123,370,209)(3,458,406,266,53,245,146,345,124,371,210)(4,459,407,267,54,246,147,346,125,372,211)(5,460,408,268,55,247,148,347,126,373,212)(6,461,409,269,56,248,149,348,127,374,213)(7,462,410,270,57,249,150,349,128,375,214)(8,463,411,271,58,250,151,350,129,376,215)(9,464,412,272,59,251,152,351,130,377,216)(10,465,413,273,60,252,153,352,131,378,217)(11,466,414,274,61,253,154,309,132,379,218)(12,467,415,275,62,254,155,310,89,380,219)(13,468,416,276,63,255,156,311,90,381,220)(14,469,417,277,64,256,157,312,91,382,177)(15,470,418,278,65,257,158,313,92,383,178)(16,471,419,279,66,258,159,314,93,384,179)(17,472,420,280,67,259,160,315,94,385,180)(18,473,421,281,68,260,161,316,95,386,181)(19,474,422,282,69,261,162,317,96,387,182)(20,475,423,283,70,262,163,318,97,388,183)(21,476,424,284,71,263,164,319,98,389,184)(22,477,425,285,72,264,165,320,99,390,185)(23,478,426,286,73,221,166,321,100,391,186)(24,479,427,287,74,222,167,322,101,392,187)(25,480,428,288,75,223,168,323,102,393,188)(26,481,429,289,76,224,169,324,103,394,189)(27,482,430,290,77,225,170,325,104,395,190)(28,483,431,291,78,226,171,326,105,396,191)(29,484,432,292,79,227,172,327,106,353,192)(30,441,433,293,80,228,173,328,107,354,193)(31,442,434,294,81,229,174,329,108,355,194)(32,443,435,295,82,230,175,330,109,356,195)(33,444,436,296,83,231,176,331,110,357,196)(34,445,437,297,84,232,133,332,111,358,197)(35,446,438,298,85,233,134,333,112,359,198)(36,447,439,299,86,234,135,334,113,360,199)(37,448,440,300,87,235,136,335,114,361,200)(38,449,397,301,88,236,137,336,115,362,201)(39,450,398,302,45,237,138,337,116,363,202)(40,451,399,303,46,238,139,338,117,364,203)(41,452,400,304,47,239,140,339,118,365,204)(42,453,401,305,48,240,141,340,119,366,205)(43,454,402,306,49,241,142,341,120,367,206)(44,455,403,307,50,242,143,342,121,368,207), (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)>;
G:=Group( (1,456,404,308,51,243,144,343,122,369,208)(2,457,405,265,52,244,145,344,123,370,209)(3,458,406,266,53,245,146,345,124,371,210)(4,459,407,267,54,246,147,346,125,372,211)(5,460,408,268,55,247,148,347,126,373,212)(6,461,409,269,56,248,149,348,127,374,213)(7,462,410,270,57,249,150,349,128,375,214)(8,463,411,271,58,250,151,350,129,376,215)(9,464,412,272,59,251,152,351,130,377,216)(10,465,413,273,60,252,153,352,131,378,217)(11,466,414,274,61,253,154,309,132,379,218)(12,467,415,275,62,254,155,310,89,380,219)(13,468,416,276,63,255,156,311,90,381,220)(14,469,417,277,64,256,157,312,91,382,177)(15,470,418,278,65,257,158,313,92,383,178)(16,471,419,279,66,258,159,314,93,384,179)(17,472,420,280,67,259,160,315,94,385,180)(18,473,421,281,68,260,161,316,95,386,181)(19,474,422,282,69,261,162,317,96,387,182)(20,475,423,283,70,262,163,318,97,388,183)(21,476,424,284,71,263,164,319,98,389,184)(22,477,425,285,72,264,165,320,99,390,185)(23,478,426,286,73,221,166,321,100,391,186)(24,479,427,287,74,222,167,322,101,392,187)(25,480,428,288,75,223,168,323,102,393,188)(26,481,429,289,76,224,169,324,103,394,189)(27,482,430,290,77,225,170,325,104,395,190)(28,483,431,291,78,226,171,326,105,396,191)(29,484,432,292,79,227,172,327,106,353,192)(30,441,433,293,80,228,173,328,107,354,193)(31,442,434,294,81,229,174,329,108,355,194)(32,443,435,295,82,230,175,330,109,356,195)(33,444,436,296,83,231,176,331,110,357,196)(34,445,437,297,84,232,133,332,111,358,197)(35,446,438,298,85,233,134,333,112,359,198)(36,447,439,299,86,234,135,334,113,360,199)(37,448,440,300,87,235,136,335,114,361,200)(38,449,397,301,88,236,137,336,115,362,201)(39,450,398,302,45,237,138,337,116,363,202)(40,451,399,303,46,238,139,338,117,364,203)(41,452,400,304,47,239,140,339,118,365,204)(42,453,401,305,48,240,141,340,119,366,205)(43,454,402,306,49,241,142,341,120,367,206)(44,455,403,307,50,242,143,342,121,368,207), (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) );
G=PermutationGroup([[(1,456,404,308,51,243,144,343,122,369,208),(2,457,405,265,52,244,145,344,123,370,209),(3,458,406,266,53,245,146,345,124,371,210),(4,459,407,267,54,246,147,346,125,372,211),(5,460,408,268,55,247,148,347,126,373,212),(6,461,409,269,56,248,149,348,127,374,213),(7,462,410,270,57,249,150,349,128,375,214),(8,463,411,271,58,250,151,350,129,376,215),(9,464,412,272,59,251,152,351,130,377,216),(10,465,413,273,60,252,153,352,131,378,217),(11,466,414,274,61,253,154,309,132,379,218),(12,467,415,275,62,254,155,310,89,380,219),(13,468,416,276,63,255,156,311,90,381,220),(14,469,417,277,64,256,157,312,91,382,177),(15,470,418,278,65,257,158,313,92,383,178),(16,471,419,279,66,258,159,314,93,384,179),(17,472,420,280,67,259,160,315,94,385,180),(18,473,421,281,68,260,161,316,95,386,181),(19,474,422,282,69,261,162,317,96,387,182),(20,475,423,283,70,262,163,318,97,388,183),(21,476,424,284,71,263,164,319,98,389,184),(22,477,425,285,72,264,165,320,99,390,185),(23,478,426,286,73,221,166,321,100,391,186),(24,479,427,287,74,222,167,322,101,392,187),(25,480,428,288,75,223,168,323,102,393,188),(26,481,429,289,76,224,169,324,103,394,189),(27,482,430,290,77,225,170,325,104,395,190),(28,483,431,291,78,226,171,326,105,396,191),(29,484,432,292,79,227,172,327,106,353,192),(30,441,433,293,80,228,173,328,107,354,193),(31,442,434,294,81,229,174,329,108,355,194),(32,443,435,295,82,230,175,330,109,356,195),(33,444,436,296,83,231,176,331,110,357,196),(34,445,437,297,84,232,133,332,111,358,197),(35,446,438,298,85,233,134,333,112,359,198),(36,447,439,299,86,234,135,334,113,360,199),(37,448,440,300,87,235,136,335,114,361,200),(38,449,397,301,88,236,137,336,115,362,201),(39,450,398,302,45,237,138,337,116,363,202),(40,451,399,303,46,238,139,338,117,364,203),(41,452,400,304,47,239,140,339,118,365,204),(42,453,401,305,48,240,141,340,119,366,205),(43,454,402,306,49,241,142,341,120,367,206),(44,455,403,307,50,242,143,342,121,368,207)], [(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)]])
484 conjugacy classes
| class | 1 | 2 | 4A | 4B | 11A | ··· | 11DP | 22A | ··· | 22DP | 44A | ··· | 44IF |
| order | 1 | 2 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
484 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C11 | C22 | C44 |
| kernel | C11×C44 | C11×C22 | C112 | C44 | C22 | C11 |
| # reps | 1 | 1 | 2 | 120 | 120 | 240 |
Matrix representation of C11×C44 ►in GL2(𝔽89) generated by
| 78 | 0 |
| 0 | 16 |
| 16 | 0 |
| 0 | 18 |
G:=sub<GL(2,GF(89))| [78,0,0,16],[16,0,0,18] >;
C11×C44 in GAP, Magma, Sage, TeX
C_{11}\times C_{44} % in TeX
G:=Group("C11xC44"); // GroupNames label
G:=SmallGroup(484,7);
// by ID
G=gap.SmallGroup(484,7);
# by ID
G:=PCGroup([4,-2,-11,-11,-2,968]);
// Polycyclic
G:=Group<a,b|a^11=b^44=1,a*b=b*a>;
// generators/relations
Export