metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic121, C121⋊C4, C242.C2, C2.D121, C22.1D11, C11.Dic11, SmallGroup(484,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C121 — Dic121 |
Generators and relations for Dic121
G = < a,b | a242=1, b2=a121, bab-1=a-1 >
Smallest permutation representation of Dic121
►Regular action on 484 points
Generators in S484
(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)
(1 269 122 390)(2 268 123 389)(3 267 124 388)(4 266 125 387)(5 265 126 386)(6 264 127 385)(7 263 128 384)(8 262 129 383)(9 261 130 382)(10 260 131 381)(11 259 132 380)(12 258 133 379)(13 257 134 378)(14 256 135 377)(15 255 136 376)(16 254 137 375)(17 253 138 374)(18 252 139 373)(19 251 140 372)(20 250 141 371)(21 249 142 370)(22 248 143 369)(23 247 144 368)(24 246 145 367)(25 245 146 366)(26 244 147 365)(27 243 148 364)(28 484 149 363)(29 483 150 362)(30 482 151 361)(31 481 152 360)(32 480 153 359)(33 479 154 358)(34 478 155 357)(35 477 156 356)(36 476 157 355)(37 475 158 354)(38 474 159 353)(39 473 160 352)(40 472 161 351)(41 471 162 350)(42 470 163 349)(43 469 164 348)(44 468 165 347)(45 467 166 346)(46 466 167 345)(47 465 168 344)(48 464 169 343)(49 463 170 342)(50 462 171 341)(51 461 172 340)(52 460 173 339)(53 459 174 338)(54 458 175 337)(55 457 176 336)(56 456 177 335)(57 455 178 334)(58 454 179 333)(59 453 180 332)(60 452 181 331)(61 451 182 330)(62 450 183 329)(63 449 184 328)(64 448 185 327)(65 447 186 326)(66 446 187 325)(67 445 188 324)(68 444 189 323)(69 443 190 322)(70 442 191 321)(71 441 192 320)(72 440 193 319)(73 439 194 318)(74 438 195 317)(75 437 196 316)(76 436 197 315)(77 435 198 314)(78 434 199 313)(79 433 200 312)(80 432 201 311)(81 431 202 310)(82 430 203 309)(83 429 204 308)(84 428 205 307)(85 427 206 306)(86 426 207 305)(87 425 208 304)(88 424 209 303)(89 423 210 302)(90 422 211 301)(91 421 212 300)(92 420 213 299)(93 419 214 298)(94 418 215 297)(95 417 216 296)(96 416 217 295)(97 415 218 294)(98 414 219 293)(99 413 220 292)(100 412 221 291)(101 411 222 290)(102 410 223 289)(103 409 224 288)(104 408 225 287)(105 407 226 286)(106 406 227 285)(107 405 228 284)(108 404 229 283)(109 403 230 282)(110 402 231 281)(111 401 232 280)(112 400 233 279)(113 399 234 278)(114 398 235 277)(115 397 236 276)(116 396 237 275)(117 395 238 274)(118 394 239 273)(119 393 240 272)(120 392 241 271)(121 391 242 270)
G:=sub<Sym(484)| (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), (1,269,122,390)(2,268,123,389)(3,267,124,388)(4,266,125,387)(5,265,126,386)(6,264,127,385)(7,263,128,384)(8,262,129,383)(9,261,130,382)(10,260,131,381)(11,259,132,380)(12,258,133,379)(13,257,134,378)(14,256,135,377)(15,255,136,376)(16,254,137,375)(17,253,138,374)(18,252,139,373)(19,251,140,372)(20,250,141,371)(21,249,142,370)(22,248,143,369)(23,247,144,368)(24,246,145,367)(25,245,146,366)(26,244,147,365)(27,243,148,364)(28,484,149,363)(29,483,150,362)(30,482,151,361)(31,481,152,360)(32,480,153,359)(33,479,154,358)(34,478,155,357)(35,477,156,356)(36,476,157,355)(37,475,158,354)(38,474,159,353)(39,473,160,352)(40,472,161,351)(41,471,162,350)(42,470,163,349)(43,469,164,348)(44,468,165,347)(45,467,166,346)(46,466,167,345)(47,465,168,344)(48,464,169,343)(49,463,170,342)(50,462,171,341)(51,461,172,340)(52,460,173,339)(53,459,174,338)(54,458,175,337)(55,457,176,336)(56,456,177,335)(57,455,178,334)(58,454,179,333)(59,453,180,332)(60,452,181,331)(61,451,182,330)(62,450,183,329)(63,449,184,328)(64,448,185,327)(65,447,186,326)(66,446,187,325)(67,445,188,324)(68,444,189,323)(69,443,190,322)(70,442,191,321)(71,441,192,320)(72,440,193,319)(73,439,194,318)(74,438,195,317)(75,437,196,316)(76,436,197,315)(77,435,198,314)(78,434,199,313)(79,433,200,312)(80,432,201,311)(81,431,202,310)(82,430,203,309)(83,429,204,308)(84,428,205,307)(85,427,206,306)(86,426,207,305)(87,425,208,304)(88,424,209,303)(89,423,210,302)(90,422,211,301)(91,421,212,300)(92,420,213,299)(93,419,214,298)(94,418,215,297)(95,417,216,296)(96,416,217,295)(97,415,218,294)(98,414,219,293)(99,413,220,292)(100,412,221,291)(101,411,222,290)(102,410,223,289)(103,409,224,288)(104,408,225,287)(105,407,226,286)(106,406,227,285)(107,405,228,284)(108,404,229,283)(109,403,230,282)(110,402,231,281)(111,401,232,280)(112,400,233,279)(113,399,234,278)(114,398,235,277)(115,397,236,276)(116,396,237,275)(117,395,238,274)(118,394,239,273)(119,393,240,272)(120,392,241,271)(121,391,242,270)>;
G:=Group( (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), (1,269,122,390)(2,268,123,389)(3,267,124,388)(4,266,125,387)(5,265,126,386)(6,264,127,385)(7,263,128,384)(8,262,129,383)(9,261,130,382)(10,260,131,381)(11,259,132,380)(12,258,133,379)(13,257,134,378)(14,256,135,377)(15,255,136,376)(16,254,137,375)(17,253,138,374)(18,252,139,373)(19,251,140,372)(20,250,141,371)(21,249,142,370)(22,248,143,369)(23,247,144,368)(24,246,145,367)(25,245,146,366)(26,244,147,365)(27,243,148,364)(28,484,149,363)(29,483,150,362)(30,482,151,361)(31,481,152,360)(32,480,153,359)(33,479,154,358)(34,478,155,357)(35,477,156,356)(36,476,157,355)(37,475,158,354)(38,474,159,353)(39,473,160,352)(40,472,161,351)(41,471,162,350)(42,470,163,349)(43,469,164,348)(44,468,165,347)(45,467,166,346)(46,466,167,345)(47,465,168,344)(48,464,169,343)(49,463,170,342)(50,462,171,341)(51,461,172,340)(52,460,173,339)(53,459,174,338)(54,458,175,337)(55,457,176,336)(56,456,177,335)(57,455,178,334)(58,454,179,333)(59,453,180,332)(60,452,181,331)(61,451,182,330)(62,450,183,329)(63,449,184,328)(64,448,185,327)(65,447,186,326)(66,446,187,325)(67,445,188,324)(68,444,189,323)(69,443,190,322)(70,442,191,321)(71,441,192,320)(72,440,193,319)(73,439,194,318)(74,438,195,317)(75,437,196,316)(76,436,197,315)(77,435,198,314)(78,434,199,313)(79,433,200,312)(80,432,201,311)(81,431,202,310)(82,430,203,309)(83,429,204,308)(84,428,205,307)(85,427,206,306)(86,426,207,305)(87,425,208,304)(88,424,209,303)(89,423,210,302)(90,422,211,301)(91,421,212,300)(92,420,213,299)(93,419,214,298)(94,418,215,297)(95,417,216,296)(96,416,217,295)(97,415,218,294)(98,414,219,293)(99,413,220,292)(100,412,221,291)(101,411,222,290)(102,410,223,289)(103,409,224,288)(104,408,225,287)(105,407,226,286)(106,406,227,285)(107,405,228,284)(108,404,229,283)(109,403,230,282)(110,402,231,281)(111,401,232,280)(112,400,233,279)(113,399,234,278)(114,398,235,277)(115,397,236,276)(116,396,237,275)(117,395,238,274)(118,394,239,273)(119,393,240,272)(120,392,241,271)(121,391,242,270) );
G=PermutationGroup([[(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)], [(1,269,122,390),(2,268,123,389),(3,267,124,388),(4,266,125,387),(5,265,126,386),(6,264,127,385),(7,263,128,384),(8,262,129,383),(9,261,130,382),(10,260,131,381),(11,259,132,380),(12,258,133,379),(13,257,134,378),(14,256,135,377),(15,255,136,376),(16,254,137,375),(17,253,138,374),(18,252,139,373),(19,251,140,372),(20,250,141,371),(21,249,142,370),(22,248,143,369),(23,247,144,368),(24,246,145,367),(25,245,146,366),(26,244,147,365),(27,243,148,364),(28,484,149,363),(29,483,150,362),(30,482,151,361),(31,481,152,360),(32,480,153,359),(33,479,154,358),(34,478,155,357),(35,477,156,356),(36,476,157,355),(37,475,158,354),(38,474,159,353),(39,473,160,352),(40,472,161,351),(41,471,162,350),(42,470,163,349),(43,469,164,348),(44,468,165,347),(45,467,166,346),(46,466,167,345),(47,465,168,344),(48,464,169,343),(49,463,170,342),(50,462,171,341),(51,461,172,340),(52,460,173,339),(53,459,174,338),(54,458,175,337),(55,457,176,336),(56,456,177,335),(57,455,178,334),(58,454,179,333),(59,453,180,332),(60,452,181,331),(61,451,182,330),(62,450,183,329),(63,449,184,328),(64,448,185,327),(65,447,186,326),(66,446,187,325),(67,445,188,324),(68,444,189,323),(69,443,190,322),(70,442,191,321),(71,441,192,320),(72,440,193,319),(73,439,194,318),(74,438,195,317),(75,437,196,316),(76,436,197,315),(77,435,198,314),(78,434,199,313),(79,433,200,312),(80,432,201,311),(81,431,202,310),(82,430,203,309),(83,429,204,308),(84,428,205,307),(85,427,206,306),(86,426,207,305),(87,425,208,304),(88,424,209,303),(89,423,210,302),(90,422,211,301),(91,421,212,300),(92,420,213,299),(93,419,214,298),(94,418,215,297),(95,417,216,296),(96,416,217,295),(97,415,218,294),(98,414,219,293),(99,413,220,292),(100,412,221,291),(101,411,222,290),(102,410,223,289),(103,409,224,288),(104,408,225,287),(105,407,226,286),(106,406,227,285),(107,405,228,284),(108,404,229,283),(109,403,230,282),(110,402,231,281),(111,401,232,280),(112,400,233,279),(113,399,234,278),(114,398,235,277),(115,397,236,276),(116,396,237,275),(117,395,238,274),(118,394,239,273),(119,393,240,272),(120,392,241,271),(121,391,242,270)]])
124 conjugacy classes
| class | 1 | 2 | 4A | 4B | 11A | ··· | 11E | 22A | ··· | 22E | 121A | ··· | 121BC | 242A | ··· | 242BC |
| order | 1 | 2 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 121 | ··· | 121 | 242 | ··· | 242 |
| size | 1 | 1 | 121 | 121 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | |
| image | C1 | C2 | C4 | D11 | Dic11 | D121 | Dic121 |
| kernel | Dic121 | C242 | C121 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 5 | 5 | 55 | 55 |
Matrix representation of Dic121 ►in GL2(𝔽1453) generated by
| 197 | 450 |
| 1003 | 875 |
| 43 | 998 |
| 36 | 1410 |
G:=sub<GL(2,GF(1453))| [197,1003,450,875],[43,36,998,1410] >;
Dic121 in GAP, Magma, Sage, TeX
{\rm Dic}_{121} % in TeX
G:=Group("Dic121"); // GroupNames label
G:=SmallGroup(484,1);
// by ID
G=gap.SmallGroup(484,1);
# by ID
G:=PCGroup([4,-2,-2,-11,-11,8,1010,1330,7043]);
// Polycyclic
G:=Group<a,b|a^242=1,b^2=a^121,b*a*b^-1=a^-1>;
// generators/relations
Export