metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic122, C61⋊Q8, C4.D61, C244.1C2, C2.3D122, C122.1C22, Dic61.1C2, SmallGroup(488,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic122
G = < a,b | a244=1, b2=a122, bab-1=a-1 >
Smallest permutation representation of Dic122
►Regular action on 488 points
Generators in S488
(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)(245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488)
(1 487 123 365)(2 486 124 364)(3 485 125 363)(4 484 126 362)(5 483 127 361)(6 482 128 360)(7 481 129 359)(8 480 130 358)(9 479 131 357)(10 478 132 356)(11 477 133 355)(12 476 134 354)(13 475 135 353)(14 474 136 352)(15 473 137 351)(16 472 138 350)(17 471 139 349)(18 470 140 348)(19 469 141 347)(20 468 142 346)(21 467 143 345)(22 466 144 344)(23 465 145 343)(24 464 146 342)(25 463 147 341)(26 462 148 340)(27 461 149 339)(28 460 150 338)(29 459 151 337)(30 458 152 336)(31 457 153 335)(32 456 154 334)(33 455 155 333)(34 454 156 332)(35 453 157 331)(36 452 158 330)(37 451 159 329)(38 450 160 328)(39 449 161 327)(40 448 162 326)(41 447 163 325)(42 446 164 324)(43 445 165 323)(44 444 166 322)(45 443 167 321)(46 442 168 320)(47 441 169 319)(48 440 170 318)(49 439 171 317)(50 438 172 316)(51 437 173 315)(52 436 174 314)(53 435 175 313)(54 434 176 312)(55 433 177 311)(56 432 178 310)(57 431 179 309)(58 430 180 308)(59 429 181 307)(60 428 182 306)(61 427 183 305)(62 426 184 304)(63 425 185 303)(64 424 186 302)(65 423 187 301)(66 422 188 300)(67 421 189 299)(68 420 190 298)(69 419 191 297)(70 418 192 296)(71 417 193 295)(72 416 194 294)(73 415 195 293)(74 414 196 292)(75 413 197 291)(76 412 198 290)(77 411 199 289)(78 410 200 288)(79 409 201 287)(80 408 202 286)(81 407 203 285)(82 406 204 284)(83 405 205 283)(84 404 206 282)(85 403 207 281)(86 402 208 280)(87 401 209 279)(88 400 210 278)(89 399 211 277)(90 398 212 276)(91 397 213 275)(92 396 214 274)(93 395 215 273)(94 394 216 272)(95 393 217 271)(96 392 218 270)(97 391 219 269)(98 390 220 268)(99 389 221 267)(100 388 222 266)(101 387 223 265)(102 386 224 264)(103 385 225 263)(104 384 226 262)(105 383 227 261)(106 382 228 260)(107 381 229 259)(108 380 230 258)(109 379 231 257)(110 378 232 256)(111 377 233 255)(112 376 234 254)(113 375 235 253)(114 374 236 252)(115 373 237 251)(116 372 238 250)(117 371 239 249)(118 370 240 248)(119 369 241 247)(120 368 242 246)(121 367 243 245)(122 366 244 488)
G:=sub<Sym(488)| (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)(245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488), (1,487,123,365)(2,486,124,364)(3,485,125,363)(4,484,126,362)(5,483,127,361)(6,482,128,360)(7,481,129,359)(8,480,130,358)(9,479,131,357)(10,478,132,356)(11,477,133,355)(12,476,134,354)(13,475,135,353)(14,474,136,352)(15,473,137,351)(16,472,138,350)(17,471,139,349)(18,470,140,348)(19,469,141,347)(20,468,142,346)(21,467,143,345)(22,466,144,344)(23,465,145,343)(24,464,146,342)(25,463,147,341)(26,462,148,340)(27,461,149,339)(28,460,150,338)(29,459,151,337)(30,458,152,336)(31,457,153,335)(32,456,154,334)(33,455,155,333)(34,454,156,332)(35,453,157,331)(36,452,158,330)(37,451,159,329)(38,450,160,328)(39,449,161,327)(40,448,162,326)(41,447,163,325)(42,446,164,324)(43,445,165,323)(44,444,166,322)(45,443,167,321)(46,442,168,320)(47,441,169,319)(48,440,170,318)(49,439,171,317)(50,438,172,316)(51,437,173,315)(52,436,174,314)(53,435,175,313)(54,434,176,312)(55,433,177,311)(56,432,178,310)(57,431,179,309)(58,430,180,308)(59,429,181,307)(60,428,182,306)(61,427,183,305)(62,426,184,304)(63,425,185,303)(64,424,186,302)(65,423,187,301)(66,422,188,300)(67,421,189,299)(68,420,190,298)(69,419,191,297)(70,418,192,296)(71,417,193,295)(72,416,194,294)(73,415,195,293)(74,414,196,292)(75,413,197,291)(76,412,198,290)(77,411,199,289)(78,410,200,288)(79,409,201,287)(80,408,202,286)(81,407,203,285)(82,406,204,284)(83,405,205,283)(84,404,206,282)(85,403,207,281)(86,402,208,280)(87,401,209,279)(88,400,210,278)(89,399,211,277)(90,398,212,276)(91,397,213,275)(92,396,214,274)(93,395,215,273)(94,394,216,272)(95,393,217,271)(96,392,218,270)(97,391,219,269)(98,390,220,268)(99,389,221,267)(100,388,222,266)(101,387,223,265)(102,386,224,264)(103,385,225,263)(104,384,226,262)(105,383,227,261)(106,382,228,260)(107,381,229,259)(108,380,230,258)(109,379,231,257)(110,378,232,256)(111,377,233,255)(112,376,234,254)(113,375,235,253)(114,374,236,252)(115,373,237,251)(116,372,238,250)(117,371,239,249)(118,370,240,248)(119,369,241,247)(120,368,242,246)(121,367,243,245)(122,366,244,488)>;
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)(245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488), (1,487,123,365)(2,486,124,364)(3,485,125,363)(4,484,126,362)(5,483,127,361)(6,482,128,360)(7,481,129,359)(8,480,130,358)(9,479,131,357)(10,478,132,356)(11,477,133,355)(12,476,134,354)(13,475,135,353)(14,474,136,352)(15,473,137,351)(16,472,138,350)(17,471,139,349)(18,470,140,348)(19,469,141,347)(20,468,142,346)(21,467,143,345)(22,466,144,344)(23,465,145,343)(24,464,146,342)(25,463,147,341)(26,462,148,340)(27,461,149,339)(28,460,150,338)(29,459,151,337)(30,458,152,336)(31,457,153,335)(32,456,154,334)(33,455,155,333)(34,454,156,332)(35,453,157,331)(36,452,158,330)(37,451,159,329)(38,450,160,328)(39,449,161,327)(40,448,162,326)(41,447,163,325)(42,446,164,324)(43,445,165,323)(44,444,166,322)(45,443,167,321)(46,442,168,320)(47,441,169,319)(48,440,170,318)(49,439,171,317)(50,438,172,316)(51,437,173,315)(52,436,174,314)(53,435,175,313)(54,434,176,312)(55,433,177,311)(56,432,178,310)(57,431,179,309)(58,430,180,308)(59,429,181,307)(60,428,182,306)(61,427,183,305)(62,426,184,304)(63,425,185,303)(64,424,186,302)(65,423,187,301)(66,422,188,300)(67,421,189,299)(68,420,190,298)(69,419,191,297)(70,418,192,296)(71,417,193,295)(72,416,194,294)(73,415,195,293)(74,414,196,292)(75,413,197,291)(76,412,198,290)(77,411,199,289)(78,410,200,288)(79,409,201,287)(80,408,202,286)(81,407,203,285)(82,406,204,284)(83,405,205,283)(84,404,206,282)(85,403,207,281)(86,402,208,280)(87,401,209,279)(88,400,210,278)(89,399,211,277)(90,398,212,276)(91,397,213,275)(92,396,214,274)(93,395,215,273)(94,394,216,272)(95,393,217,271)(96,392,218,270)(97,391,219,269)(98,390,220,268)(99,389,221,267)(100,388,222,266)(101,387,223,265)(102,386,224,264)(103,385,225,263)(104,384,226,262)(105,383,227,261)(106,382,228,260)(107,381,229,259)(108,380,230,258)(109,379,231,257)(110,378,232,256)(111,377,233,255)(112,376,234,254)(113,375,235,253)(114,374,236,252)(115,373,237,251)(116,372,238,250)(117,371,239,249)(118,370,240,248)(119,369,241,247)(120,368,242,246)(121,367,243,245)(122,366,244,488) );
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),(245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488)], [(1,487,123,365),(2,486,124,364),(3,485,125,363),(4,484,126,362),(5,483,127,361),(6,482,128,360),(7,481,129,359),(8,480,130,358),(9,479,131,357),(10,478,132,356),(11,477,133,355),(12,476,134,354),(13,475,135,353),(14,474,136,352),(15,473,137,351),(16,472,138,350),(17,471,139,349),(18,470,140,348),(19,469,141,347),(20,468,142,346),(21,467,143,345),(22,466,144,344),(23,465,145,343),(24,464,146,342),(25,463,147,341),(26,462,148,340),(27,461,149,339),(28,460,150,338),(29,459,151,337),(30,458,152,336),(31,457,153,335),(32,456,154,334),(33,455,155,333),(34,454,156,332),(35,453,157,331),(36,452,158,330),(37,451,159,329),(38,450,160,328),(39,449,161,327),(40,448,162,326),(41,447,163,325),(42,446,164,324),(43,445,165,323),(44,444,166,322),(45,443,167,321),(46,442,168,320),(47,441,169,319),(48,440,170,318),(49,439,171,317),(50,438,172,316),(51,437,173,315),(52,436,174,314),(53,435,175,313),(54,434,176,312),(55,433,177,311),(56,432,178,310),(57,431,179,309),(58,430,180,308),(59,429,181,307),(60,428,182,306),(61,427,183,305),(62,426,184,304),(63,425,185,303),(64,424,186,302),(65,423,187,301),(66,422,188,300),(67,421,189,299),(68,420,190,298),(69,419,191,297),(70,418,192,296),(71,417,193,295),(72,416,194,294),(73,415,195,293),(74,414,196,292),(75,413,197,291),(76,412,198,290),(77,411,199,289),(78,410,200,288),(79,409,201,287),(80,408,202,286),(81,407,203,285),(82,406,204,284),(83,405,205,283),(84,404,206,282),(85,403,207,281),(86,402,208,280),(87,401,209,279),(88,400,210,278),(89,399,211,277),(90,398,212,276),(91,397,213,275),(92,396,214,274),(93,395,215,273),(94,394,216,272),(95,393,217,271),(96,392,218,270),(97,391,219,269),(98,390,220,268),(99,389,221,267),(100,388,222,266),(101,387,223,265),(102,386,224,264),(103,385,225,263),(104,384,226,262),(105,383,227,261),(106,382,228,260),(107,381,229,259),(108,380,230,258),(109,379,231,257),(110,378,232,256),(111,377,233,255),(112,376,234,254),(113,375,235,253),(114,374,236,252),(115,373,237,251),(116,372,238,250),(117,371,239,249),(118,370,240,248),(119,369,241,247),(120,368,242,246),(121,367,243,245),(122,366,244,488)]])
125 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 61A | ··· | 61AD | 122A | ··· | 122AD | 244A | ··· | 244BH |
| order | 1 | 2 | 4 | 4 | 4 | 61 | ··· | 61 | 122 | ··· | 122 | 244 | ··· | 244 |
| size | 1 | 1 | 2 | 122 | 122 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D61 | D122 | Dic122 |
| kernel | Dic122 | Dic61 | C244 | C61 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 30 | 30 | 60 |
Matrix representation of Dic122 ►in GL2(𝔽733) generated by
| 610 | 444 |
| 289 | 536 |
| 107 | 570 |
| 430 | 626 |
G:=sub<GL(2,GF(733))| [610,289,444,536],[107,430,570,626] >;
Dic122 in GAP, Magma, Sage, TeX
{\rm Dic}_{122} % in TeX
G:=Group("Dic122"); // GroupNames label
G:=SmallGroup(488,4);
// by ID
G=gap.SmallGroup(488,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-61,16,49,21,7683]);
// Polycyclic
G:=Group<a,b|a^244=1,b^2=a^122,b*a*b^-1=a^-1>;
// generators/relations
Export