metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic124, C8.D31, C31⋊1Q16, C62.3D4, C248.1C2, C4.10D62, C2.5D124, Dic62.1C2, C124.10C22, SmallGroup(496,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic124
G = < a,b | a248=1, b2=a124, bab-1=a-1 >
Smallest permutation representation of Dic124
►Regular action on 496 points
Generators in S496
(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)(249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496)
(1 484 125 360)(2 483 126 359)(3 482 127 358)(4 481 128 357)(5 480 129 356)(6 479 130 355)(7 478 131 354)(8 477 132 353)(9 476 133 352)(10 475 134 351)(11 474 135 350)(12 473 136 349)(13 472 137 348)(14 471 138 347)(15 470 139 346)(16 469 140 345)(17 468 141 344)(18 467 142 343)(19 466 143 342)(20 465 144 341)(21 464 145 340)(22 463 146 339)(23 462 147 338)(24 461 148 337)(25 460 149 336)(26 459 150 335)(27 458 151 334)(28 457 152 333)(29 456 153 332)(30 455 154 331)(31 454 155 330)(32 453 156 329)(33 452 157 328)(34 451 158 327)(35 450 159 326)(36 449 160 325)(37 448 161 324)(38 447 162 323)(39 446 163 322)(40 445 164 321)(41 444 165 320)(42 443 166 319)(43 442 167 318)(44 441 168 317)(45 440 169 316)(46 439 170 315)(47 438 171 314)(48 437 172 313)(49 436 173 312)(50 435 174 311)(51 434 175 310)(52 433 176 309)(53 432 177 308)(54 431 178 307)(55 430 179 306)(56 429 180 305)(57 428 181 304)(58 427 182 303)(59 426 183 302)(60 425 184 301)(61 424 185 300)(62 423 186 299)(63 422 187 298)(64 421 188 297)(65 420 189 296)(66 419 190 295)(67 418 191 294)(68 417 192 293)(69 416 193 292)(70 415 194 291)(71 414 195 290)(72 413 196 289)(73 412 197 288)(74 411 198 287)(75 410 199 286)(76 409 200 285)(77 408 201 284)(78 407 202 283)(79 406 203 282)(80 405 204 281)(81 404 205 280)(82 403 206 279)(83 402 207 278)(84 401 208 277)(85 400 209 276)(86 399 210 275)(87 398 211 274)(88 397 212 273)(89 396 213 272)(90 395 214 271)(91 394 215 270)(92 393 216 269)(93 392 217 268)(94 391 218 267)(95 390 219 266)(96 389 220 265)(97 388 221 264)(98 387 222 263)(99 386 223 262)(100 385 224 261)(101 384 225 260)(102 383 226 259)(103 382 227 258)(104 381 228 257)(105 380 229 256)(106 379 230 255)(107 378 231 254)(108 377 232 253)(109 376 233 252)(110 375 234 251)(111 374 235 250)(112 373 236 249)(113 372 237 496)(114 371 238 495)(115 370 239 494)(116 369 240 493)(117 368 241 492)(118 367 242 491)(119 366 243 490)(120 365 244 489)(121 364 245 488)(122 363 246 487)(123 362 247 486)(124 361 248 485)
G:=sub<Sym(496)| (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)(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,484,125,360)(2,483,126,359)(3,482,127,358)(4,481,128,357)(5,480,129,356)(6,479,130,355)(7,478,131,354)(8,477,132,353)(9,476,133,352)(10,475,134,351)(11,474,135,350)(12,473,136,349)(13,472,137,348)(14,471,138,347)(15,470,139,346)(16,469,140,345)(17,468,141,344)(18,467,142,343)(19,466,143,342)(20,465,144,341)(21,464,145,340)(22,463,146,339)(23,462,147,338)(24,461,148,337)(25,460,149,336)(26,459,150,335)(27,458,151,334)(28,457,152,333)(29,456,153,332)(30,455,154,331)(31,454,155,330)(32,453,156,329)(33,452,157,328)(34,451,158,327)(35,450,159,326)(36,449,160,325)(37,448,161,324)(38,447,162,323)(39,446,163,322)(40,445,164,321)(41,444,165,320)(42,443,166,319)(43,442,167,318)(44,441,168,317)(45,440,169,316)(46,439,170,315)(47,438,171,314)(48,437,172,313)(49,436,173,312)(50,435,174,311)(51,434,175,310)(52,433,176,309)(53,432,177,308)(54,431,178,307)(55,430,179,306)(56,429,180,305)(57,428,181,304)(58,427,182,303)(59,426,183,302)(60,425,184,301)(61,424,185,300)(62,423,186,299)(63,422,187,298)(64,421,188,297)(65,420,189,296)(66,419,190,295)(67,418,191,294)(68,417,192,293)(69,416,193,292)(70,415,194,291)(71,414,195,290)(72,413,196,289)(73,412,197,288)(74,411,198,287)(75,410,199,286)(76,409,200,285)(77,408,201,284)(78,407,202,283)(79,406,203,282)(80,405,204,281)(81,404,205,280)(82,403,206,279)(83,402,207,278)(84,401,208,277)(85,400,209,276)(86,399,210,275)(87,398,211,274)(88,397,212,273)(89,396,213,272)(90,395,214,271)(91,394,215,270)(92,393,216,269)(93,392,217,268)(94,391,218,267)(95,390,219,266)(96,389,220,265)(97,388,221,264)(98,387,222,263)(99,386,223,262)(100,385,224,261)(101,384,225,260)(102,383,226,259)(103,382,227,258)(104,381,228,257)(105,380,229,256)(106,379,230,255)(107,378,231,254)(108,377,232,253)(109,376,233,252)(110,375,234,251)(111,374,235,250)(112,373,236,249)(113,372,237,496)(114,371,238,495)(115,370,239,494)(116,369,240,493)(117,368,241,492)(118,367,242,491)(119,366,243,490)(120,365,244,489)(121,364,245,488)(122,363,246,487)(123,362,247,486)(124,361,248,485)>;
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)(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,484,125,360)(2,483,126,359)(3,482,127,358)(4,481,128,357)(5,480,129,356)(6,479,130,355)(7,478,131,354)(8,477,132,353)(9,476,133,352)(10,475,134,351)(11,474,135,350)(12,473,136,349)(13,472,137,348)(14,471,138,347)(15,470,139,346)(16,469,140,345)(17,468,141,344)(18,467,142,343)(19,466,143,342)(20,465,144,341)(21,464,145,340)(22,463,146,339)(23,462,147,338)(24,461,148,337)(25,460,149,336)(26,459,150,335)(27,458,151,334)(28,457,152,333)(29,456,153,332)(30,455,154,331)(31,454,155,330)(32,453,156,329)(33,452,157,328)(34,451,158,327)(35,450,159,326)(36,449,160,325)(37,448,161,324)(38,447,162,323)(39,446,163,322)(40,445,164,321)(41,444,165,320)(42,443,166,319)(43,442,167,318)(44,441,168,317)(45,440,169,316)(46,439,170,315)(47,438,171,314)(48,437,172,313)(49,436,173,312)(50,435,174,311)(51,434,175,310)(52,433,176,309)(53,432,177,308)(54,431,178,307)(55,430,179,306)(56,429,180,305)(57,428,181,304)(58,427,182,303)(59,426,183,302)(60,425,184,301)(61,424,185,300)(62,423,186,299)(63,422,187,298)(64,421,188,297)(65,420,189,296)(66,419,190,295)(67,418,191,294)(68,417,192,293)(69,416,193,292)(70,415,194,291)(71,414,195,290)(72,413,196,289)(73,412,197,288)(74,411,198,287)(75,410,199,286)(76,409,200,285)(77,408,201,284)(78,407,202,283)(79,406,203,282)(80,405,204,281)(81,404,205,280)(82,403,206,279)(83,402,207,278)(84,401,208,277)(85,400,209,276)(86,399,210,275)(87,398,211,274)(88,397,212,273)(89,396,213,272)(90,395,214,271)(91,394,215,270)(92,393,216,269)(93,392,217,268)(94,391,218,267)(95,390,219,266)(96,389,220,265)(97,388,221,264)(98,387,222,263)(99,386,223,262)(100,385,224,261)(101,384,225,260)(102,383,226,259)(103,382,227,258)(104,381,228,257)(105,380,229,256)(106,379,230,255)(107,378,231,254)(108,377,232,253)(109,376,233,252)(110,375,234,251)(111,374,235,250)(112,373,236,249)(113,372,237,496)(114,371,238,495)(115,370,239,494)(116,369,240,493)(117,368,241,492)(118,367,242,491)(119,366,243,490)(120,365,244,489)(121,364,245,488)(122,363,246,487)(123,362,247,486)(124,361,248,485) );
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),(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496)], [(1,484,125,360),(2,483,126,359),(3,482,127,358),(4,481,128,357),(5,480,129,356),(6,479,130,355),(7,478,131,354),(8,477,132,353),(9,476,133,352),(10,475,134,351),(11,474,135,350),(12,473,136,349),(13,472,137,348),(14,471,138,347),(15,470,139,346),(16,469,140,345),(17,468,141,344),(18,467,142,343),(19,466,143,342),(20,465,144,341),(21,464,145,340),(22,463,146,339),(23,462,147,338),(24,461,148,337),(25,460,149,336),(26,459,150,335),(27,458,151,334),(28,457,152,333),(29,456,153,332),(30,455,154,331),(31,454,155,330),(32,453,156,329),(33,452,157,328),(34,451,158,327),(35,450,159,326),(36,449,160,325),(37,448,161,324),(38,447,162,323),(39,446,163,322),(40,445,164,321),(41,444,165,320),(42,443,166,319),(43,442,167,318),(44,441,168,317),(45,440,169,316),(46,439,170,315),(47,438,171,314),(48,437,172,313),(49,436,173,312),(50,435,174,311),(51,434,175,310),(52,433,176,309),(53,432,177,308),(54,431,178,307),(55,430,179,306),(56,429,180,305),(57,428,181,304),(58,427,182,303),(59,426,183,302),(60,425,184,301),(61,424,185,300),(62,423,186,299),(63,422,187,298),(64,421,188,297),(65,420,189,296),(66,419,190,295),(67,418,191,294),(68,417,192,293),(69,416,193,292),(70,415,194,291),(71,414,195,290),(72,413,196,289),(73,412,197,288),(74,411,198,287),(75,410,199,286),(76,409,200,285),(77,408,201,284),(78,407,202,283),(79,406,203,282),(80,405,204,281),(81,404,205,280),(82,403,206,279),(83,402,207,278),(84,401,208,277),(85,400,209,276),(86,399,210,275),(87,398,211,274),(88,397,212,273),(89,396,213,272),(90,395,214,271),(91,394,215,270),(92,393,216,269),(93,392,217,268),(94,391,218,267),(95,390,219,266),(96,389,220,265),(97,388,221,264),(98,387,222,263),(99,386,223,262),(100,385,224,261),(101,384,225,260),(102,383,226,259),(103,382,227,258),(104,381,228,257),(105,380,229,256),(106,379,230,255),(107,378,231,254),(108,377,232,253),(109,376,233,252),(110,375,234,251),(111,374,235,250),(112,373,236,249),(113,372,237,496),(114,371,238,495),(115,370,239,494),(116,369,240,493),(117,368,241,492),(118,367,242,491),(119,366,243,490),(120,365,244,489),(121,364,245,488),(122,363,246,487),(123,362,247,486),(124,361,248,485)]])
127 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD | 248A | ··· | 248BH |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 2 | 124 | 124 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
127 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | Q16 | D31 | D62 | D124 | Dic124 |
| kernel | Dic124 | C248 | Dic62 | C62 | C31 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 15 | 15 | 30 | 60 |
Matrix representation of Dic124 ►in GL2(𝔽1489) generated by
| 990 | 1472 |
| 17 | 108 |
| 5 | 367 |
| 714 | 1484 |
G:=sub<GL(2,GF(1489))| [990,17,1472,108],[5,714,367,1484] >;
Dic124 in GAP, Magma, Sage, TeX
{\rm Dic}_{124} % in TeX
G:=Group("Dic124"); // GroupNames label
G:=SmallGroup(496,7);
// by ID
G=gap.SmallGroup(496,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,40,61,66,182,42,12004]);
// Polycyclic
G:=Group<a,b|a^248=1,b^2=a^124,b*a*b^-1=a^-1>;
// generators/relations
Export