metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic125, C125⋊2C4, C250.C2, C2.D125, C50.1D5, C25.Dic5, C5.Dic25, C10.1D25, SmallGroup(500,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C125 — Dic125 |
Generators and relations for Dic125
G = < a,b | a250=1, b2=a125, bab-1=a-1 >
Smallest permutation representation of Dic125
►Regular action on 500 points
Generators in S500
(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)(251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500)
(1 290 126 415)(2 289 127 414)(3 288 128 413)(4 287 129 412)(5 286 130 411)(6 285 131 410)(7 284 132 409)(8 283 133 408)(9 282 134 407)(10 281 135 406)(11 280 136 405)(12 279 137 404)(13 278 138 403)(14 277 139 402)(15 276 140 401)(16 275 141 400)(17 274 142 399)(18 273 143 398)(19 272 144 397)(20 271 145 396)(21 270 146 395)(22 269 147 394)(23 268 148 393)(24 267 149 392)(25 266 150 391)(26 265 151 390)(27 264 152 389)(28 263 153 388)(29 262 154 387)(30 261 155 386)(31 260 156 385)(32 259 157 384)(33 258 158 383)(34 257 159 382)(35 256 160 381)(36 255 161 380)(37 254 162 379)(38 253 163 378)(39 252 164 377)(40 251 165 376)(41 500 166 375)(42 499 167 374)(43 498 168 373)(44 497 169 372)(45 496 170 371)(46 495 171 370)(47 494 172 369)(48 493 173 368)(49 492 174 367)(50 491 175 366)(51 490 176 365)(52 489 177 364)(53 488 178 363)(54 487 179 362)(55 486 180 361)(56 485 181 360)(57 484 182 359)(58 483 183 358)(59 482 184 357)(60 481 185 356)(61 480 186 355)(62 479 187 354)(63 478 188 353)(64 477 189 352)(65 476 190 351)(66 475 191 350)(67 474 192 349)(68 473 193 348)(69 472 194 347)(70 471 195 346)(71 470 196 345)(72 469 197 344)(73 468 198 343)(74 467 199 342)(75 466 200 341)(76 465 201 340)(77 464 202 339)(78 463 203 338)(79 462 204 337)(80 461 205 336)(81 460 206 335)(82 459 207 334)(83 458 208 333)(84 457 209 332)(85 456 210 331)(86 455 211 330)(87 454 212 329)(88 453 213 328)(89 452 214 327)(90 451 215 326)(91 450 216 325)(92 449 217 324)(93 448 218 323)(94 447 219 322)(95 446 220 321)(96 445 221 320)(97 444 222 319)(98 443 223 318)(99 442 224 317)(100 441 225 316)(101 440 226 315)(102 439 227 314)(103 438 228 313)(104 437 229 312)(105 436 230 311)(106 435 231 310)(107 434 232 309)(108 433 233 308)(109 432 234 307)(110 431 235 306)(111 430 236 305)(112 429 237 304)(113 428 238 303)(114 427 239 302)(115 426 240 301)(116 425 241 300)(117 424 242 299)(118 423 243 298)(119 422 244 297)(120 421 245 296)(121 420 246 295)(122 419 247 294)(123 418 248 293)(124 417 249 292)(125 416 250 291)
G:=sub<Sym(500)| (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)(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500), (1,290,126,415)(2,289,127,414)(3,288,128,413)(4,287,129,412)(5,286,130,411)(6,285,131,410)(7,284,132,409)(8,283,133,408)(9,282,134,407)(10,281,135,406)(11,280,136,405)(12,279,137,404)(13,278,138,403)(14,277,139,402)(15,276,140,401)(16,275,141,400)(17,274,142,399)(18,273,143,398)(19,272,144,397)(20,271,145,396)(21,270,146,395)(22,269,147,394)(23,268,148,393)(24,267,149,392)(25,266,150,391)(26,265,151,390)(27,264,152,389)(28,263,153,388)(29,262,154,387)(30,261,155,386)(31,260,156,385)(32,259,157,384)(33,258,158,383)(34,257,159,382)(35,256,160,381)(36,255,161,380)(37,254,162,379)(38,253,163,378)(39,252,164,377)(40,251,165,376)(41,500,166,375)(42,499,167,374)(43,498,168,373)(44,497,169,372)(45,496,170,371)(46,495,171,370)(47,494,172,369)(48,493,173,368)(49,492,174,367)(50,491,175,366)(51,490,176,365)(52,489,177,364)(53,488,178,363)(54,487,179,362)(55,486,180,361)(56,485,181,360)(57,484,182,359)(58,483,183,358)(59,482,184,357)(60,481,185,356)(61,480,186,355)(62,479,187,354)(63,478,188,353)(64,477,189,352)(65,476,190,351)(66,475,191,350)(67,474,192,349)(68,473,193,348)(69,472,194,347)(70,471,195,346)(71,470,196,345)(72,469,197,344)(73,468,198,343)(74,467,199,342)(75,466,200,341)(76,465,201,340)(77,464,202,339)(78,463,203,338)(79,462,204,337)(80,461,205,336)(81,460,206,335)(82,459,207,334)(83,458,208,333)(84,457,209,332)(85,456,210,331)(86,455,211,330)(87,454,212,329)(88,453,213,328)(89,452,214,327)(90,451,215,326)(91,450,216,325)(92,449,217,324)(93,448,218,323)(94,447,219,322)(95,446,220,321)(96,445,221,320)(97,444,222,319)(98,443,223,318)(99,442,224,317)(100,441,225,316)(101,440,226,315)(102,439,227,314)(103,438,228,313)(104,437,229,312)(105,436,230,311)(106,435,231,310)(107,434,232,309)(108,433,233,308)(109,432,234,307)(110,431,235,306)(111,430,236,305)(112,429,237,304)(113,428,238,303)(114,427,239,302)(115,426,240,301)(116,425,241,300)(117,424,242,299)(118,423,243,298)(119,422,244,297)(120,421,245,296)(121,420,246,295)(122,419,247,294)(123,418,248,293)(124,417,249,292)(125,416,250,291)>;
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)(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500), (1,290,126,415)(2,289,127,414)(3,288,128,413)(4,287,129,412)(5,286,130,411)(6,285,131,410)(7,284,132,409)(8,283,133,408)(9,282,134,407)(10,281,135,406)(11,280,136,405)(12,279,137,404)(13,278,138,403)(14,277,139,402)(15,276,140,401)(16,275,141,400)(17,274,142,399)(18,273,143,398)(19,272,144,397)(20,271,145,396)(21,270,146,395)(22,269,147,394)(23,268,148,393)(24,267,149,392)(25,266,150,391)(26,265,151,390)(27,264,152,389)(28,263,153,388)(29,262,154,387)(30,261,155,386)(31,260,156,385)(32,259,157,384)(33,258,158,383)(34,257,159,382)(35,256,160,381)(36,255,161,380)(37,254,162,379)(38,253,163,378)(39,252,164,377)(40,251,165,376)(41,500,166,375)(42,499,167,374)(43,498,168,373)(44,497,169,372)(45,496,170,371)(46,495,171,370)(47,494,172,369)(48,493,173,368)(49,492,174,367)(50,491,175,366)(51,490,176,365)(52,489,177,364)(53,488,178,363)(54,487,179,362)(55,486,180,361)(56,485,181,360)(57,484,182,359)(58,483,183,358)(59,482,184,357)(60,481,185,356)(61,480,186,355)(62,479,187,354)(63,478,188,353)(64,477,189,352)(65,476,190,351)(66,475,191,350)(67,474,192,349)(68,473,193,348)(69,472,194,347)(70,471,195,346)(71,470,196,345)(72,469,197,344)(73,468,198,343)(74,467,199,342)(75,466,200,341)(76,465,201,340)(77,464,202,339)(78,463,203,338)(79,462,204,337)(80,461,205,336)(81,460,206,335)(82,459,207,334)(83,458,208,333)(84,457,209,332)(85,456,210,331)(86,455,211,330)(87,454,212,329)(88,453,213,328)(89,452,214,327)(90,451,215,326)(91,450,216,325)(92,449,217,324)(93,448,218,323)(94,447,219,322)(95,446,220,321)(96,445,221,320)(97,444,222,319)(98,443,223,318)(99,442,224,317)(100,441,225,316)(101,440,226,315)(102,439,227,314)(103,438,228,313)(104,437,229,312)(105,436,230,311)(106,435,231,310)(107,434,232,309)(108,433,233,308)(109,432,234,307)(110,431,235,306)(111,430,236,305)(112,429,237,304)(113,428,238,303)(114,427,239,302)(115,426,240,301)(116,425,241,300)(117,424,242,299)(118,423,243,298)(119,422,244,297)(120,421,245,296)(121,420,246,295)(122,419,247,294)(123,418,248,293)(124,417,249,292)(125,416,250,291) );
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),(251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500)], [(1,290,126,415),(2,289,127,414),(3,288,128,413),(4,287,129,412),(5,286,130,411),(6,285,131,410),(7,284,132,409),(8,283,133,408),(9,282,134,407),(10,281,135,406),(11,280,136,405),(12,279,137,404),(13,278,138,403),(14,277,139,402),(15,276,140,401),(16,275,141,400),(17,274,142,399),(18,273,143,398),(19,272,144,397),(20,271,145,396),(21,270,146,395),(22,269,147,394),(23,268,148,393),(24,267,149,392),(25,266,150,391),(26,265,151,390),(27,264,152,389),(28,263,153,388),(29,262,154,387),(30,261,155,386),(31,260,156,385),(32,259,157,384),(33,258,158,383),(34,257,159,382),(35,256,160,381),(36,255,161,380),(37,254,162,379),(38,253,163,378),(39,252,164,377),(40,251,165,376),(41,500,166,375),(42,499,167,374),(43,498,168,373),(44,497,169,372),(45,496,170,371),(46,495,171,370),(47,494,172,369),(48,493,173,368),(49,492,174,367),(50,491,175,366),(51,490,176,365),(52,489,177,364),(53,488,178,363),(54,487,179,362),(55,486,180,361),(56,485,181,360),(57,484,182,359),(58,483,183,358),(59,482,184,357),(60,481,185,356),(61,480,186,355),(62,479,187,354),(63,478,188,353),(64,477,189,352),(65,476,190,351),(66,475,191,350),(67,474,192,349),(68,473,193,348),(69,472,194,347),(70,471,195,346),(71,470,196,345),(72,469,197,344),(73,468,198,343),(74,467,199,342),(75,466,200,341),(76,465,201,340),(77,464,202,339),(78,463,203,338),(79,462,204,337),(80,461,205,336),(81,460,206,335),(82,459,207,334),(83,458,208,333),(84,457,209,332),(85,456,210,331),(86,455,211,330),(87,454,212,329),(88,453,213,328),(89,452,214,327),(90,451,215,326),(91,450,216,325),(92,449,217,324),(93,448,218,323),(94,447,219,322),(95,446,220,321),(96,445,221,320),(97,444,222,319),(98,443,223,318),(99,442,224,317),(100,441,225,316),(101,440,226,315),(102,439,227,314),(103,438,228,313),(104,437,229,312),(105,436,230,311),(106,435,231,310),(107,434,232,309),(108,433,233,308),(109,432,234,307),(110,431,235,306),(111,430,236,305),(112,429,237,304),(113,428,238,303),(114,427,239,302),(115,426,240,301),(116,425,241,300),(117,424,242,299),(118,423,243,298),(119,422,244,297),(120,421,245,296),(121,420,246,295),(122,419,247,294),(123,418,248,293),(124,417,249,292),(125,416,250,291)]])
128 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 25A | ··· | 25J | 50A | ··· | 50J | 125A | ··· | 125AX | 250A | ··· | 250AX |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 25 | ··· | 25 | 50 | ··· | 50 | 125 | ··· | 125 | 250 | ··· | 250 |
| size | 1 | 1 | 125 | 125 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | D5 | Dic5 | D25 | Dic25 | D125 | Dic125 |
| kernel | Dic125 | C250 | C125 | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 50 | 50 |
Matrix representation of Dic125 ►in GL3(𝔽3001) generated by
| 3000 | 0 | 0 |
| 0 | 758 | 196 |
| 0 | 2805 | 658 |
| 1648 | 0 | 0 |
| 0 | 129 | 2944 |
| 0 | 2082 | 2872 |
G:=sub<GL(3,GF(3001))| [3000,0,0,0,758,2805,0,196,658],[1648,0,0,0,129,2082,0,2944,2872] >;
Dic125 in GAP, Magma, Sage, TeX
{\rm Dic}_{125} % in TeX
G:=Group("Dic125"); // GroupNames label
G:=SmallGroup(500,1);
// by ID
G=gap.SmallGroup(500,1);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,10,542,687,3603,418,10004]);
// Polycyclic
G:=Group<a,b|a^250=1,b^2=a^125,b*a*b^-1=a^-1>;
// generators/relations
Export