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G = Dic125order 500 = 22·53

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic125, C1252C4, C250.C2, C2.D125, C50.1D5, C25.Dic5, C5.Dic25, C10.1D25, SmallGroup(500,1)

Series: Derived Chief Lower central Upper central

C1C125 — Dic125
C1C5C25C125C250 — Dic125
C125 — Dic125
C1C2

Generators and relations for Dic125
 G = < a,b | a250=1, b2=a125, bab-1=a-1 >

125C4
25Dic5
5Dic25

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)
(1 364 126 489)(2 363 127 488)(3 362 128 487)(4 361 129 486)(5 360 130 485)(6 359 131 484)(7 358 132 483)(8 357 133 482)(9 356 134 481)(10 355 135 480)(11 354 136 479)(12 353 137 478)(13 352 138 477)(14 351 139 476)(15 350 140 475)(16 349 141 474)(17 348 142 473)(18 347 143 472)(19 346 144 471)(20 345 145 470)(21 344 146 469)(22 343 147 468)(23 342 148 467)(24 341 149 466)(25 340 150 465)(26 339 151 464)(27 338 152 463)(28 337 153 462)(29 336 154 461)(30 335 155 460)(31 334 156 459)(32 333 157 458)(33 332 158 457)(34 331 159 456)(35 330 160 455)(36 329 161 454)(37 328 162 453)(38 327 163 452)(39 326 164 451)(40 325 165 450)(41 324 166 449)(42 323 167 448)(43 322 168 447)(44 321 169 446)(45 320 170 445)(46 319 171 444)(47 318 172 443)(48 317 173 442)(49 316 174 441)(50 315 175 440)(51 314 176 439)(52 313 177 438)(53 312 178 437)(54 311 179 436)(55 310 180 435)(56 309 181 434)(57 308 182 433)(58 307 183 432)(59 306 184 431)(60 305 185 430)(61 304 186 429)(62 303 187 428)(63 302 188 427)(64 301 189 426)(65 300 190 425)(66 299 191 424)(67 298 192 423)(68 297 193 422)(69 296 194 421)(70 295 195 420)(71 294 196 419)(72 293 197 418)(73 292 198 417)(74 291 199 416)(75 290 200 415)(76 289 201 414)(77 288 202 413)(78 287 203 412)(79 286 204 411)(80 285 205 410)(81 284 206 409)(82 283 207 408)(83 282 208 407)(84 281 209 406)(85 280 210 405)(86 279 211 404)(87 278 212 403)(88 277 213 402)(89 276 214 401)(90 275 215 400)(91 274 216 399)(92 273 217 398)(93 272 218 397)(94 271 219 396)(95 270 220 395)(96 269 221 394)(97 268 222 393)(98 267 223 392)(99 266 224 391)(100 265 225 390)(101 264 226 389)(102 263 227 388)(103 262 228 387)(104 261 229 386)(105 260 230 385)(106 259 231 384)(107 258 232 383)(108 257 233 382)(109 256 234 381)(110 255 235 380)(111 254 236 379)(112 253 237 378)(113 252 238 377)(114 251 239 376)(115 500 240 375)(116 499 241 374)(117 498 242 373)(118 497 243 372)(119 496 244 371)(120 495 245 370)(121 494 246 369)(122 493 247 368)(123 492 248 367)(124 491 249 366)(125 490 250 365)

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), (1,364,126,489)(2,363,127,488)(3,362,128,487)(4,361,129,486)(5,360,130,485)(6,359,131,484)(7,358,132,483)(8,357,133,482)(9,356,134,481)(10,355,135,480)(11,354,136,479)(12,353,137,478)(13,352,138,477)(14,351,139,476)(15,350,140,475)(16,349,141,474)(17,348,142,473)(18,347,143,472)(19,346,144,471)(20,345,145,470)(21,344,146,469)(22,343,147,468)(23,342,148,467)(24,341,149,466)(25,340,150,465)(26,339,151,464)(27,338,152,463)(28,337,153,462)(29,336,154,461)(30,335,155,460)(31,334,156,459)(32,333,157,458)(33,332,158,457)(34,331,159,456)(35,330,160,455)(36,329,161,454)(37,328,162,453)(38,327,163,452)(39,326,164,451)(40,325,165,450)(41,324,166,449)(42,323,167,448)(43,322,168,447)(44,321,169,446)(45,320,170,445)(46,319,171,444)(47,318,172,443)(48,317,173,442)(49,316,174,441)(50,315,175,440)(51,314,176,439)(52,313,177,438)(53,312,178,437)(54,311,179,436)(55,310,180,435)(56,309,181,434)(57,308,182,433)(58,307,183,432)(59,306,184,431)(60,305,185,430)(61,304,186,429)(62,303,187,428)(63,302,188,427)(64,301,189,426)(65,300,190,425)(66,299,191,424)(67,298,192,423)(68,297,193,422)(69,296,194,421)(70,295,195,420)(71,294,196,419)(72,293,197,418)(73,292,198,417)(74,291,199,416)(75,290,200,415)(76,289,201,414)(77,288,202,413)(78,287,203,412)(79,286,204,411)(80,285,205,410)(81,284,206,409)(82,283,207,408)(83,282,208,407)(84,281,209,406)(85,280,210,405)(86,279,211,404)(87,278,212,403)(88,277,213,402)(89,276,214,401)(90,275,215,400)(91,274,216,399)(92,273,217,398)(93,272,218,397)(94,271,219,396)(95,270,220,395)(96,269,221,394)(97,268,222,393)(98,267,223,392)(99,266,224,391)(100,265,225,390)(101,264,226,389)(102,263,227,388)(103,262,228,387)(104,261,229,386)(105,260,230,385)(106,259,231,384)(107,258,232,383)(108,257,233,382)(109,256,234,381)(110,255,235,380)(111,254,236,379)(112,253,237,378)(113,252,238,377)(114,251,239,376)(115,500,240,375)(116,499,241,374)(117,498,242,373)(118,497,243,372)(119,496,244,371)(120,495,245,370)(121,494,246,369)(122,493,247,368)(123,492,248,367)(124,491,249,366)(125,490,250,365)>;

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), (1,364,126,489)(2,363,127,488)(3,362,128,487)(4,361,129,486)(5,360,130,485)(6,359,131,484)(7,358,132,483)(8,357,133,482)(9,356,134,481)(10,355,135,480)(11,354,136,479)(12,353,137,478)(13,352,138,477)(14,351,139,476)(15,350,140,475)(16,349,141,474)(17,348,142,473)(18,347,143,472)(19,346,144,471)(20,345,145,470)(21,344,146,469)(22,343,147,468)(23,342,148,467)(24,341,149,466)(25,340,150,465)(26,339,151,464)(27,338,152,463)(28,337,153,462)(29,336,154,461)(30,335,155,460)(31,334,156,459)(32,333,157,458)(33,332,158,457)(34,331,159,456)(35,330,160,455)(36,329,161,454)(37,328,162,453)(38,327,163,452)(39,326,164,451)(40,325,165,450)(41,324,166,449)(42,323,167,448)(43,322,168,447)(44,321,169,446)(45,320,170,445)(46,319,171,444)(47,318,172,443)(48,317,173,442)(49,316,174,441)(50,315,175,440)(51,314,176,439)(52,313,177,438)(53,312,178,437)(54,311,179,436)(55,310,180,435)(56,309,181,434)(57,308,182,433)(58,307,183,432)(59,306,184,431)(60,305,185,430)(61,304,186,429)(62,303,187,428)(63,302,188,427)(64,301,189,426)(65,300,190,425)(66,299,191,424)(67,298,192,423)(68,297,193,422)(69,296,194,421)(70,295,195,420)(71,294,196,419)(72,293,197,418)(73,292,198,417)(74,291,199,416)(75,290,200,415)(76,289,201,414)(77,288,202,413)(78,287,203,412)(79,286,204,411)(80,285,205,410)(81,284,206,409)(82,283,207,408)(83,282,208,407)(84,281,209,406)(85,280,210,405)(86,279,211,404)(87,278,212,403)(88,277,213,402)(89,276,214,401)(90,275,215,400)(91,274,216,399)(92,273,217,398)(93,272,218,397)(94,271,219,396)(95,270,220,395)(96,269,221,394)(97,268,222,393)(98,267,223,392)(99,266,224,391)(100,265,225,390)(101,264,226,389)(102,263,227,388)(103,262,228,387)(104,261,229,386)(105,260,230,385)(106,259,231,384)(107,258,232,383)(108,257,233,382)(109,256,234,381)(110,255,235,380)(111,254,236,379)(112,253,237,378)(113,252,238,377)(114,251,239,376)(115,500,240,375)(116,499,241,374)(117,498,242,373)(118,497,243,372)(119,496,244,371)(120,495,245,370)(121,494,246,369)(122,493,247,368)(123,492,248,367)(124,491,249,366)(125,490,250,365) );

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)], [(1,364,126,489),(2,363,127,488),(3,362,128,487),(4,361,129,486),(5,360,130,485),(6,359,131,484),(7,358,132,483),(8,357,133,482),(9,356,134,481),(10,355,135,480),(11,354,136,479),(12,353,137,478),(13,352,138,477),(14,351,139,476),(15,350,140,475),(16,349,141,474),(17,348,142,473),(18,347,143,472),(19,346,144,471),(20,345,145,470),(21,344,146,469),(22,343,147,468),(23,342,148,467),(24,341,149,466),(25,340,150,465),(26,339,151,464),(27,338,152,463),(28,337,153,462),(29,336,154,461),(30,335,155,460),(31,334,156,459),(32,333,157,458),(33,332,158,457),(34,331,159,456),(35,330,160,455),(36,329,161,454),(37,328,162,453),(38,327,163,452),(39,326,164,451),(40,325,165,450),(41,324,166,449),(42,323,167,448),(43,322,168,447),(44,321,169,446),(45,320,170,445),(46,319,171,444),(47,318,172,443),(48,317,173,442),(49,316,174,441),(50,315,175,440),(51,314,176,439),(52,313,177,438),(53,312,178,437),(54,311,179,436),(55,310,180,435),(56,309,181,434),(57,308,182,433),(58,307,183,432),(59,306,184,431),(60,305,185,430),(61,304,186,429),(62,303,187,428),(63,302,188,427),(64,301,189,426),(65,300,190,425),(66,299,191,424),(67,298,192,423),(68,297,193,422),(69,296,194,421),(70,295,195,420),(71,294,196,419),(72,293,197,418),(73,292,198,417),(74,291,199,416),(75,290,200,415),(76,289,201,414),(77,288,202,413),(78,287,203,412),(79,286,204,411),(80,285,205,410),(81,284,206,409),(82,283,207,408),(83,282,208,407),(84,281,209,406),(85,280,210,405),(86,279,211,404),(87,278,212,403),(88,277,213,402),(89,276,214,401),(90,275,215,400),(91,274,216,399),(92,273,217,398),(93,272,218,397),(94,271,219,396),(95,270,220,395),(96,269,221,394),(97,268,222,393),(98,267,223,392),(99,266,224,391),(100,265,225,390),(101,264,226,389),(102,263,227,388),(103,262,228,387),(104,261,229,386),(105,260,230,385),(106,259,231,384),(107,258,232,383),(108,257,233,382),(109,256,234,381),(110,255,235,380),(111,254,236,379),(112,253,237,378),(113,252,238,377),(114,251,239,376),(115,500,240,375),(116,499,241,374),(117,498,242,373),(118,497,243,372),(119,496,244,371),(120,495,245,370),(121,494,246,369),(122,493,247,368),(123,492,248,367),(124,491,249,366),(125,490,250,365)])

128 conjugacy classes

class 1  2 4A4B5A5B10A10B25A···25J50A···50J125A···125AX250A···250AX
order124455101025···2550···50125···125250···250
size1112512522222···22···22···22···2

128 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4D5Dic5D25Dic25D125Dic125
kernelDic125C250C125C50C25C10C5C2C1
# reps1122210105050

Matrix representation of Dic125 in GL3(𝔽3001) generated by

300000
0758196
02805658
,
164800
01292944
020822872
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

Subgroup lattice of Dic125 in TeX

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