metabelian, supersoluble, monomial, A-group
Aliases: C50.3D5, C5⋊2Dic25, C25⋊2Dic5, C10.3D25, C52.4Dic5, (C5×C25)⋊9C4, C2.(C25⋊D5), (C5×C50).3C2, (C5×C10).6D5, C10.1(C5⋊D5), C5.(C52⋊6C4), SmallGroup(500,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C25 — C50.D5 |
Generators and relations for C50.D5
G = < a,b,c | a50=b5=1, c2=a25, ab=ba, cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C50.D5
►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)
(1 439 90 233 370)(2 440 91 234 371)(3 441 92 235 372)(4 442 93 236 373)(5 443 94 237 374)(6 444 95 238 375)(7 445 96 239 376)(8 446 97 240 377)(9 447 98 241 378)(10 448 99 242 379)(11 449 100 243 380)(12 450 51 244 381)(13 401 52 245 382)(14 402 53 246 383)(15 403 54 247 384)(16 404 55 248 385)(17 405 56 249 386)(18 406 57 250 387)(19 407 58 201 388)(20 408 59 202 389)(21 409 60 203 390)(22 410 61 204 391)(23 411 62 205 392)(24 412 63 206 393)(25 413 64 207 394)(26 414 65 208 395)(27 415 66 209 396)(28 416 67 210 397)(29 417 68 211 398)(30 418 69 212 399)(31 419 70 213 400)(32 420 71 214 351)(33 421 72 215 352)(34 422 73 216 353)(35 423 74 217 354)(36 424 75 218 355)(37 425 76 219 356)(38 426 77 220 357)(39 427 78 221 358)(40 428 79 222 359)(41 429 80 223 360)(42 430 81 224 361)(43 431 82 225 362)(44 432 83 226 363)(45 433 84 227 364)(46 434 85 228 365)(47 435 86 229 366)(48 436 87 230 367)(49 437 88 231 368)(50 438 89 232 369)(101 257 475 332 188)(102 258 476 333 189)(103 259 477 334 190)(104 260 478 335 191)(105 261 479 336 192)(106 262 480 337 193)(107 263 481 338 194)(108 264 482 339 195)(109 265 483 340 196)(110 266 484 341 197)(111 267 485 342 198)(112 268 486 343 199)(113 269 487 344 200)(114 270 488 345 151)(115 271 489 346 152)(116 272 490 347 153)(117 273 491 348 154)(118 274 492 349 155)(119 275 493 350 156)(120 276 494 301 157)(121 277 495 302 158)(122 278 496 303 159)(123 279 497 304 160)(124 280 498 305 161)(125 281 499 306 162)(126 282 500 307 163)(127 283 451 308 164)(128 284 452 309 165)(129 285 453 310 166)(130 286 454 311 167)(131 287 455 312 168)(132 288 456 313 169)(133 289 457 314 170)(134 290 458 315 171)(135 291 459 316 172)(136 292 460 317 173)(137 293 461 318 174)(138 294 462 319 175)(139 295 463 320 176)(140 296 464 321 177)(141 297 465 322 178)(142 298 466 323 179)(143 299 467 324 180)(144 300 468 325 181)(145 251 469 326 182)(146 252 470 327 183)(147 253 471 328 184)(148 254 472 329 185)(149 255 473 330 186)(150 256 474 331 187)
(1 166 26 191)(2 165 27 190)(3 164 28 189)(4 163 29 188)(5 162 30 187)(6 161 31 186)(7 160 32 185)(8 159 33 184)(9 158 34 183)(10 157 35 182)(11 156 36 181)(12 155 37 180)(13 154 38 179)(14 153 39 178)(15 152 40 177)(16 151 41 176)(17 200 42 175)(18 199 43 174)(19 198 44 173)(20 197 45 172)(21 196 46 171)(22 195 47 170)(23 194 48 169)(24 193 49 168)(25 192 50 167)(51 492 76 467)(52 491 77 466)(53 490 78 465)(54 489 79 464)(55 488 80 463)(56 487 81 462)(57 486 82 461)(58 485 83 460)(59 484 84 459)(60 483 85 458)(61 482 86 457)(62 481 87 456)(63 480 88 455)(64 479 89 454)(65 478 90 453)(66 477 91 452)(67 476 92 451)(68 475 93 500)(69 474 94 499)(70 473 95 498)(71 472 96 497)(72 471 97 496)(73 470 98 495)(74 469 99 494)(75 468 100 493)(101 373 126 398)(102 372 127 397)(103 371 128 396)(104 370 129 395)(105 369 130 394)(106 368 131 393)(107 367 132 392)(108 366 133 391)(109 365 134 390)(110 364 135 389)(111 363 136 388)(112 362 137 387)(113 361 138 386)(114 360 139 385)(115 359 140 384)(116 358 141 383)(117 357 142 382)(118 356 143 381)(119 355 144 380)(120 354 145 379)(121 353 146 378)(122 352 147 377)(123 351 148 376)(124 400 149 375)(125 399 150 374)(201 267 226 292)(202 266 227 291)(203 265 228 290)(204 264 229 289)(205 263 230 288)(206 262 231 287)(207 261 232 286)(208 260 233 285)(209 259 234 284)(210 258 235 283)(211 257 236 282)(212 256 237 281)(213 255 238 280)(214 254 239 279)(215 253 240 278)(216 252 241 277)(217 251 242 276)(218 300 243 275)(219 299 244 274)(220 298 245 273)(221 297 246 272)(222 296 247 271)(223 295 248 270)(224 294 249 269)(225 293 250 268)(301 423 326 448)(302 422 327 447)(303 421 328 446)(304 420 329 445)(305 419 330 444)(306 418 331 443)(307 417 332 442)(308 416 333 441)(309 415 334 440)(310 414 335 439)(311 413 336 438)(312 412 337 437)(313 411 338 436)(314 410 339 435)(315 409 340 434)(316 408 341 433)(317 407 342 432)(318 406 343 431)(319 405 344 430)(320 404 345 429)(321 403 346 428)(322 402 347 427)(323 401 348 426)(324 450 349 425)(325 449 350 424)
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), (1,439,90,233,370)(2,440,91,234,371)(3,441,92,235,372)(4,442,93,236,373)(5,443,94,237,374)(6,444,95,238,375)(7,445,96,239,376)(8,446,97,240,377)(9,447,98,241,378)(10,448,99,242,379)(11,449,100,243,380)(12,450,51,244,381)(13,401,52,245,382)(14,402,53,246,383)(15,403,54,247,384)(16,404,55,248,385)(17,405,56,249,386)(18,406,57,250,387)(19,407,58,201,388)(20,408,59,202,389)(21,409,60,203,390)(22,410,61,204,391)(23,411,62,205,392)(24,412,63,206,393)(25,413,64,207,394)(26,414,65,208,395)(27,415,66,209,396)(28,416,67,210,397)(29,417,68,211,398)(30,418,69,212,399)(31,419,70,213,400)(32,420,71,214,351)(33,421,72,215,352)(34,422,73,216,353)(35,423,74,217,354)(36,424,75,218,355)(37,425,76,219,356)(38,426,77,220,357)(39,427,78,221,358)(40,428,79,222,359)(41,429,80,223,360)(42,430,81,224,361)(43,431,82,225,362)(44,432,83,226,363)(45,433,84,227,364)(46,434,85,228,365)(47,435,86,229,366)(48,436,87,230,367)(49,437,88,231,368)(50,438,89,232,369)(101,257,475,332,188)(102,258,476,333,189)(103,259,477,334,190)(104,260,478,335,191)(105,261,479,336,192)(106,262,480,337,193)(107,263,481,338,194)(108,264,482,339,195)(109,265,483,340,196)(110,266,484,341,197)(111,267,485,342,198)(112,268,486,343,199)(113,269,487,344,200)(114,270,488,345,151)(115,271,489,346,152)(116,272,490,347,153)(117,273,491,348,154)(118,274,492,349,155)(119,275,493,350,156)(120,276,494,301,157)(121,277,495,302,158)(122,278,496,303,159)(123,279,497,304,160)(124,280,498,305,161)(125,281,499,306,162)(126,282,500,307,163)(127,283,451,308,164)(128,284,452,309,165)(129,285,453,310,166)(130,286,454,311,167)(131,287,455,312,168)(132,288,456,313,169)(133,289,457,314,170)(134,290,458,315,171)(135,291,459,316,172)(136,292,460,317,173)(137,293,461,318,174)(138,294,462,319,175)(139,295,463,320,176)(140,296,464,321,177)(141,297,465,322,178)(142,298,466,323,179)(143,299,467,324,180)(144,300,468,325,181)(145,251,469,326,182)(146,252,470,327,183)(147,253,471,328,184)(148,254,472,329,185)(149,255,473,330,186)(150,256,474,331,187), (1,166,26,191)(2,165,27,190)(3,164,28,189)(4,163,29,188)(5,162,30,187)(6,161,31,186)(7,160,32,185)(8,159,33,184)(9,158,34,183)(10,157,35,182)(11,156,36,181)(12,155,37,180)(13,154,38,179)(14,153,39,178)(15,152,40,177)(16,151,41,176)(17,200,42,175)(18,199,43,174)(19,198,44,173)(20,197,45,172)(21,196,46,171)(22,195,47,170)(23,194,48,169)(24,193,49,168)(25,192,50,167)(51,492,76,467)(52,491,77,466)(53,490,78,465)(54,489,79,464)(55,488,80,463)(56,487,81,462)(57,486,82,461)(58,485,83,460)(59,484,84,459)(60,483,85,458)(61,482,86,457)(62,481,87,456)(63,480,88,455)(64,479,89,454)(65,478,90,453)(66,477,91,452)(67,476,92,451)(68,475,93,500)(69,474,94,499)(70,473,95,498)(71,472,96,497)(72,471,97,496)(73,470,98,495)(74,469,99,494)(75,468,100,493)(101,373,126,398)(102,372,127,397)(103,371,128,396)(104,370,129,395)(105,369,130,394)(106,368,131,393)(107,367,132,392)(108,366,133,391)(109,365,134,390)(110,364,135,389)(111,363,136,388)(112,362,137,387)(113,361,138,386)(114,360,139,385)(115,359,140,384)(116,358,141,383)(117,357,142,382)(118,356,143,381)(119,355,144,380)(120,354,145,379)(121,353,146,378)(122,352,147,377)(123,351,148,376)(124,400,149,375)(125,399,150,374)(201,267,226,292)(202,266,227,291)(203,265,228,290)(204,264,229,289)(205,263,230,288)(206,262,231,287)(207,261,232,286)(208,260,233,285)(209,259,234,284)(210,258,235,283)(211,257,236,282)(212,256,237,281)(213,255,238,280)(214,254,239,279)(215,253,240,278)(216,252,241,277)(217,251,242,276)(218,300,243,275)(219,299,244,274)(220,298,245,273)(221,297,246,272)(222,296,247,271)(223,295,248,270)(224,294,249,269)(225,293,250,268)(301,423,326,448)(302,422,327,447)(303,421,328,446)(304,420,329,445)(305,419,330,444)(306,418,331,443)(307,417,332,442)(308,416,333,441)(309,415,334,440)(310,414,335,439)(311,413,336,438)(312,412,337,437)(313,411,338,436)(314,410,339,435)(315,409,340,434)(316,408,341,433)(317,407,342,432)(318,406,343,431)(319,405,344,430)(320,404,345,429)(321,403,346,428)(322,402,347,427)(323,401,348,426)(324,450,349,425)(325,449,350,424)>;
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), (1,439,90,233,370)(2,440,91,234,371)(3,441,92,235,372)(4,442,93,236,373)(5,443,94,237,374)(6,444,95,238,375)(7,445,96,239,376)(8,446,97,240,377)(9,447,98,241,378)(10,448,99,242,379)(11,449,100,243,380)(12,450,51,244,381)(13,401,52,245,382)(14,402,53,246,383)(15,403,54,247,384)(16,404,55,248,385)(17,405,56,249,386)(18,406,57,250,387)(19,407,58,201,388)(20,408,59,202,389)(21,409,60,203,390)(22,410,61,204,391)(23,411,62,205,392)(24,412,63,206,393)(25,413,64,207,394)(26,414,65,208,395)(27,415,66,209,396)(28,416,67,210,397)(29,417,68,211,398)(30,418,69,212,399)(31,419,70,213,400)(32,420,71,214,351)(33,421,72,215,352)(34,422,73,216,353)(35,423,74,217,354)(36,424,75,218,355)(37,425,76,219,356)(38,426,77,220,357)(39,427,78,221,358)(40,428,79,222,359)(41,429,80,223,360)(42,430,81,224,361)(43,431,82,225,362)(44,432,83,226,363)(45,433,84,227,364)(46,434,85,228,365)(47,435,86,229,366)(48,436,87,230,367)(49,437,88,231,368)(50,438,89,232,369)(101,257,475,332,188)(102,258,476,333,189)(103,259,477,334,190)(104,260,478,335,191)(105,261,479,336,192)(106,262,480,337,193)(107,263,481,338,194)(108,264,482,339,195)(109,265,483,340,196)(110,266,484,341,197)(111,267,485,342,198)(112,268,486,343,199)(113,269,487,344,200)(114,270,488,345,151)(115,271,489,346,152)(116,272,490,347,153)(117,273,491,348,154)(118,274,492,349,155)(119,275,493,350,156)(120,276,494,301,157)(121,277,495,302,158)(122,278,496,303,159)(123,279,497,304,160)(124,280,498,305,161)(125,281,499,306,162)(126,282,500,307,163)(127,283,451,308,164)(128,284,452,309,165)(129,285,453,310,166)(130,286,454,311,167)(131,287,455,312,168)(132,288,456,313,169)(133,289,457,314,170)(134,290,458,315,171)(135,291,459,316,172)(136,292,460,317,173)(137,293,461,318,174)(138,294,462,319,175)(139,295,463,320,176)(140,296,464,321,177)(141,297,465,322,178)(142,298,466,323,179)(143,299,467,324,180)(144,300,468,325,181)(145,251,469,326,182)(146,252,470,327,183)(147,253,471,328,184)(148,254,472,329,185)(149,255,473,330,186)(150,256,474,331,187), (1,166,26,191)(2,165,27,190)(3,164,28,189)(4,163,29,188)(5,162,30,187)(6,161,31,186)(7,160,32,185)(8,159,33,184)(9,158,34,183)(10,157,35,182)(11,156,36,181)(12,155,37,180)(13,154,38,179)(14,153,39,178)(15,152,40,177)(16,151,41,176)(17,200,42,175)(18,199,43,174)(19,198,44,173)(20,197,45,172)(21,196,46,171)(22,195,47,170)(23,194,48,169)(24,193,49,168)(25,192,50,167)(51,492,76,467)(52,491,77,466)(53,490,78,465)(54,489,79,464)(55,488,80,463)(56,487,81,462)(57,486,82,461)(58,485,83,460)(59,484,84,459)(60,483,85,458)(61,482,86,457)(62,481,87,456)(63,480,88,455)(64,479,89,454)(65,478,90,453)(66,477,91,452)(67,476,92,451)(68,475,93,500)(69,474,94,499)(70,473,95,498)(71,472,96,497)(72,471,97,496)(73,470,98,495)(74,469,99,494)(75,468,100,493)(101,373,126,398)(102,372,127,397)(103,371,128,396)(104,370,129,395)(105,369,130,394)(106,368,131,393)(107,367,132,392)(108,366,133,391)(109,365,134,390)(110,364,135,389)(111,363,136,388)(112,362,137,387)(113,361,138,386)(114,360,139,385)(115,359,140,384)(116,358,141,383)(117,357,142,382)(118,356,143,381)(119,355,144,380)(120,354,145,379)(121,353,146,378)(122,352,147,377)(123,351,148,376)(124,400,149,375)(125,399,150,374)(201,267,226,292)(202,266,227,291)(203,265,228,290)(204,264,229,289)(205,263,230,288)(206,262,231,287)(207,261,232,286)(208,260,233,285)(209,259,234,284)(210,258,235,283)(211,257,236,282)(212,256,237,281)(213,255,238,280)(214,254,239,279)(215,253,240,278)(216,252,241,277)(217,251,242,276)(218,300,243,275)(219,299,244,274)(220,298,245,273)(221,297,246,272)(222,296,247,271)(223,295,248,270)(224,294,249,269)(225,293,250,268)(301,423,326,448)(302,422,327,447)(303,421,328,446)(304,420,329,445)(305,419,330,444)(306,418,331,443)(307,417,332,442)(308,416,333,441)(309,415,334,440)(310,414,335,439)(311,413,336,438)(312,412,337,437)(313,411,338,436)(314,410,339,435)(315,409,340,434)(316,408,341,433)(317,407,342,432)(318,406,343,431)(319,405,344,430)(320,404,345,429)(321,403,346,428)(322,402,347,427)(323,401,348,426)(324,450,349,425)(325,449,350,424) );
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)], [(1,439,90,233,370),(2,440,91,234,371),(3,441,92,235,372),(4,442,93,236,373),(5,443,94,237,374),(6,444,95,238,375),(7,445,96,239,376),(8,446,97,240,377),(9,447,98,241,378),(10,448,99,242,379),(11,449,100,243,380),(12,450,51,244,381),(13,401,52,245,382),(14,402,53,246,383),(15,403,54,247,384),(16,404,55,248,385),(17,405,56,249,386),(18,406,57,250,387),(19,407,58,201,388),(20,408,59,202,389),(21,409,60,203,390),(22,410,61,204,391),(23,411,62,205,392),(24,412,63,206,393),(25,413,64,207,394),(26,414,65,208,395),(27,415,66,209,396),(28,416,67,210,397),(29,417,68,211,398),(30,418,69,212,399),(31,419,70,213,400),(32,420,71,214,351),(33,421,72,215,352),(34,422,73,216,353),(35,423,74,217,354),(36,424,75,218,355),(37,425,76,219,356),(38,426,77,220,357),(39,427,78,221,358),(40,428,79,222,359),(41,429,80,223,360),(42,430,81,224,361),(43,431,82,225,362),(44,432,83,226,363),(45,433,84,227,364),(46,434,85,228,365),(47,435,86,229,366),(48,436,87,230,367),(49,437,88,231,368),(50,438,89,232,369),(101,257,475,332,188),(102,258,476,333,189),(103,259,477,334,190),(104,260,478,335,191),(105,261,479,336,192),(106,262,480,337,193),(107,263,481,338,194),(108,264,482,339,195),(109,265,483,340,196),(110,266,484,341,197),(111,267,485,342,198),(112,268,486,343,199),(113,269,487,344,200),(114,270,488,345,151),(115,271,489,346,152),(116,272,490,347,153),(117,273,491,348,154),(118,274,492,349,155),(119,275,493,350,156),(120,276,494,301,157),(121,277,495,302,158),(122,278,496,303,159),(123,279,497,304,160),(124,280,498,305,161),(125,281,499,306,162),(126,282,500,307,163),(127,283,451,308,164),(128,284,452,309,165),(129,285,453,310,166),(130,286,454,311,167),(131,287,455,312,168),(132,288,456,313,169),(133,289,457,314,170),(134,290,458,315,171),(135,291,459,316,172),(136,292,460,317,173),(137,293,461,318,174),(138,294,462,319,175),(139,295,463,320,176),(140,296,464,321,177),(141,297,465,322,178),(142,298,466,323,179),(143,299,467,324,180),(144,300,468,325,181),(145,251,469,326,182),(146,252,470,327,183),(147,253,471,328,184),(148,254,472,329,185),(149,255,473,330,186),(150,256,474,331,187)], [(1,166,26,191),(2,165,27,190),(3,164,28,189),(4,163,29,188),(5,162,30,187),(6,161,31,186),(7,160,32,185),(8,159,33,184),(9,158,34,183),(10,157,35,182),(11,156,36,181),(12,155,37,180),(13,154,38,179),(14,153,39,178),(15,152,40,177),(16,151,41,176),(17,200,42,175),(18,199,43,174),(19,198,44,173),(20,197,45,172),(21,196,46,171),(22,195,47,170),(23,194,48,169),(24,193,49,168),(25,192,50,167),(51,492,76,467),(52,491,77,466),(53,490,78,465),(54,489,79,464),(55,488,80,463),(56,487,81,462),(57,486,82,461),(58,485,83,460),(59,484,84,459),(60,483,85,458),(61,482,86,457),(62,481,87,456),(63,480,88,455),(64,479,89,454),(65,478,90,453),(66,477,91,452),(67,476,92,451),(68,475,93,500),(69,474,94,499),(70,473,95,498),(71,472,96,497),(72,471,97,496),(73,470,98,495),(74,469,99,494),(75,468,100,493),(101,373,126,398),(102,372,127,397),(103,371,128,396),(104,370,129,395),(105,369,130,394),(106,368,131,393),(107,367,132,392),(108,366,133,391),(109,365,134,390),(110,364,135,389),(111,363,136,388),(112,362,137,387),(113,361,138,386),(114,360,139,385),(115,359,140,384),(116,358,141,383),(117,357,142,382),(118,356,143,381),(119,355,144,380),(120,354,145,379),(121,353,146,378),(122,352,147,377),(123,351,148,376),(124,400,149,375),(125,399,150,374),(201,267,226,292),(202,266,227,291),(203,265,228,290),(204,264,229,289),(205,263,230,288),(206,262,231,287),(207,261,232,286),(208,260,233,285),(209,259,234,284),(210,258,235,283),(211,257,236,282),(212,256,237,281),(213,255,238,280),(214,254,239,279),(215,253,240,278),(216,252,241,277),(217,251,242,276),(218,300,243,275),(219,299,244,274),(220,298,245,273),(221,297,246,272),(222,296,247,271),(223,295,248,270),(224,294,249,269),(225,293,250,268),(301,423,326,448),(302,422,327,447),(303,421,328,446),(304,420,329,445),(305,419,330,444),(306,418,331,443),(307,417,332,442),(308,416,333,441),(309,415,334,440),(310,414,335,439),(311,413,336,438),(312,412,337,437),(313,411,338,436),(314,410,339,435),(315,409,340,434),(316,408,341,433),(317,407,342,432),(318,406,343,431),(319,405,344,430),(320,404,345,429),(321,403,346,428),(322,402,347,427),(323,401,348,426),(324,450,349,425),(325,449,350,424)]])
128 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5L | 10A | ··· | 10L | 25A | ··· | 25AX | 50A | ··· | 50AX |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 125 | 125 | 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 | D5 | Dic5 | Dic5 | D25 | Dic25 |
| kernel | C50.D5 | C5×C50 | C5×C25 | C50 | C5×C10 | C25 | C52 | C10 | C5 |
| # reps | 1 | 1 | 2 | 10 | 2 | 10 | 2 | 50 | 50 |
Matrix representation of C50.D5 ►in GL4(𝔽101) generated by
| 50 | 13 | 0 | 0 |
| 88 | 33 | 0 | 0 |
| 0 | 0 | 50 | 13 |
| 0 | 0 | 88 | 33 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 100 | 22 |
| 29 | 74 | 0 | 0 |
| 5 | 72 | 0 | 0 |
| 0 | 0 | 96 | 29 |
| 0 | 0 | 20 | 5 |
G:=sub<GL(4,GF(101))| [50,88,0,0,13,33,0,0,0,0,50,88,0,0,13,33],[1,0,0,0,0,1,0,0,0,0,0,100,0,0,1,22],[29,5,0,0,74,72,0,0,0,0,96,20,0,0,29,5] >;
C50.D5 in GAP, Magma, Sage, TeX
C_{50}.D_5 % in TeX
G:=Group("C50.D5"); // GroupNames label
G:=SmallGroup(500,10);
// by ID
G=gap.SmallGroup(500,10);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,10,1742,1512,1603,10004]);
// Polycyclic
G:=Group<a,b,c|a^50=b^5=1,c^2=a^25,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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