direct product, abelian, monomial
Aliases: C10×C50, SmallGroup(500,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C10×C50 |
| C1 — C10×C50 |
| C1 — C10×C50 |
Generators and relations for C10×C50
G = < a,b | a10=b50=1, ab=ba >
Subgroups: 70, all normal (8 characteristic)
C1, C2, C22, C5, C5, C10, C2×C10, C2×C10, C25, C52, C50, C5×C10, C2×C50, C102, C5×C25, C5×C50, C10×C50
Quotients: C1, C2, C22, C5, C10, C2×C10, C25, C52, C50, C5×C10, C2×C50, C102, C5×C25, C5×C50, C10×C50
►Regular action on 500 points
Generators in S500
(1 301 151 53 149 493 242 298 432 359)(2 302 152 54 150 494 243 299 433 360)(3 303 153 55 101 495 244 300 434 361)(4 304 154 56 102 496 245 251 435 362)(5 305 155 57 103 497 246 252 436 363)(6 306 156 58 104 498 247 253 437 364)(7 307 157 59 105 499 248 254 438 365)(8 308 158 60 106 500 249 255 439 366)(9 309 159 61 107 451 250 256 440 367)(10 310 160 62 108 452 201 257 441 368)(11 311 161 63 109 453 202 258 442 369)(12 312 162 64 110 454 203 259 443 370)(13 313 163 65 111 455 204 260 444 371)(14 314 164 66 112 456 205 261 445 372)(15 315 165 67 113 457 206 262 446 373)(16 316 166 68 114 458 207 263 447 374)(17 317 167 69 115 459 208 264 448 375)(18 318 168 70 116 460 209 265 449 376)(19 319 169 71 117 461 210 266 450 377)(20 320 170 72 118 462 211 267 401 378)(21 321 171 73 119 463 212 268 402 379)(22 322 172 74 120 464 213 269 403 380)(23 323 173 75 121 465 214 270 404 381)(24 324 174 76 122 466 215 271 405 382)(25 325 175 77 123 467 216 272 406 383)(26 326 176 78 124 468 217 273 407 384)(27 327 177 79 125 469 218 274 408 385)(28 328 178 80 126 470 219 275 409 386)(29 329 179 81 127 471 220 276 410 387)(30 330 180 82 128 472 221 277 411 388)(31 331 181 83 129 473 222 278 412 389)(32 332 182 84 130 474 223 279 413 390)(33 333 183 85 131 475 224 280 414 391)(34 334 184 86 132 476 225 281 415 392)(35 335 185 87 133 477 226 282 416 393)(36 336 186 88 134 478 227 283 417 394)(37 337 187 89 135 479 228 284 418 395)(38 338 188 90 136 480 229 285 419 396)(39 339 189 91 137 481 230 286 420 397)(40 340 190 92 138 482 231 287 421 398)(41 341 191 93 139 483 232 288 422 399)(42 342 192 94 140 484 233 289 423 400)(43 343 193 95 141 485 234 290 424 351)(44 344 194 96 142 486 235 291 425 352)(45 345 195 97 143 487 236 292 426 353)(46 346 196 98 144 488 237 293 427 354)(47 347 197 99 145 489 238 294 428 355)(48 348 198 100 146 490 239 295 429 356)(49 349 199 51 147 491 240 296 430 357)(50 350 200 52 148 492 241 297 431 358)
(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)
G:=sub<Sym(500)| (1,301,151,53,149,493,242,298,432,359)(2,302,152,54,150,494,243,299,433,360)(3,303,153,55,101,495,244,300,434,361)(4,304,154,56,102,496,245,251,435,362)(5,305,155,57,103,497,246,252,436,363)(6,306,156,58,104,498,247,253,437,364)(7,307,157,59,105,499,248,254,438,365)(8,308,158,60,106,500,249,255,439,366)(9,309,159,61,107,451,250,256,440,367)(10,310,160,62,108,452,201,257,441,368)(11,311,161,63,109,453,202,258,442,369)(12,312,162,64,110,454,203,259,443,370)(13,313,163,65,111,455,204,260,444,371)(14,314,164,66,112,456,205,261,445,372)(15,315,165,67,113,457,206,262,446,373)(16,316,166,68,114,458,207,263,447,374)(17,317,167,69,115,459,208,264,448,375)(18,318,168,70,116,460,209,265,449,376)(19,319,169,71,117,461,210,266,450,377)(20,320,170,72,118,462,211,267,401,378)(21,321,171,73,119,463,212,268,402,379)(22,322,172,74,120,464,213,269,403,380)(23,323,173,75,121,465,214,270,404,381)(24,324,174,76,122,466,215,271,405,382)(25,325,175,77,123,467,216,272,406,383)(26,326,176,78,124,468,217,273,407,384)(27,327,177,79,125,469,218,274,408,385)(28,328,178,80,126,470,219,275,409,386)(29,329,179,81,127,471,220,276,410,387)(30,330,180,82,128,472,221,277,411,388)(31,331,181,83,129,473,222,278,412,389)(32,332,182,84,130,474,223,279,413,390)(33,333,183,85,131,475,224,280,414,391)(34,334,184,86,132,476,225,281,415,392)(35,335,185,87,133,477,226,282,416,393)(36,336,186,88,134,478,227,283,417,394)(37,337,187,89,135,479,228,284,418,395)(38,338,188,90,136,480,229,285,419,396)(39,339,189,91,137,481,230,286,420,397)(40,340,190,92,138,482,231,287,421,398)(41,341,191,93,139,483,232,288,422,399)(42,342,192,94,140,484,233,289,423,400)(43,343,193,95,141,485,234,290,424,351)(44,344,194,96,142,486,235,291,425,352)(45,345,195,97,143,487,236,292,426,353)(46,346,196,98,144,488,237,293,427,354)(47,347,197,99,145,489,238,294,428,355)(48,348,198,100,146,490,239,295,429,356)(49,349,199,51,147,491,240,296,430,357)(50,350,200,52,148,492,241,297,431,358), (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)>;
G:=Group( (1,301,151,53,149,493,242,298,432,359)(2,302,152,54,150,494,243,299,433,360)(3,303,153,55,101,495,244,300,434,361)(4,304,154,56,102,496,245,251,435,362)(5,305,155,57,103,497,246,252,436,363)(6,306,156,58,104,498,247,253,437,364)(7,307,157,59,105,499,248,254,438,365)(8,308,158,60,106,500,249,255,439,366)(9,309,159,61,107,451,250,256,440,367)(10,310,160,62,108,452,201,257,441,368)(11,311,161,63,109,453,202,258,442,369)(12,312,162,64,110,454,203,259,443,370)(13,313,163,65,111,455,204,260,444,371)(14,314,164,66,112,456,205,261,445,372)(15,315,165,67,113,457,206,262,446,373)(16,316,166,68,114,458,207,263,447,374)(17,317,167,69,115,459,208,264,448,375)(18,318,168,70,116,460,209,265,449,376)(19,319,169,71,117,461,210,266,450,377)(20,320,170,72,118,462,211,267,401,378)(21,321,171,73,119,463,212,268,402,379)(22,322,172,74,120,464,213,269,403,380)(23,323,173,75,121,465,214,270,404,381)(24,324,174,76,122,466,215,271,405,382)(25,325,175,77,123,467,216,272,406,383)(26,326,176,78,124,468,217,273,407,384)(27,327,177,79,125,469,218,274,408,385)(28,328,178,80,126,470,219,275,409,386)(29,329,179,81,127,471,220,276,410,387)(30,330,180,82,128,472,221,277,411,388)(31,331,181,83,129,473,222,278,412,389)(32,332,182,84,130,474,223,279,413,390)(33,333,183,85,131,475,224,280,414,391)(34,334,184,86,132,476,225,281,415,392)(35,335,185,87,133,477,226,282,416,393)(36,336,186,88,134,478,227,283,417,394)(37,337,187,89,135,479,228,284,418,395)(38,338,188,90,136,480,229,285,419,396)(39,339,189,91,137,481,230,286,420,397)(40,340,190,92,138,482,231,287,421,398)(41,341,191,93,139,483,232,288,422,399)(42,342,192,94,140,484,233,289,423,400)(43,343,193,95,141,485,234,290,424,351)(44,344,194,96,142,486,235,291,425,352)(45,345,195,97,143,487,236,292,426,353)(46,346,196,98,144,488,237,293,427,354)(47,347,197,99,145,489,238,294,428,355)(48,348,198,100,146,490,239,295,429,356)(49,349,199,51,147,491,240,296,430,357)(50,350,200,52,148,492,241,297,431,358), (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) );
G=PermutationGroup([[(1,301,151,53,149,493,242,298,432,359),(2,302,152,54,150,494,243,299,433,360),(3,303,153,55,101,495,244,300,434,361),(4,304,154,56,102,496,245,251,435,362),(5,305,155,57,103,497,246,252,436,363),(6,306,156,58,104,498,247,253,437,364),(7,307,157,59,105,499,248,254,438,365),(8,308,158,60,106,500,249,255,439,366),(9,309,159,61,107,451,250,256,440,367),(10,310,160,62,108,452,201,257,441,368),(11,311,161,63,109,453,202,258,442,369),(12,312,162,64,110,454,203,259,443,370),(13,313,163,65,111,455,204,260,444,371),(14,314,164,66,112,456,205,261,445,372),(15,315,165,67,113,457,206,262,446,373),(16,316,166,68,114,458,207,263,447,374),(17,317,167,69,115,459,208,264,448,375),(18,318,168,70,116,460,209,265,449,376),(19,319,169,71,117,461,210,266,450,377),(20,320,170,72,118,462,211,267,401,378),(21,321,171,73,119,463,212,268,402,379),(22,322,172,74,120,464,213,269,403,380),(23,323,173,75,121,465,214,270,404,381),(24,324,174,76,122,466,215,271,405,382),(25,325,175,77,123,467,216,272,406,383),(26,326,176,78,124,468,217,273,407,384),(27,327,177,79,125,469,218,274,408,385),(28,328,178,80,126,470,219,275,409,386),(29,329,179,81,127,471,220,276,410,387),(30,330,180,82,128,472,221,277,411,388),(31,331,181,83,129,473,222,278,412,389),(32,332,182,84,130,474,223,279,413,390),(33,333,183,85,131,475,224,280,414,391),(34,334,184,86,132,476,225,281,415,392),(35,335,185,87,133,477,226,282,416,393),(36,336,186,88,134,478,227,283,417,394),(37,337,187,89,135,479,228,284,418,395),(38,338,188,90,136,480,229,285,419,396),(39,339,189,91,137,481,230,286,420,397),(40,340,190,92,138,482,231,287,421,398),(41,341,191,93,139,483,232,288,422,399),(42,342,192,94,140,484,233,289,423,400),(43,343,193,95,141,485,234,290,424,351),(44,344,194,96,142,486,235,291,425,352),(45,345,195,97,143,487,236,292,426,353),(46,346,196,98,144,488,237,293,427,354),(47,347,197,99,145,489,238,294,428,355),(48,348,198,100,146,490,239,295,429,356),(49,349,199,51,147,491,240,296,430,357),(50,350,200,52,148,492,241,297,431,358)], [(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)]])
500 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | ··· | 5X | 10A | ··· | 10BT | 25A | ··· | 25CV | 50A | ··· | 50KN |
| order | 1 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
500 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C5 | C10 | C10 | C25 | C50 |
| kernel | C10×C50 | C5×C50 | C2×C50 | C102 | C50 | C5×C10 | C2×C10 | C10 |
| # reps | 1 | 3 | 20 | 4 | 60 | 12 | 100 | 300 |
Matrix representation of C10×C50 ►in GL3(𝔽101) generated by
| 6 | 0 | 0 |
| 0 | 95 | 0 |
| 0 | 0 | 84 |
| 76 | 0 | 0 |
| 0 | 21 | 0 |
| 0 | 0 | 54 |
G:=sub<GL(3,GF(101))| [6,0,0,0,95,0,0,0,84],[76,0,0,0,21,0,0,0,54] >;
C10×C50 in GAP, Magma, Sage, TeX
C_{10}\times C_{50} % in TeX
G:=Group("C10xC50"); // GroupNames label
G:=SmallGroup(500,34);
// by ID
G=gap.SmallGroup(500,34);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,387]);
// Polycyclic
G:=Group<a,b|a^10=b^50=1,a*b=b*a>;
// generators/relations