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G = C10×C50order 500 = 22·53

Abelian group of type [10,50]

direct product, abelian, monomial

Aliases: C10×C50, SmallGroup(500,34)

Series: Derived Chief Lower central Upper central

C1 — C10×C50
C1C5C52C5×C25C5×C50 — 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 [×3], C22, C5, C5 [×5], C10 [×18], C2×C10, C2×C10 [×5], C25 [×5], C52, C50 [×15], C5×C10 [×3], C2×C50 [×5], C102, C5×C25, C5×C50 [×3], C10×C50
Quotients: C1, C2 [×3], C22, C5 [×6], C10 [×18], C2×C10 [×6], C25 [×5], C52, C50 [×15], C5×C10 [×3], C2×C50 [×5], C102, C5×C25, C5×C50 [×3], C10×C50

Smallest permutation representation of C10×C50
Regular action on 500 points
Generators in S500
(1 387 88 456 250 190 422 251 128 309)(2 388 89 457 201 191 423 252 129 310)(3 389 90 458 202 192 424 253 130 311)(4 390 91 459 203 193 425 254 131 312)(5 391 92 460 204 194 426 255 132 313)(6 392 93 461 205 195 427 256 133 314)(7 393 94 462 206 196 428 257 134 315)(8 394 95 463 207 197 429 258 135 316)(9 395 96 464 208 198 430 259 136 317)(10 396 97 465 209 199 431 260 137 318)(11 397 98 466 210 200 432 261 138 319)(12 398 99 467 211 151 433 262 139 320)(13 399 100 468 212 152 434 263 140 321)(14 400 51 469 213 153 435 264 141 322)(15 351 52 470 214 154 436 265 142 323)(16 352 53 471 215 155 437 266 143 324)(17 353 54 472 216 156 438 267 144 325)(18 354 55 473 217 157 439 268 145 326)(19 355 56 474 218 158 440 269 146 327)(20 356 57 475 219 159 441 270 147 328)(21 357 58 476 220 160 442 271 148 329)(22 358 59 477 221 161 443 272 149 330)(23 359 60 478 222 162 444 273 150 331)(24 360 61 479 223 163 445 274 101 332)(25 361 62 480 224 164 446 275 102 333)(26 362 63 481 225 165 447 276 103 334)(27 363 64 482 226 166 448 277 104 335)(28 364 65 483 227 167 449 278 105 336)(29 365 66 484 228 168 450 279 106 337)(30 366 67 485 229 169 401 280 107 338)(31 367 68 486 230 170 402 281 108 339)(32 368 69 487 231 171 403 282 109 340)(33 369 70 488 232 172 404 283 110 341)(34 370 71 489 233 173 405 284 111 342)(35 371 72 490 234 174 406 285 112 343)(36 372 73 491 235 175 407 286 113 344)(37 373 74 492 236 176 408 287 114 345)(38 374 75 493 237 177 409 288 115 346)(39 375 76 494 238 178 410 289 116 347)(40 376 77 495 239 179 411 290 117 348)(41 377 78 496 240 180 412 291 118 349)(42 378 79 497 241 181 413 292 119 350)(43 379 80 498 242 182 414 293 120 301)(44 380 81 499 243 183 415 294 121 302)(45 381 82 500 244 184 416 295 122 303)(46 382 83 451 245 185 417 296 123 304)(47 383 84 452 246 186 418 297 124 305)(48 384 85 453 247 187 419 298 125 306)(49 385 86 454 248 188 420 299 126 307)(50 386 87 455 249 189 421 300 127 308)
(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)| (1,387,88,456,250,190,422,251,128,309)(2,388,89,457,201,191,423,252,129,310)(3,389,90,458,202,192,424,253,130,311)(4,390,91,459,203,193,425,254,131,312)(5,391,92,460,204,194,426,255,132,313)(6,392,93,461,205,195,427,256,133,314)(7,393,94,462,206,196,428,257,134,315)(8,394,95,463,207,197,429,258,135,316)(9,395,96,464,208,198,430,259,136,317)(10,396,97,465,209,199,431,260,137,318)(11,397,98,466,210,200,432,261,138,319)(12,398,99,467,211,151,433,262,139,320)(13,399,100,468,212,152,434,263,140,321)(14,400,51,469,213,153,435,264,141,322)(15,351,52,470,214,154,436,265,142,323)(16,352,53,471,215,155,437,266,143,324)(17,353,54,472,216,156,438,267,144,325)(18,354,55,473,217,157,439,268,145,326)(19,355,56,474,218,158,440,269,146,327)(20,356,57,475,219,159,441,270,147,328)(21,357,58,476,220,160,442,271,148,329)(22,358,59,477,221,161,443,272,149,330)(23,359,60,478,222,162,444,273,150,331)(24,360,61,479,223,163,445,274,101,332)(25,361,62,480,224,164,446,275,102,333)(26,362,63,481,225,165,447,276,103,334)(27,363,64,482,226,166,448,277,104,335)(28,364,65,483,227,167,449,278,105,336)(29,365,66,484,228,168,450,279,106,337)(30,366,67,485,229,169,401,280,107,338)(31,367,68,486,230,170,402,281,108,339)(32,368,69,487,231,171,403,282,109,340)(33,369,70,488,232,172,404,283,110,341)(34,370,71,489,233,173,405,284,111,342)(35,371,72,490,234,174,406,285,112,343)(36,372,73,491,235,175,407,286,113,344)(37,373,74,492,236,176,408,287,114,345)(38,374,75,493,237,177,409,288,115,346)(39,375,76,494,238,178,410,289,116,347)(40,376,77,495,239,179,411,290,117,348)(41,377,78,496,240,180,412,291,118,349)(42,378,79,497,241,181,413,292,119,350)(43,379,80,498,242,182,414,293,120,301)(44,380,81,499,243,183,415,294,121,302)(45,381,82,500,244,184,416,295,122,303)(46,382,83,451,245,185,417,296,123,304)(47,383,84,452,246,186,418,297,124,305)(48,384,85,453,247,187,419,298,125,306)(49,385,86,454,248,188,420,299,126,307)(50,386,87,455,249,189,421,300,127,308), (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( (1,387,88,456,250,190,422,251,128,309)(2,388,89,457,201,191,423,252,129,310)(3,389,90,458,202,192,424,253,130,311)(4,390,91,459,203,193,425,254,131,312)(5,391,92,460,204,194,426,255,132,313)(6,392,93,461,205,195,427,256,133,314)(7,393,94,462,206,196,428,257,134,315)(8,394,95,463,207,197,429,258,135,316)(9,395,96,464,208,198,430,259,136,317)(10,396,97,465,209,199,431,260,137,318)(11,397,98,466,210,200,432,261,138,319)(12,398,99,467,211,151,433,262,139,320)(13,399,100,468,212,152,434,263,140,321)(14,400,51,469,213,153,435,264,141,322)(15,351,52,470,214,154,436,265,142,323)(16,352,53,471,215,155,437,266,143,324)(17,353,54,472,216,156,438,267,144,325)(18,354,55,473,217,157,439,268,145,326)(19,355,56,474,218,158,440,269,146,327)(20,356,57,475,219,159,441,270,147,328)(21,357,58,476,220,160,442,271,148,329)(22,358,59,477,221,161,443,272,149,330)(23,359,60,478,222,162,444,273,150,331)(24,360,61,479,223,163,445,274,101,332)(25,361,62,480,224,164,446,275,102,333)(26,362,63,481,225,165,447,276,103,334)(27,363,64,482,226,166,448,277,104,335)(28,364,65,483,227,167,449,278,105,336)(29,365,66,484,228,168,450,279,106,337)(30,366,67,485,229,169,401,280,107,338)(31,367,68,486,230,170,402,281,108,339)(32,368,69,487,231,171,403,282,109,340)(33,369,70,488,232,172,404,283,110,341)(34,370,71,489,233,173,405,284,111,342)(35,371,72,490,234,174,406,285,112,343)(36,372,73,491,235,175,407,286,113,344)(37,373,74,492,236,176,408,287,114,345)(38,374,75,493,237,177,409,288,115,346)(39,375,76,494,238,178,410,289,116,347)(40,376,77,495,239,179,411,290,117,348)(41,377,78,496,240,180,412,291,118,349)(42,378,79,497,241,181,413,292,119,350)(43,379,80,498,242,182,414,293,120,301)(44,380,81,499,243,183,415,294,121,302)(45,381,82,500,244,184,416,295,122,303)(46,382,83,451,245,185,417,296,123,304)(47,383,84,452,246,186,418,297,124,305)(48,384,85,453,247,187,419,298,125,306)(49,385,86,454,248,188,420,299,126,307)(50,386,87,455,249,189,421,300,127,308), (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([(1,387,88,456,250,190,422,251,128,309),(2,388,89,457,201,191,423,252,129,310),(3,389,90,458,202,192,424,253,130,311),(4,390,91,459,203,193,425,254,131,312),(5,391,92,460,204,194,426,255,132,313),(6,392,93,461,205,195,427,256,133,314),(7,393,94,462,206,196,428,257,134,315),(8,394,95,463,207,197,429,258,135,316),(9,395,96,464,208,198,430,259,136,317),(10,396,97,465,209,199,431,260,137,318),(11,397,98,466,210,200,432,261,138,319),(12,398,99,467,211,151,433,262,139,320),(13,399,100,468,212,152,434,263,140,321),(14,400,51,469,213,153,435,264,141,322),(15,351,52,470,214,154,436,265,142,323),(16,352,53,471,215,155,437,266,143,324),(17,353,54,472,216,156,438,267,144,325),(18,354,55,473,217,157,439,268,145,326),(19,355,56,474,218,158,440,269,146,327),(20,356,57,475,219,159,441,270,147,328),(21,357,58,476,220,160,442,271,148,329),(22,358,59,477,221,161,443,272,149,330),(23,359,60,478,222,162,444,273,150,331),(24,360,61,479,223,163,445,274,101,332),(25,361,62,480,224,164,446,275,102,333),(26,362,63,481,225,165,447,276,103,334),(27,363,64,482,226,166,448,277,104,335),(28,364,65,483,227,167,449,278,105,336),(29,365,66,484,228,168,450,279,106,337),(30,366,67,485,229,169,401,280,107,338),(31,367,68,486,230,170,402,281,108,339),(32,368,69,487,231,171,403,282,109,340),(33,369,70,488,232,172,404,283,110,341),(34,370,71,489,233,173,405,284,111,342),(35,371,72,490,234,174,406,285,112,343),(36,372,73,491,235,175,407,286,113,344),(37,373,74,492,236,176,408,287,114,345),(38,374,75,493,237,177,409,288,115,346),(39,375,76,494,238,178,410,289,116,347),(40,376,77,495,239,179,411,290,117,348),(41,377,78,496,240,180,412,291,118,349),(42,378,79,497,241,181,413,292,119,350),(43,379,80,498,242,182,414,293,120,301),(44,380,81,499,243,183,415,294,121,302),(45,381,82,500,244,184,416,295,122,303),(46,382,83,451,245,185,417,296,123,304),(47,383,84,452,246,186,418,297,124,305),(48,384,85,453,247,187,419,298,125,306),(49,385,86,454,248,188,420,299,126,307),(50,386,87,455,249,189,421,300,127,308)], [(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 2A2B2C5A···5X10A···10BT25A···25CV50A···50KN
order12225···510···1025···2550···50
size11111···11···11···11···1

500 irreducible representations

dim11111111
type++
imageC1C2C5C5C10C10C25C50
kernelC10×C50C5×C50C2×C50C102C50C5×C10C2×C10C10
# reps132046012100300

Matrix representation of C10×C50 in GL3(𝔽101) generated by

600
0950
0084
,
7600
0210
0054
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

׿
×
𝔽