direct product, abelian, monomial, 5-elementary
Aliases: C5×C100, SmallGroup(500,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C100 |
| C1 — C5×C100 |
| C1 — C5×C100 |
Generators and relations for C5×C100
G = < a,b | a5=b100=1, ab=ba >
Smallest permutation representation of C5×C100
►Regular action on 500 points
Generators in S500
(1 189 339 489 219)(2 190 340 490 220)(3 191 341 491 221)(4 192 342 492 222)(5 193 343 493 223)(6 194 344 494 224)(7 195 345 495 225)(8 196 346 496 226)(9 197 347 497 227)(10 198 348 498 228)(11 199 349 499 229)(12 200 350 500 230)(13 101 351 401 231)(14 102 352 402 232)(15 103 353 403 233)(16 104 354 404 234)(17 105 355 405 235)(18 106 356 406 236)(19 107 357 407 237)(20 108 358 408 238)(21 109 359 409 239)(22 110 360 410 240)(23 111 361 411 241)(24 112 362 412 242)(25 113 363 413 243)(26 114 364 414 244)(27 115 365 415 245)(28 116 366 416 246)(29 117 367 417 247)(30 118 368 418 248)(31 119 369 419 249)(32 120 370 420 250)(33 121 371 421 251)(34 122 372 422 252)(35 123 373 423 253)(36 124 374 424 254)(37 125 375 425 255)(38 126 376 426 256)(39 127 377 427 257)(40 128 378 428 258)(41 129 379 429 259)(42 130 380 430 260)(43 131 381 431 261)(44 132 382 432 262)(45 133 383 433 263)(46 134 384 434 264)(47 135 385 435 265)(48 136 386 436 266)(49 137 387 437 267)(50 138 388 438 268)(51 139 389 439 269)(52 140 390 440 270)(53 141 391 441 271)(54 142 392 442 272)(55 143 393 443 273)(56 144 394 444 274)(57 145 395 445 275)(58 146 396 446 276)(59 147 397 447 277)(60 148 398 448 278)(61 149 399 449 279)(62 150 400 450 280)(63 151 301 451 281)(64 152 302 452 282)(65 153 303 453 283)(66 154 304 454 284)(67 155 305 455 285)(68 156 306 456 286)(69 157 307 457 287)(70 158 308 458 288)(71 159 309 459 289)(72 160 310 460 290)(73 161 311 461 291)(74 162 312 462 292)(75 163 313 463 293)(76 164 314 464 294)(77 165 315 465 295)(78 166 316 466 296)(79 167 317 467 297)(80 168 318 468 298)(81 169 319 469 299)(82 170 320 470 300)(83 171 321 471 201)(84 172 322 472 202)(85 173 323 473 203)(86 174 324 474 204)(87 175 325 475 205)(88 176 326 476 206)(89 177 327 477 207)(90 178 328 478 208)(91 179 329 479 209)(92 180 330 480 210)(93 181 331 481 211)(94 182 332 482 212)(95 183 333 483 213)(96 184 334 484 214)(97 185 335 485 215)(98 186 336 486 216)(99 187 337 487 217)(100 188 338 488 218)
(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)| (1,189,339,489,219)(2,190,340,490,220)(3,191,341,491,221)(4,192,342,492,222)(5,193,343,493,223)(6,194,344,494,224)(7,195,345,495,225)(8,196,346,496,226)(9,197,347,497,227)(10,198,348,498,228)(11,199,349,499,229)(12,200,350,500,230)(13,101,351,401,231)(14,102,352,402,232)(15,103,353,403,233)(16,104,354,404,234)(17,105,355,405,235)(18,106,356,406,236)(19,107,357,407,237)(20,108,358,408,238)(21,109,359,409,239)(22,110,360,410,240)(23,111,361,411,241)(24,112,362,412,242)(25,113,363,413,243)(26,114,364,414,244)(27,115,365,415,245)(28,116,366,416,246)(29,117,367,417,247)(30,118,368,418,248)(31,119,369,419,249)(32,120,370,420,250)(33,121,371,421,251)(34,122,372,422,252)(35,123,373,423,253)(36,124,374,424,254)(37,125,375,425,255)(38,126,376,426,256)(39,127,377,427,257)(40,128,378,428,258)(41,129,379,429,259)(42,130,380,430,260)(43,131,381,431,261)(44,132,382,432,262)(45,133,383,433,263)(46,134,384,434,264)(47,135,385,435,265)(48,136,386,436,266)(49,137,387,437,267)(50,138,388,438,268)(51,139,389,439,269)(52,140,390,440,270)(53,141,391,441,271)(54,142,392,442,272)(55,143,393,443,273)(56,144,394,444,274)(57,145,395,445,275)(58,146,396,446,276)(59,147,397,447,277)(60,148,398,448,278)(61,149,399,449,279)(62,150,400,450,280)(63,151,301,451,281)(64,152,302,452,282)(65,153,303,453,283)(66,154,304,454,284)(67,155,305,455,285)(68,156,306,456,286)(69,157,307,457,287)(70,158,308,458,288)(71,159,309,459,289)(72,160,310,460,290)(73,161,311,461,291)(74,162,312,462,292)(75,163,313,463,293)(76,164,314,464,294)(77,165,315,465,295)(78,166,316,466,296)(79,167,317,467,297)(80,168,318,468,298)(81,169,319,469,299)(82,170,320,470,300)(83,171,321,471,201)(84,172,322,472,202)(85,173,323,473,203)(86,174,324,474,204)(87,175,325,475,205)(88,176,326,476,206)(89,177,327,477,207)(90,178,328,478,208)(91,179,329,479,209)(92,180,330,480,210)(93,181,331,481,211)(94,182,332,482,212)(95,183,333,483,213)(96,184,334,484,214)(97,185,335,485,215)(98,186,336,486,216)(99,187,337,487,217)(100,188,338,488,218), (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( (1,189,339,489,219)(2,190,340,490,220)(3,191,341,491,221)(4,192,342,492,222)(5,193,343,493,223)(6,194,344,494,224)(7,195,345,495,225)(8,196,346,496,226)(9,197,347,497,227)(10,198,348,498,228)(11,199,349,499,229)(12,200,350,500,230)(13,101,351,401,231)(14,102,352,402,232)(15,103,353,403,233)(16,104,354,404,234)(17,105,355,405,235)(18,106,356,406,236)(19,107,357,407,237)(20,108,358,408,238)(21,109,359,409,239)(22,110,360,410,240)(23,111,361,411,241)(24,112,362,412,242)(25,113,363,413,243)(26,114,364,414,244)(27,115,365,415,245)(28,116,366,416,246)(29,117,367,417,247)(30,118,368,418,248)(31,119,369,419,249)(32,120,370,420,250)(33,121,371,421,251)(34,122,372,422,252)(35,123,373,423,253)(36,124,374,424,254)(37,125,375,425,255)(38,126,376,426,256)(39,127,377,427,257)(40,128,378,428,258)(41,129,379,429,259)(42,130,380,430,260)(43,131,381,431,261)(44,132,382,432,262)(45,133,383,433,263)(46,134,384,434,264)(47,135,385,435,265)(48,136,386,436,266)(49,137,387,437,267)(50,138,388,438,268)(51,139,389,439,269)(52,140,390,440,270)(53,141,391,441,271)(54,142,392,442,272)(55,143,393,443,273)(56,144,394,444,274)(57,145,395,445,275)(58,146,396,446,276)(59,147,397,447,277)(60,148,398,448,278)(61,149,399,449,279)(62,150,400,450,280)(63,151,301,451,281)(64,152,302,452,282)(65,153,303,453,283)(66,154,304,454,284)(67,155,305,455,285)(68,156,306,456,286)(69,157,307,457,287)(70,158,308,458,288)(71,159,309,459,289)(72,160,310,460,290)(73,161,311,461,291)(74,162,312,462,292)(75,163,313,463,293)(76,164,314,464,294)(77,165,315,465,295)(78,166,316,466,296)(79,167,317,467,297)(80,168,318,468,298)(81,169,319,469,299)(82,170,320,470,300)(83,171,321,471,201)(84,172,322,472,202)(85,173,323,473,203)(86,174,324,474,204)(87,175,325,475,205)(88,176,326,476,206)(89,177,327,477,207)(90,178,328,478,208)(91,179,329,479,209)(92,180,330,480,210)(93,181,331,481,211)(94,182,332,482,212)(95,183,333,483,213)(96,184,334,484,214)(97,185,335,485,215)(98,186,336,486,216)(99,187,337,487,217)(100,188,338,488,218), (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([[(1,189,339,489,219),(2,190,340,490,220),(3,191,341,491,221),(4,192,342,492,222),(5,193,343,493,223),(6,194,344,494,224),(7,195,345,495,225),(8,196,346,496,226),(9,197,347,497,227),(10,198,348,498,228),(11,199,349,499,229),(12,200,350,500,230),(13,101,351,401,231),(14,102,352,402,232),(15,103,353,403,233),(16,104,354,404,234),(17,105,355,405,235),(18,106,356,406,236),(19,107,357,407,237),(20,108,358,408,238),(21,109,359,409,239),(22,110,360,410,240),(23,111,361,411,241),(24,112,362,412,242),(25,113,363,413,243),(26,114,364,414,244),(27,115,365,415,245),(28,116,366,416,246),(29,117,367,417,247),(30,118,368,418,248),(31,119,369,419,249),(32,120,370,420,250),(33,121,371,421,251),(34,122,372,422,252),(35,123,373,423,253),(36,124,374,424,254),(37,125,375,425,255),(38,126,376,426,256),(39,127,377,427,257),(40,128,378,428,258),(41,129,379,429,259),(42,130,380,430,260),(43,131,381,431,261),(44,132,382,432,262),(45,133,383,433,263),(46,134,384,434,264),(47,135,385,435,265),(48,136,386,436,266),(49,137,387,437,267),(50,138,388,438,268),(51,139,389,439,269),(52,140,390,440,270),(53,141,391,441,271),(54,142,392,442,272),(55,143,393,443,273),(56,144,394,444,274),(57,145,395,445,275),(58,146,396,446,276),(59,147,397,447,277),(60,148,398,448,278),(61,149,399,449,279),(62,150,400,450,280),(63,151,301,451,281),(64,152,302,452,282),(65,153,303,453,283),(66,154,304,454,284),(67,155,305,455,285),(68,156,306,456,286),(69,157,307,457,287),(70,158,308,458,288),(71,159,309,459,289),(72,160,310,460,290),(73,161,311,461,291),(74,162,312,462,292),(75,163,313,463,293),(76,164,314,464,294),(77,165,315,465,295),(78,166,316,466,296),(79,167,317,467,297),(80,168,318,468,298),(81,169,319,469,299),(82,170,320,470,300),(83,171,321,471,201),(84,172,322,472,202),(85,173,323,473,203),(86,174,324,474,204),(87,175,325,475,205),(88,176,326,476,206),(89,177,327,477,207),(90,178,328,478,208),(91,179,329,479,209),(92,180,330,480,210),(93,181,331,481,211),(94,182,332,482,212),(95,183,333,483,213),(96,184,334,484,214),(97,185,335,485,215),(98,186,336,486,216),(99,187,337,487,217),(100,188,338,488,218)], [(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 | 2 | 4A | 4B | 5A | ··· | 5X | 10A | ··· | 10X | 20A | ··· | 20AV | 25A | ··· | 25CV | 50A | ··· | 50CV | 100A | ··· | 100GR |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
500 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C4 | C5 | C5 | C10 | C10 | C20 | C20 | C25 | C50 | C100 |
| kernel | C5×C100 | C5×C50 | C5×C25 | C100 | C5×C20 | C50 | C5×C10 | C25 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 20 | 4 | 20 | 4 | 40 | 8 | 100 | 100 | 200 |
Matrix representation of C5×C100 ►in GL2(𝔽101) generated by
| 95 | 0 |
| 0 | 95 |
| 14 | 0 |
| 0 | 8 |
G:=sub<GL(2,GF(101))| [95,0,0,95],[14,0,0,8] >;
C5×C100 in GAP, Magma, Sage, TeX
C_5\times C_{100} % in TeX
G:=Group("C5xC100"); // GroupNames label
G:=SmallGroup(500,12);
// by ID
G=gap.SmallGroup(500,12);
# by ID
G:=PCGroup([5,-2,-5,-5,-2,-5,250,506]);
// Polycyclic
G:=Group<a,b|a^5=b^100=1,a*b=b*a>;
// generators/relations
Export