direct product, abelian, monomial, 5-elementary
Aliases: C52×C20, SmallGroup(500,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C52×C20 |
| C1 — C52×C20 |
| C1 — C52×C20 |
Generators and relations for C52×C20
G = < a,b,c | a5=b5=c20=1, ab=ba, ac=ca, bc=cb >
Subgroups: 192, all normal (6 characteristic)
C1, C2, C4, C5, C10, C20, C52, C5×C10, C5×C20, C53, C52×C10, C52×C20
Quotients: C1, C2, C4, C5, C10, C20, C52, C5×C10, C5×C20, C53, C52×C10, C52×C20
►Regular action on 500 points
Generators in S500
(1 383 462 439 296)(2 384 463 440 297)(3 385 464 421 298)(4 386 465 422 299)(5 387 466 423 300)(6 388 467 424 281)(7 389 468 425 282)(8 390 469 426 283)(9 391 470 427 284)(10 392 471 428 285)(11 393 472 429 286)(12 394 473 430 287)(13 395 474 431 288)(14 396 475 432 289)(15 397 476 433 290)(16 398 477 434 291)(17 399 478 435 292)(18 400 479 436 293)(19 381 480 437 294)(20 382 461 438 295)(21 330 452 205 139)(22 331 453 206 140)(23 332 454 207 121)(24 333 455 208 122)(25 334 456 209 123)(26 335 457 210 124)(27 336 458 211 125)(28 337 459 212 126)(29 338 460 213 127)(30 339 441 214 128)(31 340 442 215 129)(32 321 443 216 130)(33 322 444 217 131)(34 323 445 218 132)(35 324 446 219 133)(36 325 447 220 134)(37 326 448 201 135)(38 327 449 202 136)(39 328 450 203 137)(40 329 451 204 138)(41 117 313 500 366)(42 118 314 481 367)(43 119 315 482 368)(44 120 316 483 369)(45 101 317 484 370)(46 102 318 485 371)(47 103 319 486 372)(48 104 320 487 373)(49 105 301 488 374)(50 106 302 489 375)(51 107 303 490 376)(52 108 304 491 377)(53 109 305 492 378)(54 110 306 493 379)(55 111 307 494 380)(56 112 308 495 361)(57 113 309 496 362)(58 114 310 497 363)(59 115 311 498 364)(60 116 312 499 365)(61 186 245 224 163)(62 187 246 225 164)(63 188 247 226 165)(64 189 248 227 166)(65 190 249 228 167)(66 191 250 229 168)(67 192 251 230 169)(68 193 252 231 170)(69 194 253 232 171)(70 195 254 233 172)(71 196 255 234 173)(72 197 256 235 174)(73 198 257 236 175)(74 199 258 237 176)(75 200 259 238 177)(76 181 260 239 178)(77 182 241 240 179)(78 183 242 221 180)(79 184 243 222 161)(80 185 244 223 162)(81 402 349 264 143)(82 403 350 265 144)(83 404 351 266 145)(84 405 352 267 146)(85 406 353 268 147)(86 407 354 269 148)(87 408 355 270 149)(88 409 356 271 150)(89 410 357 272 151)(90 411 358 273 152)(91 412 359 274 153)(92 413 360 275 154)(93 414 341 276 155)(94 415 342 277 156)(95 416 343 278 157)(96 417 344 279 158)(97 418 345 280 159)(98 419 346 261 160)(99 420 347 262 141)(100 401 348 263 142)
(1 246 485 130 406)(2 247 486 131 407)(3 248 487 132 408)(4 249 488 133 409)(5 250 489 134 410)(6 251 490 135 411)(7 252 491 136 412)(8 253 492 137 413)(9 254 493 138 414)(10 255 494 139 415)(11 256 495 140 416)(12 257 496 121 417)(13 258 497 122 418)(14 259 498 123 419)(15 260 499 124 420)(16 241 500 125 401)(17 242 481 126 402)(18 243 482 127 403)(19 244 483 128 404)(20 245 484 129 405)(21 342 392 234 380)(22 343 393 235 361)(23 344 394 236 362)(24 345 395 237 363)(25 346 396 238 364)(26 347 397 239 365)(27 348 398 240 366)(28 349 399 221 367)(29 350 400 222 368)(30 351 381 223 369)(31 352 382 224 370)(32 353 383 225 371)(33 354 384 226 372)(34 355 385 227 373)(35 356 386 228 374)(36 357 387 229 375)(37 358 388 230 376)(38 359 389 231 377)(39 360 390 232 378)(40 341 391 233 379)(41 336 263 477 179)(42 337 264 478 180)(43 338 265 479 161)(44 339 266 480 162)(45 340 267 461 163)(46 321 268 462 164)(47 322 269 463 165)(48 323 270 464 166)(49 324 271 465 167)(50 325 272 466 168)(51 326 273 467 169)(52 327 274 468 170)(53 328 275 469 171)(54 329 276 470 172)(55 330 277 471 173)(56 331 278 472 174)(57 332 279 473 175)(58 333 280 474 176)(59 334 261 475 177)(60 335 262 476 178)(61 101 442 146 438)(62 102 443 147 439)(63 103 444 148 440)(64 104 445 149 421)(65 105 446 150 422)(66 106 447 151 423)(67 107 448 152 424)(68 108 449 153 425)(69 109 450 154 426)(70 110 451 155 427)(71 111 452 156 428)(72 112 453 157 429)(73 113 454 158 430)(74 114 455 159 431)(75 115 456 160 432)(76 116 457 141 433)(77 117 458 142 434)(78 118 459 143 435)(79 119 460 144 436)(80 120 441 145 437)(81 292 183 314 212)(82 293 184 315 213)(83 294 185 316 214)(84 295 186 317 215)(85 296 187 318 216)(86 297 188 319 217)(87 298 189 320 218)(88 299 190 301 219)(89 300 191 302 220)(90 281 192 303 201)(91 282 193 304 202)(92 283 194 305 203)(93 284 195 306 204)(94 285 196 307 205)(95 286 197 308 206)(96 287 198 309 207)(97 288 199 310 208)(98 289 200 311 209)(99 290 181 312 210)(100 291 182 313 211)
(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)| (1,383,462,439,296)(2,384,463,440,297)(3,385,464,421,298)(4,386,465,422,299)(5,387,466,423,300)(6,388,467,424,281)(7,389,468,425,282)(8,390,469,426,283)(9,391,470,427,284)(10,392,471,428,285)(11,393,472,429,286)(12,394,473,430,287)(13,395,474,431,288)(14,396,475,432,289)(15,397,476,433,290)(16,398,477,434,291)(17,399,478,435,292)(18,400,479,436,293)(19,381,480,437,294)(20,382,461,438,295)(21,330,452,205,139)(22,331,453,206,140)(23,332,454,207,121)(24,333,455,208,122)(25,334,456,209,123)(26,335,457,210,124)(27,336,458,211,125)(28,337,459,212,126)(29,338,460,213,127)(30,339,441,214,128)(31,340,442,215,129)(32,321,443,216,130)(33,322,444,217,131)(34,323,445,218,132)(35,324,446,219,133)(36,325,447,220,134)(37,326,448,201,135)(38,327,449,202,136)(39,328,450,203,137)(40,329,451,204,138)(41,117,313,500,366)(42,118,314,481,367)(43,119,315,482,368)(44,120,316,483,369)(45,101,317,484,370)(46,102,318,485,371)(47,103,319,486,372)(48,104,320,487,373)(49,105,301,488,374)(50,106,302,489,375)(51,107,303,490,376)(52,108,304,491,377)(53,109,305,492,378)(54,110,306,493,379)(55,111,307,494,380)(56,112,308,495,361)(57,113,309,496,362)(58,114,310,497,363)(59,115,311,498,364)(60,116,312,499,365)(61,186,245,224,163)(62,187,246,225,164)(63,188,247,226,165)(64,189,248,227,166)(65,190,249,228,167)(66,191,250,229,168)(67,192,251,230,169)(68,193,252,231,170)(69,194,253,232,171)(70,195,254,233,172)(71,196,255,234,173)(72,197,256,235,174)(73,198,257,236,175)(74,199,258,237,176)(75,200,259,238,177)(76,181,260,239,178)(77,182,241,240,179)(78,183,242,221,180)(79,184,243,222,161)(80,185,244,223,162)(81,402,349,264,143)(82,403,350,265,144)(83,404,351,266,145)(84,405,352,267,146)(85,406,353,268,147)(86,407,354,269,148)(87,408,355,270,149)(88,409,356,271,150)(89,410,357,272,151)(90,411,358,273,152)(91,412,359,274,153)(92,413,360,275,154)(93,414,341,276,155)(94,415,342,277,156)(95,416,343,278,157)(96,417,344,279,158)(97,418,345,280,159)(98,419,346,261,160)(99,420,347,262,141)(100,401,348,263,142), (1,246,485,130,406)(2,247,486,131,407)(3,248,487,132,408)(4,249,488,133,409)(5,250,489,134,410)(6,251,490,135,411)(7,252,491,136,412)(8,253,492,137,413)(9,254,493,138,414)(10,255,494,139,415)(11,256,495,140,416)(12,257,496,121,417)(13,258,497,122,418)(14,259,498,123,419)(15,260,499,124,420)(16,241,500,125,401)(17,242,481,126,402)(18,243,482,127,403)(19,244,483,128,404)(20,245,484,129,405)(21,342,392,234,380)(22,343,393,235,361)(23,344,394,236,362)(24,345,395,237,363)(25,346,396,238,364)(26,347,397,239,365)(27,348,398,240,366)(28,349,399,221,367)(29,350,400,222,368)(30,351,381,223,369)(31,352,382,224,370)(32,353,383,225,371)(33,354,384,226,372)(34,355,385,227,373)(35,356,386,228,374)(36,357,387,229,375)(37,358,388,230,376)(38,359,389,231,377)(39,360,390,232,378)(40,341,391,233,379)(41,336,263,477,179)(42,337,264,478,180)(43,338,265,479,161)(44,339,266,480,162)(45,340,267,461,163)(46,321,268,462,164)(47,322,269,463,165)(48,323,270,464,166)(49,324,271,465,167)(50,325,272,466,168)(51,326,273,467,169)(52,327,274,468,170)(53,328,275,469,171)(54,329,276,470,172)(55,330,277,471,173)(56,331,278,472,174)(57,332,279,473,175)(58,333,280,474,176)(59,334,261,475,177)(60,335,262,476,178)(61,101,442,146,438)(62,102,443,147,439)(63,103,444,148,440)(64,104,445,149,421)(65,105,446,150,422)(66,106,447,151,423)(67,107,448,152,424)(68,108,449,153,425)(69,109,450,154,426)(70,110,451,155,427)(71,111,452,156,428)(72,112,453,157,429)(73,113,454,158,430)(74,114,455,159,431)(75,115,456,160,432)(76,116,457,141,433)(77,117,458,142,434)(78,118,459,143,435)(79,119,460,144,436)(80,120,441,145,437)(81,292,183,314,212)(82,293,184,315,213)(83,294,185,316,214)(84,295,186,317,215)(85,296,187,318,216)(86,297,188,319,217)(87,298,189,320,218)(88,299,190,301,219)(89,300,191,302,220)(90,281,192,303,201)(91,282,193,304,202)(92,283,194,305,203)(93,284,195,306,204)(94,285,196,307,205)(95,286,197,308,206)(96,287,198,309,207)(97,288,199,310,208)(98,289,200,311,209)(99,290,181,312,210)(100,291,182,313,211), (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( (1,383,462,439,296)(2,384,463,440,297)(3,385,464,421,298)(4,386,465,422,299)(5,387,466,423,300)(6,388,467,424,281)(7,389,468,425,282)(8,390,469,426,283)(9,391,470,427,284)(10,392,471,428,285)(11,393,472,429,286)(12,394,473,430,287)(13,395,474,431,288)(14,396,475,432,289)(15,397,476,433,290)(16,398,477,434,291)(17,399,478,435,292)(18,400,479,436,293)(19,381,480,437,294)(20,382,461,438,295)(21,330,452,205,139)(22,331,453,206,140)(23,332,454,207,121)(24,333,455,208,122)(25,334,456,209,123)(26,335,457,210,124)(27,336,458,211,125)(28,337,459,212,126)(29,338,460,213,127)(30,339,441,214,128)(31,340,442,215,129)(32,321,443,216,130)(33,322,444,217,131)(34,323,445,218,132)(35,324,446,219,133)(36,325,447,220,134)(37,326,448,201,135)(38,327,449,202,136)(39,328,450,203,137)(40,329,451,204,138)(41,117,313,500,366)(42,118,314,481,367)(43,119,315,482,368)(44,120,316,483,369)(45,101,317,484,370)(46,102,318,485,371)(47,103,319,486,372)(48,104,320,487,373)(49,105,301,488,374)(50,106,302,489,375)(51,107,303,490,376)(52,108,304,491,377)(53,109,305,492,378)(54,110,306,493,379)(55,111,307,494,380)(56,112,308,495,361)(57,113,309,496,362)(58,114,310,497,363)(59,115,311,498,364)(60,116,312,499,365)(61,186,245,224,163)(62,187,246,225,164)(63,188,247,226,165)(64,189,248,227,166)(65,190,249,228,167)(66,191,250,229,168)(67,192,251,230,169)(68,193,252,231,170)(69,194,253,232,171)(70,195,254,233,172)(71,196,255,234,173)(72,197,256,235,174)(73,198,257,236,175)(74,199,258,237,176)(75,200,259,238,177)(76,181,260,239,178)(77,182,241,240,179)(78,183,242,221,180)(79,184,243,222,161)(80,185,244,223,162)(81,402,349,264,143)(82,403,350,265,144)(83,404,351,266,145)(84,405,352,267,146)(85,406,353,268,147)(86,407,354,269,148)(87,408,355,270,149)(88,409,356,271,150)(89,410,357,272,151)(90,411,358,273,152)(91,412,359,274,153)(92,413,360,275,154)(93,414,341,276,155)(94,415,342,277,156)(95,416,343,278,157)(96,417,344,279,158)(97,418,345,280,159)(98,419,346,261,160)(99,420,347,262,141)(100,401,348,263,142), (1,246,485,130,406)(2,247,486,131,407)(3,248,487,132,408)(4,249,488,133,409)(5,250,489,134,410)(6,251,490,135,411)(7,252,491,136,412)(8,253,492,137,413)(9,254,493,138,414)(10,255,494,139,415)(11,256,495,140,416)(12,257,496,121,417)(13,258,497,122,418)(14,259,498,123,419)(15,260,499,124,420)(16,241,500,125,401)(17,242,481,126,402)(18,243,482,127,403)(19,244,483,128,404)(20,245,484,129,405)(21,342,392,234,380)(22,343,393,235,361)(23,344,394,236,362)(24,345,395,237,363)(25,346,396,238,364)(26,347,397,239,365)(27,348,398,240,366)(28,349,399,221,367)(29,350,400,222,368)(30,351,381,223,369)(31,352,382,224,370)(32,353,383,225,371)(33,354,384,226,372)(34,355,385,227,373)(35,356,386,228,374)(36,357,387,229,375)(37,358,388,230,376)(38,359,389,231,377)(39,360,390,232,378)(40,341,391,233,379)(41,336,263,477,179)(42,337,264,478,180)(43,338,265,479,161)(44,339,266,480,162)(45,340,267,461,163)(46,321,268,462,164)(47,322,269,463,165)(48,323,270,464,166)(49,324,271,465,167)(50,325,272,466,168)(51,326,273,467,169)(52,327,274,468,170)(53,328,275,469,171)(54,329,276,470,172)(55,330,277,471,173)(56,331,278,472,174)(57,332,279,473,175)(58,333,280,474,176)(59,334,261,475,177)(60,335,262,476,178)(61,101,442,146,438)(62,102,443,147,439)(63,103,444,148,440)(64,104,445,149,421)(65,105,446,150,422)(66,106,447,151,423)(67,107,448,152,424)(68,108,449,153,425)(69,109,450,154,426)(70,110,451,155,427)(71,111,452,156,428)(72,112,453,157,429)(73,113,454,158,430)(74,114,455,159,431)(75,115,456,160,432)(76,116,457,141,433)(77,117,458,142,434)(78,118,459,143,435)(79,119,460,144,436)(80,120,441,145,437)(81,292,183,314,212)(82,293,184,315,213)(83,294,185,316,214)(84,295,186,317,215)(85,296,187,318,216)(86,297,188,319,217)(87,298,189,320,218)(88,299,190,301,219)(89,300,191,302,220)(90,281,192,303,201)(91,282,193,304,202)(92,283,194,305,203)(93,284,195,306,204)(94,285,196,307,205)(95,286,197,308,206)(96,287,198,309,207)(97,288,199,310,208)(98,289,200,311,209)(99,290,181,312,210)(100,291,182,313,211), (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([[(1,383,462,439,296),(2,384,463,440,297),(3,385,464,421,298),(4,386,465,422,299),(5,387,466,423,300),(6,388,467,424,281),(7,389,468,425,282),(8,390,469,426,283),(9,391,470,427,284),(10,392,471,428,285),(11,393,472,429,286),(12,394,473,430,287),(13,395,474,431,288),(14,396,475,432,289),(15,397,476,433,290),(16,398,477,434,291),(17,399,478,435,292),(18,400,479,436,293),(19,381,480,437,294),(20,382,461,438,295),(21,330,452,205,139),(22,331,453,206,140),(23,332,454,207,121),(24,333,455,208,122),(25,334,456,209,123),(26,335,457,210,124),(27,336,458,211,125),(28,337,459,212,126),(29,338,460,213,127),(30,339,441,214,128),(31,340,442,215,129),(32,321,443,216,130),(33,322,444,217,131),(34,323,445,218,132),(35,324,446,219,133),(36,325,447,220,134),(37,326,448,201,135),(38,327,449,202,136),(39,328,450,203,137),(40,329,451,204,138),(41,117,313,500,366),(42,118,314,481,367),(43,119,315,482,368),(44,120,316,483,369),(45,101,317,484,370),(46,102,318,485,371),(47,103,319,486,372),(48,104,320,487,373),(49,105,301,488,374),(50,106,302,489,375),(51,107,303,490,376),(52,108,304,491,377),(53,109,305,492,378),(54,110,306,493,379),(55,111,307,494,380),(56,112,308,495,361),(57,113,309,496,362),(58,114,310,497,363),(59,115,311,498,364),(60,116,312,499,365),(61,186,245,224,163),(62,187,246,225,164),(63,188,247,226,165),(64,189,248,227,166),(65,190,249,228,167),(66,191,250,229,168),(67,192,251,230,169),(68,193,252,231,170),(69,194,253,232,171),(70,195,254,233,172),(71,196,255,234,173),(72,197,256,235,174),(73,198,257,236,175),(74,199,258,237,176),(75,200,259,238,177),(76,181,260,239,178),(77,182,241,240,179),(78,183,242,221,180),(79,184,243,222,161),(80,185,244,223,162),(81,402,349,264,143),(82,403,350,265,144),(83,404,351,266,145),(84,405,352,267,146),(85,406,353,268,147),(86,407,354,269,148),(87,408,355,270,149),(88,409,356,271,150),(89,410,357,272,151),(90,411,358,273,152),(91,412,359,274,153),(92,413,360,275,154),(93,414,341,276,155),(94,415,342,277,156),(95,416,343,278,157),(96,417,344,279,158),(97,418,345,280,159),(98,419,346,261,160),(99,420,347,262,141),(100,401,348,263,142)], [(1,246,485,130,406),(2,247,486,131,407),(3,248,487,132,408),(4,249,488,133,409),(5,250,489,134,410),(6,251,490,135,411),(7,252,491,136,412),(8,253,492,137,413),(9,254,493,138,414),(10,255,494,139,415),(11,256,495,140,416),(12,257,496,121,417),(13,258,497,122,418),(14,259,498,123,419),(15,260,499,124,420),(16,241,500,125,401),(17,242,481,126,402),(18,243,482,127,403),(19,244,483,128,404),(20,245,484,129,405),(21,342,392,234,380),(22,343,393,235,361),(23,344,394,236,362),(24,345,395,237,363),(25,346,396,238,364),(26,347,397,239,365),(27,348,398,240,366),(28,349,399,221,367),(29,350,400,222,368),(30,351,381,223,369),(31,352,382,224,370),(32,353,383,225,371),(33,354,384,226,372),(34,355,385,227,373),(35,356,386,228,374),(36,357,387,229,375),(37,358,388,230,376),(38,359,389,231,377),(39,360,390,232,378),(40,341,391,233,379),(41,336,263,477,179),(42,337,264,478,180),(43,338,265,479,161),(44,339,266,480,162),(45,340,267,461,163),(46,321,268,462,164),(47,322,269,463,165),(48,323,270,464,166),(49,324,271,465,167),(50,325,272,466,168),(51,326,273,467,169),(52,327,274,468,170),(53,328,275,469,171),(54,329,276,470,172),(55,330,277,471,173),(56,331,278,472,174),(57,332,279,473,175),(58,333,280,474,176),(59,334,261,475,177),(60,335,262,476,178),(61,101,442,146,438),(62,102,443,147,439),(63,103,444,148,440),(64,104,445,149,421),(65,105,446,150,422),(66,106,447,151,423),(67,107,448,152,424),(68,108,449,153,425),(69,109,450,154,426),(70,110,451,155,427),(71,111,452,156,428),(72,112,453,157,429),(73,113,454,158,430),(74,114,455,159,431),(75,115,456,160,432),(76,116,457,141,433),(77,117,458,142,434),(78,118,459,143,435),(79,119,460,144,436),(80,120,441,145,437),(81,292,183,314,212),(82,293,184,315,213),(83,294,185,316,214),(84,295,186,317,215),(85,296,187,318,216),(86,297,188,319,217),(87,298,189,320,218),(88,299,190,301,219),(89,300,191,302,220),(90,281,192,303,201),(91,282,193,304,202),(92,283,194,305,203),(93,284,195,306,204),(94,285,196,307,205),(95,286,197,308,206),(96,287,198,309,207),(97,288,199,310,208),(98,289,200,311,209),(99,290,181,312,210),(100,291,182,313,211)], [(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 | ··· | 5DT | 10A | ··· | 10DT | 20A | ··· | 20IN |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
500 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C5 | C10 | C20 |
| kernel | C52×C20 | C52×C10 | C53 | C5×C20 | C5×C10 | C52 |
| # reps | 1 | 1 | 2 | 124 | 124 | 248 |
Matrix representation of C52×C20 ►in GL3(𝔽41) generated by
| 18 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 1 |
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 1 |
| 5 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(41))| [18,0,0,0,16,0,0,0,1],[18,0,0,0,18,0,0,0,1],[5,0,0,0,36,0,0,0,16] >;
C52×C20 in GAP, Magma, Sage, TeX
C_5^2\times C_{20} % in TeX
G:=Group("C5^2xC20"); // GroupNames label
G:=SmallGroup(500,40);
// by ID
G=gap.SmallGroup(500,40);
# by ID
G:=PCGroup([5,-2,-5,-5,-5,-2,1250]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^20=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations