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G = C52×C20order 500 = 22·53

Abelian group of type [5,5,20]

direct product, abelian, monomial, 5-elementary

Aliases: C52×C20, SmallGroup(500,40)

Series: Derived Chief Lower central Upper central

C1 — C52×C20
C1C2C10C5×C10C52×C10 — 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 [×31], C10 [×31], C20 [×31], C52 [×31], C5×C10 [×31], C5×C20 [×31], C53, C52×C10, C52×C20
Quotients: C1, C2, C4, C5 [×31], C10 [×31], C20 [×31], C52 [×31], C5×C10 [×31], C5×C20 [×31], C53, C52×C10, C52×C20

Smallest permutation representation of C52×C20
Regular action on 500 points
Generators in S500
(1 492 275 357 223)(2 493 276 358 224)(3 494 277 359 225)(4 495 278 360 226)(5 496 279 341 227)(6 497 280 342 228)(7 498 261 343 229)(8 499 262 344 230)(9 500 263 345 231)(10 481 264 346 232)(11 482 265 347 233)(12 483 266 348 234)(13 484 267 349 235)(14 485 268 350 236)(15 486 269 351 237)(16 487 270 352 238)(17 488 271 353 239)(18 489 272 354 240)(19 490 273 355 221)(20 491 274 356 222)(21 462 427 123 201)(22 463 428 124 202)(23 464 429 125 203)(24 465 430 126 204)(25 466 431 127 205)(26 467 432 128 206)(27 468 433 129 207)(28 469 434 130 208)(29 470 435 131 209)(30 471 436 132 210)(31 472 437 133 211)(32 473 438 134 212)(33 474 439 135 213)(34 475 440 136 214)(35 476 421 137 215)(36 477 422 138 216)(37 478 423 139 217)(38 479 424 140 218)(39 480 425 121 219)(40 461 426 122 220)(41 64 333 145 363)(42 65 334 146 364)(43 66 335 147 365)(44 67 336 148 366)(45 68 337 149 367)(46 69 338 150 368)(47 70 339 151 369)(48 71 340 152 370)(49 72 321 153 371)(50 73 322 154 372)(51 74 323 155 373)(52 75 324 156 374)(53 76 325 157 375)(54 77 326 158 376)(55 78 327 159 377)(56 79 328 160 378)(57 80 329 141 379)(58 61 330 142 380)(59 62 331 143 361)(60 63 332 144 362)(81 281 404 175 387)(82 282 405 176 388)(83 283 406 177 389)(84 284 407 178 390)(85 285 408 179 391)(86 286 409 180 392)(87 287 410 161 393)(88 288 411 162 394)(89 289 412 163 395)(90 290 413 164 396)(91 291 414 165 397)(92 292 415 166 398)(93 293 416 167 399)(94 294 417 168 400)(95 295 418 169 381)(96 296 419 170 382)(97 297 420 171 383)(98 298 401 172 384)(99 299 402 173 385)(100 300 403 174 386)(101 452 249 182 305)(102 453 250 183 306)(103 454 251 184 307)(104 455 252 185 308)(105 456 253 186 309)(106 457 254 187 310)(107 458 255 188 311)(108 459 256 189 312)(109 460 257 190 313)(110 441 258 191 314)(111 442 259 192 315)(112 443 260 193 316)(113 444 241 194 317)(114 445 242 195 318)(115 446 243 196 319)(116 447 244 197 320)(117 448 245 198 301)(118 449 246 199 302)(119 450 247 200 303)(120 451 248 181 304)
(1 165 440 80 104)(2 166 421 61 105)(3 167 422 62 106)(4 168 423 63 107)(5 169 424 64 108)(6 170 425 65 109)(7 171 426 66 110)(8 172 427 67 111)(9 173 428 68 112)(10 174 429 69 113)(11 175 430 70 114)(12 176 431 71 115)(13 177 432 72 116)(14 178 433 73 117)(15 179 434 74 118)(16 180 435 75 119)(17 161 436 76 120)(18 162 437 77 101)(19 163 438 78 102)(20 164 439 79 103)(21 366 192 344 298)(22 367 193 345 299)(23 368 194 346 300)(24 369 195 347 281)(25 370 196 348 282)(26 371 197 349 283)(27 372 198 350 284)(28 373 199 351 285)(29 374 200 352 286)(30 375 181 353 287)(31 376 182 354 288)(32 377 183 355 289)(33 378 184 356 290)(34 379 185 357 291)(35 380 186 358 292)(36 361 187 359 293)(37 362 188 360 294)(38 363 189 341 295)(39 364 190 342 296)(40 365 191 343 297)(41 312 227 418 479)(42 313 228 419 480)(43 314 229 420 461)(44 315 230 401 462)(45 316 231 402 463)(46 317 232 403 464)(47 318 233 404 465)(48 319 234 405 466)(49 320 235 406 467)(50 301 236 407 468)(51 302 237 408 469)(52 303 238 409 470)(53 304 239 410 471)(54 305 240 411 472)(55 306 221 412 473)(56 307 222 413 474)(57 308 223 414 475)(58 309 224 415 476)(59 310 225 416 477)(60 311 226 417 478)(81 204 151 242 265)(82 205 152 243 266)(83 206 153 244 267)(84 207 154 245 268)(85 208 155 246 269)(86 209 156 247 270)(87 210 157 248 271)(88 211 158 249 272)(89 212 159 250 273)(90 213 160 251 274)(91 214 141 252 275)(92 215 142 253 276)(93 216 143 254 277)(94 217 144 255 278)(95 218 145 256 279)(96 219 146 257 280)(97 220 147 258 261)(98 201 148 259 262)(99 202 149 260 263)(100 203 150 241 264)(121 334 460 497 382)(122 335 441 498 383)(123 336 442 499 384)(124 337 443 500 385)(125 338 444 481 386)(126 339 445 482 387)(127 340 446 483 388)(128 321 447 484 389)(129 322 448 485 390)(130 323 449 486 391)(131 324 450 487 392)(132 325 451 488 393)(133 326 452 489 394)(134 327 453 490 395)(135 328 454 491 396)(136 329 455 492 397)(137 330 456 493 398)(138 331 457 494 399)(139 332 458 495 400)(140 333 459 496 381)
(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)| (1,492,275,357,223)(2,493,276,358,224)(3,494,277,359,225)(4,495,278,360,226)(5,496,279,341,227)(6,497,280,342,228)(7,498,261,343,229)(8,499,262,344,230)(9,500,263,345,231)(10,481,264,346,232)(11,482,265,347,233)(12,483,266,348,234)(13,484,267,349,235)(14,485,268,350,236)(15,486,269,351,237)(16,487,270,352,238)(17,488,271,353,239)(18,489,272,354,240)(19,490,273,355,221)(20,491,274,356,222)(21,462,427,123,201)(22,463,428,124,202)(23,464,429,125,203)(24,465,430,126,204)(25,466,431,127,205)(26,467,432,128,206)(27,468,433,129,207)(28,469,434,130,208)(29,470,435,131,209)(30,471,436,132,210)(31,472,437,133,211)(32,473,438,134,212)(33,474,439,135,213)(34,475,440,136,214)(35,476,421,137,215)(36,477,422,138,216)(37,478,423,139,217)(38,479,424,140,218)(39,480,425,121,219)(40,461,426,122,220)(41,64,333,145,363)(42,65,334,146,364)(43,66,335,147,365)(44,67,336,148,366)(45,68,337,149,367)(46,69,338,150,368)(47,70,339,151,369)(48,71,340,152,370)(49,72,321,153,371)(50,73,322,154,372)(51,74,323,155,373)(52,75,324,156,374)(53,76,325,157,375)(54,77,326,158,376)(55,78,327,159,377)(56,79,328,160,378)(57,80,329,141,379)(58,61,330,142,380)(59,62,331,143,361)(60,63,332,144,362)(81,281,404,175,387)(82,282,405,176,388)(83,283,406,177,389)(84,284,407,178,390)(85,285,408,179,391)(86,286,409,180,392)(87,287,410,161,393)(88,288,411,162,394)(89,289,412,163,395)(90,290,413,164,396)(91,291,414,165,397)(92,292,415,166,398)(93,293,416,167,399)(94,294,417,168,400)(95,295,418,169,381)(96,296,419,170,382)(97,297,420,171,383)(98,298,401,172,384)(99,299,402,173,385)(100,300,403,174,386)(101,452,249,182,305)(102,453,250,183,306)(103,454,251,184,307)(104,455,252,185,308)(105,456,253,186,309)(106,457,254,187,310)(107,458,255,188,311)(108,459,256,189,312)(109,460,257,190,313)(110,441,258,191,314)(111,442,259,192,315)(112,443,260,193,316)(113,444,241,194,317)(114,445,242,195,318)(115,446,243,196,319)(116,447,244,197,320)(117,448,245,198,301)(118,449,246,199,302)(119,450,247,200,303)(120,451,248,181,304), (1,165,440,80,104)(2,166,421,61,105)(3,167,422,62,106)(4,168,423,63,107)(5,169,424,64,108)(6,170,425,65,109)(7,171,426,66,110)(8,172,427,67,111)(9,173,428,68,112)(10,174,429,69,113)(11,175,430,70,114)(12,176,431,71,115)(13,177,432,72,116)(14,178,433,73,117)(15,179,434,74,118)(16,180,435,75,119)(17,161,436,76,120)(18,162,437,77,101)(19,163,438,78,102)(20,164,439,79,103)(21,366,192,344,298)(22,367,193,345,299)(23,368,194,346,300)(24,369,195,347,281)(25,370,196,348,282)(26,371,197,349,283)(27,372,198,350,284)(28,373,199,351,285)(29,374,200,352,286)(30,375,181,353,287)(31,376,182,354,288)(32,377,183,355,289)(33,378,184,356,290)(34,379,185,357,291)(35,380,186,358,292)(36,361,187,359,293)(37,362,188,360,294)(38,363,189,341,295)(39,364,190,342,296)(40,365,191,343,297)(41,312,227,418,479)(42,313,228,419,480)(43,314,229,420,461)(44,315,230,401,462)(45,316,231,402,463)(46,317,232,403,464)(47,318,233,404,465)(48,319,234,405,466)(49,320,235,406,467)(50,301,236,407,468)(51,302,237,408,469)(52,303,238,409,470)(53,304,239,410,471)(54,305,240,411,472)(55,306,221,412,473)(56,307,222,413,474)(57,308,223,414,475)(58,309,224,415,476)(59,310,225,416,477)(60,311,226,417,478)(81,204,151,242,265)(82,205,152,243,266)(83,206,153,244,267)(84,207,154,245,268)(85,208,155,246,269)(86,209,156,247,270)(87,210,157,248,271)(88,211,158,249,272)(89,212,159,250,273)(90,213,160,251,274)(91,214,141,252,275)(92,215,142,253,276)(93,216,143,254,277)(94,217,144,255,278)(95,218,145,256,279)(96,219,146,257,280)(97,220,147,258,261)(98,201,148,259,262)(99,202,149,260,263)(100,203,150,241,264)(121,334,460,497,382)(122,335,441,498,383)(123,336,442,499,384)(124,337,443,500,385)(125,338,444,481,386)(126,339,445,482,387)(127,340,446,483,388)(128,321,447,484,389)(129,322,448,485,390)(130,323,449,486,391)(131,324,450,487,392)(132,325,451,488,393)(133,326,452,489,394)(134,327,453,490,395)(135,328,454,491,396)(136,329,455,492,397)(137,330,456,493,398)(138,331,457,494,399)(139,332,458,495,400)(140,333,459,496,381), (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( (1,492,275,357,223)(2,493,276,358,224)(3,494,277,359,225)(4,495,278,360,226)(5,496,279,341,227)(6,497,280,342,228)(7,498,261,343,229)(8,499,262,344,230)(9,500,263,345,231)(10,481,264,346,232)(11,482,265,347,233)(12,483,266,348,234)(13,484,267,349,235)(14,485,268,350,236)(15,486,269,351,237)(16,487,270,352,238)(17,488,271,353,239)(18,489,272,354,240)(19,490,273,355,221)(20,491,274,356,222)(21,462,427,123,201)(22,463,428,124,202)(23,464,429,125,203)(24,465,430,126,204)(25,466,431,127,205)(26,467,432,128,206)(27,468,433,129,207)(28,469,434,130,208)(29,470,435,131,209)(30,471,436,132,210)(31,472,437,133,211)(32,473,438,134,212)(33,474,439,135,213)(34,475,440,136,214)(35,476,421,137,215)(36,477,422,138,216)(37,478,423,139,217)(38,479,424,140,218)(39,480,425,121,219)(40,461,426,122,220)(41,64,333,145,363)(42,65,334,146,364)(43,66,335,147,365)(44,67,336,148,366)(45,68,337,149,367)(46,69,338,150,368)(47,70,339,151,369)(48,71,340,152,370)(49,72,321,153,371)(50,73,322,154,372)(51,74,323,155,373)(52,75,324,156,374)(53,76,325,157,375)(54,77,326,158,376)(55,78,327,159,377)(56,79,328,160,378)(57,80,329,141,379)(58,61,330,142,380)(59,62,331,143,361)(60,63,332,144,362)(81,281,404,175,387)(82,282,405,176,388)(83,283,406,177,389)(84,284,407,178,390)(85,285,408,179,391)(86,286,409,180,392)(87,287,410,161,393)(88,288,411,162,394)(89,289,412,163,395)(90,290,413,164,396)(91,291,414,165,397)(92,292,415,166,398)(93,293,416,167,399)(94,294,417,168,400)(95,295,418,169,381)(96,296,419,170,382)(97,297,420,171,383)(98,298,401,172,384)(99,299,402,173,385)(100,300,403,174,386)(101,452,249,182,305)(102,453,250,183,306)(103,454,251,184,307)(104,455,252,185,308)(105,456,253,186,309)(106,457,254,187,310)(107,458,255,188,311)(108,459,256,189,312)(109,460,257,190,313)(110,441,258,191,314)(111,442,259,192,315)(112,443,260,193,316)(113,444,241,194,317)(114,445,242,195,318)(115,446,243,196,319)(116,447,244,197,320)(117,448,245,198,301)(118,449,246,199,302)(119,450,247,200,303)(120,451,248,181,304), (1,165,440,80,104)(2,166,421,61,105)(3,167,422,62,106)(4,168,423,63,107)(5,169,424,64,108)(6,170,425,65,109)(7,171,426,66,110)(8,172,427,67,111)(9,173,428,68,112)(10,174,429,69,113)(11,175,430,70,114)(12,176,431,71,115)(13,177,432,72,116)(14,178,433,73,117)(15,179,434,74,118)(16,180,435,75,119)(17,161,436,76,120)(18,162,437,77,101)(19,163,438,78,102)(20,164,439,79,103)(21,366,192,344,298)(22,367,193,345,299)(23,368,194,346,300)(24,369,195,347,281)(25,370,196,348,282)(26,371,197,349,283)(27,372,198,350,284)(28,373,199,351,285)(29,374,200,352,286)(30,375,181,353,287)(31,376,182,354,288)(32,377,183,355,289)(33,378,184,356,290)(34,379,185,357,291)(35,380,186,358,292)(36,361,187,359,293)(37,362,188,360,294)(38,363,189,341,295)(39,364,190,342,296)(40,365,191,343,297)(41,312,227,418,479)(42,313,228,419,480)(43,314,229,420,461)(44,315,230,401,462)(45,316,231,402,463)(46,317,232,403,464)(47,318,233,404,465)(48,319,234,405,466)(49,320,235,406,467)(50,301,236,407,468)(51,302,237,408,469)(52,303,238,409,470)(53,304,239,410,471)(54,305,240,411,472)(55,306,221,412,473)(56,307,222,413,474)(57,308,223,414,475)(58,309,224,415,476)(59,310,225,416,477)(60,311,226,417,478)(81,204,151,242,265)(82,205,152,243,266)(83,206,153,244,267)(84,207,154,245,268)(85,208,155,246,269)(86,209,156,247,270)(87,210,157,248,271)(88,211,158,249,272)(89,212,159,250,273)(90,213,160,251,274)(91,214,141,252,275)(92,215,142,253,276)(93,216,143,254,277)(94,217,144,255,278)(95,218,145,256,279)(96,219,146,257,280)(97,220,147,258,261)(98,201,148,259,262)(99,202,149,260,263)(100,203,150,241,264)(121,334,460,497,382)(122,335,441,498,383)(123,336,442,499,384)(124,337,443,500,385)(125,338,444,481,386)(126,339,445,482,387)(127,340,446,483,388)(128,321,447,484,389)(129,322,448,485,390)(130,323,449,486,391)(131,324,450,487,392)(132,325,451,488,393)(133,326,452,489,394)(134,327,453,490,395)(135,328,454,491,396)(136,329,455,492,397)(137,330,456,493,398)(138,331,457,494,399)(139,332,458,495,400)(140,333,459,496,381), (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([(1,492,275,357,223),(2,493,276,358,224),(3,494,277,359,225),(4,495,278,360,226),(5,496,279,341,227),(6,497,280,342,228),(7,498,261,343,229),(8,499,262,344,230),(9,500,263,345,231),(10,481,264,346,232),(11,482,265,347,233),(12,483,266,348,234),(13,484,267,349,235),(14,485,268,350,236),(15,486,269,351,237),(16,487,270,352,238),(17,488,271,353,239),(18,489,272,354,240),(19,490,273,355,221),(20,491,274,356,222),(21,462,427,123,201),(22,463,428,124,202),(23,464,429,125,203),(24,465,430,126,204),(25,466,431,127,205),(26,467,432,128,206),(27,468,433,129,207),(28,469,434,130,208),(29,470,435,131,209),(30,471,436,132,210),(31,472,437,133,211),(32,473,438,134,212),(33,474,439,135,213),(34,475,440,136,214),(35,476,421,137,215),(36,477,422,138,216),(37,478,423,139,217),(38,479,424,140,218),(39,480,425,121,219),(40,461,426,122,220),(41,64,333,145,363),(42,65,334,146,364),(43,66,335,147,365),(44,67,336,148,366),(45,68,337,149,367),(46,69,338,150,368),(47,70,339,151,369),(48,71,340,152,370),(49,72,321,153,371),(50,73,322,154,372),(51,74,323,155,373),(52,75,324,156,374),(53,76,325,157,375),(54,77,326,158,376),(55,78,327,159,377),(56,79,328,160,378),(57,80,329,141,379),(58,61,330,142,380),(59,62,331,143,361),(60,63,332,144,362),(81,281,404,175,387),(82,282,405,176,388),(83,283,406,177,389),(84,284,407,178,390),(85,285,408,179,391),(86,286,409,180,392),(87,287,410,161,393),(88,288,411,162,394),(89,289,412,163,395),(90,290,413,164,396),(91,291,414,165,397),(92,292,415,166,398),(93,293,416,167,399),(94,294,417,168,400),(95,295,418,169,381),(96,296,419,170,382),(97,297,420,171,383),(98,298,401,172,384),(99,299,402,173,385),(100,300,403,174,386),(101,452,249,182,305),(102,453,250,183,306),(103,454,251,184,307),(104,455,252,185,308),(105,456,253,186,309),(106,457,254,187,310),(107,458,255,188,311),(108,459,256,189,312),(109,460,257,190,313),(110,441,258,191,314),(111,442,259,192,315),(112,443,260,193,316),(113,444,241,194,317),(114,445,242,195,318),(115,446,243,196,319),(116,447,244,197,320),(117,448,245,198,301),(118,449,246,199,302),(119,450,247,200,303),(120,451,248,181,304)], [(1,165,440,80,104),(2,166,421,61,105),(3,167,422,62,106),(4,168,423,63,107),(5,169,424,64,108),(6,170,425,65,109),(7,171,426,66,110),(8,172,427,67,111),(9,173,428,68,112),(10,174,429,69,113),(11,175,430,70,114),(12,176,431,71,115),(13,177,432,72,116),(14,178,433,73,117),(15,179,434,74,118),(16,180,435,75,119),(17,161,436,76,120),(18,162,437,77,101),(19,163,438,78,102),(20,164,439,79,103),(21,366,192,344,298),(22,367,193,345,299),(23,368,194,346,300),(24,369,195,347,281),(25,370,196,348,282),(26,371,197,349,283),(27,372,198,350,284),(28,373,199,351,285),(29,374,200,352,286),(30,375,181,353,287),(31,376,182,354,288),(32,377,183,355,289),(33,378,184,356,290),(34,379,185,357,291),(35,380,186,358,292),(36,361,187,359,293),(37,362,188,360,294),(38,363,189,341,295),(39,364,190,342,296),(40,365,191,343,297),(41,312,227,418,479),(42,313,228,419,480),(43,314,229,420,461),(44,315,230,401,462),(45,316,231,402,463),(46,317,232,403,464),(47,318,233,404,465),(48,319,234,405,466),(49,320,235,406,467),(50,301,236,407,468),(51,302,237,408,469),(52,303,238,409,470),(53,304,239,410,471),(54,305,240,411,472),(55,306,221,412,473),(56,307,222,413,474),(57,308,223,414,475),(58,309,224,415,476),(59,310,225,416,477),(60,311,226,417,478),(81,204,151,242,265),(82,205,152,243,266),(83,206,153,244,267),(84,207,154,245,268),(85,208,155,246,269),(86,209,156,247,270),(87,210,157,248,271),(88,211,158,249,272),(89,212,159,250,273),(90,213,160,251,274),(91,214,141,252,275),(92,215,142,253,276),(93,216,143,254,277),(94,217,144,255,278),(95,218,145,256,279),(96,219,146,257,280),(97,220,147,258,261),(98,201,148,259,262),(99,202,149,260,263),(100,203,150,241,264),(121,334,460,497,382),(122,335,441,498,383),(123,336,442,499,384),(124,337,443,500,385),(125,338,444,481,386),(126,339,445,482,387),(127,340,446,483,388),(128,321,447,484,389),(129,322,448,485,390),(130,323,449,486,391),(131,324,450,487,392),(132,325,451,488,393),(133,326,452,489,394),(134,327,453,490,395),(135,328,454,491,396),(136,329,455,492,397),(137,330,456,493,398),(138,331,457,494,399),(139,332,458,495,400),(140,333,459,496,381)], [(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 4A4B5A···5DT10A···10DT20A···20IN
order12445···510···1020···20
size11111···11···11···1

500 irreducible representations

dim111111
type++
imageC1C2C4C5C10C20
kernelC52×C20C52×C10C53C5×C20C5×C10C52
# reps112124124248

Matrix representation of C52×C20 in GL3(𝔽41) generated by

1800
0160
001
,
1800
0180
001
,
500
0360
0016
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

׿
×
𝔽