direct product, abelian, monomial
Aliases: C5×C102, SmallGroup(500,56)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C102 |
| C1 — C5×C102 |
| C1 — C5×C102 |
Generators and relations for C5×C102
G = < a,b,c | a5=b10=c10=1, ab=ba, ac=ca, bc=cb >
Subgroups: 320, all normal (4 characteristic)
C1, C2, C22, C5, C10, C2×C10, C52, C5×C10, C102, C53, C52×C10, C5×C102
Quotients: C1, C2, C22, C5, C10, C2×C10, C52, C5×C10, C102, C53, C52×C10, C5×C102
►Regular action on 500 points
Generators in S500
(1 82 354 52 63)(2 83 355 53 64)(3 84 356 54 65)(4 85 357 55 66)(5 86 358 56 67)(6 87 359 57 68)(7 88 360 58 69)(8 89 351 59 70)(9 90 352 60 61)(10 81 353 51 62)(11 492 230 472 263)(12 493 221 473 264)(13 494 222 474 265)(14 495 223 475 266)(15 496 224 476 267)(16 497 225 477 268)(17 498 226 478 269)(18 499 227 479 270)(19 500 228 480 261)(20 491 229 471 262)(21 392 250 372 205)(22 393 241 373 206)(23 394 242 374 207)(24 395 243 375 208)(25 396 244 376 209)(26 397 245 377 210)(27 398 246 378 201)(28 399 247 379 202)(29 400 248 380 203)(30 391 249 371 204)(31 105 75 100 150)(32 106 76 91 141)(33 107 77 92 142)(34 108 78 93 143)(35 109 79 94 144)(36 110 80 95 145)(37 101 71 96 146)(38 102 72 97 147)(39 103 73 98 148)(40 104 74 99 149)(41 288 317 340 302)(42 289 318 331 303)(43 290 319 332 304)(44 281 320 333 305)(45 282 311 334 306)(46 283 312 335 307)(47 284 313 336 308)(48 285 314 337 309)(49 286 315 338 310)(50 287 316 339 301)(111 381 139 121 364)(112 382 140 122 365)(113 383 131 123 366)(114 384 132 124 367)(115 385 133 125 368)(116 386 134 126 369)(117 387 135 127 370)(118 388 136 128 361)(119 389 137 129 362)(120 390 138 130 363)(151 298 348 196 328)(152 299 349 197 329)(153 300 350 198 330)(154 291 341 199 321)(155 292 342 200 322)(156 293 343 191 323)(157 294 344 192 324)(158 295 345 193 325)(159 296 346 194 326)(160 297 347 195 327)(161 211 443 171 423)(162 212 444 172 424)(163 213 445 173 425)(164 214 446 174 426)(165 215 447 175 427)(166 216 448 176 428)(167 217 449 177 429)(168 218 450 178 430)(169 219 441 179 421)(170 220 442 180 422)(181 439 416 409 459)(182 440 417 410 460)(183 431 418 401 451)(184 432 419 402 452)(185 433 420 403 453)(186 434 411 404 454)(187 435 412 405 455)(188 436 413 406 456)(189 437 414 407 457)(190 438 415 408 458)(231 489 466 257 275)(232 490 467 258 276)(233 481 468 259 277)(234 482 469 260 278)(235 483 470 251 279)(236 484 461 252 280)(237 485 462 253 271)(238 486 463 254 272)(239 487 464 255 273)(240 488 465 256 274)
(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)
(1 202 161 99 410 384 299 473 286 467)(2 203 162 100 401 385 300 474 287 468)(3 204 163 91 402 386 291 475 288 469)(4 205 164 92 403 387 292 476 289 470)(5 206 165 93 404 388 293 477 290 461)(6 207 166 94 405 389 294 478 281 462)(7 208 167 95 406 390 295 479 282 463)(8 209 168 96 407 381 296 480 283 464)(9 210 169 97 408 382 297 471 284 465)(10 201 170 98 409 383 298 472 285 466)(11 337 275 353 398 442 39 181 123 196)(12 338 276 354 399 443 40 182 124 197)(13 339 277 355 400 444 31 183 125 198)(14 340 278 356 391 445 32 184 126 199)(15 331 279 357 392 446 33 185 127 200)(16 332 280 358 393 447 34 186 128 191)(17 333 271 359 394 448 35 187 129 192)(18 334 272 360 395 449 36 188 130 193)(19 335 273 351 396 450 37 189 121 194)(20 336 274 352 397 441 38 190 122 195)(21 214 142 453 135 342 267 318 251 85)(22 215 143 454 136 343 268 319 252 86)(23 216 144 455 137 344 269 320 253 87)(24 217 145 456 138 345 270 311 254 88)(25 218 146 457 139 346 261 312 255 89)(26 219 147 458 140 347 262 313 256 90)(27 220 148 459 131 348 263 314 257 81)(28 211 149 460 132 349 264 315 258 82)(29 212 150 451 133 350 265 316 259 83)(30 213 141 452 134 341 266 317 260 84)(41 482 65 371 425 76 419 116 154 223)(42 483 66 372 426 77 420 117 155 224)(43 484 67 373 427 78 411 118 156 225)(44 485 68 374 428 79 412 119 157 226)(45 486 69 375 429 80 413 120 158 227)(46 487 70 376 430 71 414 111 159 228)(47 488 61 377 421 72 415 112 160 229)(48 489 62 378 422 73 416 113 151 230)(49 490 63 379 423 74 417 114 152 221)(50 481 64 380 424 75 418 115 153 222)(51 246 180 103 439 366 328 492 309 231)(52 247 171 104 440 367 329 493 310 232)(53 248 172 105 431 368 330 494 301 233)(54 249 173 106 432 369 321 495 302 234)(55 250 174 107 433 370 322 496 303 235)(56 241 175 108 434 361 323 497 304 236)(57 242 176 109 435 362 324 498 305 237)(58 243 177 110 436 363 325 499 306 238)(59 244 178 101 437 364 326 500 307 239)(60 245 179 102 438 365 327 491 308 240)
G:=sub<Sym(500)| (1,82,354,52,63)(2,83,355,53,64)(3,84,356,54,65)(4,85,357,55,66)(5,86,358,56,67)(6,87,359,57,68)(7,88,360,58,69)(8,89,351,59,70)(9,90,352,60,61)(10,81,353,51,62)(11,492,230,472,263)(12,493,221,473,264)(13,494,222,474,265)(14,495,223,475,266)(15,496,224,476,267)(16,497,225,477,268)(17,498,226,478,269)(18,499,227,479,270)(19,500,228,480,261)(20,491,229,471,262)(21,392,250,372,205)(22,393,241,373,206)(23,394,242,374,207)(24,395,243,375,208)(25,396,244,376,209)(26,397,245,377,210)(27,398,246,378,201)(28,399,247,379,202)(29,400,248,380,203)(30,391,249,371,204)(31,105,75,100,150)(32,106,76,91,141)(33,107,77,92,142)(34,108,78,93,143)(35,109,79,94,144)(36,110,80,95,145)(37,101,71,96,146)(38,102,72,97,147)(39,103,73,98,148)(40,104,74,99,149)(41,288,317,340,302)(42,289,318,331,303)(43,290,319,332,304)(44,281,320,333,305)(45,282,311,334,306)(46,283,312,335,307)(47,284,313,336,308)(48,285,314,337,309)(49,286,315,338,310)(50,287,316,339,301)(111,381,139,121,364)(112,382,140,122,365)(113,383,131,123,366)(114,384,132,124,367)(115,385,133,125,368)(116,386,134,126,369)(117,387,135,127,370)(118,388,136,128,361)(119,389,137,129,362)(120,390,138,130,363)(151,298,348,196,328)(152,299,349,197,329)(153,300,350,198,330)(154,291,341,199,321)(155,292,342,200,322)(156,293,343,191,323)(157,294,344,192,324)(158,295,345,193,325)(159,296,346,194,326)(160,297,347,195,327)(161,211,443,171,423)(162,212,444,172,424)(163,213,445,173,425)(164,214,446,174,426)(165,215,447,175,427)(166,216,448,176,428)(167,217,449,177,429)(168,218,450,178,430)(169,219,441,179,421)(170,220,442,180,422)(181,439,416,409,459)(182,440,417,410,460)(183,431,418,401,451)(184,432,419,402,452)(185,433,420,403,453)(186,434,411,404,454)(187,435,412,405,455)(188,436,413,406,456)(189,437,414,407,457)(190,438,415,408,458)(231,489,466,257,275)(232,490,467,258,276)(233,481,468,259,277)(234,482,469,260,278)(235,483,470,251,279)(236,484,461,252,280)(237,485,462,253,271)(238,486,463,254,272)(239,487,464,255,273)(240,488,465,256,274), (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), (1,202,161,99,410,384,299,473,286,467)(2,203,162,100,401,385,300,474,287,468)(3,204,163,91,402,386,291,475,288,469)(4,205,164,92,403,387,292,476,289,470)(5,206,165,93,404,388,293,477,290,461)(6,207,166,94,405,389,294,478,281,462)(7,208,167,95,406,390,295,479,282,463)(8,209,168,96,407,381,296,480,283,464)(9,210,169,97,408,382,297,471,284,465)(10,201,170,98,409,383,298,472,285,466)(11,337,275,353,398,442,39,181,123,196)(12,338,276,354,399,443,40,182,124,197)(13,339,277,355,400,444,31,183,125,198)(14,340,278,356,391,445,32,184,126,199)(15,331,279,357,392,446,33,185,127,200)(16,332,280,358,393,447,34,186,128,191)(17,333,271,359,394,448,35,187,129,192)(18,334,272,360,395,449,36,188,130,193)(19,335,273,351,396,450,37,189,121,194)(20,336,274,352,397,441,38,190,122,195)(21,214,142,453,135,342,267,318,251,85)(22,215,143,454,136,343,268,319,252,86)(23,216,144,455,137,344,269,320,253,87)(24,217,145,456,138,345,270,311,254,88)(25,218,146,457,139,346,261,312,255,89)(26,219,147,458,140,347,262,313,256,90)(27,220,148,459,131,348,263,314,257,81)(28,211,149,460,132,349,264,315,258,82)(29,212,150,451,133,350,265,316,259,83)(30,213,141,452,134,341,266,317,260,84)(41,482,65,371,425,76,419,116,154,223)(42,483,66,372,426,77,420,117,155,224)(43,484,67,373,427,78,411,118,156,225)(44,485,68,374,428,79,412,119,157,226)(45,486,69,375,429,80,413,120,158,227)(46,487,70,376,430,71,414,111,159,228)(47,488,61,377,421,72,415,112,160,229)(48,489,62,378,422,73,416,113,151,230)(49,490,63,379,423,74,417,114,152,221)(50,481,64,380,424,75,418,115,153,222)(51,246,180,103,439,366,328,492,309,231)(52,247,171,104,440,367,329,493,310,232)(53,248,172,105,431,368,330,494,301,233)(54,249,173,106,432,369,321,495,302,234)(55,250,174,107,433,370,322,496,303,235)(56,241,175,108,434,361,323,497,304,236)(57,242,176,109,435,362,324,498,305,237)(58,243,177,110,436,363,325,499,306,238)(59,244,178,101,437,364,326,500,307,239)(60,245,179,102,438,365,327,491,308,240)>;
G:=Group( (1,82,354,52,63)(2,83,355,53,64)(3,84,356,54,65)(4,85,357,55,66)(5,86,358,56,67)(6,87,359,57,68)(7,88,360,58,69)(8,89,351,59,70)(9,90,352,60,61)(10,81,353,51,62)(11,492,230,472,263)(12,493,221,473,264)(13,494,222,474,265)(14,495,223,475,266)(15,496,224,476,267)(16,497,225,477,268)(17,498,226,478,269)(18,499,227,479,270)(19,500,228,480,261)(20,491,229,471,262)(21,392,250,372,205)(22,393,241,373,206)(23,394,242,374,207)(24,395,243,375,208)(25,396,244,376,209)(26,397,245,377,210)(27,398,246,378,201)(28,399,247,379,202)(29,400,248,380,203)(30,391,249,371,204)(31,105,75,100,150)(32,106,76,91,141)(33,107,77,92,142)(34,108,78,93,143)(35,109,79,94,144)(36,110,80,95,145)(37,101,71,96,146)(38,102,72,97,147)(39,103,73,98,148)(40,104,74,99,149)(41,288,317,340,302)(42,289,318,331,303)(43,290,319,332,304)(44,281,320,333,305)(45,282,311,334,306)(46,283,312,335,307)(47,284,313,336,308)(48,285,314,337,309)(49,286,315,338,310)(50,287,316,339,301)(111,381,139,121,364)(112,382,140,122,365)(113,383,131,123,366)(114,384,132,124,367)(115,385,133,125,368)(116,386,134,126,369)(117,387,135,127,370)(118,388,136,128,361)(119,389,137,129,362)(120,390,138,130,363)(151,298,348,196,328)(152,299,349,197,329)(153,300,350,198,330)(154,291,341,199,321)(155,292,342,200,322)(156,293,343,191,323)(157,294,344,192,324)(158,295,345,193,325)(159,296,346,194,326)(160,297,347,195,327)(161,211,443,171,423)(162,212,444,172,424)(163,213,445,173,425)(164,214,446,174,426)(165,215,447,175,427)(166,216,448,176,428)(167,217,449,177,429)(168,218,450,178,430)(169,219,441,179,421)(170,220,442,180,422)(181,439,416,409,459)(182,440,417,410,460)(183,431,418,401,451)(184,432,419,402,452)(185,433,420,403,453)(186,434,411,404,454)(187,435,412,405,455)(188,436,413,406,456)(189,437,414,407,457)(190,438,415,408,458)(231,489,466,257,275)(232,490,467,258,276)(233,481,468,259,277)(234,482,469,260,278)(235,483,470,251,279)(236,484,461,252,280)(237,485,462,253,271)(238,486,463,254,272)(239,487,464,255,273)(240,488,465,256,274), (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), (1,202,161,99,410,384,299,473,286,467)(2,203,162,100,401,385,300,474,287,468)(3,204,163,91,402,386,291,475,288,469)(4,205,164,92,403,387,292,476,289,470)(5,206,165,93,404,388,293,477,290,461)(6,207,166,94,405,389,294,478,281,462)(7,208,167,95,406,390,295,479,282,463)(8,209,168,96,407,381,296,480,283,464)(9,210,169,97,408,382,297,471,284,465)(10,201,170,98,409,383,298,472,285,466)(11,337,275,353,398,442,39,181,123,196)(12,338,276,354,399,443,40,182,124,197)(13,339,277,355,400,444,31,183,125,198)(14,340,278,356,391,445,32,184,126,199)(15,331,279,357,392,446,33,185,127,200)(16,332,280,358,393,447,34,186,128,191)(17,333,271,359,394,448,35,187,129,192)(18,334,272,360,395,449,36,188,130,193)(19,335,273,351,396,450,37,189,121,194)(20,336,274,352,397,441,38,190,122,195)(21,214,142,453,135,342,267,318,251,85)(22,215,143,454,136,343,268,319,252,86)(23,216,144,455,137,344,269,320,253,87)(24,217,145,456,138,345,270,311,254,88)(25,218,146,457,139,346,261,312,255,89)(26,219,147,458,140,347,262,313,256,90)(27,220,148,459,131,348,263,314,257,81)(28,211,149,460,132,349,264,315,258,82)(29,212,150,451,133,350,265,316,259,83)(30,213,141,452,134,341,266,317,260,84)(41,482,65,371,425,76,419,116,154,223)(42,483,66,372,426,77,420,117,155,224)(43,484,67,373,427,78,411,118,156,225)(44,485,68,374,428,79,412,119,157,226)(45,486,69,375,429,80,413,120,158,227)(46,487,70,376,430,71,414,111,159,228)(47,488,61,377,421,72,415,112,160,229)(48,489,62,378,422,73,416,113,151,230)(49,490,63,379,423,74,417,114,152,221)(50,481,64,380,424,75,418,115,153,222)(51,246,180,103,439,366,328,492,309,231)(52,247,171,104,440,367,329,493,310,232)(53,248,172,105,431,368,330,494,301,233)(54,249,173,106,432,369,321,495,302,234)(55,250,174,107,433,370,322,496,303,235)(56,241,175,108,434,361,323,497,304,236)(57,242,176,109,435,362,324,498,305,237)(58,243,177,110,436,363,325,499,306,238)(59,244,178,101,437,364,326,500,307,239)(60,245,179,102,438,365,327,491,308,240) );
G=PermutationGroup([[(1,82,354,52,63),(2,83,355,53,64),(3,84,356,54,65),(4,85,357,55,66),(5,86,358,56,67),(6,87,359,57,68),(7,88,360,58,69),(8,89,351,59,70),(9,90,352,60,61),(10,81,353,51,62),(11,492,230,472,263),(12,493,221,473,264),(13,494,222,474,265),(14,495,223,475,266),(15,496,224,476,267),(16,497,225,477,268),(17,498,226,478,269),(18,499,227,479,270),(19,500,228,480,261),(20,491,229,471,262),(21,392,250,372,205),(22,393,241,373,206),(23,394,242,374,207),(24,395,243,375,208),(25,396,244,376,209),(26,397,245,377,210),(27,398,246,378,201),(28,399,247,379,202),(29,400,248,380,203),(30,391,249,371,204),(31,105,75,100,150),(32,106,76,91,141),(33,107,77,92,142),(34,108,78,93,143),(35,109,79,94,144),(36,110,80,95,145),(37,101,71,96,146),(38,102,72,97,147),(39,103,73,98,148),(40,104,74,99,149),(41,288,317,340,302),(42,289,318,331,303),(43,290,319,332,304),(44,281,320,333,305),(45,282,311,334,306),(46,283,312,335,307),(47,284,313,336,308),(48,285,314,337,309),(49,286,315,338,310),(50,287,316,339,301),(111,381,139,121,364),(112,382,140,122,365),(113,383,131,123,366),(114,384,132,124,367),(115,385,133,125,368),(116,386,134,126,369),(117,387,135,127,370),(118,388,136,128,361),(119,389,137,129,362),(120,390,138,130,363),(151,298,348,196,328),(152,299,349,197,329),(153,300,350,198,330),(154,291,341,199,321),(155,292,342,200,322),(156,293,343,191,323),(157,294,344,192,324),(158,295,345,193,325),(159,296,346,194,326),(160,297,347,195,327),(161,211,443,171,423),(162,212,444,172,424),(163,213,445,173,425),(164,214,446,174,426),(165,215,447,175,427),(166,216,448,176,428),(167,217,449,177,429),(168,218,450,178,430),(169,219,441,179,421),(170,220,442,180,422),(181,439,416,409,459),(182,440,417,410,460),(183,431,418,401,451),(184,432,419,402,452),(185,433,420,403,453),(186,434,411,404,454),(187,435,412,405,455),(188,436,413,406,456),(189,437,414,407,457),(190,438,415,408,458),(231,489,466,257,275),(232,490,467,258,276),(233,481,468,259,277),(234,482,469,260,278),(235,483,470,251,279),(236,484,461,252,280),(237,485,462,253,271),(238,486,463,254,272),(239,487,464,255,273),(240,488,465,256,274)], [(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)], [(1,202,161,99,410,384,299,473,286,467),(2,203,162,100,401,385,300,474,287,468),(3,204,163,91,402,386,291,475,288,469),(4,205,164,92,403,387,292,476,289,470),(5,206,165,93,404,388,293,477,290,461),(6,207,166,94,405,389,294,478,281,462),(7,208,167,95,406,390,295,479,282,463),(8,209,168,96,407,381,296,480,283,464),(9,210,169,97,408,382,297,471,284,465),(10,201,170,98,409,383,298,472,285,466),(11,337,275,353,398,442,39,181,123,196),(12,338,276,354,399,443,40,182,124,197),(13,339,277,355,400,444,31,183,125,198),(14,340,278,356,391,445,32,184,126,199),(15,331,279,357,392,446,33,185,127,200),(16,332,280,358,393,447,34,186,128,191),(17,333,271,359,394,448,35,187,129,192),(18,334,272,360,395,449,36,188,130,193),(19,335,273,351,396,450,37,189,121,194),(20,336,274,352,397,441,38,190,122,195),(21,214,142,453,135,342,267,318,251,85),(22,215,143,454,136,343,268,319,252,86),(23,216,144,455,137,344,269,320,253,87),(24,217,145,456,138,345,270,311,254,88),(25,218,146,457,139,346,261,312,255,89),(26,219,147,458,140,347,262,313,256,90),(27,220,148,459,131,348,263,314,257,81),(28,211,149,460,132,349,264,315,258,82),(29,212,150,451,133,350,265,316,259,83),(30,213,141,452,134,341,266,317,260,84),(41,482,65,371,425,76,419,116,154,223),(42,483,66,372,426,77,420,117,155,224),(43,484,67,373,427,78,411,118,156,225),(44,485,68,374,428,79,412,119,157,226),(45,486,69,375,429,80,413,120,158,227),(46,487,70,376,430,71,414,111,159,228),(47,488,61,377,421,72,415,112,160,229),(48,489,62,378,422,73,416,113,151,230),(49,490,63,379,423,74,417,114,152,221),(50,481,64,380,424,75,418,115,153,222),(51,246,180,103,439,366,328,492,309,231),(52,247,171,104,440,367,329,493,310,232),(53,248,172,105,431,368,330,494,301,233),(54,249,173,106,432,369,321,495,302,234),(55,250,174,107,433,370,322,496,303,235),(56,241,175,108,434,361,323,497,304,236),(57,242,176,109,435,362,324,498,305,237),(58,243,177,110,436,363,325,499,306,238),(59,244,178,101,437,364,326,500,307,239),(60,245,179,102,438,365,327,491,308,240)]])
500 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | ··· | 5DT | 10A | ··· | 10NH |
| order | 1 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
500 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C5 | C10 |
| kernel | C5×C102 | C52×C10 | C102 | C5×C10 |
| # reps | 1 | 3 | 124 | 372 |
Matrix representation of C5×C102 ►in GL3(𝔽11) generated by
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 4 |
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 6 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(11))| [1,0,0,0,9,0,0,0,4],[10,0,0,0,10,0,0,0,6],[6,0,0,0,6,0,0,0,3] >;
C5×C102 in GAP, Magma, Sage, TeX
C_5\times C_{10}^2 % in TeX
G:=Group("C5xC10^2"); // GroupNames label
G:=SmallGroup(500,56);
// by ID
G=gap.SmallGroup(500,56);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5]);
// Polycyclic
G:=Group<a,b,c|a^5=b^10=c^10=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations