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G = C5×C102order 500 = 22·53

Abelian group of type [5,10,10]

direct product, abelian, monomial

Aliases: C5×C102, SmallGroup(500,56)

Series: Derived Chief Lower central Upper central

C1 — C5×C102
C1C5C52C53C52×C10 — 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 [×3], C22, C5 [×31], C10 [×93], C2×C10 [×31], C52 [×31], C5×C10 [×93], C102 [×31], C53, C52×C10 [×3], C5×C102
Quotients: C1, C2 [×3], C22, C5 [×31], C10 [×93], C2×C10 [×31], C52 [×31], C5×C10 [×93], C102 [×31], C53, C52×C10 [×3], C5×C102

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

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

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

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

500 conjugacy classes

class 1 2A2B2C5A···5DT10A···10NH
order12225···510···10
size11111···11···1

500 irreducible representations

dim1111
type++
imageC1C2C5C10
kernelC5×C102C52×C10C102C5×C10
# reps13124372

Matrix representation of C5×C102 in GL3(𝔽11) generated by

100
090
004
,
1000
0100
006
,
600
060
003
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

׿
×
𝔽