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G = C53⋊12C4  order 500 = 22·53

3rd semidirect product of C53 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, A-group

Aliases: C53⋊12C4, C52⋊9Dic5, C10.3(C5⋊D5), (C5×C10).10D5, C2.(C53⋊C2), C5⋊2(C52⋊6C4), (C52×C10).3C2, SmallGroup(500,39)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C53 — C53⋊12C4
Chief series
C1 — C5 — C52 — C53 — C52×C10 — C53⋊12C4
Lower central
C53 — C53⋊12C4
Upper central
C1 — C2

Generators and relations for C53⋊12C4
 G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 1184 in 192 conjugacy classes, 129 normal (5 characteristic)
C1, C2, C4, C5, C10, Dic5, C52, C5×C10, C52⋊6C4, C53, C52×C10, C53⋊12C4
Quotients: C1, C2, C4, D5, Dic5, C5⋊D5, C52⋊6C4, C53⋊C2, C53⋊12C4

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

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

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

128 conjugacy classes

class 1  2 4A4B5A···5BJ10A···10BJ
order12445···510···10
size111251252···22···2

128 irreducible representations

dim11122
type+++-
imageC1C2C4D5Dic5
kernelC53⋊12C4C52×C10C53C5×C10C52
# reps1126262

Matrix representation of C53⋊12C4 ►in GL6(𝔽41)

010000
40340000
000100
00403400
000001
00004034
,
770000
34400000
00403400
007700
000001
00004034
,
010000
40340000
000100
00403400
000010
000001
,
27140000
30140000
003600
00263800
00002712
00002814

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,1,34],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,0,40,0,0,0,0,1,34],[0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,30,0,0,0,0,14,14,0,0,0,0,0,0,3,26,0,0,0,0,6,38,0,0,0,0,0,0,27,28,0,0,0,0,12,14] >;
 

C53⋊12C4 in GAP, Magma, Sage, TeX 

C_5^3\rtimes_{12}C_4
 
% in TeX
 
G:=Group("C5^3:12C4");
 
// GroupNames label
 
G:=SmallGroup(500,39);
 
// by ID
 
G=gap.SmallGroup(500,39);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1603,10004]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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