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## G = C5×C42.S3order 480 = 25·3·5

### Direct product of C5 and C42.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C42.S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C60 — C10×C3⋊C8 — C5×C42.S3
 Lower central C3 — C6 — C5×C42.S3
 Upper central C1 — C2×C20 — C4×C20

Generators and relations for C5×C42.S3
G = < a,b,c,d | a5=b6=d4=1, c4=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 116 in 80 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, C2×C8, C20, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C30, C30, C8⋊C4, C40, C2×C20, C2×C20, C2×C3⋊C8, C4×C12, C60, C60, C2×C30, C4×C20, C2×C40, C42.S3, C5×C3⋊C8, C2×C60, C2×C60, C5×C8⋊C4, C10×C3⋊C8, C4×C60, C5×C42.S3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, Dic3, D6, C42, M4(2), C20, C2×C10, C4×S3, C2×Dic3, C5×S3, C8⋊C4, C2×C20, C4.Dic3, C4×Dic3, C5×Dic3, S3×C10, C4×C20, C5×M4(2), C42.S3, S3×C20, C10×Dic3, C5×C8⋊C4, C5×C4.Dic3, Dic3×C20, C5×C42.S3

Smallest permutation representation of C5×C42.S3
Regular action on 480 points
Generators in S480
(1 191 411 167 387)(2 192 412 168 388)(3 185 413 161 389)(4 186 414 162 390)(5 187 415 163 391)(6 188 416 164 392)(7 189 409 165 385)(8 190 410 166 386)(9 233 461 209 437)(10 234 462 210 438)(11 235 463 211 439)(12 236 464 212 440)(13 237 457 213 433)(14 238 458 214 434)(15 239 459 215 435)(16 240 460 216 436)(17 344 92 320 29)(18 337 93 313 30)(19 338 94 314 31)(20 339 95 315 32)(21 340 96 316 25)(22 341 89 317 26)(23 342 90 318 27)(24 343 91 319 28)(33 261 309 57 285)(34 262 310 58 286)(35 263 311 59 287)(36 264 312 60 288)(37 257 305 61 281)(38 258 306 62 282)(39 259 307 63 283)(40 260 308 64 284)(41 269 115 65 293)(42 270 116 66 294)(43 271 117 67 295)(44 272 118 68 296)(45 265 119 69 289)(46 266 120 70 290)(47 267 113 71 291)(48 268 114 72 292)(49 273 321 73 297)(50 274 322 74 298)(51 275 323 75 299)(52 276 324 76 300)(53 277 325 77 301)(54 278 326 78 302)(55 279 327 79 303)(56 280 328 80 304)(81 129 357 105 333)(82 130 358 106 334)(83 131 359 107 335)(84 132 360 108 336)(85 133 353 109 329)(86 134 354 110 330)(87 135 355 111 331)(88 136 356 112 332)(97 363 141 121 345)(98 364 142 122 346)(99 365 143 123 347)(100 366 144 124 348)(101 367 137 125 349)(102 368 138 126 350)(103 361 139 127 351)(104 362 140 128 352)(145 369 417 169 393)(146 370 418 170 394)(147 371 419 171 395)(148 372 420 172 396)(149 373 421 173 397)(150 374 422 174 398)(151 375 423 175 399)(152 376 424 176 400)(153 381 429 177 405)(154 382 430 178 406)(155 383 431 179 407)(156 384 432 180 408)(157 377 425 181 401)(158 378 426 182 402)(159 379 427 183 403)(160 380 428 184 404)(193 241 465 217 441)(194 242 466 218 442)(195 243 467 219 443)(196 244 468 220 444)(197 245 469 221 445)(198 246 470 222 446)(199 247 471 223 447)(200 248 472 224 448)(201 253 477 225 453)(202 254 478 226 454)(203 255 479 227 455)(204 256 480 228 456)(205 249 473 229 449)(206 250 474 230 450)(207 251 475 231 451)(208 252 476 232 452)
(1 263 113 5 259 117)(2 118 260 6 114 264)(3 257 115 7 261 119)(4 120 262 8 116 258)(9 133 26 13 129 30)(10 31 130 14 27 134)(11 135 28 15 131 32)(12 25 132 16 29 136)(17 356 236 21 360 240)(18 233 353 22 237 357)(19 358 238 23 354 234)(20 235 355 24 239 359)(33 265 389 37 269 385)(34 386 270 38 390 266)(35 267 391 39 271 387)(36 388 272 40 392 268)(41 165 285 45 161 281)(42 282 162 46 286 166)(43 167 287 47 163 283)(44 284 164 48 288 168)(49 157 147 53 153 151)(50 152 154 54 148 158)(51 159 149 55 155 145)(52 146 156 56 150 160)(57 289 413 61 293 409)(58 410 294 62 414 290)(59 291 415 63 295 411)(60 412 296 64 416 292)(65 189 309 69 185 305)(66 306 186 70 310 190)(67 191 311 71 187 307)(68 308 188 72 312 192)(73 181 171 77 177 175)(74 176 178 78 172 182)(75 183 173 79 179 169)(76 170 180 80 174 184)(81 313 437 85 317 433)(82 434 318 86 438 314)(83 315 439 87 319 435)(84 436 320 88 440 316)(89 213 333 93 209 329)(90 330 210 94 334 214)(91 215 335 95 211 331)(92 332 212 96 336 216)(97 205 195 101 201 199)(98 200 202 102 196 206)(99 207 197 103 203 193)(100 194 204 104 198 208)(105 337 461 109 341 457)(106 458 342 110 462 338)(107 339 463 111 343 459)(108 460 344 112 464 340)(121 229 219 125 225 223)(122 224 226 126 220 230)(123 231 221 127 227 217)(124 218 228 128 222 232)(137 477 471 141 473 467)(138 468 474 142 472 478)(139 479 465 143 475 469)(140 470 476 144 466 480)(241 365 251 245 361 255)(242 256 362 246 252 366)(243 367 253 247 363 249)(244 250 364 248 254 368)(273 377 371 277 381 375)(274 376 382 278 372 378)(275 379 373 279 383 369)(276 370 384 280 374 380)(297 401 395 301 405 399)(298 400 406 302 396 402)(299 403 397 303 407 393)(300 394 408 304 398 404)(321 425 419 325 429 423)(322 424 430 326 420 426)(323 427 421 327 431 417)(324 418 432 328 422 428)(345 449 443 349 453 447)(346 448 454 350 444 450)(347 451 445 351 455 441)(348 442 456 352 446 452)
(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)
(1 99 275 435)(2 104 276 440)(3 101 277 437)(4 98 278 434)(5 103 279 439)(6 100 280 436)(7 97 273 433)(8 102 274 438)(9 185 367 325)(10 190 368 322)(11 187 361 327)(12 192 362 324)(13 189 363 321)(14 186 364 326)(15 191 365 323)(16 188 366 328)(17 292 466 174)(18 289 467 171)(19 294 468 176)(20 291 469 173)(21 296 470 170)(22 293 471 175)(23 290 472 172)(24 295 465 169)(25 68 246 418)(26 65 247 423)(27 70 248 420)(28 67 241 417)(29 72 242 422)(30 69 243 419)(31 66 244 424)(32 71 245 421)(33 449 157 333)(34 454 158 330)(35 451 159 335)(36 456 160 332)(37 453 153 329)(38 450 154 334)(39 455 155 331)(40 452 156 336)(41 223 399 341)(42 220 400 338)(43 217 393 343)(44 222 394 340)(45 219 395 337)(46 224 396 342)(47 221 397 339)(48 218 398 344)(49 213 385 345)(50 210 386 350)(51 215 387 347)(52 212 388 352)(53 209 389 349)(54 214 390 346)(55 211 391 351)(56 216 392 348)(57 473 181 357)(58 478 182 354)(59 475 183 359)(60 480 184 356)(61 477 177 353)(62 474 178 358)(63 479 179 355)(64 476 180 360)(73 237 409 141)(74 234 410 138)(75 239 411 143)(76 236 412 140)(77 233 413 137)(78 238 414 142)(79 235 415 139)(80 240 416 144)(81 261 205 377)(82 258 206 382)(83 263 207 379)(84 260 208 384)(85 257 201 381)(86 262 202 378)(87 259 203 383)(88 264 204 380)(89 269 447 151)(90 266 448 148)(91 271 441 145)(92 268 442 150)(93 265 443 147)(94 270 444 152)(95 267 445 149)(96 272 446 146)(105 285 229 401)(106 282 230 406)(107 287 231 403)(108 284 232 408)(109 281 225 405)(110 286 226 402)(111 283 227 407)(112 288 228 404)(113 197 373 315)(114 194 374 320)(115 199 375 317)(116 196 376 314)(117 193 369 319)(118 198 370 316)(119 195 371 313)(120 200 372 318)(121 297 457 165)(122 302 458 162)(123 299 459 167)(124 304 460 164)(125 301 461 161)(126 298 462 166)(127 303 463 163)(128 300 464 168)(129 309 249 425)(130 306 250 430)(131 311 251 427)(132 308 252 432)(133 305 253 429)(134 310 254 426)(135 307 255 431)(136 312 256 428)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A 6B 6C 8A ··· 8H 10A ··· 10L 12A ··· 12L 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 30A ··· 30L 40A ··· 40AF 60A ··· 60AV order 1 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 S3 Dic3 D6 M4(2) C4×S3 C5×S3 C4.Dic3 C5×Dic3 S3×C10 C5×M4(2) S3×C20 C5×C4.Dic3 kernel C5×C42.S3 C10×C3⋊C8 C4×C60 C5×C3⋊C8 C2×C60 C42.S3 C2×C3⋊C8 C4×C12 C3⋊C8 C2×C12 C4×C20 C2×C20 C2×C20 C30 C20 C42 C10 C2×C4 C2×C4 C6 C4 C2 # reps 1 2 1 8 4 4 8 4 32 16 1 2 1 4 4 4 8 8 4 16 16 32

Matrix representation of C5×C42.S3 in GL3(𝔽241) generated by

 1 0 0 0 87 0 0 0 87
,
 1 0 0 0 0 240 0 1 1
,
 64 0 0 0 56 69 0 13 185
,
 177 0 0 0 99 198 0 43 142
G:=sub<GL(3,GF(241))| [1,0,0,0,87,0,0,0,87],[1,0,0,0,0,1,0,240,1],[64,0,0,0,56,13,0,69,185],[177,0,0,0,99,43,0,198,142] >;

C5×C42.S3 in GAP, Magma, Sage, TeX

C_5\times C_4^2.S_3
% in TeX

G:=Group("C5xC4^2.S3");
// GroupNames label

G:=SmallGroup(480,122);
// by ID

G=gap.SmallGroup(480,122);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,1149,288,136,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^6=d^4=1,c^4=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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