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## G = C3×C10.C42order 480 = 25·3·5

### Direct product of C3 and C10.C42

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C10.C42
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C6×Dic5 — C6×C5⋊C8 — C3×C10.C42
 Lower central C5 — C10 — C3×C10.C42
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×C10.C42
G = < a,b,c,d | a3=b10=d4=1, c4=b5, ab=ba, ac=ca, ad=da, cbc-1=b3, bd=db, dcd-1=b5c >

Subgroups: 200 in 80 conjugacy classes, 48 normal (32 characteristic)
C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C30, C8⋊C4, C5⋊C8, C2×Dic5, C2×C20, C4×C12, C2×C24, C3×Dic5, C3×Dic5, C60, C2×C30, C4×Dic5, C2×C5⋊C8, C3×C8⋊C4, C3×C5⋊C8, C6×Dic5, C2×C60, C10.C42, C12×Dic5, C6×C5⋊C8, C3×C10.C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, M4(2), F5, C2×C12, C8⋊C4, C2×F5, C4×C12, C3×M4(2), C3×F5, C4.F5, C4×F5, C22.F5, C3×C8⋊C4, C6×F5, C10.C42, C3×C4.F5, C12×F5, C3×C22.F5, C3×C10.C42

Smallest permutation representation of C3×C10.C42
Regular action on 480 points
Generators in S480
(1 87 47)(2 88 48)(3 89 49)(4 90 50)(5 81 41)(6 82 42)(7 83 43)(8 84 44)(9 85 45)(10 86 46)(11 91 51)(12 92 52)(13 93 53)(14 94 54)(15 95 55)(16 96 56)(17 97 57)(18 98 58)(19 99 59)(20 100 60)(21 101 61)(22 102 62)(23 103 63)(24 104 64)(25 105 65)(26 106 66)(27 107 67)(28 108 68)(29 109 69)(30 110 70)(31 111 71)(32 112 72)(33 113 73)(34 114 74)(35 115 75)(36 116 76)(37 117 77)(38 118 78)(39 119 79)(40 120 80)(121 201 161)(122 202 162)(123 203 163)(124 204 164)(125 205 165)(126 206 166)(127 207 167)(128 208 168)(129 209 169)(130 210 170)(131 211 171)(132 212 172)(133 213 173)(134 214 174)(135 215 175)(136 216 176)(137 217 177)(138 218 178)(139 219 179)(140 220 180)(141 221 181)(142 222 182)(143 223 183)(144 224 184)(145 225 185)(146 226 186)(147 227 187)(148 228 188)(149 229 189)(150 230 190)(151 231 191)(152 232 192)(153 233 193)(154 234 194)(155 235 195)(156 236 196)(157 237 197)(158 238 198)(159 239 199)(160 240 200)(241 321 281)(242 322 282)(243 323 283)(244 324 284)(245 325 285)(246 326 286)(247 327 287)(248 328 288)(249 329 289)(250 330 290)(251 331 291)(252 332 292)(253 333 293)(254 334 294)(255 335 295)(256 336 296)(257 337 297)(258 338 298)(259 339 299)(260 340 300)(261 341 301)(262 342 302)(263 343 303)(264 344 304)(265 345 305)(266 346 306)(267 347 307)(268 348 308)(269 349 309)(270 350 310)(271 351 311)(272 352 312)(273 353 313)(274 354 314)(275 355 315)(276 356 316)(277 357 317)(278 358 318)(279 359 319)(280 360 320)(361 441 401)(362 442 402)(363 443 403)(364 444 404)(365 445 405)(366 446 406)(367 447 407)(368 448 408)(369 449 409)(370 450 410)(371 451 411)(372 452 412)(373 453 413)(374 454 414)(375 455 415)(376 456 416)(377 457 417)(378 458 418)(379 459 419)(380 460 420)(381 461 421)(382 462 422)(383 463 423)(384 464 424)(385 465 425)(386 466 426)(387 467 427)(388 468 428)(389 469 429)(390 470 430)(391 471 431)(392 472 432)(393 473 433)(394 474 434)(395 475 435)(396 476 436)(397 477 437)(398 478 438)(399 479 439)(400 480 440)
(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)
(1 264 28 246 6 269 23 241)(2 261 27 249 7 266 22 244)(3 268 26 242 8 263 21 247)(4 265 25 245 9 270 30 250)(5 262 24 248 10 267 29 243)(11 272 34 258 16 277 39 253)(12 279 33 251 17 274 38 256)(13 276 32 254 18 271 37 259)(14 273 31 257 19 278 36 252)(15 280 40 260 20 275 35 255)(41 302 64 288 46 307 69 283)(42 309 63 281 47 304 68 286)(43 306 62 284 48 301 67 289)(44 303 61 287 49 308 66 282)(45 310 70 290 50 305 65 285)(51 312 74 298 56 317 79 293)(52 319 73 291 57 314 78 296)(53 316 72 294 58 311 77 299)(54 313 71 297 59 318 76 292)(55 320 80 300 60 315 75 295)(81 342 104 328 86 347 109 323)(82 349 103 321 87 344 108 326)(83 346 102 324 88 341 107 329)(84 343 101 327 89 348 106 322)(85 350 110 330 90 345 105 325)(91 352 114 338 96 357 119 333)(92 359 113 331 97 354 118 336)(93 356 112 334 98 351 117 339)(94 353 111 337 99 358 116 332)(95 360 120 340 100 355 115 335)(121 382 144 368 126 387 149 363)(122 389 143 361 127 384 148 366)(123 386 142 364 128 381 147 369)(124 383 141 367 129 388 146 362)(125 390 150 370 130 385 145 365)(131 392 154 378 136 397 159 373)(132 399 153 371 137 394 158 376)(133 396 152 374 138 391 157 379)(134 393 151 377 139 398 156 372)(135 400 160 380 140 395 155 375)(161 422 184 408 166 427 189 403)(162 429 183 401 167 424 188 406)(163 426 182 404 168 421 187 409)(164 423 181 407 169 428 186 402)(165 430 190 410 170 425 185 405)(171 432 194 418 176 437 199 413)(172 439 193 411 177 434 198 416)(173 436 192 414 178 431 197 419)(174 433 191 417 179 438 196 412)(175 440 200 420 180 435 195 415)(201 462 224 448 206 467 229 443)(202 469 223 441 207 464 228 446)(203 466 222 444 208 461 227 449)(204 463 221 447 209 468 226 442)(205 470 230 450 210 465 225 445)(211 472 234 458 216 477 239 453)(212 479 233 451 217 474 238 456)(213 476 232 454 218 471 237 459)(214 473 231 457 219 478 236 452)(215 480 240 460 220 475 235 455)
(1 137 17 127)(2 138 18 128)(3 139 19 129)(4 140 20 130)(5 131 11 121)(6 132 12 122)(7 133 13 123)(8 134 14 124)(9 135 15 125)(10 136 16 126)(21 151 31 141)(22 152 32 142)(23 153 33 143)(24 154 34 144)(25 155 35 145)(26 156 36 146)(27 157 37 147)(28 158 38 148)(29 159 39 149)(30 160 40 150)(41 171 51 161)(42 172 52 162)(43 173 53 163)(44 174 54 164)(45 175 55 165)(46 176 56 166)(47 177 57 167)(48 178 58 168)(49 179 59 169)(50 180 60 170)(61 191 71 181)(62 192 72 182)(63 193 73 183)(64 194 74 184)(65 195 75 185)(66 196 76 186)(67 197 77 187)(68 198 78 188)(69 199 79 189)(70 200 80 190)(81 211 91 201)(82 212 92 202)(83 213 93 203)(84 214 94 204)(85 215 95 205)(86 216 96 206)(87 217 97 207)(88 218 98 208)(89 219 99 209)(90 220 100 210)(101 231 111 221)(102 232 112 222)(103 233 113 223)(104 234 114 224)(105 235 115 225)(106 236 116 226)(107 237 117 227)(108 238 118 228)(109 239 119 229)(110 240 120 230)(241 376 251 366)(242 377 252 367)(243 378 253 368)(244 379 254 369)(245 380 255 370)(246 371 256 361)(247 372 257 362)(248 373 258 363)(249 374 259 364)(250 375 260 365)(261 396 271 386)(262 397 272 387)(263 398 273 388)(264 399 274 389)(265 400 275 390)(266 391 276 381)(267 392 277 382)(268 393 278 383)(269 394 279 384)(270 395 280 385)(281 416 291 406)(282 417 292 407)(283 418 293 408)(284 419 294 409)(285 420 295 410)(286 411 296 401)(287 412 297 402)(288 413 298 403)(289 414 299 404)(290 415 300 405)(301 436 311 426)(302 437 312 427)(303 438 313 428)(304 439 314 429)(305 440 315 430)(306 431 316 421)(307 432 317 422)(308 433 318 423)(309 434 319 424)(310 435 320 425)(321 456 331 446)(322 457 332 447)(323 458 333 448)(324 459 334 449)(325 460 335 450)(326 451 336 441)(327 452 337 442)(328 453 338 443)(329 454 339 444)(330 455 340 445)(341 476 351 466)(342 477 352 467)(343 478 353 468)(344 479 354 469)(345 480 355 470)(346 471 356 461)(347 472 357 462)(348 473 358 463)(349 474 359 464)(350 475 360 465)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5 6A ··· 6F 8A ··· 8H 10A 10B 10C 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 12P 15A 15B 20A 20B 20C 20D 24A ··· 24P 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 6 ··· 6 8 ··· 8 10 10 10 12 12 12 12 12 ··· 12 12 12 12 12 15 15 20 20 20 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 2 2 5 5 5 5 10 10 4 1 ··· 1 10 ··· 10 4 4 4 2 2 2 2 5 ··· 5 10 10 10 10 4 4 4 4 4 4 10 ··· 10 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - image C1 C2 C2 C3 C4 C4 C4 C6 C6 C12 C12 C12 M4(2) C3×M4(2) F5 C2×F5 C3×F5 C4.F5 C4×F5 C22.F5 C6×F5 C3×C4.F5 C12×F5 C3×C22.F5 kernel C3×C10.C42 C12×Dic5 C6×C5⋊C8 C10.C42 C3×C5⋊C8 C6×Dic5 C2×C60 C4×Dic5 C2×C5⋊C8 C5⋊C8 C2×Dic5 C2×C20 C30 C10 C2×C12 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 2 2 8 2 2 2 4 16 4 4 4 8 1 1 2 2 2 2 2 4 4 4

Matrix representation of C3×C10.C42 in GL8(𝔽241)

 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 15
,
 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 240 1 0 0 0 0 0 0 240 0 1 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0
,
 149 184 0 0 0 0 0 0 27 92 0 0 0 0 0 0 0 0 223 24 0 0 0 0 0 0 24 18 0 0 0 0 0 0 0 0 102 62 43 18 0 0 0 0 145 80 59 120 0 0 0 0 161 182 121 163 0 0 0 0 223 225 139 179
,
 3 231 0 0 0 0 0 0 49 238 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240

G:=sub<GL(8,GF(241))| [15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,15],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[149,27,0,0,0,0,0,0,184,92,0,0,0,0,0,0,0,0,223,24,0,0,0,0,0,0,24,18,0,0,0,0,0,0,0,0,102,145,161,223,0,0,0,0,62,80,182,225,0,0,0,0,43,59,121,139,0,0,0,0,18,120,163,179],[3,49,0,0,0,0,0,0,231,238,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240] >;

C3×C10.C42 in GAP, Magma, Sage, TeX

C_3\times C_{10}.C_4^2
% in TeX

G:=Group("C3xC10.C4^2");
// GroupNames label

G:=SmallGroup(480,282);
// by ID

G=gap.SmallGroup(480,282);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,176,136,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^10=d^4=1,c^4=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations

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