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## G = C3×C10.10C42order 480 = 25·3·5

### Direct product of C3 and C10.10C42

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C10.10C42
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×C30 — C2×C6×Dic5 — C3×C10.10C42
 Lower central C5 — C10 — C3×C10.10C42
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C3×C10.10C42
G = < a,b,c,d | a3=b10=c4=d4=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b5c >

Subgroups: 384 in 152 conjugacy classes, 90 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, C12, C2×C6, C2×C6, C15, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C2.C42, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C12, C22×C12, C3×Dic5, C60, C2×C30, C2×C30, C22×Dic5, C22×C20, C3×C2.C42, C6×Dic5, C6×Dic5, C2×C60, C2×C60, C22×C30, C10.10C42, C2×C6×Dic5, C22×C60, C3×C10.10C42
Quotients:

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

156 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A 4B 4C 4D 4E ··· 4L 5A 5B 6A ··· 6N 10A ··· 10N 12A ··· 12H 12I ··· 12X 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 3 3 4 4 4 4 4 ··· 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 1 1 2 2 2 2 10 ··· 10 2 2 1 ··· 1 2 ··· 2 2 ··· 2 10 ··· 10 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

156 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 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 Q8 D5 Dic5 D10 C3×D4 C3×Q8 C3×D5 Dic10 C4×D5 D20 C5⋊D4 C3×Dic5 C6×D5 C3×Dic10 D5×C12 C3×D20 C3×C5⋊D4 kernel C3×C10.10C42 C2×C6×Dic5 C22×C60 C10.10C42 C6×Dic5 C2×C60 C22×Dic5 C22×C20 C2×Dic5 C2×C20 C2×C30 C2×C30 C22×C12 C2×C12 C22×C6 C2×C10 C2×C10 C22×C4 C2×C6 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 C22 # reps 1 2 1 2 8 4 4 2 16 8 3 1 2 4 2 6 2 4 4 8 4 8 8 4 8 16 8 16

Matrix representation of C3×C10.10C42 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 47 0 0 0 0 0 47
,
 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 3 0 0 0 0 26 41
,
 11 0 0 0 0 0 41 9 0 0 0 37 20 0 0 0 0 0 29 33 0 0 0 17 32
,
 11 0 0 0 0 0 50 59 0 0 0 0 11 0 0 0 0 0 50 0 0 0 0 60 11

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,47,0,0,0,0,0,47],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,3,26,0,0,0,0,41],[11,0,0,0,0,0,41,37,0,0,0,9,20,0,0,0,0,0,29,17,0,0,0,33,32],[11,0,0,0,0,0,50,0,0,0,0,59,11,0,0,0,0,0,50,60,0,0,0,0,11] >;

C3×C10.10C42 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC10.10C4^2");
// GroupNames label

G:=SmallGroup(480,109);
// by ID

G=gap.SmallGroup(480,109);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,176,18822]);
// Polycyclic

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

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