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## G = C15×C2.C42order 480 = 25·3·5

### Direct product of C15 and C2.C42

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C15×C2.C42
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C30 — C22×C60 — C15×C2.C42
 Lower central C1 — C2 — C15×C2.C42
 Upper central C1 — C22×C30 — C15×C2.C42

Generators and relations for C15×C2.C42
G = < a,b,c,d | a15=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 200 in 152 conjugacy classes, 104 normal (20 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, C20, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C2.C42, C2×C20, C2×C20, C22×C10, C22×C12, C60, C2×C30, C2×C30, C22×C20, C3×C2.C42, C2×C60, C2×C60, C22×C30, C5×C2.C42, C22×C60, C15×C2.C42
Quotients:

Smallest permutation representation of C15×C2.C42
Regular action on 480 points
Generators in S480
(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)
(1 384)(2 385)(3 386)(4 387)(5 388)(6 389)(7 390)(8 376)(9 377)(10 378)(11 379)(12 380)(13 381)(14 382)(15 383)(16 370)(17 371)(18 372)(19 373)(20 374)(21 375)(22 361)(23 362)(24 363)(25 364)(26 365)(27 366)(28 367)(29 368)(30 369)(31 217)(32 218)(33 219)(34 220)(35 221)(36 222)(37 223)(38 224)(39 225)(40 211)(41 212)(42 213)(43 214)(44 215)(45 216)(46 200)(47 201)(48 202)(49 203)(50 204)(51 205)(52 206)(53 207)(54 208)(55 209)(56 210)(57 196)(58 197)(59 198)(60 199)(61 398)(62 399)(63 400)(64 401)(65 402)(66 403)(67 404)(68 405)(69 391)(70 392)(71 393)(72 394)(73 395)(74 396)(75 397)(76 176)(77 177)(78 178)(79 179)(80 180)(81 166)(82 167)(83 168)(84 169)(85 170)(86 171)(87 172)(88 173)(89 174)(90 175)(91 278)(92 279)(93 280)(94 281)(95 282)(96 283)(97 284)(98 285)(99 271)(100 272)(101 273)(102 274)(103 275)(104 276)(105 277)(106 417)(107 418)(108 419)(109 420)(110 406)(111 407)(112 408)(113 409)(114 410)(115 411)(116 412)(117 413)(118 414)(119 415)(120 416)(121 307)(122 308)(123 309)(124 310)(125 311)(126 312)(127 313)(128 314)(129 315)(130 301)(131 302)(132 303)(133 304)(134 305)(135 306)(136 246)(137 247)(138 248)(139 249)(140 250)(141 251)(142 252)(143 253)(144 254)(145 255)(146 241)(147 242)(148 243)(149 244)(150 245)(151 425)(152 426)(153 427)(154 428)(155 429)(156 430)(157 431)(158 432)(159 433)(160 434)(161 435)(162 421)(163 422)(164 423)(165 424)(181 341)(182 342)(183 343)(184 344)(185 345)(186 331)(187 332)(188 333)(189 334)(190 335)(191 336)(192 337)(193 338)(194 339)(195 340)(226 257)(227 258)(228 259)(229 260)(230 261)(231 262)(232 263)(233 264)(234 265)(235 266)(236 267)(237 268)(238 269)(239 270)(240 256)(286 346)(287 347)(288 348)(289 349)(290 350)(291 351)(292 352)(293 353)(294 354)(295 355)(296 356)(297 357)(298 358)(299 359)(300 360)(316 453)(317 454)(318 455)(319 456)(320 457)(321 458)(322 459)(323 460)(324 461)(325 462)(326 463)(327 464)(328 465)(329 451)(330 452)(436 466)(437 467)(438 468)(439 469)(440 470)(441 471)(442 472)(443 473)(444 474)(445 475)(446 476)(447 477)(448 478)(449 479)(450 480)
(1 203 396 312)(2 204 397 313)(3 205 398 314)(4 206 399 315)(5 207 400 301)(6 208 401 302)(7 209 402 303)(8 210 403 304)(9 196 404 305)(10 197 405 306)(11 198 391 307)(12 199 392 308)(13 200 393 309)(14 201 394 310)(15 202 395 311)(16 44 154 356)(17 45 155 357)(18 31 156 358)(19 32 157 359)(20 33 158 360)(21 34 159 346)(22 35 160 347)(23 36 161 348)(24 37 162 349)(25 38 163 350)(26 39 164 351)(27 40 165 352)(28 41 151 353)(29 42 152 354)(30 43 153 355)(46 71 123 381)(47 72 124 382)(48 73 125 383)(49 74 126 384)(50 75 127 385)(51 61 128 386)(52 62 129 387)(53 63 130 388)(54 64 131 389)(55 65 132 390)(56 66 133 376)(57 67 134 377)(58 68 135 378)(59 69 121 379)(60 70 122 380)(76 114 466 454)(77 115 467 455)(78 116 468 456)(79 117 469 457)(80 118 470 458)(81 119 471 459)(82 120 472 460)(83 106 473 461)(84 107 474 462)(85 108 475 463)(86 109 476 464)(87 110 477 465)(88 111 478 451)(89 112 479 452)(90 113 480 453)(91 235 145 193)(92 236 146 194)(93 237 147 195)(94 238 148 181)(95 239 149 182)(96 240 150 183)(97 226 136 184)(98 227 137 185)(99 228 138 186)(100 229 139 187)(101 230 140 188)(102 231 141 189)(103 232 142 190)(104 233 143 191)(105 234 144 192)(166 415 441 322)(167 416 442 323)(168 417 443 324)(169 418 444 325)(170 419 445 326)(171 420 446 327)(172 406 447 328)(173 407 448 329)(174 408 449 330)(175 409 450 316)(176 410 436 317)(177 411 437 318)(178 412 438 319)(179 413 439 320)(180 414 440 321)(211 424 292 366)(212 425 293 367)(213 426 294 368)(214 427 295 369)(215 428 296 370)(216 429 297 371)(217 430 298 372)(218 431 299 373)(219 432 300 374)(220 433 286 375)(221 434 287 361)(222 435 288 362)(223 421 289 363)(224 422 290 364)(225 423 291 365)(241 339 279 267)(242 340 280 268)(243 341 281 269)(244 342 282 270)(245 343 283 256)(246 344 284 257)(247 345 285 258)(248 331 271 259)(249 332 272 260)(250 333 273 261)(251 334 274 262)(252 335 275 263)(253 336 276 264)(254 337 277 265)(255 338 278 266)
(1 291 411 150)(2 292 412 136)(3 293 413 137)(4 294 414 138)(5 295 415 139)(6 296 416 140)(7 297 417 141)(8 298 418 142)(9 299 419 143)(10 300 420 144)(11 286 406 145)(12 287 407 146)(13 288 408 147)(14 289 409 148)(15 290 410 149)(16 442 333 208)(17 443 334 209)(18 444 335 210)(19 445 336 196)(20 446 337 197)(21 447 338 198)(22 448 339 199)(23 449 340 200)(24 450 341 201)(25 436 342 202)(26 437 343 203)(27 438 344 204)(28 439 345 205)(29 440 331 206)(30 441 332 207)(31 462 275 66)(32 463 276 67)(33 464 277 68)(34 465 278 69)(35 451 279 70)(36 452 280 71)(37 453 281 72)(38 454 282 73)(39 455 283 74)(40 456 284 75)(41 457 285 61)(42 458 271 62)(43 459 272 63)(44 460 273 64)(45 461 274 65)(46 362 479 195)(47 363 480 181)(48 364 466 182)(49 365 467 183)(50 366 468 184)(51 367 469 185)(52 368 470 186)(53 369 471 187)(54 370 472 188)(55 371 473 189)(56 372 474 190)(57 373 475 191)(58 374 476 192)(59 375 477 193)(60 361 478 194)(76 239 125 422)(77 240 126 423)(78 226 127 424)(79 227 128 425)(80 228 129 426)(81 229 130 427)(82 230 131 428)(83 231 132 429)(84 232 133 430)(85 233 134 431)(86 234 135 432)(87 235 121 433)(88 236 122 434)(89 237 123 435)(90 238 124 421)(91 391 220 328)(92 392 221 329)(93 393 222 330)(94 394 223 316)(95 395 224 317)(96 396 225 318)(97 397 211 319)(98 398 212 320)(99 399 213 321)(100 400 214 322)(101 401 215 323)(102 402 216 324)(103 403 217 325)(104 404 218 326)(105 405 219 327)(106 251 390 357)(107 252 376 358)(108 253 377 359)(109 254 378 360)(110 255 379 346)(111 241 380 347)(112 242 381 348)(113 243 382 349)(114 244 383 350)(115 245 384 351)(116 246 385 352)(117 247 386 353)(118 248 387 354)(119 249 388 355)(120 250 389 356)(151 179 258 314)(152 180 259 315)(153 166 260 301)(154 167 261 302)(155 168 262 303)(156 169 263 304)(157 170 264 305)(158 171 265 306)(159 172 266 307)(160 173 267 308)(161 174 268 309)(162 175 269 310)(163 176 270 311)(164 177 256 312)(165 178 257 313)

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

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

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

300 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4L 5A 5B 5C 5D 6A ··· 6N 10A ··· 10AB 12A ··· 12X 15A ··· 15H 20A ··· 20AV 30A ··· 30BD 60A ··· 60CR order 1 2 ··· 2 3 3 4 ··· 4 5 5 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

300 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C5 C6 C10 C12 C15 C20 C30 C60 D4 Q8 C3×D4 C3×Q8 C5×D4 C5×Q8 D4×C15 Q8×C15 kernel C15×C2.C42 C22×C60 C5×C2.C42 C2×C60 C3×C2.C42 C22×C20 C22×C12 C2×C20 C2.C42 C2×C12 C22×C4 C2×C4 C2×C30 C2×C30 C2×C10 C2×C10 C2×C6 C2×C6 C22 C22 # reps 1 3 2 12 4 6 12 24 8 48 24 96 3 1 6 2 12 4 24 8

Matrix representation of C15×C2.C42 in GL4(𝔽61) generated by

 13 0 0 0 0 1 0 0 0 0 20 0 0 0 0 20
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 11 0 0 0 0 60 0 0 0 0 50 0 0 0 0 11
,
 60 0 0 0 0 50 0 0 0 0 0 1 0 0 60 0
G:=sub<GL(4,GF(61))| [13,0,0,0,0,1,0,0,0,0,20,0,0,0,0,20],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[11,0,0,0,0,60,0,0,0,0,50,0,0,0,0,11],[60,0,0,0,0,50,0,0,0,0,0,60,0,0,1,0] >;

C15×C2.C42 in GAP, Magma, Sage, TeX

C_{15}\times C_2.C_4^2
% in TeX

G:=Group("C15xC2.C4^2");
// GroupNames label

G:=SmallGroup(480,198);
// by ID

G=gap.SmallGroup(480,198);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,1688]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations

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