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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 [×6], C3, C4 [×6], C22, C22 [×6], C5, C6, C6 [×6], C2×C4 [×6], C2×C4 [×6], C23, C10, C10 [×6], C12 [×6], C2×C6, C2×C6 [×6], C15, C22×C4 [×3], C20 [×6], C2×C10, C2×C10 [×6], C2×C12 [×6], C2×C12 [×6], C22×C6, C30, C30 [×6], C2.C42, C2×C20 [×6], C2×C20 [×6], C22×C10, C22×C12 [×3], C60 [×6], C2×C30, C2×C30 [×6], C22×C20 [×3], C3×C2.C42, C2×C60 [×6], C2×C60 [×6], C22×C30, C5×C2.C42, C22×C60 [×3], C15×C2.C42
Quotients: C1, C2 [×3], C3, C4 [×6], C22, C5, C6 [×3], C2×C4 [×3], D4 [×3], Q8, C10 [×3], C12 [×6], C2×C6, C15, C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, C2×C12 [×3], C3×D4 [×3], C3×Q8, C30 [×3], C2.C42, C2×C20 [×3], C5×D4 [×3], C5×Q8, C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C60 [×6], C2×C30, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C3×C2.C42, C2×C60 [×3], D4×C15 [×3], Q8×C15, C5×C2.C42, C4×C60, C15×C22⋊C4 [×3], C15×C4⋊C4 [×3], C15×C2.C42

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

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

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

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

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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