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## G = Q8⋊2Dic15order 480 = 25·3·5

### 1st semidirect product of Q8 and Dic15 acting via Dic15/C30=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Q8⋊2Dic15
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C60⋊5C4 — Q8⋊2Dic15
 Lower central C15 — C30 — C60 — Q8⋊2Dic15
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for Q82Dic15
G = < a,b,c,d | a4=c30=1, b2=a2, d2=c15, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 340 in 84 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, Q8, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C30, Q8⋊C4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×C3⋊C8, C4⋊Dic3, C6×Q8, Dic15, C60, C60, C2×C30, C2×C52C8, C4⋊Dic5, Q8×C10, Q82Dic3, C153C8, C2×Dic15, C2×C60, C2×C60, Q8×C15, Q8×C15, Q8⋊Dic5, C2×C153C8, C605C4, Q8×C30, Q82Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, SD16, Q16, Dic5, D10, C2×Dic3, C3⋊D4, D15, Q8⋊C4, C2×Dic5, C5⋊D4, Q82S3, C3⋊Q16, C6.D4, Dic15, D30, Q8⋊D5, C5⋊Q16, C23.D5, Q82Dic3, C2×Dic15, C157D4, Q8⋊Dic5, Q82D15, C157Q16, C30.38D4, Q82Dic15

Smallest permutation representation of Q82Dic15
Regular action on 480 points
Generators in S480
(1 386 302 245)(2 387 303 246)(3 388 304 247)(4 389 305 248)(5 390 306 249)(6 361 307 250)(7 362 308 251)(8 363 309 252)(9 364 310 253)(10 365 311 254)(11 366 312 255)(12 367 313 256)(13 368 314 257)(14 369 315 258)(15 370 316 259)(16 371 317 260)(17 372 318 261)(18 373 319 262)(19 374 320 263)(20 375 321 264)(21 376 322 265)(22 377 323 266)(23 378 324 267)(24 379 325 268)(25 380 326 269)(26 381 327 270)(27 382 328 241)(28 383 329 242)(29 384 330 243)(30 385 301 244)(31 214 446 355)(32 215 447 356)(33 216 448 357)(34 217 449 358)(35 218 450 359)(36 219 421 360)(37 220 422 331)(38 221 423 332)(39 222 424 333)(40 223 425 334)(41 224 426 335)(42 225 427 336)(43 226 428 337)(44 227 429 338)(45 228 430 339)(46 229 431 340)(47 230 432 341)(48 231 433 342)(49 232 434 343)(50 233 435 344)(51 234 436 345)(52 235 437 346)(53 236 438 347)(54 237 439 348)(55 238 440 349)(56 239 441 350)(57 240 442 351)(58 211 443 352)(59 212 444 353)(60 213 445 354)(61 279 118 411)(62 280 119 412)(63 281 120 413)(64 282 91 414)(65 283 92 415)(66 284 93 416)(67 285 94 417)(68 286 95 418)(69 287 96 419)(70 288 97 420)(71 289 98 391)(72 290 99 392)(73 291 100 393)(74 292 101 394)(75 293 102 395)(76 294 103 396)(77 295 104 397)(78 296 105 398)(79 297 106 399)(80 298 107 400)(81 299 108 401)(82 300 109 402)(83 271 110 403)(84 272 111 404)(85 273 112 405)(86 274 113 406)(87 275 114 407)(88 276 115 408)(89 277 116 409)(90 278 117 410)(121 455 202 179)(122 456 203 180)(123 457 204 151)(124 458 205 152)(125 459 206 153)(126 460 207 154)(127 461 208 155)(128 462 209 156)(129 463 210 157)(130 464 181 158)(131 465 182 159)(132 466 183 160)(133 467 184 161)(134 468 185 162)(135 469 186 163)(136 470 187 164)(137 471 188 165)(138 472 189 166)(139 473 190 167)(140 474 191 168)(141 475 192 169)(142 476 193 170)(143 477 194 171)(144 478 195 172)(145 479 196 173)(146 480 197 174)(147 451 198 175)(148 452 199 176)(149 453 200 177)(150 454 201 178)
(1 50 302 435)(2 51 303 436)(3 52 304 437)(4 53 305 438)(5 54 306 439)(6 55 307 440)(7 56 308 441)(8 57 309 442)(9 58 310 443)(10 59 311 444)(11 60 312 445)(12 31 313 446)(13 32 314 447)(14 33 315 448)(15 34 316 449)(16 35 317 450)(17 36 318 421)(18 37 319 422)(19 38 320 423)(20 39 321 424)(21 40 322 425)(22 41 323 426)(23 42 324 427)(24 43 325 428)(25 44 326 429)(26 45 327 430)(27 46 328 431)(28 47 329 432)(29 48 330 433)(30 49 301 434)(61 194 118 143)(62 195 119 144)(63 196 120 145)(64 197 91 146)(65 198 92 147)(66 199 93 148)(67 200 94 149)(68 201 95 150)(69 202 96 121)(70 203 97 122)(71 204 98 123)(72 205 99 124)(73 206 100 125)(74 207 101 126)(75 208 102 127)(76 209 103 128)(77 210 104 129)(78 181 105 130)(79 182 106 131)(80 183 107 132)(81 184 108 133)(82 185 109 134)(83 186 110 135)(84 187 111 136)(85 188 112 137)(86 189 113 138)(87 190 114 139)(88 191 115 140)(89 192 116 141)(90 193 117 142)(151 289 457 391)(152 290 458 392)(153 291 459 393)(154 292 460 394)(155 293 461 395)(156 294 462 396)(157 295 463 397)(158 296 464 398)(159 297 465 399)(160 298 466 400)(161 299 467 401)(162 300 468 402)(163 271 469 403)(164 272 470 404)(165 273 471 405)(166 274 472 406)(167 275 473 407)(168 276 474 408)(169 277 475 409)(170 278 476 410)(171 279 477 411)(172 280 478 412)(173 281 479 413)(174 282 480 414)(175 283 451 415)(176 284 452 416)(177 285 453 417)(178 286 454 418)(179 287 455 419)(180 288 456 420)(211 364 352 253)(212 365 353 254)(213 366 354 255)(214 367 355 256)(215 368 356 257)(216 369 357 258)(217 370 358 259)(218 371 359 260)(219 372 360 261)(220 373 331 262)(221 374 332 263)(222 375 333 264)(223 376 334 265)(224 377 335 266)(225 378 336 267)(226 379 337 268)(227 380 338 269)(228 381 339 270)(229 382 340 241)(230 383 341 242)(231 384 342 243)(232 385 343 244)(233 386 344 245)(234 387 345 246)(235 388 346 247)(236 389 347 248)(237 390 348 249)(238 361 349 250)(239 362 350 251)(240 363 351 252)
(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)
(1 278 16 293)(2 277 17 292)(3 276 18 291)(4 275 19 290)(5 274 20 289)(6 273 21 288)(7 272 22 287)(8 271 23 286)(9 300 24 285)(10 299 25 284)(11 298 26 283)(12 297 27 282)(13 296 28 281)(14 295 29 280)(15 294 30 279)(31 182 46 197)(32 181 47 196)(33 210 48 195)(34 209 49 194)(35 208 50 193)(36 207 51 192)(37 206 52 191)(38 205 53 190)(39 204 54 189)(40 203 55 188)(41 202 56 187)(42 201 57 186)(43 200 58 185)(44 199 59 184)(45 198 60 183)(61 370 76 385)(62 369 77 384)(63 368 78 383)(64 367 79 382)(65 366 80 381)(66 365 81 380)(67 364 82 379)(68 363 83 378)(69 362 84 377)(70 361 85 376)(71 390 86 375)(72 389 87 374)(73 388 88 373)(74 387 89 372)(75 386 90 371)(91 256 106 241)(92 255 107 270)(93 254 108 269)(94 253 109 268)(95 252 110 267)(96 251 111 266)(97 250 112 265)(98 249 113 264)(99 248 114 263)(100 247 115 262)(101 246 116 261)(102 245 117 260)(103 244 118 259)(104 243 119 258)(105 242 120 257)(121 441 136 426)(122 440 137 425)(123 439 138 424)(124 438 139 423)(125 437 140 422)(126 436 141 421)(127 435 142 450)(128 434 143 449)(129 433 144 448)(130 432 145 447)(131 431 146 446)(132 430 147 445)(133 429 148 444)(134 428 149 443)(135 427 150 442)(151 348 166 333)(152 347 167 332)(153 346 168 331)(154 345 169 360)(155 344 170 359)(156 343 171 358)(157 342 172 357)(158 341 173 356)(159 340 174 355)(160 339 175 354)(161 338 176 353)(162 337 177 352)(163 336 178 351)(164 335 179 350)(165 334 180 349)(211 468 226 453)(212 467 227 452)(213 466 228 451)(214 465 229 480)(215 464 230 479)(216 463 231 478)(217 462 232 477)(218 461 233 476)(219 460 234 475)(220 459 235 474)(221 458 236 473)(222 457 237 472)(223 456 238 471)(224 455 239 470)(225 454 240 469)(301 411 316 396)(302 410 317 395)(303 409 318 394)(304 408 319 393)(305 407 320 392)(306 406 321 391)(307 405 322 420)(308 404 323 419)(309 403 324 418)(310 402 325 417)(311 401 326 416)(312 400 327 415)(313 399 328 414)(314 398 329 413)(315 397 330 412)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 4 4 60 60 2 2 2 2 2 30 30 30 30 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + - - + - + + - + - + - + - image C1 C2 C2 C2 C4 S3 D4 D4 D5 D6 Dic3 SD16 Q16 D10 Dic5 C3⋊D4 C3⋊D4 D15 C5⋊D4 C5⋊D4 D30 Dic15 C15⋊7D4 C15⋊7D4 Q8⋊2S3 C3⋊Q16 Q8⋊D5 C5⋊Q16 Q8⋊2D15 C15⋊7Q16 kernel Q8⋊2Dic15 C2×C15⋊3C8 C60⋊5C4 Q8×C30 Q8×C15 Q8×C10 C60 C2×C30 C6×Q8 C2×C20 C5×Q8 C30 C30 C2×C12 C3×Q8 C20 C2×C10 C2×Q8 C12 C2×C6 C2×C4 Q8 C4 C22 C10 C10 C6 C6 C2 C2 # reps 1 1 1 1 4 1 1 1 2 1 2 2 2 2 4 2 2 4 4 4 4 8 8 8 1 1 2 2 4 4

Matrix representation of Q82Dic15 in GL5(𝔽241)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 54 0 0 0 116 240
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 2 152 0 0 0 84 239
,
 240 0 0 0 0 0 211 63 0 0 0 178 68 0 0 0 0 0 1 0 0 0 0 0 1
,
 64 0 0 0 0 0 57 195 0 0 0 123 184 0 0 0 0 0 18 132 0 0 0 171 223

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,116,0,0,0,54,240],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,84,0,0,0,152,239],[240,0,0,0,0,0,211,178,0,0,0,63,68,0,0,0,0,0,1,0,0,0,0,0,1],[64,0,0,0,0,0,57,123,0,0,0,195,184,0,0,0,0,0,18,171,0,0,0,132,223] >;

Q82Dic15 in GAP, Magma, Sage, TeX

Q_8\rtimes_2{\rm Dic}_{15}
% in TeX

G:=Group("Q8:2Dic15");
// GroupNames label

G:=SmallGroup(480,195);
// by ID

G=gap.SmallGroup(480,195);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,675,346,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^30=1,b^2=a^2,d^2=c^15,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

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