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G = Q82Dic15order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.9D4, Q82Dic15, C30.18Q16, C30.34SD16, (Q8×C15)⋊7C4, (C6×Q8).1D5, C60.82(C2×C4), (C5×Q8)⋊7Dic3, (C3×Q8)⋊1Dic5, (C2×C4).39D30, (C2×C20).74D6, C6.9(Q8⋊D5), (C2×Q8).3D15, (Q8×C30).1C2, (Q8×C10).5S3, (C2×C12).75D10, (C2×C30).141D4, C33(Q8⋊Dic5), C54(Q82Dic3), C605C4.12C2, C4.2(C2×Dic15), C12.8(C2×Dic5), C6.9(C5⋊Q16), C1517(Q8⋊C4), C20.21(C3⋊D4), C4.14(C157D4), C12.23(C5⋊D4), (C2×C60).60C22, C20.29(C2×Dic3), C2.3(C157Q16), C10.9(C3⋊Q16), C10.9(Q82S3), C2.3(Q82D15), C6.17(C23.D5), C30.105(C22⋊C4), C2.6(C30.38D4), C22.18(C157D4), C10.28(C6.D4), (C2×C153C8).5C2, (C2×C6).73(C5⋊D4), (C2×C10).73(C3⋊D4), SmallGroup(480,195)

Series: Derived Chief Lower central Upper central

C1C60 — Q82Dic15
C1C5C15C30C2×C30C2×C60C605C4 — Q82Dic15
C15C30C60 — Q82Dic15
C1C22C2×C4C2×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 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×2], Q8, C10 [×3], Dic3, C12 [×2], C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20 [×2], C20 [×2], C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C30 [×3], Q8⋊C4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C2×C3⋊C8, C4⋊Dic3, C6×Q8, Dic15, C60 [×2], C60 [×2], C2×C30, C2×C52C8, C4⋊Dic5, Q8×C10, Q82Dic3, C153C8, C2×Dic15, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, Q8⋊Dic5, C2×C153C8, C605C4, Q8×C30, Q82Dic15
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, SD16, Q16, Dic5 [×2], D10, C2×Dic3, C3⋊D4 [×2], D15, Q8⋊C4, C2×Dic5, C5⋊D4 [×2], Q82S3, C3⋊Q16, C6.D4, Dic15 [×2], D30, Q8⋊D5, C5⋊Q16, C23.D5, Q82Dic3, C2×Dic15, C157D4 [×2], Q8⋊Dic5, Q82D15, C157Q16, C30.38D4, Q82Dic15

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

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222344444455666888810···1012···121515151520···2030···3060···60
size111122244606022222303030302···24···422224···42···24···4

84 irreducible representations

dim111112222222222222222222444444
type+++++++++--+-++-+-+-+-
imageC1C2C2C2C4S3D4D4D5D6Dic3SD16Q16D10Dic5C3⋊D4C3⋊D4D15C5⋊D4C5⋊D4D30Dic15C157D4C157D4Q82S3C3⋊Q16Q8⋊D5C5⋊Q16Q82D15C157Q16
kernelQ82Dic15C2×C153C8C605C4Q8×C30Q8×C15Q8×C10C60C2×C30C6×Q8C2×C20C5×Q8C30C30C2×C12C3×Q8C20C2×C10C2×Q8C12C2×C6C2×C4Q8C4C22C10C10C6C6C2C2
# reps111141112122224224444888112244

Matrix representation of Q82Dic15 in GL5(𝔽241)

10000
01000
00100
000154
000116240
,
10000
01000
00100
0002152
00084239
,
2400000
02116300
01786800
00010
00001
,
640000
05719500
012318400
00018132
000171223

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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