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## G = Q8×Dic15order 480 = 25·3·5

### Direct product of Q8 and Dic15

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×Dic15
G = < a,b,c,d | a4=c30=1, b2=a2, d2=c15, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 532 in 140 conjugacy classes, 89 normal (26 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, C2×C10, C2×Dic3, C2×C12, C3×Q8, C30, C4×Q8, C2×Dic5, C2×C20, C5×Q8, C4×Dic3, C4⋊Dic3, C6×Q8, Dic15, Dic15, C60, C2×C30, C4×Dic5, C4⋊Dic5, Q8×C10, Q8×Dic3, C2×Dic15, C2×Dic15, C2×C60, Q8×C15, Q8×Dic5, C4×Dic15, C605C4, Q8×C30, Q8×Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, Dic3, D6, C22×C4, C2×Q8, C4○D4, Dic5, D10, C2×Dic3, C22×S3, D15, C4×Q8, C2×Dic5, C22×D5, S3×Q8, Q83S3, C22×Dic3, Dic15, D30, Q8×D5, Q82D5, C22×Dic5, Q8×Dic3, C2×Dic15, C22×D15, Q8×Dic5, Q8×D15, Q83D15, C22×Dic15, Q8×Dic15

Smallest permutation representation of Q8×Dic15
Regular action on 480 points
Generators in S480
(1 342 195 461)(2 343 196 462)(3 344 197 463)(4 345 198 464)(5 346 199 465)(6 347 200 466)(7 348 201 467)(8 349 202 468)(9 350 203 469)(10 351 204 470)(11 352 205 471)(12 353 206 472)(13 354 207 473)(14 355 208 474)(15 356 209 475)(16 357 210 476)(17 358 181 477)(18 359 182 478)(19 360 183 479)(20 331 184 480)(21 332 185 451)(22 333 186 452)(23 334 187 453)(24 335 188 454)(25 336 189 455)(26 337 190 456)(27 338 191 457)(28 339 192 458)(29 340 193 459)(30 341 194 460)(31 371 171 87)(32 372 172 88)(33 373 173 89)(34 374 174 90)(35 375 175 61)(36 376 176 62)(37 377 177 63)(38 378 178 64)(39 379 179 65)(40 380 180 66)(41 381 151 67)(42 382 152 68)(43 383 153 69)(44 384 154 70)(45 385 155 71)(46 386 156 72)(47 387 157 73)(48 388 158 74)(49 389 159 75)(50 390 160 76)(51 361 161 77)(52 362 162 78)(53 363 163 79)(54 364 164 80)(55 365 165 81)(56 366 166 82)(57 367 167 83)(58 368 168 84)(59 369 169 85)(60 370 170 86)(91 215 299 126)(92 216 300 127)(93 217 271 128)(94 218 272 129)(95 219 273 130)(96 220 274 131)(97 221 275 132)(98 222 276 133)(99 223 277 134)(100 224 278 135)(101 225 279 136)(102 226 280 137)(103 227 281 138)(104 228 282 139)(105 229 283 140)(106 230 284 141)(107 231 285 142)(108 232 286 143)(109 233 287 144)(110 234 288 145)(111 235 289 146)(112 236 290 147)(113 237 291 148)(114 238 292 149)(115 239 293 150)(116 240 294 121)(117 211 295 122)(118 212 296 123)(119 213 297 124)(120 214 298 125)(241 431 311 418)(242 432 312 419)(243 433 313 420)(244 434 314 391)(245 435 315 392)(246 436 316 393)(247 437 317 394)(248 438 318 395)(249 439 319 396)(250 440 320 397)(251 441 321 398)(252 442 322 399)(253 443 323 400)(254 444 324 401)(255 445 325 402)(256 446 326 403)(257 447 327 404)(258 448 328 405)(259 449 329 406)(260 450 330 407)(261 421 301 408)(262 422 302 409)(263 423 303 410)(264 424 304 411)(265 425 305 412)(266 426 306 413)(267 427 307 414)(268 428 308 415)(269 429 309 416)(270 430 310 417)
(1 64 195 378)(2 65 196 379)(3 66 197 380)(4 67 198 381)(5 68 199 382)(6 69 200 383)(7 70 201 384)(8 71 202 385)(9 72 203 386)(10 73 204 387)(11 74 205 388)(12 75 206 389)(13 76 207 390)(14 77 208 361)(15 78 209 362)(16 79 210 363)(17 80 181 364)(18 81 182 365)(19 82 183 366)(20 83 184 367)(21 84 185 368)(22 85 186 369)(23 86 187 370)(24 87 188 371)(25 88 189 372)(26 89 190 373)(27 90 191 374)(28 61 192 375)(29 62 193 376)(30 63 194 377)(31 335 171 454)(32 336 172 455)(33 337 173 456)(34 338 174 457)(35 339 175 458)(36 340 176 459)(37 341 177 460)(38 342 178 461)(39 343 179 462)(40 344 180 463)(41 345 151 464)(42 346 152 465)(43 347 153 466)(44 348 154 467)(45 349 155 468)(46 350 156 469)(47 351 157 470)(48 352 158 471)(49 353 159 472)(50 354 160 473)(51 355 161 474)(52 356 162 475)(53 357 163 476)(54 358 164 477)(55 359 165 478)(56 360 166 479)(57 331 167 480)(58 332 168 451)(59 333 169 452)(60 334 170 453)(91 431 299 418)(92 432 300 419)(93 433 271 420)(94 434 272 391)(95 435 273 392)(96 436 274 393)(97 437 275 394)(98 438 276 395)(99 439 277 396)(100 440 278 397)(101 441 279 398)(102 442 280 399)(103 443 281 400)(104 444 282 401)(105 445 283 402)(106 446 284 403)(107 447 285 404)(108 448 286 405)(109 449 287 406)(110 450 288 407)(111 421 289 408)(112 422 290 409)(113 423 291 410)(114 424 292 411)(115 425 293 412)(116 426 294 413)(117 427 295 414)(118 428 296 415)(119 429 297 416)(120 430 298 417)(121 306 240 266)(122 307 211 267)(123 308 212 268)(124 309 213 269)(125 310 214 270)(126 311 215 241)(127 312 216 242)(128 313 217 243)(129 314 218 244)(130 315 219 245)(131 316 220 246)(132 317 221 247)(133 318 222 248)(134 319 223 249)(135 320 224 250)(136 321 225 251)(137 322 226 252)(138 323 227 253)(139 324 228 254)(140 325 229 255)(141 326 230 256)(142 327 231 257)(143 328 232 258)(144 329 233 259)(145 330 234 260)(146 301 235 261)(147 302 236 262)(148 303 237 263)(149 304 238 264)(150 305 239 265)
(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)
(1 303 16 318)(2 302 17 317)(3 301 18 316)(4 330 19 315)(5 329 20 314)(6 328 21 313)(7 327 22 312)(8 326 23 311)(9 325 24 310)(10 324 25 309)(11 323 26 308)(12 322 27 307)(13 321 28 306)(14 320 29 305)(15 319 30 304)(31 298 46 283)(32 297 47 282)(33 296 48 281)(34 295 49 280)(35 294 50 279)(36 293 51 278)(37 292 52 277)(38 291 53 276)(39 290 54 275)(40 289 55 274)(41 288 56 273)(42 287 57 272)(43 286 58 271)(44 285 59 300)(45 284 60 299)(61 240 76 225)(62 239 77 224)(63 238 78 223)(64 237 79 222)(65 236 80 221)(66 235 81 220)(67 234 82 219)(68 233 83 218)(69 232 84 217)(70 231 85 216)(71 230 86 215)(72 229 87 214)(73 228 88 213)(74 227 89 212)(75 226 90 211)(91 155 106 170)(92 154 107 169)(93 153 108 168)(94 152 109 167)(95 151 110 166)(96 180 111 165)(97 179 112 164)(98 178 113 163)(99 177 114 162)(100 176 115 161)(101 175 116 160)(102 174 117 159)(103 173 118 158)(104 172 119 157)(105 171 120 156)(121 390 136 375)(122 389 137 374)(123 388 138 373)(124 387 139 372)(125 386 140 371)(126 385 141 370)(127 384 142 369)(128 383 143 368)(129 382 144 367)(130 381 145 366)(131 380 146 365)(132 379 147 364)(133 378 148 363)(134 377 149 362)(135 376 150 361)(181 247 196 262)(182 246 197 261)(183 245 198 260)(184 244 199 259)(185 243 200 258)(186 242 201 257)(187 241 202 256)(188 270 203 255)(189 269 204 254)(190 268 205 253)(191 267 206 252)(192 266 207 251)(193 265 208 250)(194 264 209 249)(195 263 210 248)(331 391 346 406)(332 420 347 405)(333 419 348 404)(334 418 349 403)(335 417 350 402)(336 416 351 401)(337 415 352 400)(338 414 353 399)(339 413 354 398)(340 412 355 397)(341 411 356 396)(342 410 357 395)(343 409 358 394)(344 408 359 393)(345 407 360 392)(421 478 436 463)(422 477 437 462)(423 476 438 461)(424 475 439 460)(425 474 440 459)(426 473 441 458)(427 472 442 457)(428 471 443 456)(429 470 444 455)(430 469 445 454)(431 468 446 453)(432 467 447 452)(433 466 448 451)(434 465 449 480)(435 464 450 479)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 5A 5B 6A 6B 6C 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 4 ··· 4 5 5 6 6 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 ··· 2 15 15 15 15 30 ··· 30 2 2 2 2 2 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 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 Q8 D5 D6 Dic3 C4○D4 D10 Dic5 D15 D30 Dic15 S3×Q8 Q8⋊3S3 Q8×D5 Q8⋊2D5 Q8×D15 Q8⋊3D15 kernel Q8×Dic15 C4×Dic15 C60⋊5C4 Q8×C30 Q8×C15 Q8×C10 Dic15 C6×Q8 C2×C20 C5×Q8 C30 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C10 C10 C6 C6 C2 C2 # reps 1 3 3 1 8 1 2 2 3 4 2 6 8 4 12 16 1 1 2 2 4 4

Matrix representation of Q8×Dic15 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 1 12 0 0 10 60
,
 1 0 0 0 0 1 0 0 0 0 34 27 0 0 43 27
,
 33 25 0 0 36 31 0 0 0 0 1 0 0 0 0 1
,
 36 29 0 0 31 25 0 0 0 0 60 0 0 0 0 60
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,10,0,0,12,60],[1,0,0,0,0,1,0,0,0,0,34,43,0,0,27,27],[33,36,0,0,25,31,0,0,0,0,1,0,0,0,0,1],[36,31,0,0,29,25,0,0,0,0,60,0,0,0,0,60] >;

Q8×Dic15 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xDic15");
// GroupNames label

G:=SmallGroup(480,910);
// by ID

G=gap.SmallGroup(480,910);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,219,100,2693,18822]);
// Polycyclic

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

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