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## G = Q8⋊5Dic14order 448 = 26·7

### 1st semidirect product of Q8 and Dic14 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — Q8⋊5Dic14
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C4×Dic7 — Q8×Dic7 — Q8⋊5Dic14
 Lower central C7 — C2×C14 — Q8⋊5Dic14
 Upper central C1 — C22 — C4×Q8

Generators and relations for Q85Dic14
G = < a,b,c,d | a4=c28=1, b2=a2, d2=c14, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >

Subgroups: 708 in 200 conjugacy classes, 113 normal (22 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, Q83Q8, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C2×Dic14, Q8×C14, C4×Dic14, C28.6Q8, C28⋊Q8, C28.3Q8, Q8×Dic7, Q8×C28, Q85Dic14
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, C24, D14, C22×Q8, C2×C4○D4, 2- 1+4, Dic14, C22×D7, Q83Q8, C2×Dic14, C23×D7, C22×Dic14, Q8.10D14, D7×C4○D4, Q85Dic14

Smallest permutation representation of Q85Dic14
Regular action on 448 points
Generators in S448
(1 172 398 242)(2 173 399 243)(3 174 400 244)(4 175 401 245)(5 176 402 246)(6 177 403 247)(7 178 404 248)(8 179 405 249)(9 180 406 250)(10 181 407 251)(11 182 408 252)(12 183 409 225)(13 184 410 226)(14 185 411 227)(15 186 412 228)(16 187 413 229)(17 188 414 230)(18 189 415 231)(19 190 416 232)(20 191 417 233)(21 192 418 234)(22 193 419 235)(23 194 420 236)(24 195 393 237)(25 196 394 238)(26 169 395 239)(27 170 396 240)(28 171 397 241)(29 356 90 331)(30 357 91 332)(31 358 92 333)(32 359 93 334)(33 360 94 335)(34 361 95 336)(35 362 96 309)(36 363 97 310)(37 364 98 311)(38 337 99 312)(39 338 100 313)(40 339 101 314)(41 340 102 315)(42 341 103 316)(43 342 104 317)(44 343 105 318)(45 344 106 319)(46 345 107 320)(47 346 108 321)(48 347 109 322)(49 348 110 323)(50 349 111 324)(51 350 112 325)(52 351 85 326)(53 352 86 327)(54 353 87 328)(55 354 88 329)(56 355 89 330)(57 386 298 154)(58 387 299 155)(59 388 300 156)(60 389 301 157)(61 390 302 158)(62 391 303 159)(63 392 304 160)(64 365 305 161)(65 366 306 162)(66 367 307 163)(67 368 308 164)(68 369 281 165)(69 370 282 166)(70 371 283 167)(71 372 284 168)(72 373 285 141)(73 374 286 142)(74 375 287 143)(75 376 288 144)(76 377 289 145)(77 378 290 146)(78 379 291 147)(79 380 292 148)(80 381 293 149)(81 382 294 150)(82 383 295 151)(83 384 296 152)(84 385 297 153)(113 271 441 202)(114 272 442 203)(115 273 443 204)(116 274 444 205)(117 275 445 206)(118 276 446 207)(119 277 447 208)(120 278 448 209)(121 279 421 210)(122 280 422 211)(123 253 423 212)(124 254 424 213)(125 255 425 214)(126 256 426 215)(127 257 427 216)(128 258 428 217)(129 259 429 218)(130 260 430 219)(131 261 431 220)(132 262 432 221)(133 263 433 222)(134 264 434 223)(135 265 435 224)(136 266 436 197)(137 267 437 198)(138 268 438 199)(139 269 439 200)(140 270 440 201)
(1 92 398 31)(2 32 399 93)(3 94 400 33)(4 34 401 95)(5 96 402 35)(6 36 403 97)(7 98 404 37)(8 38 405 99)(9 100 406 39)(10 40 407 101)(11 102 408 41)(12 42 409 103)(13 104 410 43)(14 44 411 105)(15 106 412 45)(16 46 413 107)(17 108 414 47)(18 48 415 109)(19 110 416 49)(20 50 417 111)(21 112 418 51)(22 52 419 85)(23 86 420 53)(24 54 393 87)(25 88 394 55)(26 56 395 89)(27 90 396 29)(28 30 397 91)(57 208 298 277)(58 278 299 209)(59 210 300 279)(60 280 301 211)(61 212 302 253)(62 254 303 213)(63 214 304 255)(64 256 305 215)(65 216 306 257)(66 258 307 217)(67 218 308 259)(68 260 281 219)(69 220 282 261)(70 262 283 221)(71 222 284 263)(72 264 285 223)(73 224 286 265)(74 266 287 197)(75 198 288 267)(76 268 289 199)(77 200 290 269)(78 270 291 201)(79 202 292 271)(80 272 293 203)(81 204 294 273)(82 274 295 205)(83 206 296 275)(84 276 297 207)(113 380 441 148)(114 149 442 381)(115 382 443 150)(116 151 444 383)(117 384 445 152)(118 153 446 385)(119 386 447 154)(120 155 448 387)(121 388 421 156)(122 157 422 389)(123 390 423 158)(124 159 424 391)(125 392 425 160)(126 161 426 365)(127 366 427 162)(128 163 428 367)(129 368 429 164)(130 165 430 369)(131 370 431 166)(132 167 432 371)(133 372 433 168)(134 141 434 373)(135 374 435 142)(136 143 436 375)(137 376 437 144)(138 145 438 377)(139 378 439 146)(140 147 440 379)(169 330 239 355)(170 356 240 331)(171 332 241 357)(172 358 242 333)(173 334 243 359)(174 360 244 335)(175 336 245 361)(176 362 246 309)(177 310 247 363)(178 364 248 311)(179 312 249 337)(180 338 250 313)(181 314 251 339)(182 340 252 315)(183 316 225 341)(184 342 226 317)(185 318 227 343)(186 344 228 319)(187 320 229 345)(188 346 230 321)(189 322 231 347)(190 348 232 323)(191 324 233 349)(192 350 234 325)(193 326 235 351)(194 352 236 327)(195 328 237 353)(196 354 238 329)
(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)
(1 57 15 71)(2 84 16 70)(3 83 17 69)(4 82 18 68)(5 81 19 67)(6 80 20 66)(7 79 21 65)(8 78 22 64)(9 77 23 63)(10 76 24 62)(11 75 25 61)(12 74 26 60)(13 73 27 59)(14 72 28 58)(29 279 43 265)(30 278 44 264)(31 277 45 263)(32 276 46 262)(33 275 47 261)(34 274 48 260)(35 273 49 259)(36 272 50 258)(37 271 51 257)(38 270 52 256)(39 269 53 255)(40 268 54 254)(41 267 55 253)(42 266 56 280)(85 215 99 201)(86 214 100 200)(87 213 101 199)(88 212 102 198)(89 211 103 197)(90 210 104 224)(91 209 105 223)(92 208 106 222)(93 207 107 221)(94 206 108 220)(95 205 109 219)(96 204 110 218)(97 203 111 217)(98 202 112 216)(113 325 127 311)(114 324 128 310)(115 323 129 309)(116 322 130 336)(117 321 131 335)(118 320 132 334)(119 319 133 333)(120 318 134 332)(121 317 135 331)(122 316 136 330)(123 315 137 329)(124 314 138 328)(125 313 139 327)(126 312 140 326)(141 241 155 227)(142 240 156 226)(143 239 157 225)(144 238 158 252)(145 237 159 251)(146 236 160 250)(147 235 161 249)(148 234 162 248)(149 233 163 247)(150 232 164 246)(151 231 165 245)(152 230 166 244)(153 229 167 243)(154 228 168 242)(169 389 183 375)(170 388 184 374)(171 387 185 373)(172 386 186 372)(173 385 187 371)(174 384 188 370)(175 383 189 369)(176 382 190 368)(177 381 191 367)(178 380 192 366)(179 379 193 365)(180 378 194 392)(181 377 195 391)(182 376 196 390)(281 401 295 415)(282 400 296 414)(283 399 297 413)(284 398 298 412)(285 397 299 411)(286 396 300 410)(287 395 301 409)(288 394 302 408)(289 393 303 407)(290 420 304 406)(291 419 305 405)(292 418 306 404)(293 417 307 403)(294 416 308 402)(337 440 351 426)(338 439 352 425)(339 438 353 424)(340 437 354 423)(341 436 355 422)(342 435 356 421)(343 434 357 448)(344 433 358 447)(345 432 359 446)(346 431 360 445)(347 430 361 444)(348 429 362 443)(349 428 363 442)(350 427 364 441)

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

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

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

85 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P ··· 4U 7A 7B 7C 14A ··· 14I 28A ··· 28L 28M ··· 28AV order 1 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 ··· 2 4 4 4 14 14 14 14 28 ··· 28 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4

85 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + - - image C1 C2 C2 C2 C2 C2 C2 Q8 D7 C4○D4 D14 D14 D14 Dic14 2- 1+4 Q8.10D14 D7×C4○D4 kernel Q8⋊5Dic14 C4×Dic14 C28.6Q8 C28⋊Q8 C28.3Q8 Q8×Dic7 Q8×C28 C7×Q8 C4×Q8 Dic7 C42 C4⋊C4 C2×Q8 Q8 C14 C2 C2 # reps 1 3 3 3 3 2 1 4 3 4 9 9 3 24 1 6 6

Matrix representation of Q85Dic14 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 28 0
,
 1 0 0 0 0 1 0 0 0 0 0 17 0 0 17 0
,
 21 0 0 0 13 18 0 0 0 0 0 12 0 0 17 0
,
 10 20 0 0 8 19 0 0 0 0 28 0 0 0 0 28
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,17,0,0,17,0],[21,13,0,0,0,18,0,0,0,0,0,17,0,0,12,0],[10,8,0,0,20,19,0,0,0,0,28,0,0,0,0,28] >;

Q85Dic14 in GAP, Magma, Sage, TeX

Q_8\rtimes_5{\rm Dic}_{14}
% in TeX

G:=Group("Q8:5Dic14");
// GroupNames label

G:=SmallGroup(448,1022);
// by ID

G=gap.SmallGroup(448,1022);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,387,184,675,192,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^28=1,b^2=a^2,d^2=c^14,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽