Q8:6Dic14 - GroupNames

G = Q₈⋊₆Dic₁₄ order 448 = 2⁶·7

2^nd semidirect product of Q₈ and Dic₁₄ acting through Inn(Q₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈⋊₆Dic₁₄, C₄².₁₂₄D₁₄, C₁₄.₆₅2_-¹⁺⁴, (C₇×Q₈)⋊₇Q₈, C₇⋊₃(Q₈⋊₃Q₈), (C₄×Q₈).₁₂D₇, C₂₈.₄₆(C₂×Q₈), C₄⋊C₄.₂₉₃D₁₄, (Q₈×C₂₈).₁₃C₂, (C₂×Q₈).₁₉₈D₁₄, C₂₈⋊₂Q₈.₂₅C₂, (Q₈×Dic₇).₁₂C₂, C₄.₁₉(C₂×Dic₁₄), C₂₈.₃₃₃(C₄○D₄), C₁₄.₁₇(C₂²×Q₈), (C₂×C₁₄).₁₁₄C₂⁴, (C₄×C₂₈).₁₆₆C₂², (C₂×C₂₈).₁₆₈C₂³, C₄.₄₉(Q₈⋊₂D₇), (C₄×Dic₁₄).₂₃C₂, C₄⋊Dic₇.₄₃C₂², C₂₈.₃Q₈.₁₁C₂, (Q₈×C₁₄).₂₁₄C₂², (C₄×Dic₇).₈₃C₂², (C₂×Dic₇).₅₂C₂³, C₂.₁₉(C₂²×Dic₁₄), C₂².₁₃₉(C₂³×D₇), Dic₇⋊C₄.₁₁₅C₂², C₂.₂₂(D₄.₁₀D₁₄), (C₂×Dic₁₄).₂₄₂C₂², C₁₄.₁₁₀(C₂×C₄○D₄), C₂.₁₀(C₂×Q₈⋊₂D₇), (C₇×C₄⋊C₄).₃₄₂C₂², (C₂×C₄).₇₃₃(C₂²×D₇), SmallGroup(448,1023)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — Q₈⋊₆Dic₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×Dic₇ — C₄×Dic₇ — Q₈×Dic₇ — Q₈⋊₆Dic₁₄

Lower central

C₇ — C₂×C₁₄ — Q₈⋊₆Dic₁₄

Upper central

C₁ — C₂² — C₄×Q₈

Generators and relations for Q₈⋊₆Dic₁₄
G = < a,b,c,d | a⁴=c²⁸=1, b²=a², d²=c¹⁴, bab^-1=a^-1, ac=ca, ad=da, bc=cb, dbd^-1=a²b, dcd^-1=c^-1 >

Subgroups: 708 in 200 conjugacy classes, 115 normal (18 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, C₂₈, C₂₈, C₂×C₁₄, C₄×Q₈, C₄×Q₈, C₄².C₂, C₄⋊Q₈, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, Q₈⋊₃Q₈, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₄⋊Dic₇, C₄×C₂₈, C₇×C₄⋊C₄, C₂×Dic₁₄, Q₈×C₁₄, C₄×Dic₁₄, C₂₈⋊₂Q₈, C₂₈.₃Q₈, Q₈×Dic₇, Q₈×C₂₈, Q₈⋊₆Dic₁₄
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₇, C₂×Q₈, C₄○D₄, C₂⁴, D₁₄, C₂²×Q₈, C₂×C₄○D₄, 2_-¹⁺⁴, Dic₁₄, C₂²×D₇, Q₈⋊₃Q₈, C₂×Dic₁₄, Q₈⋊₂D₇, C₂³×D₇, C₂²×Dic₁₄, C₂×Q₈⋊₂D₇, D₄.₁₀D₁₄, Q₈⋊₆Dic₁₄

Smallest permutation representation of Q₈⋊₆Dic₁₄
►Regular action on 448 points

Generators in S₄₄₈

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

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

(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)

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

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

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

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

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

Matrix representation of Q₈⋊₆Dic₁₄ ►in GL₆(𝔽₂₉)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	1	5
0	0	0	0	17	28

28	0	0	0	0	0
0	28	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	2	26
0	0	0	0	21	27

7	0	0	0	0	0
3	25	0	0	0	0
0	0	20	5	0	0
0	0	1	9	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	25	0	0	0	0
20	20	0	0	0	0
0	0	0	11	0	0
0	0	21	0	0	0
0	0	0	0	17	27
0	0	0	0	28	12

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,17,0,0,0,0,5,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,2,21,0,0,0,0,26,27],[7,3,0,0,0,0,0,25,0,0,0,0,0,0,20,1,0,0,0,0,5,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,20,0,0,0,0,25,20,0,0,0,0,0,0,0,21,0,0,0,0,11,0,0,0,0,0,0,0,17,28,0,0,0,0,27,12] >;

Q₈⋊₆Dic₁₄ in GAP, Magma, Sage, TeX

Q_8\rtimes_6{\rm Dic}_{14}

% in TeX

G:=Group("Q8:6Dic14");

// GroupNames label

G:=SmallGroup(448,1023);

// by ID

G=gap.SmallGroup(448,1023);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,387,184,1571,192,18822]);

// 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,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;

// generators/relations

G = Q8⋊6Dic14 order 448 = 26·7

2nd semidirect product of Q8 and Dic14 acting through Inn(Q8)

G = Q₈⋊₆Dic₁₄ order 448 = 2⁶·7

2^nd semidirect product of Q₈ and Dic₁₄ acting through Inn(Q₈)