metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊1Dic14, Dic7.6SD16, (C7×Q8)⋊1Q8, C28⋊Q8.1C2, C28.5(C2×Q8), C4⋊C4.15D14, C7⋊1(Q8⋊Q8), C8⋊Dic7.6C2, (C2×C8).119D14, Dic7⋊C8.5C2, (C2×Q8).99D14, Q8⋊C4.5D7, (Q8×Dic7).3C2, C4.5(C2×Dic14), C2.14(D7×SD16), C14.27(C2×SD16), Q8⋊Dic7.1C2, C22.185(D4×D7), C4.Dic14.2C2, C28.157(C4○D4), C4.82(D4⋊2D7), (C2×C56).130C22, (C2×C28).231C23, C2.8(Q16⋊D7), (C2×Dic7).147D4, C14.11(C22⋊Q8), C4⋊Dic7.81C22, (Q8×C14).14C22, C14.53(C8.C22), (C4×Dic7).16C22, C2.16(C22⋊Dic14), (C2×C7⋊C8).27C22, (C2×C14).244(C2×D4), (C7×C4⋊C4).32C22, (C7×Q8⋊C4).5C2, (C2×C4).338(C22×D7), SmallGroup(448,325)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊Dic14
G = < a,b,c,d | a4=c28=1, b2=a2, d2=c14, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >
Subgroups: 436 in 96 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, Q8⋊Q8, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, Q8×C14, C4.Dic14, Dic7⋊C8, C8⋊Dic7, Q8⋊Dic7, C7×Q8⋊C4, C28⋊Q8, Q8×Dic7, Q8⋊Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×SD16, C8.C22, Dic14, C22×D7, Q8⋊Q8, C2×Dic14, D4×D7, D4⋊2D7, C22⋊Dic14, D7×SD16, Q16⋊D7, Q8⋊Dic14
►Regular action on 448 points
Generators in S448
(1 360 86 425)(2 426 87 361)(3 362 88 427)(4 428 89 363)(5 364 90 429)(6 430 91 337)(7 338 92 431)(8 432 93 339)(9 340 94 433)(10 434 95 341)(11 342 96 435)(12 436 97 343)(13 344 98 437)(14 438 99 345)(15 346 100 439)(16 440 101 347)(17 348 102 441)(18 442 103 349)(19 350 104 443)(20 444 105 351)(21 352 106 445)(22 446 107 353)(23 354 108 447)(24 448 109 355)(25 356 110 421)(26 422 111 357)(27 358 112 423)(28 424 85 359)(29 175 124 282)(30 283 125 176)(31 177 126 284)(32 285 127 178)(33 179 128 286)(34 287 129 180)(35 181 130 288)(36 289 131 182)(37 183 132 290)(38 291 133 184)(39 185 134 292)(40 293 135 186)(41 187 136 294)(42 295 137 188)(43 189 138 296)(44 297 139 190)(45 191 140 298)(46 299 113 192)(47 193 114 300)(48 301 115 194)(49 195 116 302)(50 303 117 196)(51 169 118 304)(52 305 119 170)(53 171 120 306)(54 307 121 172)(55 173 122 308)(56 281 123 174)(57 323 244 147)(58 148 245 324)(59 325 246 149)(60 150 247 326)(61 327 248 151)(62 152 249 328)(63 329 250 153)(64 154 251 330)(65 331 252 155)(66 156 225 332)(67 333 226 157)(68 158 227 334)(69 335 228 159)(70 160 229 336)(71 309 230 161)(72 162 231 310)(73 311 232 163)(74 164 233 312)(75 313 234 165)(76 166 235 314)(77 315 236 167)(78 168 237 316)(79 317 238 141)(80 142 239 318)(81 319 240 143)(82 144 241 320)(83 321 242 145)(84 146 243 322)(197 417 375 259)(198 260 376 418)(199 419 377 261)(200 262 378 420)(201 393 379 263)(202 264 380 394)(203 395 381 265)(204 266 382 396)(205 397 383 267)(206 268 384 398)(207 399 385 269)(208 270 386 400)(209 401 387 271)(210 272 388 402)(211 403 389 273)(212 274 390 404)(213 405 391 275)(214 276 392 406)(215 407 365 277)(216 278 366 408)(217 409 367 279)(218 280 368 410)(219 411 369 253)(220 254 370 412)(221 413 371 255)(222 256 372 414)(223 415 373 257)(224 258 374 416)
(1 402 86 272)(2 389 87 211)(3 404 88 274)(4 391 89 213)(5 406 90 276)(6 365 91 215)(7 408 92 278)(8 367 93 217)(9 410 94 280)(10 369 95 219)(11 412 96 254)(12 371 97 221)(13 414 98 256)(14 373 99 223)(15 416 100 258)(16 375 101 197)(17 418 102 260)(18 377 103 199)(19 420 104 262)(20 379 105 201)(21 394 106 264)(22 381 107 203)(23 396 108 266)(24 383 109 205)(25 398 110 268)(26 385 111 207)(27 400 112 270)(28 387 85 209)(29 246 124 59)(30 326 125 150)(31 248 126 61)(32 328 127 152)(33 250 128 63)(34 330 129 154)(35 252 130 65)(36 332 131 156)(37 226 132 67)(38 334 133 158)(39 228 134 69)(40 336 135 160)(41 230 136 71)(42 310 137 162)(43 232 138 73)(44 312 139 164)(45 234 140 75)(46 314 113 166)(47 236 114 77)(48 316 115 168)(49 238 116 79)(50 318 117 142)(51 240 118 81)(52 320 119 144)(53 242 120 83)(54 322 121 146)(55 244 122 57)(56 324 123 148)(58 281 245 174)(60 283 247 176)(62 285 249 178)(64 287 251 180)(66 289 225 182)(68 291 227 184)(70 293 229 186)(72 295 231 188)(74 297 233 190)(76 299 235 192)(78 301 237 194)(80 303 239 196)(82 305 241 170)(84 307 243 172)(141 195 317 302)(143 169 319 304)(145 171 321 306)(147 173 323 308)(149 175 325 282)(151 177 327 284)(153 179 329 286)(155 181 331 288)(157 183 333 290)(159 185 335 292)(161 187 309 294)(163 189 311 296)(165 191 313 298)(167 193 315 300)(198 348 376 441)(200 350 378 443)(202 352 380 445)(204 354 382 447)(206 356 384 421)(208 358 386 423)(210 360 388 425)(212 362 390 427)(214 364 392 429)(216 338 366 431)(218 340 368 433)(220 342 370 435)(222 344 372 437)(224 346 374 439)(253 434 411 341)(255 436 413 343)(257 438 415 345)(259 440 417 347)(261 442 419 349)(263 444 393 351)(265 446 395 353)(267 448 397 355)(269 422 399 357)(271 424 401 359)(273 426 403 361)(275 428 405 363)(277 430 407 337)(279 432 409 339)
(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)
(1 177 15 191)(2 176 16 190)(3 175 17 189)(4 174 18 188)(5 173 19 187)(6 172 20 186)(7 171 21 185)(8 170 22 184)(9 169 23 183)(10 196 24 182)(11 195 25 181)(12 194 26 180)(13 193 27 179)(14 192 28 178)(29 441 43 427)(30 440 44 426)(31 439 45 425)(32 438 46 424)(33 437 47 423)(34 436 48 422)(35 435 49 421)(36 434 50 448)(37 433 51 447)(38 432 52 446)(39 431 53 445)(40 430 54 444)(41 429 55 443)(42 428 56 442)(57 378 71 392)(58 377 72 391)(59 376 73 390)(60 375 74 389)(61 374 75 388)(62 373 76 387)(63 372 77 386)(64 371 78 385)(65 370 79 384)(66 369 80 383)(67 368 81 382)(68 367 82 381)(69 366 83 380)(70 365 84 379)(85 285 99 299)(86 284 100 298)(87 283 101 297)(88 282 102 296)(89 281 103 295)(90 308 104 294)(91 307 105 293)(92 306 106 292)(93 305 107 291)(94 304 108 290)(95 303 109 289)(96 302 110 288)(97 301 111 287)(98 300 112 286)(113 359 127 345)(114 358 128 344)(115 357 129 343)(116 356 130 342)(117 355 131 341)(118 354 132 340)(119 353 133 339)(120 352 134 338)(121 351 135 337)(122 350 136 364)(123 349 137 363)(124 348 138 362)(125 347 139 361)(126 346 140 360)(141 268 155 254)(142 267 156 253)(143 266 157 280)(144 265 158 279)(145 264 159 278)(146 263 160 277)(147 262 161 276)(148 261 162 275)(149 260 163 274)(150 259 164 273)(151 258 165 272)(152 257 166 271)(153 256 167 270)(154 255 168 269)(197 233 211 247)(198 232 212 246)(199 231 213 245)(200 230 214 244)(201 229 215 243)(202 228 216 242)(203 227 217 241)(204 226 218 240)(205 225 219 239)(206 252 220 238)(207 251 221 237)(208 250 222 236)(209 249 223 235)(210 248 224 234)(309 406 323 420)(310 405 324 419)(311 404 325 418)(312 403 326 417)(313 402 327 416)(314 401 328 415)(315 400 329 414)(316 399 330 413)(317 398 331 412)(318 397 332 411)(319 396 333 410)(320 395 334 409)(321 394 335 408)(322 393 336 407)
G:=sub<Sym(448)| (1,360,86,425)(2,426,87,361)(3,362,88,427)(4,428,89,363)(5,364,90,429)(6,430,91,337)(7,338,92,431)(8,432,93,339)(9,340,94,433)(10,434,95,341)(11,342,96,435)(12,436,97,343)(13,344,98,437)(14,438,99,345)(15,346,100,439)(16,440,101,347)(17,348,102,441)(18,442,103,349)(19,350,104,443)(20,444,105,351)(21,352,106,445)(22,446,107,353)(23,354,108,447)(24,448,109,355)(25,356,110,421)(26,422,111,357)(27,358,112,423)(28,424,85,359)(29,175,124,282)(30,283,125,176)(31,177,126,284)(32,285,127,178)(33,179,128,286)(34,287,129,180)(35,181,130,288)(36,289,131,182)(37,183,132,290)(38,291,133,184)(39,185,134,292)(40,293,135,186)(41,187,136,294)(42,295,137,188)(43,189,138,296)(44,297,139,190)(45,191,140,298)(46,299,113,192)(47,193,114,300)(48,301,115,194)(49,195,116,302)(50,303,117,196)(51,169,118,304)(52,305,119,170)(53,171,120,306)(54,307,121,172)(55,173,122,308)(56,281,123,174)(57,323,244,147)(58,148,245,324)(59,325,246,149)(60,150,247,326)(61,327,248,151)(62,152,249,328)(63,329,250,153)(64,154,251,330)(65,331,252,155)(66,156,225,332)(67,333,226,157)(68,158,227,334)(69,335,228,159)(70,160,229,336)(71,309,230,161)(72,162,231,310)(73,311,232,163)(74,164,233,312)(75,313,234,165)(76,166,235,314)(77,315,236,167)(78,168,237,316)(79,317,238,141)(80,142,239,318)(81,319,240,143)(82,144,241,320)(83,321,242,145)(84,146,243,322)(197,417,375,259)(198,260,376,418)(199,419,377,261)(200,262,378,420)(201,393,379,263)(202,264,380,394)(203,395,381,265)(204,266,382,396)(205,397,383,267)(206,268,384,398)(207,399,385,269)(208,270,386,400)(209,401,387,271)(210,272,388,402)(211,403,389,273)(212,274,390,404)(213,405,391,275)(214,276,392,406)(215,407,365,277)(216,278,366,408)(217,409,367,279)(218,280,368,410)(219,411,369,253)(220,254,370,412)(221,413,371,255)(222,256,372,414)(223,415,373,257)(224,258,374,416), (1,402,86,272)(2,389,87,211)(3,404,88,274)(4,391,89,213)(5,406,90,276)(6,365,91,215)(7,408,92,278)(8,367,93,217)(9,410,94,280)(10,369,95,219)(11,412,96,254)(12,371,97,221)(13,414,98,256)(14,373,99,223)(15,416,100,258)(16,375,101,197)(17,418,102,260)(18,377,103,199)(19,420,104,262)(20,379,105,201)(21,394,106,264)(22,381,107,203)(23,396,108,266)(24,383,109,205)(25,398,110,268)(26,385,111,207)(27,400,112,270)(28,387,85,209)(29,246,124,59)(30,326,125,150)(31,248,126,61)(32,328,127,152)(33,250,128,63)(34,330,129,154)(35,252,130,65)(36,332,131,156)(37,226,132,67)(38,334,133,158)(39,228,134,69)(40,336,135,160)(41,230,136,71)(42,310,137,162)(43,232,138,73)(44,312,139,164)(45,234,140,75)(46,314,113,166)(47,236,114,77)(48,316,115,168)(49,238,116,79)(50,318,117,142)(51,240,118,81)(52,320,119,144)(53,242,120,83)(54,322,121,146)(55,244,122,57)(56,324,123,148)(58,281,245,174)(60,283,247,176)(62,285,249,178)(64,287,251,180)(66,289,225,182)(68,291,227,184)(70,293,229,186)(72,295,231,188)(74,297,233,190)(76,299,235,192)(78,301,237,194)(80,303,239,196)(82,305,241,170)(84,307,243,172)(141,195,317,302)(143,169,319,304)(145,171,321,306)(147,173,323,308)(149,175,325,282)(151,177,327,284)(153,179,329,286)(155,181,331,288)(157,183,333,290)(159,185,335,292)(161,187,309,294)(163,189,311,296)(165,191,313,298)(167,193,315,300)(198,348,376,441)(200,350,378,443)(202,352,380,445)(204,354,382,447)(206,356,384,421)(208,358,386,423)(210,360,388,425)(212,362,390,427)(214,364,392,429)(216,338,366,431)(218,340,368,433)(220,342,370,435)(222,344,372,437)(224,346,374,439)(253,434,411,341)(255,436,413,343)(257,438,415,345)(259,440,417,347)(261,442,419,349)(263,444,393,351)(265,446,395,353)(267,448,397,355)(269,422,399,357)(271,424,401,359)(273,426,403,361)(275,428,405,363)(277,430,407,337)(279,432,409,339), (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), (1,177,15,191)(2,176,16,190)(3,175,17,189)(4,174,18,188)(5,173,19,187)(6,172,20,186)(7,171,21,185)(8,170,22,184)(9,169,23,183)(10,196,24,182)(11,195,25,181)(12,194,26,180)(13,193,27,179)(14,192,28,178)(29,441,43,427)(30,440,44,426)(31,439,45,425)(32,438,46,424)(33,437,47,423)(34,436,48,422)(35,435,49,421)(36,434,50,448)(37,433,51,447)(38,432,52,446)(39,431,53,445)(40,430,54,444)(41,429,55,443)(42,428,56,442)(57,378,71,392)(58,377,72,391)(59,376,73,390)(60,375,74,389)(61,374,75,388)(62,373,76,387)(63,372,77,386)(64,371,78,385)(65,370,79,384)(66,369,80,383)(67,368,81,382)(68,367,82,381)(69,366,83,380)(70,365,84,379)(85,285,99,299)(86,284,100,298)(87,283,101,297)(88,282,102,296)(89,281,103,295)(90,308,104,294)(91,307,105,293)(92,306,106,292)(93,305,107,291)(94,304,108,290)(95,303,109,289)(96,302,110,288)(97,301,111,287)(98,300,112,286)(113,359,127,345)(114,358,128,344)(115,357,129,343)(116,356,130,342)(117,355,131,341)(118,354,132,340)(119,353,133,339)(120,352,134,338)(121,351,135,337)(122,350,136,364)(123,349,137,363)(124,348,138,362)(125,347,139,361)(126,346,140,360)(141,268,155,254)(142,267,156,253)(143,266,157,280)(144,265,158,279)(145,264,159,278)(146,263,160,277)(147,262,161,276)(148,261,162,275)(149,260,163,274)(150,259,164,273)(151,258,165,272)(152,257,166,271)(153,256,167,270)(154,255,168,269)(197,233,211,247)(198,232,212,246)(199,231,213,245)(200,230,214,244)(201,229,215,243)(202,228,216,242)(203,227,217,241)(204,226,218,240)(205,225,219,239)(206,252,220,238)(207,251,221,237)(208,250,222,236)(209,249,223,235)(210,248,224,234)(309,406,323,420)(310,405,324,419)(311,404,325,418)(312,403,326,417)(313,402,327,416)(314,401,328,415)(315,400,329,414)(316,399,330,413)(317,398,331,412)(318,397,332,411)(319,396,333,410)(320,395,334,409)(321,394,335,408)(322,393,336,407)>;
G:=Group( (1,360,86,425)(2,426,87,361)(3,362,88,427)(4,428,89,363)(5,364,90,429)(6,430,91,337)(7,338,92,431)(8,432,93,339)(9,340,94,433)(10,434,95,341)(11,342,96,435)(12,436,97,343)(13,344,98,437)(14,438,99,345)(15,346,100,439)(16,440,101,347)(17,348,102,441)(18,442,103,349)(19,350,104,443)(20,444,105,351)(21,352,106,445)(22,446,107,353)(23,354,108,447)(24,448,109,355)(25,356,110,421)(26,422,111,357)(27,358,112,423)(28,424,85,359)(29,175,124,282)(30,283,125,176)(31,177,126,284)(32,285,127,178)(33,179,128,286)(34,287,129,180)(35,181,130,288)(36,289,131,182)(37,183,132,290)(38,291,133,184)(39,185,134,292)(40,293,135,186)(41,187,136,294)(42,295,137,188)(43,189,138,296)(44,297,139,190)(45,191,140,298)(46,299,113,192)(47,193,114,300)(48,301,115,194)(49,195,116,302)(50,303,117,196)(51,169,118,304)(52,305,119,170)(53,171,120,306)(54,307,121,172)(55,173,122,308)(56,281,123,174)(57,323,244,147)(58,148,245,324)(59,325,246,149)(60,150,247,326)(61,327,248,151)(62,152,249,328)(63,329,250,153)(64,154,251,330)(65,331,252,155)(66,156,225,332)(67,333,226,157)(68,158,227,334)(69,335,228,159)(70,160,229,336)(71,309,230,161)(72,162,231,310)(73,311,232,163)(74,164,233,312)(75,313,234,165)(76,166,235,314)(77,315,236,167)(78,168,237,316)(79,317,238,141)(80,142,239,318)(81,319,240,143)(82,144,241,320)(83,321,242,145)(84,146,243,322)(197,417,375,259)(198,260,376,418)(199,419,377,261)(200,262,378,420)(201,393,379,263)(202,264,380,394)(203,395,381,265)(204,266,382,396)(205,397,383,267)(206,268,384,398)(207,399,385,269)(208,270,386,400)(209,401,387,271)(210,272,388,402)(211,403,389,273)(212,274,390,404)(213,405,391,275)(214,276,392,406)(215,407,365,277)(216,278,366,408)(217,409,367,279)(218,280,368,410)(219,411,369,253)(220,254,370,412)(221,413,371,255)(222,256,372,414)(223,415,373,257)(224,258,374,416), (1,402,86,272)(2,389,87,211)(3,404,88,274)(4,391,89,213)(5,406,90,276)(6,365,91,215)(7,408,92,278)(8,367,93,217)(9,410,94,280)(10,369,95,219)(11,412,96,254)(12,371,97,221)(13,414,98,256)(14,373,99,223)(15,416,100,258)(16,375,101,197)(17,418,102,260)(18,377,103,199)(19,420,104,262)(20,379,105,201)(21,394,106,264)(22,381,107,203)(23,396,108,266)(24,383,109,205)(25,398,110,268)(26,385,111,207)(27,400,112,270)(28,387,85,209)(29,246,124,59)(30,326,125,150)(31,248,126,61)(32,328,127,152)(33,250,128,63)(34,330,129,154)(35,252,130,65)(36,332,131,156)(37,226,132,67)(38,334,133,158)(39,228,134,69)(40,336,135,160)(41,230,136,71)(42,310,137,162)(43,232,138,73)(44,312,139,164)(45,234,140,75)(46,314,113,166)(47,236,114,77)(48,316,115,168)(49,238,116,79)(50,318,117,142)(51,240,118,81)(52,320,119,144)(53,242,120,83)(54,322,121,146)(55,244,122,57)(56,324,123,148)(58,281,245,174)(60,283,247,176)(62,285,249,178)(64,287,251,180)(66,289,225,182)(68,291,227,184)(70,293,229,186)(72,295,231,188)(74,297,233,190)(76,299,235,192)(78,301,237,194)(80,303,239,196)(82,305,241,170)(84,307,243,172)(141,195,317,302)(143,169,319,304)(145,171,321,306)(147,173,323,308)(149,175,325,282)(151,177,327,284)(153,179,329,286)(155,181,331,288)(157,183,333,290)(159,185,335,292)(161,187,309,294)(163,189,311,296)(165,191,313,298)(167,193,315,300)(198,348,376,441)(200,350,378,443)(202,352,380,445)(204,354,382,447)(206,356,384,421)(208,358,386,423)(210,360,388,425)(212,362,390,427)(214,364,392,429)(216,338,366,431)(218,340,368,433)(220,342,370,435)(222,344,372,437)(224,346,374,439)(253,434,411,341)(255,436,413,343)(257,438,415,345)(259,440,417,347)(261,442,419,349)(263,444,393,351)(265,446,395,353)(267,448,397,355)(269,422,399,357)(271,424,401,359)(273,426,403,361)(275,428,405,363)(277,430,407,337)(279,432,409,339), (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), (1,177,15,191)(2,176,16,190)(3,175,17,189)(4,174,18,188)(5,173,19,187)(6,172,20,186)(7,171,21,185)(8,170,22,184)(9,169,23,183)(10,196,24,182)(11,195,25,181)(12,194,26,180)(13,193,27,179)(14,192,28,178)(29,441,43,427)(30,440,44,426)(31,439,45,425)(32,438,46,424)(33,437,47,423)(34,436,48,422)(35,435,49,421)(36,434,50,448)(37,433,51,447)(38,432,52,446)(39,431,53,445)(40,430,54,444)(41,429,55,443)(42,428,56,442)(57,378,71,392)(58,377,72,391)(59,376,73,390)(60,375,74,389)(61,374,75,388)(62,373,76,387)(63,372,77,386)(64,371,78,385)(65,370,79,384)(66,369,80,383)(67,368,81,382)(68,367,82,381)(69,366,83,380)(70,365,84,379)(85,285,99,299)(86,284,100,298)(87,283,101,297)(88,282,102,296)(89,281,103,295)(90,308,104,294)(91,307,105,293)(92,306,106,292)(93,305,107,291)(94,304,108,290)(95,303,109,289)(96,302,110,288)(97,301,111,287)(98,300,112,286)(113,359,127,345)(114,358,128,344)(115,357,129,343)(116,356,130,342)(117,355,131,341)(118,354,132,340)(119,353,133,339)(120,352,134,338)(121,351,135,337)(122,350,136,364)(123,349,137,363)(124,348,138,362)(125,347,139,361)(126,346,140,360)(141,268,155,254)(142,267,156,253)(143,266,157,280)(144,265,158,279)(145,264,159,278)(146,263,160,277)(147,262,161,276)(148,261,162,275)(149,260,163,274)(150,259,164,273)(151,258,165,272)(152,257,166,271)(153,256,167,270)(154,255,168,269)(197,233,211,247)(198,232,212,246)(199,231,213,245)(200,230,214,244)(201,229,215,243)(202,228,216,242)(203,227,217,241)(204,226,218,240)(205,225,219,239)(206,252,220,238)(207,251,221,237)(208,250,222,236)(209,249,223,235)(210,248,224,234)(309,406,323,420)(310,405,324,419)(311,404,325,418)(312,403,326,417)(313,402,327,416)(314,401,328,415)(315,400,329,414)(316,399,330,413)(317,398,331,412)(318,397,332,411)(319,396,333,410)(320,395,334,409)(321,394,335,408)(322,393,336,407) );
G=PermutationGroup([[(1,360,86,425),(2,426,87,361),(3,362,88,427),(4,428,89,363),(5,364,90,429),(6,430,91,337),(7,338,92,431),(8,432,93,339),(9,340,94,433),(10,434,95,341),(11,342,96,435),(12,436,97,343),(13,344,98,437),(14,438,99,345),(15,346,100,439),(16,440,101,347),(17,348,102,441),(18,442,103,349),(19,350,104,443),(20,444,105,351),(21,352,106,445),(22,446,107,353),(23,354,108,447),(24,448,109,355),(25,356,110,421),(26,422,111,357),(27,358,112,423),(28,424,85,359),(29,175,124,282),(30,283,125,176),(31,177,126,284),(32,285,127,178),(33,179,128,286),(34,287,129,180),(35,181,130,288),(36,289,131,182),(37,183,132,290),(38,291,133,184),(39,185,134,292),(40,293,135,186),(41,187,136,294),(42,295,137,188),(43,189,138,296),(44,297,139,190),(45,191,140,298),(46,299,113,192),(47,193,114,300),(48,301,115,194),(49,195,116,302),(50,303,117,196),(51,169,118,304),(52,305,119,170),(53,171,120,306),(54,307,121,172),(55,173,122,308),(56,281,123,174),(57,323,244,147),(58,148,245,324),(59,325,246,149),(60,150,247,326),(61,327,248,151),(62,152,249,328),(63,329,250,153),(64,154,251,330),(65,331,252,155),(66,156,225,332),(67,333,226,157),(68,158,227,334),(69,335,228,159),(70,160,229,336),(71,309,230,161),(72,162,231,310),(73,311,232,163),(74,164,233,312),(75,313,234,165),(76,166,235,314),(77,315,236,167),(78,168,237,316),(79,317,238,141),(80,142,239,318),(81,319,240,143),(82,144,241,320),(83,321,242,145),(84,146,243,322),(197,417,375,259),(198,260,376,418),(199,419,377,261),(200,262,378,420),(201,393,379,263),(202,264,380,394),(203,395,381,265),(204,266,382,396),(205,397,383,267),(206,268,384,398),(207,399,385,269),(208,270,386,400),(209,401,387,271),(210,272,388,402),(211,403,389,273),(212,274,390,404),(213,405,391,275),(214,276,392,406),(215,407,365,277),(216,278,366,408),(217,409,367,279),(218,280,368,410),(219,411,369,253),(220,254,370,412),(221,413,371,255),(222,256,372,414),(223,415,373,257),(224,258,374,416)], [(1,402,86,272),(2,389,87,211),(3,404,88,274),(4,391,89,213),(5,406,90,276),(6,365,91,215),(7,408,92,278),(8,367,93,217),(9,410,94,280),(10,369,95,219),(11,412,96,254),(12,371,97,221),(13,414,98,256),(14,373,99,223),(15,416,100,258),(16,375,101,197),(17,418,102,260),(18,377,103,199),(19,420,104,262),(20,379,105,201),(21,394,106,264),(22,381,107,203),(23,396,108,266),(24,383,109,205),(25,398,110,268),(26,385,111,207),(27,400,112,270),(28,387,85,209),(29,246,124,59),(30,326,125,150),(31,248,126,61),(32,328,127,152),(33,250,128,63),(34,330,129,154),(35,252,130,65),(36,332,131,156),(37,226,132,67),(38,334,133,158),(39,228,134,69),(40,336,135,160),(41,230,136,71),(42,310,137,162),(43,232,138,73),(44,312,139,164),(45,234,140,75),(46,314,113,166),(47,236,114,77),(48,316,115,168),(49,238,116,79),(50,318,117,142),(51,240,118,81),(52,320,119,144),(53,242,120,83),(54,322,121,146),(55,244,122,57),(56,324,123,148),(58,281,245,174),(60,283,247,176),(62,285,249,178),(64,287,251,180),(66,289,225,182),(68,291,227,184),(70,293,229,186),(72,295,231,188),(74,297,233,190),(76,299,235,192),(78,301,237,194),(80,303,239,196),(82,305,241,170),(84,307,243,172),(141,195,317,302),(143,169,319,304),(145,171,321,306),(147,173,323,308),(149,175,325,282),(151,177,327,284),(153,179,329,286),(155,181,331,288),(157,183,333,290),(159,185,335,292),(161,187,309,294),(163,189,311,296),(165,191,313,298),(167,193,315,300),(198,348,376,441),(200,350,378,443),(202,352,380,445),(204,354,382,447),(206,356,384,421),(208,358,386,423),(210,360,388,425),(212,362,390,427),(214,364,392,429),(216,338,366,431),(218,340,368,433),(220,342,370,435),(222,344,372,437),(224,346,374,439),(253,434,411,341),(255,436,413,343),(257,438,415,345),(259,440,417,347),(261,442,419,349),(263,444,393,351),(265,446,395,353),(267,448,397,355),(269,422,399,357),(271,424,401,359),(273,426,403,361),(275,428,405,363),(277,430,407,337),(279,432,409,339)], [(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)], [(1,177,15,191),(2,176,16,190),(3,175,17,189),(4,174,18,188),(5,173,19,187),(6,172,20,186),(7,171,21,185),(8,170,22,184),(9,169,23,183),(10,196,24,182),(11,195,25,181),(12,194,26,180),(13,193,27,179),(14,192,28,178),(29,441,43,427),(30,440,44,426),(31,439,45,425),(32,438,46,424),(33,437,47,423),(34,436,48,422),(35,435,49,421),(36,434,50,448),(37,433,51,447),(38,432,52,446),(39,431,53,445),(40,430,54,444),(41,429,55,443),(42,428,56,442),(57,378,71,392),(58,377,72,391),(59,376,73,390),(60,375,74,389),(61,374,75,388),(62,373,76,387),(63,372,77,386),(64,371,78,385),(65,370,79,384),(66,369,80,383),(67,368,81,382),(68,367,82,381),(69,366,83,380),(70,365,84,379),(85,285,99,299),(86,284,100,298),(87,283,101,297),(88,282,102,296),(89,281,103,295),(90,308,104,294),(91,307,105,293),(92,306,106,292),(93,305,107,291),(94,304,108,290),(95,303,109,289),(96,302,110,288),(97,301,111,287),(98,300,112,286),(113,359,127,345),(114,358,128,344),(115,357,129,343),(116,356,130,342),(117,355,131,341),(118,354,132,340),(119,353,133,339),(120,352,134,338),(121,351,135,337),(122,350,136,364),(123,349,137,363),(124,348,138,362),(125,347,139,361),(126,346,140,360),(141,268,155,254),(142,267,156,253),(143,266,157,280),(144,265,158,279),(145,264,159,278),(146,263,160,277),(147,262,161,276),(148,261,162,275),(149,260,163,274),(150,259,164,273),(151,258,165,272),(152,257,166,271),(153,256,167,270),(154,255,168,269),(197,233,211,247),(198,232,212,246),(199,231,213,245),(200,230,214,244),(201,229,215,243),(202,228,216,242),(203,227,217,241),(204,226,218,240),(205,225,219,239),(206,252,220,238),(207,251,221,237),(208,250,222,236),(209,249,223,235),(210,248,224,234),(309,406,323,420),(310,405,324,419),(311,404,325,418),(312,403,326,417),(313,402,327,416),(314,401,328,415),(315,400,329,414),(316,399,330,413),(317,398,331,412),(318,397,332,411),(319,396,333,410),(320,395,334,409),(321,394,335,408),(322,393,336,407)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | SD16 | C4○D4 | D14 | D14 | D14 | Dic14 | C8.C22 | D4⋊2D7 | D4×D7 | D7×SD16 | Q16⋊D7 |
| kernel | Q8⋊Dic14 | C4.Dic14 | Dic7⋊C8 | C8⋊Dic7 | Q8⋊Dic7 | C7×Q8⋊C4 | C28⋊Q8 | Q8×Dic7 | C2×Dic7 | C7×Q8 | Q8⋊C4 | Dic7 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Q8⋊Dic14 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 55 | 108 |
| 0 | 0 | 108 | 58 |
| 19 | 96 | 0 | 0 |
| 103 | 9 | 0 | 0 |
| 0 | 0 | 102 | 85 |
| 0 | 0 | 85 | 11 |
| 96 | 104 | 0 | 0 |
| 95 | 17 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,55,108,0,0,108,58],[19,103,0,0,96,9,0,0,0,0,102,85,0,0,85,11],[96,95,0,0,104,17,0,0,0,0,112,0,0,0,0,112] >;
Q8⋊Dic14 in GAP, Magma, Sage, TeX
Q_8\rtimes {\rm Dic}_{14} % in TeX
G:=Group("Q8:Dic14"); // GroupNames label
G:=SmallGroup(448,325);
// by ID
G=gap.SmallGroup(448,325);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,254,219,58,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^28=1,b^2=a^2,d^2=c^14,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations