metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.1Dic14, C7⋊1(Q8.Q8), (C7×Q8).1Q8, C28.7(C2×Q8), C4⋊C4.19D14, (C2×C8).15D14, C56⋊1C4.7C2, Dic7⋊C8.4C2, Q8⋊C4.4D7, (Q8×Dic7).5C2, C4.7(C2×Dic14), C14.67(C4○D8), (C2×C56).15C22, (C2×Q8).101D14, Q8⋊Dic7.4C2, C22.190(D4×D7), C4.Dic14.3C2, C28.3Q8.1C2, C28.159(C4○D4), C2.6(Q8.D14), C4.84(D4⋊2D7), (C2×C28).236C23, (C2×Dic7).150D4, C14.13(C22⋊Q8), C4⋊Dic7.85C22, (Q8×C14).19C22, C2.15(SD16⋊D7), C14.33(C8.C22), (C4×Dic7).20C22, C2.18(C22⋊Dic14), (C2×C7⋊C8).31C22, (C2×C14).249(C2×D4), (C7×C4⋊C4).37C22, (C7×Q8⋊C4).4C2, (C2×C4).343(C22×D7), SmallGroup(448,330)
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=a2c-1 >
Subgroups: 372 in 90 conjugacy classes, 41 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, Dic7, C28, C28, C2×C14, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C7⋊C8, C56, 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, Q8×C14, C4.Dic14, Dic7⋊C8, C56⋊1C4, Q8⋊Dic7, C7×Q8⋊C4, C28.3Q8, Q8×Dic7, Q8.Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C4○D8, C8.C22, Dic14, C22×D7, Q8.Q8, C2×Dic14, D4×D7, D4⋊2D7, C22⋊Dic14, SD16⋊D7, Q8.D14, Q8.Dic14
►Regular action on 448 points
Generators in S448
(1 265 429 406)(2 407 430 266)(3 267 431 408)(4 409 432 268)(5 269 433 410)(6 411 434 270)(7 271 435 412)(8 413 436 272)(9 273 437 414)(10 415 438 274)(11 275 439 416)(12 417 440 276)(13 277 441 418)(14 419 442 278)(15 279 443 420)(16 393 444 280)(17 253 445 394)(18 395 446 254)(19 255 447 396)(20 397 448 256)(21 257 421 398)(22 399 422 258)(23 259 423 400)(24 401 424 260)(25 261 425 402)(26 403 426 262)(27 263 427 404)(28 405 428 264)(29 356 129 326)(30 327 130 357)(31 358 131 328)(32 329 132 359)(33 360 133 330)(34 331 134 361)(35 362 135 332)(36 333 136 363)(37 364 137 334)(38 335 138 337)(39 338 139 336)(40 309 140 339)(41 340 113 310)(42 311 114 341)(43 342 115 312)(44 313 116 343)(45 344 117 314)(46 315 118 345)(47 346 119 316)(48 317 120 347)(49 348 121 318)(50 319 122 349)(51 350 123 320)(52 321 124 351)(53 352 125 322)(54 323 126 353)(55 354 127 324)(56 325 128 355)(57 201 149 290)(58 291 150 202)(59 203 151 292)(60 293 152 204)(61 205 153 294)(62 295 154 206)(63 207 155 296)(64 297 156 208)(65 209 157 298)(66 299 158 210)(67 211 159 300)(68 301 160 212)(69 213 161 302)(70 303 162 214)(71 215 163 304)(72 305 164 216)(73 217 165 306)(74 307 166 218)(75 219 167 308)(76 281 168 220)(77 221 141 282)(78 283 142 222)(79 223 143 284)(80 285 144 224)(81 197 145 286)(82 287 146 198)(83 199 147 288)(84 289 148 200)(85 175 233 368)(86 369 234 176)(87 177 235 370)(88 371 236 178)(89 179 237 372)(90 373 238 180)(91 181 239 374)(92 375 240 182)(93 183 241 376)(94 377 242 184)(95 185 243 378)(96 379 244 186)(97 187 245 380)(98 381 246 188)(99 189 247 382)(100 383 248 190)(101 191 249 384)(102 385 250 192)(103 193 251 386)(104 387 252 194)(105 195 225 388)(106 389 226 196)(107 169 227 390)(108 391 228 170)(109 171 229 392)(110 365 230 172)(111 173 231 366)(112 367 232 174)
(1 224 429 285)(2 145 430 81)(3 198 431 287)(4 147 432 83)(5 200 433 289)(6 149 434 57)(7 202 435 291)(8 151 436 59)(9 204 437 293)(10 153 438 61)(11 206 439 295)(12 155 440 63)(13 208 441 297)(14 157 442 65)(15 210 443 299)(16 159 444 67)(17 212 445 301)(18 161 446 69)(19 214 447 303)(20 163 448 71)(21 216 421 305)(22 165 422 73)(23 218 423 307)(24 167 424 75)(25 220 425 281)(26 141 426 77)(27 222 427 283)(28 143 428 79)(29 383 129 190)(30 101 130 249)(31 385 131 192)(32 103 132 251)(33 387 133 194)(34 105 134 225)(35 389 135 196)(36 107 136 227)(37 391 137 170)(38 109 138 229)(39 365 139 172)(40 111 140 231)(41 367 113 174)(42 85 114 233)(43 369 115 176)(44 87 116 235)(45 371 117 178)(46 89 118 237)(47 373 119 180)(48 91 120 239)(49 375 121 182)(50 93 122 241)(51 377 123 184)(52 95 124 243)(53 379 125 186)(54 97 126 245)(55 381 127 188)(56 99 128 247)(58 271 150 412)(60 273 152 414)(62 275 154 416)(64 277 156 418)(66 279 158 420)(68 253 160 394)(70 255 162 396)(72 257 164 398)(74 259 166 400)(76 261 168 402)(78 263 142 404)(80 265 144 406)(82 267 146 408)(84 269 148 410)(86 312 234 342)(88 314 236 344)(90 316 238 346)(92 318 240 348)(94 320 242 350)(96 322 244 352)(98 324 246 354)(100 326 248 356)(102 328 250 358)(104 330 252 360)(106 332 226 362)(108 334 228 364)(110 336 230 338)(112 310 232 340)(169 333 390 363)(171 335 392 337)(173 309 366 339)(175 311 368 341)(177 313 370 343)(179 315 372 345)(181 317 374 347)(183 319 376 349)(185 321 378 351)(187 323 380 353)(189 325 382 355)(191 327 384 357)(193 329 386 359)(195 331 388 361)(197 266 286 407)(199 268 288 409)(201 270 290 411)(203 272 292 413)(205 274 294 415)(207 276 296 417)(209 278 298 419)(211 280 300 393)(213 254 302 395)(215 256 304 397)(217 258 306 399)(219 260 308 401)(221 262 282 403)(223 264 284 405)
(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)
(1 249 15 235)(2 100 16 86)(3 247 17 233)(4 98 18 112)(5 245 19 231)(6 96 20 110)(7 243 21 229)(8 94 22 108)(9 241 23 227)(10 92 24 106)(11 239 25 225)(12 90 26 104)(13 237 27 251)(14 88 28 102)(29 300 43 286)(30 210 44 224)(31 298 45 284)(32 208 46 222)(33 296 47 282)(34 206 48 220)(35 294 49 308)(36 204 50 218)(37 292 51 306)(38 202 52 216)(39 290 53 304)(40 200 54 214)(41 288 55 302)(42 198 56 212)(57 352 71 338)(58 321 72 335)(59 350 73 364)(60 319 74 333)(61 348 75 362)(62 317 76 331)(63 346 77 360)(64 315 78 329)(65 344 79 358)(66 313 80 327)(67 342 81 356)(68 311 82 325)(69 340 83 354)(70 309 84 323)(85 431 99 445)(87 429 101 443)(89 427 103 441)(91 425 105 439)(93 423 107 437)(95 421 109 435)(97 447 111 433)(113 199 127 213)(114 287 128 301)(115 197 129 211)(116 285 130 299)(117 223 131 209)(118 283 132 297)(119 221 133 207)(120 281 134 295)(121 219 135 205)(122 307 136 293)(123 217 137 203)(124 305 138 291)(125 215 139 201)(126 303 140 289)(141 330 155 316)(142 359 156 345)(143 328 157 314)(144 357 158 343)(145 326 159 312)(146 355 160 341)(147 324 161 310)(148 353 162 339)(149 322 163 336)(150 351 164 337)(151 320 165 334)(152 349 166 363)(153 318 167 332)(154 347 168 361)(169 414 183 400)(170 272 184 258)(171 412 185 398)(172 270 186 256)(173 410 187 396)(174 268 188 254)(175 408 189 394)(176 266 190 280)(177 406 191 420)(178 264 192 278)(179 404 193 418)(180 262 194 276)(181 402 195 416)(182 260 196 274)(226 438 240 424)(228 436 242 422)(230 434 244 448)(232 432 246 446)(234 430 248 444)(236 428 250 442)(238 426 252 440)(253 368 267 382)(255 366 269 380)(257 392 271 378)(259 390 273 376)(261 388 275 374)(263 386 277 372)(265 384 279 370)(365 411 379 397)(367 409 381 395)(369 407 383 393)(371 405 385 419)(373 403 387 417)(375 401 389 415)(377 399 391 413)
G:=sub<Sym(448)| (1,265,429,406)(2,407,430,266)(3,267,431,408)(4,409,432,268)(5,269,433,410)(6,411,434,270)(7,271,435,412)(8,413,436,272)(9,273,437,414)(10,415,438,274)(11,275,439,416)(12,417,440,276)(13,277,441,418)(14,419,442,278)(15,279,443,420)(16,393,444,280)(17,253,445,394)(18,395,446,254)(19,255,447,396)(20,397,448,256)(21,257,421,398)(22,399,422,258)(23,259,423,400)(24,401,424,260)(25,261,425,402)(26,403,426,262)(27,263,427,404)(28,405,428,264)(29,356,129,326)(30,327,130,357)(31,358,131,328)(32,329,132,359)(33,360,133,330)(34,331,134,361)(35,362,135,332)(36,333,136,363)(37,364,137,334)(38,335,138,337)(39,338,139,336)(40,309,140,339)(41,340,113,310)(42,311,114,341)(43,342,115,312)(44,313,116,343)(45,344,117,314)(46,315,118,345)(47,346,119,316)(48,317,120,347)(49,348,121,318)(50,319,122,349)(51,350,123,320)(52,321,124,351)(53,352,125,322)(54,323,126,353)(55,354,127,324)(56,325,128,355)(57,201,149,290)(58,291,150,202)(59,203,151,292)(60,293,152,204)(61,205,153,294)(62,295,154,206)(63,207,155,296)(64,297,156,208)(65,209,157,298)(66,299,158,210)(67,211,159,300)(68,301,160,212)(69,213,161,302)(70,303,162,214)(71,215,163,304)(72,305,164,216)(73,217,165,306)(74,307,166,218)(75,219,167,308)(76,281,168,220)(77,221,141,282)(78,283,142,222)(79,223,143,284)(80,285,144,224)(81,197,145,286)(82,287,146,198)(83,199,147,288)(84,289,148,200)(85,175,233,368)(86,369,234,176)(87,177,235,370)(88,371,236,178)(89,179,237,372)(90,373,238,180)(91,181,239,374)(92,375,240,182)(93,183,241,376)(94,377,242,184)(95,185,243,378)(96,379,244,186)(97,187,245,380)(98,381,246,188)(99,189,247,382)(100,383,248,190)(101,191,249,384)(102,385,250,192)(103,193,251,386)(104,387,252,194)(105,195,225,388)(106,389,226,196)(107,169,227,390)(108,391,228,170)(109,171,229,392)(110,365,230,172)(111,173,231,366)(112,367,232,174), (1,224,429,285)(2,145,430,81)(3,198,431,287)(4,147,432,83)(5,200,433,289)(6,149,434,57)(7,202,435,291)(8,151,436,59)(9,204,437,293)(10,153,438,61)(11,206,439,295)(12,155,440,63)(13,208,441,297)(14,157,442,65)(15,210,443,299)(16,159,444,67)(17,212,445,301)(18,161,446,69)(19,214,447,303)(20,163,448,71)(21,216,421,305)(22,165,422,73)(23,218,423,307)(24,167,424,75)(25,220,425,281)(26,141,426,77)(27,222,427,283)(28,143,428,79)(29,383,129,190)(30,101,130,249)(31,385,131,192)(32,103,132,251)(33,387,133,194)(34,105,134,225)(35,389,135,196)(36,107,136,227)(37,391,137,170)(38,109,138,229)(39,365,139,172)(40,111,140,231)(41,367,113,174)(42,85,114,233)(43,369,115,176)(44,87,116,235)(45,371,117,178)(46,89,118,237)(47,373,119,180)(48,91,120,239)(49,375,121,182)(50,93,122,241)(51,377,123,184)(52,95,124,243)(53,379,125,186)(54,97,126,245)(55,381,127,188)(56,99,128,247)(58,271,150,412)(60,273,152,414)(62,275,154,416)(64,277,156,418)(66,279,158,420)(68,253,160,394)(70,255,162,396)(72,257,164,398)(74,259,166,400)(76,261,168,402)(78,263,142,404)(80,265,144,406)(82,267,146,408)(84,269,148,410)(86,312,234,342)(88,314,236,344)(90,316,238,346)(92,318,240,348)(94,320,242,350)(96,322,244,352)(98,324,246,354)(100,326,248,356)(102,328,250,358)(104,330,252,360)(106,332,226,362)(108,334,228,364)(110,336,230,338)(112,310,232,340)(169,333,390,363)(171,335,392,337)(173,309,366,339)(175,311,368,341)(177,313,370,343)(179,315,372,345)(181,317,374,347)(183,319,376,349)(185,321,378,351)(187,323,380,353)(189,325,382,355)(191,327,384,357)(193,329,386,359)(195,331,388,361)(197,266,286,407)(199,268,288,409)(201,270,290,411)(203,272,292,413)(205,274,294,415)(207,276,296,417)(209,278,298,419)(211,280,300,393)(213,254,302,395)(215,256,304,397)(217,258,306,399)(219,260,308,401)(221,262,282,403)(223,264,284,405), (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), (1,249,15,235)(2,100,16,86)(3,247,17,233)(4,98,18,112)(5,245,19,231)(6,96,20,110)(7,243,21,229)(8,94,22,108)(9,241,23,227)(10,92,24,106)(11,239,25,225)(12,90,26,104)(13,237,27,251)(14,88,28,102)(29,300,43,286)(30,210,44,224)(31,298,45,284)(32,208,46,222)(33,296,47,282)(34,206,48,220)(35,294,49,308)(36,204,50,218)(37,292,51,306)(38,202,52,216)(39,290,53,304)(40,200,54,214)(41,288,55,302)(42,198,56,212)(57,352,71,338)(58,321,72,335)(59,350,73,364)(60,319,74,333)(61,348,75,362)(62,317,76,331)(63,346,77,360)(64,315,78,329)(65,344,79,358)(66,313,80,327)(67,342,81,356)(68,311,82,325)(69,340,83,354)(70,309,84,323)(85,431,99,445)(87,429,101,443)(89,427,103,441)(91,425,105,439)(93,423,107,437)(95,421,109,435)(97,447,111,433)(113,199,127,213)(114,287,128,301)(115,197,129,211)(116,285,130,299)(117,223,131,209)(118,283,132,297)(119,221,133,207)(120,281,134,295)(121,219,135,205)(122,307,136,293)(123,217,137,203)(124,305,138,291)(125,215,139,201)(126,303,140,289)(141,330,155,316)(142,359,156,345)(143,328,157,314)(144,357,158,343)(145,326,159,312)(146,355,160,341)(147,324,161,310)(148,353,162,339)(149,322,163,336)(150,351,164,337)(151,320,165,334)(152,349,166,363)(153,318,167,332)(154,347,168,361)(169,414,183,400)(170,272,184,258)(171,412,185,398)(172,270,186,256)(173,410,187,396)(174,268,188,254)(175,408,189,394)(176,266,190,280)(177,406,191,420)(178,264,192,278)(179,404,193,418)(180,262,194,276)(181,402,195,416)(182,260,196,274)(226,438,240,424)(228,436,242,422)(230,434,244,448)(232,432,246,446)(234,430,248,444)(236,428,250,442)(238,426,252,440)(253,368,267,382)(255,366,269,380)(257,392,271,378)(259,390,273,376)(261,388,275,374)(263,386,277,372)(265,384,279,370)(365,411,379,397)(367,409,381,395)(369,407,383,393)(371,405,385,419)(373,403,387,417)(375,401,389,415)(377,399,391,413)>;
G:=Group( (1,265,429,406)(2,407,430,266)(3,267,431,408)(4,409,432,268)(5,269,433,410)(6,411,434,270)(7,271,435,412)(8,413,436,272)(9,273,437,414)(10,415,438,274)(11,275,439,416)(12,417,440,276)(13,277,441,418)(14,419,442,278)(15,279,443,420)(16,393,444,280)(17,253,445,394)(18,395,446,254)(19,255,447,396)(20,397,448,256)(21,257,421,398)(22,399,422,258)(23,259,423,400)(24,401,424,260)(25,261,425,402)(26,403,426,262)(27,263,427,404)(28,405,428,264)(29,356,129,326)(30,327,130,357)(31,358,131,328)(32,329,132,359)(33,360,133,330)(34,331,134,361)(35,362,135,332)(36,333,136,363)(37,364,137,334)(38,335,138,337)(39,338,139,336)(40,309,140,339)(41,340,113,310)(42,311,114,341)(43,342,115,312)(44,313,116,343)(45,344,117,314)(46,315,118,345)(47,346,119,316)(48,317,120,347)(49,348,121,318)(50,319,122,349)(51,350,123,320)(52,321,124,351)(53,352,125,322)(54,323,126,353)(55,354,127,324)(56,325,128,355)(57,201,149,290)(58,291,150,202)(59,203,151,292)(60,293,152,204)(61,205,153,294)(62,295,154,206)(63,207,155,296)(64,297,156,208)(65,209,157,298)(66,299,158,210)(67,211,159,300)(68,301,160,212)(69,213,161,302)(70,303,162,214)(71,215,163,304)(72,305,164,216)(73,217,165,306)(74,307,166,218)(75,219,167,308)(76,281,168,220)(77,221,141,282)(78,283,142,222)(79,223,143,284)(80,285,144,224)(81,197,145,286)(82,287,146,198)(83,199,147,288)(84,289,148,200)(85,175,233,368)(86,369,234,176)(87,177,235,370)(88,371,236,178)(89,179,237,372)(90,373,238,180)(91,181,239,374)(92,375,240,182)(93,183,241,376)(94,377,242,184)(95,185,243,378)(96,379,244,186)(97,187,245,380)(98,381,246,188)(99,189,247,382)(100,383,248,190)(101,191,249,384)(102,385,250,192)(103,193,251,386)(104,387,252,194)(105,195,225,388)(106,389,226,196)(107,169,227,390)(108,391,228,170)(109,171,229,392)(110,365,230,172)(111,173,231,366)(112,367,232,174), (1,224,429,285)(2,145,430,81)(3,198,431,287)(4,147,432,83)(5,200,433,289)(6,149,434,57)(7,202,435,291)(8,151,436,59)(9,204,437,293)(10,153,438,61)(11,206,439,295)(12,155,440,63)(13,208,441,297)(14,157,442,65)(15,210,443,299)(16,159,444,67)(17,212,445,301)(18,161,446,69)(19,214,447,303)(20,163,448,71)(21,216,421,305)(22,165,422,73)(23,218,423,307)(24,167,424,75)(25,220,425,281)(26,141,426,77)(27,222,427,283)(28,143,428,79)(29,383,129,190)(30,101,130,249)(31,385,131,192)(32,103,132,251)(33,387,133,194)(34,105,134,225)(35,389,135,196)(36,107,136,227)(37,391,137,170)(38,109,138,229)(39,365,139,172)(40,111,140,231)(41,367,113,174)(42,85,114,233)(43,369,115,176)(44,87,116,235)(45,371,117,178)(46,89,118,237)(47,373,119,180)(48,91,120,239)(49,375,121,182)(50,93,122,241)(51,377,123,184)(52,95,124,243)(53,379,125,186)(54,97,126,245)(55,381,127,188)(56,99,128,247)(58,271,150,412)(60,273,152,414)(62,275,154,416)(64,277,156,418)(66,279,158,420)(68,253,160,394)(70,255,162,396)(72,257,164,398)(74,259,166,400)(76,261,168,402)(78,263,142,404)(80,265,144,406)(82,267,146,408)(84,269,148,410)(86,312,234,342)(88,314,236,344)(90,316,238,346)(92,318,240,348)(94,320,242,350)(96,322,244,352)(98,324,246,354)(100,326,248,356)(102,328,250,358)(104,330,252,360)(106,332,226,362)(108,334,228,364)(110,336,230,338)(112,310,232,340)(169,333,390,363)(171,335,392,337)(173,309,366,339)(175,311,368,341)(177,313,370,343)(179,315,372,345)(181,317,374,347)(183,319,376,349)(185,321,378,351)(187,323,380,353)(189,325,382,355)(191,327,384,357)(193,329,386,359)(195,331,388,361)(197,266,286,407)(199,268,288,409)(201,270,290,411)(203,272,292,413)(205,274,294,415)(207,276,296,417)(209,278,298,419)(211,280,300,393)(213,254,302,395)(215,256,304,397)(217,258,306,399)(219,260,308,401)(221,262,282,403)(223,264,284,405), (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), (1,249,15,235)(2,100,16,86)(3,247,17,233)(4,98,18,112)(5,245,19,231)(6,96,20,110)(7,243,21,229)(8,94,22,108)(9,241,23,227)(10,92,24,106)(11,239,25,225)(12,90,26,104)(13,237,27,251)(14,88,28,102)(29,300,43,286)(30,210,44,224)(31,298,45,284)(32,208,46,222)(33,296,47,282)(34,206,48,220)(35,294,49,308)(36,204,50,218)(37,292,51,306)(38,202,52,216)(39,290,53,304)(40,200,54,214)(41,288,55,302)(42,198,56,212)(57,352,71,338)(58,321,72,335)(59,350,73,364)(60,319,74,333)(61,348,75,362)(62,317,76,331)(63,346,77,360)(64,315,78,329)(65,344,79,358)(66,313,80,327)(67,342,81,356)(68,311,82,325)(69,340,83,354)(70,309,84,323)(85,431,99,445)(87,429,101,443)(89,427,103,441)(91,425,105,439)(93,423,107,437)(95,421,109,435)(97,447,111,433)(113,199,127,213)(114,287,128,301)(115,197,129,211)(116,285,130,299)(117,223,131,209)(118,283,132,297)(119,221,133,207)(120,281,134,295)(121,219,135,205)(122,307,136,293)(123,217,137,203)(124,305,138,291)(125,215,139,201)(126,303,140,289)(141,330,155,316)(142,359,156,345)(143,328,157,314)(144,357,158,343)(145,326,159,312)(146,355,160,341)(147,324,161,310)(148,353,162,339)(149,322,163,336)(150,351,164,337)(151,320,165,334)(152,349,166,363)(153,318,167,332)(154,347,168,361)(169,414,183,400)(170,272,184,258)(171,412,185,398)(172,270,186,256)(173,410,187,396)(174,268,188,254)(175,408,189,394)(176,266,190,280)(177,406,191,420)(178,264,192,278)(179,404,193,418)(180,262,194,276)(181,402,195,416)(182,260,196,274)(226,438,240,424)(228,436,242,422)(230,434,244,448)(232,432,246,446)(234,430,248,444)(236,428,250,442)(238,426,252,440)(253,368,267,382)(255,366,269,380)(257,392,271,378)(259,390,273,376)(261,388,275,374)(263,386,277,372)(265,384,279,370)(365,411,379,397)(367,409,381,395)(369,407,383,393)(371,405,385,419)(373,403,387,417)(375,401,389,415)(377,399,391,413) );
G=PermutationGroup([[(1,265,429,406),(2,407,430,266),(3,267,431,408),(4,409,432,268),(5,269,433,410),(6,411,434,270),(7,271,435,412),(8,413,436,272),(9,273,437,414),(10,415,438,274),(11,275,439,416),(12,417,440,276),(13,277,441,418),(14,419,442,278),(15,279,443,420),(16,393,444,280),(17,253,445,394),(18,395,446,254),(19,255,447,396),(20,397,448,256),(21,257,421,398),(22,399,422,258),(23,259,423,400),(24,401,424,260),(25,261,425,402),(26,403,426,262),(27,263,427,404),(28,405,428,264),(29,356,129,326),(30,327,130,357),(31,358,131,328),(32,329,132,359),(33,360,133,330),(34,331,134,361),(35,362,135,332),(36,333,136,363),(37,364,137,334),(38,335,138,337),(39,338,139,336),(40,309,140,339),(41,340,113,310),(42,311,114,341),(43,342,115,312),(44,313,116,343),(45,344,117,314),(46,315,118,345),(47,346,119,316),(48,317,120,347),(49,348,121,318),(50,319,122,349),(51,350,123,320),(52,321,124,351),(53,352,125,322),(54,323,126,353),(55,354,127,324),(56,325,128,355),(57,201,149,290),(58,291,150,202),(59,203,151,292),(60,293,152,204),(61,205,153,294),(62,295,154,206),(63,207,155,296),(64,297,156,208),(65,209,157,298),(66,299,158,210),(67,211,159,300),(68,301,160,212),(69,213,161,302),(70,303,162,214),(71,215,163,304),(72,305,164,216),(73,217,165,306),(74,307,166,218),(75,219,167,308),(76,281,168,220),(77,221,141,282),(78,283,142,222),(79,223,143,284),(80,285,144,224),(81,197,145,286),(82,287,146,198),(83,199,147,288),(84,289,148,200),(85,175,233,368),(86,369,234,176),(87,177,235,370),(88,371,236,178),(89,179,237,372),(90,373,238,180),(91,181,239,374),(92,375,240,182),(93,183,241,376),(94,377,242,184),(95,185,243,378),(96,379,244,186),(97,187,245,380),(98,381,246,188),(99,189,247,382),(100,383,248,190),(101,191,249,384),(102,385,250,192),(103,193,251,386),(104,387,252,194),(105,195,225,388),(106,389,226,196),(107,169,227,390),(108,391,228,170),(109,171,229,392),(110,365,230,172),(111,173,231,366),(112,367,232,174)], [(1,224,429,285),(2,145,430,81),(3,198,431,287),(4,147,432,83),(5,200,433,289),(6,149,434,57),(7,202,435,291),(8,151,436,59),(9,204,437,293),(10,153,438,61),(11,206,439,295),(12,155,440,63),(13,208,441,297),(14,157,442,65),(15,210,443,299),(16,159,444,67),(17,212,445,301),(18,161,446,69),(19,214,447,303),(20,163,448,71),(21,216,421,305),(22,165,422,73),(23,218,423,307),(24,167,424,75),(25,220,425,281),(26,141,426,77),(27,222,427,283),(28,143,428,79),(29,383,129,190),(30,101,130,249),(31,385,131,192),(32,103,132,251),(33,387,133,194),(34,105,134,225),(35,389,135,196),(36,107,136,227),(37,391,137,170),(38,109,138,229),(39,365,139,172),(40,111,140,231),(41,367,113,174),(42,85,114,233),(43,369,115,176),(44,87,116,235),(45,371,117,178),(46,89,118,237),(47,373,119,180),(48,91,120,239),(49,375,121,182),(50,93,122,241),(51,377,123,184),(52,95,124,243),(53,379,125,186),(54,97,126,245),(55,381,127,188),(56,99,128,247),(58,271,150,412),(60,273,152,414),(62,275,154,416),(64,277,156,418),(66,279,158,420),(68,253,160,394),(70,255,162,396),(72,257,164,398),(74,259,166,400),(76,261,168,402),(78,263,142,404),(80,265,144,406),(82,267,146,408),(84,269,148,410),(86,312,234,342),(88,314,236,344),(90,316,238,346),(92,318,240,348),(94,320,242,350),(96,322,244,352),(98,324,246,354),(100,326,248,356),(102,328,250,358),(104,330,252,360),(106,332,226,362),(108,334,228,364),(110,336,230,338),(112,310,232,340),(169,333,390,363),(171,335,392,337),(173,309,366,339),(175,311,368,341),(177,313,370,343),(179,315,372,345),(181,317,374,347),(183,319,376,349),(185,321,378,351),(187,323,380,353),(189,325,382,355),(191,327,384,357),(193,329,386,359),(195,331,388,361),(197,266,286,407),(199,268,288,409),(201,270,290,411),(203,272,292,413),(205,274,294,415),(207,276,296,417),(209,278,298,419),(211,280,300,393),(213,254,302,395),(215,256,304,397),(217,258,306,399),(219,260,308,401),(221,262,282,403),(223,264,284,405)], [(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)], [(1,249,15,235),(2,100,16,86),(3,247,17,233),(4,98,18,112),(5,245,19,231),(6,96,20,110),(7,243,21,229),(8,94,22,108),(9,241,23,227),(10,92,24,106),(11,239,25,225),(12,90,26,104),(13,237,27,251),(14,88,28,102),(29,300,43,286),(30,210,44,224),(31,298,45,284),(32,208,46,222),(33,296,47,282),(34,206,48,220),(35,294,49,308),(36,204,50,218),(37,292,51,306),(38,202,52,216),(39,290,53,304),(40,200,54,214),(41,288,55,302),(42,198,56,212),(57,352,71,338),(58,321,72,335),(59,350,73,364),(60,319,74,333),(61,348,75,362),(62,317,76,331),(63,346,77,360),(64,315,78,329),(65,344,79,358),(66,313,80,327),(67,342,81,356),(68,311,82,325),(69,340,83,354),(70,309,84,323),(85,431,99,445),(87,429,101,443),(89,427,103,441),(91,425,105,439),(93,423,107,437),(95,421,109,435),(97,447,111,433),(113,199,127,213),(114,287,128,301),(115,197,129,211),(116,285,130,299),(117,223,131,209),(118,283,132,297),(119,221,133,207),(120,281,134,295),(121,219,135,205),(122,307,136,293),(123,217,137,203),(124,305,138,291),(125,215,139,201),(126,303,140,289),(141,330,155,316),(142,359,156,345),(143,328,157,314),(144,357,158,343),(145,326,159,312),(146,355,160,341),(147,324,161,310),(148,353,162,339),(149,322,163,336),(150,351,164,337),(151,320,165,334),(152,349,166,363),(153,318,167,332),(154,347,168,361),(169,414,183,400),(170,272,184,258),(171,412,185,398),(172,270,186,256),(173,410,187,396),(174,268,188,254),(175,408,189,394),(176,266,190,280),(177,406,191,420),(178,264,192,278),(179,404,193,418),(180,262,194,276),(181,402,195,416),(182,260,196,274),(226,438,240,424),(228,436,242,422),(230,434,244,448),(232,432,246,446),(234,430,248,444),(236,428,250,442),(238,426,252,440),(253,368,267,382),(255,366,269,380),(257,392,271,378),(259,390,273,376),(261,388,275,374),(263,386,277,372),(265,384,279,370),(365,411,379,397),(367,409,381,395),(369,407,383,393),(371,405,385,419),(373,403,387,417),(375,401,389,415),(377,399,391,413)]])
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 | C4○D4 | D14 | D14 | D14 | C4○D8 | Dic14 | C8.C22 | D4⋊2D7 | D4×D7 | SD16⋊D7 | Q8.D14 |
| kernel | Q8.Dic14 | C4.Dic14 | Dic7⋊C8 | C56⋊1C4 | Q8⋊Dic7 | C7×Q8⋊C4 | C28.3Q8 | Q8×Dic7 | C2×Dic7 | C7×Q8 | Q8⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | C14 | Q8 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Q8.Dic14 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 112 |
| 0 | 0 | 0 | 0 | 2 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 63 | 55 |
| 0 | 0 | 0 | 0 | 10 | 50 |
| 89 | 1 | 0 | 0 | 0 | 0 |
| 111 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 90 | 91 | 0 | 0 |
| 0 | 0 | 96 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 32 |
| 0 | 0 | 0 | 0 | 9 | 84 |
| 80 | 1 | 0 | 0 | 0 | 0 |
| 42 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 28 | 0 | 0 |
| 0 | 0 | 6 | 82 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,112,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,63,10,0,0,0,0,55,50],[89,111,0,0,0,0,1,33,0,0,0,0,0,0,90,96,0,0,0,0,91,23,0,0,0,0,0,0,29,9,0,0,0,0,32,84],[80,42,0,0,0,0,1,33,0,0,0,0,0,0,31,6,0,0,0,0,28,82,0,0,0,0,0,0,15,0,0,0,0,0,0,15] >;
Q8.Dic14 in GAP, Magma, Sage, TeX
Q_8.{\rm Dic}_{14} % in TeX
G:=Group("Q8.Dic14"); // GroupNames label
G:=SmallGroup(448,330);
// by ID
G=gap.SmallGroup(448,330);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,926,219,58,851,438,102,18822]);
// 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=a^2*c^-1>;
// generators/relations