metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊1Q8, Dic7.7SD16, C4.1(Q8×D7), C28⋊Q8.4C2, C28.9(C2×Q8), C4⋊C4.31D14, C7⋊3(Q8⋊Q8), C4.Q8.4D7, (C2×C8).135D14, C2.20(D7×SD16), C4.70(C4○D28), Dic7⋊C8.13C2, C14.35(C2×SD16), C14.Q16.5C2, C22.209(D4×D7), C4.Dic14.4C2, C28.166(C4○D4), (C2×C28).270C23, (C2×C56).282C22, (C2×Dic7).161D4, Dic7⋊3Q8.5C2, C14.35(C22⋊Q8), C2.12(D14⋊Q8), C28.44D4.13C2, C2.20(SD16⋊D7), C14.38(C8.C22), C4⋊Dic7.102C22, (C4×Dic7).28C22, (C2×Dic14).80C22, (C7×C4.Q8).9C2, (C2×C7⋊C8).52C22, (C2×C14).275(C2×D4), (C7×C4⋊C4).63C22, (C2×C4).373(C22×D7), SmallGroup(448,388)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, cac-1=a13, dad-1=a15, bc=cb, dbd-1=a7b, dcd-1=c-1 >
Subgroups: 460 in 96 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C4×Q8, C4⋊Q8, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, Q8⋊Q8, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×Dic14, C4.Dic14, C14.Q16, Dic7⋊C8, C28.44D4, C7×C4.Q8, Dic7⋊3Q8, C28⋊Q8, Dic14⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×SD16, C8.C22, C22×D7, Q8⋊Q8, C4○D28, D4×D7, Q8×D7, D14⋊Q8, D7×SD16, SD16⋊D7, Dic14⋊Q8
►Regular action on 448 points
Generators in S448
(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)
(1 104 15 90)(2 103 16 89)(3 102 17 88)(4 101 18 87)(5 100 19 86)(6 99 20 85)(7 98 21 112)(8 97 22 111)(9 96 23 110)(10 95 24 109)(11 94 25 108)(12 93 26 107)(13 92 27 106)(14 91 28 105)(29 319 43 333)(30 318 44 332)(31 317 45 331)(32 316 46 330)(33 315 47 329)(34 314 48 328)(35 313 49 327)(36 312 50 326)(37 311 51 325)(38 310 52 324)(39 309 53 323)(40 336 54 322)(41 335 55 321)(42 334 56 320)(57 446 71 432)(58 445 72 431)(59 444 73 430)(60 443 74 429)(61 442 75 428)(62 441 76 427)(63 440 77 426)(64 439 78 425)(65 438 79 424)(66 437 80 423)(67 436 81 422)(68 435 82 421)(69 434 83 448)(70 433 84 447)(113 223 127 209)(114 222 128 208)(115 221 129 207)(116 220 130 206)(117 219 131 205)(118 218 132 204)(119 217 133 203)(120 216 134 202)(121 215 135 201)(122 214 136 200)(123 213 137 199)(124 212 138 198)(125 211 139 197)(126 210 140 224)(141 180 155 194)(142 179 156 193)(143 178 157 192)(144 177 158 191)(145 176 159 190)(146 175 160 189)(147 174 161 188)(148 173 162 187)(149 172 163 186)(150 171 164 185)(151 170 165 184)(152 169 166 183)(153 196 167 182)(154 195 168 181)(225 349 239 363)(226 348 240 362)(227 347 241 361)(228 346 242 360)(229 345 243 359)(230 344 244 358)(231 343 245 357)(232 342 246 356)(233 341 247 355)(234 340 248 354)(235 339 249 353)(236 338 250 352)(237 337 251 351)(238 364 252 350)(253 301 267 287)(254 300 268 286)(255 299 269 285)(256 298 270 284)(257 297 271 283)(258 296 272 282)(259 295 273 281)(260 294 274 308)(261 293 275 307)(262 292 276 306)(263 291 277 305)(264 290 278 304)(265 289 279 303)(266 288 280 302)(365 420 379 406)(366 419 380 405)(367 418 381 404)(368 417 382 403)(369 416 383 402)(370 415 384 401)(371 414 385 400)(372 413 386 399)(373 412 387 398)(374 411 388 397)(375 410 389 396)(376 409 390 395)(377 408 391 394)(378 407 392 393)
(1 305 249 208)(2 290 250 221)(3 303 251 206)(4 288 252 219)(5 301 225 204)(6 286 226 217)(7 299 227 202)(8 284 228 215)(9 297 229 200)(10 282 230 213)(11 295 231 198)(12 308 232 211)(13 293 233 224)(14 306 234 209)(15 291 235 222)(16 304 236 207)(17 289 237 220)(18 302 238 205)(19 287 239 218)(20 300 240 203)(21 285 241 216)(22 298 242 201)(23 283 243 214)(24 296 244 199)(25 281 245 212)(26 294 246 197)(27 307 247 210)(28 292 248 223)(29 168 428 406)(30 153 429 419)(31 166 430 404)(32 151 431 417)(33 164 432 402)(34 149 433 415)(35 162 434 400)(36 147 435 413)(37 160 436 398)(38 145 437 411)(39 158 438 396)(40 143 439 409)(41 156 440 394)(42 141 441 407)(43 154 442 420)(44 167 443 405)(45 152 444 418)(46 165 445 403)(47 150 446 416)(48 163 447 401)(49 148 448 414)(50 161 421 399)(51 146 422 412)(52 159 423 397)(53 144 424 410)(54 157 425 395)(55 142 426 408)(56 155 427 393)(57 369 315 185)(58 382 316 170)(59 367 317 183)(60 380 318 196)(61 365 319 181)(62 378 320 194)(63 391 321 179)(64 376 322 192)(65 389 323 177)(66 374 324 190)(67 387 325 175)(68 372 326 188)(69 385 327 173)(70 370 328 186)(71 383 329 171)(72 368 330 184)(73 381 331 169)(74 366 332 182)(75 379 333 195)(76 392 334 180)(77 377 335 193)(78 390 336 178)(79 375 309 191)(80 388 310 176)(81 373 311 189)(82 386 312 174)(83 371 313 187)(84 384 314 172)(85 268 362 119)(86 253 363 132)(87 266 364 117)(88 279 337 130)(89 264 338 115)(90 277 339 128)(91 262 340 113)(92 275 341 126)(93 260 342 139)(94 273 343 124)(95 258 344 137)(96 271 345 122)(97 256 346 135)(98 269 347 120)(99 254 348 133)(100 267 349 118)(101 280 350 131)(102 265 351 116)(103 278 352 129)(104 263 353 114)(105 276 354 127)(106 261 355 140)(107 274 356 125)(108 259 357 138)(109 272 358 123)(110 257 359 136)(111 270 360 121)(112 255 361 134)
(1 311 249 81)(2 326 250 68)(3 313 251 83)(4 328 252 70)(5 315 225 57)(6 330 226 72)(7 317 227 59)(8 332 228 74)(9 319 229 61)(10 334 230 76)(11 321 231 63)(12 336 232 78)(13 323 233 65)(14 310 234 80)(15 325 235 67)(16 312 236 82)(17 327 237 69)(18 314 238 84)(19 329 239 71)(20 316 240 58)(21 331 241 73)(22 318 242 60)(23 333 243 75)(24 320 244 62)(25 335 245 77)(26 322 246 64)(27 309 247 79)(28 324 248 66)(29 352 428 103)(30 339 429 90)(31 354 430 105)(32 341 431 92)(33 356 432 107)(34 343 433 94)(35 358 434 109)(36 345 435 96)(37 360 436 111)(38 347 437 98)(39 362 438 85)(40 349 439 100)(41 364 440 87)(42 351 441 102)(43 338 442 89)(44 353 443 104)(45 340 444 91)(46 355 445 106)(47 342 446 93)(48 357 447 108)(49 344 448 95)(50 359 421 110)(51 346 422 97)(52 361 423 112)(53 348 424 99)(54 363 425 86)(55 350 426 101)(56 337 427 88)(113 152 262 418)(114 167 263 405)(115 154 264 420)(116 141 265 407)(117 156 266 394)(118 143 267 409)(119 158 268 396)(120 145 269 411)(121 160 270 398)(122 147 271 413)(123 162 272 400)(124 149 273 415)(125 164 274 402)(126 151 275 417)(127 166 276 404)(128 153 277 419)(129 168 278 406)(130 155 279 393)(131 142 280 408)(132 157 253 395)(133 144 254 410)(134 159 255 397)(135 146 256 412)(136 161 257 399)(137 148 258 414)(138 163 259 401)(139 150 260 416)(140 165 261 403)(169 285 381 216)(170 300 382 203)(171 287 383 218)(172 302 384 205)(173 289 385 220)(174 304 386 207)(175 291 387 222)(176 306 388 209)(177 293 389 224)(178 308 390 211)(179 295 391 198)(180 282 392 213)(181 297 365 200)(182 284 366 215)(183 299 367 202)(184 286 368 217)(185 301 369 204)(186 288 370 219)(187 303 371 206)(188 290 372 221)(189 305 373 208)(190 292 374 223)(191 307 375 210)(192 294 376 197)(193 281 377 212)(194 296 378 199)(195 283 379 214)(196 298 380 201)
G:=sub<Sym(448)| (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), (1,104,15,90)(2,103,16,89)(3,102,17,88)(4,101,18,87)(5,100,19,86)(6,99,20,85)(7,98,21,112)(8,97,22,111)(9,96,23,110)(10,95,24,109)(11,94,25,108)(12,93,26,107)(13,92,27,106)(14,91,28,105)(29,319,43,333)(30,318,44,332)(31,317,45,331)(32,316,46,330)(33,315,47,329)(34,314,48,328)(35,313,49,327)(36,312,50,326)(37,311,51,325)(38,310,52,324)(39,309,53,323)(40,336,54,322)(41,335,55,321)(42,334,56,320)(57,446,71,432)(58,445,72,431)(59,444,73,430)(60,443,74,429)(61,442,75,428)(62,441,76,427)(63,440,77,426)(64,439,78,425)(65,438,79,424)(66,437,80,423)(67,436,81,422)(68,435,82,421)(69,434,83,448)(70,433,84,447)(113,223,127,209)(114,222,128,208)(115,221,129,207)(116,220,130,206)(117,219,131,205)(118,218,132,204)(119,217,133,203)(120,216,134,202)(121,215,135,201)(122,214,136,200)(123,213,137,199)(124,212,138,198)(125,211,139,197)(126,210,140,224)(141,180,155,194)(142,179,156,193)(143,178,157,192)(144,177,158,191)(145,176,159,190)(146,175,160,189)(147,174,161,188)(148,173,162,187)(149,172,163,186)(150,171,164,185)(151,170,165,184)(152,169,166,183)(153,196,167,182)(154,195,168,181)(225,349,239,363)(226,348,240,362)(227,347,241,361)(228,346,242,360)(229,345,243,359)(230,344,244,358)(231,343,245,357)(232,342,246,356)(233,341,247,355)(234,340,248,354)(235,339,249,353)(236,338,250,352)(237,337,251,351)(238,364,252,350)(253,301,267,287)(254,300,268,286)(255,299,269,285)(256,298,270,284)(257,297,271,283)(258,296,272,282)(259,295,273,281)(260,294,274,308)(261,293,275,307)(262,292,276,306)(263,291,277,305)(264,290,278,304)(265,289,279,303)(266,288,280,302)(365,420,379,406)(366,419,380,405)(367,418,381,404)(368,417,382,403)(369,416,383,402)(370,415,384,401)(371,414,385,400)(372,413,386,399)(373,412,387,398)(374,411,388,397)(375,410,389,396)(376,409,390,395)(377,408,391,394)(378,407,392,393), (1,305,249,208)(2,290,250,221)(3,303,251,206)(4,288,252,219)(5,301,225,204)(6,286,226,217)(7,299,227,202)(8,284,228,215)(9,297,229,200)(10,282,230,213)(11,295,231,198)(12,308,232,211)(13,293,233,224)(14,306,234,209)(15,291,235,222)(16,304,236,207)(17,289,237,220)(18,302,238,205)(19,287,239,218)(20,300,240,203)(21,285,241,216)(22,298,242,201)(23,283,243,214)(24,296,244,199)(25,281,245,212)(26,294,246,197)(27,307,247,210)(28,292,248,223)(29,168,428,406)(30,153,429,419)(31,166,430,404)(32,151,431,417)(33,164,432,402)(34,149,433,415)(35,162,434,400)(36,147,435,413)(37,160,436,398)(38,145,437,411)(39,158,438,396)(40,143,439,409)(41,156,440,394)(42,141,441,407)(43,154,442,420)(44,167,443,405)(45,152,444,418)(46,165,445,403)(47,150,446,416)(48,163,447,401)(49,148,448,414)(50,161,421,399)(51,146,422,412)(52,159,423,397)(53,144,424,410)(54,157,425,395)(55,142,426,408)(56,155,427,393)(57,369,315,185)(58,382,316,170)(59,367,317,183)(60,380,318,196)(61,365,319,181)(62,378,320,194)(63,391,321,179)(64,376,322,192)(65,389,323,177)(66,374,324,190)(67,387,325,175)(68,372,326,188)(69,385,327,173)(70,370,328,186)(71,383,329,171)(72,368,330,184)(73,381,331,169)(74,366,332,182)(75,379,333,195)(76,392,334,180)(77,377,335,193)(78,390,336,178)(79,375,309,191)(80,388,310,176)(81,373,311,189)(82,386,312,174)(83,371,313,187)(84,384,314,172)(85,268,362,119)(86,253,363,132)(87,266,364,117)(88,279,337,130)(89,264,338,115)(90,277,339,128)(91,262,340,113)(92,275,341,126)(93,260,342,139)(94,273,343,124)(95,258,344,137)(96,271,345,122)(97,256,346,135)(98,269,347,120)(99,254,348,133)(100,267,349,118)(101,280,350,131)(102,265,351,116)(103,278,352,129)(104,263,353,114)(105,276,354,127)(106,261,355,140)(107,274,356,125)(108,259,357,138)(109,272,358,123)(110,257,359,136)(111,270,360,121)(112,255,361,134), (1,311,249,81)(2,326,250,68)(3,313,251,83)(4,328,252,70)(5,315,225,57)(6,330,226,72)(7,317,227,59)(8,332,228,74)(9,319,229,61)(10,334,230,76)(11,321,231,63)(12,336,232,78)(13,323,233,65)(14,310,234,80)(15,325,235,67)(16,312,236,82)(17,327,237,69)(18,314,238,84)(19,329,239,71)(20,316,240,58)(21,331,241,73)(22,318,242,60)(23,333,243,75)(24,320,244,62)(25,335,245,77)(26,322,246,64)(27,309,247,79)(28,324,248,66)(29,352,428,103)(30,339,429,90)(31,354,430,105)(32,341,431,92)(33,356,432,107)(34,343,433,94)(35,358,434,109)(36,345,435,96)(37,360,436,111)(38,347,437,98)(39,362,438,85)(40,349,439,100)(41,364,440,87)(42,351,441,102)(43,338,442,89)(44,353,443,104)(45,340,444,91)(46,355,445,106)(47,342,446,93)(48,357,447,108)(49,344,448,95)(50,359,421,110)(51,346,422,97)(52,361,423,112)(53,348,424,99)(54,363,425,86)(55,350,426,101)(56,337,427,88)(113,152,262,418)(114,167,263,405)(115,154,264,420)(116,141,265,407)(117,156,266,394)(118,143,267,409)(119,158,268,396)(120,145,269,411)(121,160,270,398)(122,147,271,413)(123,162,272,400)(124,149,273,415)(125,164,274,402)(126,151,275,417)(127,166,276,404)(128,153,277,419)(129,168,278,406)(130,155,279,393)(131,142,280,408)(132,157,253,395)(133,144,254,410)(134,159,255,397)(135,146,256,412)(136,161,257,399)(137,148,258,414)(138,163,259,401)(139,150,260,416)(140,165,261,403)(169,285,381,216)(170,300,382,203)(171,287,383,218)(172,302,384,205)(173,289,385,220)(174,304,386,207)(175,291,387,222)(176,306,388,209)(177,293,389,224)(178,308,390,211)(179,295,391,198)(180,282,392,213)(181,297,365,200)(182,284,366,215)(183,299,367,202)(184,286,368,217)(185,301,369,204)(186,288,370,219)(187,303,371,206)(188,290,372,221)(189,305,373,208)(190,292,374,223)(191,307,375,210)(192,294,376,197)(193,281,377,212)(194,296,378,199)(195,283,379,214)(196,298,380,201)>;
G:=Group( (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), (1,104,15,90)(2,103,16,89)(3,102,17,88)(4,101,18,87)(5,100,19,86)(6,99,20,85)(7,98,21,112)(8,97,22,111)(9,96,23,110)(10,95,24,109)(11,94,25,108)(12,93,26,107)(13,92,27,106)(14,91,28,105)(29,319,43,333)(30,318,44,332)(31,317,45,331)(32,316,46,330)(33,315,47,329)(34,314,48,328)(35,313,49,327)(36,312,50,326)(37,311,51,325)(38,310,52,324)(39,309,53,323)(40,336,54,322)(41,335,55,321)(42,334,56,320)(57,446,71,432)(58,445,72,431)(59,444,73,430)(60,443,74,429)(61,442,75,428)(62,441,76,427)(63,440,77,426)(64,439,78,425)(65,438,79,424)(66,437,80,423)(67,436,81,422)(68,435,82,421)(69,434,83,448)(70,433,84,447)(113,223,127,209)(114,222,128,208)(115,221,129,207)(116,220,130,206)(117,219,131,205)(118,218,132,204)(119,217,133,203)(120,216,134,202)(121,215,135,201)(122,214,136,200)(123,213,137,199)(124,212,138,198)(125,211,139,197)(126,210,140,224)(141,180,155,194)(142,179,156,193)(143,178,157,192)(144,177,158,191)(145,176,159,190)(146,175,160,189)(147,174,161,188)(148,173,162,187)(149,172,163,186)(150,171,164,185)(151,170,165,184)(152,169,166,183)(153,196,167,182)(154,195,168,181)(225,349,239,363)(226,348,240,362)(227,347,241,361)(228,346,242,360)(229,345,243,359)(230,344,244,358)(231,343,245,357)(232,342,246,356)(233,341,247,355)(234,340,248,354)(235,339,249,353)(236,338,250,352)(237,337,251,351)(238,364,252,350)(253,301,267,287)(254,300,268,286)(255,299,269,285)(256,298,270,284)(257,297,271,283)(258,296,272,282)(259,295,273,281)(260,294,274,308)(261,293,275,307)(262,292,276,306)(263,291,277,305)(264,290,278,304)(265,289,279,303)(266,288,280,302)(365,420,379,406)(366,419,380,405)(367,418,381,404)(368,417,382,403)(369,416,383,402)(370,415,384,401)(371,414,385,400)(372,413,386,399)(373,412,387,398)(374,411,388,397)(375,410,389,396)(376,409,390,395)(377,408,391,394)(378,407,392,393), (1,305,249,208)(2,290,250,221)(3,303,251,206)(4,288,252,219)(5,301,225,204)(6,286,226,217)(7,299,227,202)(8,284,228,215)(9,297,229,200)(10,282,230,213)(11,295,231,198)(12,308,232,211)(13,293,233,224)(14,306,234,209)(15,291,235,222)(16,304,236,207)(17,289,237,220)(18,302,238,205)(19,287,239,218)(20,300,240,203)(21,285,241,216)(22,298,242,201)(23,283,243,214)(24,296,244,199)(25,281,245,212)(26,294,246,197)(27,307,247,210)(28,292,248,223)(29,168,428,406)(30,153,429,419)(31,166,430,404)(32,151,431,417)(33,164,432,402)(34,149,433,415)(35,162,434,400)(36,147,435,413)(37,160,436,398)(38,145,437,411)(39,158,438,396)(40,143,439,409)(41,156,440,394)(42,141,441,407)(43,154,442,420)(44,167,443,405)(45,152,444,418)(46,165,445,403)(47,150,446,416)(48,163,447,401)(49,148,448,414)(50,161,421,399)(51,146,422,412)(52,159,423,397)(53,144,424,410)(54,157,425,395)(55,142,426,408)(56,155,427,393)(57,369,315,185)(58,382,316,170)(59,367,317,183)(60,380,318,196)(61,365,319,181)(62,378,320,194)(63,391,321,179)(64,376,322,192)(65,389,323,177)(66,374,324,190)(67,387,325,175)(68,372,326,188)(69,385,327,173)(70,370,328,186)(71,383,329,171)(72,368,330,184)(73,381,331,169)(74,366,332,182)(75,379,333,195)(76,392,334,180)(77,377,335,193)(78,390,336,178)(79,375,309,191)(80,388,310,176)(81,373,311,189)(82,386,312,174)(83,371,313,187)(84,384,314,172)(85,268,362,119)(86,253,363,132)(87,266,364,117)(88,279,337,130)(89,264,338,115)(90,277,339,128)(91,262,340,113)(92,275,341,126)(93,260,342,139)(94,273,343,124)(95,258,344,137)(96,271,345,122)(97,256,346,135)(98,269,347,120)(99,254,348,133)(100,267,349,118)(101,280,350,131)(102,265,351,116)(103,278,352,129)(104,263,353,114)(105,276,354,127)(106,261,355,140)(107,274,356,125)(108,259,357,138)(109,272,358,123)(110,257,359,136)(111,270,360,121)(112,255,361,134), (1,311,249,81)(2,326,250,68)(3,313,251,83)(4,328,252,70)(5,315,225,57)(6,330,226,72)(7,317,227,59)(8,332,228,74)(9,319,229,61)(10,334,230,76)(11,321,231,63)(12,336,232,78)(13,323,233,65)(14,310,234,80)(15,325,235,67)(16,312,236,82)(17,327,237,69)(18,314,238,84)(19,329,239,71)(20,316,240,58)(21,331,241,73)(22,318,242,60)(23,333,243,75)(24,320,244,62)(25,335,245,77)(26,322,246,64)(27,309,247,79)(28,324,248,66)(29,352,428,103)(30,339,429,90)(31,354,430,105)(32,341,431,92)(33,356,432,107)(34,343,433,94)(35,358,434,109)(36,345,435,96)(37,360,436,111)(38,347,437,98)(39,362,438,85)(40,349,439,100)(41,364,440,87)(42,351,441,102)(43,338,442,89)(44,353,443,104)(45,340,444,91)(46,355,445,106)(47,342,446,93)(48,357,447,108)(49,344,448,95)(50,359,421,110)(51,346,422,97)(52,361,423,112)(53,348,424,99)(54,363,425,86)(55,350,426,101)(56,337,427,88)(113,152,262,418)(114,167,263,405)(115,154,264,420)(116,141,265,407)(117,156,266,394)(118,143,267,409)(119,158,268,396)(120,145,269,411)(121,160,270,398)(122,147,271,413)(123,162,272,400)(124,149,273,415)(125,164,274,402)(126,151,275,417)(127,166,276,404)(128,153,277,419)(129,168,278,406)(130,155,279,393)(131,142,280,408)(132,157,253,395)(133,144,254,410)(134,159,255,397)(135,146,256,412)(136,161,257,399)(137,148,258,414)(138,163,259,401)(139,150,260,416)(140,165,261,403)(169,285,381,216)(170,300,382,203)(171,287,383,218)(172,302,384,205)(173,289,385,220)(174,304,386,207)(175,291,387,222)(176,306,388,209)(177,293,389,224)(178,308,390,211)(179,295,391,198)(180,282,392,213)(181,297,365,200)(182,284,366,215)(183,299,367,202)(184,286,368,217)(185,301,369,204)(186,288,370,219)(187,303,371,206)(188,290,372,221)(189,305,373,208)(190,292,374,223)(191,307,375,210)(192,294,376,197)(193,281,377,212)(194,296,378,199)(195,283,379,214)(196,298,380,201) );
G=PermutationGroup([[(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)], [(1,104,15,90),(2,103,16,89),(3,102,17,88),(4,101,18,87),(5,100,19,86),(6,99,20,85),(7,98,21,112),(8,97,22,111),(9,96,23,110),(10,95,24,109),(11,94,25,108),(12,93,26,107),(13,92,27,106),(14,91,28,105),(29,319,43,333),(30,318,44,332),(31,317,45,331),(32,316,46,330),(33,315,47,329),(34,314,48,328),(35,313,49,327),(36,312,50,326),(37,311,51,325),(38,310,52,324),(39,309,53,323),(40,336,54,322),(41,335,55,321),(42,334,56,320),(57,446,71,432),(58,445,72,431),(59,444,73,430),(60,443,74,429),(61,442,75,428),(62,441,76,427),(63,440,77,426),(64,439,78,425),(65,438,79,424),(66,437,80,423),(67,436,81,422),(68,435,82,421),(69,434,83,448),(70,433,84,447),(113,223,127,209),(114,222,128,208),(115,221,129,207),(116,220,130,206),(117,219,131,205),(118,218,132,204),(119,217,133,203),(120,216,134,202),(121,215,135,201),(122,214,136,200),(123,213,137,199),(124,212,138,198),(125,211,139,197),(126,210,140,224),(141,180,155,194),(142,179,156,193),(143,178,157,192),(144,177,158,191),(145,176,159,190),(146,175,160,189),(147,174,161,188),(148,173,162,187),(149,172,163,186),(150,171,164,185),(151,170,165,184),(152,169,166,183),(153,196,167,182),(154,195,168,181),(225,349,239,363),(226,348,240,362),(227,347,241,361),(228,346,242,360),(229,345,243,359),(230,344,244,358),(231,343,245,357),(232,342,246,356),(233,341,247,355),(234,340,248,354),(235,339,249,353),(236,338,250,352),(237,337,251,351),(238,364,252,350),(253,301,267,287),(254,300,268,286),(255,299,269,285),(256,298,270,284),(257,297,271,283),(258,296,272,282),(259,295,273,281),(260,294,274,308),(261,293,275,307),(262,292,276,306),(263,291,277,305),(264,290,278,304),(265,289,279,303),(266,288,280,302),(365,420,379,406),(366,419,380,405),(367,418,381,404),(368,417,382,403),(369,416,383,402),(370,415,384,401),(371,414,385,400),(372,413,386,399),(373,412,387,398),(374,411,388,397),(375,410,389,396),(376,409,390,395),(377,408,391,394),(378,407,392,393)], [(1,305,249,208),(2,290,250,221),(3,303,251,206),(4,288,252,219),(5,301,225,204),(6,286,226,217),(7,299,227,202),(8,284,228,215),(9,297,229,200),(10,282,230,213),(11,295,231,198),(12,308,232,211),(13,293,233,224),(14,306,234,209),(15,291,235,222),(16,304,236,207),(17,289,237,220),(18,302,238,205),(19,287,239,218),(20,300,240,203),(21,285,241,216),(22,298,242,201),(23,283,243,214),(24,296,244,199),(25,281,245,212),(26,294,246,197),(27,307,247,210),(28,292,248,223),(29,168,428,406),(30,153,429,419),(31,166,430,404),(32,151,431,417),(33,164,432,402),(34,149,433,415),(35,162,434,400),(36,147,435,413),(37,160,436,398),(38,145,437,411),(39,158,438,396),(40,143,439,409),(41,156,440,394),(42,141,441,407),(43,154,442,420),(44,167,443,405),(45,152,444,418),(46,165,445,403),(47,150,446,416),(48,163,447,401),(49,148,448,414),(50,161,421,399),(51,146,422,412),(52,159,423,397),(53,144,424,410),(54,157,425,395),(55,142,426,408),(56,155,427,393),(57,369,315,185),(58,382,316,170),(59,367,317,183),(60,380,318,196),(61,365,319,181),(62,378,320,194),(63,391,321,179),(64,376,322,192),(65,389,323,177),(66,374,324,190),(67,387,325,175),(68,372,326,188),(69,385,327,173),(70,370,328,186),(71,383,329,171),(72,368,330,184),(73,381,331,169),(74,366,332,182),(75,379,333,195),(76,392,334,180),(77,377,335,193),(78,390,336,178),(79,375,309,191),(80,388,310,176),(81,373,311,189),(82,386,312,174),(83,371,313,187),(84,384,314,172),(85,268,362,119),(86,253,363,132),(87,266,364,117),(88,279,337,130),(89,264,338,115),(90,277,339,128),(91,262,340,113),(92,275,341,126),(93,260,342,139),(94,273,343,124),(95,258,344,137),(96,271,345,122),(97,256,346,135),(98,269,347,120),(99,254,348,133),(100,267,349,118),(101,280,350,131),(102,265,351,116),(103,278,352,129),(104,263,353,114),(105,276,354,127),(106,261,355,140),(107,274,356,125),(108,259,357,138),(109,272,358,123),(110,257,359,136),(111,270,360,121),(112,255,361,134)], [(1,311,249,81),(2,326,250,68),(3,313,251,83),(4,328,252,70),(5,315,225,57),(6,330,226,72),(7,317,227,59),(8,332,228,74),(9,319,229,61),(10,334,230,76),(11,321,231,63),(12,336,232,78),(13,323,233,65),(14,310,234,80),(15,325,235,67),(16,312,236,82),(17,327,237,69),(18,314,238,84),(19,329,239,71),(20,316,240,58),(21,331,241,73),(22,318,242,60),(23,333,243,75),(24,320,244,62),(25,335,245,77),(26,322,246,64),(27,309,247,79),(28,324,248,66),(29,352,428,103),(30,339,429,90),(31,354,430,105),(32,341,431,92),(33,356,432,107),(34,343,433,94),(35,358,434,109),(36,345,435,96),(37,360,436,111),(38,347,437,98),(39,362,438,85),(40,349,439,100),(41,364,440,87),(42,351,441,102),(43,338,442,89),(44,353,443,104),(45,340,444,91),(46,355,445,106),(47,342,446,93),(48,357,447,108),(49,344,448,95),(50,359,421,110),(51,346,422,97),(52,361,423,112),(53,348,424,99),(54,363,425,86),(55,350,426,101),(56,337,427,88),(113,152,262,418),(114,167,263,405),(115,154,264,420),(116,141,265,407),(117,156,266,394),(118,143,267,409),(119,158,268,396),(120,145,269,411),(121,160,270,398),(122,147,271,413),(123,162,272,400),(124,149,273,415),(125,164,274,402),(126,151,275,417),(127,166,276,404),(128,153,277,419),(129,168,278,406),(130,155,279,393),(131,142,280,408),(132,157,253,395),(133,144,254,410),(134,159,255,397),(135,146,256,412),(136,161,257,399),(137,148,258,414),(138,163,259,401),(139,150,260,416),(140,165,261,403),(169,285,381,216),(170,300,382,203),(171,287,383,218),(172,302,384,205),(173,289,385,220),(174,304,386,207),(175,291,387,222),(176,306,388,209),(177,293,389,224),(178,308,390,211),(179,295,391,198),(180,282,392,213),(181,297,365,200),(182,284,366,215),(183,299,367,202),(184,286,368,217),(185,301,369,204),(186,288,370,219),(187,303,371,206),(188,290,372,221),(189,305,373,208),(190,292,374,223),(191,307,375,210),(192,294,376,197),(193,281,377,212),(194,296,378,199),(195,283,379,214),(196,298,380,201)]])
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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | SD16 | C4○D4 | D14 | D14 | C4○D28 | C8.C22 | Q8×D7 | D4×D7 | D7×SD16 | SD16⋊D7 |
| kernel | Dic14⋊Q8 | C4.Dic14 | C14.Q16 | Dic7⋊C8 | C28.44D4 | C7×C4.Q8 | Dic7⋊3Q8 | C28⋊Q8 | Dic14 | C2×Dic7 | C4.Q8 | Dic7 | C28 | C4⋊C4 | C2×C8 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 6 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Dic14⋊Q8 ►in GL6(𝔽113)
| 88 | 1 | 0 | 0 | 0 | 0 |
| 111 | 104 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 111 |
| 0 | 0 | 0 | 0 | 1 | 112 |
| 105 | 10 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 89 |
| 0 | 0 | 0 | 0 | 9 | 92 |
| 8 | 103 | 0 | 0 | 0 | 0 |
| 108 | 105 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 2 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 100 | 0 | 0 |
| 0 | 0 | 94 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 86 | 35 |
| 0 | 0 | 0 | 0 | 47 | 27 |
G:=sub<GL(6,GF(113))| [88,111,0,0,0,0,1,104,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,111,112],[105,5,0,0,0,0,10,8,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,21,9,0,0,0,0,89,92],[8,108,0,0,0,0,103,105,0,0,0,0,0,0,15,0,0,0,0,0,2,98,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,54,94,0,0,0,0,100,59,0,0,0,0,0,0,86,47,0,0,0,0,35,27] >;
Dic14⋊Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes Q_8 % in TeX
G:=Group("Dic14:Q8"); // GroupNames label
G:=SmallGroup(448,388);
// by ID
G=gap.SmallGroup(448,388);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,135,268,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=c^4=1,b^2=a^14,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^13,d*a*d^-1=a^15,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations