metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊4Q8, C28.14SD16, C42.42D14, C4⋊C8.10D7, C4.47(Q8×D7), C7⋊2(Q8⋊Q8), C8⋊Dic7.9C2, (C2×C4).138D28, (C2×C28).127D4, (C2×C8).134D14, C4.8(C56⋊C2), C28.106(C2×Q8), (C4×C28).77C22, C28⋊2Q8.12C2, C14.13(C2×SD16), C28.290(C4○D4), (C2×C28).761C23, (C2×C56).141C22, (C4×Dic14).13C2, C28.44D4.5C2, C22.124(C2×D28), C14.34(C22⋊Q8), C4⋊Dic7.22C22, C4.114(D4⋊2D7), C2.15(D14⋊2Q8), C2.21(C8.D14), C14.18(C8.C22), (C2×Dic14).217C22, (C7×C4⋊C8).15C2, C2.16(C2×C56⋊C2), (C2×C14).144(C2×D4), (C2×C4).706(C22×D7), SmallGroup(448,385)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊4Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=cac-1=a-1, ad=da, cbc-1=a21b, bd=db, dcd-1=c-1 >
Subgroups: 484 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C56, Dic14, Dic14, C2×Dic7, C2×C28, Q8⋊Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C2×Dic14, C28.44D4, C8⋊Dic7, C7×C4⋊C8, C4×Dic14, C28⋊2Q8, Dic14⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×SD16, C8.C22, D28, C22×D7, Q8⋊Q8, C56⋊C2, C2×D28, D4⋊2D7, Q8×D7, D14⋊2Q8, C2×C56⋊C2, C8.D14, Dic14⋊4Q8
►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)
(1 160 15 146)(2 159 16 145)(3 158 17 144)(4 157 18 143)(5 156 19 142)(6 155 20 141)(7 154 21 168)(8 153 22 167)(9 152 23 166)(10 151 24 165)(11 150 25 164)(12 149 26 163)(13 148 27 162)(14 147 28 161)(29 267 43 253)(30 266 44 280)(31 265 45 279)(32 264 46 278)(33 263 47 277)(34 262 48 276)(35 261 49 275)(36 260 50 274)(37 259 51 273)(38 258 52 272)(39 257 53 271)(40 256 54 270)(41 255 55 269)(42 254 56 268)(57 235 71 249)(58 234 72 248)(59 233 73 247)(60 232 74 246)(61 231 75 245)(62 230 76 244)(63 229 77 243)(64 228 78 242)(65 227 79 241)(66 226 80 240)(67 225 81 239)(68 252 82 238)(69 251 83 237)(70 250 84 236)(85 208 99 222)(86 207 100 221)(87 206 101 220)(88 205 102 219)(89 204 103 218)(90 203 104 217)(91 202 105 216)(92 201 106 215)(93 200 107 214)(94 199 108 213)(95 198 109 212)(96 197 110 211)(97 224 111 210)(98 223 112 209)(113 283 127 297)(114 282 128 296)(115 281 129 295)(116 308 130 294)(117 307 131 293)(118 306 132 292)(119 305 133 291)(120 304 134 290)(121 303 135 289)(122 302 136 288)(123 301 137 287)(124 300 138 286)(125 299 139 285)(126 298 140 284)(169 421 183 435)(170 448 184 434)(171 447 185 433)(172 446 186 432)(173 445 187 431)(174 444 188 430)(175 443 189 429)(176 442 190 428)(177 441 191 427)(178 440 192 426)(179 439 193 425)(180 438 194 424)(181 437 195 423)(182 436 196 422)(309 402 323 416)(310 401 324 415)(311 400 325 414)(312 399 326 413)(313 398 327 412)(314 397 328 411)(315 396 329 410)(316 395 330 409)(317 394 331 408)(318 393 332 407)(319 420 333 406)(320 419 334 405)(321 418 335 404)(322 417 336 403)(337 379 351 365)(338 378 352 392)(339 377 353 391)(340 376 354 390)(341 375 355 389)(342 374 356 388)(343 373 357 387)(344 372 358 386)(345 371 359 385)(346 370 360 384)(347 369 361 383)(348 368 362 382)(349 367 363 381)(350 366 364 380)
(1 227 253 86)(2 226 254 85)(3 225 255 112)(4 252 256 111)(5 251 257 110)(6 250 258 109)(7 249 259 108)(8 248 260 107)(9 247 261 106)(10 246 262 105)(11 245 263 104)(12 244 264 103)(13 243 265 102)(14 242 266 101)(15 241 267 100)(16 240 268 99)(17 239 269 98)(18 238 270 97)(19 237 271 96)(20 236 272 95)(21 235 273 94)(22 234 274 93)(23 233 275 92)(24 232 276 91)(25 231 277 90)(26 230 278 89)(27 229 279 88)(28 228 280 87)(29 214 160 58)(30 213 161 57)(31 212 162 84)(32 211 163 83)(33 210 164 82)(34 209 165 81)(35 208 166 80)(36 207 167 79)(37 206 168 78)(38 205 141 77)(39 204 142 76)(40 203 143 75)(41 202 144 74)(42 201 145 73)(43 200 146 72)(44 199 147 71)(45 198 148 70)(46 197 149 69)(47 224 150 68)(48 223 151 67)(49 222 152 66)(50 221 153 65)(51 220 154 64)(52 219 155 63)(53 218 156 62)(54 217 157 61)(55 216 158 60)(56 215 159 59)(113 398 367 445)(114 397 368 444)(115 396 369 443)(116 395 370 442)(117 394 371 441)(118 393 372 440)(119 420 373 439)(120 419 374 438)(121 418 375 437)(122 417 376 436)(123 416 377 435)(124 415 378 434)(125 414 379 433)(126 413 380 432)(127 412 381 431)(128 411 382 430)(129 410 383 429)(130 409 384 428)(131 408 385 427)(132 407 386 426)(133 406 387 425)(134 405 388 424)(135 404 389 423)(136 403 390 422)(137 402 391 421)(138 401 392 448)(139 400 365 447)(140 399 366 446)(169 308 309 360)(170 307 310 359)(171 306 311 358)(172 305 312 357)(173 304 313 356)(174 303 314 355)(175 302 315 354)(176 301 316 353)(177 300 317 352)(178 299 318 351)(179 298 319 350)(180 297 320 349)(181 296 321 348)(182 295 322 347)(183 294 323 346)(184 293 324 345)(185 292 325 344)(186 291 326 343)(187 290 327 342)(188 289 328 341)(189 288 329 340)(190 287 330 339)(191 286 331 338)(192 285 332 337)(193 284 333 364)(194 283 334 363)(195 282 335 362)(196 281 336 361)
(1 426 253 407)(2 427 254 408)(3 428 255 409)(4 429 256 410)(5 430 257 411)(6 431 258 412)(7 432 259 413)(8 433 260 414)(9 434 261 415)(10 435 262 416)(11 436 263 417)(12 437 264 418)(13 438 265 419)(14 439 266 420)(15 440 267 393)(16 441 268 394)(17 442 269 395)(18 443 270 396)(19 444 271 397)(20 445 272 398)(21 446 273 399)(22 447 274 400)(23 448 275 401)(24 421 276 402)(25 422 277 403)(26 423 278 404)(27 424 279 405)(28 425 280 406)(29 318 160 178)(30 319 161 179)(31 320 162 180)(32 321 163 181)(33 322 164 182)(34 323 165 183)(35 324 166 184)(36 325 167 185)(37 326 168 186)(38 327 141 187)(39 328 142 188)(40 329 143 189)(41 330 144 190)(42 331 145 191)(43 332 146 192)(44 333 147 193)(45 334 148 194)(46 335 149 195)(47 336 150 196)(48 309 151 169)(49 310 152 170)(50 311 153 171)(51 312 154 172)(52 313 155 173)(53 314 156 174)(54 315 157 175)(55 316 158 176)(56 317 159 177)(57 350 213 298)(58 351 214 299)(59 352 215 300)(60 353 216 301)(61 354 217 302)(62 355 218 303)(63 356 219 304)(64 357 220 305)(65 358 221 306)(66 359 222 307)(67 360 223 308)(68 361 224 281)(69 362 197 282)(70 363 198 283)(71 364 199 284)(72 337 200 285)(73 338 201 286)(74 339 202 287)(75 340 203 288)(76 341 204 289)(77 342 205 290)(78 343 206 291)(79 344 207 292)(80 345 208 293)(81 346 209 294)(82 347 210 295)(83 348 211 296)(84 349 212 297)(85 131 226 385)(86 132 227 386)(87 133 228 387)(88 134 229 388)(89 135 230 389)(90 136 231 390)(91 137 232 391)(92 138 233 392)(93 139 234 365)(94 140 235 366)(95 113 236 367)(96 114 237 368)(97 115 238 369)(98 116 239 370)(99 117 240 371)(100 118 241 372)(101 119 242 373)(102 120 243 374)(103 121 244 375)(104 122 245 376)(105 123 246 377)(106 124 247 378)(107 125 248 379)(108 126 249 380)(109 127 250 381)(110 128 251 382)(111 129 252 383)(112 130 225 384)
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), (1,160,15,146)(2,159,16,145)(3,158,17,144)(4,157,18,143)(5,156,19,142)(6,155,20,141)(7,154,21,168)(8,153,22,167)(9,152,23,166)(10,151,24,165)(11,150,25,164)(12,149,26,163)(13,148,27,162)(14,147,28,161)(29,267,43,253)(30,266,44,280)(31,265,45,279)(32,264,46,278)(33,263,47,277)(34,262,48,276)(35,261,49,275)(36,260,50,274)(37,259,51,273)(38,258,52,272)(39,257,53,271)(40,256,54,270)(41,255,55,269)(42,254,56,268)(57,235,71,249)(58,234,72,248)(59,233,73,247)(60,232,74,246)(61,231,75,245)(62,230,76,244)(63,229,77,243)(64,228,78,242)(65,227,79,241)(66,226,80,240)(67,225,81,239)(68,252,82,238)(69,251,83,237)(70,250,84,236)(85,208,99,222)(86,207,100,221)(87,206,101,220)(88,205,102,219)(89,204,103,218)(90,203,104,217)(91,202,105,216)(92,201,106,215)(93,200,107,214)(94,199,108,213)(95,198,109,212)(96,197,110,211)(97,224,111,210)(98,223,112,209)(113,283,127,297)(114,282,128,296)(115,281,129,295)(116,308,130,294)(117,307,131,293)(118,306,132,292)(119,305,133,291)(120,304,134,290)(121,303,135,289)(122,302,136,288)(123,301,137,287)(124,300,138,286)(125,299,139,285)(126,298,140,284)(169,421,183,435)(170,448,184,434)(171,447,185,433)(172,446,186,432)(173,445,187,431)(174,444,188,430)(175,443,189,429)(176,442,190,428)(177,441,191,427)(178,440,192,426)(179,439,193,425)(180,438,194,424)(181,437,195,423)(182,436,196,422)(309,402,323,416)(310,401,324,415)(311,400,325,414)(312,399,326,413)(313,398,327,412)(314,397,328,411)(315,396,329,410)(316,395,330,409)(317,394,331,408)(318,393,332,407)(319,420,333,406)(320,419,334,405)(321,418,335,404)(322,417,336,403)(337,379,351,365)(338,378,352,392)(339,377,353,391)(340,376,354,390)(341,375,355,389)(342,374,356,388)(343,373,357,387)(344,372,358,386)(345,371,359,385)(346,370,360,384)(347,369,361,383)(348,368,362,382)(349,367,363,381)(350,366,364,380), (1,227,253,86)(2,226,254,85)(3,225,255,112)(4,252,256,111)(5,251,257,110)(6,250,258,109)(7,249,259,108)(8,248,260,107)(9,247,261,106)(10,246,262,105)(11,245,263,104)(12,244,264,103)(13,243,265,102)(14,242,266,101)(15,241,267,100)(16,240,268,99)(17,239,269,98)(18,238,270,97)(19,237,271,96)(20,236,272,95)(21,235,273,94)(22,234,274,93)(23,233,275,92)(24,232,276,91)(25,231,277,90)(26,230,278,89)(27,229,279,88)(28,228,280,87)(29,214,160,58)(30,213,161,57)(31,212,162,84)(32,211,163,83)(33,210,164,82)(34,209,165,81)(35,208,166,80)(36,207,167,79)(37,206,168,78)(38,205,141,77)(39,204,142,76)(40,203,143,75)(41,202,144,74)(42,201,145,73)(43,200,146,72)(44,199,147,71)(45,198,148,70)(46,197,149,69)(47,224,150,68)(48,223,151,67)(49,222,152,66)(50,221,153,65)(51,220,154,64)(52,219,155,63)(53,218,156,62)(54,217,157,61)(55,216,158,60)(56,215,159,59)(113,398,367,445)(114,397,368,444)(115,396,369,443)(116,395,370,442)(117,394,371,441)(118,393,372,440)(119,420,373,439)(120,419,374,438)(121,418,375,437)(122,417,376,436)(123,416,377,435)(124,415,378,434)(125,414,379,433)(126,413,380,432)(127,412,381,431)(128,411,382,430)(129,410,383,429)(130,409,384,428)(131,408,385,427)(132,407,386,426)(133,406,387,425)(134,405,388,424)(135,404,389,423)(136,403,390,422)(137,402,391,421)(138,401,392,448)(139,400,365,447)(140,399,366,446)(169,308,309,360)(170,307,310,359)(171,306,311,358)(172,305,312,357)(173,304,313,356)(174,303,314,355)(175,302,315,354)(176,301,316,353)(177,300,317,352)(178,299,318,351)(179,298,319,350)(180,297,320,349)(181,296,321,348)(182,295,322,347)(183,294,323,346)(184,293,324,345)(185,292,325,344)(186,291,326,343)(187,290,327,342)(188,289,328,341)(189,288,329,340)(190,287,330,339)(191,286,331,338)(192,285,332,337)(193,284,333,364)(194,283,334,363)(195,282,335,362)(196,281,336,361), (1,426,253,407)(2,427,254,408)(3,428,255,409)(4,429,256,410)(5,430,257,411)(6,431,258,412)(7,432,259,413)(8,433,260,414)(9,434,261,415)(10,435,262,416)(11,436,263,417)(12,437,264,418)(13,438,265,419)(14,439,266,420)(15,440,267,393)(16,441,268,394)(17,442,269,395)(18,443,270,396)(19,444,271,397)(20,445,272,398)(21,446,273,399)(22,447,274,400)(23,448,275,401)(24,421,276,402)(25,422,277,403)(26,423,278,404)(27,424,279,405)(28,425,280,406)(29,318,160,178)(30,319,161,179)(31,320,162,180)(32,321,163,181)(33,322,164,182)(34,323,165,183)(35,324,166,184)(36,325,167,185)(37,326,168,186)(38,327,141,187)(39,328,142,188)(40,329,143,189)(41,330,144,190)(42,331,145,191)(43,332,146,192)(44,333,147,193)(45,334,148,194)(46,335,149,195)(47,336,150,196)(48,309,151,169)(49,310,152,170)(50,311,153,171)(51,312,154,172)(52,313,155,173)(53,314,156,174)(54,315,157,175)(55,316,158,176)(56,317,159,177)(57,350,213,298)(58,351,214,299)(59,352,215,300)(60,353,216,301)(61,354,217,302)(62,355,218,303)(63,356,219,304)(64,357,220,305)(65,358,221,306)(66,359,222,307)(67,360,223,308)(68,361,224,281)(69,362,197,282)(70,363,198,283)(71,364,199,284)(72,337,200,285)(73,338,201,286)(74,339,202,287)(75,340,203,288)(76,341,204,289)(77,342,205,290)(78,343,206,291)(79,344,207,292)(80,345,208,293)(81,346,209,294)(82,347,210,295)(83,348,211,296)(84,349,212,297)(85,131,226,385)(86,132,227,386)(87,133,228,387)(88,134,229,388)(89,135,230,389)(90,136,231,390)(91,137,232,391)(92,138,233,392)(93,139,234,365)(94,140,235,366)(95,113,236,367)(96,114,237,368)(97,115,238,369)(98,116,239,370)(99,117,240,371)(100,118,241,372)(101,119,242,373)(102,120,243,374)(103,121,244,375)(104,122,245,376)(105,123,246,377)(106,124,247,378)(107,125,248,379)(108,126,249,380)(109,127,250,381)(110,128,251,382)(111,129,252,383)(112,130,225,384)>;
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), (1,160,15,146)(2,159,16,145)(3,158,17,144)(4,157,18,143)(5,156,19,142)(6,155,20,141)(7,154,21,168)(8,153,22,167)(9,152,23,166)(10,151,24,165)(11,150,25,164)(12,149,26,163)(13,148,27,162)(14,147,28,161)(29,267,43,253)(30,266,44,280)(31,265,45,279)(32,264,46,278)(33,263,47,277)(34,262,48,276)(35,261,49,275)(36,260,50,274)(37,259,51,273)(38,258,52,272)(39,257,53,271)(40,256,54,270)(41,255,55,269)(42,254,56,268)(57,235,71,249)(58,234,72,248)(59,233,73,247)(60,232,74,246)(61,231,75,245)(62,230,76,244)(63,229,77,243)(64,228,78,242)(65,227,79,241)(66,226,80,240)(67,225,81,239)(68,252,82,238)(69,251,83,237)(70,250,84,236)(85,208,99,222)(86,207,100,221)(87,206,101,220)(88,205,102,219)(89,204,103,218)(90,203,104,217)(91,202,105,216)(92,201,106,215)(93,200,107,214)(94,199,108,213)(95,198,109,212)(96,197,110,211)(97,224,111,210)(98,223,112,209)(113,283,127,297)(114,282,128,296)(115,281,129,295)(116,308,130,294)(117,307,131,293)(118,306,132,292)(119,305,133,291)(120,304,134,290)(121,303,135,289)(122,302,136,288)(123,301,137,287)(124,300,138,286)(125,299,139,285)(126,298,140,284)(169,421,183,435)(170,448,184,434)(171,447,185,433)(172,446,186,432)(173,445,187,431)(174,444,188,430)(175,443,189,429)(176,442,190,428)(177,441,191,427)(178,440,192,426)(179,439,193,425)(180,438,194,424)(181,437,195,423)(182,436,196,422)(309,402,323,416)(310,401,324,415)(311,400,325,414)(312,399,326,413)(313,398,327,412)(314,397,328,411)(315,396,329,410)(316,395,330,409)(317,394,331,408)(318,393,332,407)(319,420,333,406)(320,419,334,405)(321,418,335,404)(322,417,336,403)(337,379,351,365)(338,378,352,392)(339,377,353,391)(340,376,354,390)(341,375,355,389)(342,374,356,388)(343,373,357,387)(344,372,358,386)(345,371,359,385)(346,370,360,384)(347,369,361,383)(348,368,362,382)(349,367,363,381)(350,366,364,380), (1,227,253,86)(2,226,254,85)(3,225,255,112)(4,252,256,111)(5,251,257,110)(6,250,258,109)(7,249,259,108)(8,248,260,107)(9,247,261,106)(10,246,262,105)(11,245,263,104)(12,244,264,103)(13,243,265,102)(14,242,266,101)(15,241,267,100)(16,240,268,99)(17,239,269,98)(18,238,270,97)(19,237,271,96)(20,236,272,95)(21,235,273,94)(22,234,274,93)(23,233,275,92)(24,232,276,91)(25,231,277,90)(26,230,278,89)(27,229,279,88)(28,228,280,87)(29,214,160,58)(30,213,161,57)(31,212,162,84)(32,211,163,83)(33,210,164,82)(34,209,165,81)(35,208,166,80)(36,207,167,79)(37,206,168,78)(38,205,141,77)(39,204,142,76)(40,203,143,75)(41,202,144,74)(42,201,145,73)(43,200,146,72)(44,199,147,71)(45,198,148,70)(46,197,149,69)(47,224,150,68)(48,223,151,67)(49,222,152,66)(50,221,153,65)(51,220,154,64)(52,219,155,63)(53,218,156,62)(54,217,157,61)(55,216,158,60)(56,215,159,59)(113,398,367,445)(114,397,368,444)(115,396,369,443)(116,395,370,442)(117,394,371,441)(118,393,372,440)(119,420,373,439)(120,419,374,438)(121,418,375,437)(122,417,376,436)(123,416,377,435)(124,415,378,434)(125,414,379,433)(126,413,380,432)(127,412,381,431)(128,411,382,430)(129,410,383,429)(130,409,384,428)(131,408,385,427)(132,407,386,426)(133,406,387,425)(134,405,388,424)(135,404,389,423)(136,403,390,422)(137,402,391,421)(138,401,392,448)(139,400,365,447)(140,399,366,446)(169,308,309,360)(170,307,310,359)(171,306,311,358)(172,305,312,357)(173,304,313,356)(174,303,314,355)(175,302,315,354)(176,301,316,353)(177,300,317,352)(178,299,318,351)(179,298,319,350)(180,297,320,349)(181,296,321,348)(182,295,322,347)(183,294,323,346)(184,293,324,345)(185,292,325,344)(186,291,326,343)(187,290,327,342)(188,289,328,341)(189,288,329,340)(190,287,330,339)(191,286,331,338)(192,285,332,337)(193,284,333,364)(194,283,334,363)(195,282,335,362)(196,281,336,361), (1,426,253,407)(2,427,254,408)(3,428,255,409)(4,429,256,410)(5,430,257,411)(6,431,258,412)(7,432,259,413)(8,433,260,414)(9,434,261,415)(10,435,262,416)(11,436,263,417)(12,437,264,418)(13,438,265,419)(14,439,266,420)(15,440,267,393)(16,441,268,394)(17,442,269,395)(18,443,270,396)(19,444,271,397)(20,445,272,398)(21,446,273,399)(22,447,274,400)(23,448,275,401)(24,421,276,402)(25,422,277,403)(26,423,278,404)(27,424,279,405)(28,425,280,406)(29,318,160,178)(30,319,161,179)(31,320,162,180)(32,321,163,181)(33,322,164,182)(34,323,165,183)(35,324,166,184)(36,325,167,185)(37,326,168,186)(38,327,141,187)(39,328,142,188)(40,329,143,189)(41,330,144,190)(42,331,145,191)(43,332,146,192)(44,333,147,193)(45,334,148,194)(46,335,149,195)(47,336,150,196)(48,309,151,169)(49,310,152,170)(50,311,153,171)(51,312,154,172)(52,313,155,173)(53,314,156,174)(54,315,157,175)(55,316,158,176)(56,317,159,177)(57,350,213,298)(58,351,214,299)(59,352,215,300)(60,353,216,301)(61,354,217,302)(62,355,218,303)(63,356,219,304)(64,357,220,305)(65,358,221,306)(66,359,222,307)(67,360,223,308)(68,361,224,281)(69,362,197,282)(70,363,198,283)(71,364,199,284)(72,337,200,285)(73,338,201,286)(74,339,202,287)(75,340,203,288)(76,341,204,289)(77,342,205,290)(78,343,206,291)(79,344,207,292)(80,345,208,293)(81,346,209,294)(82,347,210,295)(83,348,211,296)(84,349,212,297)(85,131,226,385)(86,132,227,386)(87,133,228,387)(88,134,229,388)(89,135,230,389)(90,136,231,390)(91,137,232,391)(92,138,233,392)(93,139,234,365)(94,140,235,366)(95,113,236,367)(96,114,237,368)(97,115,238,369)(98,116,239,370)(99,117,240,371)(100,118,241,372)(101,119,242,373)(102,120,243,374)(103,121,244,375)(104,122,245,376)(105,123,246,377)(106,124,247,378)(107,125,248,379)(108,126,249,380)(109,127,250,381)(110,128,251,382)(111,129,252,383)(112,130,225,384) );
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)], [(1,160,15,146),(2,159,16,145),(3,158,17,144),(4,157,18,143),(5,156,19,142),(6,155,20,141),(7,154,21,168),(8,153,22,167),(9,152,23,166),(10,151,24,165),(11,150,25,164),(12,149,26,163),(13,148,27,162),(14,147,28,161),(29,267,43,253),(30,266,44,280),(31,265,45,279),(32,264,46,278),(33,263,47,277),(34,262,48,276),(35,261,49,275),(36,260,50,274),(37,259,51,273),(38,258,52,272),(39,257,53,271),(40,256,54,270),(41,255,55,269),(42,254,56,268),(57,235,71,249),(58,234,72,248),(59,233,73,247),(60,232,74,246),(61,231,75,245),(62,230,76,244),(63,229,77,243),(64,228,78,242),(65,227,79,241),(66,226,80,240),(67,225,81,239),(68,252,82,238),(69,251,83,237),(70,250,84,236),(85,208,99,222),(86,207,100,221),(87,206,101,220),(88,205,102,219),(89,204,103,218),(90,203,104,217),(91,202,105,216),(92,201,106,215),(93,200,107,214),(94,199,108,213),(95,198,109,212),(96,197,110,211),(97,224,111,210),(98,223,112,209),(113,283,127,297),(114,282,128,296),(115,281,129,295),(116,308,130,294),(117,307,131,293),(118,306,132,292),(119,305,133,291),(120,304,134,290),(121,303,135,289),(122,302,136,288),(123,301,137,287),(124,300,138,286),(125,299,139,285),(126,298,140,284),(169,421,183,435),(170,448,184,434),(171,447,185,433),(172,446,186,432),(173,445,187,431),(174,444,188,430),(175,443,189,429),(176,442,190,428),(177,441,191,427),(178,440,192,426),(179,439,193,425),(180,438,194,424),(181,437,195,423),(182,436,196,422),(309,402,323,416),(310,401,324,415),(311,400,325,414),(312,399,326,413),(313,398,327,412),(314,397,328,411),(315,396,329,410),(316,395,330,409),(317,394,331,408),(318,393,332,407),(319,420,333,406),(320,419,334,405),(321,418,335,404),(322,417,336,403),(337,379,351,365),(338,378,352,392),(339,377,353,391),(340,376,354,390),(341,375,355,389),(342,374,356,388),(343,373,357,387),(344,372,358,386),(345,371,359,385),(346,370,360,384),(347,369,361,383),(348,368,362,382),(349,367,363,381),(350,366,364,380)], [(1,227,253,86),(2,226,254,85),(3,225,255,112),(4,252,256,111),(5,251,257,110),(6,250,258,109),(7,249,259,108),(8,248,260,107),(9,247,261,106),(10,246,262,105),(11,245,263,104),(12,244,264,103),(13,243,265,102),(14,242,266,101),(15,241,267,100),(16,240,268,99),(17,239,269,98),(18,238,270,97),(19,237,271,96),(20,236,272,95),(21,235,273,94),(22,234,274,93),(23,233,275,92),(24,232,276,91),(25,231,277,90),(26,230,278,89),(27,229,279,88),(28,228,280,87),(29,214,160,58),(30,213,161,57),(31,212,162,84),(32,211,163,83),(33,210,164,82),(34,209,165,81),(35,208,166,80),(36,207,167,79),(37,206,168,78),(38,205,141,77),(39,204,142,76),(40,203,143,75),(41,202,144,74),(42,201,145,73),(43,200,146,72),(44,199,147,71),(45,198,148,70),(46,197,149,69),(47,224,150,68),(48,223,151,67),(49,222,152,66),(50,221,153,65),(51,220,154,64),(52,219,155,63),(53,218,156,62),(54,217,157,61),(55,216,158,60),(56,215,159,59),(113,398,367,445),(114,397,368,444),(115,396,369,443),(116,395,370,442),(117,394,371,441),(118,393,372,440),(119,420,373,439),(120,419,374,438),(121,418,375,437),(122,417,376,436),(123,416,377,435),(124,415,378,434),(125,414,379,433),(126,413,380,432),(127,412,381,431),(128,411,382,430),(129,410,383,429),(130,409,384,428),(131,408,385,427),(132,407,386,426),(133,406,387,425),(134,405,388,424),(135,404,389,423),(136,403,390,422),(137,402,391,421),(138,401,392,448),(139,400,365,447),(140,399,366,446),(169,308,309,360),(170,307,310,359),(171,306,311,358),(172,305,312,357),(173,304,313,356),(174,303,314,355),(175,302,315,354),(176,301,316,353),(177,300,317,352),(178,299,318,351),(179,298,319,350),(180,297,320,349),(181,296,321,348),(182,295,322,347),(183,294,323,346),(184,293,324,345),(185,292,325,344),(186,291,326,343),(187,290,327,342),(188,289,328,341),(189,288,329,340),(190,287,330,339),(191,286,331,338),(192,285,332,337),(193,284,333,364),(194,283,334,363),(195,282,335,362),(196,281,336,361)], [(1,426,253,407),(2,427,254,408),(3,428,255,409),(4,429,256,410),(5,430,257,411),(6,431,258,412),(7,432,259,413),(8,433,260,414),(9,434,261,415),(10,435,262,416),(11,436,263,417),(12,437,264,418),(13,438,265,419),(14,439,266,420),(15,440,267,393),(16,441,268,394),(17,442,269,395),(18,443,270,396),(19,444,271,397),(20,445,272,398),(21,446,273,399),(22,447,274,400),(23,448,275,401),(24,421,276,402),(25,422,277,403),(26,423,278,404),(27,424,279,405),(28,425,280,406),(29,318,160,178),(30,319,161,179),(31,320,162,180),(32,321,163,181),(33,322,164,182),(34,323,165,183),(35,324,166,184),(36,325,167,185),(37,326,168,186),(38,327,141,187),(39,328,142,188),(40,329,143,189),(41,330,144,190),(42,331,145,191),(43,332,146,192),(44,333,147,193),(45,334,148,194),(46,335,149,195),(47,336,150,196),(48,309,151,169),(49,310,152,170),(50,311,153,171),(51,312,154,172),(52,313,155,173),(53,314,156,174),(54,315,157,175),(55,316,158,176),(56,317,159,177),(57,350,213,298),(58,351,214,299),(59,352,215,300),(60,353,216,301),(61,354,217,302),(62,355,218,303),(63,356,219,304),(64,357,220,305),(65,358,221,306),(66,359,222,307),(67,360,223,308),(68,361,224,281),(69,362,197,282),(70,363,198,283),(71,364,199,284),(72,337,200,285),(73,338,201,286),(74,339,202,287),(75,340,203,288),(76,341,204,289),(77,342,205,290),(78,343,206,291),(79,344,207,292),(80,345,208,293),(81,346,209,294),(82,347,210,295),(83,348,211,296),(84,349,212,297),(85,131,226,385),(86,132,227,386),(87,133,228,387),(88,134,229,388),(89,135,230,389),(90,136,231,390),(91,137,232,391),(92,138,233,392),(93,139,234,365),(94,140,235,366),(95,113,236,367),(96,114,237,368),(97,115,238,369),(98,116,239,370),(99,117,240,371),(100,118,241,372),(101,119,242,373),(102,120,243,374),(103,121,244,375),(104,122,245,376),(105,123,246,377),(106,124,247,378),(107,125,248,379),(108,126,249,380),(109,127,250,381),(110,128,251,382),(111,129,252,383),(112,130,225,384)]])
79 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 | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| 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 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | SD16 | C4○D4 | D14 | D14 | D28 | C56⋊C2 | C8.C22 | D4⋊2D7 | Q8×D7 | C8.D14 |
| kernel | Dic14⋊4Q8 | C28.44D4 | C8⋊Dic7 | C7×C4⋊C8 | C4×Dic14 | C28⋊2Q8 | Dic14 | C2×C28 | C4⋊C8 | C28 | C28 | C42 | C2×C8 | C2×C4 | C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 24 | 1 | 3 | 3 | 6 |
Matrix representation of Dic14⋊4Q8 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 111 | 0 |
| 0 | 0 | 85 | 56 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 75 | 111 |
| 0 | 0 | 101 | 38 |
| 112 | 2 | 0 | 0 |
| 112 | 1 | 0 | 0 |
| 0 | 0 | 23 | 25 |
| 0 | 0 | 6 | 90 |
| 20 | 9 | 0 | 0 |
| 81 | 93 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,111,85,0,0,0,56],[112,0,0,0,0,112,0,0,0,0,75,101,0,0,111,38],[112,112,0,0,2,1,0,0,0,0,23,6,0,0,25,90],[20,81,0,0,9,93,0,0,0,0,1,0,0,0,0,1] >;
Dic14⋊4Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes_4Q_8 % in TeX
G:=Group("Dic14:4Q8"); // GroupNames label
G:=SmallGroup(448,385);
// by ID
G=gap.SmallGroup(448,385);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,254,219,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=c^4=1,b^2=a^14,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^21*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations