metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊9Q8, C42.170D14, C14.812+ 1+4, C7⋊2Q82, C4⋊Q8.15D7, C4.18(Q8×D7), C28⋊Q8.13C2, C28.53(C2×Q8), C4⋊C4.215D14, (C2×Q8).85D14, (C2×C28).99C23, Dic7.15(C2×Q8), C14.45(C22×Q8), (C2×C14).266C24, (C4×C28).207C22, (C4×Dic14).26C2, Dic7⋊Q8.9C2, C2.85(D4⋊6D14), Dic7⋊3Q8.13C2, Dic7⋊C4.58C22, C4⋊Dic7.383C22, (Q8×C14).133C22, C22.287(C23×D7), (C2×Dic7).271C23, (C4×Dic7).158C22, (C2×Dic14).186C22, C2.28(C2×Q8×D7), (C7×C4⋊Q8).15C2, (C2×C4).91(C22×D7), (C7×C4⋊C4).209C22, SmallGroup(448,1175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊9Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, cac-1=a15, ad=da, bc=cb, dbd-1=a14b, dcd-1=c-1 >
Subgroups: 812 in 212 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×Q8, C4⋊Q8, C4⋊Q8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, Q82, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C2×Dic14, Q8×C14, C4×Dic14, Dic7⋊3Q8, C28⋊Q8, Dic7⋊Q8, C7×C4⋊Q8, Dic14⋊9Q8
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C24, D14, C22×Q8, 2+ 1+4, C22×D7, Q82, Q8×D7, C23×D7, D4⋊6D14, C2×Q8×D7, Dic14⋊9Q8
►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)
(1 201 15 215)(2 200 16 214)(3 199 17 213)(4 198 18 212)(5 197 19 211)(6 224 20 210)(7 223 21 209)(8 222 22 208)(9 221 23 207)(10 220 24 206)(11 219 25 205)(12 218 26 204)(13 217 27 203)(14 216 28 202)(29 194 43 180)(30 193 44 179)(31 192 45 178)(32 191 46 177)(33 190 47 176)(34 189 48 175)(35 188 49 174)(36 187 50 173)(37 186 51 172)(38 185 52 171)(39 184 53 170)(40 183 54 169)(41 182 55 196)(42 181 56 195)(57 367 71 381)(58 366 72 380)(59 365 73 379)(60 392 74 378)(61 391 75 377)(62 390 76 376)(63 389 77 375)(64 388 78 374)(65 387 79 373)(66 386 80 372)(67 385 81 371)(68 384 82 370)(69 383 83 369)(70 382 84 368)(85 152 99 166)(86 151 100 165)(87 150 101 164)(88 149 102 163)(89 148 103 162)(90 147 104 161)(91 146 105 160)(92 145 106 159)(93 144 107 158)(94 143 108 157)(95 142 109 156)(96 141 110 155)(97 168 111 154)(98 167 112 153)(113 340 127 354)(114 339 128 353)(115 338 129 352)(116 337 130 351)(117 364 131 350)(118 363 132 349)(119 362 133 348)(120 361 134 347)(121 360 135 346)(122 359 136 345)(123 358 137 344)(124 357 138 343)(125 356 139 342)(126 355 140 341)(225 291 239 305)(226 290 240 304)(227 289 241 303)(228 288 242 302)(229 287 243 301)(230 286 244 300)(231 285 245 299)(232 284 246 298)(233 283 247 297)(234 282 248 296)(235 281 249 295)(236 308 250 294)(237 307 251 293)(238 306 252 292)(253 405 267 419)(254 404 268 418)(255 403 269 417)(256 402 270 416)(257 401 271 415)(258 400 272 414)(259 399 273 413)(260 398 274 412)(261 397 275 411)(262 396 276 410)(263 395 277 409)(264 394 278 408)(265 393 279 407)(266 420 280 406)(309 428 323 442)(310 427 324 441)(311 426 325 440)(312 425 326 439)(313 424 327 438)(314 423 328 437)(315 422 329 436)(316 421 330 435)(317 448 331 434)(318 447 332 433)(319 446 333 432)(320 445 334 431)(321 444 335 430)(322 443 336 429)
(1 335 134 87)(2 322 135 102)(3 309 136 89)(4 324 137 104)(5 311 138 91)(6 326 139 106)(7 313 140 93)(8 328 113 108)(9 315 114 95)(10 330 115 110)(11 317 116 97)(12 332 117 112)(13 319 118 99)(14 334 119 86)(15 321 120 101)(16 336 121 88)(17 323 122 103)(18 310 123 90)(19 325 124 105)(20 312 125 92)(21 327 126 107)(22 314 127 94)(23 329 128 109)(24 316 129 96)(25 331 130 111)(26 318 131 98)(27 333 132 85)(28 320 133 100)(29 256 238 70)(30 271 239 57)(31 258 240 72)(32 273 241 59)(33 260 242 74)(34 275 243 61)(35 262 244 76)(36 277 245 63)(37 264 246 78)(38 279 247 65)(39 266 248 80)(40 253 249 67)(41 268 250 82)(42 255 251 69)(43 270 252 84)(44 257 225 71)(45 272 226 58)(46 259 227 73)(47 274 228 60)(48 261 229 75)(49 276 230 62)(50 263 231 77)(51 278 232 64)(52 265 233 79)(53 280 234 66)(54 267 235 81)(55 254 236 68)(56 269 237 83)(141 206 421 352)(142 221 422 339)(143 208 423 354)(144 223 424 341)(145 210 425 356)(146 197 426 343)(147 212 427 358)(148 199 428 345)(149 214 429 360)(150 201 430 347)(151 216 431 362)(152 203 432 349)(153 218 433 364)(154 205 434 351)(155 220 435 338)(156 207 436 353)(157 222 437 340)(158 209 438 355)(159 224 439 342)(160 211 440 357)(161 198 441 344)(162 213 442 359)(163 200 443 346)(164 215 444 361)(165 202 445 348)(166 217 446 363)(167 204 447 350)(168 219 448 337)(169 419 281 371)(170 406 282 386)(171 393 283 373)(172 408 284 388)(173 395 285 375)(174 410 286 390)(175 397 287 377)(176 412 288 392)(177 399 289 379)(178 414 290 366)(179 401 291 381)(180 416 292 368)(181 403 293 383)(182 418 294 370)(183 405 295 385)(184 420 296 372)(185 407 297 387)(186 394 298 374)(187 409 299 389)(188 396 300 376)(189 411 301 391)(190 398 302 378)(191 413 303 365)(192 400 304 380)(193 415 305 367)(194 402 306 382)(195 417 307 369)(196 404 308 384)
(1 170 134 282)(2 171 135 283)(3 172 136 284)(4 173 137 285)(5 174 138 286)(6 175 139 287)(7 176 140 288)(8 177 113 289)(9 178 114 290)(10 179 115 291)(11 180 116 292)(12 181 117 293)(13 182 118 294)(14 183 119 295)(15 184 120 296)(16 185 121 297)(17 186 122 298)(18 187 123 299)(19 188 124 300)(20 189 125 301)(21 190 126 302)(22 191 127 303)(23 192 128 304)(24 193 129 305)(25 194 130 306)(26 195 131 307)(27 196 132 308)(28 169 133 281)(29 351 238 205)(30 352 239 206)(31 353 240 207)(32 354 241 208)(33 355 242 209)(34 356 243 210)(35 357 244 211)(36 358 245 212)(37 359 246 213)(38 360 247 214)(39 361 248 215)(40 362 249 216)(41 363 250 217)(42 364 251 218)(43 337 252 219)(44 338 225 220)(45 339 226 221)(46 340 227 222)(47 341 228 223)(48 342 229 224)(49 343 230 197)(50 344 231 198)(51 345 232 199)(52 346 233 200)(53 347 234 201)(54 348 235 202)(55 349 236 203)(56 350 237 204)(57 141 271 421)(58 142 272 422)(59 143 273 423)(60 144 274 424)(61 145 275 425)(62 146 276 426)(63 147 277 427)(64 148 278 428)(65 149 279 429)(66 150 280 430)(67 151 253 431)(68 152 254 432)(69 153 255 433)(70 154 256 434)(71 155 257 435)(72 156 258 436)(73 157 259 437)(74 158 260 438)(75 159 261 439)(76 160 262 440)(77 161 263 441)(78 162 264 442)(79 163 265 443)(80 164 266 444)(81 165 267 445)(82 166 268 446)(83 167 269 447)(84 168 270 448)(85 404 333 384)(86 405 334 385)(87 406 335 386)(88 407 336 387)(89 408 309 388)(90 409 310 389)(91 410 311 390)(92 411 312 391)(93 412 313 392)(94 413 314 365)(95 414 315 366)(96 415 316 367)(97 416 317 368)(98 417 318 369)(99 418 319 370)(100 419 320 371)(101 420 321 372)(102 393 322 373)(103 394 323 374)(104 395 324 375)(105 396 325 376)(106 397 326 377)(107 398 327 378)(108 399 328 379)(109 400 329 380)(110 401 330 381)(111 402 331 382)(112 403 332 383)
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), (1,201,15,215)(2,200,16,214)(3,199,17,213)(4,198,18,212)(5,197,19,211)(6,224,20,210)(7,223,21,209)(8,222,22,208)(9,221,23,207)(10,220,24,206)(11,219,25,205)(12,218,26,204)(13,217,27,203)(14,216,28,202)(29,194,43,180)(30,193,44,179)(31,192,45,178)(32,191,46,177)(33,190,47,176)(34,189,48,175)(35,188,49,174)(36,187,50,173)(37,186,51,172)(38,185,52,171)(39,184,53,170)(40,183,54,169)(41,182,55,196)(42,181,56,195)(57,367,71,381)(58,366,72,380)(59,365,73,379)(60,392,74,378)(61,391,75,377)(62,390,76,376)(63,389,77,375)(64,388,78,374)(65,387,79,373)(66,386,80,372)(67,385,81,371)(68,384,82,370)(69,383,83,369)(70,382,84,368)(85,152,99,166)(86,151,100,165)(87,150,101,164)(88,149,102,163)(89,148,103,162)(90,147,104,161)(91,146,105,160)(92,145,106,159)(93,144,107,158)(94,143,108,157)(95,142,109,156)(96,141,110,155)(97,168,111,154)(98,167,112,153)(113,340,127,354)(114,339,128,353)(115,338,129,352)(116,337,130,351)(117,364,131,350)(118,363,132,349)(119,362,133,348)(120,361,134,347)(121,360,135,346)(122,359,136,345)(123,358,137,344)(124,357,138,343)(125,356,139,342)(126,355,140,341)(225,291,239,305)(226,290,240,304)(227,289,241,303)(228,288,242,302)(229,287,243,301)(230,286,244,300)(231,285,245,299)(232,284,246,298)(233,283,247,297)(234,282,248,296)(235,281,249,295)(236,308,250,294)(237,307,251,293)(238,306,252,292)(253,405,267,419)(254,404,268,418)(255,403,269,417)(256,402,270,416)(257,401,271,415)(258,400,272,414)(259,399,273,413)(260,398,274,412)(261,397,275,411)(262,396,276,410)(263,395,277,409)(264,394,278,408)(265,393,279,407)(266,420,280,406)(309,428,323,442)(310,427,324,441)(311,426,325,440)(312,425,326,439)(313,424,327,438)(314,423,328,437)(315,422,329,436)(316,421,330,435)(317,448,331,434)(318,447,332,433)(319,446,333,432)(320,445,334,431)(321,444,335,430)(322,443,336,429), (1,335,134,87)(2,322,135,102)(3,309,136,89)(4,324,137,104)(5,311,138,91)(6,326,139,106)(7,313,140,93)(8,328,113,108)(9,315,114,95)(10,330,115,110)(11,317,116,97)(12,332,117,112)(13,319,118,99)(14,334,119,86)(15,321,120,101)(16,336,121,88)(17,323,122,103)(18,310,123,90)(19,325,124,105)(20,312,125,92)(21,327,126,107)(22,314,127,94)(23,329,128,109)(24,316,129,96)(25,331,130,111)(26,318,131,98)(27,333,132,85)(28,320,133,100)(29,256,238,70)(30,271,239,57)(31,258,240,72)(32,273,241,59)(33,260,242,74)(34,275,243,61)(35,262,244,76)(36,277,245,63)(37,264,246,78)(38,279,247,65)(39,266,248,80)(40,253,249,67)(41,268,250,82)(42,255,251,69)(43,270,252,84)(44,257,225,71)(45,272,226,58)(46,259,227,73)(47,274,228,60)(48,261,229,75)(49,276,230,62)(50,263,231,77)(51,278,232,64)(52,265,233,79)(53,280,234,66)(54,267,235,81)(55,254,236,68)(56,269,237,83)(141,206,421,352)(142,221,422,339)(143,208,423,354)(144,223,424,341)(145,210,425,356)(146,197,426,343)(147,212,427,358)(148,199,428,345)(149,214,429,360)(150,201,430,347)(151,216,431,362)(152,203,432,349)(153,218,433,364)(154,205,434,351)(155,220,435,338)(156,207,436,353)(157,222,437,340)(158,209,438,355)(159,224,439,342)(160,211,440,357)(161,198,441,344)(162,213,442,359)(163,200,443,346)(164,215,444,361)(165,202,445,348)(166,217,446,363)(167,204,447,350)(168,219,448,337)(169,419,281,371)(170,406,282,386)(171,393,283,373)(172,408,284,388)(173,395,285,375)(174,410,286,390)(175,397,287,377)(176,412,288,392)(177,399,289,379)(178,414,290,366)(179,401,291,381)(180,416,292,368)(181,403,293,383)(182,418,294,370)(183,405,295,385)(184,420,296,372)(185,407,297,387)(186,394,298,374)(187,409,299,389)(188,396,300,376)(189,411,301,391)(190,398,302,378)(191,413,303,365)(192,400,304,380)(193,415,305,367)(194,402,306,382)(195,417,307,369)(196,404,308,384), (1,170,134,282)(2,171,135,283)(3,172,136,284)(4,173,137,285)(5,174,138,286)(6,175,139,287)(7,176,140,288)(8,177,113,289)(9,178,114,290)(10,179,115,291)(11,180,116,292)(12,181,117,293)(13,182,118,294)(14,183,119,295)(15,184,120,296)(16,185,121,297)(17,186,122,298)(18,187,123,299)(19,188,124,300)(20,189,125,301)(21,190,126,302)(22,191,127,303)(23,192,128,304)(24,193,129,305)(25,194,130,306)(26,195,131,307)(27,196,132,308)(28,169,133,281)(29,351,238,205)(30,352,239,206)(31,353,240,207)(32,354,241,208)(33,355,242,209)(34,356,243,210)(35,357,244,211)(36,358,245,212)(37,359,246,213)(38,360,247,214)(39,361,248,215)(40,362,249,216)(41,363,250,217)(42,364,251,218)(43,337,252,219)(44,338,225,220)(45,339,226,221)(46,340,227,222)(47,341,228,223)(48,342,229,224)(49,343,230,197)(50,344,231,198)(51,345,232,199)(52,346,233,200)(53,347,234,201)(54,348,235,202)(55,349,236,203)(56,350,237,204)(57,141,271,421)(58,142,272,422)(59,143,273,423)(60,144,274,424)(61,145,275,425)(62,146,276,426)(63,147,277,427)(64,148,278,428)(65,149,279,429)(66,150,280,430)(67,151,253,431)(68,152,254,432)(69,153,255,433)(70,154,256,434)(71,155,257,435)(72,156,258,436)(73,157,259,437)(74,158,260,438)(75,159,261,439)(76,160,262,440)(77,161,263,441)(78,162,264,442)(79,163,265,443)(80,164,266,444)(81,165,267,445)(82,166,268,446)(83,167,269,447)(84,168,270,448)(85,404,333,384)(86,405,334,385)(87,406,335,386)(88,407,336,387)(89,408,309,388)(90,409,310,389)(91,410,311,390)(92,411,312,391)(93,412,313,392)(94,413,314,365)(95,414,315,366)(96,415,316,367)(97,416,317,368)(98,417,318,369)(99,418,319,370)(100,419,320,371)(101,420,321,372)(102,393,322,373)(103,394,323,374)(104,395,324,375)(105,396,325,376)(106,397,326,377)(107,398,327,378)(108,399,328,379)(109,400,329,380)(110,401,330,381)(111,402,331,382)(112,403,332,383)>;
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), (1,201,15,215)(2,200,16,214)(3,199,17,213)(4,198,18,212)(5,197,19,211)(6,224,20,210)(7,223,21,209)(8,222,22,208)(9,221,23,207)(10,220,24,206)(11,219,25,205)(12,218,26,204)(13,217,27,203)(14,216,28,202)(29,194,43,180)(30,193,44,179)(31,192,45,178)(32,191,46,177)(33,190,47,176)(34,189,48,175)(35,188,49,174)(36,187,50,173)(37,186,51,172)(38,185,52,171)(39,184,53,170)(40,183,54,169)(41,182,55,196)(42,181,56,195)(57,367,71,381)(58,366,72,380)(59,365,73,379)(60,392,74,378)(61,391,75,377)(62,390,76,376)(63,389,77,375)(64,388,78,374)(65,387,79,373)(66,386,80,372)(67,385,81,371)(68,384,82,370)(69,383,83,369)(70,382,84,368)(85,152,99,166)(86,151,100,165)(87,150,101,164)(88,149,102,163)(89,148,103,162)(90,147,104,161)(91,146,105,160)(92,145,106,159)(93,144,107,158)(94,143,108,157)(95,142,109,156)(96,141,110,155)(97,168,111,154)(98,167,112,153)(113,340,127,354)(114,339,128,353)(115,338,129,352)(116,337,130,351)(117,364,131,350)(118,363,132,349)(119,362,133,348)(120,361,134,347)(121,360,135,346)(122,359,136,345)(123,358,137,344)(124,357,138,343)(125,356,139,342)(126,355,140,341)(225,291,239,305)(226,290,240,304)(227,289,241,303)(228,288,242,302)(229,287,243,301)(230,286,244,300)(231,285,245,299)(232,284,246,298)(233,283,247,297)(234,282,248,296)(235,281,249,295)(236,308,250,294)(237,307,251,293)(238,306,252,292)(253,405,267,419)(254,404,268,418)(255,403,269,417)(256,402,270,416)(257,401,271,415)(258,400,272,414)(259,399,273,413)(260,398,274,412)(261,397,275,411)(262,396,276,410)(263,395,277,409)(264,394,278,408)(265,393,279,407)(266,420,280,406)(309,428,323,442)(310,427,324,441)(311,426,325,440)(312,425,326,439)(313,424,327,438)(314,423,328,437)(315,422,329,436)(316,421,330,435)(317,448,331,434)(318,447,332,433)(319,446,333,432)(320,445,334,431)(321,444,335,430)(322,443,336,429), (1,335,134,87)(2,322,135,102)(3,309,136,89)(4,324,137,104)(5,311,138,91)(6,326,139,106)(7,313,140,93)(8,328,113,108)(9,315,114,95)(10,330,115,110)(11,317,116,97)(12,332,117,112)(13,319,118,99)(14,334,119,86)(15,321,120,101)(16,336,121,88)(17,323,122,103)(18,310,123,90)(19,325,124,105)(20,312,125,92)(21,327,126,107)(22,314,127,94)(23,329,128,109)(24,316,129,96)(25,331,130,111)(26,318,131,98)(27,333,132,85)(28,320,133,100)(29,256,238,70)(30,271,239,57)(31,258,240,72)(32,273,241,59)(33,260,242,74)(34,275,243,61)(35,262,244,76)(36,277,245,63)(37,264,246,78)(38,279,247,65)(39,266,248,80)(40,253,249,67)(41,268,250,82)(42,255,251,69)(43,270,252,84)(44,257,225,71)(45,272,226,58)(46,259,227,73)(47,274,228,60)(48,261,229,75)(49,276,230,62)(50,263,231,77)(51,278,232,64)(52,265,233,79)(53,280,234,66)(54,267,235,81)(55,254,236,68)(56,269,237,83)(141,206,421,352)(142,221,422,339)(143,208,423,354)(144,223,424,341)(145,210,425,356)(146,197,426,343)(147,212,427,358)(148,199,428,345)(149,214,429,360)(150,201,430,347)(151,216,431,362)(152,203,432,349)(153,218,433,364)(154,205,434,351)(155,220,435,338)(156,207,436,353)(157,222,437,340)(158,209,438,355)(159,224,439,342)(160,211,440,357)(161,198,441,344)(162,213,442,359)(163,200,443,346)(164,215,444,361)(165,202,445,348)(166,217,446,363)(167,204,447,350)(168,219,448,337)(169,419,281,371)(170,406,282,386)(171,393,283,373)(172,408,284,388)(173,395,285,375)(174,410,286,390)(175,397,287,377)(176,412,288,392)(177,399,289,379)(178,414,290,366)(179,401,291,381)(180,416,292,368)(181,403,293,383)(182,418,294,370)(183,405,295,385)(184,420,296,372)(185,407,297,387)(186,394,298,374)(187,409,299,389)(188,396,300,376)(189,411,301,391)(190,398,302,378)(191,413,303,365)(192,400,304,380)(193,415,305,367)(194,402,306,382)(195,417,307,369)(196,404,308,384), (1,170,134,282)(2,171,135,283)(3,172,136,284)(4,173,137,285)(5,174,138,286)(6,175,139,287)(7,176,140,288)(8,177,113,289)(9,178,114,290)(10,179,115,291)(11,180,116,292)(12,181,117,293)(13,182,118,294)(14,183,119,295)(15,184,120,296)(16,185,121,297)(17,186,122,298)(18,187,123,299)(19,188,124,300)(20,189,125,301)(21,190,126,302)(22,191,127,303)(23,192,128,304)(24,193,129,305)(25,194,130,306)(26,195,131,307)(27,196,132,308)(28,169,133,281)(29,351,238,205)(30,352,239,206)(31,353,240,207)(32,354,241,208)(33,355,242,209)(34,356,243,210)(35,357,244,211)(36,358,245,212)(37,359,246,213)(38,360,247,214)(39,361,248,215)(40,362,249,216)(41,363,250,217)(42,364,251,218)(43,337,252,219)(44,338,225,220)(45,339,226,221)(46,340,227,222)(47,341,228,223)(48,342,229,224)(49,343,230,197)(50,344,231,198)(51,345,232,199)(52,346,233,200)(53,347,234,201)(54,348,235,202)(55,349,236,203)(56,350,237,204)(57,141,271,421)(58,142,272,422)(59,143,273,423)(60,144,274,424)(61,145,275,425)(62,146,276,426)(63,147,277,427)(64,148,278,428)(65,149,279,429)(66,150,280,430)(67,151,253,431)(68,152,254,432)(69,153,255,433)(70,154,256,434)(71,155,257,435)(72,156,258,436)(73,157,259,437)(74,158,260,438)(75,159,261,439)(76,160,262,440)(77,161,263,441)(78,162,264,442)(79,163,265,443)(80,164,266,444)(81,165,267,445)(82,166,268,446)(83,167,269,447)(84,168,270,448)(85,404,333,384)(86,405,334,385)(87,406,335,386)(88,407,336,387)(89,408,309,388)(90,409,310,389)(91,410,311,390)(92,411,312,391)(93,412,313,392)(94,413,314,365)(95,414,315,366)(96,415,316,367)(97,416,317,368)(98,417,318,369)(99,418,319,370)(100,419,320,371)(101,420,321,372)(102,393,322,373)(103,394,323,374)(104,395,324,375)(105,396,325,376)(106,397,326,377)(107,398,327,378)(108,399,328,379)(109,400,329,380)(110,401,330,381)(111,402,331,382)(112,403,332,383) );
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)], [(1,201,15,215),(2,200,16,214),(3,199,17,213),(4,198,18,212),(5,197,19,211),(6,224,20,210),(7,223,21,209),(8,222,22,208),(9,221,23,207),(10,220,24,206),(11,219,25,205),(12,218,26,204),(13,217,27,203),(14,216,28,202),(29,194,43,180),(30,193,44,179),(31,192,45,178),(32,191,46,177),(33,190,47,176),(34,189,48,175),(35,188,49,174),(36,187,50,173),(37,186,51,172),(38,185,52,171),(39,184,53,170),(40,183,54,169),(41,182,55,196),(42,181,56,195),(57,367,71,381),(58,366,72,380),(59,365,73,379),(60,392,74,378),(61,391,75,377),(62,390,76,376),(63,389,77,375),(64,388,78,374),(65,387,79,373),(66,386,80,372),(67,385,81,371),(68,384,82,370),(69,383,83,369),(70,382,84,368),(85,152,99,166),(86,151,100,165),(87,150,101,164),(88,149,102,163),(89,148,103,162),(90,147,104,161),(91,146,105,160),(92,145,106,159),(93,144,107,158),(94,143,108,157),(95,142,109,156),(96,141,110,155),(97,168,111,154),(98,167,112,153),(113,340,127,354),(114,339,128,353),(115,338,129,352),(116,337,130,351),(117,364,131,350),(118,363,132,349),(119,362,133,348),(120,361,134,347),(121,360,135,346),(122,359,136,345),(123,358,137,344),(124,357,138,343),(125,356,139,342),(126,355,140,341),(225,291,239,305),(226,290,240,304),(227,289,241,303),(228,288,242,302),(229,287,243,301),(230,286,244,300),(231,285,245,299),(232,284,246,298),(233,283,247,297),(234,282,248,296),(235,281,249,295),(236,308,250,294),(237,307,251,293),(238,306,252,292),(253,405,267,419),(254,404,268,418),(255,403,269,417),(256,402,270,416),(257,401,271,415),(258,400,272,414),(259,399,273,413),(260,398,274,412),(261,397,275,411),(262,396,276,410),(263,395,277,409),(264,394,278,408),(265,393,279,407),(266,420,280,406),(309,428,323,442),(310,427,324,441),(311,426,325,440),(312,425,326,439),(313,424,327,438),(314,423,328,437),(315,422,329,436),(316,421,330,435),(317,448,331,434),(318,447,332,433),(319,446,333,432),(320,445,334,431),(321,444,335,430),(322,443,336,429)], [(1,335,134,87),(2,322,135,102),(3,309,136,89),(4,324,137,104),(5,311,138,91),(6,326,139,106),(7,313,140,93),(8,328,113,108),(9,315,114,95),(10,330,115,110),(11,317,116,97),(12,332,117,112),(13,319,118,99),(14,334,119,86),(15,321,120,101),(16,336,121,88),(17,323,122,103),(18,310,123,90),(19,325,124,105),(20,312,125,92),(21,327,126,107),(22,314,127,94),(23,329,128,109),(24,316,129,96),(25,331,130,111),(26,318,131,98),(27,333,132,85),(28,320,133,100),(29,256,238,70),(30,271,239,57),(31,258,240,72),(32,273,241,59),(33,260,242,74),(34,275,243,61),(35,262,244,76),(36,277,245,63),(37,264,246,78),(38,279,247,65),(39,266,248,80),(40,253,249,67),(41,268,250,82),(42,255,251,69),(43,270,252,84),(44,257,225,71),(45,272,226,58),(46,259,227,73),(47,274,228,60),(48,261,229,75),(49,276,230,62),(50,263,231,77),(51,278,232,64),(52,265,233,79),(53,280,234,66),(54,267,235,81),(55,254,236,68),(56,269,237,83),(141,206,421,352),(142,221,422,339),(143,208,423,354),(144,223,424,341),(145,210,425,356),(146,197,426,343),(147,212,427,358),(148,199,428,345),(149,214,429,360),(150,201,430,347),(151,216,431,362),(152,203,432,349),(153,218,433,364),(154,205,434,351),(155,220,435,338),(156,207,436,353),(157,222,437,340),(158,209,438,355),(159,224,439,342),(160,211,440,357),(161,198,441,344),(162,213,442,359),(163,200,443,346),(164,215,444,361),(165,202,445,348),(166,217,446,363),(167,204,447,350),(168,219,448,337),(169,419,281,371),(170,406,282,386),(171,393,283,373),(172,408,284,388),(173,395,285,375),(174,410,286,390),(175,397,287,377),(176,412,288,392),(177,399,289,379),(178,414,290,366),(179,401,291,381),(180,416,292,368),(181,403,293,383),(182,418,294,370),(183,405,295,385),(184,420,296,372),(185,407,297,387),(186,394,298,374),(187,409,299,389),(188,396,300,376),(189,411,301,391),(190,398,302,378),(191,413,303,365),(192,400,304,380),(193,415,305,367),(194,402,306,382),(195,417,307,369),(196,404,308,384)], [(1,170,134,282),(2,171,135,283),(3,172,136,284),(4,173,137,285),(5,174,138,286),(6,175,139,287),(7,176,140,288),(8,177,113,289),(9,178,114,290),(10,179,115,291),(11,180,116,292),(12,181,117,293),(13,182,118,294),(14,183,119,295),(15,184,120,296),(16,185,121,297),(17,186,122,298),(18,187,123,299),(19,188,124,300),(20,189,125,301),(21,190,126,302),(22,191,127,303),(23,192,128,304),(24,193,129,305),(25,194,130,306),(26,195,131,307),(27,196,132,308),(28,169,133,281),(29,351,238,205),(30,352,239,206),(31,353,240,207),(32,354,241,208),(33,355,242,209),(34,356,243,210),(35,357,244,211),(36,358,245,212),(37,359,246,213),(38,360,247,214),(39,361,248,215),(40,362,249,216),(41,363,250,217),(42,364,251,218),(43,337,252,219),(44,338,225,220),(45,339,226,221),(46,340,227,222),(47,341,228,223),(48,342,229,224),(49,343,230,197),(50,344,231,198),(51,345,232,199),(52,346,233,200),(53,347,234,201),(54,348,235,202),(55,349,236,203),(56,350,237,204),(57,141,271,421),(58,142,272,422),(59,143,273,423),(60,144,274,424),(61,145,275,425),(62,146,276,426),(63,147,277,427),(64,148,278,428),(65,149,279,429),(66,150,280,430),(67,151,253,431),(68,152,254,432),(69,153,255,433),(70,154,256,434),(71,155,257,435),(72,156,258,436),(73,157,259,437),(74,158,260,438),(75,159,261,439),(76,160,262,440),(77,161,263,441),(78,162,264,442),(79,163,265,443),(80,164,266,444),(81,165,267,445),(82,166,268,446),(83,167,269,447),(84,168,270,448),(85,404,333,384),(86,405,334,385),(87,406,335,386),(88,407,336,387),(89,408,309,388),(90,409,310,389),(91,410,311,390),(92,411,312,391),(93,412,313,392),(94,413,314,365),(95,414,315,366),(96,415,316,367),(97,416,317,368),(98,417,318,369),(99,418,319,370),(100,419,320,371),(101,420,321,372),(102,393,322,373),(103,394,323,374),(104,395,324,375),(105,396,325,376),(106,397,326,377),(107,398,327,378),(108,399,328,379),(109,400,329,380),(110,401,330,381),(111,402,331,382),(112,403,332,383)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | D14 | D14 | D14 | 2+ 1+4 | Q8×D7 | D4⋊6D14 |
| kernel | Dic14⋊9Q8 | C4×Dic14 | Dic7⋊3Q8 | C28⋊Q8 | Dic7⋊Q8 | C7×C4⋊Q8 | Dic14 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C14 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 8 | 3 | 3 | 12 | 6 | 1 | 12 | 6 |
Matrix representation of Dic14⋊9Q8 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 14 |
| 0 | 0 | 0 | 0 | 14 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 5 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 27 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 24 | 2 | 0 | 0 | 0 | 0 |
| 16 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 15 |
| 0 | 0 | 0 | 0 | 15 | 8 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,28,11,0,0,0,0,0,0,8,14,0,0,0,0,14,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,5,0,0,0,0,7,13,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[17,27,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[24,16,0,0,0,0,2,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,15,0,0,0,0,15,8] >;
Dic14⋊9Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes_9Q_8 % in TeX
G:=Group("Dic14:9Q8"); // GroupNames label
G:=SmallGroup(448,1175);
// by ID
G=gap.SmallGroup(448,1175);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,477,232,100,570,185,192,18822]);
// 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^15,a*d=d*a,b*c=c*b,d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations