metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊4Q16, C7⋊2(C4×Q16), C7⋊Q16⋊2C4, Q8.2(C4×D7), C2.1(D7×Q16), C14.36(C4×D4), C4⋊C4.144D14, (C2×C8).205D14, Q8⋊C4.9D7, (C2×Q8).98D14, C14.12(C2×Q16), (Q8×Dic7).2C2, C22.75(D4×D7), C14.44(C4○D8), C28.11(C22×C4), Dic14.3(C2×C4), (C8×Dic7).11C2, C28.Q8.1C2, C28.156(C4○D4), C4.53(D4⋊2D7), (C2×C56).191C22, (C2×C28).230C23, (C2×Dic7).203D4, Dic7⋊3Q8.2C2, C28.44D4.7C2, C4⋊Dic7.80C22, (Q8×C14).13C22, C2.3(SD16⋊3D7), C2.20(Dic7⋊4D4), (C4×Dic7).225C22, (C2×Dic14).63C22, C7⋊C8.7(C2×C4), C4.11(C2×C4×D7), (C7×Q8).2(C2×C4), (C2×C7⋊Q16).2C2, (C2×C14).243(C2×D4), (C7×C4⋊C4).31C22, (C2×C7⋊C8).215C22, (C7×Q8⋊C4).8C2, (C2×C4).337(C22×D7), SmallGroup(448,324)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊4Q16
G = < a,b,c,d | a14=c8=1, b2=a7, d2=c4, bab-1=cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 436 in 110 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×C8, Q8⋊C4, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C4×Q16, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7⋊Q16, C7×C4⋊C4, C2×C56, C2×Dic14, Q8×C14, C28.Q8, C8×Dic7, C28.44D4, C7×Q8⋊C4, Dic7⋊3Q8, C2×C7⋊Q16, Q8×Dic7, Dic7⋊4Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, Q16, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×Q16, C4○D8, C4×D7, C22×D7, C4×Q16, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, SD16⋊3D7, D7×Q16, Dic7⋊4Q16
►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)
(1 196 8 189)(2 195 9 188)(3 194 10 187)(4 193 11 186)(5 192 12 185)(6 191 13 184)(7 190 14 183)(15 33 22 40)(16 32 23 39)(17 31 24 38)(18 30 25 37)(19 29 26 36)(20 42 27 35)(21 41 28 34)(43 143 50 150)(44 142 51 149)(45 141 52 148)(46 154 53 147)(47 153 54 146)(48 152 55 145)(49 151 56 144)(57 275 64 268)(58 274 65 267)(59 273 66 280)(60 272 67 279)(61 271 68 278)(62 270 69 277)(63 269 70 276)(71 139 78 132)(72 138 79 131)(73 137 80 130)(74 136 81 129)(75 135 82 128)(76 134 83 127)(77 133 84 140)(85 330 92 323)(86 329 93 336)(87 328 94 335)(88 327 95 334)(89 326 96 333)(90 325 97 332)(91 324 98 331)(99 383 106 390)(100 382 107 389)(101 381 108 388)(102 380 109 387)(103 379 110 386)(104 392 111 385)(105 391 112 384)(113 403 120 396)(114 402 121 395)(115 401 122 394)(116 400 123 393)(117 399 124 406)(118 398 125 405)(119 397 126 404)(155 258 162 265)(156 257 163 264)(157 256 164 263)(158 255 165 262)(159 254 166 261)(160 253 167 260)(161 266 168 259)(169 431 176 424)(170 430 177 423)(171 429 178 422)(172 428 179 421)(173 427 180 434)(174 426 181 433)(175 425 182 432)(197 319 204 312)(198 318 205 311)(199 317 206 310)(200 316 207 309)(201 315 208 322)(202 314 209 321)(203 313 210 320)(211 233 218 226)(212 232 219 225)(213 231 220 238)(214 230 221 237)(215 229 222 236)(216 228 223 235)(217 227 224 234)(239 344 246 337)(240 343 247 350)(241 342 248 349)(242 341 249 348)(243 340 250 347)(244 339 251 346)(245 338 252 345)(281 378 288 371)(282 377 289 370)(283 376 290 369)(284 375 291 368)(285 374 292 367)(286 373 293 366)(287 372 294 365)(295 444 302 437)(296 443 303 436)(297 442 304 435)(298 441 305 448)(299 440 306 447)(300 439 307 446)(301 438 308 445)(351 408 358 415)(352 407 359 414)(353 420 360 413)(354 419 361 412)(355 418 362 411)(356 417 363 410)(357 416 364 409)
(1 358 440 28 282 402 56 279)(2 357 441 27 283 401 43 278)(3 356 442 26 284 400 44 277)(4 355 443 25 285 399 45 276)(5 354 444 24 286 398 46 275)(6 353 445 23 287 397 47 274)(7 352 446 22 288 396 48 273)(8 351 447 21 289 395 49 272)(9 364 448 20 290 394 50 271)(10 363 435 19 291 393 51 270)(11 362 436 18 292 406 52 269)(12 361 437 17 293 405 53 268)(13 360 438 16 294 404 54 267)(14 359 439 15 281 403 55 280)(29 368 116 149 69 187 410 297)(30 367 117 148 70 186 411 296)(31 366 118 147 57 185 412 295)(32 365 119 146 58 184 413 308)(33 378 120 145 59 183 414 307)(34 377 121 144 60 196 415 306)(35 376 122 143 61 195 416 305)(36 375 123 142 62 194 417 304)(37 374 124 141 63 193 418 303)(38 373 125 154 64 192 419 302)(39 372 126 153 65 191 420 301)(40 371 113 152 66 190 407 300)(41 370 114 151 67 189 408 299)(42 369 115 150 68 188 409 298)(71 99 429 221 95 259 346 203)(72 112 430 220 96 258 347 202)(73 111 431 219 97 257 348 201)(74 110 432 218 98 256 349 200)(75 109 433 217 85 255 350 199)(76 108 434 216 86 254 337 198)(77 107 421 215 87 253 338 197)(78 106 422 214 88 266 339 210)(79 105 423 213 89 265 340 209)(80 104 424 212 90 264 341 208)(81 103 425 211 91 263 342 207)(82 102 426 224 92 262 343 206)(83 101 427 223 93 261 344 205)(84 100 428 222 94 260 345 204)(127 381 180 235 336 159 246 311)(128 380 181 234 323 158 247 310)(129 379 182 233 324 157 248 309)(130 392 169 232 325 156 249 322)(131 391 170 231 326 155 250 321)(132 390 171 230 327 168 251 320)(133 389 172 229 328 167 252 319)(134 388 173 228 329 166 239 318)(135 387 174 227 330 165 240 317)(136 386 175 226 331 164 241 316)(137 385 176 225 332 163 242 315)(138 384 177 238 333 162 243 314)(139 383 178 237 334 161 244 313)(140 382 179 236 335 160 245 312)
(1 200 282 218)(2 199 283 217)(3 198 284 216)(4 197 285 215)(5 210 286 214)(6 209 287 213)(7 208 288 212)(8 207 289 211)(9 206 290 224)(10 205 291 223)(11 204 292 222)(12 203 293 221)(13 202 294 220)(14 201 281 219)(15 97 280 73)(16 96 267 72)(17 95 268 71)(18 94 269 84)(19 93 270 83)(20 92 271 82)(21 91 272 81)(22 90 273 80)(23 89 274 79)(24 88 275 78)(25 87 276 77)(26 86 277 76)(27 85 278 75)(28 98 279 74)(29 336 69 127)(30 335 70 140)(31 334 57 139)(32 333 58 138)(33 332 59 137)(34 331 60 136)(35 330 61 135)(36 329 62 134)(37 328 63 133)(38 327 64 132)(39 326 65 131)(40 325 66 130)(41 324 67 129)(42 323 68 128)(43 109 441 255)(44 108 442 254)(45 107 443 253)(46 106 444 266)(47 105 445 265)(48 104 446 264)(49 103 447 263)(50 102 448 262)(51 101 435 261)(52 100 436 260)(53 99 437 259)(54 112 438 258)(55 111 439 257)(56 110 440 256)(113 169 407 249)(114 182 408 248)(115 181 409 247)(116 180 410 246)(117 179 411 245)(118 178 412 244)(119 177 413 243)(120 176 414 242)(121 175 415 241)(122 174 416 240)(123 173 417 239)(124 172 418 252)(125 171 419 251)(126 170 420 250)(141 389 303 167)(142 388 304 166)(143 387 305 165)(144 386 306 164)(145 385 307 163)(146 384 308 162)(147 383 295 161)(148 382 296 160)(149 381 297 159)(150 380 298 158)(151 379 299 157)(152 392 300 156)(153 391 301 155)(154 390 302 168)(183 315 378 225)(184 314 365 238)(185 313 366 237)(186 312 367 236)(187 311 368 235)(188 310 369 234)(189 309 370 233)(190 322 371 232)(191 321 372 231)(192 320 373 230)(193 319 374 229)(194 318 375 228)(195 317 376 227)(196 316 377 226)(337 400 434 356)(338 399 421 355)(339 398 422 354)(340 397 423 353)(341 396 424 352)(342 395 425 351)(343 394 426 364)(344 393 427 363)(345 406 428 362)(346 405 429 361)(347 404 430 360)(348 403 431 359)(349 402 432 358)(350 401 433 357)
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), (1,196,8,189)(2,195,9,188)(3,194,10,187)(4,193,11,186)(5,192,12,185)(6,191,13,184)(7,190,14,183)(15,33,22,40)(16,32,23,39)(17,31,24,38)(18,30,25,37)(19,29,26,36)(20,42,27,35)(21,41,28,34)(43,143,50,150)(44,142,51,149)(45,141,52,148)(46,154,53,147)(47,153,54,146)(48,152,55,145)(49,151,56,144)(57,275,64,268)(58,274,65,267)(59,273,66,280)(60,272,67,279)(61,271,68,278)(62,270,69,277)(63,269,70,276)(71,139,78,132)(72,138,79,131)(73,137,80,130)(74,136,81,129)(75,135,82,128)(76,134,83,127)(77,133,84,140)(85,330,92,323)(86,329,93,336)(87,328,94,335)(88,327,95,334)(89,326,96,333)(90,325,97,332)(91,324,98,331)(99,383,106,390)(100,382,107,389)(101,381,108,388)(102,380,109,387)(103,379,110,386)(104,392,111,385)(105,391,112,384)(113,403,120,396)(114,402,121,395)(115,401,122,394)(116,400,123,393)(117,399,124,406)(118,398,125,405)(119,397,126,404)(155,258,162,265)(156,257,163,264)(157,256,164,263)(158,255,165,262)(159,254,166,261)(160,253,167,260)(161,266,168,259)(169,431,176,424)(170,430,177,423)(171,429,178,422)(172,428,179,421)(173,427,180,434)(174,426,181,433)(175,425,182,432)(197,319,204,312)(198,318,205,311)(199,317,206,310)(200,316,207,309)(201,315,208,322)(202,314,209,321)(203,313,210,320)(211,233,218,226)(212,232,219,225)(213,231,220,238)(214,230,221,237)(215,229,222,236)(216,228,223,235)(217,227,224,234)(239,344,246,337)(240,343,247,350)(241,342,248,349)(242,341,249,348)(243,340,250,347)(244,339,251,346)(245,338,252,345)(281,378,288,371)(282,377,289,370)(283,376,290,369)(284,375,291,368)(285,374,292,367)(286,373,293,366)(287,372,294,365)(295,444,302,437)(296,443,303,436)(297,442,304,435)(298,441,305,448)(299,440,306,447)(300,439,307,446)(301,438,308,445)(351,408,358,415)(352,407,359,414)(353,420,360,413)(354,419,361,412)(355,418,362,411)(356,417,363,410)(357,416,364,409), (1,358,440,28,282,402,56,279)(2,357,441,27,283,401,43,278)(3,356,442,26,284,400,44,277)(4,355,443,25,285,399,45,276)(5,354,444,24,286,398,46,275)(6,353,445,23,287,397,47,274)(7,352,446,22,288,396,48,273)(8,351,447,21,289,395,49,272)(9,364,448,20,290,394,50,271)(10,363,435,19,291,393,51,270)(11,362,436,18,292,406,52,269)(12,361,437,17,293,405,53,268)(13,360,438,16,294,404,54,267)(14,359,439,15,281,403,55,280)(29,368,116,149,69,187,410,297)(30,367,117,148,70,186,411,296)(31,366,118,147,57,185,412,295)(32,365,119,146,58,184,413,308)(33,378,120,145,59,183,414,307)(34,377,121,144,60,196,415,306)(35,376,122,143,61,195,416,305)(36,375,123,142,62,194,417,304)(37,374,124,141,63,193,418,303)(38,373,125,154,64,192,419,302)(39,372,126,153,65,191,420,301)(40,371,113,152,66,190,407,300)(41,370,114,151,67,189,408,299)(42,369,115,150,68,188,409,298)(71,99,429,221,95,259,346,203)(72,112,430,220,96,258,347,202)(73,111,431,219,97,257,348,201)(74,110,432,218,98,256,349,200)(75,109,433,217,85,255,350,199)(76,108,434,216,86,254,337,198)(77,107,421,215,87,253,338,197)(78,106,422,214,88,266,339,210)(79,105,423,213,89,265,340,209)(80,104,424,212,90,264,341,208)(81,103,425,211,91,263,342,207)(82,102,426,224,92,262,343,206)(83,101,427,223,93,261,344,205)(84,100,428,222,94,260,345,204)(127,381,180,235,336,159,246,311)(128,380,181,234,323,158,247,310)(129,379,182,233,324,157,248,309)(130,392,169,232,325,156,249,322)(131,391,170,231,326,155,250,321)(132,390,171,230,327,168,251,320)(133,389,172,229,328,167,252,319)(134,388,173,228,329,166,239,318)(135,387,174,227,330,165,240,317)(136,386,175,226,331,164,241,316)(137,385,176,225,332,163,242,315)(138,384,177,238,333,162,243,314)(139,383,178,237,334,161,244,313)(140,382,179,236,335,160,245,312), (1,200,282,218)(2,199,283,217)(3,198,284,216)(4,197,285,215)(5,210,286,214)(6,209,287,213)(7,208,288,212)(8,207,289,211)(9,206,290,224)(10,205,291,223)(11,204,292,222)(12,203,293,221)(13,202,294,220)(14,201,281,219)(15,97,280,73)(16,96,267,72)(17,95,268,71)(18,94,269,84)(19,93,270,83)(20,92,271,82)(21,91,272,81)(22,90,273,80)(23,89,274,79)(24,88,275,78)(25,87,276,77)(26,86,277,76)(27,85,278,75)(28,98,279,74)(29,336,69,127)(30,335,70,140)(31,334,57,139)(32,333,58,138)(33,332,59,137)(34,331,60,136)(35,330,61,135)(36,329,62,134)(37,328,63,133)(38,327,64,132)(39,326,65,131)(40,325,66,130)(41,324,67,129)(42,323,68,128)(43,109,441,255)(44,108,442,254)(45,107,443,253)(46,106,444,266)(47,105,445,265)(48,104,446,264)(49,103,447,263)(50,102,448,262)(51,101,435,261)(52,100,436,260)(53,99,437,259)(54,112,438,258)(55,111,439,257)(56,110,440,256)(113,169,407,249)(114,182,408,248)(115,181,409,247)(116,180,410,246)(117,179,411,245)(118,178,412,244)(119,177,413,243)(120,176,414,242)(121,175,415,241)(122,174,416,240)(123,173,417,239)(124,172,418,252)(125,171,419,251)(126,170,420,250)(141,389,303,167)(142,388,304,166)(143,387,305,165)(144,386,306,164)(145,385,307,163)(146,384,308,162)(147,383,295,161)(148,382,296,160)(149,381,297,159)(150,380,298,158)(151,379,299,157)(152,392,300,156)(153,391,301,155)(154,390,302,168)(183,315,378,225)(184,314,365,238)(185,313,366,237)(186,312,367,236)(187,311,368,235)(188,310,369,234)(189,309,370,233)(190,322,371,232)(191,321,372,231)(192,320,373,230)(193,319,374,229)(194,318,375,228)(195,317,376,227)(196,316,377,226)(337,400,434,356)(338,399,421,355)(339,398,422,354)(340,397,423,353)(341,396,424,352)(342,395,425,351)(343,394,426,364)(344,393,427,363)(345,406,428,362)(346,405,429,361)(347,404,430,360)(348,403,431,359)(349,402,432,358)(350,401,433,357)>;
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), (1,196,8,189)(2,195,9,188)(3,194,10,187)(4,193,11,186)(5,192,12,185)(6,191,13,184)(7,190,14,183)(15,33,22,40)(16,32,23,39)(17,31,24,38)(18,30,25,37)(19,29,26,36)(20,42,27,35)(21,41,28,34)(43,143,50,150)(44,142,51,149)(45,141,52,148)(46,154,53,147)(47,153,54,146)(48,152,55,145)(49,151,56,144)(57,275,64,268)(58,274,65,267)(59,273,66,280)(60,272,67,279)(61,271,68,278)(62,270,69,277)(63,269,70,276)(71,139,78,132)(72,138,79,131)(73,137,80,130)(74,136,81,129)(75,135,82,128)(76,134,83,127)(77,133,84,140)(85,330,92,323)(86,329,93,336)(87,328,94,335)(88,327,95,334)(89,326,96,333)(90,325,97,332)(91,324,98,331)(99,383,106,390)(100,382,107,389)(101,381,108,388)(102,380,109,387)(103,379,110,386)(104,392,111,385)(105,391,112,384)(113,403,120,396)(114,402,121,395)(115,401,122,394)(116,400,123,393)(117,399,124,406)(118,398,125,405)(119,397,126,404)(155,258,162,265)(156,257,163,264)(157,256,164,263)(158,255,165,262)(159,254,166,261)(160,253,167,260)(161,266,168,259)(169,431,176,424)(170,430,177,423)(171,429,178,422)(172,428,179,421)(173,427,180,434)(174,426,181,433)(175,425,182,432)(197,319,204,312)(198,318,205,311)(199,317,206,310)(200,316,207,309)(201,315,208,322)(202,314,209,321)(203,313,210,320)(211,233,218,226)(212,232,219,225)(213,231,220,238)(214,230,221,237)(215,229,222,236)(216,228,223,235)(217,227,224,234)(239,344,246,337)(240,343,247,350)(241,342,248,349)(242,341,249,348)(243,340,250,347)(244,339,251,346)(245,338,252,345)(281,378,288,371)(282,377,289,370)(283,376,290,369)(284,375,291,368)(285,374,292,367)(286,373,293,366)(287,372,294,365)(295,444,302,437)(296,443,303,436)(297,442,304,435)(298,441,305,448)(299,440,306,447)(300,439,307,446)(301,438,308,445)(351,408,358,415)(352,407,359,414)(353,420,360,413)(354,419,361,412)(355,418,362,411)(356,417,363,410)(357,416,364,409), (1,358,440,28,282,402,56,279)(2,357,441,27,283,401,43,278)(3,356,442,26,284,400,44,277)(4,355,443,25,285,399,45,276)(5,354,444,24,286,398,46,275)(6,353,445,23,287,397,47,274)(7,352,446,22,288,396,48,273)(8,351,447,21,289,395,49,272)(9,364,448,20,290,394,50,271)(10,363,435,19,291,393,51,270)(11,362,436,18,292,406,52,269)(12,361,437,17,293,405,53,268)(13,360,438,16,294,404,54,267)(14,359,439,15,281,403,55,280)(29,368,116,149,69,187,410,297)(30,367,117,148,70,186,411,296)(31,366,118,147,57,185,412,295)(32,365,119,146,58,184,413,308)(33,378,120,145,59,183,414,307)(34,377,121,144,60,196,415,306)(35,376,122,143,61,195,416,305)(36,375,123,142,62,194,417,304)(37,374,124,141,63,193,418,303)(38,373,125,154,64,192,419,302)(39,372,126,153,65,191,420,301)(40,371,113,152,66,190,407,300)(41,370,114,151,67,189,408,299)(42,369,115,150,68,188,409,298)(71,99,429,221,95,259,346,203)(72,112,430,220,96,258,347,202)(73,111,431,219,97,257,348,201)(74,110,432,218,98,256,349,200)(75,109,433,217,85,255,350,199)(76,108,434,216,86,254,337,198)(77,107,421,215,87,253,338,197)(78,106,422,214,88,266,339,210)(79,105,423,213,89,265,340,209)(80,104,424,212,90,264,341,208)(81,103,425,211,91,263,342,207)(82,102,426,224,92,262,343,206)(83,101,427,223,93,261,344,205)(84,100,428,222,94,260,345,204)(127,381,180,235,336,159,246,311)(128,380,181,234,323,158,247,310)(129,379,182,233,324,157,248,309)(130,392,169,232,325,156,249,322)(131,391,170,231,326,155,250,321)(132,390,171,230,327,168,251,320)(133,389,172,229,328,167,252,319)(134,388,173,228,329,166,239,318)(135,387,174,227,330,165,240,317)(136,386,175,226,331,164,241,316)(137,385,176,225,332,163,242,315)(138,384,177,238,333,162,243,314)(139,383,178,237,334,161,244,313)(140,382,179,236,335,160,245,312), (1,200,282,218)(2,199,283,217)(3,198,284,216)(4,197,285,215)(5,210,286,214)(6,209,287,213)(7,208,288,212)(8,207,289,211)(9,206,290,224)(10,205,291,223)(11,204,292,222)(12,203,293,221)(13,202,294,220)(14,201,281,219)(15,97,280,73)(16,96,267,72)(17,95,268,71)(18,94,269,84)(19,93,270,83)(20,92,271,82)(21,91,272,81)(22,90,273,80)(23,89,274,79)(24,88,275,78)(25,87,276,77)(26,86,277,76)(27,85,278,75)(28,98,279,74)(29,336,69,127)(30,335,70,140)(31,334,57,139)(32,333,58,138)(33,332,59,137)(34,331,60,136)(35,330,61,135)(36,329,62,134)(37,328,63,133)(38,327,64,132)(39,326,65,131)(40,325,66,130)(41,324,67,129)(42,323,68,128)(43,109,441,255)(44,108,442,254)(45,107,443,253)(46,106,444,266)(47,105,445,265)(48,104,446,264)(49,103,447,263)(50,102,448,262)(51,101,435,261)(52,100,436,260)(53,99,437,259)(54,112,438,258)(55,111,439,257)(56,110,440,256)(113,169,407,249)(114,182,408,248)(115,181,409,247)(116,180,410,246)(117,179,411,245)(118,178,412,244)(119,177,413,243)(120,176,414,242)(121,175,415,241)(122,174,416,240)(123,173,417,239)(124,172,418,252)(125,171,419,251)(126,170,420,250)(141,389,303,167)(142,388,304,166)(143,387,305,165)(144,386,306,164)(145,385,307,163)(146,384,308,162)(147,383,295,161)(148,382,296,160)(149,381,297,159)(150,380,298,158)(151,379,299,157)(152,392,300,156)(153,391,301,155)(154,390,302,168)(183,315,378,225)(184,314,365,238)(185,313,366,237)(186,312,367,236)(187,311,368,235)(188,310,369,234)(189,309,370,233)(190,322,371,232)(191,321,372,231)(192,320,373,230)(193,319,374,229)(194,318,375,228)(195,317,376,227)(196,316,377,226)(337,400,434,356)(338,399,421,355)(339,398,422,354)(340,397,423,353)(341,396,424,352)(342,395,425,351)(343,394,426,364)(344,393,427,363)(345,406,428,362)(346,405,429,361)(347,404,430,360)(348,403,431,359)(349,402,432,358)(350,401,433,357) );
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)], [(1,196,8,189),(2,195,9,188),(3,194,10,187),(4,193,11,186),(5,192,12,185),(6,191,13,184),(7,190,14,183),(15,33,22,40),(16,32,23,39),(17,31,24,38),(18,30,25,37),(19,29,26,36),(20,42,27,35),(21,41,28,34),(43,143,50,150),(44,142,51,149),(45,141,52,148),(46,154,53,147),(47,153,54,146),(48,152,55,145),(49,151,56,144),(57,275,64,268),(58,274,65,267),(59,273,66,280),(60,272,67,279),(61,271,68,278),(62,270,69,277),(63,269,70,276),(71,139,78,132),(72,138,79,131),(73,137,80,130),(74,136,81,129),(75,135,82,128),(76,134,83,127),(77,133,84,140),(85,330,92,323),(86,329,93,336),(87,328,94,335),(88,327,95,334),(89,326,96,333),(90,325,97,332),(91,324,98,331),(99,383,106,390),(100,382,107,389),(101,381,108,388),(102,380,109,387),(103,379,110,386),(104,392,111,385),(105,391,112,384),(113,403,120,396),(114,402,121,395),(115,401,122,394),(116,400,123,393),(117,399,124,406),(118,398,125,405),(119,397,126,404),(155,258,162,265),(156,257,163,264),(157,256,164,263),(158,255,165,262),(159,254,166,261),(160,253,167,260),(161,266,168,259),(169,431,176,424),(170,430,177,423),(171,429,178,422),(172,428,179,421),(173,427,180,434),(174,426,181,433),(175,425,182,432),(197,319,204,312),(198,318,205,311),(199,317,206,310),(200,316,207,309),(201,315,208,322),(202,314,209,321),(203,313,210,320),(211,233,218,226),(212,232,219,225),(213,231,220,238),(214,230,221,237),(215,229,222,236),(216,228,223,235),(217,227,224,234),(239,344,246,337),(240,343,247,350),(241,342,248,349),(242,341,249,348),(243,340,250,347),(244,339,251,346),(245,338,252,345),(281,378,288,371),(282,377,289,370),(283,376,290,369),(284,375,291,368),(285,374,292,367),(286,373,293,366),(287,372,294,365),(295,444,302,437),(296,443,303,436),(297,442,304,435),(298,441,305,448),(299,440,306,447),(300,439,307,446),(301,438,308,445),(351,408,358,415),(352,407,359,414),(353,420,360,413),(354,419,361,412),(355,418,362,411),(356,417,363,410),(357,416,364,409)], [(1,358,440,28,282,402,56,279),(2,357,441,27,283,401,43,278),(3,356,442,26,284,400,44,277),(4,355,443,25,285,399,45,276),(5,354,444,24,286,398,46,275),(6,353,445,23,287,397,47,274),(7,352,446,22,288,396,48,273),(8,351,447,21,289,395,49,272),(9,364,448,20,290,394,50,271),(10,363,435,19,291,393,51,270),(11,362,436,18,292,406,52,269),(12,361,437,17,293,405,53,268),(13,360,438,16,294,404,54,267),(14,359,439,15,281,403,55,280),(29,368,116,149,69,187,410,297),(30,367,117,148,70,186,411,296),(31,366,118,147,57,185,412,295),(32,365,119,146,58,184,413,308),(33,378,120,145,59,183,414,307),(34,377,121,144,60,196,415,306),(35,376,122,143,61,195,416,305),(36,375,123,142,62,194,417,304),(37,374,124,141,63,193,418,303),(38,373,125,154,64,192,419,302),(39,372,126,153,65,191,420,301),(40,371,113,152,66,190,407,300),(41,370,114,151,67,189,408,299),(42,369,115,150,68,188,409,298),(71,99,429,221,95,259,346,203),(72,112,430,220,96,258,347,202),(73,111,431,219,97,257,348,201),(74,110,432,218,98,256,349,200),(75,109,433,217,85,255,350,199),(76,108,434,216,86,254,337,198),(77,107,421,215,87,253,338,197),(78,106,422,214,88,266,339,210),(79,105,423,213,89,265,340,209),(80,104,424,212,90,264,341,208),(81,103,425,211,91,263,342,207),(82,102,426,224,92,262,343,206),(83,101,427,223,93,261,344,205),(84,100,428,222,94,260,345,204),(127,381,180,235,336,159,246,311),(128,380,181,234,323,158,247,310),(129,379,182,233,324,157,248,309),(130,392,169,232,325,156,249,322),(131,391,170,231,326,155,250,321),(132,390,171,230,327,168,251,320),(133,389,172,229,328,167,252,319),(134,388,173,228,329,166,239,318),(135,387,174,227,330,165,240,317),(136,386,175,226,331,164,241,316),(137,385,176,225,332,163,242,315),(138,384,177,238,333,162,243,314),(139,383,178,237,334,161,244,313),(140,382,179,236,335,160,245,312)], [(1,200,282,218),(2,199,283,217),(3,198,284,216),(4,197,285,215),(5,210,286,214),(6,209,287,213),(7,208,288,212),(8,207,289,211),(9,206,290,224),(10,205,291,223),(11,204,292,222),(12,203,293,221),(13,202,294,220),(14,201,281,219),(15,97,280,73),(16,96,267,72),(17,95,268,71),(18,94,269,84),(19,93,270,83),(20,92,271,82),(21,91,272,81),(22,90,273,80),(23,89,274,79),(24,88,275,78),(25,87,276,77),(26,86,277,76),(27,85,278,75),(28,98,279,74),(29,336,69,127),(30,335,70,140),(31,334,57,139),(32,333,58,138),(33,332,59,137),(34,331,60,136),(35,330,61,135),(36,329,62,134),(37,328,63,133),(38,327,64,132),(39,326,65,131),(40,325,66,130),(41,324,67,129),(42,323,68,128),(43,109,441,255),(44,108,442,254),(45,107,443,253),(46,106,444,266),(47,105,445,265),(48,104,446,264),(49,103,447,263),(50,102,448,262),(51,101,435,261),(52,100,436,260),(53,99,437,259),(54,112,438,258),(55,111,439,257),(56,110,440,256),(113,169,407,249),(114,182,408,248),(115,181,409,247),(116,180,410,246),(117,179,411,245),(118,178,412,244),(119,177,413,243),(120,176,414,242),(121,175,415,241),(122,174,416,240),(123,173,417,239),(124,172,418,252),(125,171,419,251),(126,170,420,250),(141,389,303,167),(142,388,304,166),(143,387,305,165),(144,386,306,164),(145,385,307,163),(146,384,308,162),(147,383,295,161),(148,382,296,160),(149,381,297,159),(150,380,298,158),(151,379,299,157),(152,392,300,156),(153,391,301,155),(154,390,302,168),(183,315,378,225),(184,314,365,238),(185,313,366,237),(186,312,367,236),(187,311,368,235),(188,310,369,234),(189,309,370,233),(190,322,371,232),(191,321,372,231),(192,320,373,230),(193,319,374,229),(194,318,375,228),(195,317,376,227),(196,316,377,226),(337,400,434,356),(338,399,421,355),(339,398,422,354),(340,397,423,353),(341,396,424,352),(342,395,425,351),(343,394,426,364),(344,393,427,363),(345,406,428,362),(346,405,429,361),(347,404,430,360),(348,403,431,359),(349,402,432,358),(350,401,433,357)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C4 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | D14 | C4○D8 | C4×D7 | D4⋊2D7 | D4×D7 | SD16⋊3D7 | D7×Q16 |
| kernel | Dic7⋊4Q16 | C28.Q8 | C8×Dic7 | C28.44D4 | C7×Q8⋊C4 | Dic7⋊3Q8 | C2×C7⋊Q16 | Q8×Dic7 | C7⋊Q16 | C2×Dic7 | Q8⋊C4 | Dic7 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | C14 | Q8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of Dic7⋊4Q16 ►in GL4(𝔽113) generated by
| 0 | 112 | 0 | 0 |
| 1 | 104 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 79 | 108 | 0 | 0 |
| 28 | 34 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 58 | 75 | 0 | 0 |
| 32 | 55 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 25 | 51 |
| 55 | 38 | 0 | 0 |
| 81 | 58 | 0 | 0 |
| 0 | 0 | 3 | 63 |
| 0 | 0 | 68 | 110 |
G:=sub<GL(4,GF(113))| [0,1,0,0,112,104,0,0,0,0,112,0,0,0,0,112],[79,28,0,0,108,34,0,0,0,0,98,0,0,0,0,98],[58,32,0,0,75,55,0,0,0,0,0,25,0,0,9,51],[55,81,0,0,38,58,0,0,0,0,3,68,0,0,63,110] >;
Dic7⋊4Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes_4Q_{16} % in TeX
G:=Group("Dic7:4Q16"); // GroupNames label
G:=SmallGroup(448,324);
// by ID
G=gap.SmallGroup(448,324);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,135,268,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^8=1,b^2=a^7,d^2=c^4,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations