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## G = Dic7.1Q16order 448 = 26·7

### 1st non-split extension by Dic7 of Q16 acting via Q16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — Dic7.1Q16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — C28⋊Q8 — Dic7.1Q16
 Lower central C7 — C14 — C2×C28 — Dic7.1Q16
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Dic7.1Q16
G = < a,b,c,d | a14=c8=1, b2=a7, d2=c4, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a7b, dcd-1=a7c-1 >

Subgroups: 468 in 98 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×C8, Q8⋊C4, Q8⋊C4, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C4.SD16, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×Dic14, Q8×C14, C14.Q16, C8×Dic7, C28.44D4, Q8⋊Dic7, C7×Q8⋊C4, C28⋊Q8, Dic7⋊Q8, Dic7.1Q16
Quotients: C1, C2, C22, D4, C23, D7, SD16, Q16, C2×D4, C4○D4, D14, C4.4D4, C2×SD16, C2×Q16, C22×D7, C4.SD16, C4○D28, D4×D7, D42D7, Dic7.D4, D7×SD16, D7×Q16, Dic7.1Q16

Smallest permutation representation of Dic7.1Q16
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)
(1 175 8 182)(2 174 9 181)(3 173 10 180)(4 172 11 179)(5 171 12 178)(6 170 13 177)(7 169 14 176)(15 443 22 436)(16 442 23 435)(17 441 24 448)(18 440 25 447)(19 439 26 446)(20 438 27 445)(21 437 28 444)(29 228 36 235)(30 227 37 234)(31 226 38 233)(32 225 39 232)(33 238 40 231)(34 237 41 230)(35 236 42 229)(43 92 50 85)(44 91 51 98)(45 90 52 97)(46 89 53 96)(47 88 54 95)(48 87 55 94)(49 86 56 93)(57 340 64 347)(58 339 65 346)(59 338 66 345)(60 337 67 344)(61 350 68 343)(62 349 69 342)(63 348 70 341)(71 136 78 129)(72 135 79 128)(73 134 80 127)(74 133 81 140)(75 132 82 139)(76 131 83 138)(77 130 84 137)(99 204 106 197)(100 203 107 210)(101 202 108 209)(102 201 109 208)(103 200 110 207)(104 199 111 206)(105 198 112 205)(113 329 120 336)(114 328 121 335)(115 327 122 334)(116 326 123 333)(117 325 124 332)(118 324 125 331)(119 323 126 330)(141 392 148 385)(142 391 149 384)(143 390 150 383)(144 389 151 382)(145 388 152 381)(146 387 153 380)(147 386 154 379)(155 412 162 419)(156 411 163 418)(157 410 164 417)(158 409 165 416)(159 408 166 415)(160 407 167 414)(161 420 168 413)(183 290 190 283)(184 289 191 282)(185 288 192 281)(186 287 193 294)(187 286 194 293)(188 285 195 292)(189 284 196 291)(211 268 218 275)(212 267 219 274)(213 280 220 273)(214 279 221 272)(215 278 222 271)(216 277 223 270)(217 276 224 269)(239 253 246 260)(240 266 247 259)(241 265 248 258)(242 264 249 257)(243 263 250 256)(244 262 251 255)(245 261 252 254)(295 428 302 421)(296 427 303 434)(297 426 304 433)(298 425 305 432)(299 424 306 431)(300 423 307 430)(301 422 308 429)(309 403 316 396)(310 402 317 395)(311 401 318 394)(312 400 319 393)(313 399 320 406)(314 398 321 405)(315 397 322 404)(351 378 358 371)(352 377 359 370)(353 376 360 369)(354 375 361 368)(355 374 362 367)(356 373 363 366)(357 372 364 365)
(1 265 86 400 159 133 205 63)(2 266 87 401 160 134 206 64)(3 253 88 402 161 135 207 65)(4 254 89 403 162 136 208 66)(5 255 90 404 163 137 209 67)(6 256 91 405 164 138 210 68)(7 257 92 406 165 139 197 69)(8 258 93 393 166 140 198 70)(9 259 94 394 167 127 199 57)(10 260 95 395 168 128 200 58)(11 261 96 396 155 129 201 59)(12 262 97 397 156 130 202 60)(13 263 98 398 157 131 203 61)(14 264 85 399 158 132 204 62)(15 183 433 365 145 328 271 42)(16 184 434 366 146 329 272 29)(17 185 421 367 147 330 273 30)(18 186 422 368 148 331 274 31)(19 187 423 369 149 332 275 32)(20 188 424 370 150 333 276 33)(21 189 425 371 151 334 277 34)(22 190 426 372 152 335 278 35)(23 191 427 373 153 336 279 36)(24 192 428 374 154 323 280 37)(25 193 429 375 141 324 267 38)(26 194 430 376 142 325 268 39)(27 195 431 377 143 326 269 40)(28 196 432 378 144 327 270 41)(43 320 409 82 106 349 176 249)(44 321 410 83 107 350 177 250)(45 322 411 84 108 337 178 251)(46 309 412 71 109 338 179 252)(47 310 413 72 110 339 180 239)(48 311 414 73 111 340 181 240)(49 312 415 74 112 341 182 241)(50 313 416 75 99 342 169 242)(51 314 417 76 100 343 170 243)(52 315 418 77 101 344 171 244)(53 316 419 78 102 345 172 245)(54 317 420 79 103 346 173 246)(55 318 407 80 104 347 174 247)(56 319 408 81 105 348 175 248)(113 221 235 435 282 303 363 380)(114 222 236 436 283 304 364 381)(115 223 237 437 284 305 351 382)(116 224 238 438 285 306 352 383)(117 211 225 439 286 307 353 384)(118 212 226 440 287 308 354 385)(119 213 227 441 288 295 355 386)(120 214 228 442 289 296 356 387)(121 215 229 443 290 297 357 388)(122 216 230 444 291 298 358 389)(123 217 231 445 292 299 359 390)(124 218 232 446 293 300 360 391)(125 219 233 447 294 301 361 392)(126 220 234 448 281 302 362 379)
(1 425 159 277)(2 426 160 278)(3 427 161 279)(4 428 162 280)(5 429 163 267)(6 430 164 268)(7 431 165 269)(8 432 166 270)(9 433 167 271)(10 434 168 272)(11 421 155 273)(12 422 156 274)(13 423 157 275)(14 424 158 276)(15 199 145 94)(16 200 146 95)(17 201 147 96)(18 202 148 97)(19 203 149 98)(20 204 150 85)(21 205 151 86)(22 206 152 87)(23 207 153 88)(24 208 154 89)(25 209 141 90)(26 210 142 91)(27 197 143 92)(28 198 144 93)(29 65 366 402)(30 66 367 403)(31 67 368 404)(32 68 369 405)(33 69 370 406)(34 70 371 393)(35 57 372 394)(36 58 373 395)(37 59 374 396)(38 60 375 397)(39 61 376 398)(40 62 377 399)(41 63 378 400)(42 64 365 401)(43 445 106 390)(44 446 107 391)(45 447 108 392)(46 448 109 379)(47 435 110 380)(48 436 111 381)(49 437 112 382)(50 438 99 383)(51 439 100 384)(52 440 101 385)(53 441 102 386)(54 442 103 387)(55 443 104 388)(56 444 105 389)(71 119 252 288)(72 120 239 289)(73 121 240 290)(74 122 241 291)(75 123 242 292)(76 124 243 293)(77 125 244 294)(78 126 245 281)(79 113 246 282)(80 114 247 283)(81 115 248 284)(82 116 249 285)(83 117 250 286)(84 118 251 287)(127 335 259 190)(128 336 260 191)(129 323 261 192)(130 324 262 193)(131 325 263 194)(132 326 264 195)(133 327 265 196)(134 328 266 183)(135 329 253 184)(136 330 254 185)(137 331 255 186)(138 332 256 187)(139 333 257 188)(140 334 258 189)(169 306 416 224)(170 307 417 211)(171 308 418 212)(172 295 419 213)(173 296 420 214)(174 297 407 215)(175 298 408 216)(176 299 409 217)(177 300 410 218)(178 301 411 219)(179 302 412 220)(180 303 413 221)(181 304 414 222)(182 305 415 223)(225 350 353 321)(226 337 354 322)(227 338 355 309)(228 339 356 310)(229 340 357 311)(230 341 358 312)(231 342 359 313)(232 343 360 314)(233 344 361 315)(234 345 362 316)(235 346 363 317)(236 347 364 318)(237 348 351 319)(238 349 352 320)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 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 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 8 8 14 14 14 14 56 56 2 2 2 2 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 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 D4 D7 SD16 Q16 C4○D4 D14 D14 D14 C4○D28 D4⋊2D7 D4×D7 D7×SD16 D7×Q16 kernel Dic7.1Q16 C14.Q16 C8×Dic7 C28.44D4 Q8⋊Dic7 C7×Q8⋊C4 C28⋊Q8 Dic7⋊Q8 C2×Dic7 Q8⋊C4 Dic7 Dic7 C28 C4⋊C4 C2×C8 C2×Q8 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 3 4 4 4 3 3 3 12 3 3 6 6

Matrix representation of Dic7.1Q16 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 1 112 0 0 11 103
,
 98 0 0 0 0 15 0 0 0 0 104 38 0 0 93 9
,
 69 0 0 0 0 18 0 0 0 0 15 0 0 0 0 15
,
 0 1 0 0 112 0 0 0 0 0 29 5 0 0 58 84
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,11,0,0,112,103],[98,0,0,0,0,15,0,0,0,0,104,93,0,0,38,9],[69,0,0,0,0,18,0,0,0,0,15,0,0,0,0,15],[0,112,0,0,1,0,0,0,0,0,29,58,0,0,5,84] >;

Dic7.1Q16 in GAP, Magma, Sage, TeX

{\rm Dic}_7._1Q_{16}
% in TeX

G:=Group("Dic7.1Q16");
// GroupNames label

G:=SmallGroup(448,326);
// by ID

G=gap.SmallGroup(448,326);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,422,135,184,570,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=c^8=1,b^2=a^7,d^2=c^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=a^7*c^-1>;
// generators/relations

׿
×
𝔽