Copied to
clipboard

G = Q16⋊Dic7  order 448 = 26·7

3rd semidirect product of Q16 and Dic7 acting via Dic7/C14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q16⋊3Dic7, (C7×Q16)⋊5C4, C56.30(C2×C4), C14.98(C4×D4), (C2×C8).94D14, (C2×Q16).6D7, C56⋊C4.4C2, C8.5(C2×Dic7), C7⋊5(Q16⋊C4), (C14×Q16).6C2, (Q8×Dic7).9C2, Q8.2(C2×Dic7), C8⋊Dic7.11C2, C2.15(D4×Dic7), C28.76(C22×C4), (C2×Q8).120D14, C22.119(D4×D7), C28.105(C4○D4), C4.35(D4⋊2D7), C4.6(C22×Dic7), (C2×C28).458C23, (C2×C56).149C22, C2.7(Q16⋊D7), (C2×Dic7).187D4, Q8⋊Dic7.16C2, (Q8×C14).87C22, C14.76(C8.C22), C4⋊Dic7.181C22, (C4×Dic7).54C22, (C7×Q8).9(C2×C4), (C2×C14).369(C2×D4), (C2×C7⋊C8).163C22, (C2×C4).546(C22×D7), SmallGroup(448,718)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for Q16⋊Dic7
 G = < a,b,c,d | a8=c14=1, b2=a4, d2=c7, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 388 in 108 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, Q8, C14, C14, C42, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, Q16⋊C4, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4⋊Dic7, C4⋊Dic7, C2×C56, C7×Q16, Q8×C14, C56⋊C4, C8⋊Dic7, Q8⋊Dic7, Q8×Dic7, C14×Q16, Q16⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C8.C22, C2×Dic7, C22×D7, Q16⋊C4, D4×D7, D4⋊2D7, C22×Dic7, Q16⋊D7, D4×Dic7, Q16⋊Dic7

Smallest permutation representation of Q16⋊Dic7
►Regular action on 448 points
Generators in S448
(1 155 92 400 22 365 240 292)(2 156 93 401 23 366 241 293)(3 157 94 402 24 367 242 294)(4 158 95 403 25 368 243 281)(5 159 96 404 26 369 244 282)(6 160 97 405 27 370 245 283)(7 161 98 406 28 371 246 284)(8 162 85 393 15 372 247 285)(9 163 86 394 16 373 248 286)(10 164 87 395 17 374 249 287)(11 165 88 396 18 375 250 288)(12 166 89 397 19 376 251 289)(13 167 90 398 20 377 252 290)(14 168 91 399 21 378 239 291)(29 446 83 192 53 149 418 302)(30 447 84 193 54 150 419 303)(31 448 71 194 55 151 420 304)(32 435 72 195 56 152 407 305)(33 436 73 196 43 153 408 306)(34 437 74 183 44 154 409 307)(35 438 75 184 45 141 410 308)(36 439 76 185 46 142 411 295)(37 440 77 186 47 143 412 296)(38 441 78 187 48 144 413 297)(39 442 79 188 49 145 414 298)(40 443 80 189 50 146 415 299)(41 444 81 190 51 147 416 300)(42 445 82 191 52 148 417 301)(57 363 386 333 226 313 120 426)(58 364 387 334 227 314 121 427)(59 351 388 335 228 315 122 428)(60 352 389 336 229 316 123 429)(61 353 390 323 230 317 124 430)(62 354 391 324 231 318 125 431)(63 355 392 325 232 319 126 432)(64 356 379 326 233 320 113 433)(65 357 380 327 234 321 114 434)(66 358 381 328 235 322 115 421)(67 359 382 329 236 309 116 422)(68 360 383 330 237 310 117 423)(69 361 384 331 238 311 118 424)(70 362 385 332 225 312 119 425)(99 140 267 198 253 169 220 339)(100 127 268 199 254 170 221 340)(101 128 269 200 255 171 222 341)(102 129 270 201 256 172 223 342)(103 130 271 202 257 173 224 343)(104 131 272 203 258 174 211 344)(105 132 273 204 259 175 212 345)(106 133 274 205 260 176 213 346)(107 134 275 206 261 177 214 347)(108 135 276 207 262 178 215 348)(109 136 277 208 263 179 216 349)(110 137 278 209 264 180 217 350)(111 138 279 210 265 181 218 337)(112 139 280 197 266 182 219 338)

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

(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)

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

64 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N7A7B7C8A8B8C8D14A···14I28A···28F28G···28R56A···56L
order122244444444444444777888814···1428···2828···2856···56
size111122444414141414282828282224428282···24···48···84···4

64 irreducible representations

dim11111112222224444
type+++++++++-+--+
imageC1C2C2C2C2C2C4D4D7C4○D4D14Dic7D14C8.C22D4⋊2D7D4×D7Q16⋊D7
kernelQ16⋊Dic7C56⋊C4C8⋊Dic7Q8⋊Dic7Q8×Dic7C14×Q16C7×Q16C2×Dic7C2×Q16C28C2×C8Q16C2×Q8C14C4C22C2
# reps1112218232312623312

Matrix representation of Q16⋊Dic7 ►in GL6(𝔽113)

48900000
56650000
0059108104
0010410895
005959
00104108104108
,
1770000
88960000
007787761
002618522
007613626
005228795
,
11200000
01120000
000100
001122400
000001
000011224
,
9800000
0980000
0074552064
0023399293
0093497455
0021202339

G:=sub<GL(6,GF(113))| [48,56,0,0,0,0,90,65,0,0,0,0,0,0,5,104,5,104,0,0,9,108,9,108,0,0,108,9,5,104,0,0,104,5,9,108],[17,88,0,0,0,0,7,96,0,0,0,0,0,0,77,26,7,52,0,0,87,18,61,2,0,0,7,52,36,87,0,0,61,2,26,95],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,24,0,0,0,0,0,0,0,112,0,0,0,0,1,24],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,74,23,93,21,0,0,55,39,49,20,0,0,20,92,74,23,0,0,64,93,55,39] >;
 

Q16⋊Dic7 in GAP, Magma, Sage, TeX 

Q_{16}\rtimes {\rm Dic}_7
 
% in TeX
 
G:=Group("Q16:Dic7");
 
// GroupNames label
 
G:=SmallGroup(448,718);
 
// by ID
 
G=gap.SmallGroup(448,718);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,758,219,184,851,438,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=c^14=1,b^2=a^4,d^2=c^7,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁