direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16×Dic7, C7⋊5(C4×Q16), (C7×Q16)⋊2C4, C2.5(D7×Q16), C56.19(C2×C4), C14.97(C4×D4), (C2×Q16).7D7, (C2×C8).242D14, (C14×Q16).5C2, C14.28(C2×Q16), (Q8×Dic7).8C2, (C8×Dic7).5C2, Q8.1(C2×Dic7), C56⋊1C4.16C2, C2.14(D4×Dic7), C8.10(C2×Dic7), C14.78(C4○D8), (C2×C56).94C22, C28.75(C22×C4), (C2×Q8).119D14, C22.118(D4×D7), C28.104(C4○D4), C4.34(D4⋊2D7), C2.5(Q8.D14), C4.5(C22×Dic7), (C2×C28).457C23, (C2×Dic7).214D4, Q8⋊Dic7.15C2, (Q8×C14).86C22, C4⋊Dic7.180C22, (C4×Dic7).244C22, (C7×Q8).8(C2×C4), (C2×C14).368(C2×D4), (C2×C7⋊C8).277C22, (C2×C4).545(C22×D7), SmallGroup(448,717)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×Dic7
G = < a,b,c,d | a8=c14=1, b2=a4, d2=c7, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 388 in 110 conjugacy classes, 59 normal (27 characteristic)
C1, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C4×Q16, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4⋊Dic7, C4⋊Dic7, C2×C56, C7×Q16, Q8×C14, C8×Dic7, C56⋊1C4, Q8⋊Dic7, Q8×Dic7, C14×Q16, Q16×Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, Q16, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C2×Q16, C4○D8, C2×Dic7, C22×D7, C4×Q16, D4×D7, D4⋊2D7, C22×Dic7, D7×Q16, Q8.D14, D4×Dic7, Q16×Dic7
►Regular action on 448 points
Generators in S448
(1 235 17 428 268 294 208 326)(2 236 18 429 269 281 209 327)(3 237 19 430 270 282 210 328)(4 238 20 431 271 283 197 329)(5 225 21 432 272 284 198 330)(6 226 22 433 273 285 199 331)(7 227 23 434 274 286 200 332)(8 228 24 421 275 287 201 333)(9 229 25 422 276 288 202 334)(10 230 26 423 277 289 203 335)(11 231 27 424 278 290 204 336)(12 232 28 425 279 291 205 323)(13 233 15 426 280 292 206 324)(14 234 16 427 267 293 207 325)(29 259 142 368 122 397 360 159)(30 260 143 369 123 398 361 160)(31 261 144 370 124 399 362 161)(32 262 145 371 125 400 363 162)(33 263 146 372 126 401 364 163)(34 264 147 373 113 402 351 164)(35 265 148 374 114 403 352 165)(36 266 149 375 115 404 353 166)(37 253 150 376 116 405 354 167)(38 254 151 377 117 406 355 168)(39 255 152 378 118 393 356 155)(40 256 153 365 119 394 357 156)(41 257 154 366 120 395 358 157)(42 258 141 367 121 396 359 158)(43 382 102 211 297 171 129 187)(44 383 103 212 298 172 130 188)(45 384 104 213 299 173 131 189)(46 385 105 214 300 174 132 190)(47 386 106 215 301 175 133 191)(48 387 107 216 302 176 134 192)(49 388 108 217 303 177 135 193)(50 389 109 218 304 178 136 194)(51 390 110 219 305 179 137 195)(52 391 111 220 306 180 138 196)(53 392 112 221 307 181 139 183)(54 379 99 222 308 182 140 184)(55 380 100 223 295 169 127 185)(56 381 101 224 296 170 128 186)(57 89 251 80 350 445 418 320)(58 90 252 81 337 446 419 321)(59 91 239 82 338 447 420 322)(60 92 240 83 339 448 407 309)(61 93 241 84 340 435 408 310)(62 94 242 71 341 436 409 311)(63 95 243 72 342 437 410 312)(64 96 244 73 343 438 411 313)(65 97 245 74 344 439 412 314)(66 98 246 75 345 440 413 315)(67 85 247 76 346 441 414 316)(68 86 248 77 347 442 415 317)(69 87 249 78 348 443 416 318)(70 88 250 79 349 444 417 319)
(1 45 268 299)(2 46 269 300)(3 47 270 301)(4 48 271 302)(5 49 272 303)(6 50 273 304)(7 51 274 305)(8 52 275 306)(9 53 276 307)(10 54 277 308)(11 55 278 295)(12 56 279 296)(13 43 280 297)(14 44 267 298)(15 129 206 102)(16 130 207 103)(17 131 208 104)(18 132 209 105)(19 133 210 106)(20 134 197 107)(21 135 198 108)(22 136 199 109)(23 137 200 110)(24 138 201 111)(25 139 202 112)(26 140 203 99)(27 127 204 100)(28 128 205 101)(29 95 122 437)(30 96 123 438)(31 97 124 439)(32 98 125 440)(33 85 126 441)(34 86 113 442)(35 87 114 443)(36 88 115 444)(37 89 116 445)(38 90 117 446)(39 91 118 447)(40 92 119 448)(41 93 120 435)(42 94 121 436)(57 405 350 253)(58 406 337 254)(59 393 338 255)(60 394 339 256)(61 395 340 257)(62 396 341 258)(63 397 342 259)(64 398 343 260)(65 399 344 261)(66 400 345 262)(67 401 346 263)(68 402 347 264)(69 403 348 265)(70 404 349 266)(71 141 311 359)(72 142 312 360)(73 143 313 361)(74 144 314 362)(75 145 315 363)(76 146 316 364)(77 147 317 351)(78 148 318 352)(79 149 319 353)(80 150 320 354)(81 151 321 355)(82 152 322 356)(83 153 309 357)(84 154 310 358)(155 239 378 420)(156 240 365 407)(157 241 366 408)(158 242 367 409)(159 243 368 410)(160 244 369 411)(161 245 370 412)(162 246 371 413)(163 247 372 414)(164 248 373 415)(165 249 374 416)(166 250 375 417)(167 251 376 418)(168 252 377 419)(169 336 380 424)(170 323 381 425)(171 324 382 426)(172 325 383 427)(173 326 384 428)(174 327 385 429)(175 328 386 430)(176 329 387 431)(177 330 388 432)(178 331 389 433)(179 332 390 434)(180 333 391 421)(181 334 392 422)(182 335 379 423)(183 288 221 229)(184 289 222 230)(185 290 223 231)(186 291 224 232)(187 292 211 233)(188 293 212 234)(189 294 213 235)(190 281 214 236)(191 282 215 237)(192 283 216 238)(193 284 217 225)(194 285 218 226)(195 286 219 227)(196 287 220 228)
(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)
(1 374 8 367)(2 373 9 366)(3 372 10 365)(4 371 11 378)(5 370 12 377)(6 369 13 376)(7 368 14 375)(15 405 22 398)(16 404 23 397)(17 403 24 396)(18 402 25 395)(19 401 26 394)(20 400 27 393)(21 399 28 406)(29 293 36 286)(30 292 37 285)(31 291 38 284)(32 290 39 283)(33 289 40 282)(34 288 41 281)(35 287 42 294)(43 418 50 411)(44 417 51 410)(45 416 52 409)(46 415 53 408)(47 414 54 407)(48 413 55 420)(49 412 56 419)(57 109 64 102)(58 108 65 101)(59 107 66 100)(60 106 67 99)(61 105 68 112)(62 104 69 111)(63 103 70 110)(71 173 78 180)(72 172 79 179)(73 171 80 178)(74 170 81 177)(75 169 82 176)(76 182 83 175)(77 181 84 174)(85 222 92 215)(86 221 93 214)(87 220 94 213)(88 219 95 212)(89 218 96 211)(90 217 97 224)(91 216 98 223)(113 229 120 236)(114 228 121 235)(115 227 122 234)(116 226 123 233)(117 225 124 232)(118 238 125 231)(119 237 126 230)(127 338 134 345)(128 337 135 344)(129 350 136 343)(130 349 137 342)(131 348 138 341)(132 347 139 340)(133 346 140 339)(141 326 148 333)(142 325 149 332)(143 324 150 331)(144 323 151 330)(145 336 152 329)(146 335 153 328)(147 334 154 327)(155 271 162 278)(156 270 163 277)(157 269 164 276)(158 268 165 275)(159 267 166 274)(160 280 167 273)(161 279 168 272)(183 435 190 442)(184 448 191 441)(185 447 192 440)(186 446 193 439)(187 445 194 438)(188 444 195 437)(189 443 196 436)(197 262 204 255)(198 261 205 254)(199 260 206 253)(200 259 207 266)(201 258 208 265)(202 257 209 264)(203 256 210 263)(239 302 246 295)(240 301 247 308)(241 300 248 307)(242 299 249 306)(243 298 250 305)(244 297 251 304)(245 296 252 303)(309 386 316 379)(310 385 317 392)(311 384 318 391)(312 383 319 390)(313 382 320 389)(314 381 321 388)(315 380 322 387)(351 422 358 429)(352 421 359 428)(353 434 360 427)(354 433 361 426)(355 432 362 425)(356 431 363 424)(357 430 364 423)
G:=sub<Sym(448)| (1,235,17,428,268,294,208,326)(2,236,18,429,269,281,209,327)(3,237,19,430,270,282,210,328)(4,238,20,431,271,283,197,329)(5,225,21,432,272,284,198,330)(6,226,22,433,273,285,199,331)(7,227,23,434,274,286,200,332)(8,228,24,421,275,287,201,333)(9,229,25,422,276,288,202,334)(10,230,26,423,277,289,203,335)(11,231,27,424,278,290,204,336)(12,232,28,425,279,291,205,323)(13,233,15,426,280,292,206,324)(14,234,16,427,267,293,207,325)(29,259,142,368,122,397,360,159)(30,260,143,369,123,398,361,160)(31,261,144,370,124,399,362,161)(32,262,145,371,125,400,363,162)(33,263,146,372,126,401,364,163)(34,264,147,373,113,402,351,164)(35,265,148,374,114,403,352,165)(36,266,149,375,115,404,353,166)(37,253,150,376,116,405,354,167)(38,254,151,377,117,406,355,168)(39,255,152,378,118,393,356,155)(40,256,153,365,119,394,357,156)(41,257,154,366,120,395,358,157)(42,258,141,367,121,396,359,158)(43,382,102,211,297,171,129,187)(44,383,103,212,298,172,130,188)(45,384,104,213,299,173,131,189)(46,385,105,214,300,174,132,190)(47,386,106,215,301,175,133,191)(48,387,107,216,302,176,134,192)(49,388,108,217,303,177,135,193)(50,389,109,218,304,178,136,194)(51,390,110,219,305,179,137,195)(52,391,111,220,306,180,138,196)(53,392,112,221,307,181,139,183)(54,379,99,222,308,182,140,184)(55,380,100,223,295,169,127,185)(56,381,101,224,296,170,128,186)(57,89,251,80,350,445,418,320)(58,90,252,81,337,446,419,321)(59,91,239,82,338,447,420,322)(60,92,240,83,339,448,407,309)(61,93,241,84,340,435,408,310)(62,94,242,71,341,436,409,311)(63,95,243,72,342,437,410,312)(64,96,244,73,343,438,411,313)(65,97,245,74,344,439,412,314)(66,98,246,75,345,440,413,315)(67,85,247,76,346,441,414,316)(68,86,248,77,347,442,415,317)(69,87,249,78,348,443,416,318)(70,88,250,79,349,444,417,319), (1,45,268,299)(2,46,269,300)(3,47,270,301)(4,48,271,302)(5,49,272,303)(6,50,273,304)(7,51,274,305)(8,52,275,306)(9,53,276,307)(10,54,277,308)(11,55,278,295)(12,56,279,296)(13,43,280,297)(14,44,267,298)(15,129,206,102)(16,130,207,103)(17,131,208,104)(18,132,209,105)(19,133,210,106)(20,134,197,107)(21,135,198,108)(22,136,199,109)(23,137,200,110)(24,138,201,111)(25,139,202,112)(26,140,203,99)(27,127,204,100)(28,128,205,101)(29,95,122,437)(30,96,123,438)(31,97,124,439)(32,98,125,440)(33,85,126,441)(34,86,113,442)(35,87,114,443)(36,88,115,444)(37,89,116,445)(38,90,117,446)(39,91,118,447)(40,92,119,448)(41,93,120,435)(42,94,121,436)(57,405,350,253)(58,406,337,254)(59,393,338,255)(60,394,339,256)(61,395,340,257)(62,396,341,258)(63,397,342,259)(64,398,343,260)(65,399,344,261)(66,400,345,262)(67,401,346,263)(68,402,347,264)(69,403,348,265)(70,404,349,266)(71,141,311,359)(72,142,312,360)(73,143,313,361)(74,144,314,362)(75,145,315,363)(76,146,316,364)(77,147,317,351)(78,148,318,352)(79,149,319,353)(80,150,320,354)(81,151,321,355)(82,152,322,356)(83,153,309,357)(84,154,310,358)(155,239,378,420)(156,240,365,407)(157,241,366,408)(158,242,367,409)(159,243,368,410)(160,244,369,411)(161,245,370,412)(162,246,371,413)(163,247,372,414)(164,248,373,415)(165,249,374,416)(166,250,375,417)(167,251,376,418)(168,252,377,419)(169,336,380,424)(170,323,381,425)(171,324,382,426)(172,325,383,427)(173,326,384,428)(174,327,385,429)(175,328,386,430)(176,329,387,431)(177,330,388,432)(178,331,389,433)(179,332,390,434)(180,333,391,421)(181,334,392,422)(182,335,379,423)(183,288,221,229)(184,289,222,230)(185,290,223,231)(186,291,224,232)(187,292,211,233)(188,293,212,234)(189,294,213,235)(190,281,214,236)(191,282,215,237)(192,283,216,238)(193,284,217,225)(194,285,218,226)(195,286,219,227)(196,287,220,228), (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), (1,374,8,367)(2,373,9,366)(3,372,10,365)(4,371,11,378)(5,370,12,377)(6,369,13,376)(7,368,14,375)(15,405,22,398)(16,404,23,397)(17,403,24,396)(18,402,25,395)(19,401,26,394)(20,400,27,393)(21,399,28,406)(29,293,36,286)(30,292,37,285)(31,291,38,284)(32,290,39,283)(33,289,40,282)(34,288,41,281)(35,287,42,294)(43,418,50,411)(44,417,51,410)(45,416,52,409)(46,415,53,408)(47,414,54,407)(48,413,55,420)(49,412,56,419)(57,109,64,102)(58,108,65,101)(59,107,66,100)(60,106,67,99)(61,105,68,112)(62,104,69,111)(63,103,70,110)(71,173,78,180)(72,172,79,179)(73,171,80,178)(74,170,81,177)(75,169,82,176)(76,182,83,175)(77,181,84,174)(85,222,92,215)(86,221,93,214)(87,220,94,213)(88,219,95,212)(89,218,96,211)(90,217,97,224)(91,216,98,223)(113,229,120,236)(114,228,121,235)(115,227,122,234)(116,226,123,233)(117,225,124,232)(118,238,125,231)(119,237,126,230)(127,338,134,345)(128,337,135,344)(129,350,136,343)(130,349,137,342)(131,348,138,341)(132,347,139,340)(133,346,140,339)(141,326,148,333)(142,325,149,332)(143,324,150,331)(144,323,151,330)(145,336,152,329)(146,335,153,328)(147,334,154,327)(155,271,162,278)(156,270,163,277)(157,269,164,276)(158,268,165,275)(159,267,166,274)(160,280,167,273)(161,279,168,272)(183,435,190,442)(184,448,191,441)(185,447,192,440)(186,446,193,439)(187,445,194,438)(188,444,195,437)(189,443,196,436)(197,262,204,255)(198,261,205,254)(199,260,206,253)(200,259,207,266)(201,258,208,265)(202,257,209,264)(203,256,210,263)(239,302,246,295)(240,301,247,308)(241,300,248,307)(242,299,249,306)(243,298,250,305)(244,297,251,304)(245,296,252,303)(309,386,316,379)(310,385,317,392)(311,384,318,391)(312,383,319,390)(313,382,320,389)(314,381,321,388)(315,380,322,387)(351,422,358,429)(352,421,359,428)(353,434,360,427)(354,433,361,426)(355,432,362,425)(356,431,363,424)(357,430,364,423)>;
G:=Group( (1,235,17,428,268,294,208,326)(2,236,18,429,269,281,209,327)(3,237,19,430,270,282,210,328)(4,238,20,431,271,283,197,329)(5,225,21,432,272,284,198,330)(6,226,22,433,273,285,199,331)(7,227,23,434,274,286,200,332)(8,228,24,421,275,287,201,333)(9,229,25,422,276,288,202,334)(10,230,26,423,277,289,203,335)(11,231,27,424,278,290,204,336)(12,232,28,425,279,291,205,323)(13,233,15,426,280,292,206,324)(14,234,16,427,267,293,207,325)(29,259,142,368,122,397,360,159)(30,260,143,369,123,398,361,160)(31,261,144,370,124,399,362,161)(32,262,145,371,125,400,363,162)(33,263,146,372,126,401,364,163)(34,264,147,373,113,402,351,164)(35,265,148,374,114,403,352,165)(36,266,149,375,115,404,353,166)(37,253,150,376,116,405,354,167)(38,254,151,377,117,406,355,168)(39,255,152,378,118,393,356,155)(40,256,153,365,119,394,357,156)(41,257,154,366,120,395,358,157)(42,258,141,367,121,396,359,158)(43,382,102,211,297,171,129,187)(44,383,103,212,298,172,130,188)(45,384,104,213,299,173,131,189)(46,385,105,214,300,174,132,190)(47,386,106,215,301,175,133,191)(48,387,107,216,302,176,134,192)(49,388,108,217,303,177,135,193)(50,389,109,218,304,178,136,194)(51,390,110,219,305,179,137,195)(52,391,111,220,306,180,138,196)(53,392,112,221,307,181,139,183)(54,379,99,222,308,182,140,184)(55,380,100,223,295,169,127,185)(56,381,101,224,296,170,128,186)(57,89,251,80,350,445,418,320)(58,90,252,81,337,446,419,321)(59,91,239,82,338,447,420,322)(60,92,240,83,339,448,407,309)(61,93,241,84,340,435,408,310)(62,94,242,71,341,436,409,311)(63,95,243,72,342,437,410,312)(64,96,244,73,343,438,411,313)(65,97,245,74,344,439,412,314)(66,98,246,75,345,440,413,315)(67,85,247,76,346,441,414,316)(68,86,248,77,347,442,415,317)(69,87,249,78,348,443,416,318)(70,88,250,79,349,444,417,319), (1,45,268,299)(2,46,269,300)(3,47,270,301)(4,48,271,302)(5,49,272,303)(6,50,273,304)(7,51,274,305)(8,52,275,306)(9,53,276,307)(10,54,277,308)(11,55,278,295)(12,56,279,296)(13,43,280,297)(14,44,267,298)(15,129,206,102)(16,130,207,103)(17,131,208,104)(18,132,209,105)(19,133,210,106)(20,134,197,107)(21,135,198,108)(22,136,199,109)(23,137,200,110)(24,138,201,111)(25,139,202,112)(26,140,203,99)(27,127,204,100)(28,128,205,101)(29,95,122,437)(30,96,123,438)(31,97,124,439)(32,98,125,440)(33,85,126,441)(34,86,113,442)(35,87,114,443)(36,88,115,444)(37,89,116,445)(38,90,117,446)(39,91,118,447)(40,92,119,448)(41,93,120,435)(42,94,121,436)(57,405,350,253)(58,406,337,254)(59,393,338,255)(60,394,339,256)(61,395,340,257)(62,396,341,258)(63,397,342,259)(64,398,343,260)(65,399,344,261)(66,400,345,262)(67,401,346,263)(68,402,347,264)(69,403,348,265)(70,404,349,266)(71,141,311,359)(72,142,312,360)(73,143,313,361)(74,144,314,362)(75,145,315,363)(76,146,316,364)(77,147,317,351)(78,148,318,352)(79,149,319,353)(80,150,320,354)(81,151,321,355)(82,152,322,356)(83,153,309,357)(84,154,310,358)(155,239,378,420)(156,240,365,407)(157,241,366,408)(158,242,367,409)(159,243,368,410)(160,244,369,411)(161,245,370,412)(162,246,371,413)(163,247,372,414)(164,248,373,415)(165,249,374,416)(166,250,375,417)(167,251,376,418)(168,252,377,419)(169,336,380,424)(170,323,381,425)(171,324,382,426)(172,325,383,427)(173,326,384,428)(174,327,385,429)(175,328,386,430)(176,329,387,431)(177,330,388,432)(178,331,389,433)(179,332,390,434)(180,333,391,421)(181,334,392,422)(182,335,379,423)(183,288,221,229)(184,289,222,230)(185,290,223,231)(186,291,224,232)(187,292,211,233)(188,293,212,234)(189,294,213,235)(190,281,214,236)(191,282,215,237)(192,283,216,238)(193,284,217,225)(194,285,218,226)(195,286,219,227)(196,287,220,228), (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), (1,374,8,367)(2,373,9,366)(3,372,10,365)(4,371,11,378)(5,370,12,377)(6,369,13,376)(7,368,14,375)(15,405,22,398)(16,404,23,397)(17,403,24,396)(18,402,25,395)(19,401,26,394)(20,400,27,393)(21,399,28,406)(29,293,36,286)(30,292,37,285)(31,291,38,284)(32,290,39,283)(33,289,40,282)(34,288,41,281)(35,287,42,294)(43,418,50,411)(44,417,51,410)(45,416,52,409)(46,415,53,408)(47,414,54,407)(48,413,55,420)(49,412,56,419)(57,109,64,102)(58,108,65,101)(59,107,66,100)(60,106,67,99)(61,105,68,112)(62,104,69,111)(63,103,70,110)(71,173,78,180)(72,172,79,179)(73,171,80,178)(74,170,81,177)(75,169,82,176)(76,182,83,175)(77,181,84,174)(85,222,92,215)(86,221,93,214)(87,220,94,213)(88,219,95,212)(89,218,96,211)(90,217,97,224)(91,216,98,223)(113,229,120,236)(114,228,121,235)(115,227,122,234)(116,226,123,233)(117,225,124,232)(118,238,125,231)(119,237,126,230)(127,338,134,345)(128,337,135,344)(129,350,136,343)(130,349,137,342)(131,348,138,341)(132,347,139,340)(133,346,140,339)(141,326,148,333)(142,325,149,332)(143,324,150,331)(144,323,151,330)(145,336,152,329)(146,335,153,328)(147,334,154,327)(155,271,162,278)(156,270,163,277)(157,269,164,276)(158,268,165,275)(159,267,166,274)(160,280,167,273)(161,279,168,272)(183,435,190,442)(184,448,191,441)(185,447,192,440)(186,446,193,439)(187,445,194,438)(188,444,195,437)(189,443,196,436)(197,262,204,255)(198,261,205,254)(199,260,206,253)(200,259,207,266)(201,258,208,265)(202,257,209,264)(203,256,210,263)(239,302,246,295)(240,301,247,308)(241,300,248,307)(242,299,249,306)(243,298,250,305)(244,297,251,304)(245,296,252,303)(309,386,316,379)(310,385,317,392)(311,384,318,391)(312,383,319,390)(313,382,320,389)(314,381,321,388)(315,380,322,387)(351,422,358,429)(352,421,359,428)(353,434,360,427)(354,433,361,426)(355,432,362,425)(356,431,363,424)(357,430,364,423) );
G=PermutationGroup([[(1,235,17,428,268,294,208,326),(2,236,18,429,269,281,209,327),(3,237,19,430,270,282,210,328),(4,238,20,431,271,283,197,329),(5,225,21,432,272,284,198,330),(6,226,22,433,273,285,199,331),(7,227,23,434,274,286,200,332),(8,228,24,421,275,287,201,333),(9,229,25,422,276,288,202,334),(10,230,26,423,277,289,203,335),(11,231,27,424,278,290,204,336),(12,232,28,425,279,291,205,323),(13,233,15,426,280,292,206,324),(14,234,16,427,267,293,207,325),(29,259,142,368,122,397,360,159),(30,260,143,369,123,398,361,160),(31,261,144,370,124,399,362,161),(32,262,145,371,125,400,363,162),(33,263,146,372,126,401,364,163),(34,264,147,373,113,402,351,164),(35,265,148,374,114,403,352,165),(36,266,149,375,115,404,353,166),(37,253,150,376,116,405,354,167),(38,254,151,377,117,406,355,168),(39,255,152,378,118,393,356,155),(40,256,153,365,119,394,357,156),(41,257,154,366,120,395,358,157),(42,258,141,367,121,396,359,158),(43,382,102,211,297,171,129,187),(44,383,103,212,298,172,130,188),(45,384,104,213,299,173,131,189),(46,385,105,214,300,174,132,190),(47,386,106,215,301,175,133,191),(48,387,107,216,302,176,134,192),(49,388,108,217,303,177,135,193),(50,389,109,218,304,178,136,194),(51,390,110,219,305,179,137,195),(52,391,111,220,306,180,138,196),(53,392,112,221,307,181,139,183),(54,379,99,222,308,182,140,184),(55,380,100,223,295,169,127,185),(56,381,101,224,296,170,128,186),(57,89,251,80,350,445,418,320),(58,90,252,81,337,446,419,321),(59,91,239,82,338,447,420,322),(60,92,240,83,339,448,407,309),(61,93,241,84,340,435,408,310),(62,94,242,71,341,436,409,311),(63,95,243,72,342,437,410,312),(64,96,244,73,343,438,411,313),(65,97,245,74,344,439,412,314),(66,98,246,75,345,440,413,315),(67,85,247,76,346,441,414,316),(68,86,248,77,347,442,415,317),(69,87,249,78,348,443,416,318),(70,88,250,79,349,444,417,319)], [(1,45,268,299),(2,46,269,300),(3,47,270,301),(4,48,271,302),(5,49,272,303),(6,50,273,304),(7,51,274,305),(8,52,275,306),(9,53,276,307),(10,54,277,308),(11,55,278,295),(12,56,279,296),(13,43,280,297),(14,44,267,298),(15,129,206,102),(16,130,207,103),(17,131,208,104),(18,132,209,105),(19,133,210,106),(20,134,197,107),(21,135,198,108),(22,136,199,109),(23,137,200,110),(24,138,201,111),(25,139,202,112),(26,140,203,99),(27,127,204,100),(28,128,205,101),(29,95,122,437),(30,96,123,438),(31,97,124,439),(32,98,125,440),(33,85,126,441),(34,86,113,442),(35,87,114,443),(36,88,115,444),(37,89,116,445),(38,90,117,446),(39,91,118,447),(40,92,119,448),(41,93,120,435),(42,94,121,436),(57,405,350,253),(58,406,337,254),(59,393,338,255),(60,394,339,256),(61,395,340,257),(62,396,341,258),(63,397,342,259),(64,398,343,260),(65,399,344,261),(66,400,345,262),(67,401,346,263),(68,402,347,264),(69,403,348,265),(70,404,349,266),(71,141,311,359),(72,142,312,360),(73,143,313,361),(74,144,314,362),(75,145,315,363),(76,146,316,364),(77,147,317,351),(78,148,318,352),(79,149,319,353),(80,150,320,354),(81,151,321,355),(82,152,322,356),(83,153,309,357),(84,154,310,358),(155,239,378,420),(156,240,365,407),(157,241,366,408),(158,242,367,409),(159,243,368,410),(160,244,369,411),(161,245,370,412),(162,246,371,413),(163,247,372,414),(164,248,373,415),(165,249,374,416),(166,250,375,417),(167,251,376,418),(168,252,377,419),(169,336,380,424),(170,323,381,425),(171,324,382,426),(172,325,383,427),(173,326,384,428),(174,327,385,429),(175,328,386,430),(176,329,387,431),(177,330,388,432),(178,331,389,433),(179,332,390,434),(180,333,391,421),(181,334,392,422),(182,335,379,423),(183,288,221,229),(184,289,222,230),(185,290,223,231),(186,291,224,232),(187,292,211,233),(188,293,212,234),(189,294,213,235),(190,281,214,236),(191,282,215,237),(192,283,216,238),(193,284,217,225),(194,285,218,226),(195,286,219,227),(196,287,220,228)], [(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)], [(1,374,8,367),(2,373,9,366),(3,372,10,365),(4,371,11,378),(5,370,12,377),(6,369,13,376),(7,368,14,375),(15,405,22,398),(16,404,23,397),(17,403,24,396),(18,402,25,395),(19,401,26,394),(20,400,27,393),(21,399,28,406),(29,293,36,286),(30,292,37,285),(31,291,38,284),(32,290,39,283),(33,289,40,282),(34,288,41,281),(35,287,42,294),(43,418,50,411),(44,417,51,410),(45,416,52,409),(46,415,53,408),(47,414,54,407),(48,413,55,420),(49,412,56,419),(57,109,64,102),(58,108,65,101),(59,107,66,100),(60,106,67,99),(61,105,68,112),(62,104,69,111),(63,103,70,110),(71,173,78,180),(72,172,79,179),(73,171,80,178),(74,170,81,177),(75,169,82,176),(76,182,83,175),(77,181,84,174),(85,222,92,215),(86,221,93,214),(87,220,94,213),(88,219,95,212),(89,218,96,211),(90,217,97,224),(91,216,98,223),(113,229,120,236),(114,228,121,235),(115,227,122,234),(116,226,123,233),(117,225,124,232),(118,238,125,231),(119,237,126,230),(127,338,134,345),(128,337,135,344),(129,350,136,343),(130,349,137,342),(131,348,138,341),(132,347,139,340),(133,346,140,339),(141,326,148,333),(142,325,149,332),(143,324,150,331),(144,323,151,330),(145,336,152,329),(146,335,153,328),(147,334,154,327),(155,271,162,278),(156,270,163,277),(157,269,164,276),(158,268,165,275),(159,267,166,274),(160,280,167,273),(161,279,168,272),(183,435,190,442),(184,448,191,441),(185,447,192,440),(186,446,193,439),(187,445,194,438),(188,444,195,437),(189,443,196,436),(197,262,204,255),(198,261,205,254),(199,260,206,253),(200,259,207,266),(201,258,208,265),(202,257,209,264),(203,256,210,263),(239,302,246,295),(240,301,247,308),(241,300,248,307),(242,299,249,306),(243,298,250,305),(244,297,251,304),(245,296,252,303),(309,386,316,379),(310,385,317,392),(311,384,318,391),(312,383,319,390),(313,382,320,389),(314,381,321,388),(315,380,322,387),(351,422,358,429),(352,421,359,428),(353,434,360,427),(354,433,361,426),(355,432,362,425),(356,431,363,424),(357,430,364,423)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | Q16 | C4○D4 | D14 | Dic7 | D14 | C4○D8 | D4⋊2D7 | D4×D7 | D7×Q16 | Q8.D14 |
| kernel | Q16×Dic7 | C8×Dic7 | C56⋊1C4 | Q8⋊Dic7 | Q8×Dic7 | C14×Q16 | C7×Q16 | C2×Dic7 | C2×Q16 | Dic7 | C28 | C2×C8 | Q16 | C2×Q8 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 2 | 3 | 4 | 2 | 3 | 12 | 6 | 4 | 3 | 3 | 6 | 6 |
Matrix representation of Q16×Dic7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 31 | 51 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 62 | 81 |
| 0 | 0 | 46 | 51 |
| 89 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 82 | 17 | 0 | 0 |
| 83 | 31 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,0,31,0,0,51,51],[1,0,0,0,0,1,0,0,0,0,62,46,0,0,81,51],[89,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[82,83,0,0,17,31,0,0,0,0,1,0,0,0,0,1] >;
Q16×Dic7 in GAP, Magma, Sage, TeX
Q_{16}\times {\rm Dic}_7 % in TeX
G:=Group("Q16xDic7"); // GroupNames label
G:=SmallGroup(448,717);
// by ID
G=gap.SmallGroup(448,717);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,219,184,851,438,102,18822]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations