metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊2Q16, Dic14.10D4, C4.90(D4×D7), C2.7(D7×Q16), (C2×C8).13D14, C7⋊1(C4⋊2Q16), C4⋊C4.145D14, C4.5(C4○D28), C28.114(C2×D4), Dic7⋊C8.2C2, (C2×Q8).10D14, Q8⋊C4.2D7, C14.14(C2×Q16), C28.14(C4○D4), (C2×C56).13C22, (C2×Dic28).3C2, C14.Q16.2C2, C22.187(D4×D7), C14.21(C4⋊D4), (C2×C28).233C23, Dic7⋊3Q8.3C2, Dic7⋊Q8.3C2, (C2×Dic7).148D4, (Q8×C14).16C22, C2.24(D14⋊D4), C2.14(SD16⋊D7), C14.32(C8.C22), (C4×Dic7).17C22, (C2×Dic14).65C22, (C2×C7⋊C8).28C22, (C2×C7⋊Q16).3C2, (C2×C14).246(C2×D4), (C7×C4⋊C4).34C22, (C7×Q8⋊C4).2C2, (C2×C4).340(C22×D7), SmallGroup(448,327)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊Q16
G = < a,b,c,d | a14=c8=1, b2=a7, d2=c4, bab-1=cac-1=a-1, ad=da, cbc-1=dbd-1=a7b, dcd-1=c-1 >
Subgroups: 500 in 108 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, Q16, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, Q8⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C4⋊2Q16, Dic28, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C7⋊Q16, C7×C4⋊C4, C2×C56, C2×Dic14, Q8×C14, C14.Q16, Dic7⋊C8, C7×Q8⋊C4, Dic7⋊3Q8, C2×Dic28, C2×C7⋊Q16, Dic7⋊Q8, Dic7⋊Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C4○D4, D14, C4⋊D4, C2×Q16, C8.C22, C22×D7, C4⋊2Q16, C4○D28, D4×D7, D14⋊D4, SD16⋊D7, D7×Q16, Dic7⋊Q16
►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)
(1 122 8 115)(2 121 9 114)(3 120 10 113)(4 119 11 126)(5 118 12 125)(6 117 13 124)(7 116 14 123)(15 63 22 70)(16 62 23 69)(17 61 24 68)(18 60 25 67)(19 59 26 66)(20 58 27 65)(21 57 28 64)(29 363 36 356)(30 362 37 355)(31 361 38 354)(32 360 39 353)(33 359 40 352)(34 358 41 351)(35 357 42 364)(43 440 50 447)(44 439 51 446)(45 438 52 445)(46 437 53 444)(47 436 54 443)(48 435 55 442)(49 448 56 441)(71 208 78 201)(72 207 79 200)(73 206 80 199)(74 205 81 198)(75 204 82 197)(76 203 83 210)(77 202 84 209)(85 326 92 333)(86 325 93 332)(87 324 94 331)(88 323 95 330)(89 336 96 329)(90 335 97 328)(91 334 98 327)(99 409 106 416)(100 408 107 415)(101 407 108 414)(102 420 109 413)(103 419 110 412)(104 418 111 411)(105 417 112 410)(127 251 134 244)(128 250 135 243)(129 249 136 242)(130 248 137 241)(131 247 138 240)(132 246 139 239)(133 245 140 252)(141 427 148 434)(142 426 149 433)(143 425 150 432)(144 424 151 431)(145 423 152 430)(146 422 153 429)(147 421 154 428)(155 405 162 398)(156 404 163 397)(157 403 164 396)(158 402 165 395)(159 401 166 394)(160 400 167 393)(161 399 168 406)(169 219 176 212)(170 218 177 211)(171 217 178 224)(172 216 179 223)(173 215 180 222)(174 214 181 221)(175 213 182 220)(183 228 190 235)(184 227 191 234)(185 226 192 233)(186 225 193 232)(187 238 194 231)(188 237 195 230)(189 236 196 229)(253 284 260 291)(254 283 261 290)(255 282 262 289)(256 281 263 288)(257 294 264 287)(258 293 265 286)(259 292 266 285)(267 313 274 320)(268 312 275 319)(269 311 276 318)(270 310 277 317)(271 309 278 316)(272 322 279 315)(273 321 280 314)(295 391 302 384)(296 390 303 383)(297 389 304 382)(298 388 305 381)(299 387 306 380)(300 386 307 379)(301 385 308 392)(337 371 344 378)(338 370 345 377)(339 369 346 376)(340 368 347 375)(341 367 348 374)(342 366 349 373)(343 365 350 372)
(1 141 222 343 21 249 311 407)(2 154 223 342 22 248 312 420)(3 153 224 341 23 247 313 419)(4 152 211 340 24 246 314 418)(5 151 212 339 25 245 315 417)(6 150 213 338 26 244 316 416)(7 149 214 337 27 243 317 415)(8 148 215 350 28 242 318 414)(9 147 216 349 15 241 319 413)(10 146 217 348 16 240 320 412)(11 145 218 347 17 239 321 411)(12 144 219 346 18 252 322 410)(13 143 220 345 19 251 309 409)(14 142 221 344 20 250 310 408)(29 162 291 336 200 448 303 196)(30 161 292 335 201 447 304 195)(31 160 293 334 202 446 305 194)(32 159 294 333 203 445 306 193)(33 158 281 332 204 444 307 192)(34 157 282 331 205 443 308 191)(35 156 283 330 206 442 295 190)(36 155 284 329 207 441 296 189)(37 168 285 328 208 440 297 188)(38 167 286 327 209 439 298 187)(39 166 287 326 210 438 299 186)(40 165 288 325 197 437 300 185)(41 164 289 324 198 436 301 184)(42 163 290 323 199 435 302 183)(43 389 230 355 399 259 97 78)(44 388 231 354 400 258 98 77)(45 387 232 353 401 257 85 76)(46 386 233 352 402 256 86 75)(47 385 234 351 403 255 87 74)(48 384 235 364 404 254 88 73)(49 383 236 363 405 253 89 72)(50 382 237 362 406 266 90 71)(51 381 238 361 393 265 91 84)(52 380 225 360 394 264 92 83)(53 379 226 359 395 263 93 82)(54 392 227 358 396 262 94 81)(55 391 228 357 397 261 95 80)(56 390 229 356 398 260 96 79)(57 129 276 101 122 434 173 372)(58 128 277 100 123 433 174 371)(59 127 278 99 124 432 175 370)(60 140 279 112 125 431 176 369)(61 139 280 111 126 430 177 368)(62 138 267 110 113 429 178 367)(63 137 268 109 114 428 179 366)(64 136 269 108 115 427 180 365)(65 135 270 107 116 426 181 378)(66 134 271 106 117 425 182 377)(67 133 272 105 118 424 169 376)(68 132 273 104 119 423 170 375)(69 131 274 103 120 422 171 374)(70 130 275 102 121 421 172 373)
(1 227 21 94)(2 228 22 95)(3 229 23 96)(4 230 24 97)(5 231 25 98)(6 232 26 85)(7 233 27 86)(8 234 28 87)(9 235 15 88)(10 236 16 89)(11 237 17 90)(12 238 18 91)(13 225 19 92)(14 226 20 93)(29 374 200 103)(30 375 201 104)(31 376 202 105)(32 377 203 106)(33 378 204 107)(34 365 205 108)(35 366 206 109)(36 367 207 110)(37 368 208 111)(38 369 209 112)(39 370 210 99)(40 371 197 100)(41 372 198 101)(42 373 199 102)(43 314 399 211)(44 315 400 212)(45 316 401 213)(46 317 402 214)(47 318 403 215)(48 319 404 216)(49 320 405 217)(50 321 406 218)(51 322 393 219)(52 309 394 220)(53 310 395 221)(54 311 396 222)(55 312 397 223)(56 313 398 224)(57 324 122 184)(58 325 123 185)(59 326 124 186)(60 327 125 187)(61 328 126 188)(62 329 113 189)(63 330 114 190)(64 331 115 191)(65 332 116 192)(66 333 117 193)(67 334 118 194)(68 335 119 195)(69 336 120 196)(70 323 121 183)(71 411 362 347)(72 412 363 348)(73 413 364 349)(74 414 351 350)(75 415 352 337)(76 416 353 338)(77 417 354 339)(78 418 355 340)(79 419 356 341)(80 420 357 342)(81 407 358 343)(82 408 359 344)(83 409 360 345)(84 410 361 346)(127 287 432 299)(128 288 433 300)(129 289 434 301)(130 290 421 302)(131 291 422 303)(132 292 423 304)(133 293 424 305)(134 294 425 306)(135 281 426 307)(136 282 427 308)(137 283 428 295)(138 284 429 296)(139 285 430 297)(140 286 431 298)(141 392 249 262)(142 379 250 263)(143 380 251 264)(144 381 252 265)(145 382 239 266)(146 383 240 253)(147 384 241 254)(148 385 242 255)(149 386 243 256)(150 387 244 257)(151 388 245 258)(152 389 246 259)(153 390 247 260)(154 391 248 261)(155 178 441 267)(156 179 442 268)(157 180 443 269)(158 181 444 270)(159 182 445 271)(160 169 446 272)(161 170 447 273)(162 171 448 274)(163 172 435 275)(164 173 436 276)(165 174 437 277)(166 175 438 278)(167 176 439 279)(168 177 440 280)
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), (1,122,8,115)(2,121,9,114)(3,120,10,113)(4,119,11,126)(5,118,12,125)(6,117,13,124)(7,116,14,123)(15,63,22,70)(16,62,23,69)(17,61,24,68)(18,60,25,67)(19,59,26,66)(20,58,27,65)(21,57,28,64)(29,363,36,356)(30,362,37,355)(31,361,38,354)(32,360,39,353)(33,359,40,352)(34,358,41,351)(35,357,42,364)(43,440,50,447)(44,439,51,446)(45,438,52,445)(46,437,53,444)(47,436,54,443)(48,435,55,442)(49,448,56,441)(71,208,78,201)(72,207,79,200)(73,206,80,199)(74,205,81,198)(75,204,82,197)(76,203,83,210)(77,202,84,209)(85,326,92,333)(86,325,93,332)(87,324,94,331)(88,323,95,330)(89,336,96,329)(90,335,97,328)(91,334,98,327)(99,409,106,416)(100,408,107,415)(101,407,108,414)(102,420,109,413)(103,419,110,412)(104,418,111,411)(105,417,112,410)(127,251,134,244)(128,250,135,243)(129,249,136,242)(130,248,137,241)(131,247,138,240)(132,246,139,239)(133,245,140,252)(141,427,148,434)(142,426,149,433)(143,425,150,432)(144,424,151,431)(145,423,152,430)(146,422,153,429)(147,421,154,428)(155,405,162,398)(156,404,163,397)(157,403,164,396)(158,402,165,395)(159,401,166,394)(160,400,167,393)(161,399,168,406)(169,219,176,212)(170,218,177,211)(171,217,178,224)(172,216,179,223)(173,215,180,222)(174,214,181,221)(175,213,182,220)(183,228,190,235)(184,227,191,234)(185,226,192,233)(186,225,193,232)(187,238,194,231)(188,237,195,230)(189,236,196,229)(253,284,260,291)(254,283,261,290)(255,282,262,289)(256,281,263,288)(257,294,264,287)(258,293,265,286)(259,292,266,285)(267,313,274,320)(268,312,275,319)(269,311,276,318)(270,310,277,317)(271,309,278,316)(272,322,279,315)(273,321,280,314)(295,391,302,384)(296,390,303,383)(297,389,304,382)(298,388,305,381)(299,387,306,380)(300,386,307,379)(301,385,308,392)(337,371,344,378)(338,370,345,377)(339,369,346,376)(340,368,347,375)(341,367,348,374)(342,366,349,373)(343,365,350,372), (1,141,222,343,21,249,311,407)(2,154,223,342,22,248,312,420)(3,153,224,341,23,247,313,419)(4,152,211,340,24,246,314,418)(5,151,212,339,25,245,315,417)(6,150,213,338,26,244,316,416)(7,149,214,337,27,243,317,415)(8,148,215,350,28,242,318,414)(9,147,216,349,15,241,319,413)(10,146,217,348,16,240,320,412)(11,145,218,347,17,239,321,411)(12,144,219,346,18,252,322,410)(13,143,220,345,19,251,309,409)(14,142,221,344,20,250,310,408)(29,162,291,336,200,448,303,196)(30,161,292,335,201,447,304,195)(31,160,293,334,202,446,305,194)(32,159,294,333,203,445,306,193)(33,158,281,332,204,444,307,192)(34,157,282,331,205,443,308,191)(35,156,283,330,206,442,295,190)(36,155,284,329,207,441,296,189)(37,168,285,328,208,440,297,188)(38,167,286,327,209,439,298,187)(39,166,287,326,210,438,299,186)(40,165,288,325,197,437,300,185)(41,164,289,324,198,436,301,184)(42,163,290,323,199,435,302,183)(43,389,230,355,399,259,97,78)(44,388,231,354,400,258,98,77)(45,387,232,353,401,257,85,76)(46,386,233,352,402,256,86,75)(47,385,234,351,403,255,87,74)(48,384,235,364,404,254,88,73)(49,383,236,363,405,253,89,72)(50,382,237,362,406,266,90,71)(51,381,238,361,393,265,91,84)(52,380,225,360,394,264,92,83)(53,379,226,359,395,263,93,82)(54,392,227,358,396,262,94,81)(55,391,228,357,397,261,95,80)(56,390,229,356,398,260,96,79)(57,129,276,101,122,434,173,372)(58,128,277,100,123,433,174,371)(59,127,278,99,124,432,175,370)(60,140,279,112,125,431,176,369)(61,139,280,111,126,430,177,368)(62,138,267,110,113,429,178,367)(63,137,268,109,114,428,179,366)(64,136,269,108,115,427,180,365)(65,135,270,107,116,426,181,378)(66,134,271,106,117,425,182,377)(67,133,272,105,118,424,169,376)(68,132,273,104,119,423,170,375)(69,131,274,103,120,422,171,374)(70,130,275,102,121,421,172,373), (1,227,21,94)(2,228,22,95)(3,229,23,96)(4,230,24,97)(5,231,25,98)(6,232,26,85)(7,233,27,86)(8,234,28,87)(9,235,15,88)(10,236,16,89)(11,237,17,90)(12,238,18,91)(13,225,19,92)(14,226,20,93)(29,374,200,103)(30,375,201,104)(31,376,202,105)(32,377,203,106)(33,378,204,107)(34,365,205,108)(35,366,206,109)(36,367,207,110)(37,368,208,111)(38,369,209,112)(39,370,210,99)(40,371,197,100)(41,372,198,101)(42,373,199,102)(43,314,399,211)(44,315,400,212)(45,316,401,213)(46,317,402,214)(47,318,403,215)(48,319,404,216)(49,320,405,217)(50,321,406,218)(51,322,393,219)(52,309,394,220)(53,310,395,221)(54,311,396,222)(55,312,397,223)(56,313,398,224)(57,324,122,184)(58,325,123,185)(59,326,124,186)(60,327,125,187)(61,328,126,188)(62,329,113,189)(63,330,114,190)(64,331,115,191)(65,332,116,192)(66,333,117,193)(67,334,118,194)(68,335,119,195)(69,336,120,196)(70,323,121,183)(71,411,362,347)(72,412,363,348)(73,413,364,349)(74,414,351,350)(75,415,352,337)(76,416,353,338)(77,417,354,339)(78,418,355,340)(79,419,356,341)(80,420,357,342)(81,407,358,343)(82,408,359,344)(83,409,360,345)(84,410,361,346)(127,287,432,299)(128,288,433,300)(129,289,434,301)(130,290,421,302)(131,291,422,303)(132,292,423,304)(133,293,424,305)(134,294,425,306)(135,281,426,307)(136,282,427,308)(137,283,428,295)(138,284,429,296)(139,285,430,297)(140,286,431,298)(141,392,249,262)(142,379,250,263)(143,380,251,264)(144,381,252,265)(145,382,239,266)(146,383,240,253)(147,384,241,254)(148,385,242,255)(149,386,243,256)(150,387,244,257)(151,388,245,258)(152,389,246,259)(153,390,247,260)(154,391,248,261)(155,178,441,267)(156,179,442,268)(157,180,443,269)(158,181,444,270)(159,182,445,271)(160,169,446,272)(161,170,447,273)(162,171,448,274)(163,172,435,275)(164,173,436,276)(165,174,437,277)(166,175,438,278)(167,176,439,279)(168,177,440,280)>;
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), (1,122,8,115)(2,121,9,114)(3,120,10,113)(4,119,11,126)(5,118,12,125)(6,117,13,124)(7,116,14,123)(15,63,22,70)(16,62,23,69)(17,61,24,68)(18,60,25,67)(19,59,26,66)(20,58,27,65)(21,57,28,64)(29,363,36,356)(30,362,37,355)(31,361,38,354)(32,360,39,353)(33,359,40,352)(34,358,41,351)(35,357,42,364)(43,440,50,447)(44,439,51,446)(45,438,52,445)(46,437,53,444)(47,436,54,443)(48,435,55,442)(49,448,56,441)(71,208,78,201)(72,207,79,200)(73,206,80,199)(74,205,81,198)(75,204,82,197)(76,203,83,210)(77,202,84,209)(85,326,92,333)(86,325,93,332)(87,324,94,331)(88,323,95,330)(89,336,96,329)(90,335,97,328)(91,334,98,327)(99,409,106,416)(100,408,107,415)(101,407,108,414)(102,420,109,413)(103,419,110,412)(104,418,111,411)(105,417,112,410)(127,251,134,244)(128,250,135,243)(129,249,136,242)(130,248,137,241)(131,247,138,240)(132,246,139,239)(133,245,140,252)(141,427,148,434)(142,426,149,433)(143,425,150,432)(144,424,151,431)(145,423,152,430)(146,422,153,429)(147,421,154,428)(155,405,162,398)(156,404,163,397)(157,403,164,396)(158,402,165,395)(159,401,166,394)(160,400,167,393)(161,399,168,406)(169,219,176,212)(170,218,177,211)(171,217,178,224)(172,216,179,223)(173,215,180,222)(174,214,181,221)(175,213,182,220)(183,228,190,235)(184,227,191,234)(185,226,192,233)(186,225,193,232)(187,238,194,231)(188,237,195,230)(189,236,196,229)(253,284,260,291)(254,283,261,290)(255,282,262,289)(256,281,263,288)(257,294,264,287)(258,293,265,286)(259,292,266,285)(267,313,274,320)(268,312,275,319)(269,311,276,318)(270,310,277,317)(271,309,278,316)(272,322,279,315)(273,321,280,314)(295,391,302,384)(296,390,303,383)(297,389,304,382)(298,388,305,381)(299,387,306,380)(300,386,307,379)(301,385,308,392)(337,371,344,378)(338,370,345,377)(339,369,346,376)(340,368,347,375)(341,367,348,374)(342,366,349,373)(343,365,350,372), (1,141,222,343,21,249,311,407)(2,154,223,342,22,248,312,420)(3,153,224,341,23,247,313,419)(4,152,211,340,24,246,314,418)(5,151,212,339,25,245,315,417)(6,150,213,338,26,244,316,416)(7,149,214,337,27,243,317,415)(8,148,215,350,28,242,318,414)(9,147,216,349,15,241,319,413)(10,146,217,348,16,240,320,412)(11,145,218,347,17,239,321,411)(12,144,219,346,18,252,322,410)(13,143,220,345,19,251,309,409)(14,142,221,344,20,250,310,408)(29,162,291,336,200,448,303,196)(30,161,292,335,201,447,304,195)(31,160,293,334,202,446,305,194)(32,159,294,333,203,445,306,193)(33,158,281,332,204,444,307,192)(34,157,282,331,205,443,308,191)(35,156,283,330,206,442,295,190)(36,155,284,329,207,441,296,189)(37,168,285,328,208,440,297,188)(38,167,286,327,209,439,298,187)(39,166,287,326,210,438,299,186)(40,165,288,325,197,437,300,185)(41,164,289,324,198,436,301,184)(42,163,290,323,199,435,302,183)(43,389,230,355,399,259,97,78)(44,388,231,354,400,258,98,77)(45,387,232,353,401,257,85,76)(46,386,233,352,402,256,86,75)(47,385,234,351,403,255,87,74)(48,384,235,364,404,254,88,73)(49,383,236,363,405,253,89,72)(50,382,237,362,406,266,90,71)(51,381,238,361,393,265,91,84)(52,380,225,360,394,264,92,83)(53,379,226,359,395,263,93,82)(54,392,227,358,396,262,94,81)(55,391,228,357,397,261,95,80)(56,390,229,356,398,260,96,79)(57,129,276,101,122,434,173,372)(58,128,277,100,123,433,174,371)(59,127,278,99,124,432,175,370)(60,140,279,112,125,431,176,369)(61,139,280,111,126,430,177,368)(62,138,267,110,113,429,178,367)(63,137,268,109,114,428,179,366)(64,136,269,108,115,427,180,365)(65,135,270,107,116,426,181,378)(66,134,271,106,117,425,182,377)(67,133,272,105,118,424,169,376)(68,132,273,104,119,423,170,375)(69,131,274,103,120,422,171,374)(70,130,275,102,121,421,172,373), (1,227,21,94)(2,228,22,95)(3,229,23,96)(4,230,24,97)(5,231,25,98)(6,232,26,85)(7,233,27,86)(8,234,28,87)(9,235,15,88)(10,236,16,89)(11,237,17,90)(12,238,18,91)(13,225,19,92)(14,226,20,93)(29,374,200,103)(30,375,201,104)(31,376,202,105)(32,377,203,106)(33,378,204,107)(34,365,205,108)(35,366,206,109)(36,367,207,110)(37,368,208,111)(38,369,209,112)(39,370,210,99)(40,371,197,100)(41,372,198,101)(42,373,199,102)(43,314,399,211)(44,315,400,212)(45,316,401,213)(46,317,402,214)(47,318,403,215)(48,319,404,216)(49,320,405,217)(50,321,406,218)(51,322,393,219)(52,309,394,220)(53,310,395,221)(54,311,396,222)(55,312,397,223)(56,313,398,224)(57,324,122,184)(58,325,123,185)(59,326,124,186)(60,327,125,187)(61,328,126,188)(62,329,113,189)(63,330,114,190)(64,331,115,191)(65,332,116,192)(66,333,117,193)(67,334,118,194)(68,335,119,195)(69,336,120,196)(70,323,121,183)(71,411,362,347)(72,412,363,348)(73,413,364,349)(74,414,351,350)(75,415,352,337)(76,416,353,338)(77,417,354,339)(78,418,355,340)(79,419,356,341)(80,420,357,342)(81,407,358,343)(82,408,359,344)(83,409,360,345)(84,410,361,346)(127,287,432,299)(128,288,433,300)(129,289,434,301)(130,290,421,302)(131,291,422,303)(132,292,423,304)(133,293,424,305)(134,294,425,306)(135,281,426,307)(136,282,427,308)(137,283,428,295)(138,284,429,296)(139,285,430,297)(140,286,431,298)(141,392,249,262)(142,379,250,263)(143,380,251,264)(144,381,252,265)(145,382,239,266)(146,383,240,253)(147,384,241,254)(148,385,242,255)(149,386,243,256)(150,387,244,257)(151,388,245,258)(152,389,246,259)(153,390,247,260)(154,391,248,261)(155,178,441,267)(156,179,442,268)(157,180,443,269)(158,181,444,270)(159,182,445,271)(160,169,446,272)(161,170,447,273)(162,171,448,274)(163,172,435,275)(164,173,436,276)(165,174,437,277)(166,175,438,278)(167,176,439,279)(168,177,440,280) );
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)], [(1,122,8,115),(2,121,9,114),(3,120,10,113),(4,119,11,126),(5,118,12,125),(6,117,13,124),(7,116,14,123),(15,63,22,70),(16,62,23,69),(17,61,24,68),(18,60,25,67),(19,59,26,66),(20,58,27,65),(21,57,28,64),(29,363,36,356),(30,362,37,355),(31,361,38,354),(32,360,39,353),(33,359,40,352),(34,358,41,351),(35,357,42,364),(43,440,50,447),(44,439,51,446),(45,438,52,445),(46,437,53,444),(47,436,54,443),(48,435,55,442),(49,448,56,441),(71,208,78,201),(72,207,79,200),(73,206,80,199),(74,205,81,198),(75,204,82,197),(76,203,83,210),(77,202,84,209),(85,326,92,333),(86,325,93,332),(87,324,94,331),(88,323,95,330),(89,336,96,329),(90,335,97,328),(91,334,98,327),(99,409,106,416),(100,408,107,415),(101,407,108,414),(102,420,109,413),(103,419,110,412),(104,418,111,411),(105,417,112,410),(127,251,134,244),(128,250,135,243),(129,249,136,242),(130,248,137,241),(131,247,138,240),(132,246,139,239),(133,245,140,252),(141,427,148,434),(142,426,149,433),(143,425,150,432),(144,424,151,431),(145,423,152,430),(146,422,153,429),(147,421,154,428),(155,405,162,398),(156,404,163,397),(157,403,164,396),(158,402,165,395),(159,401,166,394),(160,400,167,393),(161,399,168,406),(169,219,176,212),(170,218,177,211),(171,217,178,224),(172,216,179,223),(173,215,180,222),(174,214,181,221),(175,213,182,220),(183,228,190,235),(184,227,191,234),(185,226,192,233),(186,225,193,232),(187,238,194,231),(188,237,195,230),(189,236,196,229),(253,284,260,291),(254,283,261,290),(255,282,262,289),(256,281,263,288),(257,294,264,287),(258,293,265,286),(259,292,266,285),(267,313,274,320),(268,312,275,319),(269,311,276,318),(270,310,277,317),(271,309,278,316),(272,322,279,315),(273,321,280,314),(295,391,302,384),(296,390,303,383),(297,389,304,382),(298,388,305,381),(299,387,306,380),(300,386,307,379),(301,385,308,392),(337,371,344,378),(338,370,345,377),(339,369,346,376),(340,368,347,375),(341,367,348,374),(342,366,349,373),(343,365,350,372)], [(1,141,222,343,21,249,311,407),(2,154,223,342,22,248,312,420),(3,153,224,341,23,247,313,419),(4,152,211,340,24,246,314,418),(5,151,212,339,25,245,315,417),(6,150,213,338,26,244,316,416),(7,149,214,337,27,243,317,415),(8,148,215,350,28,242,318,414),(9,147,216,349,15,241,319,413),(10,146,217,348,16,240,320,412),(11,145,218,347,17,239,321,411),(12,144,219,346,18,252,322,410),(13,143,220,345,19,251,309,409),(14,142,221,344,20,250,310,408),(29,162,291,336,200,448,303,196),(30,161,292,335,201,447,304,195),(31,160,293,334,202,446,305,194),(32,159,294,333,203,445,306,193),(33,158,281,332,204,444,307,192),(34,157,282,331,205,443,308,191),(35,156,283,330,206,442,295,190),(36,155,284,329,207,441,296,189),(37,168,285,328,208,440,297,188),(38,167,286,327,209,439,298,187),(39,166,287,326,210,438,299,186),(40,165,288,325,197,437,300,185),(41,164,289,324,198,436,301,184),(42,163,290,323,199,435,302,183),(43,389,230,355,399,259,97,78),(44,388,231,354,400,258,98,77),(45,387,232,353,401,257,85,76),(46,386,233,352,402,256,86,75),(47,385,234,351,403,255,87,74),(48,384,235,364,404,254,88,73),(49,383,236,363,405,253,89,72),(50,382,237,362,406,266,90,71),(51,381,238,361,393,265,91,84),(52,380,225,360,394,264,92,83),(53,379,226,359,395,263,93,82),(54,392,227,358,396,262,94,81),(55,391,228,357,397,261,95,80),(56,390,229,356,398,260,96,79),(57,129,276,101,122,434,173,372),(58,128,277,100,123,433,174,371),(59,127,278,99,124,432,175,370),(60,140,279,112,125,431,176,369),(61,139,280,111,126,430,177,368),(62,138,267,110,113,429,178,367),(63,137,268,109,114,428,179,366),(64,136,269,108,115,427,180,365),(65,135,270,107,116,426,181,378),(66,134,271,106,117,425,182,377),(67,133,272,105,118,424,169,376),(68,132,273,104,119,423,170,375),(69,131,274,103,120,422,171,374),(70,130,275,102,121,421,172,373)], [(1,227,21,94),(2,228,22,95),(3,229,23,96),(4,230,24,97),(5,231,25,98),(6,232,26,85),(7,233,27,86),(8,234,28,87),(9,235,15,88),(10,236,16,89),(11,237,17,90),(12,238,18,91),(13,225,19,92),(14,226,20,93),(29,374,200,103),(30,375,201,104),(31,376,202,105),(32,377,203,106),(33,378,204,107),(34,365,205,108),(35,366,206,109),(36,367,207,110),(37,368,208,111),(38,369,209,112),(39,370,210,99),(40,371,197,100),(41,372,198,101),(42,373,199,102),(43,314,399,211),(44,315,400,212),(45,316,401,213),(46,317,402,214),(47,318,403,215),(48,319,404,216),(49,320,405,217),(50,321,406,218),(51,322,393,219),(52,309,394,220),(53,310,395,221),(54,311,396,222),(55,312,397,223),(56,313,398,224),(57,324,122,184),(58,325,123,185),(59,326,124,186),(60,327,125,187),(61,328,126,188),(62,329,113,189),(63,330,114,190),(64,331,115,191),(65,332,116,192),(66,333,117,193),(67,334,118,194),(68,335,119,195),(69,336,120,196),(70,323,121,183),(71,411,362,347),(72,412,363,348),(73,413,364,349),(74,414,351,350),(75,415,352,337),(76,416,353,338),(77,417,354,339),(78,418,355,340),(79,419,356,341),(80,420,357,342),(81,407,358,343),(82,408,359,344),(83,409,360,345),(84,410,361,346),(127,287,432,299),(128,288,433,300),(129,289,434,301),(130,290,421,302),(131,291,422,303),(132,292,423,304),(133,293,424,305),(134,294,425,306),(135,281,426,307),(136,282,427,308),(137,283,428,295),(138,284,429,296),(139,285,430,297),(140,286,431,298),(141,392,249,262),(142,379,250,263),(143,380,251,264),(144,381,252,265),(145,382,239,266),(146,383,240,253),(147,384,241,254),(148,385,242,255),(149,386,243,256),(150,387,244,257),(151,388,245,258),(152,389,246,259),(153,390,247,260),(154,391,248,261),(155,178,441,267),(156,179,442,268),(157,180,443,269),(158,181,444,270),(159,182,445,271),(160,169,446,272),(161,170,447,273),(162,171,448,274),(163,172,435,275),(164,173,436,276),(165,174,437,277),(166,175,438,278),(167,176,439,279),(168,177,440,280)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | D14 | C4○D28 | C8.C22 | D4×D7 | D4×D7 | SD16⋊D7 | D7×Q16 |
| kernel | Dic7⋊Q16 | C14.Q16 | Dic7⋊C8 | C7×Q8⋊C4 | Dic7⋊3Q8 | C2×Dic28 | C2×C7⋊Q16 | Dic7⋊Q8 | Dic14 | C2×Dic7 | Q8⋊C4 | Dic7 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Dic7⋊Q16 ►in GL6(𝔽113)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 17 | 0 | 0 | 0 | 0 |
| 90 | 77 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 3 | 0 | 0 |
| 0 | 0 | 8 | 75 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 77 | 96 | 0 | 0 | 0 | 0 |
| 23 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 65 | 0 | 0 |
| 0 | 0 | 85 | 89 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 62 |
| 0 | 0 | 0 | 0 | 82 | 62 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 36 | 0 | 0 |
| 0 | 0 | 82 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 110 | 24 |
| 0 | 0 | 0 | 0 | 9 | 3 |
G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,24,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,90,0,0,0,0,17,77,0,0,0,0,0,0,38,8,0,0,0,0,3,75,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[77,23,0,0,0,0,96,36,0,0,0,0,0,0,24,85,0,0,0,0,65,89,0,0,0,0,0,0,0,82,0,0,0,0,62,62],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,82,0,0,0,0,36,10,0,0,0,0,0,0,110,9,0,0,0,0,24,3] >;
Dic7⋊Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes Q_{16} % in TeX
G:=Group("Dic7:Q16"); // GroupNames label
G:=SmallGroup(448,327);
// by ID
G=gap.SmallGroup(448,327);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,422,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^8=1,b^2=a^7,d^2=c^4,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations