metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊3Q16, (C7×Q8).7D4, C7⋊5(C4⋊2Q16), (C2×C8).36D14, (C2×Q16).4D7, C2.17(D7×Q16), C28.184(C2×D4), (C14×Q16).8C2, C14.27(C2×Q16), (Q8×Dic7).7C2, Q8.2(C7⋊D4), Dic7⋊C8.12C2, (C2×Q8).118D14, C22.275(D4×D7), C28.103(C4○D4), C4.13(D4⋊2D7), (C2×C28).456C23, (C2×C56).250C22, Dic7⋊Q8.6C2, (C2×Dic7).186D4, Q8⋊Dic7.14C2, (Q8×C14).85C22, C14.119(C4⋊D4), C2.26(Q16⋊D7), C28.44D4.12C2, C14.75(C8.C22), C4⋊Dic7.179C22, (C4×Dic7).53C22, C2.28(Dic7⋊D4), (C2×Dic14).131C22, C4.46(C2×C7⋊D4), (C2×C7⋊Q16).9C2, (C2×C14).367(C2×D4), (C2×C7⋊C8).162C22, (C2×C4).544(C22×D7), SmallGroup(448,716)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊3Q16
G = < a,b,c,d | a14=c8=1, b2=a7, d2=c4, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a7b, dcd-1=c-1 >
Subgroups: 452 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C2×Q16, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C4⋊2Q16, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7⋊Q16, C2×C56, C7×Q16, C2×Dic14, Q8×C14, Dic7⋊C8, C28.44D4, Q8⋊Dic7, C2×C7⋊Q16, Dic7⋊Q8, Q8×Dic7, C14×Q16, Dic7⋊3Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C4○D4, D14, C4⋊D4, C2×Q16, C8.C22, C7⋊D4, C22×D7, C4⋊2Q16, D4×D7, D4⋊2D7, C2×C7⋊D4, D7×Q16, Q16⋊D7, Dic7⋊D4, Dic7⋊3Q16
►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)
(1 308 8 301)(2 307 9 300)(3 306 10 299)(4 305 11 298)(5 304 12 297)(6 303 13 296)(7 302 14 295)(15 222 22 215)(16 221 23 214)(17 220 24 213)(18 219 25 212)(19 218 26 211)(20 217 27 224)(21 216 28 223)(29 413 36 420)(30 412 37 419)(31 411 38 418)(32 410 39 417)(33 409 40 416)(34 408 41 415)(35 407 42 414)(43 61 50 68)(44 60 51 67)(45 59 52 66)(46 58 53 65)(47 57 54 64)(48 70 55 63)(49 69 56 62)(71 352 78 359)(72 351 79 358)(73 364 80 357)(74 363 81 356)(75 362 82 355)(76 361 83 354)(77 360 84 353)(85 201 92 208)(86 200 93 207)(87 199 94 206)(88 198 95 205)(89 197 96 204)(90 210 97 203)(91 209 98 202)(99 191 106 184)(100 190 107 183)(101 189 108 196)(102 188 109 195)(103 187 110 194)(104 186 111 193)(105 185 112 192)(113 165 120 158)(114 164 121 157)(115 163 122 156)(116 162 123 155)(117 161 124 168)(118 160 125 167)(119 159 126 166)(127 333 134 326)(128 332 135 325)(129 331 136 324)(130 330 137 323)(131 329 138 336)(132 328 139 335)(133 327 140 334)(141 278 148 271)(142 277 149 270)(143 276 150 269)(144 275 151 268)(145 274 152 267)(146 273 153 280)(147 272 154 279)(169 250 176 243)(170 249 177 242)(171 248 178 241)(172 247 179 240)(173 246 180 239)(174 245 181 252)(175 244 182 251)(225 387 232 380)(226 386 233 379)(227 385 234 392)(228 384 235 391)(229 383 236 390)(230 382 237 389)(231 381 238 388)(253 368 260 375)(254 367 261 374)(255 366 262 373)(256 365 263 372)(257 378 264 371)(258 377 265 370)(259 376 266 369)(281 430 288 423)(282 429 289 422)(283 428 290 421)(284 427 291 434)(285 426 292 433)(286 425 293 432)(287 424 294 431)(309 337 316 344)(310 350 317 343)(311 349 318 342)(312 348 319 341)(313 347 320 340)(314 346 321 339)(315 345 322 338)(393 441 400 448)(394 440 401 447)(395 439 402 446)(396 438 403 445)(397 437 404 444)(398 436 405 443)(399 435 406 442)
(1 391 340 164 144 354 284 93)(2 392 341 165 145 355 285 94)(3 379 342 166 146 356 286 95)(4 380 343 167 147 357 287 96)(5 381 344 168 148 358 288 97)(6 382 345 155 149 359 289 98)(7 383 346 156 150 360 290 85)(8 384 347 157 151 361 291 86)(9 385 348 158 152 362 292 87)(10 386 349 159 153 363 293 88)(11 387 350 160 154 364 294 89)(12 388 337 161 141 351 281 90)(13 389 338 162 142 352 282 91)(14 390 339 163 143 353 283 92)(15 105 443 131 43 419 254 178)(16 106 444 132 44 420 255 179)(17 107 445 133 45 407 256 180)(18 108 446 134 46 408 257 181)(19 109 447 135 47 409 258 182)(20 110 448 136 48 410 259 169)(21 111 435 137 49 411 260 170)(22 112 436 138 50 412 261 171)(23 99 437 139 51 413 262 172)(24 100 438 140 52 414 263 173)(25 101 439 127 53 415 264 174)(26 102 440 128 54 416 265 175)(27 103 441 129 55 417 266 176)(28 104 442 130 56 418 253 177)(29 373 240 214 184 404 328 67)(30 374 241 215 185 405 329 68)(31 375 242 216 186 406 330 69)(32 376 243 217 187 393 331 70)(33 377 244 218 188 394 332 57)(34 378 245 219 189 395 333 58)(35 365 246 220 190 396 334 59)(36 366 247 221 191 397 335 60)(37 367 248 222 192 398 336 61)(38 368 249 223 193 399 323 62)(39 369 250 224 194 400 324 63)(40 370 251 211 195 401 325 64)(41 371 252 212 196 402 326 65)(42 372 239 213 183 403 327 66)(71 429 202 296 237 315 116 277)(72 430 203 297 238 316 117 278)(73 431 204 298 225 317 118 279)(74 432 205 299 226 318 119 280)(75 433 206 300 227 319 120 267)(76 434 207 301 228 320 121 268)(77 421 208 302 229 321 122 269)(78 422 209 303 230 322 123 270)(79 423 210 304 231 309 124 271)(80 424 197 305 232 310 125 272)(81 425 198 306 233 311 126 273)(82 426 199 307 234 312 113 274)(83 427 200 308 235 313 114 275)(84 428 201 295 236 314 115 276)
(1 26 144 54)(2 27 145 55)(3 28 146 56)(4 15 147 43)(5 16 148 44)(6 17 149 45)(7 18 150 46)(8 19 151 47)(9 20 152 48)(10 21 153 49)(11 22 154 50)(12 23 141 51)(13 24 142 52)(14 25 143 53)(29 210 184 124)(30 197 185 125)(31 198 186 126)(32 199 187 113)(33 200 188 114)(34 201 189 115)(35 202 190 116)(36 203 191 117)(37 204 192 118)(38 205 193 119)(39 206 194 120)(40 207 195 121)(41 208 196 122)(42 209 183 123)(57 308 218 275)(58 295 219 276)(59 296 220 277)(60 297 221 278)(61 298 222 279)(62 299 223 280)(63 300 224 267)(64 301 211 268)(65 302 212 269)(66 303 213 270)(67 304 214 271)(68 305 215 272)(69 306 216 273)(70 307 217 274)(71 334 237 246)(72 335 238 247)(73 336 225 248)(74 323 226 249)(75 324 227 250)(76 325 228 251)(77 326 229 252)(78 327 230 239)(79 328 231 240)(80 329 232 241)(81 330 233 242)(82 331 234 243)(83 332 235 244)(84 333 236 245)(85 108 156 408)(86 109 157 409)(87 110 158 410)(88 111 159 411)(89 112 160 412)(90 99 161 413)(91 100 162 414)(92 101 163 415)(93 102 164 416)(94 103 165 417)(95 104 166 418)(96 105 167 419)(97 106 168 420)(98 107 155 407)(127 390 174 353)(128 391 175 354)(129 392 176 355)(130 379 177 356)(131 380 178 357)(132 381 179 358)(133 382 180 359)(134 383 181 360)(135 384 182 361)(136 385 169 362)(137 386 170 363)(138 387 171 364)(139 388 172 351)(140 389 173 352)(253 286 442 342)(254 287 443 343)(255 288 444 344)(256 289 445 345)(257 290 446 346)(258 291 447 347)(259 292 448 348)(260 293 435 349)(261 294 436 350)(262 281 437 337)(263 282 438 338)(264 283 439 339)(265 284 440 340)(266 285 441 341)(309 373 423 404)(310 374 424 405)(311 375 425 406)(312 376 426 393)(313 377 427 394)(314 378 428 395)(315 365 429 396)(316 366 430 397)(317 367 431 398)(318 368 432 399)(319 369 433 400)(320 370 434 401)(321 371 421 402)(322 372 422 403)
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), (1,308,8,301)(2,307,9,300)(3,306,10,299)(4,305,11,298)(5,304,12,297)(6,303,13,296)(7,302,14,295)(15,222,22,215)(16,221,23,214)(17,220,24,213)(18,219,25,212)(19,218,26,211)(20,217,27,224)(21,216,28,223)(29,413,36,420)(30,412,37,419)(31,411,38,418)(32,410,39,417)(33,409,40,416)(34,408,41,415)(35,407,42,414)(43,61,50,68)(44,60,51,67)(45,59,52,66)(46,58,53,65)(47,57,54,64)(48,70,55,63)(49,69,56,62)(71,352,78,359)(72,351,79,358)(73,364,80,357)(74,363,81,356)(75,362,82,355)(76,361,83,354)(77,360,84,353)(85,201,92,208)(86,200,93,207)(87,199,94,206)(88,198,95,205)(89,197,96,204)(90,210,97,203)(91,209,98,202)(99,191,106,184)(100,190,107,183)(101,189,108,196)(102,188,109,195)(103,187,110,194)(104,186,111,193)(105,185,112,192)(113,165,120,158)(114,164,121,157)(115,163,122,156)(116,162,123,155)(117,161,124,168)(118,160,125,167)(119,159,126,166)(127,333,134,326)(128,332,135,325)(129,331,136,324)(130,330,137,323)(131,329,138,336)(132,328,139,335)(133,327,140,334)(141,278,148,271)(142,277,149,270)(143,276,150,269)(144,275,151,268)(145,274,152,267)(146,273,153,280)(147,272,154,279)(169,250,176,243)(170,249,177,242)(171,248,178,241)(172,247,179,240)(173,246,180,239)(174,245,181,252)(175,244,182,251)(225,387,232,380)(226,386,233,379)(227,385,234,392)(228,384,235,391)(229,383,236,390)(230,382,237,389)(231,381,238,388)(253,368,260,375)(254,367,261,374)(255,366,262,373)(256,365,263,372)(257,378,264,371)(258,377,265,370)(259,376,266,369)(281,430,288,423)(282,429,289,422)(283,428,290,421)(284,427,291,434)(285,426,292,433)(286,425,293,432)(287,424,294,431)(309,337,316,344)(310,350,317,343)(311,349,318,342)(312,348,319,341)(313,347,320,340)(314,346,321,339)(315,345,322,338)(393,441,400,448)(394,440,401,447)(395,439,402,446)(396,438,403,445)(397,437,404,444)(398,436,405,443)(399,435,406,442), (1,391,340,164,144,354,284,93)(2,392,341,165,145,355,285,94)(3,379,342,166,146,356,286,95)(4,380,343,167,147,357,287,96)(5,381,344,168,148,358,288,97)(6,382,345,155,149,359,289,98)(7,383,346,156,150,360,290,85)(8,384,347,157,151,361,291,86)(9,385,348,158,152,362,292,87)(10,386,349,159,153,363,293,88)(11,387,350,160,154,364,294,89)(12,388,337,161,141,351,281,90)(13,389,338,162,142,352,282,91)(14,390,339,163,143,353,283,92)(15,105,443,131,43,419,254,178)(16,106,444,132,44,420,255,179)(17,107,445,133,45,407,256,180)(18,108,446,134,46,408,257,181)(19,109,447,135,47,409,258,182)(20,110,448,136,48,410,259,169)(21,111,435,137,49,411,260,170)(22,112,436,138,50,412,261,171)(23,99,437,139,51,413,262,172)(24,100,438,140,52,414,263,173)(25,101,439,127,53,415,264,174)(26,102,440,128,54,416,265,175)(27,103,441,129,55,417,266,176)(28,104,442,130,56,418,253,177)(29,373,240,214,184,404,328,67)(30,374,241,215,185,405,329,68)(31,375,242,216,186,406,330,69)(32,376,243,217,187,393,331,70)(33,377,244,218,188,394,332,57)(34,378,245,219,189,395,333,58)(35,365,246,220,190,396,334,59)(36,366,247,221,191,397,335,60)(37,367,248,222,192,398,336,61)(38,368,249,223,193,399,323,62)(39,369,250,224,194,400,324,63)(40,370,251,211,195,401,325,64)(41,371,252,212,196,402,326,65)(42,372,239,213,183,403,327,66)(71,429,202,296,237,315,116,277)(72,430,203,297,238,316,117,278)(73,431,204,298,225,317,118,279)(74,432,205,299,226,318,119,280)(75,433,206,300,227,319,120,267)(76,434,207,301,228,320,121,268)(77,421,208,302,229,321,122,269)(78,422,209,303,230,322,123,270)(79,423,210,304,231,309,124,271)(80,424,197,305,232,310,125,272)(81,425,198,306,233,311,126,273)(82,426,199,307,234,312,113,274)(83,427,200,308,235,313,114,275)(84,428,201,295,236,314,115,276), (1,26,144,54)(2,27,145,55)(3,28,146,56)(4,15,147,43)(5,16,148,44)(6,17,149,45)(7,18,150,46)(8,19,151,47)(9,20,152,48)(10,21,153,49)(11,22,154,50)(12,23,141,51)(13,24,142,52)(14,25,143,53)(29,210,184,124)(30,197,185,125)(31,198,186,126)(32,199,187,113)(33,200,188,114)(34,201,189,115)(35,202,190,116)(36,203,191,117)(37,204,192,118)(38,205,193,119)(39,206,194,120)(40,207,195,121)(41,208,196,122)(42,209,183,123)(57,308,218,275)(58,295,219,276)(59,296,220,277)(60,297,221,278)(61,298,222,279)(62,299,223,280)(63,300,224,267)(64,301,211,268)(65,302,212,269)(66,303,213,270)(67,304,214,271)(68,305,215,272)(69,306,216,273)(70,307,217,274)(71,334,237,246)(72,335,238,247)(73,336,225,248)(74,323,226,249)(75,324,227,250)(76,325,228,251)(77,326,229,252)(78,327,230,239)(79,328,231,240)(80,329,232,241)(81,330,233,242)(82,331,234,243)(83,332,235,244)(84,333,236,245)(85,108,156,408)(86,109,157,409)(87,110,158,410)(88,111,159,411)(89,112,160,412)(90,99,161,413)(91,100,162,414)(92,101,163,415)(93,102,164,416)(94,103,165,417)(95,104,166,418)(96,105,167,419)(97,106,168,420)(98,107,155,407)(127,390,174,353)(128,391,175,354)(129,392,176,355)(130,379,177,356)(131,380,178,357)(132,381,179,358)(133,382,180,359)(134,383,181,360)(135,384,182,361)(136,385,169,362)(137,386,170,363)(138,387,171,364)(139,388,172,351)(140,389,173,352)(253,286,442,342)(254,287,443,343)(255,288,444,344)(256,289,445,345)(257,290,446,346)(258,291,447,347)(259,292,448,348)(260,293,435,349)(261,294,436,350)(262,281,437,337)(263,282,438,338)(264,283,439,339)(265,284,440,340)(266,285,441,341)(309,373,423,404)(310,374,424,405)(311,375,425,406)(312,376,426,393)(313,377,427,394)(314,378,428,395)(315,365,429,396)(316,366,430,397)(317,367,431,398)(318,368,432,399)(319,369,433,400)(320,370,434,401)(321,371,421,402)(322,372,422,403)>;
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), (1,308,8,301)(2,307,9,300)(3,306,10,299)(4,305,11,298)(5,304,12,297)(6,303,13,296)(7,302,14,295)(15,222,22,215)(16,221,23,214)(17,220,24,213)(18,219,25,212)(19,218,26,211)(20,217,27,224)(21,216,28,223)(29,413,36,420)(30,412,37,419)(31,411,38,418)(32,410,39,417)(33,409,40,416)(34,408,41,415)(35,407,42,414)(43,61,50,68)(44,60,51,67)(45,59,52,66)(46,58,53,65)(47,57,54,64)(48,70,55,63)(49,69,56,62)(71,352,78,359)(72,351,79,358)(73,364,80,357)(74,363,81,356)(75,362,82,355)(76,361,83,354)(77,360,84,353)(85,201,92,208)(86,200,93,207)(87,199,94,206)(88,198,95,205)(89,197,96,204)(90,210,97,203)(91,209,98,202)(99,191,106,184)(100,190,107,183)(101,189,108,196)(102,188,109,195)(103,187,110,194)(104,186,111,193)(105,185,112,192)(113,165,120,158)(114,164,121,157)(115,163,122,156)(116,162,123,155)(117,161,124,168)(118,160,125,167)(119,159,126,166)(127,333,134,326)(128,332,135,325)(129,331,136,324)(130,330,137,323)(131,329,138,336)(132,328,139,335)(133,327,140,334)(141,278,148,271)(142,277,149,270)(143,276,150,269)(144,275,151,268)(145,274,152,267)(146,273,153,280)(147,272,154,279)(169,250,176,243)(170,249,177,242)(171,248,178,241)(172,247,179,240)(173,246,180,239)(174,245,181,252)(175,244,182,251)(225,387,232,380)(226,386,233,379)(227,385,234,392)(228,384,235,391)(229,383,236,390)(230,382,237,389)(231,381,238,388)(253,368,260,375)(254,367,261,374)(255,366,262,373)(256,365,263,372)(257,378,264,371)(258,377,265,370)(259,376,266,369)(281,430,288,423)(282,429,289,422)(283,428,290,421)(284,427,291,434)(285,426,292,433)(286,425,293,432)(287,424,294,431)(309,337,316,344)(310,350,317,343)(311,349,318,342)(312,348,319,341)(313,347,320,340)(314,346,321,339)(315,345,322,338)(393,441,400,448)(394,440,401,447)(395,439,402,446)(396,438,403,445)(397,437,404,444)(398,436,405,443)(399,435,406,442), (1,391,340,164,144,354,284,93)(2,392,341,165,145,355,285,94)(3,379,342,166,146,356,286,95)(4,380,343,167,147,357,287,96)(5,381,344,168,148,358,288,97)(6,382,345,155,149,359,289,98)(7,383,346,156,150,360,290,85)(8,384,347,157,151,361,291,86)(9,385,348,158,152,362,292,87)(10,386,349,159,153,363,293,88)(11,387,350,160,154,364,294,89)(12,388,337,161,141,351,281,90)(13,389,338,162,142,352,282,91)(14,390,339,163,143,353,283,92)(15,105,443,131,43,419,254,178)(16,106,444,132,44,420,255,179)(17,107,445,133,45,407,256,180)(18,108,446,134,46,408,257,181)(19,109,447,135,47,409,258,182)(20,110,448,136,48,410,259,169)(21,111,435,137,49,411,260,170)(22,112,436,138,50,412,261,171)(23,99,437,139,51,413,262,172)(24,100,438,140,52,414,263,173)(25,101,439,127,53,415,264,174)(26,102,440,128,54,416,265,175)(27,103,441,129,55,417,266,176)(28,104,442,130,56,418,253,177)(29,373,240,214,184,404,328,67)(30,374,241,215,185,405,329,68)(31,375,242,216,186,406,330,69)(32,376,243,217,187,393,331,70)(33,377,244,218,188,394,332,57)(34,378,245,219,189,395,333,58)(35,365,246,220,190,396,334,59)(36,366,247,221,191,397,335,60)(37,367,248,222,192,398,336,61)(38,368,249,223,193,399,323,62)(39,369,250,224,194,400,324,63)(40,370,251,211,195,401,325,64)(41,371,252,212,196,402,326,65)(42,372,239,213,183,403,327,66)(71,429,202,296,237,315,116,277)(72,430,203,297,238,316,117,278)(73,431,204,298,225,317,118,279)(74,432,205,299,226,318,119,280)(75,433,206,300,227,319,120,267)(76,434,207,301,228,320,121,268)(77,421,208,302,229,321,122,269)(78,422,209,303,230,322,123,270)(79,423,210,304,231,309,124,271)(80,424,197,305,232,310,125,272)(81,425,198,306,233,311,126,273)(82,426,199,307,234,312,113,274)(83,427,200,308,235,313,114,275)(84,428,201,295,236,314,115,276), (1,26,144,54)(2,27,145,55)(3,28,146,56)(4,15,147,43)(5,16,148,44)(6,17,149,45)(7,18,150,46)(8,19,151,47)(9,20,152,48)(10,21,153,49)(11,22,154,50)(12,23,141,51)(13,24,142,52)(14,25,143,53)(29,210,184,124)(30,197,185,125)(31,198,186,126)(32,199,187,113)(33,200,188,114)(34,201,189,115)(35,202,190,116)(36,203,191,117)(37,204,192,118)(38,205,193,119)(39,206,194,120)(40,207,195,121)(41,208,196,122)(42,209,183,123)(57,308,218,275)(58,295,219,276)(59,296,220,277)(60,297,221,278)(61,298,222,279)(62,299,223,280)(63,300,224,267)(64,301,211,268)(65,302,212,269)(66,303,213,270)(67,304,214,271)(68,305,215,272)(69,306,216,273)(70,307,217,274)(71,334,237,246)(72,335,238,247)(73,336,225,248)(74,323,226,249)(75,324,227,250)(76,325,228,251)(77,326,229,252)(78,327,230,239)(79,328,231,240)(80,329,232,241)(81,330,233,242)(82,331,234,243)(83,332,235,244)(84,333,236,245)(85,108,156,408)(86,109,157,409)(87,110,158,410)(88,111,159,411)(89,112,160,412)(90,99,161,413)(91,100,162,414)(92,101,163,415)(93,102,164,416)(94,103,165,417)(95,104,166,418)(96,105,167,419)(97,106,168,420)(98,107,155,407)(127,390,174,353)(128,391,175,354)(129,392,176,355)(130,379,177,356)(131,380,178,357)(132,381,179,358)(133,382,180,359)(134,383,181,360)(135,384,182,361)(136,385,169,362)(137,386,170,363)(138,387,171,364)(139,388,172,351)(140,389,173,352)(253,286,442,342)(254,287,443,343)(255,288,444,344)(256,289,445,345)(257,290,446,346)(258,291,447,347)(259,292,448,348)(260,293,435,349)(261,294,436,350)(262,281,437,337)(263,282,438,338)(264,283,439,339)(265,284,440,340)(266,285,441,341)(309,373,423,404)(310,374,424,405)(311,375,425,406)(312,376,426,393)(313,377,427,394)(314,378,428,395)(315,365,429,396)(316,366,430,397)(317,367,431,398)(318,368,432,399)(319,369,433,400)(320,370,434,401)(321,371,421,402)(322,372,422,403) );
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)], [(1,308,8,301),(2,307,9,300),(3,306,10,299),(4,305,11,298),(5,304,12,297),(6,303,13,296),(7,302,14,295),(15,222,22,215),(16,221,23,214),(17,220,24,213),(18,219,25,212),(19,218,26,211),(20,217,27,224),(21,216,28,223),(29,413,36,420),(30,412,37,419),(31,411,38,418),(32,410,39,417),(33,409,40,416),(34,408,41,415),(35,407,42,414),(43,61,50,68),(44,60,51,67),(45,59,52,66),(46,58,53,65),(47,57,54,64),(48,70,55,63),(49,69,56,62),(71,352,78,359),(72,351,79,358),(73,364,80,357),(74,363,81,356),(75,362,82,355),(76,361,83,354),(77,360,84,353),(85,201,92,208),(86,200,93,207),(87,199,94,206),(88,198,95,205),(89,197,96,204),(90,210,97,203),(91,209,98,202),(99,191,106,184),(100,190,107,183),(101,189,108,196),(102,188,109,195),(103,187,110,194),(104,186,111,193),(105,185,112,192),(113,165,120,158),(114,164,121,157),(115,163,122,156),(116,162,123,155),(117,161,124,168),(118,160,125,167),(119,159,126,166),(127,333,134,326),(128,332,135,325),(129,331,136,324),(130,330,137,323),(131,329,138,336),(132,328,139,335),(133,327,140,334),(141,278,148,271),(142,277,149,270),(143,276,150,269),(144,275,151,268),(145,274,152,267),(146,273,153,280),(147,272,154,279),(169,250,176,243),(170,249,177,242),(171,248,178,241),(172,247,179,240),(173,246,180,239),(174,245,181,252),(175,244,182,251),(225,387,232,380),(226,386,233,379),(227,385,234,392),(228,384,235,391),(229,383,236,390),(230,382,237,389),(231,381,238,388),(253,368,260,375),(254,367,261,374),(255,366,262,373),(256,365,263,372),(257,378,264,371),(258,377,265,370),(259,376,266,369),(281,430,288,423),(282,429,289,422),(283,428,290,421),(284,427,291,434),(285,426,292,433),(286,425,293,432),(287,424,294,431),(309,337,316,344),(310,350,317,343),(311,349,318,342),(312,348,319,341),(313,347,320,340),(314,346,321,339),(315,345,322,338),(393,441,400,448),(394,440,401,447),(395,439,402,446),(396,438,403,445),(397,437,404,444),(398,436,405,443),(399,435,406,442)], [(1,391,340,164,144,354,284,93),(2,392,341,165,145,355,285,94),(3,379,342,166,146,356,286,95),(4,380,343,167,147,357,287,96),(5,381,344,168,148,358,288,97),(6,382,345,155,149,359,289,98),(7,383,346,156,150,360,290,85),(8,384,347,157,151,361,291,86),(9,385,348,158,152,362,292,87),(10,386,349,159,153,363,293,88),(11,387,350,160,154,364,294,89),(12,388,337,161,141,351,281,90),(13,389,338,162,142,352,282,91),(14,390,339,163,143,353,283,92),(15,105,443,131,43,419,254,178),(16,106,444,132,44,420,255,179),(17,107,445,133,45,407,256,180),(18,108,446,134,46,408,257,181),(19,109,447,135,47,409,258,182),(20,110,448,136,48,410,259,169),(21,111,435,137,49,411,260,170),(22,112,436,138,50,412,261,171),(23,99,437,139,51,413,262,172),(24,100,438,140,52,414,263,173),(25,101,439,127,53,415,264,174),(26,102,440,128,54,416,265,175),(27,103,441,129,55,417,266,176),(28,104,442,130,56,418,253,177),(29,373,240,214,184,404,328,67),(30,374,241,215,185,405,329,68),(31,375,242,216,186,406,330,69),(32,376,243,217,187,393,331,70),(33,377,244,218,188,394,332,57),(34,378,245,219,189,395,333,58),(35,365,246,220,190,396,334,59),(36,366,247,221,191,397,335,60),(37,367,248,222,192,398,336,61),(38,368,249,223,193,399,323,62),(39,369,250,224,194,400,324,63),(40,370,251,211,195,401,325,64),(41,371,252,212,196,402,326,65),(42,372,239,213,183,403,327,66),(71,429,202,296,237,315,116,277),(72,430,203,297,238,316,117,278),(73,431,204,298,225,317,118,279),(74,432,205,299,226,318,119,280),(75,433,206,300,227,319,120,267),(76,434,207,301,228,320,121,268),(77,421,208,302,229,321,122,269),(78,422,209,303,230,322,123,270),(79,423,210,304,231,309,124,271),(80,424,197,305,232,310,125,272),(81,425,198,306,233,311,126,273),(82,426,199,307,234,312,113,274),(83,427,200,308,235,313,114,275),(84,428,201,295,236,314,115,276)], [(1,26,144,54),(2,27,145,55),(3,28,146,56),(4,15,147,43),(5,16,148,44),(6,17,149,45),(7,18,150,46),(8,19,151,47),(9,20,152,48),(10,21,153,49),(11,22,154,50),(12,23,141,51),(13,24,142,52),(14,25,143,53),(29,210,184,124),(30,197,185,125),(31,198,186,126),(32,199,187,113),(33,200,188,114),(34,201,189,115),(35,202,190,116),(36,203,191,117),(37,204,192,118),(38,205,193,119),(39,206,194,120),(40,207,195,121),(41,208,196,122),(42,209,183,123),(57,308,218,275),(58,295,219,276),(59,296,220,277),(60,297,221,278),(61,298,222,279),(62,299,223,280),(63,300,224,267),(64,301,211,268),(65,302,212,269),(66,303,213,270),(67,304,214,271),(68,305,215,272),(69,306,216,273),(70,307,217,274),(71,334,237,246),(72,335,238,247),(73,336,225,248),(74,323,226,249),(75,324,227,250),(76,325,228,251),(77,326,229,252),(78,327,230,239),(79,328,231,240),(80,329,232,241),(81,330,233,242),(82,331,234,243),(83,332,235,244),(84,333,236,245),(85,108,156,408),(86,109,157,409),(87,110,158,410),(88,111,159,411),(89,112,160,412),(90,99,161,413),(91,100,162,414),(92,101,163,415),(93,102,164,416),(94,103,165,417),(95,104,166,418),(96,105,167,419),(97,106,168,420),(98,107,155,407),(127,390,174,353),(128,391,175,354),(129,392,176,355),(130,379,177,356),(131,380,178,357),(132,381,179,358),(133,382,180,359),(134,383,181,360),(135,384,182,361),(136,385,169,362),(137,386,170,363),(138,387,171,364),(139,388,172,351),(140,389,173,352),(253,286,442,342),(254,287,443,343),(255,288,444,344),(256,289,445,345),(257,290,446,346),(258,291,447,347),(259,292,448,348),(260,293,435,349),(261,294,436,350),(262,281,437,337),(263,282,438,338),(264,283,439,339),(265,284,440,340),(266,285,441,341),(309,373,423,404),(310,374,424,405),(311,375,425,406),(312,376,426,393),(313,377,427,394),(314,378,428,395),(315,365,429,396),(316,366,430,397),(317,367,431,398),(318,368,432,399),(319,369,433,400),(320,370,434,401),(321,371,421,402),(322,372,422,403)]])
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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | C7⋊D4 | C8.C22 | D4⋊2D7 | D4×D7 | D7×Q16 | Q16⋊D7 |
| kernel | Dic7⋊3Q16 | Dic7⋊C8 | C28.44D4 | Q8⋊Dic7 | C2×C7⋊Q16 | Dic7⋊Q8 | Q8×Dic7 | C14×Q16 | C2×Dic7 | C7×Q8 | C2×Q16 | Dic7 | C28 | C2×C8 | C2×Q8 | Q8 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Dic7⋊3Q16 ►in GL4(𝔽113) generated by
| 24 | 112 | 0 | 0 |
| 100 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 104 | 66 | 0 | 0 |
| 33 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 29 | 12 | 0 | 0 |
| 43 | 84 | 0 | 0 |
| 0 | 0 | 0 | 109 |
| 0 | 0 | 85 | 51 |
| 84 | 101 | 0 | 0 |
| 70 | 29 | 0 | 0 |
| 0 | 0 | 25 | 22 |
| 0 | 0 | 64 | 88 |
G:=sub<GL(4,GF(113))| [24,100,0,0,112,10,0,0,0,0,1,0,0,0,0,1],[104,33,0,0,66,9,0,0,0,0,1,0,0,0,0,1],[29,43,0,0,12,84,0,0,0,0,0,85,0,0,109,51],[84,70,0,0,101,29,0,0,0,0,25,64,0,0,22,88] >;
Dic7⋊3Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes_3Q_{16} % in TeX
G:=Group("Dic7:3Q16"); // GroupNames label
G:=SmallGroup(448,716);
// by ID
G=gap.SmallGroup(448,716);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,422,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^8=1,b^2=a^7,d^2=c^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations