metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2Dic14, Dic7.3Q16, (C7×Q8)⋊2Q8, C28⋊Q8.3C2, C2.8(D7×Q16), C28.6(C2×Q8), C4⋊C4.17D14, C7⋊1(C4.Q16), (C2×C8).14D14, C56⋊1C4.6C2, Dic7⋊C8.3C2, Q8⋊C4.3D7, C14.15(C2×Q16), (Q8×Dic7).4C2, C4.6(C2×Dic14), (C2×C56).14C22, (C2×Q8).100D14, C28.Q8.2C2, Q8⋊Dic7.3C2, C22.188(D4×D7), C28.158(C4○D4), C4.83(D4⋊2D7), C2.14(D56⋊C2), C14.59(C8⋊C22), (C2×C28).234C23, (C2×Dic7).149D4, C14.12(C22⋊Q8), C4⋊Dic7.83C22, (Q8×C14).17C22, (C4×Dic7).18C22, C2.17(C22⋊Dic14), (C2×C7⋊C8).29C22, (C2×C14).247(C2×D4), (C7×C4⋊C4).35C22, (C7×Q8⋊C4).3C2, (C2×C4).341(C22×D7), SmallGroup(448,328)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7.Q16
G = < a,b,c,d | a14=c8=1, b2=a7, d2=a7c4, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a7b, dcd-1=a7c-1 >
Subgroups: 436 in 96 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, Q8⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C4.Q16, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, Q8×C14, C28.Q8, Dic7⋊C8, C56⋊1C4, Q8⋊Dic7, C7×Q8⋊C4, C28⋊Q8, Q8×Dic7, Dic7.Q16
Quotients: C1, C2, C22, D4, Q8, C23, D7, Q16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×Q16, C8⋊C22, Dic14, C22×D7, C4.Q16, C2×Dic14, D4×D7, D4⋊2D7, C22⋊Dic14, D56⋊C2, 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 26 8 19)(2 25 9 18)(3 24 10 17)(4 23 11 16)(5 22 12 15)(6 21 13 28)(7 20 14 27)(29 150 36 143)(30 149 37 142)(31 148 38 141)(32 147 39 154)(33 146 40 153)(34 145 41 152)(35 144 42 151)(43 292 50 285)(44 291 51 284)(45 290 52 283)(46 289 53 282)(47 288 54 281)(48 287 55 294)(49 286 56 293)(57 327 64 334)(58 326 65 333)(59 325 66 332)(60 324 67 331)(61 323 68 330)(62 336 69 329)(63 335 70 328)(71 140 78 133)(72 139 79 132)(73 138 80 131)(74 137 81 130)(75 136 82 129)(76 135 83 128)(77 134 84 127)(85 388 92 381)(86 387 93 380)(87 386 94 379)(88 385 95 392)(89 384 96 391)(90 383 97 390)(91 382 98 389)(99 416 106 409)(100 415 107 408)(101 414 108 407)(102 413 109 420)(103 412 110 419)(104 411 111 418)(105 410 112 417)(113 307 120 300)(114 306 121 299)(115 305 122 298)(116 304 123 297)(117 303 124 296)(118 302 125 295)(119 301 126 308)(155 275 162 268)(156 274 163 267)(157 273 164 280)(158 272 165 279)(159 271 166 278)(160 270 167 277)(161 269 168 276)(169 439 176 446)(170 438 177 445)(171 437 178 444)(172 436 179 443)(173 435 180 442)(174 448 181 441)(175 447 182 440)(183 225 190 232)(184 238 191 231)(185 237 192 230)(186 236 193 229)(187 235 194 228)(188 234 195 227)(189 233 196 226)(197 348 204 341)(198 347 205 340)(199 346 206 339)(200 345 207 338)(201 344 208 337)(202 343 209 350)(203 342 210 349)(211 428 218 421)(212 427 219 434)(213 426 220 433)(214 425 221 432)(215 424 222 431)(216 423 223 430)(217 422 224 429)(239 372 246 365)(240 371 247 378)(241 370 248 377)(242 369 249 376)(243 368 250 375)(244 367 251 374)(245 366 252 373)(253 404 260 397)(254 403 261 396)(255 402 262 395)(256 401 263 394)(257 400 264 393)(258 399 265 406)(259 398 266 405)(309 360 316 353)(310 359 317 352)(311 358 318 351)(312 357 319 364)(313 356 320 363)(314 355 321 362)(315 354 322 361)
(1 30 315 236 375 328 379 197)(2 31 316 237 376 329 380 198)(3 32 317 238 377 330 381 199)(4 33 318 225 378 331 382 200)(5 34 319 226 365 332 383 201)(6 35 320 227 366 333 384 202)(7 36 321 228 367 334 385 203)(8 37 322 229 368 335 386 204)(9 38 309 230 369 336 387 205)(10 39 310 231 370 323 388 206)(11 40 311 232 371 324 389 207)(12 41 312 233 372 325 390 208)(13 42 313 234 373 326 391 209)(14 29 314 235 374 327 392 210)(15 145 357 189 246 59 90 344)(16 146 358 190 247 60 91 345)(17 147 359 191 248 61 92 346)(18 148 360 192 249 62 93 347)(19 149 361 193 250 63 94 348)(20 150 362 194 251 64 95 349)(21 151 363 195 252 65 96 350)(22 152 364 196 239 66 97 337)(23 153 351 183 240 67 98 338)(24 154 352 184 241 68 85 339)(25 141 353 185 242 69 86 340)(26 142 354 186 243 70 87 341)(27 143 355 187 244 57 88 342)(28 144 356 188 245 58 89 343)(43 163 265 71 177 117 102 223)(44 164 266 72 178 118 103 224)(45 165 253 73 179 119 104 211)(46 166 254 74 180 120 105 212)(47 167 255 75 181 121 106 213)(48 168 256 76 182 122 107 214)(49 155 257 77 169 123 108 215)(50 156 258 78 170 124 109 216)(51 157 259 79 171 125 110 217)(52 158 260 80 172 126 111 218)(53 159 261 81 173 113 112 219)(54 160 262 82 174 114 99 220)(55 161 263 83 175 115 100 221)(56 162 264 84 176 116 101 222)(127 439 304 407 431 286 268 400)(128 440 305 408 432 287 269 401)(129 441 306 409 433 288 270 402)(130 442 307 410 434 289 271 403)(131 443 308 411 421 290 272 404)(132 444 295 412 422 291 273 405)(133 445 296 413 423 292 274 406)(134 446 297 414 424 293 275 393)(135 447 298 415 425 294 276 394)(136 448 299 416 426 281 277 395)(137 435 300 417 427 282 278 396)(138 436 301 418 428 283 279 397)(139 437 302 419 429 284 280 398)(140 438 303 420 430 285 267 399)
(1 99 368 255)(2 100 369 256)(3 101 370 257)(4 102 371 258)(5 103 372 259)(6 104 373 260)(7 105 374 261)(8 106 375 262)(9 107 376 263)(10 108 377 264)(11 109 378 265)(12 110 365 266)(13 111 366 253)(14 112 367 254)(15 412 239 398)(16 413 240 399)(17 414 241 400)(18 415 242 401)(19 416 243 402)(20 417 244 403)(21 418 245 404)(22 419 246 405)(23 420 247 406)(24 407 248 393)(25 408 249 394)(26 409 250 395)(27 410 251 396)(28 411 252 397)(29 120 334 159)(30 121 335 160)(31 122 336 161)(32 123 323 162)(33 124 324 163)(34 125 325 164)(35 126 326 165)(36 113 327 166)(37 114 328 167)(38 115 329 168)(39 116 330 155)(40 117 331 156)(41 118 332 157)(42 119 333 158)(43 311 170 382)(44 312 171 383)(45 313 172 384)(46 314 173 385)(47 315 174 386)(48 316 175 387)(49 317 176 388)(50 318 177 389)(51 319 178 390)(52 320 179 391)(53 321 180 392)(54 322 181 379)(55 309 182 380)(56 310 169 381)(57 278 150 307)(58 279 151 308)(59 280 152 295)(60 267 153 296)(61 268 154 297)(62 269 141 298)(63 270 142 299)(64 271 143 300)(65 272 144 301)(66 273 145 302)(67 274 146 303)(68 275 147 304)(69 276 148 305)(70 277 149 306)(71 200 216 232)(72 201 217 233)(73 202 218 234)(74 203 219 235)(75 204 220 236)(76 205 221 237)(77 206 222 238)(78 207 223 225)(79 208 224 226)(80 209 211 227)(81 210 212 228)(82 197 213 229)(83 198 214 230)(84 199 215 231)(85 286 359 446)(86 287 360 447)(87 288 361 448)(88 289 362 435)(89 290 363 436)(90 291 364 437)(91 292 351 438)(92 293 352 439)(93 294 353 440)(94 281 354 441)(95 282 355 442)(96 283 356 443)(97 284 357 444)(98 285 358 445)(127 339 424 191)(128 340 425 192)(129 341 426 193)(130 342 427 194)(131 343 428 195)(132 344 429 196)(133 345 430 183)(134 346 431 184)(135 347 432 185)(136 348 433 186)(137 349 434 187)(138 350 421 188)(139 337 422 189)(140 338 423 190)
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), (1,26,8,19)(2,25,9,18)(3,24,10,17)(4,23,11,16)(5,22,12,15)(6,21,13,28)(7,20,14,27)(29,150,36,143)(30,149,37,142)(31,148,38,141)(32,147,39,154)(33,146,40,153)(34,145,41,152)(35,144,42,151)(43,292,50,285)(44,291,51,284)(45,290,52,283)(46,289,53,282)(47,288,54,281)(48,287,55,294)(49,286,56,293)(57,327,64,334)(58,326,65,333)(59,325,66,332)(60,324,67,331)(61,323,68,330)(62,336,69,329)(63,335,70,328)(71,140,78,133)(72,139,79,132)(73,138,80,131)(74,137,81,130)(75,136,82,129)(76,135,83,128)(77,134,84,127)(85,388,92,381)(86,387,93,380)(87,386,94,379)(88,385,95,392)(89,384,96,391)(90,383,97,390)(91,382,98,389)(99,416,106,409)(100,415,107,408)(101,414,108,407)(102,413,109,420)(103,412,110,419)(104,411,111,418)(105,410,112,417)(113,307,120,300)(114,306,121,299)(115,305,122,298)(116,304,123,297)(117,303,124,296)(118,302,125,295)(119,301,126,308)(155,275,162,268)(156,274,163,267)(157,273,164,280)(158,272,165,279)(159,271,166,278)(160,270,167,277)(161,269,168,276)(169,439,176,446)(170,438,177,445)(171,437,178,444)(172,436,179,443)(173,435,180,442)(174,448,181,441)(175,447,182,440)(183,225,190,232)(184,238,191,231)(185,237,192,230)(186,236,193,229)(187,235,194,228)(188,234,195,227)(189,233,196,226)(197,348,204,341)(198,347,205,340)(199,346,206,339)(200,345,207,338)(201,344,208,337)(202,343,209,350)(203,342,210,349)(211,428,218,421)(212,427,219,434)(213,426,220,433)(214,425,221,432)(215,424,222,431)(216,423,223,430)(217,422,224,429)(239,372,246,365)(240,371,247,378)(241,370,248,377)(242,369,249,376)(243,368,250,375)(244,367,251,374)(245,366,252,373)(253,404,260,397)(254,403,261,396)(255,402,262,395)(256,401,263,394)(257,400,264,393)(258,399,265,406)(259,398,266,405)(309,360,316,353)(310,359,317,352)(311,358,318,351)(312,357,319,364)(313,356,320,363)(314,355,321,362)(315,354,322,361), (1,30,315,236,375,328,379,197)(2,31,316,237,376,329,380,198)(3,32,317,238,377,330,381,199)(4,33,318,225,378,331,382,200)(5,34,319,226,365,332,383,201)(6,35,320,227,366,333,384,202)(7,36,321,228,367,334,385,203)(8,37,322,229,368,335,386,204)(9,38,309,230,369,336,387,205)(10,39,310,231,370,323,388,206)(11,40,311,232,371,324,389,207)(12,41,312,233,372,325,390,208)(13,42,313,234,373,326,391,209)(14,29,314,235,374,327,392,210)(15,145,357,189,246,59,90,344)(16,146,358,190,247,60,91,345)(17,147,359,191,248,61,92,346)(18,148,360,192,249,62,93,347)(19,149,361,193,250,63,94,348)(20,150,362,194,251,64,95,349)(21,151,363,195,252,65,96,350)(22,152,364,196,239,66,97,337)(23,153,351,183,240,67,98,338)(24,154,352,184,241,68,85,339)(25,141,353,185,242,69,86,340)(26,142,354,186,243,70,87,341)(27,143,355,187,244,57,88,342)(28,144,356,188,245,58,89,343)(43,163,265,71,177,117,102,223)(44,164,266,72,178,118,103,224)(45,165,253,73,179,119,104,211)(46,166,254,74,180,120,105,212)(47,167,255,75,181,121,106,213)(48,168,256,76,182,122,107,214)(49,155,257,77,169,123,108,215)(50,156,258,78,170,124,109,216)(51,157,259,79,171,125,110,217)(52,158,260,80,172,126,111,218)(53,159,261,81,173,113,112,219)(54,160,262,82,174,114,99,220)(55,161,263,83,175,115,100,221)(56,162,264,84,176,116,101,222)(127,439,304,407,431,286,268,400)(128,440,305,408,432,287,269,401)(129,441,306,409,433,288,270,402)(130,442,307,410,434,289,271,403)(131,443,308,411,421,290,272,404)(132,444,295,412,422,291,273,405)(133,445,296,413,423,292,274,406)(134,446,297,414,424,293,275,393)(135,447,298,415,425,294,276,394)(136,448,299,416,426,281,277,395)(137,435,300,417,427,282,278,396)(138,436,301,418,428,283,279,397)(139,437,302,419,429,284,280,398)(140,438,303,420,430,285,267,399), (1,99,368,255)(2,100,369,256)(3,101,370,257)(4,102,371,258)(5,103,372,259)(6,104,373,260)(7,105,374,261)(8,106,375,262)(9,107,376,263)(10,108,377,264)(11,109,378,265)(12,110,365,266)(13,111,366,253)(14,112,367,254)(15,412,239,398)(16,413,240,399)(17,414,241,400)(18,415,242,401)(19,416,243,402)(20,417,244,403)(21,418,245,404)(22,419,246,405)(23,420,247,406)(24,407,248,393)(25,408,249,394)(26,409,250,395)(27,410,251,396)(28,411,252,397)(29,120,334,159)(30,121,335,160)(31,122,336,161)(32,123,323,162)(33,124,324,163)(34,125,325,164)(35,126,326,165)(36,113,327,166)(37,114,328,167)(38,115,329,168)(39,116,330,155)(40,117,331,156)(41,118,332,157)(42,119,333,158)(43,311,170,382)(44,312,171,383)(45,313,172,384)(46,314,173,385)(47,315,174,386)(48,316,175,387)(49,317,176,388)(50,318,177,389)(51,319,178,390)(52,320,179,391)(53,321,180,392)(54,322,181,379)(55,309,182,380)(56,310,169,381)(57,278,150,307)(58,279,151,308)(59,280,152,295)(60,267,153,296)(61,268,154,297)(62,269,141,298)(63,270,142,299)(64,271,143,300)(65,272,144,301)(66,273,145,302)(67,274,146,303)(68,275,147,304)(69,276,148,305)(70,277,149,306)(71,200,216,232)(72,201,217,233)(73,202,218,234)(74,203,219,235)(75,204,220,236)(76,205,221,237)(77,206,222,238)(78,207,223,225)(79,208,224,226)(80,209,211,227)(81,210,212,228)(82,197,213,229)(83,198,214,230)(84,199,215,231)(85,286,359,446)(86,287,360,447)(87,288,361,448)(88,289,362,435)(89,290,363,436)(90,291,364,437)(91,292,351,438)(92,293,352,439)(93,294,353,440)(94,281,354,441)(95,282,355,442)(96,283,356,443)(97,284,357,444)(98,285,358,445)(127,339,424,191)(128,340,425,192)(129,341,426,193)(130,342,427,194)(131,343,428,195)(132,344,429,196)(133,345,430,183)(134,346,431,184)(135,347,432,185)(136,348,433,186)(137,349,434,187)(138,350,421,188)(139,337,422,189)(140,338,423,190)>;
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), (1,26,8,19)(2,25,9,18)(3,24,10,17)(4,23,11,16)(5,22,12,15)(6,21,13,28)(7,20,14,27)(29,150,36,143)(30,149,37,142)(31,148,38,141)(32,147,39,154)(33,146,40,153)(34,145,41,152)(35,144,42,151)(43,292,50,285)(44,291,51,284)(45,290,52,283)(46,289,53,282)(47,288,54,281)(48,287,55,294)(49,286,56,293)(57,327,64,334)(58,326,65,333)(59,325,66,332)(60,324,67,331)(61,323,68,330)(62,336,69,329)(63,335,70,328)(71,140,78,133)(72,139,79,132)(73,138,80,131)(74,137,81,130)(75,136,82,129)(76,135,83,128)(77,134,84,127)(85,388,92,381)(86,387,93,380)(87,386,94,379)(88,385,95,392)(89,384,96,391)(90,383,97,390)(91,382,98,389)(99,416,106,409)(100,415,107,408)(101,414,108,407)(102,413,109,420)(103,412,110,419)(104,411,111,418)(105,410,112,417)(113,307,120,300)(114,306,121,299)(115,305,122,298)(116,304,123,297)(117,303,124,296)(118,302,125,295)(119,301,126,308)(155,275,162,268)(156,274,163,267)(157,273,164,280)(158,272,165,279)(159,271,166,278)(160,270,167,277)(161,269,168,276)(169,439,176,446)(170,438,177,445)(171,437,178,444)(172,436,179,443)(173,435,180,442)(174,448,181,441)(175,447,182,440)(183,225,190,232)(184,238,191,231)(185,237,192,230)(186,236,193,229)(187,235,194,228)(188,234,195,227)(189,233,196,226)(197,348,204,341)(198,347,205,340)(199,346,206,339)(200,345,207,338)(201,344,208,337)(202,343,209,350)(203,342,210,349)(211,428,218,421)(212,427,219,434)(213,426,220,433)(214,425,221,432)(215,424,222,431)(216,423,223,430)(217,422,224,429)(239,372,246,365)(240,371,247,378)(241,370,248,377)(242,369,249,376)(243,368,250,375)(244,367,251,374)(245,366,252,373)(253,404,260,397)(254,403,261,396)(255,402,262,395)(256,401,263,394)(257,400,264,393)(258,399,265,406)(259,398,266,405)(309,360,316,353)(310,359,317,352)(311,358,318,351)(312,357,319,364)(313,356,320,363)(314,355,321,362)(315,354,322,361), (1,30,315,236,375,328,379,197)(2,31,316,237,376,329,380,198)(3,32,317,238,377,330,381,199)(4,33,318,225,378,331,382,200)(5,34,319,226,365,332,383,201)(6,35,320,227,366,333,384,202)(7,36,321,228,367,334,385,203)(8,37,322,229,368,335,386,204)(9,38,309,230,369,336,387,205)(10,39,310,231,370,323,388,206)(11,40,311,232,371,324,389,207)(12,41,312,233,372,325,390,208)(13,42,313,234,373,326,391,209)(14,29,314,235,374,327,392,210)(15,145,357,189,246,59,90,344)(16,146,358,190,247,60,91,345)(17,147,359,191,248,61,92,346)(18,148,360,192,249,62,93,347)(19,149,361,193,250,63,94,348)(20,150,362,194,251,64,95,349)(21,151,363,195,252,65,96,350)(22,152,364,196,239,66,97,337)(23,153,351,183,240,67,98,338)(24,154,352,184,241,68,85,339)(25,141,353,185,242,69,86,340)(26,142,354,186,243,70,87,341)(27,143,355,187,244,57,88,342)(28,144,356,188,245,58,89,343)(43,163,265,71,177,117,102,223)(44,164,266,72,178,118,103,224)(45,165,253,73,179,119,104,211)(46,166,254,74,180,120,105,212)(47,167,255,75,181,121,106,213)(48,168,256,76,182,122,107,214)(49,155,257,77,169,123,108,215)(50,156,258,78,170,124,109,216)(51,157,259,79,171,125,110,217)(52,158,260,80,172,126,111,218)(53,159,261,81,173,113,112,219)(54,160,262,82,174,114,99,220)(55,161,263,83,175,115,100,221)(56,162,264,84,176,116,101,222)(127,439,304,407,431,286,268,400)(128,440,305,408,432,287,269,401)(129,441,306,409,433,288,270,402)(130,442,307,410,434,289,271,403)(131,443,308,411,421,290,272,404)(132,444,295,412,422,291,273,405)(133,445,296,413,423,292,274,406)(134,446,297,414,424,293,275,393)(135,447,298,415,425,294,276,394)(136,448,299,416,426,281,277,395)(137,435,300,417,427,282,278,396)(138,436,301,418,428,283,279,397)(139,437,302,419,429,284,280,398)(140,438,303,420,430,285,267,399), (1,99,368,255)(2,100,369,256)(3,101,370,257)(4,102,371,258)(5,103,372,259)(6,104,373,260)(7,105,374,261)(8,106,375,262)(9,107,376,263)(10,108,377,264)(11,109,378,265)(12,110,365,266)(13,111,366,253)(14,112,367,254)(15,412,239,398)(16,413,240,399)(17,414,241,400)(18,415,242,401)(19,416,243,402)(20,417,244,403)(21,418,245,404)(22,419,246,405)(23,420,247,406)(24,407,248,393)(25,408,249,394)(26,409,250,395)(27,410,251,396)(28,411,252,397)(29,120,334,159)(30,121,335,160)(31,122,336,161)(32,123,323,162)(33,124,324,163)(34,125,325,164)(35,126,326,165)(36,113,327,166)(37,114,328,167)(38,115,329,168)(39,116,330,155)(40,117,331,156)(41,118,332,157)(42,119,333,158)(43,311,170,382)(44,312,171,383)(45,313,172,384)(46,314,173,385)(47,315,174,386)(48,316,175,387)(49,317,176,388)(50,318,177,389)(51,319,178,390)(52,320,179,391)(53,321,180,392)(54,322,181,379)(55,309,182,380)(56,310,169,381)(57,278,150,307)(58,279,151,308)(59,280,152,295)(60,267,153,296)(61,268,154,297)(62,269,141,298)(63,270,142,299)(64,271,143,300)(65,272,144,301)(66,273,145,302)(67,274,146,303)(68,275,147,304)(69,276,148,305)(70,277,149,306)(71,200,216,232)(72,201,217,233)(73,202,218,234)(74,203,219,235)(75,204,220,236)(76,205,221,237)(77,206,222,238)(78,207,223,225)(79,208,224,226)(80,209,211,227)(81,210,212,228)(82,197,213,229)(83,198,214,230)(84,199,215,231)(85,286,359,446)(86,287,360,447)(87,288,361,448)(88,289,362,435)(89,290,363,436)(90,291,364,437)(91,292,351,438)(92,293,352,439)(93,294,353,440)(94,281,354,441)(95,282,355,442)(96,283,356,443)(97,284,357,444)(98,285,358,445)(127,339,424,191)(128,340,425,192)(129,341,426,193)(130,342,427,194)(131,343,428,195)(132,344,429,196)(133,345,430,183)(134,346,431,184)(135,347,432,185)(136,348,433,186)(137,349,434,187)(138,350,421,188)(139,337,422,189)(140,338,423,190) );
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)], [(1,26,8,19),(2,25,9,18),(3,24,10,17),(4,23,11,16),(5,22,12,15),(6,21,13,28),(7,20,14,27),(29,150,36,143),(30,149,37,142),(31,148,38,141),(32,147,39,154),(33,146,40,153),(34,145,41,152),(35,144,42,151),(43,292,50,285),(44,291,51,284),(45,290,52,283),(46,289,53,282),(47,288,54,281),(48,287,55,294),(49,286,56,293),(57,327,64,334),(58,326,65,333),(59,325,66,332),(60,324,67,331),(61,323,68,330),(62,336,69,329),(63,335,70,328),(71,140,78,133),(72,139,79,132),(73,138,80,131),(74,137,81,130),(75,136,82,129),(76,135,83,128),(77,134,84,127),(85,388,92,381),(86,387,93,380),(87,386,94,379),(88,385,95,392),(89,384,96,391),(90,383,97,390),(91,382,98,389),(99,416,106,409),(100,415,107,408),(101,414,108,407),(102,413,109,420),(103,412,110,419),(104,411,111,418),(105,410,112,417),(113,307,120,300),(114,306,121,299),(115,305,122,298),(116,304,123,297),(117,303,124,296),(118,302,125,295),(119,301,126,308),(155,275,162,268),(156,274,163,267),(157,273,164,280),(158,272,165,279),(159,271,166,278),(160,270,167,277),(161,269,168,276),(169,439,176,446),(170,438,177,445),(171,437,178,444),(172,436,179,443),(173,435,180,442),(174,448,181,441),(175,447,182,440),(183,225,190,232),(184,238,191,231),(185,237,192,230),(186,236,193,229),(187,235,194,228),(188,234,195,227),(189,233,196,226),(197,348,204,341),(198,347,205,340),(199,346,206,339),(200,345,207,338),(201,344,208,337),(202,343,209,350),(203,342,210,349),(211,428,218,421),(212,427,219,434),(213,426,220,433),(214,425,221,432),(215,424,222,431),(216,423,223,430),(217,422,224,429),(239,372,246,365),(240,371,247,378),(241,370,248,377),(242,369,249,376),(243,368,250,375),(244,367,251,374),(245,366,252,373),(253,404,260,397),(254,403,261,396),(255,402,262,395),(256,401,263,394),(257,400,264,393),(258,399,265,406),(259,398,266,405),(309,360,316,353),(310,359,317,352),(311,358,318,351),(312,357,319,364),(313,356,320,363),(314,355,321,362),(315,354,322,361)], [(1,30,315,236,375,328,379,197),(2,31,316,237,376,329,380,198),(3,32,317,238,377,330,381,199),(4,33,318,225,378,331,382,200),(5,34,319,226,365,332,383,201),(6,35,320,227,366,333,384,202),(7,36,321,228,367,334,385,203),(8,37,322,229,368,335,386,204),(9,38,309,230,369,336,387,205),(10,39,310,231,370,323,388,206),(11,40,311,232,371,324,389,207),(12,41,312,233,372,325,390,208),(13,42,313,234,373,326,391,209),(14,29,314,235,374,327,392,210),(15,145,357,189,246,59,90,344),(16,146,358,190,247,60,91,345),(17,147,359,191,248,61,92,346),(18,148,360,192,249,62,93,347),(19,149,361,193,250,63,94,348),(20,150,362,194,251,64,95,349),(21,151,363,195,252,65,96,350),(22,152,364,196,239,66,97,337),(23,153,351,183,240,67,98,338),(24,154,352,184,241,68,85,339),(25,141,353,185,242,69,86,340),(26,142,354,186,243,70,87,341),(27,143,355,187,244,57,88,342),(28,144,356,188,245,58,89,343),(43,163,265,71,177,117,102,223),(44,164,266,72,178,118,103,224),(45,165,253,73,179,119,104,211),(46,166,254,74,180,120,105,212),(47,167,255,75,181,121,106,213),(48,168,256,76,182,122,107,214),(49,155,257,77,169,123,108,215),(50,156,258,78,170,124,109,216),(51,157,259,79,171,125,110,217),(52,158,260,80,172,126,111,218),(53,159,261,81,173,113,112,219),(54,160,262,82,174,114,99,220),(55,161,263,83,175,115,100,221),(56,162,264,84,176,116,101,222),(127,439,304,407,431,286,268,400),(128,440,305,408,432,287,269,401),(129,441,306,409,433,288,270,402),(130,442,307,410,434,289,271,403),(131,443,308,411,421,290,272,404),(132,444,295,412,422,291,273,405),(133,445,296,413,423,292,274,406),(134,446,297,414,424,293,275,393),(135,447,298,415,425,294,276,394),(136,448,299,416,426,281,277,395),(137,435,300,417,427,282,278,396),(138,436,301,418,428,283,279,397),(139,437,302,419,429,284,280,398),(140,438,303,420,430,285,267,399)], [(1,99,368,255),(2,100,369,256),(3,101,370,257),(4,102,371,258),(5,103,372,259),(6,104,373,260),(7,105,374,261),(8,106,375,262),(9,107,376,263),(10,108,377,264),(11,109,378,265),(12,110,365,266),(13,111,366,253),(14,112,367,254),(15,412,239,398),(16,413,240,399),(17,414,241,400),(18,415,242,401),(19,416,243,402),(20,417,244,403),(21,418,245,404),(22,419,246,405),(23,420,247,406),(24,407,248,393),(25,408,249,394),(26,409,250,395),(27,410,251,396),(28,411,252,397),(29,120,334,159),(30,121,335,160),(31,122,336,161),(32,123,323,162),(33,124,324,163),(34,125,325,164),(35,126,326,165),(36,113,327,166),(37,114,328,167),(38,115,329,168),(39,116,330,155),(40,117,331,156),(41,118,332,157),(42,119,333,158),(43,311,170,382),(44,312,171,383),(45,313,172,384),(46,314,173,385),(47,315,174,386),(48,316,175,387),(49,317,176,388),(50,318,177,389),(51,319,178,390),(52,320,179,391),(53,321,180,392),(54,322,181,379),(55,309,182,380),(56,310,169,381),(57,278,150,307),(58,279,151,308),(59,280,152,295),(60,267,153,296),(61,268,154,297),(62,269,141,298),(63,270,142,299),(64,271,143,300),(65,272,144,301),(66,273,145,302),(67,274,146,303),(68,275,147,304),(69,276,148,305),(70,277,149,306),(71,200,216,232),(72,201,217,233),(73,202,218,234),(74,203,219,235),(75,204,220,236),(76,205,221,237),(77,206,222,238),(78,207,223,225),(79,208,224,226),(80,209,211,227),(81,210,212,228),(82,197,213,229),(83,198,214,230),(84,199,215,231),(85,286,359,446),(86,287,360,447),(87,288,361,448),(88,289,362,435),(89,290,363,436),(90,291,364,437),(91,292,351,438),(92,293,352,439),(93,294,353,440),(94,281,354,441),(95,282,355,442),(96,283,356,443),(97,284,357,444),(98,285,358,445),(127,339,424,191),(128,340,425,192),(129,341,426,193),(130,342,427,194),(131,343,428,195),(132,344,429,196),(133,345,430,183),(134,346,431,184),(135,347,432,185),(136,348,433,186),(137,349,434,187),(138,350,421,188),(139,337,422,189),(140,338,423,190)]])
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 | Q8 | D7 | Q16 | C4○D4 | D14 | D14 | D14 | Dic14 | C8⋊C22 | D4⋊2D7 | D4×D7 | D56⋊C2 | D7×Q16 |
| kernel | Dic7.Q16 | C28.Q8 | Dic7⋊C8 | C56⋊1C4 | Q8⋊Dic7 | C7×Q8⋊C4 | C28⋊Q8 | Q8×Dic7 | C2×Dic7 | C7×Q8 | Q8⋊C4 | Dic7 | C28 | C4⋊C4 | 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 | 3 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Dic7.Q16 ►in GL6(𝔽113)
| 112 | 1 | 0 | 0 | 0 | 0 |
| 32 | 80 | 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 |
| 9 | 25 | 0 | 0 | 0 | 0 |
| 42 | 104 | 0 | 0 | 0 | 0 |
| 0 | 0 | 74 | 88 | 0 | 0 |
| 0 | 0 | 88 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 93 | 0 | 0 |
| 0 | 0 | 93 | 77 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 82 |
| 0 | 0 | 0 | 0 | 31 | 31 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 95 |
| 0 | 0 | 0 | 0 | 95 | 63 |
G:=sub<GL(6,GF(113))| [112,32,0,0,0,0,1,80,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,42,0,0,0,0,25,104,0,0,0,0,0,0,74,88,0,0,0,0,88,39,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,93,0,0,0,0,93,77,0,0,0,0,0,0,31,31,0,0,0,0,82,31],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,50,95,0,0,0,0,95,63] >;
Dic7.Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_7.Q_{16} % in TeX
G:=Group("Dic7.Q16"); // GroupNames label
G:=SmallGroup(448,328);
// by ID
G=gap.SmallGroup(448,328);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,254,219,226,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^8=1,b^2=a^7,d^2=a^7*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=a^7*c^-1>;
// generators/relations