metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊C16, C56.88D4, C56.17Q8, C8.16Dic14, C14.1M5(2), C28.16M4(2), C7⋊2(C4⋊C16), C2.4(D7×C16), (C2×C16).1D7, C14.8(C4⋊C8), C14.4(C2×C16), (C2×C112).1C2, C28.40(C4⋊C4), C22.9(C8×D7), (C2×C8).333D14, C8.48(C7⋊D4), (C2×Dic7).2C8, (C8×Dic7).8C2, C4.14(C8⋊D7), C2.1(C16⋊D7), C2.1(Dic7⋊C8), (C4×Dic7).10C4, C4.27(Dic7⋊C4), (C2×C56).399C22, (C2×C7⋊C8).12C4, (C2×C7⋊C16).9C2, (C2×C14).10(C2×C8), (C2×C4).167(C4×D7), (C2×C28).241(C2×C4), SmallGroup(448,58)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊C16
G = < a,b,c | a14=c16=1, b2=a7, bab-1=a-1, ac=ca, cbc-1=a7b >
Smallest permutation representation of Dic7⋊C16
►Regular action on 448 points
Generators in S448
(1 353 327 219 349 275 417 35 208 101 265 312 300 164)(2 354 328 220 350 276 418 36 193 102 266 313 301 165)(3 355 329 221 351 277 419 37 194 103 267 314 302 166)(4 356 330 222 352 278 420 38 195 104 268 315 303 167)(5 357 331 223 337 279 421 39 196 105 269 316 304 168)(6 358 332 224 338 280 422 40 197 106 270 317 289 169)(7 359 333 209 339 281 423 41 198 107 271 318 290 170)(8 360 334 210 340 282 424 42 199 108 272 319 291 171)(9 361 335 211 341 283 425 43 200 109 257 320 292 172)(10 362 336 212 342 284 426 44 201 110 258 305 293 173)(11 363 321 213 343 285 427 45 202 111 259 306 294 174)(12 364 322 214 344 286 428 46 203 112 260 307 295 175)(13 365 323 215 345 287 429 47 204 97 261 308 296 176)(14 366 324 216 346 288 430 48 205 98 262 309 297 161)(15 367 325 217 347 273 431 33 206 99 263 310 298 162)(16 368 326 218 348 274 432 34 207 100 264 311 299 163)(17 402 252 63 400 66 439 149 186 96 379 226 121 137)(18 403 253 64 385 67 440 150 187 81 380 227 122 138)(19 404 254 49 386 68 441 151 188 82 381 228 123 139)(20 405 255 50 387 69 442 152 189 83 382 229 124 140)(21 406 256 51 388 70 443 153 190 84 383 230 125 141)(22 407 241 52 389 71 444 154 191 85 384 231 126 142)(23 408 242 53 390 72 445 155 192 86 369 232 127 143)(24 409 243 54 391 73 446 156 177 87 370 233 128 144)(25 410 244 55 392 74 447 157 178 88 371 234 113 129)(26 411 245 56 393 75 448 158 179 89 372 235 114 130)(27 412 246 57 394 76 433 159 180 90 373 236 115 131)(28 413 247 58 395 77 434 160 181 91 374 237 116 132)(29 414 248 59 396 78 435 145 182 92 375 238 117 133)(30 415 249 60 397 79 436 146 183 93 376 239 118 134)(31 416 250 61 398 80 437 147 184 94 377 240 119 135)(32 401 251 62 399 65 438 148 185 95 378 225 120 136)
(1 25 35 157)(2 158 36 26)(3 27 37 159)(4 160 38 28)(5 29 39 145)(6 146 40 30)(7 31 41 147)(8 148 42 32)(9 17 43 149)(10 150 44 18)(11 19 45 151)(12 152 46 20)(13 21 47 153)(14 154 48 22)(15 23 33 155)(16 156 34 24)(49 343 381 306)(50 307 382 344)(51 345 383 308)(52 309 384 346)(53 347 369 310)(54 311 370 348)(55 349 371 312)(56 313 372 350)(57 351 373 314)(58 315 374 352)(59 337 375 316)(60 317 376 338)(61 339 377 318)(62 319 378 340)(63 341 379 320)(64 305 380 342)(65 108 120 334)(66 335 121 109)(67 110 122 336)(68 321 123 111)(69 112 124 322)(70 323 125 97)(71 98 126 324)(72 325 127 99)(73 100 128 326)(74 327 113 101)(75 102 114 328)(76 329 115 103)(77 104 116 330)(78 331 117 105)(79 106 118 332)(80 333 119 107)(81 284 253 293)(82 294 254 285)(83 286 255 295)(84 296 256 287)(85 288 241 297)(86 298 242 273)(87 274 243 299)(88 300 244 275)(89 276 245 301)(90 302 246 277)(91 278 247 303)(92 304 248 279)(93 280 249 289)(94 290 250 281)(95 282 251 291)(96 292 252 283)(129 208 447 353)(130 354 448 193)(131 194 433 355)(132 356 434 195)(133 196 435 357)(134 358 436 197)(135 198 437 359)(136 360 438 199)(137 200 439 361)(138 362 440 201)(139 202 441 363)(140 364 442 203)(141 204 443 365)(142 366 444 205)(143 206 445 367)(144 368 446 207)(161 191 430 407)(162 408 431 192)(163 177 432 409)(164 410 417 178)(165 179 418 411)(166 412 419 180)(167 181 420 413)(168 414 421 182)(169 183 422 415)(170 416 423 184)(171 185 424 401)(172 402 425 186)(173 187 426 403)(174 404 427 188)(175 189 428 405)(176 406 429 190)(209 240 271 398)(210 399 272 225)(211 226 257 400)(212 385 258 227)(213 228 259 386)(214 387 260 229)(215 230 261 388)(216 389 262 231)(217 232 263 390)(218 391 264 233)(219 234 265 392)(220 393 266 235)(221 236 267 394)(222 395 268 237)(223 238 269 396)(224 397 270 239)
(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)
G:=sub<Sym(448)| (1,353,327,219,349,275,417,35,208,101,265,312,300,164)(2,354,328,220,350,276,418,36,193,102,266,313,301,165)(3,355,329,221,351,277,419,37,194,103,267,314,302,166)(4,356,330,222,352,278,420,38,195,104,268,315,303,167)(5,357,331,223,337,279,421,39,196,105,269,316,304,168)(6,358,332,224,338,280,422,40,197,106,270,317,289,169)(7,359,333,209,339,281,423,41,198,107,271,318,290,170)(8,360,334,210,340,282,424,42,199,108,272,319,291,171)(9,361,335,211,341,283,425,43,200,109,257,320,292,172)(10,362,336,212,342,284,426,44,201,110,258,305,293,173)(11,363,321,213,343,285,427,45,202,111,259,306,294,174)(12,364,322,214,344,286,428,46,203,112,260,307,295,175)(13,365,323,215,345,287,429,47,204,97,261,308,296,176)(14,366,324,216,346,288,430,48,205,98,262,309,297,161)(15,367,325,217,347,273,431,33,206,99,263,310,298,162)(16,368,326,218,348,274,432,34,207,100,264,311,299,163)(17,402,252,63,400,66,439,149,186,96,379,226,121,137)(18,403,253,64,385,67,440,150,187,81,380,227,122,138)(19,404,254,49,386,68,441,151,188,82,381,228,123,139)(20,405,255,50,387,69,442,152,189,83,382,229,124,140)(21,406,256,51,388,70,443,153,190,84,383,230,125,141)(22,407,241,52,389,71,444,154,191,85,384,231,126,142)(23,408,242,53,390,72,445,155,192,86,369,232,127,143)(24,409,243,54,391,73,446,156,177,87,370,233,128,144)(25,410,244,55,392,74,447,157,178,88,371,234,113,129)(26,411,245,56,393,75,448,158,179,89,372,235,114,130)(27,412,246,57,394,76,433,159,180,90,373,236,115,131)(28,413,247,58,395,77,434,160,181,91,374,237,116,132)(29,414,248,59,396,78,435,145,182,92,375,238,117,133)(30,415,249,60,397,79,436,146,183,93,376,239,118,134)(31,416,250,61,398,80,437,147,184,94,377,240,119,135)(32,401,251,62,399,65,438,148,185,95,378,225,120,136), (1,25,35,157)(2,158,36,26)(3,27,37,159)(4,160,38,28)(5,29,39,145)(6,146,40,30)(7,31,41,147)(8,148,42,32)(9,17,43,149)(10,150,44,18)(11,19,45,151)(12,152,46,20)(13,21,47,153)(14,154,48,22)(15,23,33,155)(16,156,34,24)(49,343,381,306)(50,307,382,344)(51,345,383,308)(52,309,384,346)(53,347,369,310)(54,311,370,348)(55,349,371,312)(56,313,372,350)(57,351,373,314)(58,315,374,352)(59,337,375,316)(60,317,376,338)(61,339,377,318)(62,319,378,340)(63,341,379,320)(64,305,380,342)(65,108,120,334)(66,335,121,109)(67,110,122,336)(68,321,123,111)(69,112,124,322)(70,323,125,97)(71,98,126,324)(72,325,127,99)(73,100,128,326)(74,327,113,101)(75,102,114,328)(76,329,115,103)(77,104,116,330)(78,331,117,105)(79,106,118,332)(80,333,119,107)(81,284,253,293)(82,294,254,285)(83,286,255,295)(84,296,256,287)(85,288,241,297)(86,298,242,273)(87,274,243,299)(88,300,244,275)(89,276,245,301)(90,302,246,277)(91,278,247,303)(92,304,248,279)(93,280,249,289)(94,290,250,281)(95,282,251,291)(96,292,252,283)(129,208,447,353)(130,354,448,193)(131,194,433,355)(132,356,434,195)(133,196,435,357)(134,358,436,197)(135,198,437,359)(136,360,438,199)(137,200,439,361)(138,362,440,201)(139,202,441,363)(140,364,442,203)(141,204,443,365)(142,366,444,205)(143,206,445,367)(144,368,446,207)(161,191,430,407)(162,408,431,192)(163,177,432,409)(164,410,417,178)(165,179,418,411)(166,412,419,180)(167,181,420,413)(168,414,421,182)(169,183,422,415)(170,416,423,184)(171,185,424,401)(172,402,425,186)(173,187,426,403)(174,404,427,188)(175,189,428,405)(176,406,429,190)(209,240,271,398)(210,399,272,225)(211,226,257,400)(212,385,258,227)(213,228,259,386)(214,387,260,229)(215,230,261,388)(216,389,262,231)(217,232,263,390)(218,391,264,233)(219,234,265,392)(220,393,266,235)(221,236,267,394)(222,395,268,237)(223,238,269,396)(224,397,270,239), (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)>;
G:=Group( (1,353,327,219,349,275,417,35,208,101,265,312,300,164)(2,354,328,220,350,276,418,36,193,102,266,313,301,165)(3,355,329,221,351,277,419,37,194,103,267,314,302,166)(4,356,330,222,352,278,420,38,195,104,268,315,303,167)(5,357,331,223,337,279,421,39,196,105,269,316,304,168)(6,358,332,224,338,280,422,40,197,106,270,317,289,169)(7,359,333,209,339,281,423,41,198,107,271,318,290,170)(8,360,334,210,340,282,424,42,199,108,272,319,291,171)(9,361,335,211,341,283,425,43,200,109,257,320,292,172)(10,362,336,212,342,284,426,44,201,110,258,305,293,173)(11,363,321,213,343,285,427,45,202,111,259,306,294,174)(12,364,322,214,344,286,428,46,203,112,260,307,295,175)(13,365,323,215,345,287,429,47,204,97,261,308,296,176)(14,366,324,216,346,288,430,48,205,98,262,309,297,161)(15,367,325,217,347,273,431,33,206,99,263,310,298,162)(16,368,326,218,348,274,432,34,207,100,264,311,299,163)(17,402,252,63,400,66,439,149,186,96,379,226,121,137)(18,403,253,64,385,67,440,150,187,81,380,227,122,138)(19,404,254,49,386,68,441,151,188,82,381,228,123,139)(20,405,255,50,387,69,442,152,189,83,382,229,124,140)(21,406,256,51,388,70,443,153,190,84,383,230,125,141)(22,407,241,52,389,71,444,154,191,85,384,231,126,142)(23,408,242,53,390,72,445,155,192,86,369,232,127,143)(24,409,243,54,391,73,446,156,177,87,370,233,128,144)(25,410,244,55,392,74,447,157,178,88,371,234,113,129)(26,411,245,56,393,75,448,158,179,89,372,235,114,130)(27,412,246,57,394,76,433,159,180,90,373,236,115,131)(28,413,247,58,395,77,434,160,181,91,374,237,116,132)(29,414,248,59,396,78,435,145,182,92,375,238,117,133)(30,415,249,60,397,79,436,146,183,93,376,239,118,134)(31,416,250,61,398,80,437,147,184,94,377,240,119,135)(32,401,251,62,399,65,438,148,185,95,378,225,120,136), (1,25,35,157)(2,158,36,26)(3,27,37,159)(4,160,38,28)(5,29,39,145)(6,146,40,30)(7,31,41,147)(8,148,42,32)(9,17,43,149)(10,150,44,18)(11,19,45,151)(12,152,46,20)(13,21,47,153)(14,154,48,22)(15,23,33,155)(16,156,34,24)(49,343,381,306)(50,307,382,344)(51,345,383,308)(52,309,384,346)(53,347,369,310)(54,311,370,348)(55,349,371,312)(56,313,372,350)(57,351,373,314)(58,315,374,352)(59,337,375,316)(60,317,376,338)(61,339,377,318)(62,319,378,340)(63,341,379,320)(64,305,380,342)(65,108,120,334)(66,335,121,109)(67,110,122,336)(68,321,123,111)(69,112,124,322)(70,323,125,97)(71,98,126,324)(72,325,127,99)(73,100,128,326)(74,327,113,101)(75,102,114,328)(76,329,115,103)(77,104,116,330)(78,331,117,105)(79,106,118,332)(80,333,119,107)(81,284,253,293)(82,294,254,285)(83,286,255,295)(84,296,256,287)(85,288,241,297)(86,298,242,273)(87,274,243,299)(88,300,244,275)(89,276,245,301)(90,302,246,277)(91,278,247,303)(92,304,248,279)(93,280,249,289)(94,290,250,281)(95,282,251,291)(96,292,252,283)(129,208,447,353)(130,354,448,193)(131,194,433,355)(132,356,434,195)(133,196,435,357)(134,358,436,197)(135,198,437,359)(136,360,438,199)(137,200,439,361)(138,362,440,201)(139,202,441,363)(140,364,442,203)(141,204,443,365)(142,366,444,205)(143,206,445,367)(144,368,446,207)(161,191,430,407)(162,408,431,192)(163,177,432,409)(164,410,417,178)(165,179,418,411)(166,412,419,180)(167,181,420,413)(168,414,421,182)(169,183,422,415)(170,416,423,184)(171,185,424,401)(172,402,425,186)(173,187,426,403)(174,404,427,188)(175,189,428,405)(176,406,429,190)(209,240,271,398)(210,399,272,225)(211,226,257,400)(212,385,258,227)(213,228,259,386)(214,387,260,229)(215,230,261,388)(216,389,262,231)(217,232,263,390)(218,391,264,233)(219,234,265,392)(220,393,266,235)(221,236,267,394)(222,395,268,237)(223,238,269,396)(224,397,270,239), (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) );
G=PermutationGroup([[(1,353,327,219,349,275,417,35,208,101,265,312,300,164),(2,354,328,220,350,276,418,36,193,102,266,313,301,165),(3,355,329,221,351,277,419,37,194,103,267,314,302,166),(4,356,330,222,352,278,420,38,195,104,268,315,303,167),(5,357,331,223,337,279,421,39,196,105,269,316,304,168),(6,358,332,224,338,280,422,40,197,106,270,317,289,169),(7,359,333,209,339,281,423,41,198,107,271,318,290,170),(8,360,334,210,340,282,424,42,199,108,272,319,291,171),(9,361,335,211,341,283,425,43,200,109,257,320,292,172),(10,362,336,212,342,284,426,44,201,110,258,305,293,173),(11,363,321,213,343,285,427,45,202,111,259,306,294,174),(12,364,322,214,344,286,428,46,203,112,260,307,295,175),(13,365,323,215,345,287,429,47,204,97,261,308,296,176),(14,366,324,216,346,288,430,48,205,98,262,309,297,161),(15,367,325,217,347,273,431,33,206,99,263,310,298,162),(16,368,326,218,348,274,432,34,207,100,264,311,299,163),(17,402,252,63,400,66,439,149,186,96,379,226,121,137),(18,403,253,64,385,67,440,150,187,81,380,227,122,138),(19,404,254,49,386,68,441,151,188,82,381,228,123,139),(20,405,255,50,387,69,442,152,189,83,382,229,124,140),(21,406,256,51,388,70,443,153,190,84,383,230,125,141),(22,407,241,52,389,71,444,154,191,85,384,231,126,142),(23,408,242,53,390,72,445,155,192,86,369,232,127,143),(24,409,243,54,391,73,446,156,177,87,370,233,128,144),(25,410,244,55,392,74,447,157,178,88,371,234,113,129),(26,411,245,56,393,75,448,158,179,89,372,235,114,130),(27,412,246,57,394,76,433,159,180,90,373,236,115,131),(28,413,247,58,395,77,434,160,181,91,374,237,116,132),(29,414,248,59,396,78,435,145,182,92,375,238,117,133),(30,415,249,60,397,79,436,146,183,93,376,239,118,134),(31,416,250,61,398,80,437,147,184,94,377,240,119,135),(32,401,251,62,399,65,438,148,185,95,378,225,120,136)], [(1,25,35,157),(2,158,36,26),(3,27,37,159),(4,160,38,28),(5,29,39,145),(6,146,40,30),(7,31,41,147),(8,148,42,32),(9,17,43,149),(10,150,44,18),(11,19,45,151),(12,152,46,20),(13,21,47,153),(14,154,48,22),(15,23,33,155),(16,156,34,24),(49,343,381,306),(50,307,382,344),(51,345,383,308),(52,309,384,346),(53,347,369,310),(54,311,370,348),(55,349,371,312),(56,313,372,350),(57,351,373,314),(58,315,374,352),(59,337,375,316),(60,317,376,338),(61,339,377,318),(62,319,378,340),(63,341,379,320),(64,305,380,342),(65,108,120,334),(66,335,121,109),(67,110,122,336),(68,321,123,111),(69,112,124,322),(70,323,125,97),(71,98,126,324),(72,325,127,99),(73,100,128,326),(74,327,113,101),(75,102,114,328),(76,329,115,103),(77,104,116,330),(78,331,117,105),(79,106,118,332),(80,333,119,107),(81,284,253,293),(82,294,254,285),(83,286,255,295),(84,296,256,287),(85,288,241,297),(86,298,242,273),(87,274,243,299),(88,300,244,275),(89,276,245,301),(90,302,246,277),(91,278,247,303),(92,304,248,279),(93,280,249,289),(94,290,250,281),(95,282,251,291),(96,292,252,283),(129,208,447,353),(130,354,448,193),(131,194,433,355),(132,356,434,195),(133,196,435,357),(134,358,436,197),(135,198,437,359),(136,360,438,199),(137,200,439,361),(138,362,440,201),(139,202,441,363),(140,364,442,203),(141,204,443,365),(142,366,444,205),(143,206,445,367),(144,368,446,207),(161,191,430,407),(162,408,431,192),(163,177,432,409),(164,410,417,178),(165,179,418,411),(166,412,419,180),(167,181,420,413),(168,414,421,182),(169,183,422,415),(170,416,423,184),(171,185,424,401),(172,402,425,186),(173,187,426,403),(174,404,427,188),(175,189,428,405),(176,406,429,190),(209,240,271,398),(210,399,272,225),(211,226,257,400),(212,385,258,227),(213,228,259,386),(214,387,260,229),(215,230,261,388),(216,389,262,231),(217,232,263,390),(218,391,264,233),(219,234,265,392),(220,393,266,235),(221,236,267,394),(222,395,268,237),(223,238,269,396),(224,397,270,239)], [(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)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 16A | ··· | 16H | 16I | ··· | 16P | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C16 | D4 | Q8 | D7 | M4(2) | D14 | M5(2) | Dic14 | C7⋊D4 | C4×D7 | C8⋊D7 | C8×D7 | D7×C16 | C16⋊D7 |
| kernel | Dic7⋊C16 | C2×C7⋊C16 | C8×Dic7 | C2×C112 | C2×C7⋊C8 | C4×Dic7 | C2×Dic7 | Dic7 | C56 | C56 | C2×C16 | C28 | C2×C8 | C14 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 16 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 24 | 24 |
Matrix representation of Dic7⋊C16 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 0 | 112 |
| 0 | 1 | 104 |
| 112 | 0 | 0 |
| 0 | 36 | 9 |
| 0 | 107 | 77 |
| 48 | 0 | 0 |
| 0 | 53 | 51 |
| 0 | 62 | 60 |
G:=sub<GL(3,GF(113))| [1,0,0,0,0,1,0,112,104],[112,0,0,0,36,107,0,9,77],[48,0,0,0,53,62,0,51,60] >;
Dic7⋊C16 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes C_{16} % in TeX
G:=Group("Dic7:C16"); // GroupNames label
G:=SmallGroup(448,58);
// by ID
G=gap.SmallGroup(448,58);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,141,36,100,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^14=c^16=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations
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