metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊3C8, C42.197D14, C7⋊2(C8×Q8), C7⋊C8⋊12Q8, C4.4(C8×D7), C4⋊C8.13D7, C4.52(Q8×D7), C28.10(C2×C8), C14.17(C4×Q8), (C2×C8).213D14, Dic7⋊C8.8C2, C4⋊Dic7.16C4, C14.8(C22×C8), C28.110(C2×Q8), Dic7.2(C2×C8), C14.26(C8○D4), Dic7⋊C4.11C4, C2.3(D28.C4), (C4×C28).56C22, (C4×Dic14).9C2, (C8×Dic7).14C2, C28.302(C4○D4), (C2×C56).206C22, (C2×C28).827C23, (C2×Dic14).12C4, C4.128(D4⋊2D7), C2.2(Dic7⋊3Q8), (C4×Dic7).274C22, (C4×C7⋊C8).5C2, C2.10(D7×C2×C8), (C7×C4⋊C8).17C2, (C2×C4).70(C4×D7), C22.45(C2×C4×D7), (C2×C28).156(C2×C4), (C2×C7⋊C8).330C22, (C2×C14).82(C22×C4), (C2×Dic7).52(C2×C4), (C2×C4).769(C22×D7), SmallGroup(448,364)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊C8
G = < a,b,c | a28=c8=1, b2=a14, bab-1=a-1, cac-1=a15, bc=cb >
Subgroups: 324 in 102 conjugacy classes, 61 normal (31 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, Dic7, C28, C28, C28, C2×C14, C4×C8, C4⋊C8, C4⋊C8, C4×Q8, C7⋊C8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C8×Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C4×C7⋊C8, C8×Dic7, Dic7⋊C8, C7×C4⋊C8, C4×Dic14, Dic14⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, D7, C2×C8, C22×C4, C2×Q8, C4○D4, D14, C4×Q8, C22×C8, C8○D4, C4×D7, C22×D7, C8×Q8, C8×D7, C2×C4×D7, D4⋊2D7, Q8×D7, Dic7⋊3Q8, D7×C2×C8, D28.C4, Dic14⋊C8
►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 72 15 58)(2 71 16 57)(3 70 17 84)(4 69 18 83)(5 68 19 82)(6 67 20 81)(7 66 21 80)(8 65 22 79)(9 64 23 78)(10 63 24 77)(11 62 25 76)(12 61 26 75)(13 60 27 74)(14 59 28 73)(29 183 43 169)(30 182 44 196)(31 181 45 195)(32 180 46 194)(33 179 47 193)(34 178 48 192)(35 177 49 191)(36 176 50 190)(37 175 51 189)(38 174 52 188)(39 173 53 187)(40 172 54 186)(41 171 55 185)(42 170 56 184)(85 401 99 415)(86 400 100 414)(87 399 101 413)(88 398 102 412)(89 397 103 411)(90 396 104 410)(91 395 105 409)(92 394 106 408)(93 393 107 407)(94 420 108 406)(95 419 109 405)(96 418 110 404)(97 417 111 403)(98 416 112 402)(113 143 127 157)(114 142 128 156)(115 141 129 155)(116 168 130 154)(117 167 131 153)(118 166 132 152)(119 165 133 151)(120 164 134 150)(121 163 135 149)(122 162 136 148)(123 161 137 147)(124 160 138 146)(125 159 139 145)(126 158 140 144)(197 424 211 438)(198 423 212 437)(199 422 213 436)(200 421 214 435)(201 448 215 434)(202 447 216 433)(203 446 217 432)(204 445 218 431)(205 444 219 430)(206 443 220 429)(207 442 221 428)(208 441 222 427)(209 440 223 426)(210 439 224 425)(225 323 239 309)(226 322 240 336)(227 321 241 335)(228 320 242 334)(229 319 243 333)(230 318 244 332)(231 317 245 331)(232 316 246 330)(233 315 247 329)(234 314 248 328)(235 313 249 327)(236 312 250 326)(237 311 251 325)(238 310 252 324)(253 308 267 294)(254 307 268 293)(255 306 269 292)(256 305 270 291)(257 304 271 290)(258 303 272 289)(259 302 273 288)(260 301 274 287)(261 300 275 286)(262 299 276 285)(263 298 277 284)(264 297 278 283)(265 296 279 282)(266 295 280 281)(337 367 351 381)(338 366 352 380)(339 365 353 379)(340 392 354 378)(341 391 355 377)(342 390 356 376)(343 389 357 375)(344 388 358 374)(345 387 359 373)(346 386 360 372)(347 385 361 371)(348 384 362 370)(349 383 363 369)(350 382 364 368)
(1 309 219 114 109 392 296 182)(2 324 220 129 110 379 297 169)(3 311 221 116 111 366 298 184)(4 326 222 131 112 381 299 171)(5 313 223 118 85 368 300 186)(6 328 224 133 86 383 301 173)(7 315 197 120 87 370 302 188)(8 330 198 135 88 385 303 175)(9 317 199 122 89 372 304 190)(10 332 200 137 90 387 305 177)(11 319 201 124 91 374 306 192)(12 334 202 139 92 389 307 179)(13 321 203 126 93 376 308 194)(14 336 204 113 94 391 281 181)(15 323 205 128 95 378 282 196)(16 310 206 115 96 365 283 183)(17 325 207 130 97 380 284 170)(18 312 208 117 98 367 285 185)(19 327 209 132 99 382 286 172)(20 314 210 119 100 369 287 187)(21 329 211 134 101 384 288 174)(22 316 212 121 102 371 289 189)(23 331 213 136 103 386 290 176)(24 318 214 123 104 373 291 191)(25 333 215 138 105 388 292 178)(26 320 216 125 106 375 293 193)(27 335 217 140 107 390 294 180)(28 322 218 127 108 377 295 195)(29 71 238 429 155 404 339 278)(30 58 239 444 156 419 340 265)(31 73 240 431 157 406 341 280)(32 60 241 446 158 393 342 267)(33 75 242 433 159 408 343 254)(34 62 243 448 160 395 344 269)(35 77 244 435 161 410 345 256)(36 64 245 422 162 397 346 271)(37 79 246 437 163 412 347 258)(38 66 247 424 164 399 348 273)(39 81 248 439 165 414 349 260)(40 68 249 426 166 401 350 275)(41 83 250 441 167 416 351 262)(42 70 251 428 168 403 352 277)(43 57 252 443 141 418 353 264)(44 72 225 430 142 405 354 279)(45 59 226 445 143 420 355 266)(46 74 227 432 144 407 356 253)(47 61 228 447 145 394 357 268)(48 76 229 434 146 409 358 255)(49 63 230 421 147 396 359 270)(50 78 231 436 148 411 360 257)(51 65 232 423 149 398 361 272)(52 80 233 438 150 413 362 259)(53 67 234 425 151 400 363 274)(54 82 235 440 152 415 364 261)(55 69 236 427 153 402 337 276)(56 84 237 442 154 417 338 263)
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), (1,72,15,58)(2,71,16,57)(3,70,17,84)(4,69,18,83)(5,68,19,82)(6,67,20,81)(7,66,21,80)(8,65,22,79)(9,64,23,78)(10,63,24,77)(11,62,25,76)(12,61,26,75)(13,60,27,74)(14,59,28,73)(29,183,43,169)(30,182,44,196)(31,181,45,195)(32,180,46,194)(33,179,47,193)(34,178,48,192)(35,177,49,191)(36,176,50,190)(37,175,51,189)(38,174,52,188)(39,173,53,187)(40,172,54,186)(41,171,55,185)(42,170,56,184)(85,401,99,415)(86,400,100,414)(87,399,101,413)(88,398,102,412)(89,397,103,411)(90,396,104,410)(91,395,105,409)(92,394,106,408)(93,393,107,407)(94,420,108,406)(95,419,109,405)(96,418,110,404)(97,417,111,403)(98,416,112,402)(113,143,127,157)(114,142,128,156)(115,141,129,155)(116,168,130,154)(117,167,131,153)(118,166,132,152)(119,165,133,151)(120,164,134,150)(121,163,135,149)(122,162,136,148)(123,161,137,147)(124,160,138,146)(125,159,139,145)(126,158,140,144)(197,424,211,438)(198,423,212,437)(199,422,213,436)(200,421,214,435)(201,448,215,434)(202,447,216,433)(203,446,217,432)(204,445,218,431)(205,444,219,430)(206,443,220,429)(207,442,221,428)(208,441,222,427)(209,440,223,426)(210,439,224,425)(225,323,239,309)(226,322,240,336)(227,321,241,335)(228,320,242,334)(229,319,243,333)(230,318,244,332)(231,317,245,331)(232,316,246,330)(233,315,247,329)(234,314,248,328)(235,313,249,327)(236,312,250,326)(237,311,251,325)(238,310,252,324)(253,308,267,294)(254,307,268,293)(255,306,269,292)(256,305,270,291)(257,304,271,290)(258,303,272,289)(259,302,273,288)(260,301,274,287)(261,300,275,286)(262,299,276,285)(263,298,277,284)(264,297,278,283)(265,296,279,282)(266,295,280,281)(337,367,351,381)(338,366,352,380)(339,365,353,379)(340,392,354,378)(341,391,355,377)(342,390,356,376)(343,389,357,375)(344,388,358,374)(345,387,359,373)(346,386,360,372)(347,385,361,371)(348,384,362,370)(349,383,363,369)(350,382,364,368), (1,309,219,114,109,392,296,182)(2,324,220,129,110,379,297,169)(3,311,221,116,111,366,298,184)(4,326,222,131,112,381,299,171)(5,313,223,118,85,368,300,186)(6,328,224,133,86,383,301,173)(7,315,197,120,87,370,302,188)(8,330,198,135,88,385,303,175)(9,317,199,122,89,372,304,190)(10,332,200,137,90,387,305,177)(11,319,201,124,91,374,306,192)(12,334,202,139,92,389,307,179)(13,321,203,126,93,376,308,194)(14,336,204,113,94,391,281,181)(15,323,205,128,95,378,282,196)(16,310,206,115,96,365,283,183)(17,325,207,130,97,380,284,170)(18,312,208,117,98,367,285,185)(19,327,209,132,99,382,286,172)(20,314,210,119,100,369,287,187)(21,329,211,134,101,384,288,174)(22,316,212,121,102,371,289,189)(23,331,213,136,103,386,290,176)(24,318,214,123,104,373,291,191)(25,333,215,138,105,388,292,178)(26,320,216,125,106,375,293,193)(27,335,217,140,107,390,294,180)(28,322,218,127,108,377,295,195)(29,71,238,429,155,404,339,278)(30,58,239,444,156,419,340,265)(31,73,240,431,157,406,341,280)(32,60,241,446,158,393,342,267)(33,75,242,433,159,408,343,254)(34,62,243,448,160,395,344,269)(35,77,244,435,161,410,345,256)(36,64,245,422,162,397,346,271)(37,79,246,437,163,412,347,258)(38,66,247,424,164,399,348,273)(39,81,248,439,165,414,349,260)(40,68,249,426,166,401,350,275)(41,83,250,441,167,416,351,262)(42,70,251,428,168,403,352,277)(43,57,252,443,141,418,353,264)(44,72,225,430,142,405,354,279)(45,59,226,445,143,420,355,266)(46,74,227,432,144,407,356,253)(47,61,228,447,145,394,357,268)(48,76,229,434,146,409,358,255)(49,63,230,421,147,396,359,270)(50,78,231,436,148,411,360,257)(51,65,232,423,149,398,361,272)(52,80,233,438,150,413,362,259)(53,67,234,425,151,400,363,274)(54,82,235,440,152,415,364,261)(55,69,236,427,153,402,337,276)(56,84,237,442,154,417,338,263)>;
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), (1,72,15,58)(2,71,16,57)(3,70,17,84)(4,69,18,83)(5,68,19,82)(6,67,20,81)(7,66,21,80)(8,65,22,79)(9,64,23,78)(10,63,24,77)(11,62,25,76)(12,61,26,75)(13,60,27,74)(14,59,28,73)(29,183,43,169)(30,182,44,196)(31,181,45,195)(32,180,46,194)(33,179,47,193)(34,178,48,192)(35,177,49,191)(36,176,50,190)(37,175,51,189)(38,174,52,188)(39,173,53,187)(40,172,54,186)(41,171,55,185)(42,170,56,184)(85,401,99,415)(86,400,100,414)(87,399,101,413)(88,398,102,412)(89,397,103,411)(90,396,104,410)(91,395,105,409)(92,394,106,408)(93,393,107,407)(94,420,108,406)(95,419,109,405)(96,418,110,404)(97,417,111,403)(98,416,112,402)(113,143,127,157)(114,142,128,156)(115,141,129,155)(116,168,130,154)(117,167,131,153)(118,166,132,152)(119,165,133,151)(120,164,134,150)(121,163,135,149)(122,162,136,148)(123,161,137,147)(124,160,138,146)(125,159,139,145)(126,158,140,144)(197,424,211,438)(198,423,212,437)(199,422,213,436)(200,421,214,435)(201,448,215,434)(202,447,216,433)(203,446,217,432)(204,445,218,431)(205,444,219,430)(206,443,220,429)(207,442,221,428)(208,441,222,427)(209,440,223,426)(210,439,224,425)(225,323,239,309)(226,322,240,336)(227,321,241,335)(228,320,242,334)(229,319,243,333)(230,318,244,332)(231,317,245,331)(232,316,246,330)(233,315,247,329)(234,314,248,328)(235,313,249,327)(236,312,250,326)(237,311,251,325)(238,310,252,324)(253,308,267,294)(254,307,268,293)(255,306,269,292)(256,305,270,291)(257,304,271,290)(258,303,272,289)(259,302,273,288)(260,301,274,287)(261,300,275,286)(262,299,276,285)(263,298,277,284)(264,297,278,283)(265,296,279,282)(266,295,280,281)(337,367,351,381)(338,366,352,380)(339,365,353,379)(340,392,354,378)(341,391,355,377)(342,390,356,376)(343,389,357,375)(344,388,358,374)(345,387,359,373)(346,386,360,372)(347,385,361,371)(348,384,362,370)(349,383,363,369)(350,382,364,368), (1,309,219,114,109,392,296,182)(2,324,220,129,110,379,297,169)(3,311,221,116,111,366,298,184)(4,326,222,131,112,381,299,171)(5,313,223,118,85,368,300,186)(6,328,224,133,86,383,301,173)(7,315,197,120,87,370,302,188)(8,330,198,135,88,385,303,175)(9,317,199,122,89,372,304,190)(10,332,200,137,90,387,305,177)(11,319,201,124,91,374,306,192)(12,334,202,139,92,389,307,179)(13,321,203,126,93,376,308,194)(14,336,204,113,94,391,281,181)(15,323,205,128,95,378,282,196)(16,310,206,115,96,365,283,183)(17,325,207,130,97,380,284,170)(18,312,208,117,98,367,285,185)(19,327,209,132,99,382,286,172)(20,314,210,119,100,369,287,187)(21,329,211,134,101,384,288,174)(22,316,212,121,102,371,289,189)(23,331,213,136,103,386,290,176)(24,318,214,123,104,373,291,191)(25,333,215,138,105,388,292,178)(26,320,216,125,106,375,293,193)(27,335,217,140,107,390,294,180)(28,322,218,127,108,377,295,195)(29,71,238,429,155,404,339,278)(30,58,239,444,156,419,340,265)(31,73,240,431,157,406,341,280)(32,60,241,446,158,393,342,267)(33,75,242,433,159,408,343,254)(34,62,243,448,160,395,344,269)(35,77,244,435,161,410,345,256)(36,64,245,422,162,397,346,271)(37,79,246,437,163,412,347,258)(38,66,247,424,164,399,348,273)(39,81,248,439,165,414,349,260)(40,68,249,426,166,401,350,275)(41,83,250,441,167,416,351,262)(42,70,251,428,168,403,352,277)(43,57,252,443,141,418,353,264)(44,72,225,430,142,405,354,279)(45,59,226,445,143,420,355,266)(46,74,227,432,144,407,356,253)(47,61,228,447,145,394,357,268)(48,76,229,434,146,409,358,255)(49,63,230,421,147,396,359,270)(50,78,231,436,148,411,360,257)(51,65,232,423,149,398,361,272)(52,80,233,438,150,413,362,259)(53,67,234,425,151,400,363,274)(54,82,235,440,152,415,364,261)(55,69,236,427,153,402,337,276)(56,84,237,442,154,417,338,263) );
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)], [(1,72,15,58),(2,71,16,57),(3,70,17,84),(4,69,18,83),(5,68,19,82),(6,67,20,81),(7,66,21,80),(8,65,22,79),(9,64,23,78),(10,63,24,77),(11,62,25,76),(12,61,26,75),(13,60,27,74),(14,59,28,73),(29,183,43,169),(30,182,44,196),(31,181,45,195),(32,180,46,194),(33,179,47,193),(34,178,48,192),(35,177,49,191),(36,176,50,190),(37,175,51,189),(38,174,52,188),(39,173,53,187),(40,172,54,186),(41,171,55,185),(42,170,56,184),(85,401,99,415),(86,400,100,414),(87,399,101,413),(88,398,102,412),(89,397,103,411),(90,396,104,410),(91,395,105,409),(92,394,106,408),(93,393,107,407),(94,420,108,406),(95,419,109,405),(96,418,110,404),(97,417,111,403),(98,416,112,402),(113,143,127,157),(114,142,128,156),(115,141,129,155),(116,168,130,154),(117,167,131,153),(118,166,132,152),(119,165,133,151),(120,164,134,150),(121,163,135,149),(122,162,136,148),(123,161,137,147),(124,160,138,146),(125,159,139,145),(126,158,140,144),(197,424,211,438),(198,423,212,437),(199,422,213,436),(200,421,214,435),(201,448,215,434),(202,447,216,433),(203,446,217,432),(204,445,218,431),(205,444,219,430),(206,443,220,429),(207,442,221,428),(208,441,222,427),(209,440,223,426),(210,439,224,425),(225,323,239,309),(226,322,240,336),(227,321,241,335),(228,320,242,334),(229,319,243,333),(230,318,244,332),(231,317,245,331),(232,316,246,330),(233,315,247,329),(234,314,248,328),(235,313,249,327),(236,312,250,326),(237,311,251,325),(238,310,252,324),(253,308,267,294),(254,307,268,293),(255,306,269,292),(256,305,270,291),(257,304,271,290),(258,303,272,289),(259,302,273,288),(260,301,274,287),(261,300,275,286),(262,299,276,285),(263,298,277,284),(264,297,278,283),(265,296,279,282),(266,295,280,281),(337,367,351,381),(338,366,352,380),(339,365,353,379),(340,392,354,378),(341,391,355,377),(342,390,356,376),(343,389,357,375),(344,388,358,374),(345,387,359,373),(346,386,360,372),(347,385,361,371),(348,384,362,370),(349,383,363,369),(350,382,364,368)], [(1,309,219,114,109,392,296,182),(2,324,220,129,110,379,297,169),(3,311,221,116,111,366,298,184),(4,326,222,131,112,381,299,171),(5,313,223,118,85,368,300,186),(6,328,224,133,86,383,301,173),(7,315,197,120,87,370,302,188),(8,330,198,135,88,385,303,175),(9,317,199,122,89,372,304,190),(10,332,200,137,90,387,305,177),(11,319,201,124,91,374,306,192),(12,334,202,139,92,389,307,179),(13,321,203,126,93,376,308,194),(14,336,204,113,94,391,281,181),(15,323,205,128,95,378,282,196),(16,310,206,115,96,365,283,183),(17,325,207,130,97,380,284,170),(18,312,208,117,98,367,285,185),(19,327,209,132,99,382,286,172),(20,314,210,119,100,369,287,187),(21,329,211,134,101,384,288,174),(22,316,212,121,102,371,289,189),(23,331,213,136,103,386,290,176),(24,318,214,123,104,373,291,191),(25,333,215,138,105,388,292,178),(26,320,216,125,106,375,293,193),(27,335,217,140,107,390,294,180),(28,322,218,127,108,377,295,195),(29,71,238,429,155,404,339,278),(30,58,239,444,156,419,340,265),(31,73,240,431,157,406,341,280),(32,60,241,446,158,393,342,267),(33,75,242,433,159,408,343,254),(34,62,243,448,160,395,344,269),(35,77,244,435,161,410,345,256),(36,64,245,422,162,397,346,271),(37,79,246,437,163,412,347,258),(38,66,247,424,164,399,348,273),(39,81,248,439,165,414,349,260),(40,68,249,426,166,401,350,275),(41,83,250,441,167,416,351,262),(42,70,251,428,168,403,352,277),(43,57,252,443,141,418,353,264),(44,72,225,430,142,405,354,279),(45,59,226,445,143,420,355,266),(46,74,227,432,144,407,356,253),(47,61,228,447,145,394,357,268),(48,76,229,434,146,409,358,255),(49,63,230,421,147,396,359,270),(50,78,231,436,148,411,360,257),(51,65,232,423,149,398,361,272),(52,80,233,438,150,413,362,259),(53,67,234,425,151,400,363,274),(54,82,235,440,152,415,364,261),(55,69,236,427,153,402,337,276),(56,84,237,442,154,417,338,263)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | Q8 | D7 | C4○D4 | D14 | D14 | C8○D4 | C4×D7 | C8×D7 | D4⋊2D7 | Q8×D7 | D28.C4 |
| kernel | Dic14⋊C8 | C4×C7⋊C8 | C8×Dic7 | Dic7⋊C8 | C7×C4⋊C8 | C4×Dic14 | Dic7⋊C4 | C4⋊Dic7 | C2×Dic14 | Dic14 | C7⋊C8 | C4⋊C8 | C28 | C42 | C2×C8 | C14 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 16 | 2 | 3 | 2 | 3 | 6 | 4 | 12 | 24 | 3 | 3 | 6 |
Matrix representation of Dic14⋊C8 ►in GL4(𝔽113) generated by
| 0 | 112 | 0 | 0 |
| 1 | 104 | 0 | 0 |
| 0 | 0 | 53 | 36 |
| 0 | 0 | 104 | 60 |
| 47 | 60 | 0 | 0 |
| 31 | 66 | 0 | 0 |
| 0 | 0 | 44 | 8 |
| 0 | 0 | 111 | 69 |
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 106 |
| 0 | 0 | 30 | 95 |
G:=sub<GL(4,GF(113))| [0,1,0,0,112,104,0,0,0,0,53,104,0,0,36,60],[47,31,0,0,60,66,0,0,0,0,44,111,0,0,8,69],[18,0,0,0,0,18,0,0,0,0,18,30,0,0,106,95] >;
Dic14⋊C8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes C_8 % in TeX
G:=Group("Dic14:C8"); // GroupNames label
G:=SmallGroup(448,364);
// by ID
G=gap.SmallGroup(448,364);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,64,135,142,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=c^8=1,b^2=a^14,b*a*b^-1=a^-1,c*a*c^-1=a^15,b*c=c*b>;
// generators/relations