metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊1Q8, C8⋊1Dic14, C42.13D14, C7⋊1(C8⋊Q8), (C2×C8).52D14, (C2×C4).23D28, (C2×C28).34D4, C8⋊C4.1D7, C14.5(C4⋊Q8), C28.72(C2×Q8), C8⋊Dic7.2C2, (C4×C28).1C22, C28⋊2Q8.5C2, C56⋊1C4.10C2, C2.6(C8⋊D14), C14.1(C8⋊C22), (C2×C56).53C22, C2.9(C28⋊2Q8), C4.38(C2×Dic14), C22.95(C2×D28), C4⋊Dic7.7C22, C28.6Q8.2C2, (C2×C28).729C23, C2.6(C8.D14), C14.1(C8.C22), (C7×C8⋊C4).1C2, (C2×C14).112(C2×D4), (C2×C4).673(C22×D7), SmallGroup(448,236)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊Dic14
G = < a,b,c | a8=b28=1, c2=b14, bab-1=a5, cac-1=a-1, cbc-1=b-1 >
Subgroups: 452 in 90 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C56, Dic14, C2×Dic7, C2×C28, C8⋊Q8, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C8⋊Dic7, C56⋊1C4, C7×C8⋊C4, C28⋊2Q8, C28.6Q8, C8⋊Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, C8⋊C22, C8.C22, Dic14, D28, C22×D7, C8⋊Q8, C2×Dic14, C2×D28, C28⋊2Q8, C8⋊D14, C8.D14, C8⋊Dic14
►Regular action on 448 points
Generators in S448
(1 364 192 240 149 135 89 294)(2 136 193 295 150 337 90 241)(3 338 194 242 151 137 91 296)(4 138 195 297 152 339 92 243)(5 340 196 244 153 139 93 298)(6 140 169 299 154 341 94 245)(7 342 170 246 155 113 95 300)(8 114 171 301 156 343 96 247)(9 344 172 248 157 115 97 302)(10 116 173 303 158 345 98 249)(11 346 174 250 159 117 99 304)(12 118 175 305 160 347 100 251)(13 348 176 252 161 119 101 306)(14 120 177 307 162 349 102 225)(15 350 178 226 163 121 103 308)(16 122 179 281 164 351 104 227)(17 352 180 228 165 123 105 282)(18 124 181 283 166 353 106 229)(19 354 182 230 167 125 107 284)(20 126 183 285 168 355 108 231)(21 356 184 232 141 127 109 286)(22 128 185 287 142 357 110 233)(23 358 186 234 143 129 111 288)(24 130 187 289 144 359 112 235)(25 360 188 236 145 131 85 290)(26 132 189 291 146 361 86 237)(27 362 190 238 147 133 87 292)(28 134 191 293 148 363 88 239)(29 220 391 58 272 313 443 411)(30 314 392 412 273 221 444 59)(31 222 365 60 274 315 445 413)(32 316 366 414 275 223 446 61)(33 224 367 62 276 317 447 415)(34 318 368 416 277 197 448 63)(35 198 369 64 278 319 421 417)(36 320 370 418 279 199 422 65)(37 200 371 66 280 321 423 419)(38 322 372 420 253 201 424 67)(39 202 373 68 254 323 425 393)(40 324 374 394 255 203 426 69)(41 204 375 70 256 325 427 395)(42 326 376 396 257 205 428 71)(43 206 377 72 258 327 429 397)(44 328 378 398 259 207 430 73)(45 208 379 74 260 329 431 399)(46 330 380 400 261 209 432 75)(47 210 381 76 262 331 433 401)(48 332 382 402 263 211 434 77)(49 212 383 78 264 333 435 403)(50 334 384 404 265 213 436 79)(51 214 385 80 266 335 437 405)(52 336 386 406 267 215 438 81)(53 216 387 82 268 309 439 407)(54 310 388 408 269 217 440 83)(55 218 389 84 270 311 441 409)(56 312 390 410 271 219 442 57)
(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)
(1 255 15 269)(2 254 16 268)(3 253 17 267)(4 280 18 266)(5 279 19 265)(6 278 20 264)(7 277 21 263)(8 276 22 262)(9 275 23 261)(10 274 24 260)(11 273 25 259)(12 272 26 258)(13 271 27 257)(14 270 28 256)(29 146 43 160)(30 145 44 159)(31 144 45 158)(32 143 46 157)(33 142 47 156)(34 141 48 155)(35 168 49 154)(36 167 50 153)(37 166 51 152)(38 165 52 151)(39 164 53 150)(40 163 54 149)(41 162 55 148)(42 161 56 147)(57 133 71 119)(58 132 72 118)(59 131 73 117)(60 130 74 116)(61 129 75 115)(62 128 76 114)(63 127 77 113)(64 126 78 140)(65 125 79 139)(66 124 80 138)(67 123 81 137)(68 122 82 136)(69 121 83 135)(70 120 84 134)(85 430 99 444)(86 429 100 443)(87 428 101 442)(88 427 102 441)(89 426 103 440)(90 425 104 439)(91 424 105 438)(92 423 106 437)(93 422 107 436)(94 421 108 435)(95 448 109 434)(96 447 110 433)(97 446 111 432)(98 445 112 431)(169 369 183 383)(170 368 184 382)(171 367 185 381)(172 366 186 380)(173 365 187 379)(174 392 188 378)(175 391 189 377)(176 390 190 376)(177 389 191 375)(178 388 192 374)(179 387 193 373)(180 386 194 372)(181 385 195 371)(182 384 196 370)(197 286 211 300)(198 285 212 299)(199 284 213 298)(200 283 214 297)(201 282 215 296)(202 281 216 295)(203 308 217 294)(204 307 218 293)(205 306 219 292)(206 305 220 291)(207 304 221 290)(208 303 222 289)(209 302 223 288)(210 301 224 287)(225 311 239 325)(226 310 240 324)(227 309 241 323)(228 336 242 322)(229 335 243 321)(230 334 244 320)(231 333 245 319)(232 332 246 318)(233 331 247 317)(234 330 248 316)(235 329 249 315)(236 328 250 314)(237 327 251 313)(238 326 252 312)(337 393 351 407)(338 420 352 406)(339 419 353 405)(340 418 354 404)(341 417 355 403)(342 416 356 402)(343 415 357 401)(344 414 358 400)(345 413 359 399)(346 412 360 398)(347 411 361 397)(348 410 362 396)(349 409 363 395)(350 408 364 394)
G:=sub<Sym(448)| (1,364,192,240,149,135,89,294)(2,136,193,295,150,337,90,241)(3,338,194,242,151,137,91,296)(4,138,195,297,152,339,92,243)(5,340,196,244,153,139,93,298)(6,140,169,299,154,341,94,245)(7,342,170,246,155,113,95,300)(8,114,171,301,156,343,96,247)(9,344,172,248,157,115,97,302)(10,116,173,303,158,345,98,249)(11,346,174,250,159,117,99,304)(12,118,175,305,160,347,100,251)(13,348,176,252,161,119,101,306)(14,120,177,307,162,349,102,225)(15,350,178,226,163,121,103,308)(16,122,179,281,164,351,104,227)(17,352,180,228,165,123,105,282)(18,124,181,283,166,353,106,229)(19,354,182,230,167,125,107,284)(20,126,183,285,168,355,108,231)(21,356,184,232,141,127,109,286)(22,128,185,287,142,357,110,233)(23,358,186,234,143,129,111,288)(24,130,187,289,144,359,112,235)(25,360,188,236,145,131,85,290)(26,132,189,291,146,361,86,237)(27,362,190,238,147,133,87,292)(28,134,191,293,148,363,88,239)(29,220,391,58,272,313,443,411)(30,314,392,412,273,221,444,59)(31,222,365,60,274,315,445,413)(32,316,366,414,275,223,446,61)(33,224,367,62,276,317,447,415)(34,318,368,416,277,197,448,63)(35,198,369,64,278,319,421,417)(36,320,370,418,279,199,422,65)(37,200,371,66,280,321,423,419)(38,322,372,420,253,201,424,67)(39,202,373,68,254,323,425,393)(40,324,374,394,255,203,426,69)(41,204,375,70,256,325,427,395)(42,326,376,396,257,205,428,71)(43,206,377,72,258,327,429,397)(44,328,378,398,259,207,430,73)(45,208,379,74,260,329,431,399)(46,330,380,400,261,209,432,75)(47,210,381,76,262,331,433,401)(48,332,382,402,263,211,434,77)(49,212,383,78,264,333,435,403)(50,334,384,404,265,213,436,79)(51,214,385,80,266,335,437,405)(52,336,386,406,267,215,438,81)(53,216,387,82,268,309,439,407)(54,310,388,408,269,217,440,83)(55,218,389,84,270,311,441,409)(56,312,390,410,271,219,442,57), (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), (1,255,15,269)(2,254,16,268)(3,253,17,267)(4,280,18,266)(5,279,19,265)(6,278,20,264)(7,277,21,263)(8,276,22,262)(9,275,23,261)(10,274,24,260)(11,273,25,259)(12,272,26,258)(13,271,27,257)(14,270,28,256)(29,146,43,160)(30,145,44,159)(31,144,45,158)(32,143,46,157)(33,142,47,156)(34,141,48,155)(35,168,49,154)(36,167,50,153)(37,166,51,152)(38,165,52,151)(39,164,53,150)(40,163,54,149)(41,162,55,148)(42,161,56,147)(57,133,71,119)(58,132,72,118)(59,131,73,117)(60,130,74,116)(61,129,75,115)(62,128,76,114)(63,127,77,113)(64,126,78,140)(65,125,79,139)(66,124,80,138)(67,123,81,137)(68,122,82,136)(69,121,83,135)(70,120,84,134)(85,430,99,444)(86,429,100,443)(87,428,101,442)(88,427,102,441)(89,426,103,440)(90,425,104,439)(91,424,105,438)(92,423,106,437)(93,422,107,436)(94,421,108,435)(95,448,109,434)(96,447,110,433)(97,446,111,432)(98,445,112,431)(169,369,183,383)(170,368,184,382)(171,367,185,381)(172,366,186,380)(173,365,187,379)(174,392,188,378)(175,391,189,377)(176,390,190,376)(177,389,191,375)(178,388,192,374)(179,387,193,373)(180,386,194,372)(181,385,195,371)(182,384,196,370)(197,286,211,300)(198,285,212,299)(199,284,213,298)(200,283,214,297)(201,282,215,296)(202,281,216,295)(203,308,217,294)(204,307,218,293)(205,306,219,292)(206,305,220,291)(207,304,221,290)(208,303,222,289)(209,302,223,288)(210,301,224,287)(225,311,239,325)(226,310,240,324)(227,309,241,323)(228,336,242,322)(229,335,243,321)(230,334,244,320)(231,333,245,319)(232,332,246,318)(233,331,247,317)(234,330,248,316)(235,329,249,315)(236,328,250,314)(237,327,251,313)(238,326,252,312)(337,393,351,407)(338,420,352,406)(339,419,353,405)(340,418,354,404)(341,417,355,403)(342,416,356,402)(343,415,357,401)(344,414,358,400)(345,413,359,399)(346,412,360,398)(347,411,361,397)(348,410,362,396)(349,409,363,395)(350,408,364,394)>;
G:=Group( (1,364,192,240,149,135,89,294)(2,136,193,295,150,337,90,241)(3,338,194,242,151,137,91,296)(4,138,195,297,152,339,92,243)(5,340,196,244,153,139,93,298)(6,140,169,299,154,341,94,245)(7,342,170,246,155,113,95,300)(8,114,171,301,156,343,96,247)(9,344,172,248,157,115,97,302)(10,116,173,303,158,345,98,249)(11,346,174,250,159,117,99,304)(12,118,175,305,160,347,100,251)(13,348,176,252,161,119,101,306)(14,120,177,307,162,349,102,225)(15,350,178,226,163,121,103,308)(16,122,179,281,164,351,104,227)(17,352,180,228,165,123,105,282)(18,124,181,283,166,353,106,229)(19,354,182,230,167,125,107,284)(20,126,183,285,168,355,108,231)(21,356,184,232,141,127,109,286)(22,128,185,287,142,357,110,233)(23,358,186,234,143,129,111,288)(24,130,187,289,144,359,112,235)(25,360,188,236,145,131,85,290)(26,132,189,291,146,361,86,237)(27,362,190,238,147,133,87,292)(28,134,191,293,148,363,88,239)(29,220,391,58,272,313,443,411)(30,314,392,412,273,221,444,59)(31,222,365,60,274,315,445,413)(32,316,366,414,275,223,446,61)(33,224,367,62,276,317,447,415)(34,318,368,416,277,197,448,63)(35,198,369,64,278,319,421,417)(36,320,370,418,279,199,422,65)(37,200,371,66,280,321,423,419)(38,322,372,420,253,201,424,67)(39,202,373,68,254,323,425,393)(40,324,374,394,255,203,426,69)(41,204,375,70,256,325,427,395)(42,326,376,396,257,205,428,71)(43,206,377,72,258,327,429,397)(44,328,378,398,259,207,430,73)(45,208,379,74,260,329,431,399)(46,330,380,400,261,209,432,75)(47,210,381,76,262,331,433,401)(48,332,382,402,263,211,434,77)(49,212,383,78,264,333,435,403)(50,334,384,404,265,213,436,79)(51,214,385,80,266,335,437,405)(52,336,386,406,267,215,438,81)(53,216,387,82,268,309,439,407)(54,310,388,408,269,217,440,83)(55,218,389,84,270,311,441,409)(56,312,390,410,271,219,442,57), (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), (1,255,15,269)(2,254,16,268)(3,253,17,267)(4,280,18,266)(5,279,19,265)(6,278,20,264)(7,277,21,263)(8,276,22,262)(9,275,23,261)(10,274,24,260)(11,273,25,259)(12,272,26,258)(13,271,27,257)(14,270,28,256)(29,146,43,160)(30,145,44,159)(31,144,45,158)(32,143,46,157)(33,142,47,156)(34,141,48,155)(35,168,49,154)(36,167,50,153)(37,166,51,152)(38,165,52,151)(39,164,53,150)(40,163,54,149)(41,162,55,148)(42,161,56,147)(57,133,71,119)(58,132,72,118)(59,131,73,117)(60,130,74,116)(61,129,75,115)(62,128,76,114)(63,127,77,113)(64,126,78,140)(65,125,79,139)(66,124,80,138)(67,123,81,137)(68,122,82,136)(69,121,83,135)(70,120,84,134)(85,430,99,444)(86,429,100,443)(87,428,101,442)(88,427,102,441)(89,426,103,440)(90,425,104,439)(91,424,105,438)(92,423,106,437)(93,422,107,436)(94,421,108,435)(95,448,109,434)(96,447,110,433)(97,446,111,432)(98,445,112,431)(169,369,183,383)(170,368,184,382)(171,367,185,381)(172,366,186,380)(173,365,187,379)(174,392,188,378)(175,391,189,377)(176,390,190,376)(177,389,191,375)(178,388,192,374)(179,387,193,373)(180,386,194,372)(181,385,195,371)(182,384,196,370)(197,286,211,300)(198,285,212,299)(199,284,213,298)(200,283,214,297)(201,282,215,296)(202,281,216,295)(203,308,217,294)(204,307,218,293)(205,306,219,292)(206,305,220,291)(207,304,221,290)(208,303,222,289)(209,302,223,288)(210,301,224,287)(225,311,239,325)(226,310,240,324)(227,309,241,323)(228,336,242,322)(229,335,243,321)(230,334,244,320)(231,333,245,319)(232,332,246,318)(233,331,247,317)(234,330,248,316)(235,329,249,315)(236,328,250,314)(237,327,251,313)(238,326,252,312)(337,393,351,407)(338,420,352,406)(339,419,353,405)(340,418,354,404)(341,417,355,403)(342,416,356,402)(343,415,357,401)(344,414,358,400)(345,413,359,399)(346,412,360,398)(347,411,361,397)(348,410,362,396)(349,409,363,395)(350,408,364,394) );
G=PermutationGroup([[(1,364,192,240,149,135,89,294),(2,136,193,295,150,337,90,241),(3,338,194,242,151,137,91,296),(4,138,195,297,152,339,92,243),(5,340,196,244,153,139,93,298),(6,140,169,299,154,341,94,245),(7,342,170,246,155,113,95,300),(8,114,171,301,156,343,96,247),(9,344,172,248,157,115,97,302),(10,116,173,303,158,345,98,249),(11,346,174,250,159,117,99,304),(12,118,175,305,160,347,100,251),(13,348,176,252,161,119,101,306),(14,120,177,307,162,349,102,225),(15,350,178,226,163,121,103,308),(16,122,179,281,164,351,104,227),(17,352,180,228,165,123,105,282),(18,124,181,283,166,353,106,229),(19,354,182,230,167,125,107,284),(20,126,183,285,168,355,108,231),(21,356,184,232,141,127,109,286),(22,128,185,287,142,357,110,233),(23,358,186,234,143,129,111,288),(24,130,187,289,144,359,112,235),(25,360,188,236,145,131,85,290),(26,132,189,291,146,361,86,237),(27,362,190,238,147,133,87,292),(28,134,191,293,148,363,88,239),(29,220,391,58,272,313,443,411),(30,314,392,412,273,221,444,59),(31,222,365,60,274,315,445,413),(32,316,366,414,275,223,446,61),(33,224,367,62,276,317,447,415),(34,318,368,416,277,197,448,63),(35,198,369,64,278,319,421,417),(36,320,370,418,279,199,422,65),(37,200,371,66,280,321,423,419),(38,322,372,420,253,201,424,67),(39,202,373,68,254,323,425,393),(40,324,374,394,255,203,426,69),(41,204,375,70,256,325,427,395),(42,326,376,396,257,205,428,71),(43,206,377,72,258,327,429,397),(44,328,378,398,259,207,430,73),(45,208,379,74,260,329,431,399),(46,330,380,400,261,209,432,75),(47,210,381,76,262,331,433,401),(48,332,382,402,263,211,434,77),(49,212,383,78,264,333,435,403),(50,334,384,404,265,213,436,79),(51,214,385,80,266,335,437,405),(52,336,386,406,267,215,438,81),(53,216,387,82,268,309,439,407),(54,310,388,408,269,217,440,83),(55,218,389,84,270,311,441,409),(56,312,390,410,271,219,442,57)], [(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)], [(1,255,15,269),(2,254,16,268),(3,253,17,267),(4,280,18,266),(5,279,19,265),(6,278,20,264),(7,277,21,263),(8,276,22,262),(9,275,23,261),(10,274,24,260),(11,273,25,259),(12,272,26,258),(13,271,27,257),(14,270,28,256),(29,146,43,160),(30,145,44,159),(31,144,45,158),(32,143,46,157),(33,142,47,156),(34,141,48,155),(35,168,49,154),(36,167,50,153),(37,166,51,152),(38,165,52,151),(39,164,53,150),(40,163,54,149),(41,162,55,148),(42,161,56,147),(57,133,71,119),(58,132,72,118),(59,131,73,117),(60,130,74,116),(61,129,75,115),(62,128,76,114),(63,127,77,113),(64,126,78,140),(65,125,79,139),(66,124,80,138),(67,123,81,137),(68,122,82,136),(69,121,83,135),(70,120,84,134),(85,430,99,444),(86,429,100,443),(87,428,101,442),(88,427,102,441),(89,426,103,440),(90,425,104,439),(91,424,105,438),(92,423,106,437),(93,422,107,436),(94,421,108,435),(95,448,109,434),(96,447,110,433),(97,446,111,432),(98,445,112,431),(169,369,183,383),(170,368,184,382),(171,367,185,381),(172,366,186,380),(173,365,187,379),(174,392,188,378),(175,391,189,377),(176,390,190,376),(177,389,191,375),(178,388,192,374),(179,387,193,373),(180,386,194,372),(181,385,195,371),(182,384,196,370),(197,286,211,300),(198,285,212,299),(199,284,213,298),(200,283,214,297),(201,282,215,296),(202,281,216,295),(203,308,217,294),(204,307,218,293),(205,306,219,292),(206,305,220,291),(207,304,221,290),(208,303,222,289),(209,302,223,288),(210,301,224,287),(225,311,239,325),(226,310,240,324),(227,309,241,323),(228,336,242,322),(229,335,243,321),(230,334,244,320),(231,333,245,319),(232,332,246,318),(233,331,247,317),(234,330,248,316),(235,329,249,315),(236,328,250,314),(237,327,251,313),(238,326,252,312),(337,393,351,407),(338,420,352,406),(339,419,353,405),(340,418,354,404),(341,417,355,403),(342,416,356,402),(343,415,357,401),(344,414,358,400),(345,413,359,399),(346,412,360,398),(347,411,361,397),(348,410,362,396),(349,409,363,395),(350,408,364,394)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 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 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | D14 | D14 | Dic14 | D28 | C8⋊C22 | C8.C22 | C8⋊D14 | C8.D14 |
| kernel | C8⋊Dic14 | C8⋊Dic7 | C56⋊1C4 | C7×C8⋊C4 | C28⋊2Q8 | C28.6Q8 | C56 | C2×C28 | C8⋊C4 | C42 | C2×C8 | C8 | C2×C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 24 | 12 | 1 | 1 | 6 | 6 |
Matrix representation of C8⋊Dic14 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 48 | 52 | 111 |
| 0 | 0 | 65 | 1 | 2 | 34 |
| 0 | 0 | 2 | 61 | 92 | 65 |
| 0 | 0 | 52 | 99 | 48 | 112 |
| 42 | 41 | 0 | 0 | 0 | 0 |
| 92 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 111 | 79 | 34 | 42 |
| 0 | 0 | 34 | 31 | 71 | 73 |
| 0 | 0 | 65 | 1 | 2 | 34 |
| 0 | 0 | 112 | 74 | 79 | 82 |
| 72 | 54 | 0 | 0 | 0 | 0 |
| 40 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 37 | 106 | 93 |
| 0 | 0 | 54 | 86 | 30 | 7 |
| 0 | 0 | 7 | 20 | 79 | 8 |
| 0 | 0 | 83 | 106 | 41 | 34 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,65,2,52,0,0,48,1,61,99,0,0,52,2,92,48,0,0,111,34,65,112],[42,92,0,0,0,0,41,71,0,0,0,0,0,0,111,34,65,112,0,0,79,31,1,74,0,0,34,71,2,79,0,0,42,73,34,82],[72,40,0,0,0,0,54,41,0,0,0,0,0,0,27,54,7,83,0,0,37,86,20,106,0,0,106,30,79,41,0,0,93,7,8,34] >;
C8⋊Dic14 in GAP, Magma, Sage, TeX
C_8\rtimes {\rm Dic}_{14} % in TeX
G:=Group("C8:Dic14"); // GroupNames label
G:=SmallGroup(448,236);
// by ID
G=gap.SmallGroup(448,236);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,254,387,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=1,c^2=b^14,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations