metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.3Q8, C28.4Q16, C14.6SD32, C8.5Dic14, C7⋊C16⋊6C4, C56.8(C2×C4), C8.25(C4×D7), C7⋊1(C16⋊4C4), C28.3(C4⋊C4), (C2×C28).90D4, (C2×C14).32D8, C2.D8.2D7, (C2×C8).220D14, C2.1(D8.D7), C56⋊1C4.12C2, C14.3(C2.D8), C4.2(C7⋊Q16), C4.3(Dic7⋊C4), (C2×C56).72C22, C2.1(C7⋊SD32), C22.13(D4⋊D7), C2.4(C28.Q8), (C2×C7⋊C16).3C2, (C7×C2.D8).2C2, (C2×C4).114(C7⋊D4), SmallGroup(448,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.5Dic14
G = < a,b,c | a8=b28=1, c2=ab14, bab-1=a-1, ac=ca, cbc-1=a5b-1 >
Smallest permutation representation of C8.5Dic14
►Regular action on 448 points
Generators in S448
(1 152 376 125 301 239 276 339)(2 340 277 240 302 126 377 153)(3 154 378 127 303 241 278 341)(4 342 279 242 304 128 379 155)(5 156 380 129 305 243 280 343)(6 344 253 244 306 130 381 157)(7 158 382 131 307 245 254 345)(8 346 255 246 308 132 383 159)(9 160 384 133 281 247 256 347)(10 348 257 248 282 134 385 161)(11 162 386 135 283 249 258 349)(12 350 259 250 284 136 387 163)(13 164 388 137 285 251 260 351)(14 352 261 252 286 138 389 165)(15 166 390 139 287 225 262 353)(16 354 263 226 288 140 391 167)(17 168 392 113 289 227 264 355)(18 356 265 228 290 114 365 141)(19 142 366 115 291 229 266 357)(20 358 267 230 292 116 367 143)(21 144 368 117 293 231 268 359)(22 360 269 232 294 118 369 145)(23 146 370 119 295 233 270 361)(24 362 271 234 296 120 371 147)(25 148 372 121 297 235 272 363)(26 364 273 236 298 122 373 149)(27 150 374 123 299 237 274 337)(28 338 275 238 300 124 375 151)(29 213 66 415 446 321 106 187)(30 188 107 322 447 416 67 214)(31 215 68 417 448 323 108 189)(32 190 109 324 421 418 69 216)(33 217 70 419 422 325 110 191)(34 192 111 326 423 420 71 218)(35 219 72 393 424 327 112 193)(36 194 85 328 425 394 73 220)(37 221 74 395 426 329 86 195)(38 196 87 330 427 396 75 222)(39 223 76 397 428 331 88 169)(40 170 89 332 429 398 77 224)(41 197 78 399 430 333 90 171)(42 172 91 334 431 400 79 198)(43 199 80 401 432 335 92 173)(44 174 93 336 433 402 81 200)(45 201 82 403 434 309 94 175)(46 176 95 310 435 404 83 202)(47 203 84 405 436 311 96 177)(48 178 97 312 437 406 57 204)(49 205 58 407 438 313 98 179)(50 180 99 314 439 408 59 206)(51 207 60 409 440 315 100 181)(52 182 101 316 441 410 61 208)(53 209 62 411 442 317 102 183)(54 184 103 318 443 412 63 210)(55 211 64 413 444 319 104 185)(56 186 105 320 445 414 65 212)
(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)
(1 444 166 333 376 104 139 171 301 55 225 197 276 64 353 399)(2 184 354 89 277 318 226 429 302 412 140 77 377 210 167 40)(3 442 168 331 378 102 113 169 303 53 227 223 278 62 355 397)(4 182 356 87 279 316 228 427 304 410 114 75 379 208 141 38)(5 440 142 329 380 100 115 195 305 51 229 221 280 60 357 395)(6 180 358 85 253 314 230 425 306 408 116 73 381 206 143 36)(7 438 144 327 382 98 117 193 307 49 231 219 254 58 359 393)(8 178 360 111 255 312 232 423 308 406 118 71 383 204 145 34)(9 436 146 325 384 96 119 191 281 47 233 217 256 84 361 419)(10 176 362 109 257 310 234 421 282 404 120 69 385 202 147 32)(11 434 148 323 386 94 121 189 283 45 235 215 258 82 363 417)(12 174 364 107 259 336 236 447 284 402 122 67 387 200 149 30)(13 432 150 321 388 92 123 187 285 43 237 213 260 80 337 415)(14 172 338 105 261 334 238 445 286 400 124 65 389 198 151 56)(15 430 152 319 390 90 125 185 287 41 239 211 262 78 339 413)(16 170 340 103 263 332 240 443 288 398 126 63 391 224 153 54)(17 428 154 317 392 88 127 183 289 39 241 209 264 76 341 411)(18 196 342 101 265 330 242 441 290 396 128 61 365 222 155 52)(19 426 156 315 366 86 129 181 291 37 243 207 266 74 343 409)(20 194 344 99 267 328 244 439 292 394 130 59 367 220 157 50)(21 424 158 313 368 112 131 179 293 35 245 205 268 72 345 407)(22 192 346 97 269 326 246 437 294 420 132 57 369 218 159 48)(23 422 160 311 370 110 133 177 295 33 247 203 270 70 347 405)(24 190 348 95 271 324 248 435 296 418 134 83 371 216 161 46)(25 448 162 309 372 108 135 175 297 31 249 201 272 68 349 403)(26 188 350 93 273 322 250 433 298 416 136 81 373 214 163 44)(27 446 164 335 374 106 137 173 299 29 251 199 274 66 351 401)(28 186 352 91 275 320 252 431 300 414 138 79 375 212 165 42)
G:=sub<Sym(448)| (1,152,376,125,301,239,276,339)(2,340,277,240,302,126,377,153)(3,154,378,127,303,241,278,341)(4,342,279,242,304,128,379,155)(5,156,380,129,305,243,280,343)(6,344,253,244,306,130,381,157)(7,158,382,131,307,245,254,345)(8,346,255,246,308,132,383,159)(9,160,384,133,281,247,256,347)(10,348,257,248,282,134,385,161)(11,162,386,135,283,249,258,349)(12,350,259,250,284,136,387,163)(13,164,388,137,285,251,260,351)(14,352,261,252,286,138,389,165)(15,166,390,139,287,225,262,353)(16,354,263,226,288,140,391,167)(17,168,392,113,289,227,264,355)(18,356,265,228,290,114,365,141)(19,142,366,115,291,229,266,357)(20,358,267,230,292,116,367,143)(21,144,368,117,293,231,268,359)(22,360,269,232,294,118,369,145)(23,146,370,119,295,233,270,361)(24,362,271,234,296,120,371,147)(25,148,372,121,297,235,272,363)(26,364,273,236,298,122,373,149)(27,150,374,123,299,237,274,337)(28,338,275,238,300,124,375,151)(29,213,66,415,446,321,106,187)(30,188,107,322,447,416,67,214)(31,215,68,417,448,323,108,189)(32,190,109,324,421,418,69,216)(33,217,70,419,422,325,110,191)(34,192,111,326,423,420,71,218)(35,219,72,393,424,327,112,193)(36,194,85,328,425,394,73,220)(37,221,74,395,426,329,86,195)(38,196,87,330,427,396,75,222)(39,223,76,397,428,331,88,169)(40,170,89,332,429,398,77,224)(41,197,78,399,430,333,90,171)(42,172,91,334,431,400,79,198)(43,199,80,401,432,335,92,173)(44,174,93,336,433,402,81,200)(45,201,82,403,434,309,94,175)(46,176,95,310,435,404,83,202)(47,203,84,405,436,311,96,177)(48,178,97,312,437,406,57,204)(49,205,58,407,438,313,98,179)(50,180,99,314,439,408,59,206)(51,207,60,409,440,315,100,181)(52,182,101,316,441,410,61,208)(53,209,62,411,442,317,102,183)(54,184,103,318,443,412,63,210)(55,211,64,413,444,319,104,185)(56,186,105,320,445,414,65,212), (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), (1,444,166,333,376,104,139,171,301,55,225,197,276,64,353,399)(2,184,354,89,277,318,226,429,302,412,140,77,377,210,167,40)(3,442,168,331,378,102,113,169,303,53,227,223,278,62,355,397)(4,182,356,87,279,316,228,427,304,410,114,75,379,208,141,38)(5,440,142,329,380,100,115,195,305,51,229,221,280,60,357,395)(6,180,358,85,253,314,230,425,306,408,116,73,381,206,143,36)(7,438,144,327,382,98,117,193,307,49,231,219,254,58,359,393)(8,178,360,111,255,312,232,423,308,406,118,71,383,204,145,34)(9,436,146,325,384,96,119,191,281,47,233,217,256,84,361,419)(10,176,362,109,257,310,234,421,282,404,120,69,385,202,147,32)(11,434,148,323,386,94,121,189,283,45,235,215,258,82,363,417)(12,174,364,107,259,336,236,447,284,402,122,67,387,200,149,30)(13,432,150,321,388,92,123,187,285,43,237,213,260,80,337,415)(14,172,338,105,261,334,238,445,286,400,124,65,389,198,151,56)(15,430,152,319,390,90,125,185,287,41,239,211,262,78,339,413)(16,170,340,103,263,332,240,443,288,398,126,63,391,224,153,54)(17,428,154,317,392,88,127,183,289,39,241,209,264,76,341,411)(18,196,342,101,265,330,242,441,290,396,128,61,365,222,155,52)(19,426,156,315,366,86,129,181,291,37,243,207,266,74,343,409)(20,194,344,99,267,328,244,439,292,394,130,59,367,220,157,50)(21,424,158,313,368,112,131,179,293,35,245,205,268,72,345,407)(22,192,346,97,269,326,246,437,294,420,132,57,369,218,159,48)(23,422,160,311,370,110,133,177,295,33,247,203,270,70,347,405)(24,190,348,95,271,324,248,435,296,418,134,83,371,216,161,46)(25,448,162,309,372,108,135,175,297,31,249,201,272,68,349,403)(26,188,350,93,273,322,250,433,298,416,136,81,373,214,163,44)(27,446,164,335,374,106,137,173,299,29,251,199,274,66,351,401)(28,186,352,91,275,320,252,431,300,414,138,79,375,212,165,42)>;
G:=Group( (1,152,376,125,301,239,276,339)(2,340,277,240,302,126,377,153)(3,154,378,127,303,241,278,341)(4,342,279,242,304,128,379,155)(5,156,380,129,305,243,280,343)(6,344,253,244,306,130,381,157)(7,158,382,131,307,245,254,345)(8,346,255,246,308,132,383,159)(9,160,384,133,281,247,256,347)(10,348,257,248,282,134,385,161)(11,162,386,135,283,249,258,349)(12,350,259,250,284,136,387,163)(13,164,388,137,285,251,260,351)(14,352,261,252,286,138,389,165)(15,166,390,139,287,225,262,353)(16,354,263,226,288,140,391,167)(17,168,392,113,289,227,264,355)(18,356,265,228,290,114,365,141)(19,142,366,115,291,229,266,357)(20,358,267,230,292,116,367,143)(21,144,368,117,293,231,268,359)(22,360,269,232,294,118,369,145)(23,146,370,119,295,233,270,361)(24,362,271,234,296,120,371,147)(25,148,372,121,297,235,272,363)(26,364,273,236,298,122,373,149)(27,150,374,123,299,237,274,337)(28,338,275,238,300,124,375,151)(29,213,66,415,446,321,106,187)(30,188,107,322,447,416,67,214)(31,215,68,417,448,323,108,189)(32,190,109,324,421,418,69,216)(33,217,70,419,422,325,110,191)(34,192,111,326,423,420,71,218)(35,219,72,393,424,327,112,193)(36,194,85,328,425,394,73,220)(37,221,74,395,426,329,86,195)(38,196,87,330,427,396,75,222)(39,223,76,397,428,331,88,169)(40,170,89,332,429,398,77,224)(41,197,78,399,430,333,90,171)(42,172,91,334,431,400,79,198)(43,199,80,401,432,335,92,173)(44,174,93,336,433,402,81,200)(45,201,82,403,434,309,94,175)(46,176,95,310,435,404,83,202)(47,203,84,405,436,311,96,177)(48,178,97,312,437,406,57,204)(49,205,58,407,438,313,98,179)(50,180,99,314,439,408,59,206)(51,207,60,409,440,315,100,181)(52,182,101,316,441,410,61,208)(53,209,62,411,442,317,102,183)(54,184,103,318,443,412,63,210)(55,211,64,413,444,319,104,185)(56,186,105,320,445,414,65,212), (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), (1,444,166,333,376,104,139,171,301,55,225,197,276,64,353,399)(2,184,354,89,277,318,226,429,302,412,140,77,377,210,167,40)(3,442,168,331,378,102,113,169,303,53,227,223,278,62,355,397)(4,182,356,87,279,316,228,427,304,410,114,75,379,208,141,38)(5,440,142,329,380,100,115,195,305,51,229,221,280,60,357,395)(6,180,358,85,253,314,230,425,306,408,116,73,381,206,143,36)(7,438,144,327,382,98,117,193,307,49,231,219,254,58,359,393)(8,178,360,111,255,312,232,423,308,406,118,71,383,204,145,34)(9,436,146,325,384,96,119,191,281,47,233,217,256,84,361,419)(10,176,362,109,257,310,234,421,282,404,120,69,385,202,147,32)(11,434,148,323,386,94,121,189,283,45,235,215,258,82,363,417)(12,174,364,107,259,336,236,447,284,402,122,67,387,200,149,30)(13,432,150,321,388,92,123,187,285,43,237,213,260,80,337,415)(14,172,338,105,261,334,238,445,286,400,124,65,389,198,151,56)(15,430,152,319,390,90,125,185,287,41,239,211,262,78,339,413)(16,170,340,103,263,332,240,443,288,398,126,63,391,224,153,54)(17,428,154,317,392,88,127,183,289,39,241,209,264,76,341,411)(18,196,342,101,265,330,242,441,290,396,128,61,365,222,155,52)(19,426,156,315,366,86,129,181,291,37,243,207,266,74,343,409)(20,194,344,99,267,328,244,439,292,394,130,59,367,220,157,50)(21,424,158,313,368,112,131,179,293,35,245,205,268,72,345,407)(22,192,346,97,269,326,246,437,294,420,132,57,369,218,159,48)(23,422,160,311,370,110,133,177,295,33,247,203,270,70,347,405)(24,190,348,95,271,324,248,435,296,418,134,83,371,216,161,46)(25,448,162,309,372,108,135,175,297,31,249,201,272,68,349,403)(26,188,350,93,273,322,250,433,298,416,136,81,373,214,163,44)(27,446,164,335,374,106,137,173,299,29,251,199,274,66,351,401)(28,186,352,91,275,320,252,431,300,414,138,79,375,212,165,42) );
G=PermutationGroup([[(1,152,376,125,301,239,276,339),(2,340,277,240,302,126,377,153),(3,154,378,127,303,241,278,341),(4,342,279,242,304,128,379,155),(5,156,380,129,305,243,280,343),(6,344,253,244,306,130,381,157),(7,158,382,131,307,245,254,345),(8,346,255,246,308,132,383,159),(9,160,384,133,281,247,256,347),(10,348,257,248,282,134,385,161),(11,162,386,135,283,249,258,349),(12,350,259,250,284,136,387,163),(13,164,388,137,285,251,260,351),(14,352,261,252,286,138,389,165),(15,166,390,139,287,225,262,353),(16,354,263,226,288,140,391,167),(17,168,392,113,289,227,264,355),(18,356,265,228,290,114,365,141),(19,142,366,115,291,229,266,357),(20,358,267,230,292,116,367,143),(21,144,368,117,293,231,268,359),(22,360,269,232,294,118,369,145),(23,146,370,119,295,233,270,361),(24,362,271,234,296,120,371,147),(25,148,372,121,297,235,272,363),(26,364,273,236,298,122,373,149),(27,150,374,123,299,237,274,337),(28,338,275,238,300,124,375,151),(29,213,66,415,446,321,106,187),(30,188,107,322,447,416,67,214),(31,215,68,417,448,323,108,189),(32,190,109,324,421,418,69,216),(33,217,70,419,422,325,110,191),(34,192,111,326,423,420,71,218),(35,219,72,393,424,327,112,193),(36,194,85,328,425,394,73,220),(37,221,74,395,426,329,86,195),(38,196,87,330,427,396,75,222),(39,223,76,397,428,331,88,169),(40,170,89,332,429,398,77,224),(41,197,78,399,430,333,90,171),(42,172,91,334,431,400,79,198),(43,199,80,401,432,335,92,173),(44,174,93,336,433,402,81,200),(45,201,82,403,434,309,94,175),(46,176,95,310,435,404,83,202),(47,203,84,405,436,311,96,177),(48,178,97,312,437,406,57,204),(49,205,58,407,438,313,98,179),(50,180,99,314,439,408,59,206),(51,207,60,409,440,315,100,181),(52,182,101,316,441,410,61,208),(53,209,62,411,442,317,102,183),(54,184,103,318,443,412,63,210),(55,211,64,413,444,319,104,185),(56,186,105,320,445,414,65,212)], [(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)], [(1,444,166,333,376,104,139,171,301,55,225,197,276,64,353,399),(2,184,354,89,277,318,226,429,302,412,140,77,377,210,167,40),(3,442,168,331,378,102,113,169,303,53,227,223,278,62,355,397),(4,182,356,87,279,316,228,427,304,410,114,75,379,208,141,38),(5,440,142,329,380,100,115,195,305,51,229,221,280,60,357,395),(6,180,358,85,253,314,230,425,306,408,116,73,381,206,143,36),(7,438,144,327,382,98,117,193,307,49,231,219,254,58,359,393),(8,178,360,111,255,312,232,423,308,406,118,71,383,204,145,34),(9,436,146,325,384,96,119,191,281,47,233,217,256,84,361,419),(10,176,362,109,257,310,234,421,282,404,120,69,385,202,147,32),(11,434,148,323,386,94,121,189,283,45,235,215,258,82,363,417),(12,174,364,107,259,336,236,447,284,402,122,67,387,200,149,30),(13,432,150,321,388,92,123,187,285,43,237,213,260,80,337,415),(14,172,338,105,261,334,238,445,286,400,124,65,389,198,151,56),(15,430,152,319,390,90,125,185,287,41,239,211,262,78,339,413),(16,170,340,103,263,332,240,443,288,398,126,63,391,224,153,54),(17,428,154,317,392,88,127,183,289,39,241,209,264,76,341,411),(18,196,342,101,265,330,242,441,290,396,128,61,365,222,155,52),(19,426,156,315,366,86,129,181,291,37,243,207,266,74,343,409),(20,194,344,99,267,328,244,439,292,394,130,59,367,220,157,50),(21,424,158,313,368,112,131,179,293,35,245,205,268,72,345,407),(22,192,346,97,269,326,246,437,294,420,132,57,369,218,159,48),(23,422,160,311,370,110,133,177,295,33,247,203,270,70,347,405),(24,190,348,95,271,324,248,435,296,418,134,83,371,216,161,46),(25,448,162,309,372,108,135,175,297,31,249,201,272,68,349,403),(26,188,350,93,273,322,250,433,298,416,136,81,373,214,163,44),(27,446,164,335,374,106,137,173,299,29,251,199,274,66,351,401),(28,186,352,91,275,320,252,431,300,414,138,79,375,212,165,42)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | + | - | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D7 | Q16 | D8 | D14 | SD32 | Dic14 | C4×D7 | C7⋊D4 | C7⋊Q16 | D4⋊D7 | D8.D7 | C7⋊SD32 |
| kernel | C8.5Dic14 | C2×C7⋊C16 | C56⋊1C4 | C7×C2.D8 | C7⋊C16 | C56 | C2×C28 | C2.D8 | C28 | C2×C14 | C2×C8 | C14 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 8 | 6 | 6 | 6 | 3 | 3 | 6 | 6 |
Matrix representation of C8.5Dic14 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 62 | 4 |
| 0 | 0 | 0 | 28 | 0 |
| 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 49 | 15 |
| 0 | 0 | 0 | 96 | 64 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 76 | 53 | 0 | 0 |
| 0 | 68 | 37 | 0 | 0 |
| 0 | 0 | 0 | 9 | 49 |
| 0 | 0 | 0 | 4 | 97 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,62,28,0,0,0,4,0],[15,0,0,0,0,0,0,1,0,0,0,112,34,0,0,0,0,0,49,96,0,0,0,15,64],[1,0,0,0,0,0,76,68,0,0,0,53,37,0,0,0,0,0,9,4,0,0,0,49,97] >;
C8.5Dic14 in GAP, Magma, Sage, TeX
C_8._5{\rm Dic}_{14} % in TeX
G:=Group("C8.5Dic14"); // GroupNames label
G:=SmallGroup(448,47);
// by ID
G=gap.SmallGroup(448,47);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,589,36,346,192,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=1,c^2=a*b^14,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1>;
// generators/relations
Export