metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊1C8, C28.16Q16, C4.8Dic28, C28.27SD16, C42.246D14, C28.12M4(2), C7⋊1(Q8⋊C8), C4.6(C8×D7), (C4×C8).2D7, C14.3C4≀C2, (C4×C56).2C2, C28.16(C2×C8), C4⋊Dic7.4C4, C28⋊C8.1C2, C2.3(D14⋊C8), (C2×C4).158D28, (C2×C28).435D4, C4.4(C8⋊D7), C4.14(C56⋊C2), C14.1(C22⋊C8), (C2×Dic14).5C4, (C4×Dic14).1C2, C2.1(Dic14⋊C4), (C4×C28).317C22, C14.5(Q8⋊C4), C22.30(D14⋊C4), C2.1(C28.44D4), (C2×C4).95(C4×D7), (C2×C28).215(C2×C4), (C2×C4).204(C7⋊D4), (C2×C14).35(C22⋊C4), SmallGroup(448,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.8Dic28
G = < a,b,c | a4=b56=1, c2=a-1b28, ab=ba, ac=ca, cbc-1=ab-1 >
Subgroups: 292 in 70 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C4×C8, C4⋊C8, C4×Q8, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×C28, Q8⋊C8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28⋊C8, C4×C56, C4×Dic14, C4.8Dic28
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), SD16, Q16, D14, C22⋊C8, Q8⋊C4, C4≀C2, C4×D7, D28, C7⋊D4, Q8⋊C8, C8×D7, C8⋊D7, C56⋊C2, Dic28, D14⋊C4, Dic14⋊C4, C28.44D4, D14⋊C8, C4.8Dic28
►Regular action on 448 points
Generators in S448
(1 99 168 178)(2 100 113 179)(3 101 114 180)(4 102 115 181)(5 103 116 182)(6 104 117 183)(7 105 118 184)(8 106 119 185)(9 107 120 186)(10 108 121 187)(11 109 122 188)(12 110 123 189)(13 111 124 190)(14 112 125 191)(15 57 126 192)(16 58 127 193)(17 59 128 194)(18 60 129 195)(19 61 130 196)(20 62 131 197)(21 63 132 198)(22 64 133 199)(23 65 134 200)(24 66 135 201)(25 67 136 202)(26 68 137 203)(27 69 138 204)(28 70 139 205)(29 71 140 206)(30 72 141 207)(31 73 142 208)(32 74 143 209)(33 75 144 210)(34 76 145 211)(35 77 146 212)(36 78 147 213)(37 79 148 214)(38 80 149 215)(39 81 150 216)(40 82 151 217)(41 83 152 218)(42 84 153 219)(43 85 154 220)(44 86 155 221)(45 87 156 222)(46 88 157 223)(47 89 158 224)(48 90 159 169)(49 91 160 170)(50 92 161 171)(51 93 162 172)(52 94 163 173)(53 95 164 174)(54 96 165 175)(55 97 166 176)(56 98 167 177)(225 292 383 438)(226 293 384 439)(227 294 385 440)(228 295 386 441)(229 296 387 442)(230 297 388 443)(231 298 389 444)(232 299 390 445)(233 300 391 446)(234 301 392 447)(235 302 337 448)(236 303 338 393)(237 304 339 394)(238 305 340 395)(239 306 341 396)(240 307 342 397)(241 308 343 398)(242 309 344 399)(243 310 345 400)(244 311 346 401)(245 312 347 402)(246 313 348 403)(247 314 349 404)(248 315 350 405)(249 316 351 406)(250 317 352 407)(251 318 353 408)(252 319 354 409)(253 320 355 410)(254 321 356 411)(255 322 357 412)(256 323 358 413)(257 324 359 414)(258 325 360 415)(259 326 361 416)(260 327 362 417)(261 328 363 418)(262 329 364 419)(263 330 365 420)(264 331 366 421)(265 332 367 422)(266 333 368 423)(267 334 369 424)(268 335 370 425)(269 336 371 426)(270 281 372 427)(271 282 373 428)(272 283 374 429)(273 284 375 430)(274 285 376 431)(275 286 377 432)(276 287 378 433)(277 288 379 434)(278 289 380 435)(279 290 381 436)(280 291 382 437)
(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)
(1 438 206 355 168 292 71 253)(2 280 207 409 113 382 72 319)(3 290 208 251 114 436 73 353)(4 380 209 317 115 278 74 407)(5 434 210 351 116 288 75 249)(6 276 211 405 117 378 76 315)(7 286 212 247 118 432 77 349)(8 376 213 313 119 274 78 403)(9 430 214 347 120 284 79 245)(10 272 215 401 121 374 80 311)(11 282 216 243 122 428 81 345)(12 372 217 309 123 270 82 399)(13 426 218 343 124 336 83 241)(14 268 219 397 125 370 84 307)(15 334 220 239 126 424 85 341)(16 368 221 305 127 266 86 395)(17 422 222 339 128 332 87 237)(18 264 223 393 129 366 88 303)(19 330 224 235 130 420 89 337)(20 364 169 301 131 262 90 447)(21 418 170 391 132 328 91 233)(22 260 171 445 133 362 92 299)(23 326 172 231 134 416 93 389)(24 360 173 297 135 258 94 443)(25 414 174 387 136 324 95 229)(26 256 175 441 137 358 96 295)(27 322 176 227 138 412 97 385)(28 356 177 293 139 254 98 439)(29 410 178 383 140 320 99 225)(30 252 179 437 141 354 100 291)(31 318 180 279 142 408 101 381)(32 352 181 289 143 250 102 435)(33 406 182 379 144 316 103 277)(34 248 183 433 145 350 104 287)(35 314 184 275 146 404 105 377)(36 348 185 285 147 246 106 431)(37 402 186 375 148 312 107 273)(38 244 187 429 149 346 108 283)(39 310 188 271 150 400 109 373)(40 344 189 281 151 242 110 427)(41 398 190 371 152 308 111 269)(42 240 191 425 153 342 112 335)(43 306 192 267 154 396 57 369)(44 340 193 333 155 238 58 423)(45 394 194 367 156 304 59 265)(46 236 195 421 157 338 60 331)(47 302 196 263 158 448 61 365)(48 392 197 329 159 234 62 419)(49 446 198 363 160 300 63 261)(50 232 199 417 161 390 64 327)(51 298 200 259 162 444 65 361)(52 388 201 325 163 230 66 415)(53 442 202 359 164 296 67 257)(54 228 203 413 165 386 68 323)(55 294 204 255 166 440 69 357)(56 384 205 321 167 226 70 411)
G:=sub<Sym(448)| (1,99,168,178)(2,100,113,179)(3,101,114,180)(4,102,115,181)(5,103,116,182)(6,104,117,183)(7,105,118,184)(8,106,119,185)(9,107,120,186)(10,108,121,187)(11,109,122,188)(12,110,123,189)(13,111,124,190)(14,112,125,191)(15,57,126,192)(16,58,127,193)(17,59,128,194)(18,60,129,195)(19,61,130,196)(20,62,131,197)(21,63,132,198)(22,64,133,199)(23,65,134,200)(24,66,135,201)(25,67,136,202)(26,68,137,203)(27,69,138,204)(28,70,139,205)(29,71,140,206)(30,72,141,207)(31,73,142,208)(32,74,143,209)(33,75,144,210)(34,76,145,211)(35,77,146,212)(36,78,147,213)(37,79,148,214)(38,80,149,215)(39,81,150,216)(40,82,151,217)(41,83,152,218)(42,84,153,219)(43,85,154,220)(44,86,155,221)(45,87,156,222)(46,88,157,223)(47,89,158,224)(48,90,159,169)(49,91,160,170)(50,92,161,171)(51,93,162,172)(52,94,163,173)(53,95,164,174)(54,96,165,175)(55,97,166,176)(56,98,167,177)(225,292,383,438)(226,293,384,439)(227,294,385,440)(228,295,386,441)(229,296,387,442)(230,297,388,443)(231,298,389,444)(232,299,390,445)(233,300,391,446)(234,301,392,447)(235,302,337,448)(236,303,338,393)(237,304,339,394)(238,305,340,395)(239,306,341,396)(240,307,342,397)(241,308,343,398)(242,309,344,399)(243,310,345,400)(244,311,346,401)(245,312,347,402)(246,313,348,403)(247,314,349,404)(248,315,350,405)(249,316,351,406)(250,317,352,407)(251,318,353,408)(252,319,354,409)(253,320,355,410)(254,321,356,411)(255,322,357,412)(256,323,358,413)(257,324,359,414)(258,325,360,415)(259,326,361,416)(260,327,362,417)(261,328,363,418)(262,329,364,419)(263,330,365,420)(264,331,366,421)(265,332,367,422)(266,333,368,423)(267,334,369,424)(268,335,370,425)(269,336,371,426)(270,281,372,427)(271,282,373,428)(272,283,374,429)(273,284,375,430)(274,285,376,431)(275,286,377,432)(276,287,378,433)(277,288,379,434)(278,289,380,435)(279,290,381,436)(280,291,382,437), (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), (1,438,206,355,168,292,71,253)(2,280,207,409,113,382,72,319)(3,290,208,251,114,436,73,353)(4,380,209,317,115,278,74,407)(5,434,210,351,116,288,75,249)(6,276,211,405,117,378,76,315)(7,286,212,247,118,432,77,349)(8,376,213,313,119,274,78,403)(9,430,214,347,120,284,79,245)(10,272,215,401,121,374,80,311)(11,282,216,243,122,428,81,345)(12,372,217,309,123,270,82,399)(13,426,218,343,124,336,83,241)(14,268,219,397,125,370,84,307)(15,334,220,239,126,424,85,341)(16,368,221,305,127,266,86,395)(17,422,222,339,128,332,87,237)(18,264,223,393,129,366,88,303)(19,330,224,235,130,420,89,337)(20,364,169,301,131,262,90,447)(21,418,170,391,132,328,91,233)(22,260,171,445,133,362,92,299)(23,326,172,231,134,416,93,389)(24,360,173,297,135,258,94,443)(25,414,174,387,136,324,95,229)(26,256,175,441,137,358,96,295)(27,322,176,227,138,412,97,385)(28,356,177,293,139,254,98,439)(29,410,178,383,140,320,99,225)(30,252,179,437,141,354,100,291)(31,318,180,279,142,408,101,381)(32,352,181,289,143,250,102,435)(33,406,182,379,144,316,103,277)(34,248,183,433,145,350,104,287)(35,314,184,275,146,404,105,377)(36,348,185,285,147,246,106,431)(37,402,186,375,148,312,107,273)(38,244,187,429,149,346,108,283)(39,310,188,271,150,400,109,373)(40,344,189,281,151,242,110,427)(41,398,190,371,152,308,111,269)(42,240,191,425,153,342,112,335)(43,306,192,267,154,396,57,369)(44,340,193,333,155,238,58,423)(45,394,194,367,156,304,59,265)(46,236,195,421,157,338,60,331)(47,302,196,263,158,448,61,365)(48,392,197,329,159,234,62,419)(49,446,198,363,160,300,63,261)(50,232,199,417,161,390,64,327)(51,298,200,259,162,444,65,361)(52,388,201,325,163,230,66,415)(53,442,202,359,164,296,67,257)(54,228,203,413,165,386,68,323)(55,294,204,255,166,440,69,357)(56,384,205,321,167,226,70,411)>;
G:=Group( (1,99,168,178)(2,100,113,179)(3,101,114,180)(4,102,115,181)(5,103,116,182)(6,104,117,183)(7,105,118,184)(8,106,119,185)(9,107,120,186)(10,108,121,187)(11,109,122,188)(12,110,123,189)(13,111,124,190)(14,112,125,191)(15,57,126,192)(16,58,127,193)(17,59,128,194)(18,60,129,195)(19,61,130,196)(20,62,131,197)(21,63,132,198)(22,64,133,199)(23,65,134,200)(24,66,135,201)(25,67,136,202)(26,68,137,203)(27,69,138,204)(28,70,139,205)(29,71,140,206)(30,72,141,207)(31,73,142,208)(32,74,143,209)(33,75,144,210)(34,76,145,211)(35,77,146,212)(36,78,147,213)(37,79,148,214)(38,80,149,215)(39,81,150,216)(40,82,151,217)(41,83,152,218)(42,84,153,219)(43,85,154,220)(44,86,155,221)(45,87,156,222)(46,88,157,223)(47,89,158,224)(48,90,159,169)(49,91,160,170)(50,92,161,171)(51,93,162,172)(52,94,163,173)(53,95,164,174)(54,96,165,175)(55,97,166,176)(56,98,167,177)(225,292,383,438)(226,293,384,439)(227,294,385,440)(228,295,386,441)(229,296,387,442)(230,297,388,443)(231,298,389,444)(232,299,390,445)(233,300,391,446)(234,301,392,447)(235,302,337,448)(236,303,338,393)(237,304,339,394)(238,305,340,395)(239,306,341,396)(240,307,342,397)(241,308,343,398)(242,309,344,399)(243,310,345,400)(244,311,346,401)(245,312,347,402)(246,313,348,403)(247,314,349,404)(248,315,350,405)(249,316,351,406)(250,317,352,407)(251,318,353,408)(252,319,354,409)(253,320,355,410)(254,321,356,411)(255,322,357,412)(256,323,358,413)(257,324,359,414)(258,325,360,415)(259,326,361,416)(260,327,362,417)(261,328,363,418)(262,329,364,419)(263,330,365,420)(264,331,366,421)(265,332,367,422)(266,333,368,423)(267,334,369,424)(268,335,370,425)(269,336,371,426)(270,281,372,427)(271,282,373,428)(272,283,374,429)(273,284,375,430)(274,285,376,431)(275,286,377,432)(276,287,378,433)(277,288,379,434)(278,289,380,435)(279,290,381,436)(280,291,382,437), (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), (1,438,206,355,168,292,71,253)(2,280,207,409,113,382,72,319)(3,290,208,251,114,436,73,353)(4,380,209,317,115,278,74,407)(5,434,210,351,116,288,75,249)(6,276,211,405,117,378,76,315)(7,286,212,247,118,432,77,349)(8,376,213,313,119,274,78,403)(9,430,214,347,120,284,79,245)(10,272,215,401,121,374,80,311)(11,282,216,243,122,428,81,345)(12,372,217,309,123,270,82,399)(13,426,218,343,124,336,83,241)(14,268,219,397,125,370,84,307)(15,334,220,239,126,424,85,341)(16,368,221,305,127,266,86,395)(17,422,222,339,128,332,87,237)(18,264,223,393,129,366,88,303)(19,330,224,235,130,420,89,337)(20,364,169,301,131,262,90,447)(21,418,170,391,132,328,91,233)(22,260,171,445,133,362,92,299)(23,326,172,231,134,416,93,389)(24,360,173,297,135,258,94,443)(25,414,174,387,136,324,95,229)(26,256,175,441,137,358,96,295)(27,322,176,227,138,412,97,385)(28,356,177,293,139,254,98,439)(29,410,178,383,140,320,99,225)(30,252,179,437,141,354,100,291)(31,318,180,279,142,408,101,381)(32,352,181,289,143,250,102,435)(33,406,182,379,144,316,103,277)(34,248,183,433,145,350,104,287)(35,314,184,275,146,404,105,377)(36,348,185,285,147,246,106,431)(37,402,186,375,148,312,107,273)(38,244,187,429,149,346,108,283)(39,310,188,271,150,400,109,373)(40,344,189,281,151,242,110,427)(41,398,190,371,152,308,111,269)(42,240,191,425,153,342,112,335)(43,306,192,267,154,396,57,369)(44,340,193,333,155,238,58,423)(45,394,194,367,156,304,59,265)(46,236,195,421,157,338,60,331)(47,302,196,263,158,448,61,365)(48,392,197,329,159,234,62,419)(49,446,198,363,160,300,63,261)(50,232,199,417,161,390,64,327)(51,298,200,259,162,444,65,361)(52,388,201,325,163,230,66,415)(53,442,202,359,164,296,67,257)(54,228,203,413,165,386,68,323)(55,294,204,255,166,440,69,357)(56,384,205,321,167,226,70,411) );
G=PermutationGroup([[(1,99,168,178),(2,100,113,179),(3,101,114,180),(4,102,115,181),(5,103,116,182),(6,104,117,183),(7,105,118,184),(8,106,119,185),(9,107,120,186),(10,108,121,187),(11,109,122,188),(12,110,123,189),(13,111,124,190),(14,112,125,191),(15,57,126,192),(16,58,127,193),(17,59,128,194),(18,60,129,195),(19,61,130,196),(20,62,131,197),(21,63,132,198),(22,64,133,199),(23,65,134,200),(24,66,135,201),(25,67,136,202),(26,68,137,203),(27,69,138,204),(28,70,139,205),(29,71,140,206),(30,72,141,207),(31,73,142,208),(32,74,143,209),(33,75,144,210),(34,76,145,211),(35,77,146,212),(36,78,147,213),(37,79,148,214),(38,80,149,215),(39,81,150,216),(40,82,151,217),(41,83,152,218),(42,84,153,219),(43,85,154,220),(44,86,155,221),(45,87,156,222),(46,88,157,223),(47,89,158,224),(48,90,159,169),(49,91,160,170),(50,92,161,171),(51,93,162,172),(52,94,163,173),(53,95,164,174),(54,96,165,175),(55,97,166,176),(56,98,167,177),(225,292,383,438),(226,293,384,439),(227,294,385,440),(228,295,386,441),(229,296,387,442),(230,297,388,443),(231,298,389,444),(232,299,390,445),(233,300,391,446),(234,301,392,447),(235,302,337,448),(236,303,338,393),(237,304,339,394),(238,305,340,395),(239,306,341,396),(240,307,342,397),(241,308,343,398),(242,309,344,399),(243,310,345,400),(244,311,346,401),(245,312,347,402),(246,313,348,403),(247,314,349,404),(248,315,350,405),(249,316,351,406),(250,317,352,407),(251,318,353,408),(252,319,354,409),(253,320,355,410),(254,321,356,411),(255,322,357,412),(256,323,358,413),(257,324,359,414),(258,325,360,415),(259,326,361,416),(260,327,362,417),(261,328,363,418),(262,329,364,419),(263,330,365,420),(264,331,366,421),(265,332,367,422),(266,333,368,423),(267,334,369,424),(268,335,370,425),(269,336,371,426),(270,281,372,427),(271,282,373,428),(272,283,374,429),(273,284,375,430),(274,285,376,431),(275,286,377,432),(276,287,378,433),(277,288,379,434),(278,289,380,435),(279,290,381,436),(280,291,382,437)], [(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)], [(1,438,206,355,168,292,71,253),(2,280,207,409,113,382,72,319),(3,290,208,251,114,436,73,353),(4,380,209,317,115,278,74,407),(5,434,210,351,116,288,75,249),(6,276,211,405,117,378,76,315),(7,286,212,247,118,432,77,349),(8,376,213,313,119,274,78,403),(9,430,214,347,120,284,79,245),(10,272,215,401,121,374,80,311),(11,282,216,243,122,428,81,345),(12,372,217,309,123,270,82,399),(13,426,218,343,124,336,83,241),(14,268,219,397,125,370,84,307),(15,334,220,239,126,424,85,341),(16,368,221,305,127,266,86,395),(17,422,222,339,128,332,87,237),(18,264,223,393,129,366,88,303),(19,330,224,235,130,420,89,337),(20,364,169,301,131,262,90,447),(21,418,170,391,132,328,91,233),(22,260,171,445,133,362,92,299),(23,326,172,231,134,416,93,389),(24,360,173,297,135,258,94,443),(25,414,174,387,136,324,95,229),(26,256,175,441,137,358,96,295),(27,322,176,227,138,412,97,385),(28,356,177,293,139,254,98,439),(29,410,178,383,140,320,99,225),(30,252,179,437,141,354,100,291),(31,318,180,279,142,408,101,381),(32,352,181,289,143,250,102,435),(33,406,182,379,144,316,103,277),(34,248,183,433,145,350,104,287),(35,314,184,275,146,404,105,377),(36,348,185,285,147,246,106,431),(37,402,186,375,148,312,107,273),(38,244,187,429,149,346,108,283),(39,310,188,271,150,400,109,373),(40,344,189,281,151,242,110,427),(41,398,190,371,152,308,111,269),(42,240,191,425,153,342,112,335),(43,306,192,267,154,396,57,369),(44,340,193,333,155,238,58,423),(45,394,194,367,156,304,59,265),(46,236,195,421,157,338,60,331),(47,302,196,263,158,448,61,365),(48,392,197,329,159,234,62,419),(49,446,198,363,160,300,63,261),(50,232,199,417,161,390,64,327),(51,298,200,259,162,444,65,361),(52,388,201,325,163,230,66,415),(53,442,202,359,164,296,67,257),(54,228,203,413,165,386,68,323),(55,294,204,255,166,440,69,357),(56,384,205,321,167,226,70,411)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D7 | M4(2) | SD16 | Q16 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | C8×D7 | C8⋊D7 | C56⋊C2 | Dic28 | Dic14⋊C4 |
| kernel | C4.8Dic28 | C28⋊C8 | C4×C56 | C4×Dic14 | C4⋊Dic7 | C2×Dic14 | Dic14 | C2×C28 | C4×C8 | C28 | C28 | C28 | C42 | C14 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 24 |
Matrix representation of C4.8Dic28 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 95 | 0 | 0 |
| 0 | 8 | 63 |
| 0 | 98 | 23 |
| 18 | 0 | 0 |
| 0 | 61 | 17 |
| 0 | 67 | 52 |
G:=sub<GL(3,GF(113))| [98,0,0,0,112,0,0,0,112],[95,0,0,0,8,98,0,63,23],[18,0,0,0,61,67,0,17,52] >;
C4.8Dic28 in GAP, Magma, Sage, TeX
C_4._8{\rm Dic}_{28} % in TeX
G:=Group("C4.8Dic28"); // GroupNames label
G:=SmallGroup(448,13);
// by ID
G=gap.SmallGroup(448,13);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,85,92,422,100,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^56=1,c^2=a^-1*b^28,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations