metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.26D4, Dic7⋊1Q16, C7⋊C8.21D4, C4.27(D4×D7), C28.51(C2×D4), C7⋊3(C4⋊Q16), (C2×Q16).3D7, C2.16(D7×Q16), (C2×C8).241D14, C8.18(C7⋊D4), (C2×Q8).60D14, (C14×Q16).4C2, C14.26(C2×Q16), (C8×Dic7).4C2, (C2×C56).93C22, C22.274(D4×D7), C2.24(C28⋊D4), C14.33(C4⋊1D4), (C2×C28).455C23, (C2×Dic28).11C2, Dic7⋊Q8.5C2, (C2×Dic7).116D4, (Q8×C14).84C22, (C4×Dic7).243C22, (C2×Dic14).130C22, C4.14(C2×C7⋊D4), (C2×C7⋊Q16).8C2, (C2×C14).366(C2×D4), (C2×C7⋊C8).276C22, (C2×C4).543(C22×D7), SmallGroup(448,715)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.26D4
G = < a,b,c | a56=b4=1, c2=a28, bab-1=a41, cac-1=a-1, cbc-1=b-1 >
Subgroups: 548 in 122 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×C8, C4⋊Q8, C2×Q16, C2×Q16, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C4⋊Q16, Dic28, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C7⋊Q16, C2×C56, C7×Q16, C2×Dic14, Q8×C14, C8×Dic7, C2×Dic28, C2×C7⋊Q16, Dic7⋊Q8, C14×Q16, C56.26D4
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C4⋊1D4, C2×Q16, C7⋊D4, C22×D7, C4⋊Q16, D4×D7, C2×C7⋊D4, D7×Q16, C28⋊D4, C56.26D4
►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)
(1 199 96 157)(2 184 97 142)(3 169 98 127)(4 210 99 168)(5 195 100 153)(6 180 101 138)(7 221 102 123)(8 206 103 164)(9 191 104 149)(10 176 105 134)(11 217 106 119)(12 202 107 160)(13 187 108 145)(14 172 109 130)(15 213 110 115)(16 198 111 156)(17 183 112 141)(18 224 57 126)(19 209 58 167)(20 194 59 152)(21 179 60 137)(22 220 61 122)(23 205 62 163)(24 190 63 148)(25 175 64 133)(26 216 65 118)(27 201 66 159)(28 186 67 144)(29 171 68 129)(30 212 69 114)(31 197 70 155)(32 182 71 140)(33 223 72 125)(34 208 73 166)(35 193 74 151)(36 178 75 136)(37 219 76 121)(38 204 77 162)(39 189 78 147)(40 174 79 132)(41 215 80 117)(42 200 81 158)(43 185 82 143)(44 170 83 128)(45 211 84 113)(46 196 85 154)(47 181 86 139)(48 222 87 124)(49 207 88 165)(50 192 89 150)(51 177 90 135)(52 218 91 120)(53 203 92 161)(54 188 93 146)(55 173 94 131)(56 214 95 116)(225 410 315 357)(226 395 316 342)(227 436 317 383)(228 421 318 368)(229 406 319 353)(230 447 320 338)(231 432 321 379)(232 417 322 364)(233 402 323 349)(234 443 324 390)(235 428 325 375)(236 413 326 360)(237 398 327 345)(238 439 328 386)(239 424 329 371)(240 409 330 356)(241 394 331 341)(242 435 332 382)(243 420 333 367)(244 405 334 352)(245 446 335 337)(246 431 336 378)(247 416 281 363)(248 401 282 348)(249 442 283 389)(250 427 284 374)(251 412 285 359)(252 397 286 344)(253 438 287 385)(254 423 288 370)(255 408 289 355)(256 393 290 340)(257 434 291 381)(258 419 292 366)(259 404 293 351)(260 445 294 392)(261 430 295 377)(262 415 296 362)(263 400 297 347)(264 441 298 388)(265 426 299 373)(266 411 300 358)(267 396 301 343)(268 437 302 384)(269 422 303 369)(270 407 304 354)(271 448 305 339)(272 433 306 380)(273 418 307 365)(274 403 308 350)(275 444 309 391)(276 429 310 376)(277 414 311 361)(278 399 312 346)(279 440 313 387)(280 425 314 372)
(1 426 29 398)(2 425 30 397)(3 424 31 396)(4 423 32 395)(5 422 33 394)(6 421 34 393)(7 420 35 448)(8 419 36 447)(9 418 37 446)(10 417 38 445)(11 416 39 444)(12 415 40 443)(13 414 41 442)(14 413 42 441)(15 412 43 440)(16 411 44 439)(17 410 45 438)(18 409 46 437)(19 408 47 436)(20 407 48 435)(21 406 49 434)(22 405 50 433)(23 404 51 432)(24 403 52 431)(25 402 53 430)(26 401 54 429)(27 400 55 428)(28 399 56 427)(57 356 85 384)(58 355 86 383)(59 354 87 382)(60 353 88 381)(61 352 89 380)(62 351 90 379)(63 350 91 378)(64 349 92 377)(65 348 93 376)(66 347 94 375)(67 346 95 374)(68 345 96 373)(69 344 97 372)(70 343 98 371)(71 342 99 370)(72 341 100 369)(73 340 101 368)(74 339 102 367)(75 338 103 366)(76 337 104 365)(77 392 105 364)(78 391 106 363)(79 390 107 362)(80 389 108 361)(81 388 109 360)(82 387 110 359)(83 386 111 358)(84 385 112 357)(113 287 141 315)(114 286 142 314)(115 285 143 313)(116 284 144 312)(117 283 145 311)(118 282 146 310)(119 281 147 309)(120 336 148 308)(121 335 149 307)(122 334 150 306)(123 333 151 305)(124 332 152 304)(125 331 153 303)(126 330 154 302)(127 329 155 301)(128 328 156 300)(129 327 157 299)(130 326 158 298)(131 325 159 297)(132 324 160 296)(133 323 161 295)(134 322 162 294)(135 321 163 293)(136 320 164 292)(137 319 165 291)(138 318 166 290)(139 317 167 289)(140 316 168 288)(169 239 197 267)(170 238 198 266)(171 237 199 265)(172 236 200 264)(173 235 201 263)(174 234 202 262)(175 233 203 261)(176 232 204 260)(177 231 205 259)(178 230 206 258)(179 229 207 257)(180 228 208 256)(181 227 209 255)(182 226 210 254)(183 225 211 253)(184 280 212 252)(185 279 213 251)(186 278 214 250)(187 277 215 249)(188 276 216 248)(189 275 217 247)(190 274 218 246)(191 273 219 245)(192 272 220 244)(193 271 221 243)(194 270 222 242)(195 269 223 241)(196 268 224 240)
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), (1,199,96,157)(2,184,97,142)(3,169,98,127)(4,210,99,168)(5,195,100,153)(6,180,101,138)(7,221,102,123)(8,206,103,164)(9,191,104,149)(10,176,105,134)(11,217,106,119)(12,202,107,160)(13,187,108,145)(14,172,109,130)(15,213,110,115)(16,198,111,156)(17,183,112,141)(18,224,57,126)(19,209,58,167)(20,194,59,152)(21,179,60,137)(22,220,61,122)(23,205,62,163)(24,190,63,148)(25,175,64,133)(26,216,65,118)(27,201,66,159)(28,186,67,144)(29,171,68,129)(30,212,69,114)(31,197,70,155)(32,182,71,140)(33,223,72,125)(34,208,73,166)(35,193,74,151)(36,178,75,136)(37,219,76,121)(38,204,77,162)(39,189,78,147)(40,174,79,132)(41,215,80,117)(42,200,81,158)(43,185,82,143)(44,170,83,128)(45,211,84,113)(46,196,85,154)(47,181,86,139)(48,222,87,124)(49,207,88,165)(50,192,89,150)(51,177,90,135)(52,218,91,120)(53,203,92,161)(54,188,93,146)(55,173,94,131)(56,214,95,116)(225,410,315,357)(226,395,316,342)(227,436,317,383)(228,421,318,368)(229,406,319,353)(230,447,320,338)(231,432,321,379)(232,417,322,364)(233,402,323,349)(234,443,324,390)(235,428,325,375)(236,413,326,360)(237,398,327,345)(238,439,328,386)(239,424,329,371)(240,409,330,356)(241,394,331,341)(242,435,332,382)(243,420,333,367)(244,405,334,352)(245,446,335,337)(246,431,336,378)(247,416,281,363)(248,401,282,348)(249,442,283,389)(250,427,284,374)(251,412,285,359)(252,397,286,344)(253,438,287,385)(254,423,288,370)(255,408,289,355)(256,393,290,340)(257,434,291,381)(258,419,292,366)(259,404,293,351)(260,445,294,392)(261,430,295,377)(262,415,296,362)(263,400,297,347)(264,441,298,388)(265,426,299,373)(266,411,300,358)(267,396,301,343)(268,437,302,384)(269,422,303,369)(270,407,304,354)(271,448,305,339)(272,433,306,380)(273,418,307,365)(274,403,308,350)(275,444,309,391)(276,429,310,376)(277,414,311,361)(278,399,312,346)(279,440,313,387)(280,425,314,372), (1,426,29,398)(2,425,30,397)(3,424,31,396)(4,423,32,395)(5,422,33,394)(6,421,34,393)(7,420,35,448)(8,419,36,447)(9,418,37,446)(10,417,38,445)(11,416,39,444)(12,415,40,443)(13,414,41,442)(14,413,42,441)(15,412,43,440)(16,411,44,439)(17,410,45,438)(18,409,46,437)(19,408,47,436)(20,407,48,435)(21,406,49,434)(22,405,50,433)(23,404,51,432)(24,403,52,431)(25,402,53,430)(26,401,54,429)(27,400,55,428)(28,399,56,427)(57,356,85,384)(58,355,86,383)(59,354,87,382)(60,353,88,381)(61,352,89,380)(62,351,90,379)(63,350,91,378)(64,349,92,377)(65,348,93,376)(66,347,94,375)(67,346,95,374)(68,345,96,373)(69,344,97,372)(70,343,98,371)(71,342,99,370)(72,341,100,369)(73,340,101,368)(74,339,102,367)(75,338,103,366)(76,337,104,365)(77,392,105,364)(78,391,106,363)(79,390,107,362)(80,389,108,361)(81,388,109,360)(82,387,110,359)(83,386,111,358)(84,385,112,357)(113,287,141,315)(114,286,142,314)(115,285,143,313)(116,284,144,312)(117,283,145,311)(118,282,146,310)(119,281,147,309)(120,336,148,308)(121,335,149,307)(122,334,150,306)(123,333,151,305)(124,332,152,304)(125,331,153,303)(126,330,154,302)(127,329,155,301)(128,328,156,300)(129,327,157,299)(130,326,158,298)(131,325,159,297)(132,324,160,296)(133,323,161,295)(134,322,162,294)(135,321,163,293)(136,320,164,292)(137,319,165,291)(138,318,166,290)(139,317,167,289)(140,316,168,288)(169,239,197,267)(170,238,198,266)(171,237,199,265)(172,236,200,264)(173,235,201,263)(174,234,202,262)(175,233,203,261)(176,232,204,260)(177,231,205,259)(178,230,206,258)(179,229,207,257)(180,228,208,256)(181,227,209,255)(182,226,210,254)(183,225,211,253)(184,280,212,252)(185,279,213,251)(186,278,214,250)(187,277,215,249)(188,276,216,248)(189,275,217,247)(190,274,218,246)(191,273,219,245)(192,272,220,244)(193,271,221,243)(194,270,222,242)(195,269,223,241)(196,268,224,240)>;
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), (1,199,96,157)(2,184,97,142)(3,169,98,127)(4,210,99,168)(5,195,100,153)(6,180,101,138)(7,221,102,123)(8,206,103,164)(9,191,104,149)(10,176,105,134)(11,217,106,119)(12,202,107,160)(13,187,108,145)(14,172,109,130)(15,213,110,115)(16,198,111,156)(17,183,112,141)(18,224,57,126)(19,209,58,167)(20,194,59,152)(21,179,60,137)(22,220,61,122)(23,205,62,163)(24,190,63,148)(25,175,64,133)(26,216,65,118)(27,201,66,159)(28,186,67,144)(29,171,68,129)(30,212,69,114)(31,197,70,155)(32,182,71,140)(33,223,72,125)(34,208,73,166)(35,193,74,151)(36,178,75,136)(37,219,76,121)(38,204,77,162)(39,189,78,147)(40,174,79,132)(41,215,80,117)(42,200,81,158)(43,185,82,143)(44,170,83,128)(45,211,84,113)(46,196,85,154)(47,181,86,139)(48,222,87,124)(49,207,88,165)(50,192,89,150)(51,177,90,135)(52,218,91,120)(53,203,92,161)(54,188,93,146)(55,173,94,131)(56,214,95,116)(225,410,315,357)(226,395,316,342)(227,436,317,383)(228,421,318,368)(229,406,319,353)(230,447,320,338)(231,432,321,379)(232,417,322,364)(233,402,323,349)(234,443,324,390)(235,428,325,375)(236,413,326,360)(237,398,327,345)(238,439,328,386)(239,424,329,371)(240,409,330,356)(241,394,331,341)(242,435,332,382)(243,420,333,367)(244,405,334,352)(245,446,335,337)(246,431,336,378)(247,416,281,363)(248,401,282,348)(249,442,283,389)(250,427,284,374)(251,412,285,359)(252,397,286,344)(253,438,287,385)(254,423,288,370)(255,408,289,355)(256,393,290,340)(257,434,291,381)(258,419,292,366)(259,404,293,351)(260,445,294,392)(261,430,295,377)(262,415,296,362)(263,400,297,347)(264,441,298,388)(265,426,299,373)(266,411,300,358)(267,396,301,343)(268,437,302,384)(269,422,303,369)(270,407,304,354)(271,448,305,339)(272,433,306,380)(273,418,307,365)(274,403,308,350)(275,444,309,391)(276,429,310,376)(277,414,311,361)(278,399,312,346)(279,440,313,387)(280,425,314,372), (1,426,29,398)(2,425,30,397)(3,424,31,396)(4,423,32,395)(5,422,33,394)(6,421,34,393)(7,420,35,448)(8,419,36,447)(9,418,37,446)(10,417,38,445)(11,416,39,444)(12,415,40,443)(13,414,41,442)(14,413,42,441)(15,412,43,440)(16,411,44,439)(17,410,45,438)(18,409,46,437)(19,408,47,436)(20,407,48,435)(21,406,49,434)(22,405,50,433)(23,404,51,432)(24,403,52,431)(25,402,53,430)(26,401,54,429)(27,400,55,428)(28,399,56,427)(57,356,85,384)(58,355,86,383)(59,354,87,382)(60,353,88,381)(61,352,89,380)(62,351,90,379)(63,350,91,378)(64,349,92,377)(65,348,93,376)(66,347,94,375)(67,346,95,374)(68,345,96,373)(69,344,97,372)(70,343,98,371)(71,342,99,370)(72,341,100,369)(73,340,101,368)(74,339,102,367)(75,338,103,366)(76,337,104,365)(77,392,105,364)(78,391,106,363)(79,390,107,362)(80,389,108,361)(81,388,109,360)(82,387,110,359)(83,386,111,358)(84,385,112,357)(113,287,141,315)(114,286,142,314)(115,285,143,313)(116,284,144,312)(117,283,145,311)(118,282,146,310)(119,281,147,309)(120,336,148,308)(121,335,149,307)(122,334,150,306)(123,333,151,305)(124,332,152,304)(125,331,153,303)(126,330,154,302)(127,329,155,301)(128,328,156,300)(129,327,157,299)(130,326,158,298)(131,325,159,297)(132,324,160,296)(133,323,161,295)(134,322,162,294)(135,321,163,293)(136,320,164,292)(137,319,165,291)(138,318,166,290)(139,317,167,289)(140,316,168,288)(169,239,197,267)(170,238,198,266)(171,237,199,265)(172,236,200,264)(173,235,201,263)(174,234,202,262)(175,233,203,261)(176,232,204,260)(177,231,205,259)(178,230,206,258)(179,229,207,257)(180,228,208,256)(181,227,209,255)(182,226,210,254)(183,225,211,253)(184,280,212,252)(185,279,213,251)(186,278,214,250)(187,277,215,249)(188,276,216,248)(189,275,217,247)(190,274,218,246)(191,273,219,245)(192,272,220,244)(193,271,221,243)(194,270,222,242)(195,269,223,241)(196,268,224,240) );
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)], [(1,199,96,157),(2,184,97,142),(3,169,98,127),(4,210,99,168),(5,195,100,153),(6,180,101,138),(7,221,102,123),(8,206,103,164),(9,191,104,149),(10,176,105,134),(11,217,106,119),(12,202,107,160),(13,187,108,145),(14,172,109,130),(15,213,110,115),(16,198,111,156),(17,183,112,141),(18,224,57,126),(19,209,58,167),(20,194,59,152),(21,179,60,137),(22,220,61,122),(23,205,62,163),(24,190,63,148),(25,175,64,133),(26,216,65,118),(27,201,66,159),(28,186,67,144),(29,171,68,129),(30,212,69,114),(31,197,70,155),(32,182,71,140),(33,223,72,125),(34,208,73,166),(35,193,74,151),(36,178,75,136),(37,219,76,121),(38,204,77,162),(39,189,78,147),(40,174,79,132),(41,215,80,117),(42,200,81,158),(43,185,82,143),(44,170,83,128),(45,211,84,113),(46,196,85,154),(47,181,86,139),(48,222,87,124),(49,207,88,165),(50,192,89,150),(51,177,90,135),(52,218,91,120),(53,203,92,161),(54,188,93,146),(55,173,94,131),(56,214,95,116),(225,410,315,357),(226,395,316,342),(227,436,317,383),(228,421,318,368),(229,406,319,353),(230,447,320,338),(231,432,321,379),(232,417,322,364),(233,402,323,349),(234,443,324,390),(235,428,325,375),(236,413,326,360),(237,398,327,345),(238,439,328,386),(239,424,329,371),(240,409,330,356),(241,394,331,341),(242,435,332,382),(243,420,333,367),(244,405,334,352),(245,446,335,337),(246,431,336,378),(247,416,281,363),(248,401,282,348),(249,442,283,389),(250,427,284,374),(251,412,285,359),(252,397,286,344),(253,438,287,385),(254,423,288,370),(255,408,289,355),(256,393,290,340),(257,434,291,381),(258,419,292,366),(259,404,293,351),(260,445,294,392),(261,430,295,377),(262,415,296,362),(263,400,297,347),(264,441,298,388),(265,426,299,373),(266,411,300,358),(267,396,301,343),(268,437,302,384),(269,422,303,369),(270,407,304,354),(271,448,305,339),(272,433,306,380),(273,418,307,365),(274,403,308,350),(275,444,309,391),(276,429,310,376),(277,414,311,361),(278,399,312,346),(279,440,313,387),(280,425,314,372)], [(1,426,29,398),(2,425,30,397),(3,424,31,396),(4,423,32,395),(5,422,33,394),(6,421,34,393),(7,420,35,448),(8,419,36,447),(9,418,37,446),(10,417,38,445),(11,416,39,444),(12,415,40,443),(13,414,41,442),(14,413,42,441),(15,412,43,440),(16,411,44,439),(17,410,45,438),(18,409,46,437),(19,408,47,436),(20,407,48,435),(21,406,49,434),(22,405,50,433),(23,404,51,432),(24,403,52,431),(25,402,53,430),(26,401,54,429),(27,400,55,428),(28,399,56,427),(57,356,85,384),(58,355,86,383),(59,354,87,382),(60,353,88,381),(61,352,89,380),(62,351,90,379),(63,350,91,378),(64,349,92,377),(65,348,93,376),(66,347,94,375),(67,346,95,374),(68,345,96,373),(69,344,97,372),(70,343,98,371),(71,342,99,370),(72,341,100,369),(73,340,101,368),(74,339,102,367),(75,338,103,366),(76,337,104,365),(77,392,105,364),(78,391,106,363),(79,390,107,362),(80,389,108,361),(81,388,109,360),(82,387,110,359),(83,386,111,358),(84,385,112,357),(113,287,141,315),(114,286,142,314),(115,285,143,313),(116,284,144,312),(117,283,145,311),(118,282,146,310),(119,281,147,309),(120,336,148,308),(121,335,149,307),(122,334,150,306),(123,333,151,305),(124,332,152,304),(125,331,153,303),(126,330,154,302),(127,329,155,301),(128,328,156,300),(129,327,157,299),(130,326,158,298),(131,325,159,297),(132,324,160,296),(133,323,161,295),(134,322,162,294),(135,321,163,293),(136,320,164,292),(137,319,165,291),(138,318,166,290),(139,317,167,289),(140,316,168,288),(169,239,197,267),(170,238,198,266),(171,237,199,265),(172,236,200,264),(173,235,201,263),(174,234,202,262),(175,233,203,261),(176,232,204,260),(177,231,205,259),(178,230,206,258),(179,229,207,257),(180,228,208,256),(181,227,209,255),(182,226,210,254),(183,225,211,253),(184,280,212,252),(185,279,213,251),(186,278,214,250),(187,277,215,249),(188,276,216,248),(189,275,217,247),(190,274,218,246),(191,273,219,245),(192,272,220,244),(193,271,221,243),(194,270,222,242),(195,269,223,241),(196,268,224,240)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| 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 | 2 | 2 | 8 | 8 | 14 | 14 | 14 | 14 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 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 | D4 | D4 | D4 | D7 | Q16 | D14 | D14 | C7⋊D4 | D4×D7 | D4×D7 | D7×Q16 |
| kernel | C56.26D4 | C8×Dic7 | C2×Dic28 | C2×C7⋊Q16 | Dic7⋊Q8 | C14×Q16 | C7⋊C8 | C56 | C2×Dic7 | C2×Q16 | Dic7 | C2×C8 | C2×Q8 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 3 | 8 | 3 | 6 | 12 | 3 | 3 | 12 |
Matrix representation of C56.26D4 ►in GL4(𝔽113) generated by
| 79 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 51 | 85 |
| 0 | 0 | 109 | 0 |
| 33 | 17 | 0 | 0 |
| 9 | 80 | 0 | 0 |
| 0 | 0 | 112 | 72 |
| 0 | 0 | 91 | 1 |
| 43 | 91 | 0 | 0 |
| 84 | 70 | 0 | 0 |
| 0 | 0 | 95 | 107 |
| 0 | 0 | 73 | 18 |
G:=sub<GL(4,GF(113))| [79,112,0,0,1,0,0,0,0,0,51,109,0,0,85,0],[33,9,0,0,17,80,0,0,0,0,112,91,0,0,72,1],[43,84,0,0,91,70,0,0,0,0,95,73,0,0,107,18] >;
C56.26D4 in GAP, Magma, Sage, TeX
C_{56}._{26}D_4 % in TeX
G:=Group("C56.26D4"); // GroupNames label
G:=SmallGroup(448,715);
// by ID
G=gap.SmallGroup(448,715);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,232,422,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=a^41,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations