direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic56, C14⋊1Q32, C4.8D56, C56.62D4, C8.12D28, C28.33D8, C16.17D14, C56.58C23, C22.14D56, C112.17C22, Dic28.8C22, C7⋊1(C2×Q32), (C2×C16).4D7, (C2×C112).6C2, (C2×C4).86D28, (C2×C14).21D8, C14.12(C2×D8), C4.39(C2×D28), C2.14(C2×D56), C28.282(C2×D4), (C2×C8).305D14, (C2×C28).382D4, C8.48(C22×D7), (C2×Dic28).5C2, (C2×C56).378C22, SmallGroup(448,439)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic56
G = < a,b,c | a2=b112=1, c2=b56, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 500 in 82 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C14, C16, C2×C8, Q16, C2×Q8, Dic7, C28, C2×C14, C2×C16, Q32, C2×Q16, C56, Dic14, C2×Dic7, C2×C28, C2×Q32, C112, Dic28, Dic28, C2×C56, C2×Dic14, Dic56, C2×C112, C2×Dic28, C2×Dic56
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, Q32, C2×D8, D28, C22×D7, C2×Q32, D56, C2×D28, Dic56, C2×D56, C2×Dic56
►Regular action on 448 points
Generators in S448
(1 257)(2 258)(3 259)(4 260)(5 261)(6 262)(7 263)(8 264)(9 265)(10 266)(11 267)(12 268)(13 269)(14 270)(15 271)(16 272)(17 273)(18 274)(19 275)(20 276)(21 277)(22 278)(23 279)(24 280)(25 281)(26 282)(27 283)(28 284)(29 285)(30 286)(31 287)(32 288)(33 289)(34 290)(35 291)(36 292)(37 293)(38 294)(39 295)(40 296)(41 297)(42 298)(43 299)(44 300)(45 301)(46 302)(47 303)(48 304)(49 305)(50 306)(51 307)(52 308)(53 309)(54 310)(55 311)(56 312)(57 313)(58 314)(59 315)(60 316)(61 317)(62 318)(63 319)(64 320)(65 321)(66 322)(67 323)(68 324)(69 325)(70 326)(71 327)(72 328)(73 329)(74 330)(75 331)(76 332)(77 333)(78 334)(79 335)(80 336)(81 225)(82 226)(83 227)(84 228)(85 229)(86 230)(87 231)(88 232)(89 233)(90 234)(91 235)(92 236)(93 237)(94 238)(95 239)(96 240)(97 241)(98 242)(99 243)(100 244)(101 245)(102 246)(103 247)(104 248)(105 249)(106 250)(107 251)(108 252)(109 253)(110 254)(111 255)(112 256)(113 419)(114 420)(115 421)(116 422)(117 423)(118 424)(119 425)(120 426)(121 427)(122 428)(123 429)(124 430)(125 431)(126 432)(127 433)(128 434)(129 435)(130 436)(131 437)(132 438)(133 439)(134 440)(135 441)(136 442)(137 443)(138 444)(139 445)(140 446)(141 447)(142 448)(143 337)(144 338)(145 339)(146 340)(147 341)(148 342)(149 343)(150 344)(151 345)(152 346)(153 347)(154 348)(155 349)(156 350)(157 351)(158 352)(159 353)(160 354)(161 355)(162 356)(163 357)(164 358)(165 359)(166 360)(167 361)(168 362)(169 363)(170 364)(171 365)(172 366)(173 367)(174 368)(175 369)(176 370)(177 371)(178 372)(179 373)(180 374)(181 375)(182 376)(183 377)(184 378)(185 379)(186 380)(187 381)(188 382)(189 383)(190 384)(191 385)(192 386)(193 387)(194 388)(195 389)(196 390)(197 391)(198 392)(199 393)(200 394)(201 395)(202 396)(203 397)(204 398)(205 399)(206 400)(207 401)(208 402)(209 403)(210 404)(211 405)(212 406)(213 407)(214 408)(215 409)(216 410)(217 411)(218 412)(219 413)(220 414)(221 415)(222 416)(223 417)(224 418)
(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)
(1 220 57 164)(2 219 58 163)(3 218 59 162)(4 217 60 161)(5 216 61 160)(6 215 62 159)(7 214 63 158)(8 213 64 157)(9 212 65 156)(10 211 66 155)(11 210 67 154)(12 209 68 153)(13 208 69 152)(14 207 70 151)(15 206 71 150)(16 205 72 149)(17 204 73 148)(18 203 74 147)(19 202 75 146)(20 201 76 145)(21 200 77 144)(22 199 78 143)(23 198 79 142)(24 197 80 141)(25 196 81 140)(26 195 82 139)(27 194 83 138)(28 193 84 137)(29 192 85 136)(30 191 86 135)(31 190 87 134)(32 189 88 133)(33 188 89 132)(34 187 90 131)(35 186 91 130)(36 185 92 129)(37 184 93 128)(38 183 94 127)(39 182 95 126)(40 181 96 125)(41 180 97 124)(42 179 98 123)(43 178 99 122)(44 177 100 121)(45 176 101 120)(46 175 102 119)(47 174 103 118)(48 173 104 117)(49 172 105 116)(50 171 106 115)(51 170 107 114)(52 169 108 113)(53 168 109 224)(54 167 110 223)(55 166 111 222)(56 165 112 221)(225 446 281 390)(226 445 282 389)(227 444 283 388)(228 443 284 387)(229 442 285 386)(230 441 286 385)(231 440 287 384)(232 439 288 383)(233 438 289 382)(234 437 290 381)(235 436 291 380)(236 435 292 379)(237 434 293 378)(238 433 294 377)(239 432 295 376)(240 431 296 375)(241 430 297 374)(242 429 298 373)(243 428 299 372)(244 427 300 371)(245 426 301 370)(246 425 302 369)(247 424 303 368)(248 423 304 367)(249 422 305 366)(250 421 306 365)(251 420 307 364)(252 419 308 363)(253 418 309 362)(254 417 310 361)(255 416 311 360)(256 415 312 359)(257 414 313 358)(258 413 314 357)(259 412 315 356)(260 411 316 355)(261 410 317 354)(262 409 318 353)(263 408 319 352)(264 407 320 351)(265 406 321 350)(266 405 322 349)(267 404 323 348)(268 403 324 347)(269 402 325 346)(270 401 326 345)(271 400 327 344)(272 399 328 343)(273 398 329 342)(274 397 330 341)(275 396 331 340)(276 395 332 339)(277 394 333 338)(278 393 334 337)(279 392 335 448)(280 391 336 447)
G:=sub<Sym(448)| (1,257)(2,258)(3,259)(4,260)(5,261)(6,262)(7,263)(8,264)(9,265)(10,266)(11,267)(12,268)(13,269)(14,270)(15,271)(16,272)(17,273)(18,274)(19,275)(20,276)(21,277)(22,278)(23,279)(24,280)(25,281)(26,282)(27,283)(28,284)(29,285)(30,286)(31,287)(32,288)(33,289)(34,290)(35,291)(36,292)(37,293)(38,294)(39,295)(40,296)(41,297)(42,298)(43,299)(44,300)(45,301)(46,302)(47,303)(48,304)(49,305)(50,306)(51,307)(52,308)(53,309)(54,310)(55,311)(56,312)(57,313)(58,314)(59,315)(60,316)(61,317)(62,318)(63,319)(64,320)(65,321)(66,322)(67,323)(68,324)(69,325)(70,326)(71,327)(72,328)(73,329)(74,330)(75,331)(76,332)(77,333)(78,334)(79,335)(80,336)(81,225)(82,226)(83,227)(84,228)(85,229)(86,230)(87,231)(88,232)(89,233)(90,234)(91,235)(92,236)(93,237)(94,238)(95,239)(96,240)(97,241)(98,242)(99,243)(100,244)(101,245)(102,246)(103,247)(104,248)(105,249)(106,250)(107,251)(108,252)(109,253)(110,254)(111,255)(112,256)(113,419)(114,420)(115,421)(116,422)(117,423)(118,424)(119,425)(120,426)(121,427)(122,428)(123,429)(124,430)(125,431)(126,432)(127,433)(128,434)(129,435)(130,436)(131,437)(132,438)(133,439)(134,440)(135,441)(136,442)(137,443)(138,444)(139,445)(140,446)(141,447)(142,448)(143,337)(144,338)(145,339)(146,340)(147,341)(148,342)(149,343)(150,344)(151,345)(152,346)(153,347)(154,348)(155,349)(156,350)(157,351)(158,352)(159,353)(160,354)(161,355)(162,356)(163,357)(164,358)(165,359)(166,360)(167,361)(168,362)(169,363)(170,364)(171,365)(172,366)(173,367)(174,368)(175,369)(176,370)(177,371)(178,372)(179,373)(180,374)(181,375)(182,376)(183,377)(184,378)(185,379)(186,380)(187,381)(188,382)(189,383)(190,384)(191,385)(192,386)(193,387)(194,388)(195,389)(196,390)(197,391)(198,392)(199,393)(200,394)(201,395)(202,396)(203,397)(204,398)(205,399)(206,400)(207,401)(208,402)(209,403)(210,404)(211,405)(212,406)(213,407)(214,408)(215,409)(216,410)(217,411)(218,412)(219,413)(220,414)(221,415)(222,416)(223,417)(224,418), (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), (1,220,57,164)(2,219,58,163)(3,218,59,162)(4,217,60,161)(5,216,61,160)(6,215,62,159)(7,214,63,158)(8,213,64,157)(9,212,65,156)(10,211,66,155)(11,210,67,154)(12,209,68,153)(13,208,69,152)(14,207,70,151)(15,206,71,150)(16,205,72,149)(17,204,73,148)(18,203,74,147)(19,202,75,146)(20,201,76,145)(21,200,77,144)(22,199,78,143)(23,198,79,142)(24,197,80,141)(25,196,81,140)(26,195,82,139)(27,194,83,138)(28,193,84,137)(29,192,85,136)(30,191,86,135)(31,190,87,134)(32,189,88,133)(33,188,89,132)(34,187,90,131)(35,186,91,130)(36,185,92,129)(37,184,93,128)(38,183,94,127)(39,182,95,126)(40,181,96,125)(41,180,97,124)(42,179,98,123)(43,178,99,122)(44,177,100,121)(45,176,101,120)(46,175,102,119)(47,174,103,118)(48,173,104,117)(49,172,105,116)(50,171,106,115)(51,170,107,114)(52,169,108,113)(53,168,109,224)(54,167,110,223)(55,166,111,222)(56,165,112,221)(225,446,281,390)(226,445,282,389)(227,444,283,388)(228,443,284,387)(229,442,285,386)(230,441,286,385)(231,440,287,384)(232,439,288,383)(233,438,289,382)(234,437,290,381)(235,436,291,380)(236,435,292,379)(237,434,293,378)(238,433,294,377)(239,432,295,376)(240,431,296,375)(241,430,297,374)(242,429,298,373)(243,428,299,372)(244,427,300,371)(245,426,301,370)(246,425,302,369)(247,424,303,368)(248,423,304,367)(249,422,305,366)(250,421,306,365)(251,420,307,364)(252,419,308,363)(253,418,309,362)(254,417,310,361)(255,416,311,360)(256,415,312,359)(257,414,313,358)(258,413,314,357)(259,412,315,356)(260,411,316,355)(261,410,317,354)(262,409,318,353)(263,408,319,352)(264,407,320,351)(265,406,321,350)(266,405,322,349)(267,404,323,348)(268,403,324,347)(269,402,325,346)(270,401,326,345)(271,400,327,344)(272,399,328,343)(273,398,329,342)(274,397,330,341)(275,396,331,340)(276,395,332,339)(277,394,333,338)(278,393,334,337)(279,392,335,448)(280,391,336,447)>;
G:=Group( (1,257)(2,258)(3,259)(4,260)(5,261)(6,262)(7,263)(8,264)(9,265)(10,266)(11,267)(12,268)(13,269)(14,270)(15,271)(16,272)(17,273)(18,274)(19,275)(20,276)(21,277)(22,278)(23,279)(24,280)(25,281)(26,282)(27,283)(28,284)(29,285)(30,286)(31,287)(32,288)(33,289)(34,290)(35,291)(36,292)(37,293)(38,294)(39,295)(40,296)(41,297)(42,298)(43,299)(44,300)(45,301)(46,302)(47,303)(48,304)(49,305)(50,306)(51,307)(52,308)(53,309)(54,310)(55,311)(56,312)(57,313)(58,314)(59,315)(60,316)(61,317)(62,318)(63,319)(64,320)(65,321)(66,322)(67,323)(68,324)(69,325)(70,326)(71,327)(72,328)(73,329)(74,330)(75,331)(76,332)(77,333)(78,334)(79,335)(80,336)(81,225)(82,226)(83,227)(84,228)(85,229)(86,230)(87,231)(88,232)(89,233)(90,234)(91,235)(92,236)(93,237)(94,238)(95,239)(96,240)(97,241)(98,242)(99,243)(100,244)(101,245)(102,246)(103,247)(104,248)(105,249)(106,250)(107,251)(108,252)(109,253)(110,254)(111,255)(112,256)(113,419)(114,420)(115,421)(116,422)(117,423)(118,424)(119,425)(120,426)(121,427)(122,428)(123,429)(124,430)(125,431)(126,432)(127,433)(128,434)(129,435)(130,436)(131,437)(132,438)(133,439)(134,440)(135,441)(136,442)(137,443)(138,444)(139,445)(140,446)(141,447)(142,448)(143,337)(144,338)(145,339)(146,340)(147,341)(148,342)(149,343)(150,344)(151,345)(152,346)(153,347)(154,348)(155,349)(156,350)(157,351)(158,352)(159,353)(160,354)(161,355)(162,356)(163,357)(164,358)(165,359)(166,360)(167,361)(168,362)(169,363)(170,364)(171,365)(172,366)(173,367)(174,368)(175,369)(176,370)(177,371)(178,372)(179,373)(180,374)(181,375)(182,376)(183,377)(184,378)(185,379)(186,380)(187,381)(188,382)(189,383)(190,384)(191,385)(192,386)(193,387)(194,388)(195,389)(196,390)(197,391)(198,392)(199,393)(200,394)(201,395)(202,396)(203,397)(204,398)(205,399)(206,400)(207,401)(208,402)(209,403)(210,404)(211,405)(212,406)(213,407)(214,408)(215,409)(216,410)(217,411)(218,412)(219,413)(220,414)(221,415)(222,416)(223,417)(224,418), (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), (1,220,57,164)(2,219,58,163)(3,218,59,162)(4,217,60,161)(5,216,61,160)(6,215,62,159)(7,214,63,158)(8,213,64,157)(9,212,65,156)(10,211,66,155)(11,210,67,154)(12,209,68,153)(13,208,69,152)(14,207,70,151)(15,206,71,150)(16,205,72,149)(17,204,73,148)(18,203,74,147)(19,202,75,146)(20,201,76,145)(21,200,77,144)(22,199,78,143)(23,198,79,142)(24,197,80,141)(25,196,81,140)(26,195,82,139)(27,194,83,138)(28,193,84,137)(29,192,85,136)(30,191,86,135)(31,190,87,134)(32,189,88,133)(33,188,89,132)(34,187,90,131)(35,186,91,130)(36,185,92,129)(37,184,93,128)(38,183,94,127)(39,182,95,126)(40,181,96,125)(41,180,97,124)(42,179,98,123)(43,178,99,122)(44,177,100,121)(45,176,101,120)(46,175,102,119)(47,174,103,118)(48,173,104,117)(49,172,105,116)(50,171,106,115)(51,170,107,114)(52,169,108,113)(53,168,109,224)(54,167,110,223)(55,166,111,222)(56,165,112,221)(225,446,281,390)(226,445,282,389)(227,444,283,388)(228,443,284,387)(229,442,285,386)(230,441,286,385)(231,440,287,384)(232,439,288,383)(233,438,289,382)(234,437,290,381)(235,436,291,380)(236,435,292,379)(237,434,293,378)(238,433,294,377)(239,432,295,376)(240,431,296,375)(241,430,297,374)(242,429,298,373)(243,428,299,372)(244,427,300,371)(245,426,301,370)(246,425,302,369)(247,424,303,368)(248,423,304,367)(249,422,305,366)(250,421,306,365)(251,420,307,364)(252,419,308,363)(253,418,309,362)(254,417,310,361)(255,416,311,360)(256,415,312,359)(257,414,313,358)(258,413,314,357)(259,412,315,356)(260,411,316,355)(261,410,317,354)(262,409,318,353)(263,408,319,352)(264,407,320,351)(265,406,321,350)(266,405,322,349)(267,404,323,348)(268,403,324,347)(269,402,325,346)(270,401,326,345)(271,400,327,344)(272,399,328,343)(273,398,329,342)(274,397,330,341)(275,396,331,340)(276,395,332,339)(277,394,333,338)(278,393,334,337)(279,392,335,448)(280,391,336,447) );
G=PermutationGroup([[(1,257),(2,258),(3,259),(4,260),(5,261),(6,262),(7,263),(8,264),(9,265),(10,266),(11,267),(12,268),(13,269),(14,270),(15,271),(16,272),(17,273),(18,274),(19,275),(20,276),(21,277),(22,278),(23,279),(24,280),(25,281),(26,282),(27,283),(28,284),(29,285),(30,286),(31,287),(32,288),(33,289),(34,290),(35,291),(36,292),(37,293),(38,294),(39,295),(40,296),(41,297),(42,298),(43,299),(44,300),(45,301),(46,302),(47,303),(48,304),(49,305),(50,306),(51,307),(52,308),(53,309),(54,310),(55,311),(56,312),(57,313),(58,314),(59,315),(60,316),(61,317),(62,318),(63,319),(64,320),(65,321),(66,322),(67,323),(68,324),(69,325),(70,326),(71,327),(72,328),(73,329),(74,330),(75,331),(76,332),(77,333),(78,334),(79,335),(80,336),(81,225),(82,226),(83,227),(84,228),(85,229),(86,230),(87,231),(88,232),(89,233),(90,234),(91,235),(92,236),(93,237),(94,238),(95,239),(96,240),(97,241),(98,242),(99,243),(100,244),(101,245),(102,246),(103,247),(104,248),(105,249),(106,250),(107,251),(108,252),(109,253),(110,254),(111,255),(112,256),(113,419),(114,420),(115,421),(116,422),(117,423),(118,424),(119,425),(120,426),(121,427),(122,428),(123,429),(124,430),(125,431),(126,432),(127,433),(128,434),(129,435),(130,436),(131,437),(132,438),(133,439),(134,440),(135,441),(136,442),(137,443),(138,444),(139,445),(140,446),(141,447),(142,448),(143,337),(144,338),(145,339),(146,340),(147,341),(148,342),(149,343),(150,344),(151,345),(152,346),(153,347),(154,348),(155,349),(156,350),(157,351),(158,352),(159,353),(160,354),(161,355),(162,356),(163,357),(164,358),(165,359),(166,360),(167,361),(168,362),(169,363),(170,364),(171,365),(172,366),(173,367),(174,368),(175,369),(176,370),(177,371),(178,372),(179,373),(180,374),(181,375),(182,376),(183,377),(184,378),(185,379),(186,380),(187,381),(188,382),(189,383),(190,384),(191,385),(192,386),(193,387),(194,388),(195,389),(196,390),(197,391),(198,392),(199,393),(200,394),(201,395),(202,396),(203,397),(204,398),(205,399),(206,400),(207,401),(208,402),(209,403),(210,404),(211,405),(212,406),(213,407),(214,408),(215,409),(216,410),(217,411),(218,412),(219,413),(220,414),(221,415),(222,416),(223,417),(224,418)], [(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)], [(1,220,57,164),(2,219,58,163),(3,218,59,162),(4,217,60,161),(5,216,61,160),(6,215,62,159),(7,214,63,158),(8,213,64,157),(9,212,65,156),(10,211,66,155),(11,210,67,154),(12,209,68,153),(13,208,69,152),(14,207,70,151),(15,206,71,150),(16,205,72,149),(17,204,73,148),(18,203,74,147),(19,202,75,146),(20,201,76,145),(21,200,77,144),(22,199,78,143),(23,198,79,142),(24,197,80,141),(25,196,81,140),(26,195,82,139),(27,194,83,138),(28,193,84,137),(29,192,85,136),(30,191,86,135),(31,190,87,134),(32,189,88,133),(33,188,89,132),(34,187,90,131),(35,186,91,130),(36,185,92,129),(37,184,93,128),(38,183,94,127),(39,182,95,126),(40,181,96,125),(41,180,97,124),(42,179,98,123),(43,178,99,122),(44,177,100,121),(45,176,101,120),(46,175,102,119),(47,174,103,118),(48,173,104,117),(49,172,105,116),(50,171,106,115),(51,170,107,114),(52,169,108,113),(53,168,109,224),(54,167,110,223),(55,166,111,222),(56,165,112,221),(225,446,281,390),(226,445,282,389),(227,444,283,388),(228,443,284,387),(229,442,285,386),(230,441,286,385),(231,440,287,384),(232,439,288,383),(233,438,289,382),(234,437,290,381),(235,436,291,380),(236,435,292,379),(237,434,293,378),(238,433,294,377),(239,432,295,376),(240,431,296,375),(241,430,297,374),(242,429,298,373),(243,428,299,372),(244,427,300,371),(245,426,301,370),(246,425,302,369),(247,424,303,368),(248,423,304,367),(249,422,305,366),(250,421,306,365),(251,420,307,364),(252,419,308,363),(253,418,309,362),(254,417,310,361),(255,416,311,360),(256,415,312,359),(257,414,313,358),(258,413,314,357),(259,412,315,356),(260,411,316,355),(261,410,317,354),(262,409,318,353),(263,408,319,352),(264,407,320,351),(265,406,321,350),(266,405,322,349),(267,404,323,348),(268,403,324,347),(269,402,325,346),(270,401,326,345),(271,400,327,344),(272,399,328,343),(273,398,329,342),(274,397,330,341),(275,396,331,340),(276,395,332,339),(277,394,333,338),(278,393,334,337),(279,392,335,448),(280,391,336,447)]])
118 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 | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | Q32 | D28 | D28 | D56 | D56 | Dic56 |
| kernel | C2×Dic56 | Dic56 | C2×C112 | C2×Dic28 | C56 | C2×C28 | C2×C16 | C28 | C2×C14 | C16 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C2×Dic56 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 81 | 63 |
| 0 | 50 | 11 |
| 1 | 0 | 0 |
| 0 | 26 | 101 |
| 0 | 47 | 87 |
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[1,0,0,0,81,50,0,63,11],[1,0,0,0,26,47,0,101,87] >;
C2×Dic56 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{56} % in TeX
G:=Group("C2xDic56"); // GroupNames label
G:=SmallGroup(448,439);
// by ID
G=gap.SmallGroup(448,439);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,254,142,675,192,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^112=1,c^2=b^56,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations