direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C56⋊1C4, C22.15D56, C23.57D28, C22.6Dic28, (C2×C56)⋊9C4, C56⋊29(C2×C4), (C2×C8)⋊5Dic7, C8⋊7(C2×Dic7), C2.2(C2×D56), (C2×C14).22D8, C14.15(C2×D8), C14⋊2(C2.D8), (C2×C4).94D28, C28.36(C4⋊C4), C28.74(C2×Q8), (C2×C28).56Q8, (C22×C8).8D7, (C2×C8).306D14, (C2×C28).387D4, C14.10(C2×Q16), (C2×C14).10Q16, C2.3(C2×Dic28), (C22×C56).14C2, C4.17(C4⋊Dic7), (C2×C4).49Dic14, C4.40(C2×Dic14), C22.51(C2×D28), (C2×C56).379C22, C28.170(C22×C4), (C2×C28).764C23, (C22×C14).136D4, (C22×C4).424D14, C4.24(C22×Dic7), C4⋊Dic7.280C22, C22.22(C4⋊Dic7), (C22×C28).516C22, C7⋊3(C2×C2.D8), C14.45(C2×C4⋊C4), C2.11(C2×C4⋊Dic7), (C2×C14).50(C4⋊C4), (C2×C28).299(C2×C4), (C2×C14).154(C2×D4), (C2×C4⋊Dic7).23C2, (C2×C4).82(C2×Dic7), (C2×C4).711(C22×D7), SmallGroup(448,639)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C56⋊1C4
G = < a,b,c | a2=b56=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 548 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2.D8, C2×C4⋊C4, C22×C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C2.D8, C4⋊Dic7, C4⋊Dic7, C2×C56, C22×Dic7, C22×C28, C56⋊1C4, C2×C4⋊Dic7, C22×C56, C2×C56⋊1C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, Dic7, D14, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, Dic14, D28, C2×Dic7, C22×D7, C2×C2.D8, D56, Dic28, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C56⋊1C4, C2×D56, C2×Dic28, C2×C4⋊Dic7, C2×C56⋊1C4
►Regular action on 448 points
Generators in S448
(1 439)(2 440)(3 441)(4 442)(5 443)(6 444)(7 445)(8 446)(9 447)(10 448)(11 393)(12 394)(13 395)(14 396)(15 397)(16 398)(17 399)(18 400)(19 401)(20 402)(21 403)(22 404)(23 405)(24 406)(25 407)(26 408)(27 409)(28 410)(29 411)(30 412)(31 413)(32 414)(33 415)(34 416)(35 417)(36 418)(37 419)(38 420)(39 421)(40 422)(41 423)(42 424)(43 425)(44 426)(45 427)(46 428)(47 429)(48 430)(49 431)(50 432)(51 433)(52 434)(53 435)(54 436)(55 437)(56 438)(57 196)(58 197)(59 198)(60 199)(61 200)(62 201)(63 202)(64 203)(65 204)(66 205)(67 206)(68 207)(69 208)(70 209)(71 210)(72 211)(73 212)(74 213)(75 214)(76 215)(77 216)(78 217)(79 218)(80 219)(81 220)(82 221)(83 222)(84 223)(85 224)(86 169)(87 170)(88 171)(89 172)(90 173)(91 174)(92 175)(93 176)(94 177)(95 178)(96 179)(97 180)(98 181)(99 182)(100 183)(101 184)(102 185)(103 186)(104 187)(105 188)(106 189)(107 190)(108 191)(109 192)(110 193)(111 194)(112 195)(113 359)(114 360)(115 361)(116 362)(117 363)(118 364)(119 365)(120 366)(121 367)(122 368)(123 369)(124 370)(125 371)(126 372)(127 373)(128 374)(129 375)(130 376)(131 377)(132 378)(133 379)(134 380)(135 381)(136 382)(137 383)(138 384)(139 385)(140 386)(141 387)(142 388)(143 389)(144 390)(145 391)(146 392)(147 337)(148 338)(149 339)(150 340)(151 341)(152 342)(153 343)(154 344)(155 345)(156 346)(157 347)(158 348)(159 349)(160 350)(161 351)(162 352)(163 353)(164 354)(165 355)(166 356)(167 357)(168 358)(225 297)(226 298)(227 299)(228 300)(229 301)(230 302)(231 303)(232 304)(233 305)(234 306)(235 307)(236 308)(237 309)(238 310)(239 311)(240 312)(241 313)(242 314)(243 315)(244 316)(245 317)(246 318)(247 319)(248 320)(249 321)(250 322)(251 323)(252 324)(253 325)(254 326)(255 327)(256 328)(257 329)(258 330)(259 331)(260 332)(261 333)(262 334)(263 335)(264 336)(265 281)(266 282)(267 283)(268 284)(269 285)(270 286)(271 287)(272 288)(273 289)(274 290)(275 291)(276 292)(277 293)(278 294)(279 295)(280 296)
(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)
(1 120 299 210)(2 119 300 209)(3 118 301 208)(4 117 302 207)(5 116 303 206)(6 115 304 205)(7 114 305 204)(8 113 306 203)(9 168 307 202)(10 167 308 201)(11 166 309 200)(12 165 310 199)(13 164 311 198)(14 163 312 197)(15 162 313 196)(16 161 314 195)(17 160 315 194)(18 159 316 193)(19 158 317 192)(20 157 318 191)(21 156 319 190)(22 155 320 189)(23 154 321 188)(24 153 322 187)(25 152 323 186)(26 151 324 185)(27 150 325 184)(28 149 326 183)(29 148 327 182)(30 147 328 181)(31 146 329 180)(32 145 330 179)(33 144 331 178)(34 143 332 177)(35 142 333 176)(36 141 334 175)(37 140 335 174)(38 139 336 173)(39 138 281 172)(40 137 282 171)(41 136 283 170)(42 135 284 169)(43 134 285 224)(44 133 286 223)(45 132 287 222)(46 131 288 221)(47 130 289 220)(48 129 290 219)(49 128 291 218)(50 127 292 217)(51 126 293 216)(52 125 294 215)(53 124 295 214)(54 123 296 213)(55 122 297 212)(56 121 298 211)(57 397 352 241)(58 396 353 240)(59 395 354 239)(60 394 355 238)(61 393 356 237)(62 448 357 236)(63 447 358 235)(64 446 359 234)(65 445 360 233)(66 444 361 232)(67 443 362 231)(68 442 363 230)(69 441 364 229)(70 440 365 228)(71 439 366 227)(72 438 367 226)(73 437 368 225)(74 436 369 280)(75 435 370 279)(76 434 371 278)(77 433 372 277)(78 432 373 276)(79 431 374 275)(80 430 375 274)(81 429 376 273)(82 428 377 272)(83 427 378 271)(84 426 379 270)(85 425 380 269)(86 424 381 268)(87 423 382 267)(88 422 383 266)(89 421 384 265)(90 420 385 264)(91 419 386 263)(92 418 387 262)(93 417 388 261)(94 416 389 260)(95 415 390 259)(96 414 391 258)(97 413 392 257)(98 412 337 256)(99 411 338 255)(100 410 339 254)(101 409 340 253)(102 408 341 252)(103 407 342 251)(104 406 343 250)(105 405 344 249)(106 404 345 248)(107 403 346 247)(108 402 347 246)(109 401 348 245)(110 400 349 244)(111 399 350 243)(112 398 351 242)
G:=sub<Sym(448)| (1,439)(2,440)(3,441)(4,442)(5,443)(6,444)(7,445)(8,446)(9,447)(10,448)(11,393)(12,394)(13,395)(14,396)(15,397)(16,398)(17,399)(18,400)(19,401)(20,402)(21,403)(22,404)(23,405)(24,406)(25,407)(26,408)(27,409)(28,410)(29,411)(30,412)(31,413)(32,414)(33,415)(34,416)(35,417)(36,418)(37,419)(38,420)(39,421)(40,422)(41,423)(42,424)(43,425)(44,426)(45,427)(46,428)(47,429)(48,430)(49,431)(50,432)(51,433)(52,434)(53,435)(54,436)(55,437)(56,438)(57,196)(58,197)(59,198)(60,199)(61,200)(62,201)(63,202)(64,203)(65,204)(66,205)(67,206)(68,207)(69,208)(70,209)(71,210)(72,211)(73,212)(74,213)(75,214)(76,215)(77,216)(78,217)(79,218)(80,219)(81,220)(82,221)(83,222)(84,223)(85,224)(86,169)(87,170)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(97,180)(98,181)(99,182)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)(108,191)(109,192)(110,193)(111,194)(112,195)(113,359)(114,360)(115,361)(116,362)(117,363)(118,364)(119,365)(120,366)(121,367)(122,368)(123,369)(124,370)(125,371)(126,372)(127,373)(128,374)(129,375)(130,376)(131,377)(132,378)(133,379)(134,380)(135,381)(136,382)(137,383)(138,384)(139,385)(140,386)(141,387)(142,388)(143,389)(144,390)(145,391)(146,392)(147,337)(148,338)(149,339)(150,340)(151,341)(152,342)(153,343)(154,344)(155,345)(156,346)(157,347)(158,348)(159,349)(160,350)(161,351)(162,352)(163,353)(164,354)(165,355)(166,356)(167,357)(168,358)(225,297)(226,298)(227,299)(228,300)(229,301)(230,302)(231,303)(232,304)(233,305)(234,306)(235,307)(236,308)(237,309)(238,310)(239,311)(240,312)(241,313)(242,314)(243,315)(244,316)(245,317)(246,318)(247,319)(248,320)(249,321)(250,322)(251,323)(252,324)(253,325)(254,326)(255,327)(256,328)(257,329)(258,330)(259,331)(260,332)(261,333)(262,334)(263,335)(264,336)(265,281)(266,282)(267,283)(268,284)(269,285)(270,286)(271,287)(272,288)(273,289)(274,290)(275,291)(276,292)(277,293)(278,294)(279,295)(280,296), (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), (1,120,299,210)(2,119,300,209)(3,118,301,208)(4,117,302,207)(5,116,303,206)(6,115,304,205)(7,114,305,204)(8,113,306,203)(9,168,307,202)(10,167,308,201)(11,166,309,200)(12,165,310,199)(13,164,311,198)(14,163,312,197)(15,162,313,196)(16,161,314,195)(17,160,315,194)(18,159,316,193)(19,158,317,192)(20,157,318,191)(21,156,319,190)(22,155,320,189)(23,154,321,188)(24,153,322,187)(25,152,323,186)(26,151,324,185)(27,150,325,184)(28,149,326,183)(29,148,327,182)(30,147,328,181)(31,146,329,180)(32,145,330,179)(33,144,331,178)(34,143,332,177)(35,142,333,176)(36,141,334,175)(37,140,335,174)(38,139,336,173)(39,138,281,172)(40,137,282,171)(41,136,283,170)(42,135,284,169)(43,134,285,224)(44,133,286,223)(45,132,287,222)(46,131,288,221)(47,130,289,220)(48,129,290,219)(49,128,291,218)(50,127,292,217)(51,126,293,216)(52,125,294,215)(53,124,295,214)(54,123,296,213)(55,122,297,212)(56,121,298,211)(57,397,352,241)(58,396,353,240)(59,395,354,239)(60,394,355,238)(61,393,356,237)(62,448,357,236)(63,447,358,235)(64,446,359,234)(65,445,360,233)(66,444,361,232)(67,443,362,231)(68,442,363,230)(69,441,364,229)(70,440,365,228)(71,439,366,227)(72,438,367,226)(73,437,368,225)(74,436,369,280)(75,435,370,279)(76,434,371,278)(77,433,372,277)(78,432,373,276)(79,431,374,275)(80,430,375,274)(81,429,376,273)(82,428,377,272)(83,427,378,271)(84,426,379,270)(85,425,380,269)(86,424,381,268)(87,423,382,267)(88,422,383,266)(89,421,384,265)(90,420,385,264)(91,419,386,263)(92,418,387,262)(93,417,388,261)(94,416,389,260)(95,415,390,259)(96,414,391,258)(97,413,392,257)(98,412,337,256)(99,411,338,255)(100,410,339,254)(101,409,340,253)(102,408,341,252)(103,407,342,251)(104,406,343,250)(105,405,344,249)(106,404,345,248)(107,403,346,247)(108,402,347,246)(109,401,348,245)(110,400,349,244)(111,399,350,243)(112,398,351,242)>;
G:=Group( (1,439)(2,440)(3,441)(4,442)(5,443)(6,444)(7,445)(8,446)(9,447)(10,448)(11,393)(12,394)(13,395)(14,396)(15,397)(16,398)(17,399)(18,400)(19,401)(20,402)(21,403)(22,404)(23,405)(24,406)(25,407)(26,408)(27,409)(28,410)(29,411)(30,412)(31,413)(32,414)(33,415)(34,416)(35,417)(36,418)(37,419)(38,420)(39,421)(40,422)(41,423)(42,424)(43,425)(44,426)(45,427)(46,428)(47,429)(48,430)(49,431)(50,432)(51,433)(52,434)(53,435)(54,436)(55,437)(56,438)(57,196)(58,197)(59,198)(60,199)(61,200)(62,201)(63,202)(64,203)(65,204)(66,205)(67,206)(68,207)(69,208)(70,209)(71,210)(72,211)(73,212)(74,213)(75,214)(76,215)(77,216)(78,217)(79,218)(80,219)(81,220)(82,221)(83,222)(84,223)(85,224)(86,169)(87,170)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(97,180)(98,181)(99,182)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)(108,191)(109,192)(110,193)(111,194)(112,195)(113,359)(114,360)(115,361)(116,362)(117,363)(118,364)(119,365)(120,366)(121,367)(122,368)(123,369)(124,370)(125,371)(126,372)(127,373)(128,374)(129,375)(130,376)(131,377)(132,378)(133,379)(134,380)(135,381)(136,382)(137,383)(138,384)(139,385)(140,386)(141,387)(142,388)(143,389)(144,390)(145,391)(146,392)(147,337)(148,338)(149,339)(150,340)(151,341)(152,342)(153,343)(154,344)(155,345)(156,346)(157,347)(158,348)(159,349)(160,350)(161,351)(162,352)(163,353)(164,354)(165,355)(166,356)(167,357)(168,358)(225,297)(226,298)(227,299)(228,300)(229,301)(230,302)(231,303)(232,304)(233,305)(234,306)(235,307)(236,308)(237,309)(238,310)(239,311)(240,312)(241,313)(242,314)(243,315)(244,316)(245,317)(246,318)(247,319)(248,320)(249,321)(250,322)(251,323)(252,324)(253,325)(254,326)(255,327)(256,328)(257,329)(258,330)(259,331)(260,332)(261,333)(262,334)(263,335)(264,336)(265,281)(266,282)(267,283)(268,284)(269,285)(270,286)(271,287)(272,288)(273,289)(274,290)(275,291)(276,292)(277,293)(278,294)(279,295)(280,296), (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), (1,120,299,210)(2,119,300,209)(3,118,301,208)(4,117,302,207)(5,116,303,206)(6,115,304,205)(7,114,305,204)(8,113,306,203)(9,168,307,202)(10,167,308,201)(11,166,309,200)(12,165,310,199)(13,164,311,198)(14,163,312,197)(15,162,313,196)(16,161,314,195)(17,160,315,194)(18,159,316,193)(19,158,317,192)(20,157,318,191)(21,156,319,190)(22,155,320,189)(23,154,321,188)(24,153,322,187)(25,152,323,186)(26,151,324,185)(27,150,325,184)(28,149,326,183)(29,148,327,182)(30,147,328,181)(31,146,329,180)(32,145,330,179)(33,144,331,178)(34,143,332,177)(35,142,333,176)(36,141,334,175)(37,140,335,174)(38,139,336,173)(39,138,281,172)(40,137,282,171)(41,136,283,170)(42,135,284,169)(43,134,285,224)(44,133,286,223)(45,132,287,222)(46,131,288,221)(47,130,289,220)(48,129,290,219)(49,128,291,218)(50,127,292,217)(51,126,293,216)(52,125,294,215)(53,124,295,214)(54,123,296,213)(55,122,297,212)(56,121,298,211)(57,397,352,241)(58,396,353,240)(59,395,354,239)(60,394,355,238)(61,393,356,237)(62,448,357,236)(63,447,358,235)(64,446,359,234)(65,445,360,233)(66,444,361,232)(67,443,362,231)(68,442,363,230)(69,441,364,229)(70,440,365,228)(71,439,366,227)(72,438,367,226)(73,437,368,225)(74,436,369,280)(75,435,370,279)(76,434,371,278)(77,433,372,277)(78,432,373,276)(79,431,374,275)(80,430,375,274)(81,429,376,273)(82,428,377,272)(83,427,378,271)(84,426,379,270)(85,425,380,269)(86,424,381,268)(87,423,382,267)(88,422,383,266)(89,421,384,265)(90,420,385,264)(91,419,386,263)(92,418,387,262)(93,417,388,261)(94,416,389,260)(95,415,390,259)(96,414,391,258)(97,413,392,257)(98,412,337,256)(99,411,338,255)(100,410,339,254)(101,409,340,253)(102,408,341,252)(103,407,342,251)(104,406,343,250)(105,405,344,249)(106,404,345,248)(107,403,346,247)(108,402,347,246)(109,401,348,245)(110,400,349,244)(111,399,350,243)(112,398,351,242) );
G=PermutationGroup([[(1,439),(2,440),(3,441),(4,442),(5,443),(6,444),(7,445),(8,446),(9,447),(10,448),(11,393),(12,394),(13,395),(14,396),(15,397),(16,398),(17,399),(18,400),(19,401),(20,402),(21,403),(22,404),(23,405),(24,406),(25,407),(26,408),(27,409),(28,410),(29,411),(30,412),(31,413),(32,414),(33,415),(34,416),(35,417),(36,418),(37,419),(38,420),(39,421),(40,422),(41,423),(42,424),(43,425),(44,426),(45,427),(46,428),(47,429),(48,430),(49,431),(50,432),(51,433),(52,434),(53,435),(54,436),(55,437),(56,438),(57,196),(58,197),(59,198),(60,199),(61,200),(62,201),(63,202),(64,203),(65,204),(66,205),(67,206),(68,207),(69,208),(70,209),(71,210),(72,211),(73,212),(74,213),(75,214),(76,215),(77,216),(78,217),(79,218),(80,219),(81,220),(82,221),(83,222),(84,223),(85,224),(86,169),(87,170),(88,171),(89,172),(90,173),(91,174),(92,175),(93,176),(94,177),(95,178),(96,179),(97,180),(98,181),(99,182),(100,183),(101,184),(102,185),(103,186),(104,187),(105,188),(106,189),(107,190),(108,191),(109,192),(110,193),(111,194),(112,195),(113,359),(114,360),(115,361),(116,362),(117,363),(118,364),(119,365),(120,366),(121,367),(122,368),(123,369),(124,370),(125,371),(126,372),(127,373),(128,374),(129,375),(130,376),(131,377),(132,378),(133,379),(134,380),(135,381),(136,382),(137,383),(138,384),(139,385),(140,386),(141,387),(142,388),(143,389),(144,390),(145,391),(146,392),(147,337),(148,338),(149,339),(150,340),(151,341),(152,342),(153,343),(154,344),(155,345),(156,346),(157,347),(158,348),(159,349),(160,350),(161,351),(162,352),(163,353),(164,354),(165,355),(166,356),(167,357),(168,358),(225,297),(226,298),(227,299),(228,300),(229,301),(230,302),(231,303),(232,304),(233,305),(234,306),(235,307),(236,308),(237,309),(238,310),(239,311),(240,312),(241,313),(242,314),(243,315),(244,316),(245,317),(246,318),(247,319),(248,320),(249,321),(250,322),(251,323),(252,324),(253,325),(254,326),(255,327),(256,328),(257,329),(258,330),(259,331),(260,332),(261,333),(262,334),(263,335),(264,336),(265,281),(266,282),(267,283),(268,284),(269,285),(270,286),(271,287),(272,288),(273,289),(274,290),(275,291),(276,292),(277,293),(278,294),(279,295),(280,296)], [(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)], [(1,120,299,210),(2,119,300,209),(3,118,301,208),(4,117,302,207),(5,116,303,206),(6,115,304,205),(7,114,305,204),(8,113,306,203),(9,168,307,202),(10,167,308,201),(11,166,309,200),(12,165,310,199),(13,164,311,198),(14,163,312,197),(15,162,313,196),(16,161,314,195),(17,160,315,194),(18,159,316,193),(19,158,317,192),(20,157,318,191),(21,156,319,190),(22,155,320,189),(23,154,321,188),(24,153,322,187),(25,152,323,186),(26,151,324,185),(27,150,325,184),(28,149,326,183),(29,148,327,182),(30,147,328,181),(31,146,329,180),(32,145,330,179),(33,144,331,178),(34,143,332,177),(35,142,333,176),(36,141,334,175),(37,140,335,174),(38,139,336,173),(39,138,281,172),(40,137,282,171),(41,136,283,170),(42,135,284,169),(43,134,285,224),(44,133,286,223),(45,132,287,222),(46,131,288,221),(47,130,289,220),(48,129,290,219),(49,128,291,218),(50,127,292,217),(51,126,293,216),(52,125,294,215),(53,124,295,214),(54,123,296,213),(55,122,297,212),(56,121,298,211),(57,397,352,241),(58,396,353,240),(59,395,354,239),(60,394,355,238),(61,393,356,237),(62,448,357,236),(63,447,358,235),(64,446,359,234),(65,445,360,233),(66,444,361,232),(67,443,362,231),(68,442,363,230),(69,441,364,229),(70,440,365,228),(71,439,366,227),(72,438,367,226),(73,437,368,225),(74,436,369,280),(75,435,370,279),(76,434,371,278),(77,433,372,277),(78,432,373,276),(79,431,374,275),(80,430,375,274),(81,429,376,273),(82,428,377,272),(83,427,378,271),(84,426,379,270),(85,425,380,269),(86,424,381,268),(87,423,382,267),(88,422,383,266),(89,421,384,265),(90,420,385,264),(91,419,386,263),(92,418,387,262),(93,417,388,261),(94,416,389,260),(95,415,390,259),(96,414,391,258),(97,413,392,257),(98,412,337,256),(99,411,338,255),(100,410,339,254),(101,409,340,253),(102,408,341,252),(103,407,342,251),(104,406,343,250),(105,405,344,249),(106,404,345,248),(107,403,346,247),(108,402,347,246),(109,401,348,245),(110,400,349,244),(111,399,350,243),(112,398,351,242)]])
124 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | + | - | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D7 | D8 | Q16 | Dic7 | D14 | D14 | Dic14 | D28 | D28 | D56 | Dic28 |
| kernel | C2×C56⋊1C4 | C56⋊1C4 | C2×C4⋊Dic7 | C22×C56 | C2×C56 | C2×C28 | C2×C28 | C22×C14 | C22×C8 | C2×C14 | C2×C14 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 1 | 3 | 4 | 4 | 12 | 6 | 3 | 12 | 6 | 6 | 24 | 24 |
Matrix representation of C2×C56⋊1C4 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 105 | 69 |
| 0 | 0 | 0 | 14 | 91 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 104 | 101 | 0 | 0 |
| 0 | 101 | 9 | 0 | 0 |
| 0 | 0 | 0 | 73 | 7 |
| 0 | 0 | 0 | 46 | 40 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,112,0,0,0,0,0,0,105,14,0,0,0,69,91],[112,0,0,0,0,0,104,101,0,0,0,101,9,0,0,0,0,0,73,46,0,0,0,7,40] >;
C2×C56⋊1C4 in GAP, Magma, Sage, TeX
C_2\times C_{56}\rtimes_1C_4 % in TeX
G:=Group("C2xC56:1C4"); // GroupNames label
G:=SmallGroup(448,639);
// by ID
G=gap.SmallGroup(448,639);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,268,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^56=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations