direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C56⋊C4, C28.26C42, (C2×C56)⋊13C4, C56⋊32(C2×C4), (C2×C8)⋊8Dic7, C8⋊9(C2×Dic7), C14⋊2(C8⋊C4), (C2×C8).327D14, C4.21(C4×Dic7), (C22×C8).15D7, C23.62(C4×D7), (C2×C14).25C42, (C22×C56).24C2, C14.23(C2×C42), (C4×Dic7).21C4, (C2×C56).414C22, (C2×C28).855C23, C28.138(C22×C4), (C22×C4).462D14, (C2×C14).12M4(2), C14.11(C2×M4(2)), C4.32(C22×Dic7), C22.21(C4×Dic7), C22.10(C8⋊D7), (C22×Dic7).12C4, (C22×C28).558C22, (C4×Dic7).280C22, (C2×C7⋊C8)⋊13C4, C7⋊C8⋊27(C2×C4), C7⋊3(C2×C8⋊C4), C4.112(C2×C4×D7), C2.4(C2×C8⋊D7), C2.11(C2×C4×Dic7), C22.58(C2×C4×D7), (C2×C4).183(C4×D7), (C22×C7⋊C8).20C2, (C2×C4×Dic7).38C2, (C2×C28).253(C2×C4), (C2×C7⋊C8).318C22, (C22×C14).90(C2×C4), (C2×Dic7).98(C2×C4), (C2×C4).103(C2×Dic7), (C2×C4).797(C22×D7), (C2×C14).125(C22×C4), SmallGroup(448,634)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C56⋊C4
G = < a,b,c | a2=b56=c4=1, ab=ba, ac=ca, cbc-1=b13 >
Subgroups: 420 in 146 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C8⋊C4, C2×C42, C22×C8, C22×C8, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C8⋊C4, C2×C7⋊C8, C4×Dic7, C2×C56, C22×Dic7, C22×C28, C56⋊C4, C22×C7⋊C8, C2×C4×Dic7, C22×C56, C2×C56⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, M4(2), C22×C4, Dic7, D14, C8⋊C4, C2×C42, C2×M4(2), C4×D7, C2×Dic7, C22×D7, C2×C8⋊C4, C8⋊D7, C4×Dic7, C2×C4×D7, C22×Dic7, C56⋊C4, C2×C8⋊D7, C2×C4×Dic7, C2×C56⋊C4
►Regular action on 448 points
Generators in S448
(1 198)(2 199)(3 200)(4 201)(5 202)(6 203)(7 204)(8 205)(9 206)(10 207)(11 208)(12 209)(13 210)(14 211)(15 212)(16 213)(17 214)(18 215)(19 216)(20 217)(21 218)(22 219)(23 220)(24 221)(25 222)(26 223)(27 224)(28 169)(29 170)(30 171)(31 172)(32 173)(33 174)(34 175)(35 176)(36 177)(37 178)(38 179)(39 180)(40 181)(41 182)(42 183)(43 184)(44 185)(45 186)(46 187)(47 188)(48 189)(49 190)(50 191)(51 192)(52 193)(53 194)(54 195)(55 196)(56 197)(57 327)(58 328)(59 329)(60 330)(61 331)(62 332)(63 333)(64 334)(65 335)(66 336)(67 281)(68 282)(69 283)(70 284)(71 285)(72 286)(73 287)(74 288)(75 289)(76 290)(77 291)(78 292)(79 293)(80 294)(81 295)(82 296)(83 297)(84 298)(85 299)(86 300)(87 301)(88 302)(89 303)(90 304)(91 305)(92 306)(93 307)(94 308)(95 309)(96 310)(97 311)(98 312)(99 313)(100 314)(101 315)(102 316)(103 317)(104 318)(105 319)(106 320)(107 321)(108 322)(109 323)(110 324)(111 325)(112 326)(113 371)(114 372)(115 373)(116 374)(117 375)(118 376)(119 377)(120 378)(121 379)(122 380)(123 381)(124 382)(125 383)(126 384)(127 385)(128 386)(129 387)(130 388)(131 389)(132 390)(133 391)(134 392)(135 337)(136 338)(137 339)(138 340)(139 341)(140 342)(141 343)(142 344)(143 345)(144 346)(145 347)(146 348)(147 349)(148 350)(149 351)(150 352)(151 353)(152 354)(153 355)(154 356)(155 357)(156 358)(157 359)(158 360)(159 361)(160 362)(161 363)(162 364)(163 365)(164 366)(165 367)(166 368)(167 369)(168 370)(225 420)(226 421)(227 422)(228 423)(229 424)(230 425)(231 426)(232 427)(233 428)(234 429)(235 430)(236 431)(237 432)(238 433)(239 434)(240 435)(241 436)(242 437)(243 438)(244 439)(245 440)(246 441)(247 442)(248 443)(249 444)(250 445)(251 446)(252 447)(253 448)(254 393)(255 394)(256 395)(257 396)(258 397)(259 398)(260 399)(261 400)(262 401)(263 402)(264 403)(265 404)(266 405)(267 406)(268 407)(269 408)(270 409)(271 410)(272 411)(273 412)(274 413)(275 414)(276 415)(277 416)(278 417)(279 418)(280 419)
(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)
(1 282 151 393)(2 295 152 406)(3 308 153 419)(4 321 154 432)(5 334 155 445)(6 291 156 402)(7 304 157 415)(8 317 158 428)(9 330 159 441)(10 287 160 398)(11 300 161 411)(12 313 162 424)(13 326 163 437)(14 283 164 394)(15 296 165 407)(16 309 166 420)(17 322 167 433)(18 335 168 446)(19 292 113 403)(20 305 114 416)(21 318 115 429)(22 331 116 442)(23 288 117 399)(24 301 118 412)(25 314 119 425)(26 327 120 438)(27 284 121 395)(28 297 122 408)(29 310 123 421)(30 323 124 434)(31 336 125 447)(32 293 126 404)(33 306 127 417)(34 319 128 430)(35 332 129 443)(36 289 130 400)(37 302 131 413)(38 315 132 426)(39 328 133 439)(40 285 134 396)(41 298 135 409)(42 311 136 422)(43 324 137 435)(44 281 138 448)(45 294 139 405)(46 307 140 418)(47 320 141 431)(48 333 142 444)(49 290 143 401)(50 303 144 414)(51 316 145 427)(52 329 146 440)(53 286 147 397)(54 299 148 410)(55 312 149 423)(56 325 150 436)(57 378 243 223)(58 391 244 180)(59 348 245 193)(60 361 246 206)(61 374 247 219)(62 387 248 176)(63 344 249 189)(64 357 250 202)(65 370 251 215)(66 383 252 172)(67 340 253 185)(68 353 254 198)(69 366 255 211)(70 379 256 224)(71 392 257 181)(72 349 258 194)(73 362 259 207)(74 375 260 220)(75 388 261 177)(76 345 262 190)(77 358 263 203)(78 371 264 216)(79 384 265 173)(80 341 266 186)(81 354 267 199)(82 367 268 212)(83 380 269 169)(84 337 270 182)(85 350 271 195)(86 363 272 208)(87 376 273 221)(88 389 274 178)(89 346 275 191)(90 359 276 204)(91 372 277 217)(92 385 278 174)(93 342 279 187)(94 355 280 200)(95 368 225 213)(96 381 226 170)(97 338 227 183)(98 351 228 196)(99 364 229 209)(100 377 230 222)(101 390 231 179)(102 347 232 192)(103 360 233 205)(104 373 234 218)(105 386 235 175)(106 343 236 188)(107 356 237 201)(108 369 238 214)(109 382 239 171)(110 339 240 184)(111 352 241 197)(112 365 242 210)
G:=sub<Sym(448)| (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,211)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,169)(29,170)(30,171)(31,172)(32,173)(33,174)(34,175)(35,176)(36,177)(37,178)(38,179)(39,180)(40,181)(41,182)(42,183)(43,184)(44,185)(45,186)(46,187)(47,188)(48,189)(49,190)(50,191)(51,192)(52,193)(53,194)(54,195)(55,196)(56,197)(57,327)(58,328)(59,329)(60,330)(61,331)(62,332)(63,333)(64,334)(65,335)(66,336)(67,281)(68,282)(69,283)(70,284)(71,285)(72,286)(73,287)(74,288)(75,289)(76,290)(77,291)(78,292)(79,293)(80,294)(81,295)(82,296)(83,297)(84,298)(85,299)(86,300)(87,301)(88,302)(89,303)(90,304)(91,305)(92,306)(93,307)(94,308)(95,309)(96,310)(97,311)(98,312)(99,313)(100,314)(101,315)(102,316)(103,317)(104,318)(105,319)(106,320)(107,321)(108,322)(109,323)(110,324)(111,325)(112,326)(113,371)(114,372)(115,373)(116,374)(117,375)(118,376)(119,377)(120,378)(121,379)(122,380)(123,381)(124,382)(125,383)(126,384)(127,385)(128,386)(129,387)(130,388)(131,389)(132,390)(133,391)(134,392)(135,337)(136,338)(137,339)(138,340)(139,341)(140,342)(141,343)(142,344)(143,345)(144,346)(145,347)(146,348)(147,349)(148,350)(149,351)(150,352)(151,353)(152,354)(153,355)(154,356)(155,357)(156,358)(157,359)(158,360)(159,361)(160,362)(161,363)(162,364)(163,365)(164,366)(165,367)(166,368)(167,369)(168,370)(225,420)(226,421)(227,422)(228,423)(229,424)(230,425)(231,426)(232,427)(233,428)(234,429)(235,430)(236,431)(237,432)(238,433)(239,434)(240,435)(241,436)(242,437)(243,438)(244,439)(245,440)(246,441)(247,442)(248,443)(249,444)(250,445)(251,446)(252,447)(253,448)(254,393)(255,394)(256,395)(257,396)(258,397)(259,398)(260,399)(261,400)(262,401)(263,402)(264,403)(265,404)(266,405)(267,406)(268,407)(269,408)(270,409)(271,410)(272,411)(273,412)(274,413)(275,414)(276,415)(277,416)(278,417)(279,418)(280,419), (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), (1,282,151,393)(2,295,152,406)(3,308,153,419)(4,321,154,432)(5,334,155,445)(6,291,156,402)(7,304,157,415)(8,317,158,428)(9,330,159,441)(10,287,160,398)(11,300,161,411)(12,313,162,424)(13,326,163,437)(14,283,164,394)(15,296,165,407)(16,309,166,420)(17,322,167,433)(18,335,168,446)(19,292,113,403)(20,305,114,416)(21,318,115,429)(22,331,116,442)(23,288,117,399)(24,301,118,412)(25,314,119,425)(26,327,120,438)(27,284,121,395)(28,297,122,408)(29,310,123,421)(30,323,124,434)(31,336,125,447)(32,293,126,404)(33,306,127,417)(34,319,128,430)(35,332,129,443)(36,289,130,400)(37,302,131,413)(38,315,132,426)(39,328,133,439)(40,285,134,396)(41,298,135,409)(42,311,136,422)(43,324,137,435)(44,281,138,448)(45,294,139,405)(46,307,140,418)(47,320,141,431)(48,333,142,444)(49,290,143,401)(50,303,144,414)(51,316,145,427)(52,329,146,440)(53,286,147,397)(54,299,148,410)(55,312,149,423)(56,325,150,436)(57,378,243,223)(58,391,244,180)(59,348,245,193)(60,361,246,206)(61,374,247,219)(62,387,248,176)(63,344,249,189)(64,357,250,202)(65,370,251,215)(66,383,252,172)(67,340,253,185)(68,353,254,198)(69,366,255,211)(70,379,256,224)(71,392,257,181)(72,349,258,194)(73,362,259,207)(74,375,260,220)(75,388,261,177)(76,345,262,190)(77,358,263,203)(78,371,264,216)(79,384,265,173)(80,341,266,186)(81,354,267,199)(82,367,268,212)(83,380,269,169)(84,337,270,182)(85,350,271,195)(86,363,272,208)(87,376,273,221)(88,389,274,178)(89,346,275,191)(90,359,276,204)(91,372,277,217)(92,385,278,174)(93,342,279,187)(94,355,280,200)(95,368,225,213)(96,381,226,170)(97,338,227,183)(98,351,228,196)(99,364,229,209)(100,377,230,222)(101,390,231,179)(102,347,232,192)(103,360,233,205)(104,373,234,218)(105,386,235,175)(106,343,236,188)(107,356,237,201)(108,369,238,214)(109,382,239,171)(110,339,240,184)(111,352,241,197)(112,365,242,210)>;
G:=Group( (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,211)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,169)(29,170)(30,171)(31,172)(32,173)(33,174)(34,175)(35,176)(36,177)(37,178)(38,179)(39,180)(40,181)(41,182)(42,183)(43,184)(44,185)(45,186)(46,187)(47,188)(48,189)(49,190)(50,191)(51,192)(52,193)(53,194)(54,195)(55,196)(56,197)(57,327)(58,328)(59,329)(60,330)(61,331)(62,332)(63,333)(64,334)(65,335)(66,336)(67,281)(68,282)(69,283)(70,284)(71,285)(72,286)(73,287)(74,288)(75,289)(76,290)(77,291)(78,292)(79,293)(80,294)(81,295)(82,296)(83,297)(84,298)(85,299)(86,300)(87,301)(88,302)(89,303)(90,304)(91,305)(92,306)(93,307)(94,308)(95,309)(96,310)(97,311)(98,312)(99,313)(100,314)(101,315)(102,316)(103,317)(104,318)(105,319)(106,320)(107,321)(108,322)(109,323)(110,324)(111,325)(112,326)(113,371)(114,372)(115,373)(116,374)(117,375)(118,376)(119,377)(120,378)(121,379)(122,380)(123,381)(124,382)(125,383)(126,384)(127,385)(128,386)(129,387)(130,388)(131,389)(132,390)(133,391)(134,392)(135,337)(136,338)(137,339)(138,340)(139,341)(140,342)(141,343)(142,344)(143,345)(144,346)(145,347)(146,348)(147,349)(148,350)(149,351)(150,352)(151,353)(152,354)(153,355)(154,356)(155,357)(156,358)(157,359)(158,360)(159,361)(160,362)(161,363)(162,364)(163,365)(164,366)(165,367)(166,368)(167,369)(168,370)(225,420)(226,421)(227,422)(228,423)(229,424)(230,425)(231,426)(232,427)(233,428)(234,429)(235,430)(236,431)(237,432)(238,433)(239,434)(240,435)(241,436)(242,437)(243,438)(244,439)(245,440)(246,441)(247,442)(248,443)(249,444)(250,445)(251,446)(252,447)(253,448)(254,393)(255,394)(256,395)(257,396)(258,397)(259,398)(260,399)(261,400)(262,401)(263,402)(264,403)(265,404)(266,405)(267,406)(268,407)(269,408)(270,409)(271,410)(272,411)(273,412)(274,413)(275,414)(276,415)(277,416)(278,417)(279,418)(280,419), (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), (1,282,151,393)(2,295,152,406)(3,308,153,419)(4,321,154,432)(5,334,155,445)(6,291,156,402)(7,304,157,415)(8,317,158,428)(9,330,159,441)(10,287,160,398)(11,300,161,411)(12,313,162,424)(13,326,163,437)(14,283,164,394)(15,296,165,407)(16,309,166,420)(17,322,167,433)(18,335,168,446)(19,292,113,403)(20,305,114,416)(21,318,115,429)(22,331,116,442)(23,288,117,399)(24,301,118,412)(25,314,119,425)(26,327,120,438)(27,284,121,395)(28,297,122,408)(29,310,123,421)(30,323,124,434)(31,336,125,447)(32,293,126,404)(33,306,127,417)(34,319,128,430)(35,332,129,443)(36,289,130,400)(37,302,131,413)(38,315,132,426)(39,328,133,439)(40,285,134,396)(41,298,135,409)(42,311,136,422)(43,324,137,435)(44,281,138,448)(45,294,139,405)(46,307,140,418)(47,320,141,431)(48,333,142,444)(49,290,143,401)(50,303,144,414)(51,316,145,427)(52,329,146,440)(53,286,147,397)(54,299,148,410)(55,312,149,423)(56,325,150,436)(57,378,243,223)(58,391,244,180)(59,348,245,193)(60,361,246,206)(61,374,247,219)(62,387,248,176)(63,344,249,189)(64,357,250,202)(65,370,251,215)(66,383,252,172)(67,340,253,185)(68,353,254,198)(69,366,255,211)(70,379,256,224)(71,392,257,181)(72,349,258,194)(73,362,259,207)(74,375,260,220)(75,388,261,177)(76,345,262,190)(77,358,263,203)(78,371,264,216)(79,384,265,173)(80,341,266,186)(81,354,267,199)(82,367,268,212)(83,380,269,169)(84,337,270,182)(85,350,271,195)(86,363,272,208)(87,376,273,221)(88,389,274,178)(89,346,275,191)(90,359,276,204)(91,372,277,217)(92,385,278,174)(93,342,279,187)(94,355,280,200)(95,368,225,213)(96,381,226,170)(97,338,227,183)(98,351,228,196)(99,364,229,209)(100,377,230,222)(101,390,231,179)(102,347,232,192)(103,360,233,205)(104,373,234,218)(105,386,235,175)(106,343,236,188)(107,356,237,201)(108,369,238,214)(109,382,239,171)(110,339,240,184)(111,352,241,197)(112,365,242,210) );
G=PermutationGroup([[(1,198),(2,199),(3,200),(4,201),(5,202),(6,203),(7,204),(8,205),(9,206),(10,207),(11,208),(12,209),(13,210),(14,211),(15,212),(16,213),(17,214),(18,215),(19,216),(20,217),(21,218),(22,219),(23,220),(24,221),(25,222),(26,223),(27,224),(28,169),(29,170),(30,171),(31,172),(32,173),(33,174),(34,175),(35,176),(36,177),(37,178),(38,179),(39,180),(40,181),(41,182),(42,183),(43,184),(44,185),(45,186),(46,187),(47,188),(48,189),(49,190),(50,191),(51,192),(52,193),(53,194),(54,195),(55,196),(56,197),(57,327),(58,328),(59,329),(60,330),(61,331),(62,332),(63,333),(64,334),(65,335),(66,336),(67,281),(68,282),(69,283),(70,284),(71,285),(72,286),(73,287),(74,288),(75,289),(76,290),(77,291),(78,292),(79,293),(80,294),(81,295),(82,296),(83,297),(84,298),(85,299),(86,300),(87,301),(88,302),(89,303),(90,304),(91,305),(92,306),(93,307),(94,308),(95,309),(96,310),(97,311),(98,312),(99,313),(100,314),(101,315),(102,316),(103,317),(104,318),(105,319),(106,320),(107,321),(108,322),(109,323),(110,324),(111,325),(112,326),(113,371),(114,372),(115,373),(116,374),(117,375),(118,376),(119,377),(120,378),(121,379),(122,380),(123,381),(124,382),(125,383),(126,384),(127,385),(128,386),(129,387),(130,388),(131,389),(132,390),(133,391),(134,392),(135,337),(136,338),(137,339),(138,340),(139,341),(140,342),(141,343),(142,344),(143,345),(144,346),(145,347),(146,348),(147,349),(148,350),(149,351),(150,352),(151,353),(152,354),(153,355),(154,356),(155,357),(156,358),(157,359),(158,360),(159,361),(160,362),(161,363),(162,364),(163,365),(164,366),(165,367),(166,368),(167,369),(168,370),(225,420),(226,421),(227,422),(228,423),(229,424),(230,425),(231,426),(232,427),(233,428),(234,429),(235,430),(236,431),(237,432),(238,433),(239,434),(240,435),(241,436),(242,437),(243,438),(244,439),(245,440),(246,441),(247,442),(248,443),(249,444),(250,445),(251,446),(252,447),(253,448),(254,393),(255,394),(256,395),(257,396),(258,397),(259,398),(260,399),(261,400),(262,401),(263,402),(264,403),(265,404),(266,405),(267,406),(268,407),(269,408),(270,409),(271,410),(272,411),(273,412),(274,413),(275,414),(276,415),(277,416),(278,417),(279,418),(280,419)], [(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)], [(1,282,151,393),(2,295,152,406),(3,308,153,419),(4,321,154,432),(5,334,155,445),(6,291,156,402),(7,304,157,415),(8,317,158,428),(9,330,159,441),(10,287,160,398),(11,300,161,411),(12,313,162,424),(13,326,163,437),(14,283,164,394),(15,296,165,407),(16,309,166,420),(17,322,167,433),(18,335,168,446),(19,292,113,403),(20,305,114,416),(21,318,115,429),(22,331,116,442),(23,288,117,399),(24,301,118,412),(25,314,119,425),(26,327,120,438),(27,284,121,395),(28,297,122,408),(29,310,123,421),(30,323,124,434),(31,336,125,447),(32,293,126,404),(33,306,127,417),(34,319,128,430),(35,332,129,443),(36,289,130,400),(37,302,131,413),(38,315,132,426),(39,328,133,439),(40,285,134,396),(41,298,135,409),(42,311,136,422),(43,324,137,435),(44,281,138,448),(45,294,139,405),(46,307,140,418),(47,320,141,431),(48,333,142,444),(49,290,143,401),(50,303,144,414),(51,316,145,427),(52,329,146,440),(53,286,147,397),(54,299,148,410),(55,312,149,423),(56,325,150,436),(57,378,243,223),(58,391,244,180),(59,348,245,193),(60,361,246,206),(61,374,247,219),(62,387,248,176),(63,344,249,189),(64,357,250,202),(65,370,251,215),(66,383,252,172),(67,340,253,185),(68,353,254,198),(69,366,255,211),(70,379,256,224),(71,392,257,181),(72,349,258,194),(73,362,259,207),(74,375,260,220),(75,388,261,177),(76,345,262,190),(77,358,263,203),(78,371,264,216),(79,384,265,173),(80,341,266,186),(81,354,267,199),(82,367,268,212),(83,380,269,169),(84,337,270,182),(85,350,271,195),(86,363,272,208),(87,376,273,221),(88,389,274,178),(89,346,275,191),(90,359,276,204),(91,372,277,217),(92,385,278,174),(93,342,279,187),(94,355,280,200),(95,368,225,213),(96,381,226,170),(97,338,227,183),(98,351,228,196),(99,364,229,209),(100,377,230,222),(101,390,231,179),(102,347,232,192),(103,360,233,205),(104,373,234,218),(105,386,235,175),(106,343,236,188),(107,356,237,201),(108,369,238,214),(109,382,239,171),(110,339,240,184),(111,352,241,197),(112,365,242,210)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D7 | M4(2) | Dic7 | D14 | D14 | C4×D7 | C4×D7 | C8⋊D7 |
| kernel | C2×C56⋊C4 | C56⋊C4 | C22×C7⋊C8 | C2×C4×Dic7 | C22×C56 | C2×C7⋊C8 | C4×Dic7 | C2×C56 | C22×Dic7 | C22×C8 | C2×C14 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 8 | 4 | 3 | 8 | 12 | 6 | 3 | 18 | 6 | 48 |
Matrix representation of C2×C56⋊C4 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 89 | 86 |
| 0 | 0 | 88 | 29 |
| 1 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 4 | 58 |
| 0 | 0 | 64 | 109 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,15,0,0,0,0,89,88,0,0,86,29],[1,0,0,0,0,98,0,0,0,0,4,64,0,0,58,109] >;
C2×C56⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_{56}\rtimes C_4 % in TeX
G:=Group("C2xC56:C4"); // GroupNames label
G:=SmallGroup(448,634);
// by ID
G=gap.SmallGroup(448,634);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,758,100,102,18822]);
// 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^13>;
// generators/relations