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## G = C2×C4×C56order 448 = 26·7

### Abelian group of type [2,4,56]

Aliases: C2×C4×C56, SmallGroup(448,810)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4×C56
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — C2×C56 — C4×C56 — C2×C4×C56
 Lower central C1 — C2×C4×C56
 Upper central C1 — C2×C4×C56

Generators and relations for C2×C4×C56
G = < a,b,c | a2=b4=c56=1, ab=ba, ac=ca, bc=cb >

Subgroups: 162, all normal (18 characteristic)
C1, C2, C2 [×6], C4 [×12], C22, C22 [×6], C7, C8 [×8], C2×C4 [×2], C2×C4 [×16], C23, C14, C14 [×6], C42 [×4], C2×C8 [×12], C22×C4, C22×C4 [×2], C28 [×12], C2×C14, C2×C14 [×6], C4×C8 [×4], C2×C42, C22×C8 [×2], C56 [×8], C2×C28 [×2], C2×C28 [×16], C22×C14, C2×C4×C8, C4×C28 [×4], C2×C56 [×12], C22×C28, C22×C28 [×2], C4×C56 [×4], C2×C4×C28, C22×C56 [×2], C2×C4×C56
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C7, C8 [×8], C2×C4 [×18], C23, C14 [×7], C42 [×4], C2×C8 [×12], C22×C4 [×3], C28 [×12], C2×C14 [×7], C4×C8 [×4], C2×C42, C22×C8 [×2], C56 [×8], C2×C28 [×18], C22×C14, C2×C4×C8, C4×C28 [×4], C2×C56 [×12], C22×C28 [×3], C4×C56 [×4], C2×C4×C28, C22×C56 [×2], C2×C4×C56

Smallest permutation representation of C2×C4×C56
Regular action on 448 points
Generators in S448
(1 303)(2 304)(3 305)(4 306)(5 307)(6 308)(7 309)(8 310)(9 311)(10 312)(11 313)(12 314)(13 315)(14 316)(15 317)(16 318)(17 319)(18 320)(19 321)(20 322)(21 323)(22 324)(23 325)(24 326)(25 327)(26 328)(27 329)(28 330)(29 331)(30 332)(31 333)(32 334)(33 335)(34 336)(35 281)(36 282)(37 283)(38 284)(39 285)(40 286)(41 287)(42 288)(43 289)(44 290)(45 291)(46 292)(47 293)(48 294)(49 295)(50 296)(51 297)(52 298)(53 299)(54 300)(55 301)(56 302)(57 446)(58 447)(59 448)(60 393)(61 394)(62 395)(63 396)(64 397)(65 398)(66 399)(67 400)(68 401)(69 402)(70 403)(71 404)(72 405)(73 406)(74 407)(75 408)(76 409)(77 410)(78 411)(79 412)(80 413)(81 414)(82 415)(83 416)(84 417)(85 418)(86 419)(87 420)(88 421)(89 422)(90 423)(91 424)(92 425)(93 426)(94 427)(95 428)(96 429)(97 430)(98 431)(99 432)(100 433)(101 434)(102 435)(103 436)(104 437)(105 438)(106 439)(107 440)(108 441)(109 442)(110 443)(111 444)(112 445)(113 256)(114 257)(115 258)(116 259)(117 260)(118 261)(119 262)(120 263)(121 264)(122 265)(123 266)(124 267)(125 268)(126 269)(127 270)(128 271)(129 272)(130 273)(131 274)(132 275)(133 276)(134 277)(135 278)(136 279)(137 280)(138 225)(139 226)(140 227)(141 228)(142 229)(143 230)(144 231)(145 232)(146 233)(147 234)(148 235)(149 236)(150 237)(151 238)(152 239)(153 240)(154 241)(155 242)(156 243)(157 244)(158 245)(159 246)(160 247)(161 248)(162 249)(163 250)(164 251)(165 252)(166 253)(167 254)(168 255)(169 371)(170 372)(171 373)(172 374)(173 375)(174 376)(175 377)(176 378)(177 379)(178 380)(179 381)(180 382)(181 383)(182 384)(183 385)(184 386)(185 387)(186 388)(187 389)(188 390)(189 391)(190 392)(191 337)(192 338)(193 339)(194 340)(195 341)(196 342)(197 343)(198 344)(199 345)(200 346)(201 347)(202 348)(203 349)(204 350)(205 351)(206 352)(207 353)(208 354)(209 355)(210 356)(211 357)(212 358)(213 359)(214 360)(215 361)(216 362)(217 363)(218 364)(219 365)(220 366)(221 367)(222 368)(223 369)(224 370)
(1 120 393 392)(2 121 394 337)(3 122 395 338)(4 123 396 339)(5 124 397 340)(6 125 398 341)(7 126 399 342)(8 127 400 343)(9 128 401 344)(10 129 402 345)(11 130 403 346)(12 131 404 347)(13 132 405 348)(14 133 406 349)(15 134 407 350)(16 135 408 351)(17 136 409 352)(18 137 410 353)(19 138 411 354)(20 139 412 355)(21 140 413 356)(22 141 414 357)(23 142 415 358)(24 143 416 359)(25 144 417 360)(26 145 418 361)(27 146 419 362)(28 147 420 363)(29 148 421 364)(30 149 422 365)(31 150 423 366)(32 151 424 367)(33 152 425 368)(34 153 426 369)(35 154 427 370)(36 155 428 371)(37 156 429 372)(38 157 430 373)(39 158 431 374)(40 159 432 375)(41 160 433 376)(42 161 434 377)(43 162 435 378)(44 163 436 379)(45 164 437 380)(46 165 438 381)(47 166 439 382)(48 167 440 383)(49 168 441 384)(50 113 442 385)(51 114 443 386)(52 115 444 387)(53 116 445 388)(54 117 446 389)(55 118 447 390)(56 119 448 391)(57 187 300 260)(58 188 301 261)(59 189 302 262)(60 190 303 263)(61 191 304 264)(62 192 305 265)(63 193 306 266)(64 194 307 267)(65 195 308 268)(66 196 309 269)(67 197 310 270)(68 198 311 271)(69 199 312 272)(70 200 313 273)(71 201 314 274)(72 202 315 275)(73 203 316 276)(74 204 317 277)(75 205 318 278)(76 206 319 279)(77 207 320 280)(78 208 321 225)(79 209 322 226)(80 210 323 227)(81 211 324 228)(82 212 325 229)(83 213 326 230)(84 214 327 231)(85 215 328 232)(86 216 329 233)(87 217 330 234)(88 218 331 235)(89 219 332 236)(90 220 333 237)(91 221 334 238)(92 222 335 239)(93 223 336 240)(94 224 281 241)(95 169 282 242)(96 170 283 243)(97 171 284 244)(98 172 285 245)(99 173 286 246)(100 174 287 247)(101 175 288 248)(102 176 289 249)(103 177 290 250)(104 178 291 251)(105 179 292 252)(106 180 293 253)(107 181 294 254)(108 182 295 255)(109 183 296 256)(110 184 297 257)(111 185 298 258)(112 186 299 259)
(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)

G:=sub<Sym(448)| (1,303)(2,304)(3,305)(4,306)(5,307)(6,308)(7,309)(8,310)(9,311)(10,312)(11,313)(12,314)(13,315)(14,316)(15,317)(16,318)(17,319)(18,320)(19,321)(20,322)(21,323)(22,324)(23,325)(24,326)(25,327)(26,328)(27,329)(28,330)(29,331)(30,332)(31,333)(32,334)(33,335)(34,336)(35,281)(36,282)(37,283)(38,284)(39,285)(40,286)(41,287)(42,288)(43,289)(44,290)(45,291)(46,292)(47,293)(48,294)(49,295)(50,296)(51,297)(52,298)(53,299)(54,300)(55,301)(56,302)(57,446)(58,447)(59,448)(60,393)(61,394)(62,395)(63,396)(64,397)(65,398)(66,399)(67,400)(68,401)(69,402)(70,403)(71,404)(72,405)(73,406)(74,407)(75,408)(76,409)(77,410)(78,411)(79,412)(80,413)(81,414)(82,415)(83,416)(84,417)(85,418)(86,419)(87,420)(88,421)(89,422)(90,423)(91,424)(92,425)(93,426)(94,427)(95,428)(96,429)(97,430)(98,431)(99,432)(100,433)(101,434)(102,435)(103,436)(104,437)(105,438)(106,439)(107,440)(108,441)(109,442)(110,443)(111,444)(112,445)(113,256)(114,257)(115,258)(116,259)(117,260)(118,261)(119,262)(120,263)(121,264)(122,265)(123,266)(124,267)(125,268)(126,269)(127,270)(128,271)(129,272)(130,273)(131,274)(132,275)(133,276)(134,277)(135,278)(136,279)(137,280)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231)(145,232)(146,233)(147,234)(148,235)(149,236)(150,237)(151,238)(152,239)(153,240)(154,241)(155,242)(156,243)(157,244)(158,245)(159,246)(160,247)(161,248)(162,249)(163,250)(164,251)(165,252)(166,253)(167,254)(168,255)(169,371)(170,372)(171,373)(172,374)(173,375)(174,376)(175,377)(176,378)(177,379)(178,380)(179,381)(180,382)(181,383)(182,384)(183,385)(184,386)(185,387)(186,388)(187,389)(188,390)(189,391)(190,392)(191,337)(192,338)(193,339)(194,340)(195,341)(196,342)(197,343)(198,344)(199,345)(200,346)(201,347)(202,348)(203,349)(204,350)(205,351)(206,352)(207,353)(208,354)(209,355)(210,356)(211,357)(212,358)(213,359)(214,360)(215,361)(216,362)(217,363)(218,364)(219,365)(220,366)(221,367)(222,368)(223,369)(224,370), (1,120,393,392)(2,121,394,337)(3,122,395,338)(4,123,396,339)(5,124,397,340)(6,125,398,341)(7,126,399,342)(8,127,400,343)(9,128,401,344)(10,129,402,345)(11,130,403,346)(12,131,404,347)(13,132,405,348)(14,133,406,349)(15,134,407,350)(16,135,408,351)(17,136,409,352)(18,137,410,353)(19,138,411,354)(20,139,412,355)(21,140,413,356)(22,141,414,357)(23,142,415,358)(24,143,416,359)(25,144,417,360)(26,145,418,361)(27,146,419,362)(28,147,420,363)(29,148,421,364)(30,149,422,365)(31,150,423,366)(32,151,424,367)(33,152,425,368)(34,153,426,369)(35,154,427,370)(36,155,428,371)(37,156,429,372)(38,157,430,373)(39,158,431,374)(40,159,432,375)(41,160,433,376)(42,161,434,377)(43,162,435,378)(44,163,436,379)(45,164,437,380)(46,165,438,381)(47,166,439,382)(48,167,440,383)(49,168,441,384)(50,113,442,385)(51,114,443,386)(52,115,444,387)(53,116,445,388)(54,117,446,389)(55,118,447,390)(56,119,448,391)(57,187,300,260)(58,188,301,261)(59,189,302,262)(60,190,303,263)(61,191,304,264)(62,192,305,265)(63,193,306,266)(64,194,307,267)(65,195,308,268)(66,196,309,269)(67,197,310,270)(68,198,311,271)(69,199,312,272)(70,200,313,273)(71,201,314,274)(72,202,315,275)(73,203,316,276)(74,204,317,277)(75,205,318,278)(76,206,319,279)(77,207,320,280)(78,208,321,225)(79,209,322,226)(80,210,323,227)(81,211,324,228)(82,212,325,229)(83,213,326,230)(84,214,327,231)(85,215,328,232)(86,216,329,233)(87,217,330,234)(88,218,331,235)(89,219,332,236)(90,220,333,237)(91,221,334,238)(92,222,335,239)(93,223,336,240)(94,224,281,241)(95,169,282,242)(96,170,283,243)(97,171,284,244)(98,172,285,245)(99,173,286,246)(100,174,287,247)(101,175,288,248)(102,176,289,249)(103,177,290,250)(104,178,291,251)(105,179,292,252)(106,180,293,253)(107,181,294,254)(108,182,295,255)(109,183,296,256)(110,184,297,257)(111,185,298,258)(112,186,299,259), (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)>;

G:=Group( (1,303)(2,304)(3,305)(4,306)(5,307)(6,308)(7,309)(8,310)(9,311)(10,312)(11,313)(12,314)(13,315)(14,316)(15,317)(16,318)(17,319)(18,320)(19,321)(20,322)(21,323)(22,324)(23,325)(24,326)(25,327)(26,328)(27,329)(28,330)(29,331)(30,332)(31,333)(32,334)(33,335)(34,336)(35,281)(36,282)(37,283)(38,284)(39,285)(40,286)(41,287)(42,288)(43,289)(44,290)(45,291)(46,292)(47,293)(48,294)(49,295)(50,296)(51,297)(52,298)(53,299)(54,300)(55,301)(56,302)(57,446)(58,447)(59,448)(60,393)(61,394)(62,395)(63,396)(64,397)(65,398)(66,399)(67,400)(68,401)(69,402)(70,403)(71,404)(72,405)(73,406)(74,407)(75,408)(76,409)(77,410)(78,411)(79,412)(80,413)(81,414)(82,415)(83,416)(84,417)(85,418)(86,419)(87,420)(88,421)(89,422)(90,423)(91,424)(92,425)(93,426)(94,427)(95,428)(96,429)(97,430)(98,431)(99,432)(100,433)(101,434)(102,435)(103,436)(104,437)(105,438)(106,439)(107,440)(108,441)(109,442)(110,443)(111,444)(112,445)(113,256)(114,257)(115,258)(116,259)(117,260)(118,261)(119,262)(120,263)(121,264)(122,265)(123,266)(124,267)(125,268)(126,269)(127,270)(128,271)(129,272)(130,273)(131,274)(132,275)(133,276)(134,277)(135,278)(136,279)(137,280)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231)(145,232)(146,233)(147,234)(148,235)(149,236)(150,237)(151,238)(152,239)(153,240)(154,241)(155,242)(156,243)(157,244)(158,245)(159,246)(160,247)(161,248)(162,249)(163,250)(164,251)(165,252)(166,253)(167,254)(168,255)(169,371)(170,372)(171,373)(172,374)(173,375)(174,376)(175,377)(176,378)(177,379)(178,380)(179,381)(180,382)(181,383)(182,384)(183,385)(184,386)(185,387)(186,388)(187,389)(188,390)(189,391)(190,392)(191,337)(192,338)(193,339)(194,340)(195,341)(196,342)(197,343)(198,344)(199,345)(200,346)(201,347)(202,348)(203,349)(204,350)(205,351)(206,352)(207,353)(208,354)(209,355)(210,356)(211,357)(212,358)(213,359)(214,360)(215,361)(216,362)(217,363)(218,364)(219,365)(220,366)(221,367)(222,368)(223,369)(224,370), (1,120,393,392)(2,121,394,337)(3,122,395,338)(4,123,396,339)(5,124,397,340)(6,125,398,341)(7,126,399,342)(8,127,400,343)(9,128,401,344)(10,129,402,345)(11,130,403,346)(12,131,404,347)(13,132,405,348)(14,133,406,349)(15,134,407,350)(16,135,408,351)(17,136,409,352)(18,137,410,353)(19,138,411,354)(20,139,412,355)(21,140,413,356)(22,141,414,357)(23,142,415,358)(24,143,416,359)(25,144,417,360)(26,145,418,361)(27,146,419,362)(28,147,420,363)(29,148,421,364)(30,149,422,365)(31,150,423,366)(32,151,424,367)(33,152,425,368)(34,153,426,369)(35,154,427,370)(36,155,428,371)(37,156,429,372)(38,157,430,373)(39,158,431,374)(40,159,432,375)(41,160,433,376)(42,161,434,377)(43,162,435,378)(44,163,436,379)(45,164,437,380)(46,165,438,381)(47,166,439,382)(48,167,440,383)(49,168,441,384)(50,113,442,385)(51,114,443,386)(52,115,444,387)(53,116,445,388)(54,117,446,389)(55,118,447,390)(56,119,448,391)(57,187,300,260)(58,188,301,261)(59,189,302,262)(60,190,303,263)(61,191,304,264)(62,192,305,265)(63,193,306,266)(64,194,307,267)(65,195,308,268)(66,196,309,269)(67,197,310,270)(68,198,311,271)(69,199,312,272)(70,200,313,273)(71,201,314,274)(72,202,315,275)(73,203,316,276)(74,204,317,277)(75,205,318,278)(76,206,319,279)(77,207,320,280)(78,208,321,225)(79,209,322,226)(80,210,323,227)(81,211,324,228)(82,212,325,229)(83,213,326,230)(84,214,327,231)(85,215,328,232)(86,216,329,233)(87,217,330,234)(88,218,331,235)(89,219,332,236)(90,220,333,237)(91,221,334,238)(92,222,335,239)(93,223,336,240)(94,224,281,241)(95,169,282,242)(96,170,283,243)(97,171,284,244)(98,172,285,245)(99,173,286,246)(100,174,287,247)(101,175,288,248)(102,176,289,249)(103,177,290,250)(104,178,291,251)(105,179,292,252)(106,180,293,253)(107,181,294,254)(108,182,295,255)(109,183,296,256)(110,184,297,257)(111,185,298,258)(112,186,299,259), (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) );

G=PermutationGroup([(1,303),(2,304),(3,305),(4,306),(5,307),(6,308),(7,309),(8,310),(9,311),(10,312),(11,313),(12,314),(13,315),(14,316),(15,317),(16,318),(17,319),(18,320),(19,321),(20,322),(21,323),(22,324),(23,325),(24,326),(25,327),(26,328),(27,329),(28,330),(29,331),(30,332),(31,333),(32,334),(33,335),(34,336),(35,281),(36,282),(37,283),(38,284),(39,285),(40,286),(41,287),(42,288),(43,289),(44,290),(45,291),(46,292),(47,293),(48,294),(49,295),(50,296),(51,297),(52,298),(53,299),(54,300),(55,301),(56,302),(57,446),(58,447),(59,448),(60,393),(61,394),(62,395),(63,396),(64,397),(65,398),(66,399),(67,400),(68,401),(69,402),(70,403),(71,404),(72,405),(73,406),(74,407),(75,408),(76,409),(77,410),(78,411),(79,412),(80,413),(81,414),(82,415),(83,416),(84,417),(85,418),(86,419),(87,420),(88,421),(89,422),(90,423),(91,424),(92,425),(93,426),(94,427),(95,428),(96,429),(97,430),(98,431),(99,432),(100,433),(101,434),(102,435),(103,436),(104,437),(105,438),(106,439),(107,440),(108,441),(109,442),(110,443),(111,444),(112,445),(113,256),(114,257),(115,258),(116,259),(117,260),(118,261),(119,262),(120,263),(121,264),(122,265),(123,266),(124,267),(125,268),(126,269),(127,270),(128,271),(129,272),(130,273),(131,274),(132,275),(133,276),(134,277),(135,278),(136,279),(137,280),(138,225),(139,226),(140,227),(141,228),(142,229),(143,230),(144,231),(145,232),(146,233),(147,234),(148,235),(149,236),(150,237),(151,238),(152,239),(153,240),(154,241),(155,242),(156,243),(157,244),(158,245),(159,246),(160,247),(161,248),(162,249),(163,250),(164,251),(165,252),(166,253),(167,254),(168,255),(169,371),(170,372),(171,373),(172,374),(173,375),(174,376),(175,377),(176,378),(177,379),(178,380),(179,381),(180,382),(181,383),(182,384),(183,385),(184,386),(185,387),(186,388),(187,389),(188,390),(189,391),(190,392),(191,337),(192,338),(193,339),(194,340),(195,341),(196,342),(197,343),(198,344),(199,345),(200,346),(201,347),(202,348),(203,349),(204,350),(205,351),(206,352),(207,353),(208,354),(209,355),(210,356),(211,357),(212,358),(213,359),(214,360),(215,361),(216,362),(217,363),(218,364),(219,365),(220,366),(221,367),(222,368),(223,369),(224,370)], [(1,120,393,392),(2,121,394,337),(3,122,395,338),(4,123,396,339),(5,124,397,340),(6,125,398,341),(7,126,399,342),(8,127,400,343),(9,128,401,344),(10,129,402,345),(11,130,403,346),(12,131,404,347),(13,132,405,348),(14,133,406,349),(15,134,407,350),(16,135,408,351),(17,136,409,352),(18,137,410,353),(19,138,411,354),(20,139,412,355),(21,140,413,356),(22,141,414,357),(23,142,415,358),(24,143,416,359),(25,144,417,360),(26,145,418,361),(27,146,419,362),(28,147,420,363),(29,148,421,364),(30,149,422,365),(31,150,423,366),(32,151,424,367),(33,152,425,368),(34,153,426,369),(35,154,427,370),(36,155,428,371),(37,156,429,372),(38,157,430,373),(39,158,431,374),(40,159,432,375),(41,160,433,376),(42,161,434,377),(43,162,435,378),(44,163,436,379),(45,164,437,380),(46,165,438,381),(47,166,439,382),(48,167,440,383),(49,168,441,384),(50,113,442,385),(51,114,443,386),(52,115,444,387),(53,116,445,388),(54,117,446,389),(55,118,447,390),(56,119,448,391),(57,187,300,260),(58,188,301,261),(59,189,302,262),(60,190,303,263),(61,191,304,264),(62,192,305,265),(63,193,306,266),(64,194,307,267),(65,195,308,268),(66,196,309,269),(67,197,310,270),(68,198,311,271),(69,199,312,272),(70,200,313,273),(71,201,314,274),(72,202,315,275),(73,203,316,276),(74,204,317,277),(75,205,318,278),(76,206,319,279),(77,207,320,280),(78,208,321,225),(79,209,322,226),(80,210,323,227),(81,211,324,228),(82,212,325,229),(83,213,326,230),(84,214,327,231),(85,215,328,232),(86,216,329,233),(87,217,330,234),(88,218,331,235),(89,219,332,236),(90,220,333,237),(91,221,334,238),(92,222,335,239),(93,223,336,240),(94,224,281,241),(95,169,282,242),(96,170,283,243),(97,171,284,244),(98,172,285,245),(99,173,286,246),(100,174,287,247),(101,175,288,248),(102,176,289,249),(103,177,290,250),(104,178,291,251),(105,179,292,252),(106,180,293,253),(107,181,294,254),(108,182,295,255),(109,183,296,256),(110,184,297,257),(111,185,298,258),(112,186,299,259)], [(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)])

448 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 7A ··· 7F 8A ··· 8AF 14A ··· 14AP 28A ··· 28EN 56A ··· 56GJ order 1 2 ··· 2 4 ··· 4 7 ··· 7 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

448 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + + image C1 C2 C2 C2 C4 C4 C4 C7 C8 C14 C14 C14 C28 C28 C28 C56 kernel C2×C4×C56 C4×C56 C2×C4×C28 C22×C56 C4×C28 C2×C56 C22×C28 C2×C4×C8 C2×C28 C4×C8 C2×C42 C22×C8 C42 C2×C8 C22×C4 C2×C4 # reps 1 4 1 2 4 16 4 6 32 24 6 12 24 96 24 192

Matrix representation of C2×C4×C56 in GL3(𝔽113) generated by

 112 0 0 0 1 0 0 0 1
,
 15 0 0 0 15 0 0 0 15
,
 99 0 0 0 98 0 0 0 63
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[15,0,0,0,15,0,0,0,15],[99,0,0,0,98,0,0,0,63] >;

C2×C4×C56 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{56}
% in TeX

G:=Group("C2xC4xC56");
// GroupNames label

G:=SmallGroup(448,810);
// by ID

G=gap.SmallGroup(448,810);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,792,172]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^56=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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