direct product, abelian, monomial, 2-elementary
Aliases: C2×C4×C56, SmallGroup(448,810)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C4×C56 |
| C1 — C2×C4×C56 |
| 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, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C22×C4, C22×C4, C28, C2×C14, C2×C14, C4×C8, C2×C42, C22×C8, C56, C2×C28, C2×C28, C22×C14, C2×C4×C8, C4×C28, C2×C56, C22×C28, C22×C28, C4×C56, C2×C4×C28, C22×C56, C2×C4×C56
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C23, C14, C42, C2×C8, C22×C4, C28, C2×C14, C4×C8, C2×C42, C22×C8, C56, C2×C28, C22×C14, C2×C4×C8, C4×C28, C2×C56, C22×C28, C4×C56, C2×C4×C28, C22×C56, C2×C4×C56
►Regular action on 448 points
Generators in S448
(1 310)(2 311)(3 312)(4 313)(5 314)(6 315)(7 316)(8 317)(9 318)(10 319)(11 320)(12 321)(13 322)(14 323)(15 324)(16 325)(17 326)(18 327)(19 328)(20 329)(21 330)(22 331)(23 332)(24 333)(25 334)(26 335)(27 336)(28 281)(29 282)(30 283)(31 284)(32 285)(33 286)(34 287)(35 288)(36 289)(37 290)(38 291)(39 292)(40 293)(41 294)(42 295)(43 296)(44 297)(45 298)(46 299)(47 300)(48 301)(49 302)(50 303)(51 304)(52 305)(53 306)(54 307)(55 308)(56 309)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 141)(75 142)(76 143)(77 144)(78 145)(79 146)(80 147)(81 148)(82 149)(83 150)(84 151)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(91 158)(92 159)(93 160)(94 161)(95 162)(96 163)(97 164)(98 165)(99 166)(100 167)(101 168)(102 113)(103 114)(104 115)(105 116)(106 117)(107 118)(108 119)(109 120)(110 121)(111 122)(112 123)(169 265)(170 266)(171 267)(172 268)(173 269)(174 270)(175 271)(176 272)(177 273)(178 274)(179 275)(180 276)(181 277)(182 278)(183 279)(184 280)(185 225)(186 226)(187 227)(188 228)(189 229)(190 230)(191 231)(192 232)(193 233)(194 234)(195 235)(196 236)(197 237)(198 238)(199 239)(200 240)(201 241)(202 242)(203 243)(204 244)(205 245)(206 246)(207 247)(208 248)(209 249)(210 250)(211 251)(212 252)(213 253)(214 254)(215 255)(216 256)(217 257)(218 258)(219 259)(220 260)(221 261)(222 262)(223 263)(224 264)(337 440)(338 441)(339 442)(340 443)(341 444)(342 445)(343 446)(344 447)(345 448)(346 393)(347 394)(348 395)(349 396)(350 397)(351 398)(352 399)(353 400)(354 401)(355 402)(356 403)(357 404)(358 405)(359 406)(360 407)(361 408)(362 409)(363 410)(364 411)(365 412)(366 413)(367 414)(368 415)(369 416)(370 417)(371 418)(372 419)(373 420)(374 421)(375 422)(376 423)(377 424)(378 425)(379 426)(380 427)(381 428)(382 429)(383 430)(384 431)(385 432)(386 433)(387 434)(388 435)(389 436)(390 437)(391 438)(392 439)
(1 263 160 413)(2 264 161 414)(3 265 162 415)(4 266 163 416)(5 267 164 417)(6 268 165 418)(7 269 166 419)(8 270 167 420)(9 271 168 421)(10 272 113 422)(11 273 114 423)(12 274 115 424)(13 275 116 425)(14 276 117 426)(15 277 118 427)(16 278 119 428)(17 279 120 429)(18 280 121 430)(19 225 122 431)(20 226 123 432)(21 227 124 433)(22 228 125 434)(23 229 126 435)(24 230 127 436)(25 231 128 437)(26 232 129 438)(27 233 130 439)(28 234 131 440)(29 235 132 441)(30 236 133 442)(31 237 134 443)(32 238 135 444)(33 239 136 445)(34 240 137 446)(35 241 138 447)(36 242 139 448)(37 243 140 393)(38 244 141 394)(39 245 142 395)(40 246 143 396)(41 247 144 397)(42 248 145 398)(43 249 146 399)(44 250 147 400)(45 251 148 401)(46 252 149 402)(47 253 150 403)(48 254 151 404)(49 255 152 405)(50 256 153 406)(51 257 154 407)(52 258 155 408)(53 259 156 409)(54 260 157 410)(55 261 158 411)(56 262 159 412)(57 386 330 187)(58 387 331 188)(59 388 332 189)(60 389 333 190)(61 390 334 191)(62 391 335 192)(63 392 336 193)(64 337 281 194)(65 338 282 195)(66 339 283 196)(67 340 284 197)(68 341 285 198)(69 342 286 199)(70 343 287 200)(71 344 288 201)(72 345 289 202)(73 346 290 203)(74 347 291 204)(75 348 292 205)(76 349 293 206)(77 350 294 207)(78 351 295 208)(79 352 296 209)(80 353 297 210)(81 354 298 211)(82 355 299 212)(83 356 300 213)(84 357 301 214)(85 358 302 215)(86 359 303 216)(87 360 304 217)(88 361 305 218)(89 362 306 219)(90 363 307 220)(91 364 308 221)(92 365 309 222)(93 366 310 223)(94 367 311 224)(95 368 312 169)(96 369 313 170)(97 370 314 171)(98 371 315 172)(99 372 316 173)(100 373 317 174)(101 374 318 175)(102 375 319 176)(103 376 320 177)(104 377 321 178)(105 378 322 179)(106 379 323 180)(107 380 324 181)(108 381 325 182)(109 382 326 183)(110 383 327 184)(111 384 328 185)(112 385 329 186)
(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)| (1,310)(2,311)(3,312)(4,313)(5,314)(6,315)(7,316)(8,317)(9,318)(10,319)(11,320)(12,321)(13,322)(14,323)(15,324)(16,325)(17,326)(18,327)(19,328)(20,329)(21,330)(22,331)(23,332)(24,333)(25,334)(26,335)(27,336)(28,281)(29,282)(30,283)(31,284)(32,285)(33,286)(34,287)(35,288)(36,289)(37,290)(38,291)(39,292)(40,293)(41,294)(42,295)(43,296)(44,297)(45,298)(46,299)(47,300)(48,301)(49,302)(50,303)(51,304)(52,305)(53,306)(54,307)(55,308)(56,309)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,158)(92,159)(93,160)(94,161)(95,162)(96,163)(97,164)(98,165)(99,166)(100,167)(101,168)(102,113)(103,114)(104,115)(105,116)(106,117)(107,118)(108,119)(109,120)(110,121)(111,122)(112,123)(169,265)(170,266)(171,267)(172,268)(173,269)(174,270)(175,271)(176,272)(177,273)(178,274)(179,275)(180,276)(181,277)(182,278)(183,279)(184,280)(185,225)(186,226)(187,227)(188,228)(189,229)(190,230)(191,231)(192,232)(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)(200,240)(201,241)(202,242)(203,243)(204,244)(205,245)(206,246)(207,247)(208,248)(209,249)(210,250)(211,251)(212,252)(213,253)(214,254)(215,255)(216,256)(217,257)(218,258)(219,259)(220,260)(221,261)(222,262)(223,263)(224,264)(337,440)(338,441)(339,442)(340,443)(341,444)(342,445)(343,446)(344,447)(345,448)(346,393)(347,394)(348,395)(349,396)(350,397)(351,398)(352,399)(353,400)(354,401)(355,402)(356,403)(357,404)(358,405)(359,406)(360,407)(361,408)(362,409)(363,410)(364,411)(365,412)(366,413)(367,414)(368,415)(369,416)(370,417)(371,418)(372,419)(373,420)(374,421)(375,422)(376,423)(377,424)(378,425)(379,426)(380,427)(381,428)(382,429)(383,430)(384,431)(385,432)(386,433)(387,434)(388,435)(389,436)(390,437)(391,438)(392,439), (1,263,160,413)(2,264,161,414)(3,265,162,415)(4,266,163,416)(5,267,164,417)(6,268,165,418)(7,269,166,419)(8,270,167,420)(9,271,168,421)(10,272,113,422)(11,273,114,423)(12,274,115,424)(13,275,116,425)(14,276,117,426)(15,277,118,427)(16,278,119,428)(17,279,120,429)(18,280,121,430)(19,225,122,431)(20,226,123,432)(21,227,124,433)(22,228,125,434)(23,229,126,435)(24,230,127,436)(25,231,128,437)(26,232,129,438)(27,233,130,439)(28,234,131,440)(29,235,132,441)(30,236,133,442)(31,237,134,443)(32,238,135,444)(33,239,136,445)(34,240,137,446)(35,241,138,447)(36,242,139,448)(37,243,140,393)(38,244,141,394)(39,245,142,395)(40,246,143,396)(41,247,144,397)(42,248,145,398)(43,249,146,399)(44,250,147,400)(45,251,148,401)(46,252,149,402)(47,253,150,403)(48,254,151,404)(49,255,152,405)(50,256,153,406)(51,257,154,407)(52,258,155,408)(53,259,156,409)(54,260,157,410)(55,261,158,411)(56,262,159,412)(57,386,330,187)(58,387,331,188)(59,388,332,189)(60,389,333,190)(61,390,334,191)(62,391,335,192)(63,392,336,193)(64,337,281,194)(65,338,282,195)(66,339,283,196)(67,340,284,197)(68,341,285,198)(69,342,286,199)(70,343,287,200)(71,344,288,201)(72,345,289,202)(73,346,290,203)(74,347,291,204)(75,348,292,205)(76,349,293,206)(77,350,294,207)(78,351,295,208)(79,352,296,209)(80,353,297,210)(81,354,298,211)(82,355,299,212)(83,356,300,213)(84,357,301,214)(85,358,302,215)(86,359,303,216)(87,360,304,217)(88,361,305,218)(89,362,306,219)(90,363,307,220)(91,364,308,221)(92,365,309,222)(93,366,310,223)(94,367,311,224)(95,368,312,169)(96,369,313,170)(97,370,314,171)(98,371,315,172)(99,372,316,173)(100,373,317,174)(101,374,318,175)(102,375,319,176)(103,376,320,177)(104,377,321,178)(105,378,322,179)(106,379,323,180)(107,380,324,181)(108,381,325,182)(109,382,326,183)(110,383,327,184)(111,384,328,185)(112,385,329,186), (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( (1,310)(2,311)(3,312)(4,313)(5,314)(6,315)(7,316)(8,317)(9,318)(10,319)(11,320)(12,321)(13,322)(14,323)(15,324)(16,325)(17,326)(18,327)(19,328)(20,329)(21,330)(22,331)(23,332)(24,333)(25,334)(26,335)(27,336)(28,281)(29,282)(30,283)(31,284)(32,285)(33,286)(34,287)(35,288)(36,289)(37,290)(38,291)(39,292)(40,293)(41,294)(42,295)(43,296)(44,297)(45,298)(46,299)(47,300)(48,301)(49,302)(50,303)(51,304)(52,305)(53,306)(54,307)(55,308)(56,309)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,158)(92,159)(93,160)(94,161)(95,162)(96,163)(97,164)(98,165)(99,166)(100,167)(101,168)(102,113)(103,114)(104,115)(105,116)(106,117)(107,118)(108,119)(109,120)(110,121)(111,122)(112,123)(169,265)(170,266)(171,267)(172,268)(173,269)(174,270)(175,271)(176,272)(177,273)(178,274)(179,275)(180,276)(181,277)(182,278)(183,279)(184,280)(185,225)(186,226)(187,227)(188,228)(189,229)(190,230)(191,231)(192,232)(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)(200,240)(201,241)(202,242)(203,243)(204,244)(205,245)(206,246)(207,247)(208,248)(209,249)(210,250)(211,251)(212,252)(213,253)(214,254)(215,255)(216,256)(217,257)(218,258)(219,259)(220,260)(221,261)(222,262)(223,263)(224,264)(337,440)(338,441)(339,442)(340,443)(341,444)(342,445)(343,446)(344,447)(345,448)(346,393)(347,394)(348,395)(349,396)(350,397)(351,398)(352,399)(353,400)(354,401)(355,402)(356,403)(357,404)(358,405)(359,406)(360,407)(361,408)(362,409)(363,410)(364,411)(365,412)(366,413)(367,414)(368,415)(369,416)(370,417)(371,418)(372,419)(373,420)(374,421)(375,422)(376,423)(377,424)(378,425)(379,426)(380,427)(381,428)(382,429)(383,430)(384,431)(385,432)(386,433)(387,434)(388,435)(389,436)(390,437)(391,438)(392,439), (1,263,160,413)(2,264,161,414)(3,265,162,415)(4,266,163,416)(5,267,164,417)(6,268,165,418)(7,269,166,419)(8,270,167,420)(9,271,168,421)(10,272,113,422)(11,273,114,423)(12,274,115,424)(13,275,116,425)(14,276,117,426)(15,277,118,427)(16,278,119,428)(17,279,120,429)(18,280,121,430)(19,225,122,431)(20,226,123,432)(21,227,124,433)(22,228,125,434)(23,229,126,435)(24,230,127,436)(25,231,128,437)(26,232,129,438)(27,233,130,439)(28,234,131,440)(29,235,132,441)(30,236,133,442)(31,237,134,443)(32,238,135,444)(33,239,136,445)(34,240,137,446)(35,241,138,447)(36,242,139,448)(37,243,140,393)(38,244,141,394)(39,245,142,395)(40,246,143,396)(41,247,144,397)(42,248,145,398)(43,249,146,399)(44,250,147,400)(45,251,148,401)(46,252,149,402)(47,253,150,403)(48,254,151,404)(49,255,152,405)(50,256,153,406)(51,257,154,407)(52,258,155,408)(53,259,156,409)(54,260,157,410)(55,261,158,411)(56,262,159,412)(57,386,330,187)(58,387,331,188)(59,388,332,189)(60,389,333,190)(61,390,334,191)(62,391,335,192)(63,392,336,193)(64,337,281,194)(65,338,282,195)(66,339,283,196)(67,340,284,197)(68,341,285,198)(69,342,286,199)(70,343,287,200)(71,344,288,201)(72,345,289,202)(73,346,290,203)(74,347,291,204)(75,348,292,205)(76,349,293,206)(77,350,294,207)(78,351,295,208)(79,352,296,209)(80,353,297,210)(81,354,298,211)(82,355,299,212)(83,356,300,213)(84,357,301,214)(85,358,302,215)(86,359,303,216)(87,360,304,217)(88,361,305,218)(89,362,306,219)(90,363,307,220)(91,364,308,221)(92,365,309,222)(93,366,310,223)(94,367,311,224)(95,368,312,169)(96,369,313,170)(97,370,314,171)(98,371,315,172)(99,372,316,173)(100,373,317,174)(101,374,318,175)(102,375,319,176)(103,376,320,177)(104,377,321,178)(105,378,322,179)(106,379,323,180)(107,380,324,181)(108,381,325,182)(109,382,326,183)(110,383,327,184)(111,384,328,185)(112,385,329,186), (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([[(1,310),(2,311),(3,312),(4,313),(5,314),(6,315),(7,316),(8,317),(9,318),(10,319),(11,320),(12,321),(13,322),(14,323),(15,324),(16,325),(17,326),(18,327),(19,328),(20,329),(21,330),(22,331),(23,332),(24,333),(25,334),(26,335),(27,336),(28,281),(29,282),(30,283),(31,284),(32,285),(33,286),(34,287),(35,288),(36,289),(37,290),(38,291),(39,292),(40,293),(41,294),(42,295),(43,296),(44,297),(45,298),(46,299),(47,300),(48,301),(49,302),(50,303),(51,304),(52,305),(53,306),(54,307),(55,308),(56,309),(57,124),(58,125),(59,126),(60,127),(61,128),(62,129),(63,130),(64,131),(65,132),(66,133),(67,134),(68,135),(69,136),(70,137),(71,138),(72,139),(73,140),(74,141),(75,142),(76,143),(77,144),(78,145),(79,146),(80,147),(81,148),(82,149),(83,150),(84,151),(85,152),(86,153),(87,154),(88,155),(89,156),(90,157),(91,158),(92,159),(93,160),(94,161),(95,162),(96,163),(97,164),(98,165),(99,166),(100,167),(101,168),(102,113),(103,114),(104,115),(105,116),(106,117),(107,118),(108,119),(109,120),(110,121),(111,122),(112,123),(169,265),(170,266),(171,267),(172,268),(173,269),(174,270),(175,271),(176,272),(177,273),(178,274),(179,275),(180,276),(181,277),(182,278),(183,279),(184,280),(185,225),(186,226),(187,227),(188,228),(189,229),(190,230),(191,231),(192,232),(193,233),(194,234),(195,235),(196,236),(197,237),(198,238),(199,239),(200,240),(201,241),(202,242),(203,243),(204,244),(205,245),(206,246),(207,247),(208,248),(209,249),(210,250),(211,251),(212,252),(213,253),(214,254),(215,255),(216,256),(217,257),(218,258),(219,259),(220,260),(221,261),(222,262),(223,263),(224,264),(337,440),(338,441),(339,442),(340,443),(341,444),(342,445),(343,446),(344,447),(345,448),(346,393),(347,394),(348,395),(349,396),(350,397),(351,398),(352,399),(353,400),(354,401),(355,402),(356,403),(357,404),(358,405),(359,406),(360,407),(361,408),(362,409),(363,410),(364,411),(365,412),(366,413),(367,414),(368,415),(369,416),(370,417),(371,418),(372,419),(373,420),(374,421),(375,422),(376,423),(377,424),(378,425),(379,426),(380,427),(381,428),(382,429),(383,430),(384,431),(385,432),(386,433),(387,434),(388,435),(389,436),(390,437),(391,438),(392,439)], [(1,263,160,413),(2,264,161,414),(3,265,162,415),(4,266,163,416),(5,267,164,417),(6,268,165,418),(7,269,166,419),(8,270,167,420),(9,271,168,421),(10,272,113,422),(11,273,114,423),(12,274,115,424),(13,275,116,425),(14,276,117,426),(15,277,118,427),(16,278,119,428),(17,279,120,429),(18,280,121,430),(19,225,122,431),(20,226,123,432),(21,227,124,433),(22,228,125,434),(23,229,126,435),(24,230,127,436),(25,231,128,437),(26,232,129,438),(27,233,130,439),(28,234,131,440),(29,235,132,441),(30,236,133,442),(31,237,134,443),(32,238,135,444),(33,239,136,445),(34,240,137,446),(35,241,138,447),(36,242,139,448),(37,243,140,393),(38,244,141,394),(39,245,142,395),(40,246,143,396),(41,247,144,397),(42,248,145,398),(43,249,146,399),(44,250,147,400),(45,251,148,401),(46,252,149,402),(47,253,150,403),(48,254,151,404),(49,255,152,405),(50,256,153,406),(51,257,154,407),(52,258,155,408),(53,259,156,409),(54,260,157,410),(55,261,158,411),(56,262,159,412),(57,386,330,187),(58,387,331,188),(59,388,332,189),(60,389,333,190),(61,390,334,191),(62,391,335,192),(63,392,336,193),(64,337,281,194),(65,338,282,195),(66,339,283,196),(67,340,284,197),(68,341,285,198),(69,342,286,199),(70,343,287,200),(71,344,288,201),(72,345,289,202),(73,346,290,203),(74,347,291,204),(75,348,292,205),(76,349,293,206),(77,350,294,207),(78,351,295,208),(79,352,296,209),(80,353,297,210),(81,354,298,211),(82,355,299,212),(83,356,300,213),(84,357,301,214),(85,358,302,215),(86,359,303,216),(87,360,304,217),(88,361,305,218),(89,362,306,219),(90,363,307,220),(91,364,308,221),(92,365,309,222),(93,366,310,223),(94,367,311,224),(95,368,312,169),(96,369,313,170),(97,370,314,171),(98,371,315,172),(99,372,316,173),(100,373,317,174),(101,374,318,175),(102,375,319,176),(103,376,320,177),(104,377,321,178),(105,378,322,179),(106,379,323,180),(107,380,324,181),(108,381,325,182),(109,382,326,183),(110,383,327,184),(111,384,328,185),(112,385,329,186)], [(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