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G = C23×C56order 448 = 26·7

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

Aliases: C23×C56, SmallGroup(448,1348)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C56
 Chief series C1 — C2 — C4 — C28 — C56 — C2×C56 — C22×C56 — C23×C56
 Lower central C1 — C23×C56
 Upper central C1 — C23×C56

Generators and relations for C23×C56
G = < a,b,c,d | a2=b2=c2=d56=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 338, all normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C23, C14, C14, C2×C8, C22×C4, C24, C28, C28, C2×C14, C22×C8, C23×C4, C56, C2×C28, C22×C14, C23×C8, C2×C56, C22×C28, C23×C14, C22×C56, C23×C28, C23×C56
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C23, C14, C2×C8, C22×C4, C24, C28, C2×C14, C22×C8, C23×C4, C56, C2×C28, C22×C14, C23×C8, C2×C56, C22×C28, C23×C14, C22×C56, C23×C28, C23×C56

Smallest permutation representation of C23×C56
Regular action on 448 points
Generators in S448
(1 346)(2 347)(3 348)(4 349)(5 350)(6 351)(7 352)(8 353)(9 354)(10 355)(11 356)(12 357)(13 358)(14 359)(15 360)(16 361)(17 362)(18 363)(19 364)(20 365)(21 366)(22 367)(23 368)(24 369)(25 370)(26 371)(27 372)(28 373)(29 374)(30 375)(31 376)(32 377)(33 378)(34 379)(35 380)(36 381)(37 382)(38 383)(39 384)(40 385)(41 386)(42 387)(43 388)(44 389)(45 390)(46 391)(47 392)(48 337)(49 338)(50 339)(51 340)(52 341)(53 342)(54 343)(55 344)(56 345)(57 147)(58 148)(59 149)(60 150)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 161)(72 162)(73 163)(74 164)(75 165)(76 166)(77 167)(78 168)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 121)(88 122)(89 123)(90 124)(91 125)(92 126)(93 127)(94 128)(95 129)(96 130)(97 131)(98 132)(99 133)(100 134)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 141)(108 142)(109 143)(110 144)(111 145)(112 146)(169 290)(170 291)(171 292)(172 293)(173 294)(174 295)(175 296)(176 297)(177 298)(178 299)(179 300)(180 301)(181 302)(182 303)(183 304)(184 305)(185 306)(186 307)(187 308)(188 309)(189 310)(190 311)(191 312)(192 313)(193 314)(194 315)(195 316)(196 317)(197 318)(198 319)(199 320)(200 321)(201 322)(202 323)(203 324)(204 325)(205 326)(206 327)(207 328)(208 329)(209 330)(210 331)(211 332)(212 333)(213 334)(214 335)(215 336)(216 281)(217 282)(218 283)(219 284)(220 285)(221 286)(222 287)(223 288)(224 289)(225 395)(226 396)(227 397)(228 398)(229 399)(230 400)(231 401)(232 402)(233 403)(234 404)(235 405)(236 406)(237 407)(238 408)(239 409)(240 410)(241 411)(242 412)(243 413)(244 414)(245 415)(246 416)(247 417)(248 418)(249 419)(250 420)(251 421)(252 422)(253 423)(254 424)(255 425)(256 426)(257 427)(258 428)(259 429)(260 430)(261 431)(262 432)(263 433)(264 434)(265 435)(266 436)(267 437)(268 438)(269 439)(270 440)(271 441)(272 442)(273 443)(274 444)(275 445)(276 446)(277 447)(278 448)(279 393)(280 394)
(1 235)(2 236)(3 237)(4 238)(5 239)(6 240)(7 241)(8 242)(9 243)(10 244)(11 245)(12 246)(13 247)(14 248)(15 249)(16 250)(17 251)(18 252)(19 253)(20 254)(21 255)(22 256)(23 257)(24 258)(25 259)(26 260)(27 261)(28 262)(29 263)(30 264)(31 265)(32 266)(33 267)(34 268)(35 269)(36 270)(37 271)(38 272)(39 273)(40 274)(41 275)(42 276)(43 277)(44 278)(45 279)(46 280)(47 225)(48 226)(49 227)(50 228)(51 229)(52 230)(53 231)(54 232)(55 233)(56 234)(57 295)(58 296)(59 297)(60 298)(61 299)(62 300)(63 301)(64 302)(65 303)(66 304)(67 305)(68 306)(69 307)(70 308)(71 309)(72 310)(73 311)(74 312)(75 313)(76 314)(77 315)(78 316)(79 317)(80 318)(81 319)(82 320)(83 321)(84 322)(85 323)(86 324)(87 325)(88 326)(89 327)(90 328)(91 329)(92 330)(93 331)(94 332)(95 333)(96 334)(97 335)(98 336)(99 281)(100 282)(101 283)(102 284)(103 285)(104 286)(105 287)(106 288)(107 289)(108 290)(109 291)(110 292)(111 293)(112 294)(113 196)(114 197)(115 198)(116 199)(117 200)(118 201)(119 202)(120 203)(121 204)(122 205)(123 206)(124 207)(125 208)(126 209)(127 210)(128 211)(129 212)(130 213)(131 214)(132 215)(133 216)(134 217)(135 218)(136 219)(137 220)(138 221)(139 222)(140 223)(141 224)(142 169)(143 170)(144 171)(145 172)(146 173)(147 174)(148 175)(149 176)(150 177)(151 178)(152 179)(153 180)(154 181)(155 182)(156 183)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 190)(164 191)(165 192)(166 193)(167 194)(168 195)(337 396)(338 397)(339 398)(340 399)(341 400)(342 401)(343 402)(344 403)(345 404)(346 405)(347 406)(348 407)(349 408)(350 409)(351 410)(352 411)(353 412)(354 413)(355 414)(356 415)(357 416)(358 417)(359 418)(360 419)(361 420)(362 421)(363 422)(364 423)(365 424)(366 425)(367 426)(368 427)(369 428)(370 429)(371 430)(372 431)(373 432)(374 433)(375 434)(376 435)(377 436)(378 437)(379 438)(380 439)(381 440)(382 441)(383 442)(384 443)(385 444)(386 445)(387 446)(388 447)(389 448)(390 393)(391 394)(392 395)
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(31 157)(32 158)(33 159)(34 160)(35 161)(36 162)(37 163)(38 164)(39 165)(40 166)(41 167)(42 168)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 366)(58 367)(59 368)(60 369)(61 370)(62 371)(63 372)(64 373)(65 374)(66 375)(67 376)(68 377)(69 378)(70 379)(71 380)(72 381)(73 382)(74 383)(75 384)(76 385)(77 386)(78 387)(79 388)(80 389)(81 390)(82 391)(83 392)(84 337)(85 338)(86 339)(87 340)(88 341)(89 342)(90 343)(91 344)(92 345)(93 346)(94 347)(95 348)(96 349)(97 350)(98 351)(99 352)(100 353)(101 354)(102 355)(103 356)(104 357)(105 358)(106 359)(107 360)(108 361)(109 362)(110 363)(111 364)(112 365)(169 250)(170 251)(171 252)(172 253)(173 254)(174 255)(175 256)(176 257)(177 258)(178 259)(179 260)(180 261)(181 262)(182 263)(183 264)(184 265)(185 266)(186 267)(187 268)(188 269)(189 270)(190 271)(191 272)(192 273)(193 274)(194 275)(195 276)(196 277)(197 278)(198 279)(199 280)(200 225)(201 226)(202 227)(203 228)(204 229)(205 230)(206 231)(207 232)(208 233)(209 234)(210 235)(211 236)(212 237)(213 238)(214 239)(215 240)(216 241)(217 242)(218 243)(219 244)(220 245)(221 246)(222 247)(223 248)(224 249)(281 411)(282 412)(283 413)(284 414)(285 415)(286 416)(287 417)(288 418)(289 419)(290 420)(291 421)(292 422)(293 423)(294 424)(295 425)(296 426)(297 427)(298 428)(299 429)(300 430)(301 431)(302 432)(303 433)(304 434)(305 435)(306 436)(307 437)(308 438)(309 439)(310 440)(311 441)(312 442)(313 443)(314 444)(315 445)(316 446)(317 447)(318 448)(319 393)(320 394)(321 395)(322 396)(323 397)(324 398)(325 399)(326 400)(327 401)(328 402)(329 403)(330 404)(331 405)(332 406)(333 407)(334 408)(335 409)(336 410)
(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)| (1,346)(2,347)(3,348)(4,349)(5,350)(6,351)(7,352)(8,353)(9,354)(10,355)(11,356)(12,357)(13,358)(14,359)(15,360)(16,361)(17,362)(18,363)(19,364)(20,365)(21,366)(22,367)(23,368)(24,369)(25,370)(26,371)(27,372)(28,373)(29,374)(30,375)(31,376)(32,377)(33,378)(34,379)(35,380)(36,381)(37,382)(38,383)(39,384)(40,385)(41,386)(42,387)(43,388)(44,389)(45,390)(46,391)(47,392)(48,337)(49,338)(50,339)(51,340)(52,341)(53,342)(54,343)(55,344)(56,345)(57,147)(58,148)(59,149)(60,150)(61,151)(62,152)(63,153)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,163)(74,164)(75,165)(76,166)(77,167)(78,168)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(169,290)(170,291)(171,292)(172,293)(173,294)(174,295)(175,296)(176,297)(177,298)(178,299)(179,300)(180,301)(181,302)(182,303)(183,304)(184,305)(185,306)(186,307)(187,308)(188,309)(189,310)(190,311)(191,312)(192,313)(193,314)(194,315)(195,316)(196,317)(197,318)(198,319)(199,320)(200,321)(201,322)(202,323)(203,324)(204,325)(205,326)(206,327)(207,328)(208,329)(209,330)(210,331)(211,332)(212,333)(213,334)(214,335)(215,336)(216,281)(217,282)(218,283)(219,284)(220,285)(221,286)(222,287)(223,288)(224,289)(225,395)(226,396)(227,397)(228,398)(229,399)(230,400)(231,401)(232,402)(233,403)(234,404)(235,405)(236,406)(237,407)(238,408)(239,409)(240,410)(241,411)(242,412)(243,413)(244,414)(245,415)(246,416)(247,417)(248,418)(249,419)(250,420)(251,421)(252,422)(253,423)(254,424)(255,425)(256,426)(257,427)(258,428)(259,429)(260,430)(261,431)(262,432)(263,433)(264,434)(265,435)(266,436)(267,437)(268,438)(269,439)(270,440)(271,441)(272,442)(273,443)(274,444)(275,445)(276,446)(277,447)(278,448)(279,393)(280,394), (1,235)(2,236)(3,237)(4,238)(5,239)(6,240)(7,241)(8,242)(9,243)(10,244)(11,245)(12,246)(13,247)(14,248)(15,249)(16,250)(17,251)(18,252)(19,253)(20,254)(21,255)(22,256)(23,257)(24,258)(25,259)(26,260)(27,261)(28,262)(29,263)(30,264)(31,265)(32,266)(33,267)(34,268)(35,269)(36,270)(37,271)(38,272)(39,273)(40,274)(41,275)(42,276)(43,277)(44,278)(45,279)(46,280)(47,225)(48,226)(49,227)(50,228)(51,229)(52,230)(53,231)(54,232)(55,233)(56,234)(57,295)(58,296)(59,297)(60,298)(61,299)(62,300)(63,301)(64,302)(65,303)(66,304)(67,305)(68,306)(69,307)(70,308)(71,309)(72,310)(73,311)(74,312)(75,313)(76,314)(77,315)(78,316)(79,317)(80,318)(81,319)(82,320)(83,321)(84,322)(85,323)(86,324)(87,325)(88,326)(89,327)(90,328)(91,329)(92,330)(93,331)(94,332)(95,333)(96,334)(97,335)(98,336)(99,281)(100,282)(101,283)(102,284)(103,285)(104,286)(105,287)(106,288)(107,289)(108,290)(109,291)(110,292)(111,293)(112,294)(113,196)(114,197)(115,198)(116,199)(117,200)(118,201)(119,202)(120,203)(121,204)(122,205)(123,206)(124,207)(125,208)(126,209)(127,210)(128,211)(129,212)(130,213)(131,214)(132,215)(133,216)(134,217)(135,218)(136,219)(137,220)(138,221)(139,222)(140,223)(141,224)(142,169)(143,170)(144,171)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(337,396)(338,397)(339,398)(340,399)(341,400)(342,401)(343,402)(344,403)(345,404)(346,405)(347,406)(348,407)(349,408)(350,409)(351,410)(352,411)(353,412)(354,413)(355,414)(356,415)(357,416)(358,417)(359,418)(360,419)(361,420)(362,421)(363,422)(364,423)(365,424)(366,425)(367,426)(368,427)(369,428)(370,429)(371,430)(372,431)(373,432)(374,433)(375,434)(376,435)(377,436)(378,437)(379,438)(380,439)(381,440)(382,441)(383,442)(384,443)(385,444)(386,445)(387,446)(388,447)(389,448)(390,393)(391,394)(392,395), (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,157)(32,158)(33,159)(34,160)(35,161)(36,162)(37,163)(38,164)(39,165)(40,166)(41,167)(42,168)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,366)(58,367)(59,368)(60,369)(61,370)(62,371)(63,372)(64,373)(65,374)(66,375)(67,376)(68,377)(69,378)(70,379)(71,380)(72,381)(73,382)(74,383)(75,384)(76,385)(77,386)(78,387)(79,388)(80,389)(81,390)(82,391)(83,392)(84,337)(85,338)(86,339)(87,340)(88,341)(89,342)(90,343)(91,344)(92,345)(93,346)(94,347)(95,348)(96,349)(97,350)(98,351)(99,352)(100,353)(101,354)(102,355)(103,356)(104,357)(105,358)(106,359)(107,360)(108,361)(109,362)(110,363)(111,364)(112,365)(169,250)(170,251)(171,252)(172,253)(173,254)(174,255)(175,256)(176,257)(177,258)(178,259)(179,260)(180,261)(181,262)(182,263)(183,264)(184,265)(185,266)(186,267)(187,268)(188,269)(189,270)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)(196,277)(197,278)(198,279)(199,280)(200,225)(201,226)(202,227)(203,228)(204,229)(205,230)(206,231)(207,232)(208,233)(209,234)(210,235)(211,236)(212,237)(213,238)(214,239)(215,240)(216,241)(217,242)(218,243)(219,244)(220,245)(221,246)(222,247)(223,248)(224,249)(281,411)(282,412)(283,413)(284,414)(285,415)(286,416)(287,417)(288,418)(289,419)(290,420)(291,421)(292,422)(293,423)(294,424)(295,425)(296,426)(297,427)(298,428)(299,429)(300,430)(301,431)(302,432)(303,433)(304,434)(305,435)(306,436)(307,437)(308,438)(309,439)(310,440)(311,441)(312,442)(313,443)(314,444)(315,445)(316,446)(317,447)(318,448)(319,393)(320,394)(321,395)(322,396)(323,397)(324,398)(325,399)(326,400)(327,401)(328,402)(329,403)(330,404)(331,405)(332,406)(333,407)(334,408)(335,409)(336,410), (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( (1,346)(2,347)(3,348)(4,349)(5,350)(6,351)(7,352)(8,353)(9,354)(10,355)(11,356)(12,357)(13,358)(14,359)(15,360)(16,361)(17,362)(18,363)(19,364)(20,365)(21,366)(22,367)(23,368)(24,369)(25,370)(26,371)(27,372)(28,373)(29,374)(30,375)(31,376)(32,377)(33,378)(34,379)(35,380)(36,381)(37,382)(38,383)(39,384)(40,385)(41,386)(42,387)(43,388)(44,389)(45,390)(46,391)(47,392)(48,337)(49,338)(50,339)(51,340)(52,341)(53,342)(54,343)(55,344)(56,345)(57,147)(58,148)(59,149)(60,150)(61,151)(62,152)(63,153)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,163)(74,164)(75,165)(76,166)(77,167)(78,168)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(169,290)(170,291)(171,292)(172,293)(173,294)(174,295)(175,296)(176,297)(177,298)(178,299)(179,300)(180,301)(181,302)(182,303)(183,304)(184,305)(185,306)(186,307)(187,308)(188,309)(189,310)(190,311)(191,312)(192,313)(193,314)(194,315)(195,316)(196,317)(197,318)(198,319)(199,320)(200,321)(201,322)(202,323)(203,324)(204,325)(205,326)(206,327)(207,328)(208,329)(209,330)(210,331)(211,332)(212,333)(213,334)(214,335)(215,336)(216,281)(217,282)(218,283)(219,284)(220,285)(221,286)(222,287)(223,288)(224,289)(225,395)(226,396)(227,397)(228,398)(229,399)(230,400)(231,401)(232,402)(233,403)(234,404)(235,405)(236,406)(237,407)(238,408)(239,409)(240,410)(241,411)(242,412)(243,413)(244,414)(245,415)(246,416)(247,417)(248,418)(249,419)(250,420)(251,421)(252,422)(253,423)(254,424)(255,425)(256,426)(257,427)(258,428)(259,429)(260,430)(261,431)(262,432)(263,433)(264,434)(265,435)(266,436)(267,437)(268,438)(269,439)(270,440)(271,441)(272,442)(273,443)(274,444)(275,445)(276,446)(277,447)(278,448)(279,393)(280,394), (1,235)(2,236)(3,237)(4,238)(5,239)(6,240)(7,241)(8,242)(9,243)(10,244)(11,245)(12,246)(13,247)(14,248)(15,249)(16,250)(17,251)(18,252)(19,253)(20,254)(21,255)(22,256)(23,257)(24,258)(25,259)(26,260)(27,261)(28,262)(29,263)(30,264)(31,265)(32,266)(33,267)(34,268)(35,269)(36,270)(37,271)(38,272)(39,273)(40,274)(41,275)(42,276)(43,277)(44,278)(45,279)(46,280)(47,225)(48,226)(49,227)(50,228)(51,229)(52,230)(53,231)(54,232)(55,233)(56,234)(57,295)(58,296)(59,297)(60,298)(61,299)(62,300)(63,301)(64,302)(65,303)(66,304)(67,305)(68,306)(69,307)(70,308)(71,309)(72,310)(73,311)(74,312)(75,313)(76,314)(77,315)(78,316)(79,317)(80,318)(81,319)(82,320)(83,321)(84,322)(85,323)(86,324)(87,325)(88,326)(89,327)(90,328)(91,329)(92,330)(93,331)(94,332)(95,333)(96,334)(97,335)(98,336)(99,281)(100,282)(101,283)(102,284)(103,285)(104,286)(105,287)(106,288)(107,289)(108,290)(109,291)(110,292)(111,293)(112,294)(113,196)(114,197)(115,198)(116,199)(117,200)(118,201)(119,202)(120,203)(121,204)(122,205)(123,206)(124,207)(125,208)(126,209)(127,210)(128,211)(129,212)(130,213)(131,214)(132,215)(133,216)(134,217)(135,218)(136,219)(137,220)(138,221)(139,222)(140,223)(141,224)(142,169)(143,170)(144,171)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(337,396)(338,397)(339,398)(340,399)(341,400)(342,401)(343,402)(344,403)(345,404)(346,405)(347,406)(348,407)(349,408)(350,409)(351,410)(352,411)(353,412)(354,413)(355,414)(356,415)(357,416)(358,417)(359,418)(360,419)(361,420)(362,421)(363,422)(364,423)(365,424)(366,425)(367,426)(368,427)(369,428)(370,429)(371,430)(372,431)(373,432)(374,433)(375,434)(376,435)(377,436)(378,437)(379,438)(380,439)(381,440)(382,441)(383,442)(384,443)(385,444)(386,445)(387,446)(388,447)(389,448)(390,393)(391,394)(392,395), (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,157)(32,158)(33,159)(34,160)(35,161)(36,162)(37,163)(38,164)(39,165)(40,166)(41,167)(42,168)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,366)(58,367)(59,368)(60,369)(61,370)(62,371)(63,372)(64,373)(65,374)(66,375)(67,376)(68,377)(69,378)(70,379)(71,380)(72,381)(73,382)(74,383)(75,384)(76,385)(77,386)(78,387)(79,388)(80,389)(81,390)(82,391)(83,392)(84,337)(85,338)(86,339)(87,340)(88,341)(89,342)(90,343)(91,344)(92,345)(93,346)(94,347)(95,348)(96,349)(97,350)(98,351)(99,352)(100,353)(101,354)(102,355)(103,356)(104,357)(105,358)(106,359)(107,360)(108,361)(109,362)(110,363)(111,364)(112,365)(169,250)(170,251)(171,252)(172,253)(173,254)(174,255)(175,256)(176,257)(177,258)(178,259)(179,260)(180,261)(181,262)(182,263)(183,264)(184,265)(185,266)(186,267)(187,268)(188,269)(189,270)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)(196,277)(197,278)(198,279)(199,280)(200,225)(201,226)(202,227)(203,228)(204,229)(205,230)(206,231)(207,232)(208,233)(209,234)(210,235)(211,236)(212,237)(213,238)(214,239)(215,240)(216,241)(217,242)(218,243)(219,244)(220,245)(221,246)(222,247)(223,248)(224,249)(281,411)(282,412)(283,413)(284,414)(285,415)(286,416)(287,417)(288,418)(289,419)(290,420)(291,421)(292,422)(293,423)(294,424)(295,425)(296,426)(297,427)(298,428)(299,429)(300,430)(301,431)(302,432)(303,433)(304,434)(305,435)(306,436)(307,437)(308,438)(309,439)(310,440)(311,441)(312,442)(313,443)(314,444)(315,445)(316,446)(317,447)(318,448)(319,393)(320,394)(321,395)(322,396)(323,397)(324,398)(325,399)(326,400)(327,401)(328,402)(329,403)(330,404)(331,405)(332,406)(333,407)(334,408)(335,409)(336,410), (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([[(1,346),(2,347),(3,348),(4,349),(5,350),(6,351),(7,352),(8,353),(9,354),(10,355),(11,356),(12,357),(13,358),(14,359),(15,360),(16,361),(17,362),(18,363),(19,364),(20,365),(21,366),(22,367),(23,368),(24,369),(25,370),(26,371),(27,372),(28,373),(29,374),(30,375),(31,376),(32,377),(33,378),(34,379),(35,380),(36,381),(37,382),(38,383),(39,384),(40,385),(41,386),(42,387),(43,388),(44,389),(45,390),(46,391),(47,392),(48,337),(49,338),(50,339),(51,340),(52,341),(53,342),(54,343),(55,344),(56,345),(57,147),(58,148),(59,149),(60,150),(61,151),(62,152),(63,153),(64,154),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,161),(72,162),(73,163),(74,164),(75,165),(76,166),(77,167),(78,168),(79,113),(80,114),(81,115),(82,116),(83,117),(84,118),(85,119),(86,120),(87,121),(88,122),(89,123),(90,124),(91,125),(92,126),(93,127),(94,128),(95,129),(96,130),(97,131),(98,132),(99,133),(100,134),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,141),(108,142),(109,143),(110,144),(111,145),(112,146),(169,290),(170,291),(171,292),(172,293),(173,294),(174,295),(175,296),(176,297),(177,298),(178,299),(179,300),(180,301),(181,302),(182,303),(183,304),(184,305),(185,306),(186,307),(187,308),(188,309),(189,310),(190,311),(191,312),(192,313),(193,314),(194,315),(195,316),(196,317),(197,318),(198,319),(199,320),(200,321),(201,322),(202,323),(203,324),(204,325),(205,326),(206,327),(207,328),(208,329),(209,330),(210,331),(211,332),(212,333),(213,334),(214,335),(215,336),(216,281),(217,282),(218,283),(219,284),(220,285),(221,286),(222,287),(223,288),(224,289),(225,395),(226,396),(227,397),(228,398),(229,399),(230,400),(231,401),(232,402),(233,403),(234,404),(235,405),(236,406),(237,407),(238,408),(239,409),(240,410),(241,411),(242,412),(243,413),(244,414),(245,415),(246,416),(247,417),(248,418),(249,419),(250,420),(251,421),(252,422),(253,423),(254,424),(255,425),(256,426),(257,427),(258,428),(259,429),(260,430),(261,431),(262,432),(263,433),(264,434),(265,435),(266,436),(267,437),(268,438),(269,439),(270,440),(271,441),(272,442),(273,443),(274,444),(275,445),(276,446),(277,447),(278,448),(279,393),(280,394)], [(1,235),(2,236),(3,237),(4,238),(5,239),(6,240),(7,241),(8,242),(9,243),(10,244),(11,245),(12,246),(13,247),(14,248),(15,249),(16,250),(17,251),(18,252),(19,253),(20,254),(21,255),(22,256),(23,257),(24,258),(25,259),(26,260),(27,261),(28,262),(29,263),(30,264),(31,265),(32,266),(33,267),(34,268),(35,269),(36,270),(37,271),(38,272),(39,273),(40,274),(41,275),(42,276),(43,277),(44,278),(45,279),(46,280),(47,225),(48,226),(49,227),(50,228),(51,229),(52,230),(53,231),(54,232),(55,233),(56,234),(57,295),(58,296),(59,297),(60,298),(61,299),(62,300),(63,301),(64,302),(65,303),(66,304),(67,305),(68,306),(69,307),(70,308),(71,309),(72,310),(73,311),(74,312),(75,313),(76,314),(77,315),(78,316),(79,317),(80,318),(81,319),(82,320),(83,321),(84,322),(85,323),(86,324),(87,325),(88,326),(89,327),(90,328),(91,329),(92,330),(93,331),(94,332),(95,333),(96,334),(97,335),(98,336),(99,281),(100,282),(101,283),(102,284),(103,285),(104,286),(105,287),(106,288),(107,289),(108,290),(109,291),(110,292),(111,293),(112,294),(113,196),(114,197),(115,198),(116,199),(117,200),(118,201),(119,202),(120,203),(121,204),(122,205),(123,206),(124,207),(125,208),(126,209),(127,210),(128,211),(129,212),(130,213),(131,214),(132,215),(133,216),(134,217),(135,218),(136,219),(137,220),(138,221),(139,222),(140,223),(141,224),(142,169),(143,170),(144,171),(145,172),(146,173),(147,174),(148,175),(149,176),(150,177),(151,178),(152,179),(153,180),(154,181),(155,182),(156,183),(157,184),(158,185),(159,186),(160,187),(161,188),(162,189),(163,190),(164,191),(165,192),(166,193),(167,194),(168,195),(337,396),(338,397),(339,398),(340,399),(341,400),(342,401),(343,402),(344,403),(345,404),(346,405),(347,406),(348,407),(349,408),(350,409),(351,410),(352,411),(353,412),(354,413),(355,414),(356,415),(357,416),(358,417),(359,418),(360,419),(361,420),(362,421),(363,422),(364,423),(365,424),(366,425),(367,426),(368,427),(369,428),(370,429),(371,430),(372,431),(373,432),(374,433),(375,434),(376,435),(377,436),(378,437),(379,438),(380,439),(381,440),(382,441),(383,442),(384,443),(385,444),(386,445),(387,446),(388,447),(389,448),(390,393),(391,394),(392,395)], [(1,127),(2,128),(3,129),(4,130),(5,131),(6,132),(7,133),(8,134),(9,135),(10,136),(11,137),(12,138),(13,139),(14,140),(15,141),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,151),(26,152),(27,153),(28,154),(29,155),(30,156),(31,157),(32,158),(33,159),(34,160),(35,161),(36,162),(37,163),(38,164),(39,165),(40,166),(41,167),(42,168),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,366),(58,367),(59,368),(60,369),(61,370),(62,371),(63,372),(64,373),(65,374),(66,375),(67,376),(68,377),(69,378),(70,379),(71,380),(72,381),(73,382),(74,383),(75,384),(76,385),(77,386),(78,387),(79,388),(80,389),(81,390),(82,391),(83,392),(84,337),(85,338),(86,339),(87,340),(88,341),(89,342),(90,343),(91,344),(92,345),(93,346),(94,347),(95,348),(96,349),(97,350),(98,351),(99,352),(100,353),(101,354),(102,355),(103,356),(104,357),(105,358),(106,359),(107,360),(108,361),(109,362),(110,363),(111,364),(112,365),(169,250),(170,251),(171,252),(172,253),(173,254),(174,255),(175,256),(176,257),(177,258),(178,259),(179,260),(180,261),(181,262),(182,263),(183,264),(184,265),(185,266),(186,267),(187,268),(188,269),(189,270),(190,271),(191,272),(192,273),(193,274),(194,275),(195,276),(196,277),(197,278),(198,279),(199,280),(200,225),(201,226),(202,227),(203,228),(204,229),(205,230),(206,231),(207,232),(208,233),(209,234),(210,235),(211,236),(212,237),(213,238),(214,239),(215,240),(216,241),(217,242),(218,243),(219,244),(220,245),(221,246),(222,247),(223,248),(224,249),(281,411),(282,412),(283,413),(284,414),(285,415),(286,416),(287,417),(288,418),(289,419),(290,420),(291,421),(292,422),(293,423),(294,424),(295,425),(296,426),(297,427),(298,428),(299,429),(300,430),(301,431),(302,432),(303,433),(304,434),(305,435),(306,436),(307,437),(308,438),(309,439),(310,440),(311,441),(312,442),(313,443),(314,444),(315,445),(316,446),(317,447),(318,448),(319,393),(320,394),(321,395),(322,396),(323,397),(324,398),(325,399),(326,400),(327,401),(328,402),(329,403),(330,404),(331,405),(332,406),(333,407),(334,408),(335,409),(336,410)], [(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 ··· 2O 4A ··· 4P 7A ··· 7F 8A ··· 8AF 14A ··· 14CL 28A ··· 28CR 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 type + + + image C1 C2 C2 C4 C4 C7 C8 C14 C14 C28 C28 C56 kernel C23×C56 C22×C56 C23×C28 C22×C28 C23×C14 C23×C8 C22×C14 C22×C8 C23×C4 C22×C4 C24 C23 # reps 1 14 1 14 2 6 32 84 6 84 12 192

Matrix representation of C23×C56 in GL4(𝔽113) generated by

 1 0 0 0 0 112 0 0 0 0 112 0 0 0 0 1
,
 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 1
,
 98 0 0 0 0 62 0 0 0 0 15 0 0 0 0 81
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,1],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,1],[98,0,0,0,0,62,0,0,0,0,15,0,0,0,0,81] >;

C23×C56 in GAP, Magma, Sage, TeX

C_2^3\times C_{56}
% in TeX

G:=Group("C2^3xC56");
// GroupNames label

G:=SmallGroup(448,1348);
// by ID

G=gap.SmallGroup(448,1348);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,124]);
// Polycyclic

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

׿
×
𝔽