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G = C23×C56order 448 = 26·7

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

direct product, abelian, monomial, 2-elementary

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

Series: Derived Chief Lower central Upper central

C1 — C23×C56
C1C2C4C28C56C2×C56C22×C56 — C23×C56
C1 — C23×C56
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 [×14], C4, C4 [×7], C22 [×35], C7, C8 [×8], C2×C4 [×28], C23 [×15], C14, C14 [×14], C2×C8 [×28], C22×C4 [×14], C24, C28, C28 [×7], C2×C14 [×35], C22×C8 [×14], C23×C4, C56 [×8], C2×C28 [×28], C22×C14 [×15], C23×C8, C2×C56 [×28], C22×C28 [×14], C23×C14, C22×C56 [×14], C23×C28, C23×C56
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C7, C8 [×8], C2×C4 [×28], C23 [×15], C14 [×15], C2×C8 [×28], C22×C4 [×14], C24, C28 [×8], C2×C14 [×35], C22×C8 [×14], C23×C4, C56 [×8], C2×C28 [×28], C22×C14 [×15], C23×C8, C2×C56 [×28], C22×C28 [×14], C23×C14, C22×C56 [×14], C23×C28, C23×C56

Smallest permutation representation of C23×C56
Regular action on 448 points
Generators in S448
(1 418)(2 419)(3 420)(4 421)(5 422)(6 423)(7 424)(8 425)(9 426)(10 427)(11 428)(12 429)(13 430)(14 431)(15 432)(16 433)(17 434)(18 435)(19 436)(20 437)(21 438)(22 439)(23 440)(24 441)(25 442)(26 443)(27 444)(28 445)(29 446)(30 447)(31 448)(32 393)(33 394)(34 395)(35 396)(36 397)(37 398)(38 399)(39 400)(40 401)(41 402)(42 403)(43 404)(44 405)(45 406)(46 407)(47 408)(48 409)(49 410)(50 411)(51 412)(52 413)(53 414)(54 415)(55 416)(56 417)(57 160)(58 161)(59 162)(60 163)(61 164)(62 165)(63 166)(64 167)(65 168)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)(73 120)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(81 128)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(169 245)(170 246)(171 247)(172 248)(173 249)(174 250)(175 251)(176 252)(177 253)(178 254)(179 255)(180 256)(181 257)(182 258)(183 259)(184 260)(185 261)(186 262)(187 263)(188 264)(189 265)(190 266)(191 267)(192 268)(193 269)(194 270)(195 271)(196 272)(197 273)(198 274)(199 275)(200 276)(201 277)(202 278)(203 279)(204 280)(205 225)(206 226)(207 227)(208 228)(209 229)(210 230)(211 231)(212 232)(213 233)(214 234)(215 235)(216 236)(217 237)(218 238)(219 239)(220 240)(221 241)(222 242)(223 243)(224 244)(281 385)(282 386)(283 387)(284 388)(285 389)(286 390)(287 391)(288 392)(289 337)(290 338)(291 339)(292 340)(293 341)(294 342)(295 343)(296 344)(297 345)(298 346)(299 347)(300 348)(301 349)(302 350)(303 351)(304 352)(305 353)(306 354)(307 355)(308 356)(309 357)(310 358)(311 359)(312 360)(313 361)(314 362)(315 363)(316 364)(317 365)(318 366)(319 367)(320 368)(321 369)(322 370)(323 371)(324 372)(325 373)(326 374)(327 375)(328 376)(329 377)(330 378)(331 379)(332 380)(333 381)(334 382)(335 383)(336 384)
(1 220)(2 221)(3 222)(4 223)(5 224)(6 169)(7 170)(8 171)(9 172)(10 173)(11 174)(12 175)(13 176)(14 177)(15 178)(16 179)(17 180)(18 181)(19 182)(20 183)(21 184)(22 185)(23 186)(24 187)(25 188)(26 189)(27 190)(28 191)(29 192)(30 193)(31 194)(32 195)(33 196)(34 197)(35 198)(36 199)(37 200)(38 201)(39 202)(40 203)(41 204)(42 205)(43 206)(44 207)(45 208)(46 209)(47 210)(48 211)(49 212)(50 213)(51 214)(52 215)(53 216)(54 217)(55 218)(56 219)(57 363)(58 364)(59 365)(60 366)(61 367)(62 368)(63 369)(64 370)(65 371)(66 372)(67 373)(68 374)(69 375)(70 376)(71 377)(72 378)(73 379)(74 380)(75 381)(76 382)(77 383)(78 384)(79 385)(80 386)(81 387)(82 388)(83 389)(84 390)(85 391)(86 392)(87 337)(88 338)(89 339)(90 340)(91 341)(92 342)(93 343)(94 344)(95 345)(96 346)(97 347)(98 348)(99 349)(100 350)(101 351)(102 352)(103 353)(104 354)(105 355)(106 356)(107 357)(108 358)(109 359)(110 360)(111 361)(112 362)(113 324)(114 325)(115 326)(116 327)(117 328)(118 329)(119 330)(120 331)(121 332)(122 333)(123 334)(124 335)(125 336)(126 281)(127 282)(128 283)(129 284)(130 285)(131 286)(132 287)(133 288)(134 289)(135 290)(136 291)(137 292)(138 293)(139 294)(140 295)(141 296)(142 297)(143 298)(144 299)(145 300)(146 301)(147 302)(148 303)(149 304)(150 305)(151 306)(152 307)(153 308)(154 309)(155 310)(156 311)(157 312)(158 313)(159 314)(160 315)(161 316)(162 317)(163 318)(164 319)(165 320)(166 321)(167 322)(168 323)(225 403)(226 404)(227 405)(228 406)(229 407)(230 408)(231 409)(232 410)(233 411)(234 412)(235 413)(236 414)(237 415)(238 416)(239 417)(240 418)(241 419)(242 420)(243 421)(244 422)(245 423)(246 424)(247 425)(248 426)(249 427)(250 428)(251 429)(252 430)(253 431)(254 432)(255 433)(256 434)(257 435)(258 436)(259 437)(260 438)(261 439)(262 440)(263 441)(264 442)(265 443)(266 444)(267 445)(268 446)(269 447)(270 448)(271 393)(272 394)(273 395)(274 396)(275 397)(276 398)(277 399)(278 400)(279 401)(280 402)
(1 349)(2 350)(3 351)(4 352)(5 353)(6 354)(7 355)(8 356)(9 357)(10 358)(11 359)(12 360)(13 361)(14 362)(15 363)(16 364)(17 365)(18 366)(19 367)(20 368)(21 369)(22 370)(23 371)(24 372)(25 373)(26 374)(27 375)(28 376)(29 377)(30 378)(31 379)(32 380)(33 381)(34 382)(35 383)(36 384)(37 385)(38 386)(39 387)(40 388)(41 389)(42 390)(43 391)(44 392)(45 337)(46 338)(47 339)(48 340)(49 341)(50 342)(51 343)(52 344)(53 345)(54 346)(55 347)(56 348)(57 178)(58 179)(59 180)(60 181)(61 182)(62 183)(63 184)(64 185)(65 186)(66 187)(67 188)(68 189)(69 190)(70 191)(71 192)(72 193)(73 194)(74 195)(75 196)(76 197)(77 198)(78 199)(79 200)(80 201)(81 202)(82 203)(83 204)(84 205)(85 206)(86 207)(87 208)(88 209)(89 210)(90 211)(91 212)(92 213)(93 214)(94 215)(95 216)(96 217)(97 218)(98 219)(99 220)(100 221)(101 222)(102 223)(103 224)(104 169)(105 170)(106 171)(107 172)(108 173)(109 174)(110 175)(111 176)(112 177)(113 263)(114 264)(115 265)(116 266)(117 267)(118 268)(119 269)(120 270)(121 271)(122 272)(123 273)(124 274)(125 275)(126 276)(127 277)(128 278)(129 279)(130 280)(131 225)(132 226)(133 227)(134 228)(135 229)(136 230)(137 231)(138 232)(139 233)(140 234)(141 235)(142 236)(143 237)(144 238)(145 239)(146 240)(147 241)(148 242)(149 243)(150 244)(151 245)(152 246)(153 247)(154 248)(155 249)(156 250)(157 251)(158 252)(159 253)(160 254)(161 255)(162 256)(163 257)(164 258)(165 259)(166 260)(167 261)(168 262)(281 398)(282 399)(283 400)(284 401)(285 402)(286 403)(287 404)(288 405)(289 406)(290 407)(291 408)(292 409)(293 410)(294 411)(295 412)(296 413)(297 414)(298 415)(299 416)(300 417)(301 418)(302 419)(303 420)(304 421)(305 422)(306 423)(307 424)(308 425)(309 426)(310 427)(311 428)(312 429)(313 430)(314 431)(315 432)(316 433)(317 434)(318 435)(319 436)(320 437)(321 438)(322 439)(323 440)(324 441)(325 442)(326 443)(327 444)(328 445)(329 446)(330 447)(331 448)(332 393)(333 394)(334 395)(335 396)(336 397)
(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)| (1,418)(2,419)(3,420)(4,421)(5,422)(6,423)(7,424)(8,425)(9,426)(10,427)(11,428)(12,429)(13,430)(14,431)(15,432)(16,433)(17,434)(18,435)(19,436)(20,437)(21,438)(22,439)(23,440)(24,441)(25,442)(26,443)(27,444)(28,445)(29,446)(30,447)(31,448)(32,393)(33,394)(34,395)(35,396)(36,397)(37,398)(38,399)(39,400)(40,401)(41,402)(42,403)(43,404)(44,405)(45,406)(46,407)(47,408)(48,409)(49,410)(50,411)(51,412)(52,413)(53,414)(54,415)(55,416)(56,417)(57,160)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,113)(67,114)(68,115)(69,116)(70,117)(71,118)(72,119)(73,120)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,127)(81,128)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(169,245)(170,246)(171,247)(172,248)(173,249)(174,250)(175,251)(176,252)(177,253)(178,254)(179,255)(180,256)(181,257)(182,258)(183,259)(184,260)(185,261)(186,262)(187,263)(188,264)(189,265)(190,266)(191,267)(192,268)(193,269)(194,270)(195,271)(196,272)(197,273)(198,274)(199,275)(200,276)(201,277)(202,278)(203,279)(204,280)(205,225)(206,226)(207,227)(208,228)(209,229)(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)(216,236)(217,237)(218,238)(219,239)(220,240)(221,241)(222,242)(223,243)(224,244)(281,385)(282,386)(283,387)(284,388)(285,389)(286,390)(287,391)(288,392)(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)(296,344)(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)(304,352)(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)(312,360)(313,361)(314,362)(315,363)(316,364)(317,365)(318,366)(319,367)(320,368)(321,369)(322,370)(323,371)(324,372)(325,373)(326,374)(327,375)(328,376)(329,377)(330,378)(331,379)(332,380)(333,381)(334,382)(335,383)(336,384), (1,220)(2,221)(3,222)(4,223)(5,224)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,193)(31,194)(32,195)(33,196)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,209)(47,210)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,363)(58,364)(59,365)(60,366)(61,367)(62,368)(63,369)(64,370)(65,371)(66,372)(67,373)(68,374)(69,375)(70,376)(71,377)(72,378)(73,379)(74,380)(75,381)(76,382)(77,383)(78,384)(79,385)(80,386)(81,387)(82,388)(83,389)(84,390)(85,391)(86,392)(87,337)(88,338)(89,339)(90,340)(91,341)(92,342)(93,343)(94,344)(95,345)(96,346)(97,347)(98,348)(99,349)(100,350)(101,351)(102,352)(103,353)(104,354)(105,355)(106,356)(107,357)(108,358)(109,359)(110,360)(111,361)(112,362)(113,324)(114,325)(115,326)(116,327)(117,328)(118,329)(119,330)(120,331)(121,332)(122,333)(123,334)(124,335)(125,336)(126,281)(127,282)(128,283)(129,284)(130,285)(131,286)(132,287)(133,288)(134,289)(135,290)(136,291)(137,292)(138,293)(139,294)(140,295)(141,296)(142,297)(143,298)(144,299)(145,300)(146,301)(147,302)(148,303)(149,304)(150,305)(151,306)(152,307)(153,308)(154,309)(155,310)(156,311)(157,312)(158,313)(159,314)(160,315)(161,316)(162,317)(163,318)(164,319)(165,320)(166,321)(167,322)(168,323)(225,403)(226,404)(227,405)(228,406)(229,407)(230,408)(231,409)(232,410)(233,411)(234,412)(235,413)(236,414)(237,415)(238,416)(239,417)(240,418)(241,419)(242,420)(243,421)(244,422)(245,423)(246,424)(247,425)(248,426)(249,427)(250,428)(251,429)(252,430)(253,431)(254,432)(255,433)(256,434)(257,435)(258,436)(259,437)(260,438)(261,439)(262,440)(263,441)(264,442)(265,443)(266,444)(267,445)(268,446)(269,447)(270,448)(271,393)(272,394)(273,395)(274,396)(275,397)(276,398)(277,399)(278,400)(279,401)(280,402), (1,349)(2,350)(3,351)(4,352)(5,353)(6,354)(7,355)(8,356)(9,357)(10,358)(11,359)(12,360)(13,361)(14,362)(15,363)(16,364)(17,365)(18,366)(19,367)(20,368)(21,369)(22,370)(23,371)(24,372)(25,373)(26,374)(27,375)(28,376)(29,377)(30,378)(31,379)(32,380)(33,381)(34,382)(35,383)(36,384)(37,385)(38,386)(39,387)(40,388)(41,389)(42,390)(43,391)(44,392)(45,337)(46,338)(47,339)(48,340)(49,341)(50,342)(51,343)(52,344)(53,345)(54,346)(55,347)(56,348)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,184)(64,185)(65,186)(66,187)(67,188)(68,189)(69,190)(70,191)(71,192)(72,193)(73,194)(74,195)(75,196)(76,197)(77,198)(78,199)(79,200)(80,201)(81,202)(82,203)(83,204)(84,205)(85,206)(86,207)(87,208)(88,209)(89,210)(90,211)(91,212)(92,213)(93,214)(94,215)(95,216)(96,217)(97,218)(98,219)(99,220)(100,221)(101,222)(102,223)(103,224)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,263)(114,264)(115,265)(116,266)(117,267)(118,268)(119,269)(120,270)(121,271)(122,272)(123,273)(124,274)(125,275)(126,276)(127,277)(128,278)(129,279)(130,280)(131,225)(132,226)(133,227)(134,228)(135,229)(136,230)(137,231)(138,232)(139,233)(140,234)(141,235)(142,236)(143,237)(144,238)(145,239)(146,240)(147,241)(148,242)(149,243)(150,244)(151,245)(152,246)(153,247)(154,248)(155,249)(156,250)(157,251)(158,252)(159,253)(160,254)(161,255)(162,256)(163,257)(164,258)(165,259)(166,260)(167,261)(168,262)(281,398)(282,399)(283,400)(284,401)(285,402)(286,403)(287,404)(288,405)(289,406)(290,407)(291,408)(292,409)(293,410)(294,411)(295,412)(296,413)(297,414)(298,415)(299,416)(300,417)(301,418)(302,419)(303,420)(304,421)(305,422)(306,423)(307,424)(308,425)(309,426)(310,427)(311,428)(312,429)(313,430)(314,431)(315,432)(316,433)(317,434)(318,435)(319,436)(320,437)(321,438)(322,439)(323,440)(324,441)(325,442)(326,443)(327,444)(328,445)(329,446)(330,447)(331,448)(332,393)(333,394)(334,395)(335,396)(336,397), (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( (1,418)(2,419)(3,420)(4,421)(5,422)(6,423)(7,424)(8,425)(9,426)(10,427)(11,428)(12,429)(13,430)(14,431)(15,432)(16,433)(17,434)(18,435)(19,436)(20,437)(21,438)(22,439)(23,440)(24,441)(25,442)(26,443)(27,444)(28,445)(29,446)(30,447)(31,448)(32,393)(33,394)(34,395)(35,396)(36,397)(37,398)(38,399)(39,400)(40,401)(41,402)(42,403)(43,404)(44,405)(45,406)(46,407)(47,408)(48,409)(49,410)(50,411)(51,412)(52,413)(53,414)(54,415)(55,416)(56,417)(57,160)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,113)(67,114)(68,115)(69,116)(70,117)(71,118)(72,119)(73,120)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,127)(81,128)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(169,245)(170,246)(171,247)(172,248)(173,249)(174,250)(175,251)(176,252)(177,253)(178,254)(179,255)(180,256)(181,257)(182,258)(183,259)(184,260)(185,261)(186,262)(187,263)(188,264)(189,265)(190,266)(191,267)(192,268)(193,269)(194,270)(195,271)(196,272)(197,273)(198,274)(199,275)(200,276)(201,277)(202,278)(203,279)(204,280)(205,225)(206,226)(207,227)(208,228)(209,229)(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)(216,236)(217,237)(218,238)(219,239)(220,240)(221,241)(222,242)(223,243)(224,244)(281,385)(282,386)(283,387)(284,388)(285,389)(286,390)(287,391)(288,392)(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)(296,344)(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)(304,352)(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)(312,360)(313,361)(314,362)(315,363)(316,364)(317,365)(318,366)(319,367)(320,368)(321,369)(322,370)(323,371)(324,372)(325,373)(326,374)(327,375)(328,376)(329,377)(330,378)(331,379)(332,380)(333,381)(334,382)(335,383)(336,384), (1,220)(2,221)(3,222)(4,223)(5,224)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,193)(31,194)(32,195)(33,196)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,209)(47,210)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,363)(58,364)(59,365)(60,366)(61,367)(62,368)(63,369)(64,370)(65,371)(66,372)(67,373)(68,374)(69,375)(70,376)(71,377)(72,378)(73,379)(74,380)(75,381)(76,382)(77,383)(78,384)(79,385)(80,386)(81,387)(82,388)(83,389)(84,390)(85,391)(86,392)(87,337)(88,338)(89,339)(90,340)(91,341)(92,342)(93,343)(94,344)(95,345)(96,346)(97,347)(98,348)(99,349)(100,350)(101,351)(102,352)(103,353)(104,354)(105,355)(106,356)(107,357)(108,358)(109,359)(110,360)(111,361)(112,362)(113,324)(114,325)(115,326)(116,327)(117,328)(118,329)(119,330)(120,331)(121,332)(122,333)(123,334)(124,335)(125,336)(126,281)(127,282)(128,283)(129,284)(130,285)(131,286)(132,287)(133,288)(134,289)(135,290)(136,291)(137,292)(138,293)(139,294)(140,295)(141,296)(142,297)(143,298)(144,299)(145,300)(146,301)(147,302)(148,303)(149,304)(150,305)(151,306)(152,307)(153,308)(154,309)(155,310)(156,311)(157,312)(158,313)(159,314)(160,315)(161,316)(162,317)(163,318)(164,319)(165,320)(166,321)(167,322)(168,323)(225,403)(226,404)(227,405)(228,406)(229,407)(230,408)(231,409)(232,410)(233,411)(234,412)(235,413)(236,414)(237,415)(238,416)(239,417)(240,418)(241,419)(242,420)(243,421)(244,422)(245,423)(246,424)(247,425)(248,426)(249,427)(250,428)(251,429)(252,430)(253,431)(254,432)(255,433)(256,434)(257,435)(258,436)(259,437)(260,438)(261,439)(262,440)(263,441)(264,442)(265,443)(266,444)(267,445)(268,446)(269,447)(270,448)(271,393)(272,394)(273,395)(274,396)(275,397)(276,398)(277,399)(278,400)(279,401)(280,402), (1,349)(2,350)(3,351)(4,352)(5,353)(6,354)(7,355)(8,356)(9,357)(10,358)(11,359)(12,360)(13,361)(14,362)(15,363)(16,364)(17,365)(18,366)(19,367)(20,368)(21,369)(22,370)(23,371)(24,372)(25,373)(26,374)(27,375)(28,376)(29,377)(30,378)(31,379)(32,380)(33,381)(34,382)(35,383)(36,384)(37,385)(38,386)(39,387)(40,388)(41,389)(42,390)(43,391)(44,392)(45,337)(46,338)(47,339)(48,340)(49,341)(50,342)(51,343)(52,344)(53,345)(54,346)(55,347)(56,348)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,184)(64,185)(65,186)(66,187)(67,188)(68,189)(69,190)(70,191)(71,192)(72,193)(73,194)(74,195)(75,196)(76,197)(77,198)(78,199)(79,200)(80,201)(81,202)(82,203)(83,204)(84,205)(85,206)(86,207)(87,208)(88,209)(89,210)(90,211)(91,212)(92,213)(93,214)(94,215)(95,216)(96,217)(97,218)(98,219)(99,220)(100,221)(101,222)(102,223)(103,224)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,263)(114,264)(115,265)(116,266)(117,267)(118,268)(119,269)(120,270)(121,271)(122,272)(123,273)(124,274)(125,275)(126,276)(127,277)(128,278)(129,279)(130,280)(131,225)(132,226)(133,227)(134,228)(135,229)(136,230)(137,231)(138,232)(139,233)(140,234)(141,235)(142,236)(143,237)(144,238)(145,239)(146,240)(147,241)(148,242)(149,243)(150,244)(151,245)(152,246)(153,247)(154,248)(155,249)(156,250)(157,251)(158,252)(159,253)(160,254)(161,255)(162,256)(163,257)(164,258)(165,259)(166,260)(167,261)(168,262)(281,398)(282,399)(283,400)(284,401)(285,402)(286,403)(287,404)(288,405)(289,406)(290,407)(291,408)(292,409)(293,410)(294,411)(295,412)(296,413)(297,414)(298,415)(299,416)(300,417)(301,418)(302,419)(303,420)(304,421)(305,422)(306,423)(307,424)(308,425)(309,426)(310,427)(311,428)(312,429)(313,430)(314,431)(315,432)(316,433)(317,434)(318,435)(319,436)(320,437)(321,438)(322,439)(323,440)(324,441)(325,442)(326,443)(327,444)(328,445)(329,446)(330,447)(331,448)(332,393)(333,394)(334,395)(335,396)(336,397), (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([(1,418),(2,419),(3,420),(4,421),(5,422),(6,423),(7,424),(8,425),(9,426),(10,427),(11,428),(12,429),(13,430),(14,431),(15,432),(16,433),(17,434),(18,435),(19,436),(20,437),(21,438),(22,439),(23,440),(24,441),(25,442),(26,443),(27,444),(28,445),(29,446),(30,447),(31,448),(32,393),(33,394),(34,395),(35,396),(36,397),(37,398),(38,399),(39,400),(40,401),(41,402),(42,403),(43,404),(44,405),(45,406),(46,407),(47,408),(48,409),(49,410),(50,411),(51,412),(52,413),(53,414),(54,415),(55,416),(56,417),(57,160),(58,161),(59,162),(60,163),(61,164),(62,165),(63,166),(64,167),(65,168),(66,113),(67,114),(68,115),(69,116),(70,117),(71,118),(72,119),(73,120),(74,121),(75,122),(76,123),(77,124),(78,125),(79,126),(80,127),(81,128),(82,129),(83,130),(84,131),(85,132),(86,133),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143),(97,144),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159),(169,245),(170,246),(171,247),(172,248),(173,249),(174,250),(175,251),(176,252),(177,253),(178,254),(179,255),(180,256),(181,257),(182,258),(183,259),(184,260),(185,261),(186,262),(187,263),(188,264),(189,265),(190,266),(191,267),(192,268),(193,269),(194,270),(195,271),(196,272),(197,273),(198,274),(199,275),(200,276),(201,277),(202,278),(203,279),(204,280),(205,225),(206,226),(207,227),(208,228),(209,229),(210,230),(211,231),(212,232),(213,233),(214,234),(215,235),(216,236),(217,237),(218,238),(219,239),(220,240),(221,241),(222,242),(223,243),(224,244),(281,385),(282,386),(283,387),(284,388),(285,389),(286,390),(287,391),(288,392),(289,337),(290,338),(291,339),(292,340),(293,341),(294,342),(295,343),(296,344),(297,345),(298,346),(299,347),(300,348),(301,349),(302,350),(303,351),(304,352),(305,353),(306,354),(307,355),(308,356),(309,357),(310,358),(311,359),(312,360),(313,361),(314,362),(315,363),(316,364),(317,365),(318,366),(319,367),(320,368),(321,369),(322,370),(323,371),(324,372),(325,373),(326,374),(327,375),(328,376),(329,377),(330,378),(331,379),(332,380),(333,381),(334,382),(335,383),(336,384)], [(1,220),(2,221),(3,222),(4,223),(5,224),(6,169),(7,170),(8,171),(9,172),(10,173),(11,174),(12,175),(13,176),(14,177),(15,178),(16,179),(17,180),(18,181),(19,182),(20,183),(21,184),(22,185),(23,186),(24,187),(25,188),(26,189),(27,190),(28,191),(29,192),(30,193),(31,194),(32,195),(33,196),(34,197),(35,198),(36,199),(37,200),(38,201),(39,202),(40,203),(41,204),(42,205),(43,206),(44,207),(45,208),(46,209),(47,210),(48,211),(49,212),(50,213),(51,214),(52,215),(53,216),(54,217),(55,218),(56,219),(57,363),(58,364),(59,365),(60,366),(61,367),(62,368),(63,369),(64,370),(65,371),(66,372),(67,373),(68,374),(69,375),(70,376),(71,377),(72,378),(73,379),(74,380),(75,381),(76,382),(77,383),(78,384),(79,385),(80,386),(81,387),(82,388),(83,389),(84,390),(85,391),(86,392),(87,337),(88,338),(89,339),(90,340),(91,341),(92,342),(93,343),(94,344),(95,345),(96,346),(97,347),(98,348),(99,349),(100,350),(101,351),(102,352),(103,353),(104,354),(105,355),(106,356),(107,357),(108,358),(109,359),(110,360),(111,361),(112,362),(113,324),(114,325),(115,326),(116,327),(117,328),(118,329),(119,330),(120,331),(121,332),(122,333),(123,334),(124,335),(125,336),(126,281),(127,282),(128,283),(129,284),(130,285),(131,286),(132,287),(133,288),(134,289),(135,290),(136,291),(137,292),(138,293),(139,294),(140,295),(141,296),(142,297),(143,298),(144,299),(145,300),(146,301),(147,302),(148,303),(149,304),(150,305),(151,306),(152,307),(153,308),(154,309),(155,310),(156,311),(157,312),(158,313),(159,314),(160,315),(161,316),(162,317),(163,318),(164,319),(165,320),(166,321),(167,322),(168,323),(225,403),(226,404),(227,405),(228,406),(229,407),(230,408),(231,409),(232,410),(233,411),(234,412),(235,413),(236,414),(237,415),(238,416),(239,417),(240,418),(241,419),(242,420),(243,421),(244,422),(245,423),(246,424),(247,425),(248,426),(249,427),(250,428),(251,429),(252,430),(253,431),(254,432),(255,433),(256,434),(257,435),(258,436),(259,437),(260,438),(261,439),(262,440),(263,441),(264,442),(265,443),(266,444),(267,445),(268,446),(269,447),(270,448),(271,393),(272,394),(273,395),(274,396),(275,397),(276,398),(277,399),(278,400),(279,401),(280,402)], [(1,349),(2,350),(3,351),(4,352),(5,353),(6,354),(7,355),(8,356),(9,357),(10,358),(11,359),(12,360),(13,361),(14,362),(15,363),(16,364),(17,365),(18,366),(19,367),(20,368),(21,369),(22,370),(23,371),(24,372),(25,373),(26,374),(27,375),(28,376),(29,377),(30,378),(31,379),(32,380),(33,381),(34,382),(35,383),(36,384),(37,385),(38,386),(39,387),(40,388),(41,389),(42,390),(43,391),(44,392),(45,337),(46,338),(47,339),(48,340),(49,341),(50,342),(51,343),(52,344),(53,345),(54,346),(55,347),(56,348),(57,178),(58,179),(59,180),(60,181),(61,182),(62,183),(63,184),(64,185),(65,186),(66,187),(67,188),(68,189),(69,190),(70,191),(71,192),(72,193),(73,194),(74,195),(75,196),(76,197),(77,198),(78,199),(79,200),(80,201),(81,202),(82,203),(83,204),(84,205),(85,206),(86,207),(87,208),(88,209),(89,210),(90,211),(91,212),(92,213),(93,214),(94,215),(95,216),(96,217),(97,218),(98,219),(99,220),(100,221),(101,222),(102,223),(103,224),(104,169),(105,170),(106,171),(107,172),(108,173),(109,174),(110,175),(111,176),(112,177),(113,263),(114,264),(115,265),(116,266),(117,267),(118,268),(119,269),(120,270),(121,271),(122,272),(123,273),(124,274),(125,275),(126,276),(127,277),(128,278),(129,279),(130,280),(131,225),(132,226),(133,227),(134,228),(135,229),(136,230),(137,231),(138,232),(139,233),(140,234),(141,235),(142,236),(143,237),(144,238),(145,239),(146,240),(147,241),(148,242),(149,243),(150,244),(151,245),(152,246),(153,247),(154,248),(155,249),(156,250),(157,251),(158,252),(159,253),(160,254),(161,255),(162,256),(163,257),(164,258),(165,259),(166,260),(167,261),(168,262),(281,398),(282,399),(283,400),(284,401),(285,402),(286,403),(287,404),(288,405),(289,406),(290,407),(291,408),(292,409),(293,410),(294,411),(295,412),(296,413),(297,414),(298,415),(299,416),(300,417),(301,418),(302,419),(303,420),(304,421),(305,422),(306,423),(307,424),(308,425),(309,426),(310,427),(311,428),(312,429),(313,430),(314,431),(315,432),(316,433),(317,434),(318,435),(319,436),(320,437),(321,438),(322,439),(323,440),(324,441),(325,442),(326,443),(327,444),(328,445),(329,446),(330,447),(331,448),(332,393),(333,394),(334,395),(335,396),(336,397)], [(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···2O4A···4P7A···7F8A···8AF14A···14CL28A···28CR56A···56GJ
order12···24···47···78···814···1428···2856···56
size11···11···11···11···11···11···11···1

448 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C7C8C14C14C28C28C56
kernelC23×C56C22×C56C23×C28C22×C28C23×C14C23×C8C22×C14C22×C8C23×C4C22×C4C24C23
# reps11411426328468412192

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

1000
011200
001120
0001
,
1000
011200
0010
000112
,
112000
011200
001120
0001
,
98000
06200
00150
00081
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