direct product, abelian, monomial, 2-elementary
Aliases: C23×C56, SmallGroup(448,1348)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C56 |
| C1 — C23×C56 |
| C1 — C23×C56 |
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
Generators and relations
G = < a,b,c,d | a2=b2=c2=d56=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
►Regular action on 448 points
Generators in S448
(1 416)(2 417)(3 418)(4 419)(5 420)(6 421)(7 422)(8 423)(9 424)(10 425)(11 426)(12 427)(13 428)(14 429)(15 430)(16 431)(17 432)(18 433)(19 434)(20 435)(21 436)(22 437)(23 438)(24 439)(25 440)(26 441)(27 442)(28 443)(29 444)(30 445)(31 446)(32 447)(33 448)(34 393)(35 394)(36 395)(37 396)(38 397)(39 398)(40 399)(41 400)(42 401)(43 402)(44 403)(45 404)(46 405)(47 406)(48 407)(49 408)(50 409)(51 410)(52 411)(53 412)(54 413)(55 414)(56 415)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 161)(88 162)(89 163)(90 164)(91 165)(92 166)(93 167)(94 168)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)(112 130)(169 330)(170 331)(171 332)(172 333)(173 334)(174 335)(175 336)(176 281)(177 282)(178 283)(179 284)(180 285)(181 286)(182 287)(183 288)(184 289)(185 290)(186 291)(187 292)(188 293)(189 294)(190 295)(191 296)(192 297)(193 298)(194 299)(195 300)(196 301)(197 302)(198 303)(199 304)(200 305)(201 306)(202 307)(203 308)(204 309)(205 310)(206 311)(207 312)(208 313)(209 314)(210 315)(211 316)(212 317)(213 318)(214 319)(215 320)(216 321)(217 322)(218 323)(219 324)(220 325)(221 326)(222 327)(223 328)(224 329)(225 353)(226 354)(227 355)(228 356)(229 357)(230 358)(231 359)(232 360)(233 361)(234 362)(235 363)(236 364)(237 365)(238 366)(239 367)(240 368)(241 369)(242 370)(243 371)(244 372)(245 373)(246 374)(247 375)(248 376)(249 377)(250 378)(251 379)(252 380)(253 381)(254 382)(255 383)(256 384)(257 385)(258 386)(259 387)(260 388)(261 389)(262 390)(263 391)(264 392)(265 337)(266 338)(267 339)(268 340)(269 341)(270 342)(271 343)(272 344)(273 345)(274 346)(275 347)(276 348)(277 349)(278 350)(279 351)(280 352)
(1 249)(2 250)(3 251)(4 252)(5 253)(6 254)(7 255)(8 256)(9 257)(10 258)(11 259)(12 260)(13 261)(14 262)(15 263)(16 264)(17 265)(18 266)(19 267)(20 268)(21 269)(22 270)(23 271)(24 272)(25 273)(26 274)(27 275)(28 276)(29 277)(30 278)(31 279)(32 280)(33 225)(34 226)(35 227)(36 228)(37 229)(38 230)(39 231)(40 232)(41 233)(42 234)(43 235)(44 236)(45 237)(46 238)(47 239)(48 240)(49 241)(50 242)(51 243)(52 244)(53 245)(54 246)(55 247)(56 248)(57 289)(58 290)(59 291)(60 292)(61 293)(62 294)(63 295)(64 296)(65 297)(66 298)(67 299)(68 300)(69 301)(70 302)(71 303)(72 304)(73 305)(74 306)(75 307)(76 308)(77 309)(78 310)(79 311)(80 312)(81 313)(82 314)(83 315)(84 316)(85 317)(86 318)(87 319)(88 320)(89 321)(90 322)(91 323)(92 324)(93 325)(94 326)(95 327)(96 328)(97 329)(98 330)(99 331)(100 332)(101 333)(102 334)(103 335)(104 336)(105 281)(106 282)(107 283)(108 284)(109 285)(110 286)(111 287)(112 288)(113 222)(114 223)(115 224)(116 169)(117 170)(118 171)(119 172)(120 173)(121 174)(122 175)(123 176)(124 177)(125 178)(126 179)(127 180)(128 181)(129 182)(130 183)(131 184)(132 185)(133 186)(134 187)(135 188)(136 189)(137 190)(138 191)(139 192)(140 193)(141 194)(142 195)(143 196)(144 197)(145 198)(146 199)(147 200)(148 201)(149 202)(150 203)(151 204)(152 205)(153 206)(154 207)(155 208)(156 209)(157 210)(158 211)(159 212)(160 213)(161 214)(162 215)(163 216)(164 217)(165 218)(166 219)(167 220)(168 221)(337 432)(338 433)(339 434)(340 435)(341 436)(342 437)(343 438)(344 439)(345 440)(346 441)(347 442)(348 443)(349 444)(350 445)(351 446)(352 447)(353 448)(354 393)(355 394)(356 395)(357 396)(358 397)(359 398)(360 399)(361 400)(362 401)(363 402)(364 403)(365 404)(366 405)(367 406)(368 407)(369 408)(370 409)(371 410)(372 411)(373 412)(374 413)(375 414)(376 415)(377 416)(378 417)(379 418)(380 419)(381 420)(382 421)(383 422)(384 423)(385 424)(386 425)(387 426)(388 427)(389 428)(390 429)(391 430)(392 431)
(1 335)(2 336)(3 281)(4 282)(5 283)(6 284)(7 285)(8 286)(9 287)(10 288)(11 289)(12 290)(13 291)(14 292)(15 293)(16 294)(17 295)(18 296)(19 297)(20 298)(21 299)(22 300)(23 301)(24 302)(25 303)(26 304)(27 305)(28 306)(29 307)(30 308)(31 309)(32 310)(33 311)(34 312)(35 313)(36 314)(37 315)(38 316)(39 317)(40 318)(41 319)(42 320)(43 321)(44 322)(45 323)(46 324)(47 325)(48 326)(49 327)(50 328)(51 329)(52 330)(53 331)(54 332)(55 333)(56 334)(57 259)(58 260)(59 261)(60 262)(61 263)(62 264)(63 265)(64 266)(65 267)(66 268)(67 269)(68 270)(69 271)(70 272)(71 273)(72 274)(73 275)(74 276)(75 277)(76 278)(77 279)(78 280)(79 225)(80 226)(81 227)(82 228)(83 229)(84 230)(85 231)(86 232)(87 233)(88 234)(89 235)(90 236)(91 237)(92 238)(93 239)(94 240)(95 241)(96 242)(97 243)(98 244)(99 245)(100 246)(101 247)(102 248)(103 249)(104 250)(105 251)(106 252)(107 253)(108 254)(109 255)(110 256)(111 257)(112 258)(113 369)(114 370)(115 371)(116 372)(117 373)(118 374)(119 375)(120 376)(121 377)(122 378)(123 379)(124 380)(125 381)(126 382)(127 383)(128 384)(129 385)(130 386)(131 387)(132 388)(133 389)(134 390)(135 391)(136 392)(137 337)(138 338)(139 339)(140 340)(141 341)(142 342)(143 343)(144 344)(145 345)(146 346)(147 347)(148 348)(149 349)(150 350)(151 351)(152 352)(153 353)(154 354)(155 355)(156 356)(157 357)(158 358)(159 359)(160 360)(161 361)(162 362)(163 363)(164 364)(165 365)(166 366)(167 367)(168 368)(169 411)(170 412)(171 413)(172 414)(173 415)(174 416)(175 417)(176 418)(177 419)(178 420)(179 421)(180 422)(181 423)(182 424)(183 425)(184 426)(185 427)(186 428)(187 429)(188 430)(189 431)(190 432)(191 433)(192 434)(193 435)(194 436)(195 437)(196 438)(197 439)(198 440)(199 441)(200 442)(201 443)(202 444)(203 445)(204 446)(205 447)(206 448)(207 393)(208 394)(209 395)(210 396)(211 397)(212 398)(213 399)(214 400)(215 401)(216 402)(217 403)(218 404)(219 405)(220 406)(221 407)(222 408)(223 409)(224 410)
(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)| (1,416)(2,417)(3,418)(4,419)(5,420)(6,421)(7,422)(8,423)(9,424)(10,425)(11,426)(12,427)(13,428)(14,429)(15,430)(16,431)(17,432)(18,433)(19,434)(20,435)(21,436)(22,437)(23,438)(24,439)(25,440)(26,441)(27,442)(28,443)(29,444)(30,445)(31,446)(32,447)(33,448)(34,393)(35,394)(36,395)(37,396)(38,397)(39,398)(40,399)(41,400)(42,401)(43,402)(44,403)(45,404)(46,405)(47,406)(48,407)(49,408)(50,409)(51,410)(52,411)(53,412)(54,413)(55,414)(56,415)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,161)(88,162)(89,163)(90,164)(91,165)(92,166)(93,167)(94,168)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(169,330)(170,331)(171,332)(172,333)(173,334)(174,335)(175,336)(176,281)(177,282)(178,283)(179,284)(180,285)(181,286)(182,287)(183,288)(184,289)(185,290)(186,291)(187,292)(188,293)(189,294)(190,295)(191,296)(192,297)(193,298)(194,299)(195,300)(196,301)(197,302)(198,303)(199,304)(200,305)(201,306)(202,307)(203,308)(204,309)(205,310)(206,311)(207,312)(208,313)(209,314)(210,315)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,323)(219,324)(220,325)(221,326)(222,327)(223,328)(224,329)(225,353)(226,354)(227,355)(228,356)(229,357)(230,358)(231,359)(232,360)(233,361)(234,362)(235,363)(236,364)(237,365)(238,366)(239,367)(240,368)(241,369)(242,370)(243,371)(244,372)(245,373)(246,374)(247,375)(248,376)(249,377)(250,378)(251,379)(252,380)(253,381)(254,382)(255,383)(256,384)(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)(264,392)(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,343)(272,344)(273,345)(274,346)(275,347)(276,348)(277,349)(278,350)(279,351)(280,352), (1,249)(2,250)(3,251)(4,252)(5,253)(6,254)(7,255)(8,256)(9,257)(10,258)(11,259)(12,260)(13,261)(14,262)(15,263)(16,264)(17,265)(18,266)(19,267)(20,268)(21,269)(22,270)(23,271)(24,272)(25,273)(26,274)(27,275)(28,276)(29,277)(30,278)(31,279)(32,280)(33,225)(34,226)(35,227)(36,228)(37,229)(38,230)(39,231)(40,232)(41,233)(42,234)(43,235)(44,236)(45,237)(46,238)(47,239)(48,240)(49,241)(50,242)(51,243)(52,244)(53,245)(54,246)(55,247)(56,248)(57,289)(58,290)(59,291)(60,292)(61,293)(62,294)(63,295)(64,296)(65,297)(66,298)(67,299)(68,300)(69,301)(70,302)(71,303)(72,304)(73,305)(74,306)(75,307)(76,308)(77,309)(78,310)(79,311)(80,312)(81,313)(82,314)(83,315)(84,316)(85,317)(86,318)(87,319)(88,320)(89,321)(90,322)(91,323)(92,324)(93,325)(94,326)(95,327)(96,328)(97,329)(98,330)(99,331)(100,332)(101,333)(102,334)(103,335)(104,336)(105,281)(106,282)(107,283)(108,284)(109,285)(110,286)(111,287)(112,288)(113,222)(114,223)(115,224)(116,169)(117,170)(118,171)(119,172)(120,173)(121,174)(122,175)(123,176)(124,177)(125,178)(126,179)(127,180)(128,181)(129,182)(130,183)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193)(141,194)(142,195)(143,196)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,204)(152,205)(153,206)(154,207)(155,208)(156,209)(157,210)(158,211)(159,212)(160,213)(161,214)(162,215)(163,216)(164,217)(165,218)(166,219)(167,220)(168,221)(337,432)(338,433)(339,434)(340,435)(341,436)(342,437)(343,438)(344,439)(345,440)(346,441)(347,442)(348,443)(349,444)(350,445)(351,446)(352,447)(353,448)(354,393)(355,394)(356,395)(357,396)(358,397)(359,398)(360,399)(361,400)(362,401)(363,402)(364,403)(365,404)(366,405)(367,406)(368,407)(369,408)(370,409)(371,410)(372,411)(373,412)(374,413)(375,414)(376,415)(377,416)(378,417)(379,418)(380,419)(381,420)(382,421)(383,422)(384,423)(385,424)(386,425)(387,426)(388,427)(389,428)(390,429)(391,430)(392,431), (1,335)(2,336)(3,281)(4,282)(5,283)(6,284)(7,285)(8,286)(9,287)(10,288)(11,289)(12,290)(13,291)(14,292)(15,293)(16,294)(17,295)(18,296)(19,297)(20,298)(21,299)(22,300)(23,301)(24,302)(25,303)(26,304)(27,305)(28,306)(29,307)(30,308)(31,309)(32,310)(33,311)(34,312)(35,313)(36,314)(37,315)(38,316)(39,317)(40,318)(41,319)(42,320)(43,321)(44,322)(45,323)(46,324)(47,325)(48,326)(49,327)(50,328)(51,329)(52,330)(53,331)(54,332)(55,333)(56,334)(57,259)(58,260)(59,261)(60,262)(61,263)(62,264)(63,265)(64,266)(65,267)(66,268)(67,269)(68,270)(69,271)(70,272)(71,273)(72,274)(73,275)(74,276)(75,277)(76,278)(77,279)(78,280)(79,225)(80,226)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,241)(96,242)(97,243)(98,244)(99,245)(100,246)(101,247)(102,248)(103,249)(104,250)(105,251)(106,252)(107,253)(108,254)(109,255)(110,256)(111,257)(112,258)(113,369)(114,370)(115,371)(116,372)(117,373)(118,374)(119,375)(120,376)(121,377)(122,378)(123,379)(124,380)(125,381)(126,382)(127,383)(128,384)(129,385)(130,386)(131,387)(132,388)(133,389)(134,390)(135,391)(136,392)(137,337)(138,338)(139,339)(140,340)(141,341)(142,342)(143,343)(144,344)(145,345)(146,346)(147,347)(148,348)(149,349)(150,350)(151,351)(152,352)(153,353)(154,354)(155,355)(156,356)(157,357)(158,358)(159,359)(160,360)(161,361)(162,362)(163,363)(164,364)(165,365)(166,366)(167,367)(168,368)(169,411)(170,412)(171,413)(172,414)(173,415)(174,416)(175,417)(176,418)(177,419)(178,420)(179,421)(180,422)(181,423)(182,424)(183,425)(184,426)(185,427)(186,428)(187,429)(188,430)(189,431)(190,432)(191,433)(192,434)(193,435)(194,436)(195,437)(196,438)(197,439)(198,440)(199,441)(200,442)(201,443)(202,444)(203,445)(204,446)(205,447)(206,448)(207,393)(208,394)(209,395)(210,396)(211,397)(212,398)(213,399)(214,400)(215,401)(216,402)(217,403)(218,404)(219,405)(220,406)(221,407)(222,408)(223,409)(224,410), (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( (1,416)(2,417)(3,418)(4,419)(5,420)(6,421)(7,422)(8,423)(9,424)(10,425)(11,426)(12,427)(13,428)(14,429)(15,430)(16,431)(17,432)(18,433)(19,434)(20,435)(21,436)(22,437)(23,438)(24,439)(25,440)(26,441)(27,442)(28,443)(29,444)(30,445)(31,446)(32,447)(33,448)(34,393)(35,394)(36,395)(37,396)(38,397)(39,398)(40,399)(41,400)(42,401)(43,402)(44,403)(45,404)(46,405)(47,406)(48,407)(49,408)(50,409)(51,410)(52,411)(53,412)(54,413)(55,414)(56,415)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,161)(88,162)(89,163)(90,164)(91,165)(92,166)(93,167)(94,168)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(169,330)(170,331)(171,332)(172,333)(173,334)(174,335)(175,336)(176,281)(177,282)(178,283)(179,284)(180,285)(181,286)(182,287)(183,288)(184,289)(185,290)(186,291)(187,292)(188,293)(189,294)(190,295)(191,296)(192,297)(193,298)(194,299)(195,300)(196,301)(197,302)(198,303)(199,304)(200,305)(201,306)(202,307)(203,308)(204,309)(205,310)(206,311)(207,312)(208,313)(209,314)(210,315)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,323)(219,324)(220,325)(221,326)(222,327)(223,328)(224,329)(225,353)(226,354)(227,355)(228,356)(229,357)(230,358)(231,359)(232,360)(233,361)(234,362)(235,363)(236,364)(237,365)(238,366)(239,367)(240,368)(241,369)(242,370)(243,371)(244,372)(245,373)(246,374)(247,375)(248,376)(249,377)(250,378)(251,379)(252,380)(253,381)(254,382)(255,383)(256,384)(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)(264,392)(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,343)(272,344)(273,345)(274,346)(275,347)(276,348)(277,349)(278,350)(279,351)(280,352), (1,249)(2,250)(3,251)(4,252)(5,253)(6,254)(7,255)(8,256)(9,257)(10,258)(11,259)(12,260)(13,261)(14,262)(15,263)(16,264)(17,265)(18,266)(19,267)(20,268)(21,269)(22,270)(23,271)(24,272)(25,273)(26,274)(27,275)(28,276)(29,277)(30,278)(31,279)(32,280)(33,225)(34,226)(35,227)(36,228)(37,229)(38,230)(39,231)(40,232)(41,233)(42,234)(43,235)(44,236)(45,237)(46,238)(47,239)(48,240)(49,241)(50,242)(51,243)(52,244)(53,245)(54,246)(55,247)(56,248)(57,289)(58,290)(59,291)(60,292)(61,293)(62,294)(63,295)(64,296)(65,297)(66,298)(67,299)(68,300)(69,301)(70,302)(71,303)(72,304)(73,305)(74,306)(75,307)(76,308)(77,309)(78,310)(79,311)(80,312)(81,313)(82,314)(83,315)(84,316)(85,317)(86,318)(87,319)(88,320)(89,321)(90,322)(91,323)(92,324)(93,325)(94,326)(95,327)(96,328)(97,329)(98,330)(99,331)(100,332)(101,333)(102,334)(103,335)(104,336)(105,281)(106,282)(107,283)(108,284)(109,285)(110,286)(111,287)(112,288)(113,222)(114,223)(115,224)(116,169)(117,170)(118,171)(119,172)(120,173)(121,174)(122,175)(123,176)(124,177)(125,178)(126,179)(127,180)(128,181)(129,182)(130,183)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193)(141,194)(142,195)(143,196)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,204)(152,205)(153,206)(154,207)(155,208)(156,209)(157,210)(158,211)(159,212)(160,213)(161,214)(162,215)(163,216)(164,217)(165,218)(166,219)(167,220)(168,221)(337,432)(338,433)(339,434)(340,435)(341,436)(342,437)(343,438)(344,439)(345,440)(346,441)(347,442)(348,443)(349,444)(350,445)(351,446)(352,447)(353,448)(354,393)(355,394)(356,395)(357,396)(358,397)(359,398)(360,399)(361,400)(362,401)(363,402)(364,403)(365,404)(366,405)(367,406)(368,407)(369,408)(370,409)(371,410)(372,411)(373,412)(374,413)(375,414)(376,415)(377,416)(378,417)(379,418)(380,419)(381,420)(382,421)(383,422)(384,423)(385,424)(386,425)(387,426)(388,427)(389,428)(390,429)(391,430)(392,431), (1,335)(2,336)(3,281)(4,282)(5,283)(6,284)(7,285)(8,286)(9,287)(10,288)(11,289)(12,290)(13,291)(14,292)(15,293)(16,294)(17,295)(18,296)(19,297)(20,298)(21,299)(22,300)(23,301)(24,302)(25,303)(26,304)(27,305)(28,306)(29,307)(30,308)(31,309)(32,310)(33,311)(34,312)(35,313)(36,314)(37,315)(38,316)(39,317)(40,318)(41,319)(42,320)(43,321)(44,322)(45,323)(46,324)(47,325)(48,326)(49,327)(50,328)(51,329)(52,330)(53,331)(54,332)(55,333)(56,334)(57,259)(58,260)(59,261)(60,262)(61,263)(62,264)(63,265)(64,266)(65,267)(66,268)(67,269)(68,270)(69,271)(70,272)(71,273)(72,274)(73,275)(74,276)(75,277)(76,278)(77,279)(78,280)(79,225)(80,226)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,241)(96,242)(97,243)(98,244)(99,245)(100,246)(101,247)(102,248)(103,249)(104,250)(105,251)(106,252)(107,253)(108,254)(109,255)(110,256)(111,257)(112,258)(113,369)(114,370)(115,371)(116,372)(117,373)(118,374)(119,375)(120,376)(121,377)(122,378)(123,379)(124,380)(125,381)(126,382)(127,383)(128,384)(129,385)(130,386)(131,387)(132,388)(133,389)(134,390)(135,391)(136,392)(137,337)(138,338)(139,339)(140,340)(141,341)(142,342)(143,343)(144,344)(145,345)(146,346)(147,347)(148,348)(149,349)(150,350)(151,351)(152,352)(153,353)(154,354)(155,355)(156,356)(157,357)(158,358)(159,359)(160,360)(161,361)(162,362)(163,363)(164,364)(165,365)(166,366)(167,367)(168,368)(169,411)(170,412)(171,413)(172,414)(173,415)(174,416)(175,417)(176,418)(177,419)(178,420)(179,421)(180,422)(181,423)(182,424)(183,425)(184,426)(185,427)(186,428)(187,429)(188,430)(189,431)(190,432)(191,433)(192,434)(193,435)(194,436)(195,437)(196,438)(197,439)(198,440)(199,441)(200,442)(201,443)(202,444)(203,445)(204,446)(205,447)(206,448)(207,393)(208,394)(209,395)(210,396)(211,397)(212,398)(213,399)(214,400)(215,401)(216,402)(217,403)(218,404)(219,405)(220,406)(221,407)(222,408)(223,409)(224,410), (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([(1,416),(2,417),(3,418),(4,419),(5,420),(6,421),(7,422),(8,423),(9,424),(10,425),(11,426),(12,427),(13,428),(14,429),(15,430),(16,431),(17,432),(18,433),(19,434),(20,435),(21,436),(22,437),(23,438),(24,439),(25,440),(26,441),(27,442),(28,443),(29,444),(30,445),(31,446),(32,447),(33,448),(34,393),(35,394),(36,395),(37,396),(38,397),(39,398),(40,399),(41,400),(42,401),(43,402),(44,403),(45,404),(46,405),(47,406),(48,407),(49,408),(50,409),(51,410),(52,411),(53,412),(54,413),(55,414),(56,415),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,145),(72,146),(73,147),(74,148),(75,149),(76,150),(77,151),(78,152),(79,153),(80,154),(81,155),(82,156),(83,157),(84,158),(85,159),(86,160),(87,161),(88,162),(89,163),(90,164),(91,165),(92,166),(93,167),(94,168),(95,113),(96,114),(97,115),(98,116),(99,117),(100,118),(101,119),(102,120),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129),(112,130),(169,330),(170,331),(171,332),(172,333),(173,334),(174,335),(175,336),(176,281),(177,282),(178,283),(179,284),(180,285),(181,286),(182,287),(183,288),(184,289),(185,290),(186,291),(187,292),(188,293),(189,294),(190,295),(191,296),(192,297),(193,298),(194,299),(195,300),(196,301),(197,302),(198,303),(199,304),(200,305),(201,306),(202,307),(203,308),(204,309),(205,310),(206,311),(207,312),(208,313),(209,314),(210,315),(211,316),(212,317),(213,318),(214,319),(215,320),(216,321),(217,322),(218,323),(219,324),(220,325),(221,326),(222,327),(223,328),(224,329),(225,353),(226,354),(227,355),(228,356),(229,357),(230,358),(231,359),(232,360),(233,361),(234,362),(235,363),(236,364),(237,365),(238,366),(239,367),(240,368),(241,369),(242,370),(243,371),(244,372),(245,373),(246,374),(247,375),(248,376),(249,377),(250,378),(251,379),(252,380),(253,381),(254,382),(255,383),(256,384),(257,385),(258,386),(259,387),(260,388),(261,389),(262,390),(263,391),(264,392),(265,337),(266,338),(267,339),(268,340),(269,341),(270,342),(271,343),(272,344),(273,345),(274,346),(275,347),(276,348),(277,349),(278,350),(279,351),(280,352)], [(1,249),(2,250),(3,251),(4,252),(5,253),(6,254),(7,255),(8,256),(9,257),(10,258),(11,259),(12,260),(13,261),(14,262),(15,263),(16,264),(17,265),(18,266),(19,267),(20,268),(21,269),(22,270),(23,271),(24,272),(25,273),(26,274),(27,275),(28,276),(29,277),(30,278),(31,279),(32,280),(33,225),(34,226),(35,227),(36,228),(37,229),(38,230),(39,231),(40,232),(41,233),(42,234),(43,235),(44,236),(45,237),(46,238),(47,239),(48,240),(49,241),(50,242),(51,243),(52,244),(53,245),(54,246),(55,247),(56,248),(57,289),(58,290),(59,291),(60,292),(61,293),(62,294),(63,295),(64,296),(65,297),(66,298),(67,299),(68,300),(69,301),(70,302),(71,303),(72,304),(73,305),(74,306),(75,307),(76,308),(77,309),(78,310),(79,311),(80,312),(81,313),(82,314),(83,315),(84,316),(85,317),(86,318),(87,319),(88,320),(89,321),(90,322),(91,323),(92,324),(93,325),(94,326),(95,327),(96,328),(97,329),(98,330),(99,331),(100,332),(101,333),(102,334),(103,335),(104,336),(105,281),(106,282),(107,283),(108,284),(109,285),(110,286),(111,287),(112,288),(113,222),(114,223),(115,224),(116,169),(117,170),(118,171),(119,172),(120,173),(121,174),(122,175),(123,176),(124,177),(125,178),(126,179),(127,180),(128,181),(129,182),(130,183),(131,184),(132,185),(133,186),(134,187),(135,188),(136,189),(137,190),(138,191),(139,192),(140,193),(141,194),(142,195),(143,196),(144,197),(145,198),(146,199),(147,200),(148,201),(149,202),(150,203),(151,204),(152,205),(153,206),(154,207),(155,208),(156,209),(157,210),(158,211),(159,212),(160,213),(161,214),(162,215),(163,216),(164,217),(165,218),(166,219),(167,220),(168,221),(337,432),(338,433),(339,434),(340,435),(341,436),(342,437),(343,438),(344,439),(345,440),(346,441),(347,442),(348,443),(349,444),(350,445),(351,446),(352,447),(353,448),(354,393),(355,394),(356,395),(357,396),(358,397),(359,398),(360,399),(361,400),(362,401),(363,402),(364,403),(365,404),(366,405),(367,406),(368,407),(369,408),(370,409),(371,410),(372,411),(373,412),(374,413),(375,414),(376,415),(377,416),(378,417),(379,418),(380,419),(381,420),(382,421),(383,422),(384,423),(385,424),(386,425),(387,426),(388,427),(389,428),(390,429),(391,430),(392,431)], [(1,335),(2,336),(3,281),(4,282),(5,283),(6,284),(7,285),(8,286),(9,287),(10,288),(11,289),(12,290),(13,291),(14,292),(15,293),(16,294),(17,295),(18,296),(19,297),(20,298),(21,299),(22,300),(23,301),(24,302),(25,303),(26,304),(27,305),(28,306),(29,307),(30,308),(31,309),(32,310),(33,311),(34,312),(35,313),(36,314),(37,315),(38,316),(39,317),(40,318),(41,319),(42,320),(43,321),(44,322),(45,323),(46,324),(47,325),(48,326),(49,327),(50,328),(51,329),(52,330),(53,331),(54,332),(55,333),(56,334),(57,259),(58,260),(59,261),(60,262),(61,263),(62,264),(63,265),(64,266),(65,267),(66,268),(67,269),(68,270),(69,271),(70,272),(71,273),(72,274),(73,275),(74,276),(75,277),(76,278),(77,279),(78,280),(79,225),(80,226),(81,227),(82,228),(83,229),(84,230),(85,231),(86,232),(87,233),(88,234),(89,235),(90,236),(91,237),(92,238),(93,239),(94,240),(95,241),(96,242),(97,243),(98,244),(99,245),(100,246),(101,247),(102,248),(103,249),(104,250),(105,251),(106,252),(107,253),(108,254),(109,255),(110,256),(111,257),(112,258),(113,369),(114,370),(115,371),(116,372),(117,373),(118,374),(119,375),(120,376),(121,377),(122,378),(123,379),(124,380),(125,381),(126,382),(127,383),(128,384),(129,385),(130,386),(131,387),(132,388),(133,389),(134,390),(135,391),(136,392),(137,337),(138,338),(139,339),(140,340),(141,341),(142,342),(143,343),(144,344),(145,345),(146,346),(147,347),(148,348),(149,349),(150,350),(151,351),(152,352),(153,353),(154,354),(155,355),(156,356),(157,357),(158,358),(159,359),(160,360),(161,361),(162,362),(163,363),(164,364),(165,365),(166,366),(167,367),(168,368),(169,411),(170,412),(171,413),(172,414),(173,415),(174,416),(175,417),(176,418),(177,419),(178,420),(179,421),(180,422),(181,423),(182,424),(183,425),(184,426),(185,427),(186,428),(187,429),(188,430),(189,431),(190,432),(191,433),(192,434),(193,435),(194,436),(195,437),(196,438),(197,439),(198,440),(199,441),(200,442),(201,443),(202,444),(203,445),(204,446),(205,447),(206,448),(207,393),(208,394),(209,395),(210,396),(211,397),(212,398),(213,399),(214,400),(215,401),(216,402),(217,403),(218,404),(219,405),(220,406),(221,407),(222,408),(223,409),(224,410)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀