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G = C23×C52order 416 = 25·13

Abelian group of type [2,2,2,52]

direct product, abelian, monomial, 2-elementary

Aliases: C23×C52, SmallGroup(416,227)

Series: Derived Chief Lower central Upper central

C1 — C23×C52
C1C2C26C52C2×C52C22×C52 — C23×C52
C1 — C23×C52
C1 — C23×C52

Generators and relations for C23×C52
 G = < a,b,c,d | a2=b2=c2=d52=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 236, all normal (8 characteristic)
C1, C2, C2 [×14], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C13, C22×C4 [×14], C24, C26, C26 [×14], C23×C4, C52 [×8], C2×C26 [×35], C2×C52 [×28], C22×C26 [×15], C22×C52 [×14], C23×C26, C23×C52
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C13, C22×C4 [×14], C24, C26 [×15], C23×C4, C52 [×8], C2×C26 [×35], C2×C52 [×28], C22×C26 [×15], C22×C52 [×14], C23×C26, C23×C52

Smallest permutation representation of C23×C52
Regular action on 416 points
Generators in S416
(1 386)(2 387)(3 388)(4 389)(5 390)(6 391)(7 392)(8 393)(9 394)(10 395)(11 396)(12 397)(13 398)(14 399)(15 400)(16 401)(17 402)(18 403)(19 404)(20 405)(21 406)(22 407)(23 408)(24 409)(25 410)(26 411)(27 412)(28 413)(29 414)(30 415)(31 416)(32 365)(33 366)(34 367)(35 368)(36 369)(37 370)(38 371)(39 372)(40 373)(41 374)(42 375)(43 376)(44 377)(45 378)(46 379)(47 380)(48 381)(49 382)(50 383)(51 384)(52 385)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)(82 134)(83 135)(84 136)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 155)(104 156)(157 302)(158 303)(159 304)(160 305)(161 306)(162 307)(163 308)(164 309)(165 310)(166 311)(167 312)(168 261)(169 262)(170 263)(171 264)(172 265)(173 266)(174 267)(175 268)(176 269)(177 270)(178 271)(179 272)(180 273)(181 274)(182 275)(183 276)(184 277)(185 278)(186 279)(187 280)(188 281)(189 282)(190 283)(191 284)(192 285)(193 286)(194 287)(195 288)(196 289)(197 290)(198 291)(199 292)(200 293)(201 294)(202 295)(203 296)(204 297)(205 298)(206 299)(207 300)(208 301)(209 340)(210 341)(211 342)(212 343)(213 344)(214 345)(215 346)(216 347)(217 348)(218 349)(219 350)(220 351)(221 352)(222 353)(223 354)(224 355)(225 356)(226 357)(227 358)(228 359)(229 360)(230 361)(231 362)(232 363)(233 364)(234 313)(235 314)(236 315)(237 316)(238 317)(239 318)(240 319)(241 320)(242 321)(243 322)(244 323)(245 324)(246 325)(247 326)(248 327)(249 328)(250 329)(251 330)(252 331)(253 332)(254 333)(255 334)(256 335)(257 336)(258 337)(259 338)(260 339)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 376)(54 377)(55 378)(56 379)(57 380)(58 381)(59 382)(60 383)(61 384)(62 385)(63 386)(64 387)(65 388)(66 389)(67 390)(68 391)(69 392)(70 393)(71 394)(72 395)(73 396)(74 397)(75 398)(76 399)(77 400)(78 401)(79 402)(80 403)(81 404)(82 405)(83 406)(84 407)(85 408)(86 409)(87 410)(88 411)(89 412)(90 413)(91 414)(92 415)(93 416)(94 365)(95 366)(96 367)(97 368)(98 369)(99 370)(100 371)(101 372)(102 373)(103 374)(104 375)(157 226)(158 227)(159 228)(160 229)(161 230)(162 231)(163 232)(164 233)(165 234)(166 235)(167 236)(168 237)(169 238)(170 239)(171 240)(172 241)(173 242)(174 243)(175 244)(176 245)(177 246)(178 247)(179 248)(180 249)(181 250)(182 251)(183 252)(184 253)(185 254)(186 255)(187 256)(188 257)(189 258)(190 259)(191 260)(192 209)(193 210)(194 211)(195 212)(196 213)(197 214)(198 215)(199 216)(200 217)(201 218)(202 219)(203 220)(204 221)(205 222)(206 223)(207 224)(208 225)(261 316)(262 317)(263 318)(264 319)(265 320)(266 321)(267 322)(268 323)(269 324)(270 325)(271 326)(272 327)(273 328)(274 329)(275 330)(276 331)(277 332)(278 333)(279 334)(280 335)(281 336)(282 337)(283 338)(284 339)(285 340)(286 341)(287 342)(288 343)(289 344)(290 345)(291 346)(292 347)(293 348)(294 349)(295 350)(296 351)(297 352)(298 353)(299 354)(300 355)(301 356)(302 357)(303 358)(304 359)(305 360)(306 361)(307 362)(308 363)(309 364)(310 313)(311 314)(312 315)
(1 160)(2 161)(3 162)(4 163)(5 164)(6 165)(7 166)(8 167)(9 168)(10 169)(11 170)(12 171)(13 172)(14 173)(15 174)(16 175)(17 176)(18 177)(19 178)(20 179)(21 180)(22 181)(23 182)(24 183)(25 184)(26 185)(27 186)(28 187)(29 188)(30 189)(31 190)(32 191)(33 192)(34 193)(35 194)(36 195)(37 196)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 208)(50 157)(51 158)(52 159)(53 350)(54 351)(55 352)(56 353)(57 354)(58 355)(59 356)(60 357)(61 358)(62 359)(63 360)(64 361)(65 362)(66 363)(67 364)(68 313)(69 314)(70 315)(71 316)(72 317)(73 318)(74 319)(75 320)(76 321)(77 322)(78 323)(79 324)(80 325)(81 326)(82 327)(83 328)(84 329)(85 330)(86 331)(87 332)(88 333)(89 334)(90 335)(91 336)(92 337)(93 338)(94 339)(95 340)(96 341)(97 342)(98 343)(99 344)(100 345)(101 346)(102 347)(103 348)(104 349)(105 219)(106 220)(107 221)(108 222)(109 223)(110 224)(111 225)(112 226)(113 227)(114 228)(115 229)(116 230)(117 231)(118 232)(119 233)(120 234)(121 235)(122 236)(123 237)(124 238)(125 239)(126 240)(127 241)(128 242)(129 243)(130 244)(131 245)(132 246)(133 247)(134 248)(135 249)(136 250)(137 251)(138 252)(139 253)(140 254)(141 255)(142 256)(143 257)(144 258)(145 259)(146 260)(147 209)(148 210)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)(155 217)(156 218)(261 394)(262 395)(263 396)(264 397)(265 398)(266 399)(267 400)(268 401)(269 402)(270 403)(271 404)(272 405)(273 406)(274 407)(275 408)(276 409)(277 410)(278 411)(279 412)(280 413)(281 414)(282 415)(283 416)(284 365)(285 366)(286 367)(287 368)(288 369)(289 370)(290 371)(291 372)(292 373)(293 374)(294 375)(295 376)(296 377)(297 378)(298 379)(299 380)(300 381)(301 382)(302 383)(303 384)(304 385)(305 386)(306 387)(307 388)(308 389)(309 390)(310 391)(311 392)(312 393)
(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)

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

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

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

416 conjugacy classes

class 1 2A···2O4A···4P13A···13L26A···26FX52A···52GJ
order12···24···413···1326···2652···52
size11···11···11···11···11···1

416 irreducible representations

dim11111111
type+++
imageC1C2C2C4C13C26C26C52
kernelC23×C52C22×C52C23×C26C22×C26C23×C4C22×C4C24C23
# reps1141161216812192

Matrix representation of C23×C52 in GL4(𝔽53) generated by

1000
05200
00520
0001
,
52000
05200
00520
0001
,
52000
05200
0010
00052
,
52000
04300
0010
0002
G:=sub<GL(4,GF(53))| [1,0,0,0,0,52,0,0,0,0,52,0,0,0,0,1],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,1],[52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,52],[52,0,0,0,0,43,0,0,0,0,1,0,0,0,0,2] >;

C23×C52 in GAP, Magma, Sage, TeX

C_2^3\times C_{52}
% in TeX

G:=Group("C2^3xC52");
// GroupNames label

G:=SmallGroup(416,227);
// by ID

G=gap.SmallGroup(416,227);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-13,-2,1248]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^52=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