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G = C23×C52  order 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

Derived series
C1 — C23×C52
Chief series
C1 — C2 — C26 — C52 — C2×C52 — C22×C52 — C23×C52
Lower central
C1 — C23×C52
Upper central
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, C4, C22, C2×C4, C23, C13, C22×C4, C24, C26, C26, C23×C4, C52, C2×C26, C2×C52, C22×C26, C22×C52, C23×C26, C23×C52
Quotients: C1, C2, C4, C22, C2×C4, C23, C13, C22×C4, C24, C26, C23×C4, C52, C2×C26, C2×C52, C22×C26, C22×C52, C23×C26, C23×C52

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

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

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

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

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
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