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G = C23×C50order 400 = 24·52

Abelian group of type [2,2,2,50]

direct product, abelian, monomial, 2-elementary

Aliases: C23×C50, SmallGroup(400,55)

Series: Derived Chief Lower central Upper central

C1 — C23×C50
C1C5C25C50C2×C50C22×C50 — C23×C50
C1 — C23×C50
C1 — C23×C50

Generators and relations for C23×C50
 G = < a,b,c,d | a2=b2=c2=d50=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 201, all normal (6 characteristic)
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C24, C2×C10 [×35], C25, C22×C10 [×15], C50 [×15], C23×C10, C2×C50 [×35], C22×C50 [×15], C23×C50
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C24, C2×C10 [×35], C25, C22×C10 [×15], C50 [×15], C23×C10, C2×C50 [×35], C22×C50 [×15], C23×C50

Smallest permutation representation of C23×C50
Regular action on 400 points
Generators in S400
(1 201)(2 202)(3 203)(4 204)(5 205)(6 206)(7 207)(8 208)(9 209)(10 210)(11 211)(12 212)(13 213)(14 214)(15 215)(16 216)(17 217)(18 218)(19 219)(20 220)(21 221)(22 222)(23 223)(24 224)(25 225)(26 226)(27 227)(28 228)(29 229)(30 230)(31 231)(32 232)(33 233)(34 234)(35 235)(36 236)(37 237)(38 238)(39 239)(40 240)(41 241)(42 242)(43 243)(44 244)(45 245)(46 246)(47 247)(48 248)(49 249)(50 250)(51 299)(52 300)(53 251)(54 252)(55 253)(56 254)(57 255)(58 256)(59 257)(60 258)(61 259)(62 260)(63 261)(64 262)(65 263)(66 264)(67 265)(68 266)(69 267)(70 268)(71 269)(72 270)(73 271)(74 272)(75 273)(76 274)(77 275)(78 276)(79 277)(80 278)(81 279)(82 280)(83 281)(84 282)(85 283)(86 284)(87 285)(88 286)(89 287)(90 288)(91 289)(92 290)(93 291)(94 292)(95 293)(96 294)(97 295)(98 296)(99 297)(100 298)(101 301)(102 302)(103 303)(104 304)(105 305)(106 306)(107 307)(108 308)(109 309)(110 310)(111 311)(112 312)(113 313)(114 314)(115 315)(116 316)(117 317)(118 318)(119 319)(120 320)(121 321)(122 322)(123 323)(124 324)(125 325)(126 326)(127 327)(128 328)(129 329)(130 330)(131 331)(132 332)(133 333)(134 334)(135 335)(136 336)(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)(151 364)(152 365)(153 366)(154 367)(155 368)(156 369)(157 370)(158 371)(159 372)(160 373)(161 374)(162 375)(163 376)(164 377)(165 378)(166 379)(167 380)(168 381)(169 382)(170 383)(171 384)(172 385)(173 386)(174 387)(175 388)(176 389)(177 390)(178 391)(179 392)(180 393)(181 394)(182 395)(183 396)(184 397)(185 398)(186 399)(187 400)(188 351)(189 352)(190 353)(191 354)(192 355)(193 356)(194 357)(195 358)(196 359)(197 360)(198 361)(199 362)(200 363)
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 130)(38 131)(39 132)(40 133)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(49 142)(50 143)(51 199)(52 200)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 159)(62 160)(63 161)(64 162)(65 163)(66 164)(67 165)(68 166)(69 167)(70 168)(71 169)(72 170)(73 171)(74 172)(75 173)(76 174)(77 175)(78 176)(79 177)(80 178)(81 179)(82 180)(83 181)(84 182)(85 183)(86 184)(87 185)(88 186)(89 187)(90 188)(91 189)(92 190)(93 191)(94 192)(95 193)(96 194)(97 195)(98 196)(99 197)(100 198)(201 344)(202 345)(203 346)(204 347)(205 348)(206 349)(207 350)(208 301)(209 302)(210 303)(211 304)(212 305)(213 306)(214 307)(215 308)(216 309)(217 310)(218 311)(219 312)(220 313)(221 314)(222 315)(223 316)(224 317)(225 318)(226 319)(227 320)(228 321)(229 322)(230 323)(231 324)(232 325)(233 326)(234 327)(235 328)(236 329)(237 330)(238 331)(239 332)(240 333)(241 334)(242 335)(243 336)(244 337)(245 338)(246 339)(247 340)(248 341)(249 342)(250 343)(251 364)(252 365)(253 366)(254 367)(255 368)(256 369)(257 370)(258 371)(259 372)(260 373)(261 374)(262 375)(263 376)(264 377)(265 378)(266 379)(267 380)(268 381)(269 382)(270 383)(271 384)(272 385)(273 386)(274 387)(275 388)(276 389)(277 390)(278 391)(279 392)(280 393)(281 394)(282 395)(283 396)(284 397)(285 398)(286 399)(287 400)(288 351)(289 352)(290 353)(291 354)(292 355)(293 356)(294 357)(295 358)(296 359)(297 360)(298 361)(299 362)(300 363)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(101 199)(102 200)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 169)(122 170)(123 171)(124 172)(125 173)(126 174)(127 175)(128 176)(129 177)(130 178)(131 179)(132 180)(133 181)(134 182)(135 183)(136 184)(137 185)(138 186)(139 187)(140 188)(141 189)(142 190)(143 191)(144 192)(145 193)(146 194)(147 195)(148 196)(149 197)(150 198)(201 292)(202 293)(203 294)(204 295)(205 296)(206 297)(207 298)(208 299)(209 300)(210 251)(211 252)(212 253)(213 254)(214 255)(215 256)(216 257)(217 258)(218 259)(219 260)(220 261)(221 262)(222 263)(223 264)(224 265)(225 266)(226 267)(227 268)(228 269)(229 270)(230 271)(231 272)(232 273)(233 274)(234 275)(235 276)(236 277)(237 278)(238 279)(239 280)(240 281)(241 282)(242 283)(243 284)(244 285)(245 286)(246 287)(247 288)(248 289)(249 290)(250 291)(301 362)(302 363)(303 364)(304 365)(305 366)(306 367)(307 368)(308 369)(309 370)(310 371)(311 372)(312 373)(313 374)(314 375)(315 376)(316 377)(317 378)(318 379)(319 380)(320 381)(321 382)(322 383)(323 384)(324 385)(325 386)(326 387)(327 388)(328 389)(329 390)(330 391)(331 392)(332 393)(333 394)(334 395)(335 396)(336 397)(337 398)(338 399)(339 400)(340 351)(341 352)(342 353)(343 354)(344 355)(345 356)(346 357)(347 358)(348 359)(349 360)(350 361)
(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)

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

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

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

400 conjugacy classes

class 1 2A···2O5A5B5C5D10A···10BH25A···25T50A···50KN
order12···2555510···1025···2550···50
size11···111111···11···11···1

400 irreducible representations

dim111111
type++
imageC1C2C5C10C25C50
kernelC23×C50C22×C50C23×C10C22×C10C24C23
# reps11546020300

Matrix representation of C23×C50 in GL4(𝔽101) generated by

1000
010000
0010
000100
,
100000
0100
001000
0001
,
100000
010000
0010
0001
,
13000
01700
00870
000100
G:=sub<GL(4,GF(101))| [1,0,0,0,0,100,0,0,0,0,1,0,0,0,0,100],[100,0,0,0,0,1,0,0,0,0,100,0,0,0,0,1],[100,0,0,0,0,100,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,17,0,0,0,0,87,0,0,0,0,100] >;

C23×C50 in GAP, Magma, Sage, TeX

C_2^3\times C_{50}
% in TeX

G:=Group("C2^3xC50");
// GroupNames label

G:=SmallGroup(400,55);
// by ID

G=gap.SmallGroup(400,55);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,178]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^50=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