Copied to
clipboard

## G = C2×C4×C52order 416 = 25·13

### Abelian group of type [2,4,52]

Aliases: C2×C4×C52, SmallGroup(416,175)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4×C52
 Chief series C1 — C2 — C22 — C2×C26 — C2×C52 — C4×C52 — C2×C4×C52
 Lower central C1 — C2×C4×C52
 Upper central C1 — C2×C4×C52

Generators and relations for C2×C4×C52
G = < a,b,c | a2=b4=c52=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (8 characteristic)
C1, C2, C4, C22, C22, C2×C4, C23, C13, C42, C22×C4, C26, C2×C42, C52, C2×C26, C2×C26, C2×C52, C22×C26, C4×C52, C22×C52, C2×C4×C52
Quotients: C1, C2, C4, C22, C2×C4, C23, C13, C42, C22×C4, C26, C2×C42, C52, C2×C26, C2×C52, C22×C26, C4×C52, C22×C52, C2×C4×C52

Smallest permutation representation of C2×C4×C52
Regular action on 416 points
Generators in S416
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 145)(9 146)(10 147)(11 148)(12 149)(13 150)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 304)(54 305)(55 306)(56 307)(57 308)(58 309)(59 310)(60 311)(61 312)(62 261)(63 262)(64 263)(65 264)(66 265)(67 266)(68 267)(69 268)(70 269)(71 270)(72 271)(73 272)(74 273)(75 274)(76 275)(77 276)(78 277)(79 278)(80 279)(81 280)(82 281)(83 282)(84 283)(85 284)(86 285)(87 286)(88 287)(89 288)(90 289)(91 290)(92 291)(93 292)(94 293)(95 294)(96 295)(97 296)(98 297)(99 298)(100 299)(101 300)(102 301)(103 302)(104 303)(157 325)(158 326)(159 327)(160 328)(161 329)(162 330)(163 331)(164 332)(165 333)(166 334)(167 335)(168 336)(169 337)(170 338)(171 339)(172 340)(173 341)(174 342)(175 343)(176 344)(177 345)(178 346)(179 347)(180 348)(181 349)(182 350)(183 351)(184 352)(185 353)(186 354)(187 355)(188 356)(189 357)(190 358)(191 359)(192 360)(193 361)(194 362)(195 363)(196 364)(197 313)(198 314)(199 315)(200 316)(201 317)(202 318)(203 319)(204 320)(205 321)(206 322)(207 323)(208 324)(209 394)(210 395)(211 396)(212 397)(213 398)(214 399)(215 400)(216 401)(217 402)(218 403)(219 404)(220 405)(221 406)(222 407)(223 408)(224 409)(225 410)(226 411)(227 412)(228 413)(229 414)(230 415)(231 416)(232 365)(233 366)(234 367)(235 368)(236 369)(237 370)(238 371)(239 372)(240 373)(241 374)(242 375)(243 376)(244 377)(245 378)(246 379)(247 380)(248 381)(249 382)(250 383)(251 384)(252 385)(253 386)(254 387)(255 388)(256 389)(257 390)(258 391)(259 392)(260 393)
(1 71 400 179)(2 72 401 180)(3 73 402 181)(4 74 403 182)(5 75 404 183)(6 76 405 184)(7 77 406 185)(8 78 407 186)(9 79 408 187)(10 80 409 188)(11 81 410 189)(12 82 411 190)(13 83 412 191)(14 84 413 192)(15 85 414 193)(16 86 415 194)(17 87 416 195)(18 88 365 196)(19 89 366 197)(20 90 367 198)(21 91 368 199)(22 92 369 200)(23 93 370 201)(24 94 371 202)(25 95 372 203)(26 96 373 204)(27 97 374 205)(28 98 375 206)(29 99 376 207)(30 100 377 208)(31 101 378 157)(32 102 379 158)(33 103 380 159)(34 104 381 160)(35 53 382 161)(36 54 383 162)(37 55 384 163)(38 56 385 164)(39 57 386 165)(40 58 387 166)(41 59 388 167)(42 60 389 168)(43 61 390 169)(44 62 391 170)(45 63 392 171)(46 64 393 172)(47 65 394 173)(48 66 395 174)(49 67 396 175)(50 68 397 176)(51 69 398 177)(52 70 399 178)(105 289 234 314)(106 290 235 315)(107 291 236 316)(108 292 237 317)(109 293 238 318)(110 294 239 319)(111 295 240 320)(112 296 241 321)(113 297 242 322)(114 298 243 323)(115 299 244 324)(116 300 245 325)(117 301 246 326)(118 302 247 327)(119 303 248 328)(120 304 249 329)(121 305 250 330)(122 306 251 331)(123 307 252 332)(124 308 253 333)(125 309 254 334)(126 310 255 335)(127 311 256 336)(128 312 257 337)(129 261 258 338)(130 262 259 339)(131 263 260 340)(132 264 209 341)(133 265 210 342)(134 266 211 343)(135 267 212 344)(136 268 213 345)(137 269 214 346)(138 270 215 347)(139 271 216 348)(140 272 217 349)(141 273 218 350)(142 274 219 351)(143 275 220 352)(144 276 221 353)(145 277 222 354)(146 278 223 355)(147 279 224 356)(148 280 225 357)(149 281 226 358)(150 282 227 359)(151 283 228 360)(152 284 229 361)(153 285 230 362)(154 286 231 363)(155 287 232 364)(156 288 233 313)
(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)| (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,304)(54,305)(55,306)(56,307)(57,308)(58,309)(59,310)(60,311)(61,312)(62,261)(63,262)(64,263)(65,264)(66,265)(67,266)(68,267)(69,268)(70,269)(71,270)(72,271)(73,272)(74,273)(75,274)(76,275)(77,276)(78,277)(79,278)(80,279)(81,280)(82,281)(83,282)(84,283)(85,284)(86,285)(87,286)(88,287)(89,288)(90,289)(91,290)(92,291)(93,292)(94,293)(95,294)(96,295)(97,296)(98,297)(99,298)(100,299)(101,300)(102,301)(103,302)(104,303)(157,325)(158,326)(159,327)(160,328)(161,329)(162,330)(163,331)(164,332)(165,333)(166,334)(167,335)(168,336)(169,337)(170,338)(171,339)(172,340)(173,341)(174,342)(175,343)(176,344)(177,345)(178,346)(179,347)(180,348)(181,349)(182,350)(183,351)(184,352)(185,353)(186,354)(187,355)(188,356)(189,357)(190,358)(191,359)(192,360)(193,361)(194,362)(195,363)(196,364)(197,313)(198,314)(199,315)(200,316)(201,317)(202,318)(203,319)(204,320)(205,321)(206,322)(207,323)(208,324)(209,394)(210,395)(211,396)(212,397)(213,398)(214,399)(215,400)(216,401)(217,402)(218,403)(219,404)(220,405)(221,406)(222,407)(223,408)(224,409)(225,410)(226,411)(227,412)(228,413)(229,414)(230,415)(231,416)(232,365)(233,366)(234,367)(235,368)(236,369)(237,370)(238,371)(239,372)(240,373)(241,374)(242,375)(243,376)(244,377)(245,378)(246,379)(247,380)(248,381)(249,382)(250,383)(251,384)(252,385)(253,386)(254,387)(255,388)(256,389)(257,390)(258,391)(259,392)(260,393), (1,71,400,179)(2,72,401,180)(3,73,402,181)(4,74,403,182)(5,75,404,183)(6,76,405,184)(7,77,406,185)(8,78,407,186)(9,79,408,187)(10,80,409,188)(11,81,410,189)(12,82,411,190)(13,83,412,191)(14,84,413,192)(15,85,414,193)(16,86,415,194)(17,87,416,195)(18,88,365,196)(19,89,366,197)(20,90,367,198)(21,91,368,199)(22,92,369,200)(23,93,370,201)(24,94,371,202)(25,95,372,203)(26,96,373,204)(27,97,374,205)(28,98,375,206)(29,99,376,207)(30,100,377,208)(31,101,378,157)(32,102,379,158)(33,103,380,159)(34,104,381,160)(35,53,382,161)(36,54,383,162)(37,55,384,163)(38,56,385,164)(39,57,386,165)(40,58,387,166)(41,59,388,167)(42,60,389,168)(43,61,390,169)(44,62,391,170)(45,63,392,171)(46,64,393,172)(47,65,394,173)(48,66,395,174)(49,67,396,175)(50,68,397,176)(51,69,398,177)(52,70,399,178)(105,289,234,314)(106,290,235,315)(107,291,236,316)(108,292,237,317)(109,293,238,318)(110,294,239,319)(111,295,240,320)(112,296,241,321)(113,297,242,322)(114,298,243,323)(115,299,244,324)(116,300,245,325)(117,301,246,326)(118,302,247,327)(119,303,248,328)(120,304,249,329)(121,305,250,330)(122,306,251,331)(123,307,252,332)(124,308,253,333)(125,309,254,334)(126,310,255,335)(127,311,256,336)(128,312,257,337)(129,261,258,338)(130,262,259,339)(131,263,260,340)(132,264,209,341)(133,265,210,342)(134,266,211,343)(135,267,212,344)(136,268,213,345)(137,269,214,346)(138,270,215,347)(139,271,216,348)(140,272,217,349)(141,273,218,350)(142,274,219,351)(143,275,220,352)(144,276,221,353)(145,277,222,354)(146,278,223,355)(147,279,224,356)(148,280,225,357)(149,281,226,358)(150,282,227,359)(151,283,228,360)(152,284,229,361)(153,285,230,362)(154,286,231,363)(155,287,232,364)(156,288,233,313), (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( (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,304)(54,305)(55,306)(56,307)(57,308)(58,309)(59,310)(60,311)(61,312)(62,261)(63,262)(64,263)(65,264)(66,265)(67,266)(68,267)(69,268)(70,269)(71,270)(72,271)(73,272)(74,273)(75,274)(76,275)(77,276)(78,277)(79,278)(80,279)(81,280)(82,281)(83,282)(84,283)(85,284)(86,285)(87,286)(88,287)(89,288)(90,289)(91,290)(92,291)(93,292)(94,293)(95,294)(96,295)(97,296)(98,297)(99,298)(100,299)(101,300)(102,301)(103,302)(104,303)(157,325)(158,326)(159,327)(160,328)(161,329)(162,330)(163,331)(164,332)(165,333)(166,334)(167,335)(168,336)(169,337)(170,338)(171,339)(172,340)(173,341)(174,342)(175,343)(176,344)(177,345)(178,346)(179,347)(180,348)(181,349)(182,350)(183,351)(184,352)(185,353)(186,354)(187,355)(188,356)(189,357)(190,358)(191,359)(192,360)(193,361)(194,362)(195,363)(196,364)(197,313)(198,314)(199,315)(200,316)(201,317)(202,318)(203,319)(204,320)(205,321)(206,322)(207,323)(208,324)(209,394)(210,395)(211,396)(212,397)(213,398)(214,399)(215,400)(216,401)(217,402)(218,403)(219,404)(220,405)(221,406)(222,407)(223,408)(224,409)(225,410)(226,411)(227,412)(228,413)(229,414)(230,415)(231,416)(232,365)(233,366)(234,367)(235,368)(236,369)(237,370)(238,371)(239,372)(240,373)(241,374)(242,375)(243,376)(244,377)(245,378)(246,379)(247,380)(248,381)(249,382)(250,383)(251,384)(252,385)(253,386)(254,387)(255,388)(256,389)(257,390)(258,391)(259,392)(260,393), (1,71,400,179)(2,72,401,180)(3,73,402,181)(4,74,403,182)(5,75,404,183)(6,76,405,184)(7,77,406,185)(8,78,407,186)(9,79,408,187)(10,80,409,188)(11,81,410,189)(12,82,411,190)(13,83,412,191)(14,84,413,192)(15,85,414,193)(16,86,415,194)(17,87,416,195)(18,88,365,196)(19,89,366,197)(20,90,367,198)(21,91,368,199)(22,92,369,200)(23,93,370,201)(24,94,371,202)(25,95,372,203)(26,96,373,204)(27,97,374,205)(28,98,375,206)(29,99,376,207)(30,100,377,208)(31,101,378,157)(32,102,379,158)(33,103,380,159)(34,104,381,160)(35,53,382,161)(36,54,383,162)(37,55,384,163)(38,56,385,164)(39,57,386,165)(40,58,387,166)(41,59,388,167)(42,60,389,168)(43,61,390,169)(44,62,391,170)(45,63,392,171)(46,64,393,172)(47,65,394,173)(48,66,395,174)(49,67,396,175)(50,68,397,176)(51,69,398,177)(52,70,399,178)(105,289,234,314)(106,290,235,315)(107,291,236,316)(108,292,237,317)(109,293,238,318)(110,294,239,319)(111,295,240,320)(112,296,241,321)(113,297,242,322)(114,298,243,323)(115,299,244,324)(116,300,245,325)(117,301,246,326)(118,302,247,327)(119,303,248,328)(120,304,249,329)(121,305,250,330)(122,306,251,331)(123,307,252,332)(124,308,253,333)(125,309,254,334)(126,310,255,335)(127,311,256,336)(128,312,257,337)(129,261,258,338)(130,262,259,339)(131,263,260,340)(132,264,209,341)(133,265,210,342)(134,266,211,343)(135,267,212,344)(136,268,213,345)(137,269,214,346)(138,270,215,347)(139,271,216,348)(140,272,217,349)(141,273,218,350)(142,274,219,351)(143,275,220,352)(144,276,221,353)(145,277,222,354)(146,278,223,355)(147,279,224,356)(148,280,225,357)(149,281,226,358)(150,282,227,359)(151,283,228,360)(152,284,229,361)(153,285,230,362)(154,286,231,363)(155,287,232,364)(156,288,233,313), (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([[(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,145),(9,146),(10,147),(11,148),(12,149),(13,150),(14,151),(15,152),(16,153),(17,154),(18,155),(19,156),(20,105),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,113),(29,114),(30,115),(31,116),(32,117),(33,118),(34,119),(35,120),(36,121),(37,122),(38,123),(39,124),(40,125),(41,126),(42,127),(43,128),(44,129),(45,130),(46,131),(47,132),(48,133),(49,134),(50,135),(51,136),(52,137),(53,304),(54,305),(55,306),(56,307),(57,308),(58,309),(59,310),(60,311),(61,312),(62,261),(63,262),(64,263),(65,264),(66,265),(67,266),(68,267),(69,268),(70,269),(71,270),(72,271),(73,272),(74,273),(75,274),(76,275),(77,276),(78,277),(79,278),(80,279),(81,280),(82,281),(83,282),(84,283),(85,284),(86,285),(87,286),(88,287),(89,288),(90,289),(91,290),(92,291),(93,292),(94,293),(95,294),(96,295),(97,296),(98,297),(99,298),(100,299),(101,300),(102,301),(103,302),(104,303),(157,325),(158,326),(159,327),(160,328),(161,329),(162,330),(163,331),(164,332),(165,333),(166,334),(167,335),(168,336),(169,337),(170,338),(171,339),(172,340),(173,341),(174,342),(175,343),(176,344),(177,345),(178,346),(179,347),(180,348),(181,349),(182,350),(183,351),(184,352),(185,353),(186,354),(187,355),(188,356),(189,357),(190,358),(191,359),(192,360),(193,361),(194,362),(195,363),(196,364),(197,313),(198,314),(199,315),(200,316),(201,317),(202,318),(203,319),(204,320),(205,321),(206,322),(207,323),(208,324),(209,394),(210,395),(211,396),(212,397),(213,398),(214,399),(215,400),(216,401),(217,402),(218,403),(219,404),(220,405),(221,406),(222,407),(223,408),(224,409),(225,410),(226,411),(227,412),(228,413),(229,414),(230,415),(231,416),(232,365),(233,366),(234,367),(235,368),(236,369),(237,370),(238,371),(239,372),(240,373),(241,374),(242,375),(243,376),(244,377),(245,378),(246,379),(247,380),(248,381),(249,382),(250,383),(251,384),(252,385),(253,386),(254,387),(255,388),(256,389),(257,390),(258,391),(259,392),(260,393)], [(1,71,400,179),(2,72,401,180),(3,73,402,181),(4,74,403,182),(5,75,404,183),(6,76,405,184),(7,77,406,185),(8,78,407,186),(9,79,408,187),(10,80,409,188),(11,81,410,189),(12,82,411,190),(13,83,412,191),(14,84,413,192),(15,85,414,193),(16,86,415,194),(17,87,416,195),(18,88,365,196),(19,89,366,197),(20,90,367,198),(21,91,368,199),(22,92,369,200),(23,93,370,201),(24,94,371,202),(25,95,372,203),(26,96,373,204),(27,97,374,205),(28,98,375,206),(29,99,376,207),(30,100,377,208),(31,101,378,157),(32,102,379,158),(33,103,380,159),(34,104,381,160),(35,53,382,161),(36,54,383,162),(37,55,384,163),(38,56,385,164),(39,57,386,165),(40,58,387,166),(41,59,388,167),(42,60,389,168),(43,61,390,169),(44,62,391,170),(45,63,392,171),(46,64,393,172),(47,65,394,173),(48,66,395,174),(49,67,396,175),(50,68,397,176),(51,69,398,177),(52,70,399,178),(105,289,234,314),(106,290,235,315),(107,291,236,316),(108,292,237,317),(109,293,238,318),(110,294,239,319),(111,295,240,320),(112,296,241,321),(113,297,242,322),(114,298,243,323),(115,299,244,324),(116,300,245,325),(117,301,246,326),(118,302,247,327),(119,303,248,328),(120,304,249,329),(121,305,250,330),(122,306,251,331),(123,307,252,332),(124,308,253,333),(125,309,254,334),(126,310,255,335),(127,311,256,336),(128,312,257,337),(129,261,258,338),(130,262,259,339),(131,263,260,340),(132,264,209,341),(133,265,210,342),(134,266,211,343),(135,267,212,344),(136,268,213,345),(137,269,214,346),(138,270,215,347),(139,271,216,348),(140,272,217,349),(141,273,218,350),(142,274,219,351),(143,275,220,352),(144,276,221,353),(145,277,222,354),(146,278,223,355),(147,279,224,356),(148,280,225,357),(149,281,226,358),(150,282,227,359),(151,283,228,360),(152,284,229,361),(153,285,230,362),(154,286,231,363),(155,287,232,364),(156,288,233,313)], [(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 ··· 2G 4A ··· 4X 13A ··· 13L 26A ··· 26CF 52A ··· 52KB order 1 2 ··· 2 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

416 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C13 C26 C26 C52 kernel C2×C4×C52 C4×C52 C22×C52 C2×C52 C2×C42 C42 C22×C4 C2×C4 # reps 1 4 3 24 12 48 36 288

Matrix representation of C2×C4×C52 in GL3(𝔽53) generated by

 52 0 0 0 1 0 0 0 52
,
 23 0 0 0 1 0 0 0 1
,
 36 0 0 0 5 0 0 0 45
G:=sub<GL(3,GF(53))| [52,0,0,0,1,0,0,0,52],[23,0,0,0,1,0,0,0,1],[36,0,0,0,5,0,0,0,45] >;

C2×C4×C52 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{52}
% in TeX

G:=Group("C2xC4xC52");
// GroupNames label

G:=SmallGroup(416,175);
// by ID

G=gap.SmallGroup(416,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,1255]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^52=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽