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G = C2×C4×C52order 416 = 25·13

Abelian group of type [2,4,52]

direct product, abelian, monomial, 2-elementary

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

Series: Derived Chief Lower central Upper central

C1 — C2×C4×C52
C1C2C22C2×C26C2×C52C4×C52 — C2×C4×C52
C1 — C2×C4×C52
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 [×7], C4 [×12], C22, C22 [×6], C2×C4 [×18], C23, C13, C42 [×4], C22×C4 [×3], C26 [×7], C2×C42, C52 [×12], C2×C26, C2×C26 [×6], C2×C52 [×18], C22×C26, C4×C52 [×4], C22×C52 [×3], C2×C4×C52
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C13, C42 [×4], C22×C4 [×3], C26 [×7], C2×C42, C52 [×12], C2×C26 [×7], C2×C52 [×18], C22×C26, C4×C52 [×4], C22×C52 [×3], C2×C4×C52

Smallest permutation representation of C2×C4×C52
Regular action on 416 points
Generators in S416
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 53)(45 54)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(105 280)(106 281)(107 282)(108 283)(109 284)(110 285)(111 286)(112 287)(113 288)(114 289)(115 290)(116 291)(117 292)(118 293)(119 294)(120 295)(121 296)(122 297)(123 298)(124 299)(125 300)(126 301)(127 302)(128 303)(129 304)(130 305)(131 306)(132 307)(133 308)(134 309)(135 310)(136 311)(137 312)(138 261)(139 262)(140 263)(141 264)(142 265)(143 266)(144 267)(145 268)(146 269)(147 270)(148 271)(149 272)(150 273)(151 274)(152 275)(153 276)(154 277)(155 278)(156 279)(157 251)(158 252)(159 253)(160 254)(161 255)(162 256)(163 257)(164 258)(165 259)(166 260)(167 209)(168 210)(169 211)(170 212)(171 213)(172 214)(173 215)(174 216)(175 217)(176 218)(177 219)(178 220)(179 221)(180 222)(181 223)(182 224)(183 225)(184 226)(185 227)(186 228)(187 229)(188 230)(189 231)(190 232)(191 233)(192 234)(193 235)(194 236)(195 237)(196 238)(197 239)(198 240)(199 241)(200 242)(201 243)(202 244)(203 245)(204 246)(205 247)(206 248)(207 249)(208 250)(313 412)(314 413)(315 414)(316 415)(317 416)(318 365)(319 366)(320 367)(321 368)(322 369)(323 370)(324 371)(325 372)(326 373)(327 374)(328 375)(329 376)(330 377)(331 378)(332 379)(333 380)(334 381)(335 382)(336 383)(337 384)(338 385)(339 386)(340 387)(341 388)(342 389)(343 390)(344 391)(345 392)(346 393)(347 394)(348 395)(349 396)(350 397)(351 398)(352 399)(353 400)(354 401)(355 402)(356 403)(357 404)(358 405)(359 406)(360 407)(361 408)(362 409)(363 410)(364 411)
(1 373 183 301)(2 374 184 302)(3 375 185 303)(4 376 186 304)(5 377 187 305)(6 378 188 306)(7 379 189 307)(8 380 190 308)(9 381 191 309)(10 382 192 310)(11 383 193 311)(12 384 194 312)(13 385 195 261)(14 386 196 262)(15 387 197 263)(16 388 198 264)(17 389 199 265)(18 390 200 266)(19 391 201 267)(20 392 202 268)(21 393 203 269)(22 394 204 270)(23 395 205 271)(24 396 206 272)(25 397 207 273)(26 398 208 274)(27 399 157 275)(28 400 158 276)(29 401 159 277)(30 402 160 278)(31 403 161 279)(32 404 162 280)(33 405 163 281)(34 406 164 282)(35 407 165 283)(36 408 166 284)(37 409 167 285)(38 410 168 286)(39 411 169 287)(40 412 170 288)(41 413 171 289)(42 414 172 290)(43 415 173 291)(44 416 174 292)(45 365 175 293)(46 366 176 294)(47 367 177 295)(48 368 178 296)(49 369 179 297)(50 370 180 298)(51 371 181 299)(52 372 182 300)(53 317 216 117)(54 318 217 118)(55 319 218 119)(56 320 219 120)(57 321 220 121)(58 322 221 122)(59 323 222 123)(60 324 223 124)(61 325 224 125)(62 326 225 126)(63 327 226 127)(64 328 227 128)(65 329 228 129)(66 330 229 130)(67 331 230 131)(68 332 231 132)(69 333 232 133)(70 334 233 134)(71 335 234 135)(72 336 235 136)(73 337 236 137)(74 338 237 138)(75 339 238 139)(76 340 239 140)(77 341 240 141)(78 342 241 142)(79 343 242 143)(80 344 243 144)(81 345 244 145)(82 346 245 146)(83 347 246 147)(84 348 247 148)(85 349 248 149)(86 350 249 150)(87 351 250 151)(88 352 251 152)(89 353 252 153)(90 354 253 154)(91 355 254 155)(92 356 255 156)(93 357 256 105)(94 358 257 106)(95 359 258 107)(96 360 259 108)(97 361 260 109)(98 362 209 110)(99 363 210 111)(100 364 211 112)(101 313 212 113)(102 314 213 114)(103 315 214 115)(104 316 215 116)
(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)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(105,280)(106,281)(107,282)(108,283)(109,284)(110,285)(111,286)(112,287)(113,288)(114,289)(115,290)(116,291)(117,292)(118,293)(119,294)(120,295)(121,296)(122,297)(123,298)(124,299)(125,300)(126,301)(127,302)(128,303)(129,304)(130,305)(131,306)(132,307)(133,308)(134,309)(135,310)(136,311)(137,312)(138,261)(139,262)(140,263)(141,264)(142,265)(143,266)(144,267)(145,268)(146,269)(147,270)(148,271)(149,272)(150,273)(151,274)(152,275)(153,276)(154,277)(155,278)(156,279)(157,251)(158,252)(159,253)(160,254)(161,255)(162,256)(163,257)(164,258)(165,259)(166,260)(167,209)(168,210)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222)(181,223)(182,224)(183,225)(184,226)(185,227)(186,228)(187,229)(188,230)(189,231)(190,232)(191,233)(192,234)(193,235)(194,236)(195,237)(196,238)(197,239)(198,240)(199,241)(200,242)(201,243)(202,244)(203,245)(204,246)(205,247)(206,248)(207,249)(208,250)(313,412)(314,413)(315,414)(316,415)(317,416)(318,365)(319,366)(320,367)(321,368)(322,369)(323,370)(324,371)(325,372)(326,373)(327,374)(328,375)(329,376)(330,377)(331,378)(332,379)(333,380)(334,381)(335,382)(336,383)(337,384)(338,385)(339,386)(340,387)(341,388)(342,389)(343,390)(344,391)(345,392)(346,393)(347,394)(348,395)(349,396)(350,397)(351,398)(352,399)(353,400)(354,401)(355,402)(356,403)(357,404)(358,405)(359,406)(360,407)(361,408)(362,409)(363,410)(364,411), (1,373,183,301)(2,374,184,302)(3,375,185,303)(4,376,186,304)(5,377,187,305)(6,378,188,306)(7,379,189,307)(8,380,190,308)(9,381,191,309)(10,382,192,310)(11,383,193,311)(12,384,194,312)(13,385,195,261)(14,386,196,262)(15,387,197,263)(16,388,198,264)(17,389,199,265)(18,390,200,266)(19,391,201,267)(20,392,202,268)(21,393,203,269)(22,394,204,270)(23,395,205,271)(24,396,206,272)(25,397,207,273)(26,398,208,274)(27,399,157,275)(28,400,158,276)(29,401,159,277)(30,402,160,278)(31,403,161,279)(32,404,162,280)(33,405,163,281)(34,406,164,282)(35,407,165,283)(36,408,166,284)(37,409,167,285)(38,410,168,286)(39,411,169,287)(40,412,170,288)(41,413,171,289)(42,414,172,290)(43,415,173,291)(44,416,174,292)(45,365,175,293)(46,366,176,294)(47,367,177,295)(48,368,178,296)(49,369,179,297)(50,370,180,298)(51,371,181,299)(52,372,182,300)(53,317,216,117)(54,318,217,118)(55,319,218,119)(56,320,219,120)(57,321,220,121)(58,322,221,122)(59,323,222,123)(60,324,223,124)(61,325,224,125)(62,326,225,126)(63,327,226,127)(64,328,227,128)(65,329,228,129)(66,330,229,130)(67,331,230,131)(68,332,231,132)(69,333,232,133)(70,334,233,134)(71,335,234,135)(72,336,235,136)(73,337,236,137)(74,338,237,138)(75,339,238,139)(76,340,239,140)(77,341,240,141)(78,342,241,142)(79,343,242,143)(80,344,243,144)(81,345,244,145)(82,346,245,146)(83,347,246,147)(84,348,247,148)(85,349,248,149)(86,350,249,150)(87,351,250,151)(88,352,251,152)(89,353,252,153)(90,354,253,154)(91,355,254,155)(92,356,255,156)(93,357,256,105)(94,358,257,106)(95,359,258,107)(96,360,259,108)(97,361,260,109)(98,362,209,110)(99,363,210,111)(100,364,211,112)(101,313,212,113)(102,314,213,114)(103,315,214,115)(104,316,215,116), (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( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(105,280)(106,281)(107,282)(108,283)(109,284)(110,285)(111,286)(112,287)(113,288)(114,289)(115,290)(116,291)(117,292)(118,293)(119,294)(120,295)(121,296)(122,297)(123,298)(124,299)(125,300)(126,301)(127,302)(128,303)(129,304)(130,305)(131,306)(132,307)(133,308)(134,309)(135,310)(136,311)(137,312)(138,261)(139,262)(140,263)(141,264)(142,265)(143,266)(144,267)(145,268)(146,269)(147,270)(148,271)(149,272)(150,273)(151,274)(152,275)(153,276)(154,277)(155,278)(156,279)(157,251)(158,252)(159,253)(160,254)(161,255)(162,256)(163,257)(164,258)(165,259)(166,260)(167,209)(168,210)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222)(181,223)(182,224)(183,225)(184,226)(185,227)(186,228)(187,229)(188,230)(189,231)(190,232)(191,233)(192,234)(193,235)(194,236)(195,237)(196,238)(197,239)(198,240)(199,241)(200,242)(201,243)(202,244)(203,245)(204,246)(205,247)(206,248)(207,249)(208,250)(313,412)(314,413)(315,414)(316,415)(317,416)(318,365)(319,366)(320,367)(321,368)(322,369)(323,370)(324,371)(325,372)(326,373)(327,374)(328,375)(329,376)(330,377)(331,378)(332,379)(333,380)(334,381)(335,382)(336,383)(337,384)(338,385)(339,386)(340,387)(341,388)(342,389)(343,390)(344,391)(345,392)(346,393)(347,394)(348,395)(349,396)(350,397)(351,398)(352,399)(353,400)(354,401)(355,402)(356,403)(357,404)(358,405)(359,406)(360,407)(361,408)(362,409)(363,410)(364,411), (1,373,183,301)(2,374,184,302)(3,375,185,303)(4,376,186,304)(5,377,187,305)(6,378,188,306)(7,379,189,307)(8,380,190,308)(9,381,191,309)(10,382,192,310)(11,383,193,311)(12,384,194,312)(13,385,195,261)(14,386,196,262)(15,387,197,263)(16,388,198,264)(17,389,199,265)(18,390,200,266)(19,391,201,267)(20,392,202,268)(21,393,203,269)(22,394,204,270)(23,395,205,271)(24,396,206,272)(25,397,207,273)(26,398,208,274)(27,399,157,275)(28,400,158,276)(29,401,159,277)(30,402,160,278)(31,403,161,279)(32,404,162,280)(33,405,163,281)(34,406,164,282)(35,407,165,283)(36,408,166,284)(37,409,167,285)(38,410,168,286)(39,411,169,287)(40,412,170,288)(41,413,171,289)(42,414,172,290)(43,415,173,291)(44,416,174,292)(45,365,175,293)(46,366,176,294)(47,367,177,295)(48,368,178,296)(49,369,179,297)(50,370,180,298)(51,371,181,299)(52,372,182,300)(53,317,216,117)(54,318,217,118)(55,319,218,119)(56,320,219,120)(57,321,220,121)(58,322,221,122)(59,323,222,123)(60,324,223,124)(61,325,224,125)(62,326,225,126)(63,327,226,127)(64,328,227,128)(65,329,228,129)(66,330,229,130)(67,331,230,131)(68,332,231,132)(69,333,232,133)(70,334,233,134)(71,335,234,135)(72,336,235,136)(73,337,236,137)(74,338,237,138)(75,339,238,139)(76,340,239,140)(77,341,240,141)(78,342,241,142)(79,343,242,143)(80,344,243,144)(81,345,244,145)(82,346,245,146)(83,347,246,147)(84,348,247,148)(85,349,248,149)(86,350,249,150)(87,351,250,151)(88,352,251,152)(89,353,252,153)(90,354,253,154)(91,355,254,155)(92,356,255,156)(93,357,256,105)(94,358,257,106)(95,359,258,107)(96,360,259,108)(97,361,260,109)(98,362,209,110)(99,363,210,111)(100,364,211,112)(101,313,212,113)(102,314,213,114)(103,315,214,115)(104,316,215,116), (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([(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,101),(41,102),(42,103),(43,104),(44,53),(45,54),(46,55),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(105,280),(106,281),(107,282),(108,283),(109,284),(110,285),(111,286),(112,287),(113,288),(114,289),(115,290),(116,291),(117,292),(118,293),(119,294),(120,295),(121,296),(122,297),(123,298),(124,299),(125,300),(126,301),(127,302),(128,303),(129,304),(130,305),(131,306),(132,307),(133,308),(134,309),(135,310),(136,311),(137,312),(138,261),(139,262),(140,263),(141,264),(142,265),(143,266),(144,267),(145,268),(146,269),(147,270),(148,271),(149,272),(150,273),(151,274),(152,275),(153,276),(154,277),(155,278),(156,279),(157,251),(158,252),(159,253),(160,254),(161,255),(162,256),(163,257),(164,258),(165,259),(166,260),(167,209),(168,210),(169,211),(170,212),(171,213),(172,214),(173,215),(174,216),(175,217),(176,218),(177,219),(178,220),(179,221),(180,222),(181,223),(182,224),(183,225),(184,226),(185,227),(186,228),(187,229),(188,230),(189,231),(190,232),(191,233),(192,234),(193,235),(194,236),(195,237),(196,238),(197,239),(198,240),(199,241),(200,242),(201,243),(202,244),(203,245),(204,246),(205,247),(206,248),(207,249),(208,250),(313,412),(314,413),(315,414),(316,415),(317,416),(318,365),(319,366),(320,367),(321,368),(322,369),(323,370),(324,371),(325,372),(326,373),(327,374),(328,375),(329,376),(330,377),(331,378),(332,379),(333,380),(334,381),(335,382),(336,383),(337,384),(338,385),(339,386),(340,387),(341,388),(342,389),(343,390),(344,391),(345,392),(346,393),(347,394),(348,395),(349,396),(350,397),(351,398),(352,399),(353,400),(354,401),(355,402),(356,403),(357,404),(358,405),(359,406),(360,407),(361,408),(362,409),(363,410),(364,411)], [(1,373,183,301),(2,374,184,302),(3,375,185,303),(4,376,186,304),(5,377,187,305),(6,378,188,306),(7,379,189,307),(8,380,190,308),(9,381,191,309),(10,382,192,310),(11,383,193,311),(12,384,194,312),(13,385,195,261),(14,386,196,262),(15,387,197,263),(16,388,198,264),(17,389,199,265),(18,390,200,266),(19,391,201,267),(20,392,202,268),(21,393,203,269),(22,394,204,270),(23,395,205,271),(24,396,206,272),(25,397,207,273),(26,398,208,274),(27,399,157,275),(28,400,158,276),(29,401,159,277),(30,402,160,278),(31,403,161,279),(32,404,162,280),(33,405,163,281),(34,406,164,282),(35,407,165,283),(36,408,166,284),(37,409,167,285),(38,410,168,286),(39,411,169,287),(40,412,170,288),(41,413,171,289),(42,414,172,290),(43,415,173,291),(44,416,174,292),(45,365,175,293),(46,366,176,294),(47,367,177,295),(48,368,178,296),(49,369,179,297),(50,370,180,298),(51,371,181,299),(52,372,182,300),(53,317,216,117),(54,318,217,118),(55,319,218,119),(56,320,219,120),(57,321,220,121),(58,322,221,122),(59,323,222,123),(60,324,223,124),(61,325,224,125),(62,326,225,126),(63,327,226,127),(64,328,227,128),(65,329,228,129),(66,330,229,130),(67,331,230,131),(68,332,231,132),(69,333,232,133),(70,334,233,134),(71,335,234,135),(72,336,235,136),(73,337,236,137),(74,338,237,138),(75,339,238,139),(76,340,239,140),(77,341,240,141),(78,342,241,142),(79,343,242,143),(80,344,243,144),(81,345,244,145),(82,346,245,146),(83,347,246,147),(84,348,247,148),(85,349,248,149),(86,350,249,150),(87,351,250,151),(88,352,251,152),(89,353,252,153),(90,354,253,154),(91,355,254,155),(92,356,255,156),(93,357,256,105),(94,358,257,106),(95,359,258,107),(96,360,259,108),(97,361,260,109),(98,362,209,110),(99,363,210,111),(100,364,211,112),(101,313,212,113),(102,314,213,114),(103,315,214,115),(104,316,215,116)], [(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···2G4A···4X13A···13L26A···26CF52A···52KB
order12···24···413···1326···2652···52
size11···11···11···11···11···1

416 irreducible representations

dim11111111
type+++
imageC1C2C2C4C13C26C26C52
kernelC2×C4×C52C4×C52C22×C52C2×C52C2×C42C42C22×C4C2×C4
# reps14324124836288

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

5200
010
0052
,
2300
010
001
,
3600
050
0045
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