direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C52⋊3C4, C22.15D52, C23.30D26, C22.5Dic26, C26⋊3(C4⋊C4), (C2×C52)⋊10C4, C52⋊10(C2×C4), C2.2(C2×D52), (C2×C26).6Q8, C26.9(C2×Q8), (C2×C4)⋊3Dic13, C4⋊2(C2×Dic13), C26.15(C2×D4), (C2×C26).20D4, (C2×C4).84D26, (C22×C52).7C2, C2.3(C2×Dic26), (C22×C4).6D13, (C2×C52).92C22, (C2×C26).43C23, C26.36(C22×C4), C2.4(C22×Dic13), (C22×C26).35C22, (C22×Dic13).5C2, C22.14(C2×Dic13), C22.21(C22×D13), (C2×Dic13).37C22, C13⋊4(C2×C4⋊C4), (C2×C26).54(C2×C4), SmallGroup(416,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C52⋊3C4
G = < a,b,c | a2=b52=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 432 in 92 conjugacy classes, 65 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C13, C4⋊C4, C22×C4, C22×C4, C26, C26, C2×C4⋊C4, Dic13, C52, C2×C26, C2×C26, C2×Dic13, C2×Dic13, C2×C52, C22×C26, C52⋊3C4, C22×Dic13, C22×C52, C2×C52⋊3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, D13, C2×C4⋊C4, Dic13, D26, Dic26, D52, C2×Dic13, C22×D13, C52⋊3C4, C2×Dic26, C2×D52, C22×Dic13, C2×C52⋊3C4
►Regular action on 416 points
Generators in S416
(1 311)(2 312)(3 261)(4 262)(5 263)(6 264)(7 265)(8 266)(9 267)(10 268)(11 269)(12 270)(13 271)(14 272)(15 273)(16 274)(17 275)(18 276)(19 277)(20 278)(21 279)(22 280)(23 281)(24 282)(25 283)(26 284)(27 285)(28 286)(29 287)(30 288)(31 289)(32 290)(33 291)(34 292)(35 293)(36 294)(37 295)(38 296)(39 297)(40 298)(41 299)(42 300)(43 301)(44 302)(45 303)(46 304)(47 305)(48 306)(49 307)(50 308)(51 309)(52 310)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 155)(82 156)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(157 221)(158 222)(159 223)(160 224)(161 225)(162 226)(163 227)(164 228)(165 229)(166 230)(167 231)(168 232)(169 233)(170 234)(171 235)(172 236)(173 237)(174 238)(175 239)(176 240)(177 241)(178 242)(179 243)(180 244)(181 245)(182 246)(183 247)(184 248)(185 249)(186 250)(187 251)(188 252)(189 253)(190 254)(191 255)(192 256)(193 257)(194 258)(195 259)(196 260)(197 209)(198 210)(199 211)(200 212)(201 213)(202 214)(203 215)(204 216)(205 217)(206 218)(207 219)(208 220)(313 400)(314 401)(315 402)(316 403)(317 404)(318 405)(319 406)(320 407)(321 408)(322 409)(323 410)(324 411)(325 412)(326 413)(327 414)(328 415)(329 416)(330 365)(331 366)(332 367)(333 368)(334 369)(335 370)(336 371)(337 372)(338 373)(339 374)(340 375)(341 376)(342 377)(343 378)(344 379)(345 380)(346 381)(347 382)(348 383)(349 384)(350 385)(351 386)(352 387)(353 388)(354 389)(355 390)(356 391)(357 392)(358 393)(359 394)(360 395)(361 396)(362 397)(363 398)(364 399)
(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)
(1 172 374 111)(2 171 375 110)(3 170 376 109)(4 169 377 108)(5 168 378 107)(6 167 379 106)(7 166 380 105)(8 165 381 156)(9 164 382 155)(10 163 383 154)(11 162 384 153)(12 161 385 152)(13 160 386 151)(14 159 387 150)(15 158 388 149)(16 157 389 148)(17 208 390 147)(18 207 391 146)(19 206 392 145)(20 205 393 144)(21 204 394 143)(22 203 395 142)(23 202 396 141)(24 201 397 140)(25 200 398 139)(26 199 399 138)(27 198 400 137)(28 197 401 136)(29 196 402 135)(30 195 403 134)(31 194 404 133)(32 193 405 132)(33 192 406 131)(34 191 407 130)(35 190 408 129)(36 189 409 128)(37 188 410 127)(38 187 411 126)(39 186 412 125)(40 185 413 124)(41 184 414 123)(42 183 415 122)(43 182 416 121)(44 181 365 120)(45 180 366 119)(46 179 367 118)(47 178 368 117)(48 177 369 116)(49 176 370 115)(50 175 371 114)(51 174 372 113)(52 173 373 112)(53 295 252 323)(54 294 253 322)(55 293 254 321)(56 292 255 320)(57 291 256 319)(58 290 257 318)(59 289 258 317)(60 288 259 316)(61 287 260 315)(62 286 209 314)(63 285 210 313)(64 284 211 364)(65 283 212 363)(66 282 213 362)(67 281 214 361)(68 280 215 360)(69 279 216 359)(70 278 217 358)(71 277 218 357)(72 276 219 356)(73 275 220 355)(74 274 221 354)(75 273 222 353)(76 272 223 352)(77 271 224 351)(78 270 225 350)(79 269 226 349)(80 268 227 348)(81 267 228 347)(82 266 229 346)(83 265 230 345)(84 264 231 344)(85 263 232 343)(86 262 233 342)(87 261 234 341)(88 312 235 340)(89 311 236 339)(90 310 237 338)(91 309 238 337)(92 308 239 336)(93 307 240 335)(94 306 241 334)(95 305 242 333)(96 304 243 332)(97 303 244 331)(98 302 245 330)(99 301 246 329)(100 300 247 328)(101 299 248 327)(102 298 249 326)(103 297 250 325)(104 296 251 324)
G:=sub<Sym(416)| (1,311)(2,312)(3,261)(4,262)(5,263)(6,264)(7,265)(8,266)(9,267)(10,268)(11,269)(12,270)(13,271)(14,272)(15,273)(16,274)(17,275)(18,276)(19,277)(20,278)(21,279)(22,280)(23,281)(24,282)(25,283)(26,284)(27,285)(28,286)(29,287)(30,288)(31,289)(32,290)(33,291)(34,292)(35,293)(36,294)(37,295)(38,296)(39,297)(40,298)(41,299)(42,300)(43,301)(44,302)(45,303)(46,304)(47,305)(48,306)(49,307)(50,308)(51,309)(52,310)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(157,221)(158,222)(159,223)(160,224)(161,225)(162,226)(163,227)(164,228)(165,229)(166,230)(167,231)(168,232)(169,233)(170,234)(171,235)(172,236)(173,237)(174,238)(175,239)(176,240)(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)(184,248)(185,249)(186,250)(187,251)(188,252)(189,253)(190,254)(191,255)(192,256)(193,257)(194,258)(195,259)(196,260)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218)(207,219)(208,220)(313,400)(314,401)(315,402)(316,403)(317,404)(318,405)(319,406)(320,407)(321,408)(322,409)(323,410)(324,411)(325,412)(326,413)(327,414)(328,415)(329,416)(330,365)(331,366)(332,367)(333,368)(334,369)(335,370)(336,371)(337,372)(338,373)(339,374)(340,375)(341,376)(342,377)(343,378)(344,379)(345,380)(346,381)(347,382)(348,383)(349,384)(350,385)(351,386)(352,387)(353,388)(354,389)(355,390)(356,391)(357,392)(358,393)(359,394)(360,395)(361,396)(362,397)(363,398)(364,399), (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), (1,172,374,111)(2,171,375,110)(3,170,376,109)(4,169,377,108)(5,168,378,107)(6,167,379,106)(7,166,380,105)(8,165,381,156)(9,164,382,155)(10,163,383,154)(11,162,384,153)(12,161,385,152)(13,160,386,151)(14,159,387,150)(15,158,388,149)(16,157,389,148)(17,208,390,147)(18,207,391,146)(19,206,392,145)(20,205,393,144)(21,204,394,143)(22,203,395,142)(23,202,396,141)(24,201,397,140)(25,200,398,139)(26,199,399,138)(27,198,400,137)(28,197,401,136)(29,196,402,135)(30,195,403,134)(31,194,404,133)(32,193,405,132)(33,192,406,131)(34,191,407,130)(35,190,408,129)(36,189,409,128)(37,188,410,127)(38,187,411,126)(39,186,412,125)(40,185,413,124)(41,184,414,123)(42,183,415,122)(43,182,416,121)(44,181,365,120)(45,180,366,119)(46,179,367,118)(47,178,368,117)(48,177,369,116)(49,176,370,115)(50,175,371,114)(51,174,372,113)(52,173,373,112)(53,295,252,323)(54,294,253,322)(55,293,254,321)(56,292,255,320)(57,291,256,319)(58,290,257,318)(59,289,258,317)(60,288,259,316)(61,287,260,315)(62,286,209,314)(63,285,210,313)(64,284,211,364)(65,283,212,363)(66,282,213,362)(67,281,214,361)(68,280,215,360)(69,279,216,359)(70,278,217,358)(71,277,218,357)(72,276,219,356)(73,275,220,355)(74,274,221,354)(75,273,222,353)(76,272,223,352)(77,271,224,351)(78,270,225,350)(79,269,226,349)(80,268,227,348)(81,267,228,347)(82,266,229,346)(83,265,230,345)(84,264,231,344)(85,263,232,343)(86,262,233,342)(87,261,234,341)(88,312,235,340)(89,311,236,339)(90,310,237,338)(91,309,238,337)(92,308,239,336)(93,307,240,335)(94,306,241,334)(95,305,242,333)(96,304,243,332)(97,303,244,331)(98,302,245,330)(99,301,246,329)(100,300,247,328)(101,299,248,327)(102,298,249,326)(103,297,250,325)(104,296,251,324)>;
G:=Group( (1,311)(2,312)(3,261)(4,262)(5,263)(6,264)(7,265)(8,266)(9,267)(10,268)(11,269)(12,270)(13,271)(14,272)(15,273)(16,274)(17,275)(18,276)(19,277)(20,278)(21,279)(22,280)(23,281)(24,282)(25,283)(26,284)(27,285)(28,286)(29,287)(30,288)(31,289)(32,290)(33,291)(34,292)(35,293)(36,294)(37,295)(38,296)(39,297)(40,298)(41,299)(42,300)(43,301)(44,302)(45,303)(46,304)(47,305)(48,306)(49,307)(50,308)(51,309)(52,310)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(157,221)(158,222)(159,223)(160,224)(161,225)(162,226)(163,227)(164,228)(165,229)(166,230)(167,231)(168,232)(169,233)(170,234)(171,235)(172,236)(173,237)(174,238)(175,239)(176,240)(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)(184,248)(185,249)(186,250)(187,251)(188,252)(189,253)(190,254)(191,255)(192,256)(193,257)(194,258)(195,259)(196,260)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218)(207,219)(208,220)(313,400)(314,401)(315,402)(316,403)(317,404)(318,405)(319,406)(320,407)(321,408)(322,409)(323,410)(324,411)(325,412)(326,413)(327,414)(328,415)(329,416)(330,365)(331,366)(332,367)(333,368)(334,369)(335,370)(336,371)(337,372)(338,373)(339,374)(340,375)(341,376)(342,377)(343,378)(344,379)(345,380)(346,381)(347,382)(348,383)(349,384)(350,385)(351,386)(352,387)(353,388)(354,389)(355,390)(356,391)(357,392)(358,393)(359,394)(360,395)(361,396)(362,397)(363,398)(364,399), (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), (1,172,374,111)(2,171,375,110)(3,170,376,109)(4,169,377,108)(5,168,378,107)(6,167,379,106)(7,166,380,105)(8,165,381,156)(9,164,382,155)(10,163,383,154)(11,162,384,153)(12,161,385,152)(13,160,386,151)(14,159,387,150)(15,158,388,149)(16,157,389,148)(17,208,390,147)(18,207,391,146)(19,206,392,145)(20,205,393,144)(21,204,394,143)(22,203,395,142)(23,202,396,141)(24,201,397,140)(25,200,398,139)(26,199,399,138)(27,198,400,137)(28,197,401,136)(29,196,402,135)(30,195,403,134)(31,194,404,133)(32,193,405,132)(33,192,406,131)(34,191,407,130)(35,190,408,129)(36,189,409,128)(37,188,410,127)(38,187,411,126)(39,186,412,125)(40,185,413,124)(41,184,414,123)(42,183,415,122)(43,182,416,121)(44,181,365,120)(45,180,366,119)(46,179,367,118)(47,178,368,117)(48,177,369,116)(49,176,370,115)(50,175,371,114)(51,174,372,113)(52,173,373,112)(53,295,252,323)(54,294,253,322)(55,293,254,321)(56,292,255,320)(57,291,256,319)(58,290,257,318)(59,289,258,317)(60,288,259,316)(61,287,260,315)(62,286,209,314)(63,285,210,313)(64,284,211,364)(65,283,212,363)(66,282,213,362)(67,281,214,361)(68,280,215,360)(69,279,216,359)(70,278,217,358)(71,277,218,357)(72,276,219,356)(73,275,220,355)(74,274,221,354)(75,273,222,353)(76,272,223,352)(77,271,224,351)(78,270,225,350)(79,269,226,349)(80,268,227,348)(81,267,228,347)(82,266,229,346)(83,265,230,345)(84,264,231,344)(85,263,232,343)(86,262,233,342)(87,261,234,341)(88,312,235,340)(89,311,236,339)(90,310,237,338)(91,309,238,337)(92,308,239,336)(93,307,240,335)(94,306,241,334)(95,305,242,333)(96,304,243,332)(97,303,244,331)(98,302,245,330)(99,301,246,329)(100,300,247,328)(101,299,248,327)(102,298,249,326)(103,297,250,325)(104,296,251,324) );
G=PermutationGroup([[(1,311),(2,312),(3,261),(4,262),(5,263),(6,264),(7,265),(8,266),(9,267),(10,268),(11,269),(12,270),(13,271),(14,272),(15,273),(16,274),(17,275),(18,276),(19,277),(20,278),(21,279),(22,280),(23,281),(24,282),(25,283),(26,284),(27,285),(28,286),(29,287),(30,288),(31,289),(32,290),(33,291),(34,292),(35,293),(36,294),(37,295),(38,296),(39,297),(40,298),(41,299),(42,300),(43,301),(44,302),(45,303),(46,304),(47,305),(48,306),(49,307),(50,308),(51,309),(52,310),(53,127),(54,128),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,145),(72,146),(73,147),(74,148),(75,149),(76,150),(77,151),(78,152),(79,153),(80,154),(81,155),(82,156),(83,105),(84,106),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(157,221),(158,222),(159,223),(160,224),(161,225),(162,226),(163,227),(164,228),(165,229),(166,230),(167,231),(168,232),(169,233),(170,234),(171,235),(172,236),(173,237),(174,238),(175,239),(176,240),(177,241),(178,242),(179,243),(180,244),(181,245),(182,246),(183,247),(184,248),(185,249),(186,250),(187,251),(188,252),(189,253),(190,254),(191,255),(192,256),(193,257),(194,258),(195,259),(196,260),(197,209),(198,210),(199,211),(200,212),(201,213),(202,214),(203,215),(204,216),(205,217),(206,218),(207,219),(208,220),(313,400),(314,401),(315,402),(316,403),(317,404),(318,405),(319,406),(320,407),(321,408),(322,409),(323,410),(324,411),(325,412),(326,413),(327,414),(328,415),(329,416),(330,365),(331,366),(332,367),(333,368),(334,369),(335,370),(336,371),(337,372),(338,373),(339,374),(340,375),(341,376),(342,377),(343,378),(344,379),(345,380),(346,381),(347,382),(348,383),(349,384),(350,385),(351,386),(352,387),(353,388),(354,389),(355,390),(356,391),(357,392),(358,393),(359,394),(360,395),(361,396),(362,397),(363,398),(364,399)], [(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)], [(1,172,374,111),(2,171,375,110),(3,170,376,109),(4,169,377,108),(5,168,378,107),(6,167,379,106),(7,166,380,105),(8,165,381,156),(9,164,382,155),(10,163,383,154),(11,162,384,153),(12,161,385,152),(13,160,386,151),(14,159,387,150),(15,158,388,149),(16,157,389,148),(17,208,390,147),(18,207,391,146),(19,206,392,145),(20,205,393,144),(21,204,394,143),(22,203,395,142),(23,202,396,141),(24,201,397,140),(25,200,398,139),(26,199,399,138),(27,198,400,137),(28,197,401,136),(29,196,402,135),(30,195,403,134),(31,194,404,133),(32,193,405,132),(33,192,406,131),(34,191,407,130),(35,190,408,129),(36,189,409,128),(37,188,410,127),(38,187,411,126),(39,186,412,125),(40,185,413,124),(41,184,414,123),(42,183,415,122),(43,182,416,121),(44,181,365,120),(45,180,366,119),(46,179,367,118),(47,178,368,117),(48,177,369,116),(49,176,370,115),(50,175,371,114),(51,174,372,113),(52,173,373,112),(53,295,252,323),(54,294,253,322),(55,293,254,321),(56,292,255,320),(57,291,256,319),(58,290,257,318),(59,289,258,317),(60,288,259,316),(61,287,260,315),(62,286,209,314),(63,285,210,313),(64,284,211,364),(65,283,212,363),(66,282,213,362),(67,281,214,361),(68,280,215,360),(69,279,216,359),(70,278,217,358),(71,277,218,357),(72,276,219,356),(73,275,220,355),(74,274,221,354),(75,273,222,353),(76,272,223,352),(77,271,224,351),(78,270,225,350),(79,269,226,349),(80,268,227,348),(81,267,228,347),(82,266,229,346),(83,265,230,345),(84,264,231,344),(85,263,232,343),(86,262,233,342),(87,261,234,341),(88,312,235,340),(89,311,236,339),(90,310,237,338),(91,309,238,337),(92,308,239,336),(93,307,240,335),(94,306,241,334),(95,305,242,333),(96,304,243,332),(97,303,244,331),(98,302,245,330),(99,301,246,329),(100,300,247,328),(101,299,248,327),(102,298,249,326),(103,297,250,325),(104,296,251,324)]])
116 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 13A | ··· | 13F | 26A | ··· | 26AP | 52A | ··· | 52AV |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D13 | Dic13 | D26 | D26 | Dic26 | D52 |
| kernel | C2×C52⋊3C4 | C52⋊3C4 | C22×Dic13 | C22×C52 | C2×C52 | C2×C26 | C2×C26 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 2 | 6 | 24 | 12 | 6 | 24 | 24 |
Matrix representation of C2×C52⋊3C4 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 17 | 37 |
| 1 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 9 | 49 |
| 0 | 0 | 20 | 44 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,0,17,0,0,28,37],[1,0,0,0,0,23,0,0,0,0,9,20,0,0,49,44] >;
C2×C52⋊3C4 in GAP, Magma, Sage, TeX
C_2\times C_{52}\rtimes_3C_4 % in TeX
G:=Group("C2xC52:3C4"); // GroupNames label
G:=SmallGroup(416,146);
// by ID
G=gap.SmallGroup(416,146);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,86,13829]);
// Polycyclic
G:=Group<a,b,c|a^2=b^52=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations