direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×Dic13, C26⋊2C42, C23.28D26, (C2×C52)⋊12C4, C52⋊11(C2×C4), C13⋊3(C2×C42), (C2×C4).101D26, (C22×C52).13C2, (C2×C26).40C23, C26.35(C22×C4), (C22×C4).10D13, C22.15(C4×D13), (C2×C52).113C22, C2.2(C22×Dic13), (C22×C26).32C22, C22.13(C2×Dic13), C22.19(C22×D13), (C22×Dic13).10C2, (C2×Dic13).64C22, C2.3(C2×C4×D13), (C2×C26).53(C2×C4), SmallGroup(416,143)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C4×Dic13 |
Generators and relations for C2×C4×Dic13
G = < a,b,c,d | a2=b4=c26=1, d2=c13, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 432 in 108 conjugacy classes, 81 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C13, C42, C22×C4, C22×C4, C26, C26, C2×C42, Dic13, C52, C2×C26, C2×C26, C2×Dic13, C2×C52, C22×C26, C4×Dic13, C22×Dic13, C22×C52, C2×C4×Dic13
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, D13, C2×C42, Dic13, D26, C4×D13, C2×Dic13, C22×D13, C4×Dic13, C2×C4×D13, C22×Dic13, C2×C4×Dic13
►Regular action on 416 points
Generators in S416
(1 371)(2 372)(3 373)(4 374)(5 375)(6 376)(7 377)(8 378)(9 379)(10 380)(11 381)(12 382)(13 383)(14 384)(15 385)(16 386)(17 387)(18 388)(19 389)(20 390)(21 365)(22 366)(23 367)(24 368)(25 369)(26 370)(27 275)(28 276)(29 277)(30 278)(31 279)(32 280)(33 281)(34 282)(35 283)(36 284)(37 285)(38 286)(39 261)(40 262)(41 263)(42 264)(43 265)(44 266)(45 267)(46 268)(47 269)(48 270)(49 271)(50 272)(51 273)(52 274)(53 171)(54 172)(55 173)(56 174)(57 175)(58 176)(59 177)(60 178)(61 179)(62 180)(63 181)(64 182)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 169)(78 170)(79 204)(80 205)(81 206)(82 207)(83 208)(84 183)(85 184)(86 185)(87 186)(88 187)(89 188)(90 189)(91 190)(92 191)(93 192)(94 193)(95 194)(96 195)(97 196)(98 197)(99 198)(100 199)(101 200)(102 201)(103 202)(104 203)(105 405)(106 406)(107 407)(108 408)(109 409)(110 410)(111 411)(112 412)(113 413)(114 414)(115 415)(116 416)(117 391)(118 392)(119 393)(120 394)(121 395)(122 396)(123 397)(124 398)(125 399)(126 400)(127 401)(128 402)(129 403)(130 404)(131 325)(132 326)(133 327)(134 328)(135 329)(136 330)(137 331)(138 332)(139 333)(140 334)(141 335)(142 336)(143 337)(144 338)(145 313)(146 314)(147 315)(148 316)(149 317)(150 318)(151 319)(152 320)(153 321)(154 322)(155 323)(156 324)(209 358)(210 359)(211 360)(212 361)(213 362)(214 363)(215 364)(216 339)(217 340)(218 341)(219 342)(220 343)(221 344)(222 345)(223 346)(224 347)(225 348)(226 349)(227 350)(228 351)(229 352)(230 353)(231 354)(232 355)(233 356)(234 357)(235 311)(236 312)(237 287)(238 288)(239 289)(240 290)(241 291)(242 292)(243 293)(244 294)(245 295)(246 296)(247 297)(248 298)(249 299)(250 300)(251 301)(252 302)(253 303)(254 304)(255 305)(256 306)(257 307)(258 308)(259 309)(260 310)
(1 71 391 194)(2 72 392 195)(3 73 393 196)(4 74 394 197)(5 75 395 198)(6 76 396 199)(7 77 397 200)(8 78 398 201)(9 53 399 202)(10 54 400 203)(11 55 401 204)(12 56 402 205)(13 57 403 206)(14 58 404 207)(15 59 405 208)(16 60 406 183)(17 61 407 184)(18 62 408 185)(19 63 409 186)(20 64 410 187)(21 65 411 188)(22 66 412 189)(23 67 413 190)(24 68 414 191)(25 69 415 192)(26 70 416 193)(27 356 133 256)(28 357 134 257)(29 358 135 258)(30 359 136 259)(31 360 137 260)(32 361 138 235)(33 362 139 236)(34 363 140 237)(35 364 141 238)(36 339 142 239)(37 340 143 240)(38 341 144 241)(39 342 145 242)(40 343 146 243)(41 344 147 244)(42 345 148 245)(43 346 149 246)(44 347 150 247)(45 348 151 248)(46 349 152 249)(47 350 153 250)(48 351 154 251)(49 352 155 252)(50 353 156 253)(51 354 131 254)(52 355 132 255)(79 381 173 127)(80 382 174 128)(81 383 175 129)(82 384 176 130)(83 385 177 105)(84 386 178 106)(85 387 179 107)(86 388 180 108)(87 389 181 109)(88 390 182 110)(89 365 157 111)(90 366 158 112)(91 367 159 113)(92 368 160 114)(93 369 161 115)(94 370 162 116)(95 371 163 117)(96 372 164 118)(97 373 165 119)(98 374 166 120)(99 375 167 121)(100 376 168 122)(101 377 169 123)(102 378 170 124)(103 379 171 125)(104 380 172 126)(209 329 308 277)(210 330 309 278)(211 331 310 279)(212 332 311 280)(213 333 312 281)(214 334 287 282)(215 335 288 283)(216 336 289 284)(217 337 290 285)(218 338 291 286)(219 313 292 261)(220 314 293 262)(221 315 294 263)(222 316 295 264)(223 317 296 265)(224 318 297 266)(225 319 298 267)(226 320 299 268)(227 321 300 269)(228 322 301 270)(229 323 302 271)(230 324 303 272)(231 325 304 273)(232 326 305 274)(233 327 306 275)(234 328 307 276)
(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)
(1 145 14 132)(2 144 15 131)(3 143 16 156)(4 142 17 155)(5 141 18 154)(6 140 19 153)(7 139 20 152)(8 138 21 151)(9 137 22 150)(10 136 23 149)(11 135 24 148)(12 134 25 147)(13 133 26 146)(27 416 40 403)(28 415 41 402)(29 414 42 401)(30 413 43 400)(31 412 44 399)(32 411 45 398)(33 410 46 397)(34 409 47 396)(35 408 48 395)(36 407 49 394)(37 406 50 393)(38 405 51 392)(39 404 52 391)(53 260 66 247)(54 259 67 246)(55 258 68 245)(56 257 69 244)(57 256 70 243)(58 255 71 242)(59 254 72 241)(60 253 73 240)(61 252 74 239)(62 251 75 238)(63 250 76 237)(64 249 77 236)(65 248 78 235)(79 209 92 222)(80 234 93 221)(81 233 94 220)(82 232 95 219)(83 231 96 218)(84 230 97 217)(85 229 98 216)(86 228 99 215)(87 227 100 214)(88 226 101 213)(89 225 102 212)(90 224 103 211)(91 223 104 210)(105 273 118 286)(106 272 119 285)(107 271 120 284)(108 270 121 283)(109 269 122 282)(110 268 123 281)(111 267 124 280)(112 266 125 279)(113 265 126 278)(114 264 127 277)(115 263 128 276)(116 262 129 275)(117 261 130 274)(157 298 170 311)(158 297 171 310)(159 296 172 309)(160 295 173 308)(161 294 174 307)(162 293 175 306)(163 292 176 305)(164 291 177 304)(165 290 178 303)(166 289 179 302)(167 288 180 301)(168 287 181 300)(169 312 182 299)(183 353 196 340)(184 352 197 339)(185 351 198 364)(186 350 199 363)(187 349 200 362)(188 348 201 361)(189 347 202 360)(190 346 203 359)(191 345 204 358)(192 344 205 357)(193 343 206 356)(194 342 207 355)(195 341 208 354)(313 384 326 371)(314 383 327 370)(315 382 328 369)(316 381 329 368)(317 380 330 367)(318 379 331 366)(319 378 332 365)(320 377 333 390)(321 376 334 389)(322 375 335 388)(323 374 336 387)(324 373 337 386)(325 372 338 385)
G:=sub<Sym(416)| (1,371)(2,372)(3,373)(4,374)(5,375)(6,376)(7,377)(8,378)(9,379)(10,380)(11,381)(12,382)(13,383)(14,384)(15,385)(16,386)(17,387)(18,388)(19,389)(20,390)(21,365)(22,366)(23,367)(24,368)(25,369)(26,370)(27,275)(28,276)(29,277)(30,278)(31,279)(32,280)(33,281)(34,282)(35,283)(36,284)(37,285)(38,286)(39,261)(40,262)(41,263)(42,264)(43,265)(44,266)(45,267)(46,268)(47,269)(48,270)(49,271)(50,272)(51,273)(52,274)(53,171)(54,172)(55,173)(56,174)(57,175)(58,176)(59,177)(60,178)(61,179)(62,180)(63,181)(64,182)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,204)(80,205)(81,206)(82,207)(83,208)(84,183)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,191)(93,192)(94,193)(95,194)(96,195)(97,196)(98,197)(99,198)(100,199)(101,200)(102,201)(103,202)(104,203)(105,405)(106,406)(107,407)(108,408)(109,409)(110,410)(111,411)(112,412)(113,413)(114,414)(115,415)(116,416)(117,391)(118,392)(119,393)(120,394)(121,395)(122,396)(123,397)(124,398)(125,399)(126,400)(127,401)(128,402)(129,403)(130,404)(131,325)(132,326)(133,327)(134,328)(135,329)(136,330)(137,331)(138,332)(139,333)(140,334)(141,335)(142,336)(143,337)(144,338)(145,313)(146,314)(147,315)(148,316)(149,317)(150,318)(151,319)(152,320)(153,321)(154,322)(155,323)(156,324)(209,358)(210,359)(211,360)(212,361)(213,362)(214,363)(215,364)(216,339)(217,340)(218,341)(219,342)(220,343)(221,344)(222,345)(223,346)(224,347)(225,348)(226,349)(227,350)(228,351)(229,352)(230,353)(231,354)(232,355)(233,356)(234,357)(235,311)(236,312)(237,287)(238,288)(239,289)(240,290)(241,291)(242,292)(243,293)(244,294)(245,295)(246,296)(247,297)(248,298)(249,299)(250,300)(251,301)(252,302)(253,303)(254,304)(255,305)(256,306)(257,307)(258,308)(259,309)(260,310), (1,71,391,194)(2,72,392,195)(3,73,393,196)(4,74,394,197)(5,75,395,198)(6,76,396,199)(7,77,397,200)(8,78,398,201)(9,53,399,202)(10,54,400,203)(11,55,401,204)(12,56,402,205)(13,57,403,206)(14,58,404,207)(15,59,405,208)(16,60,406,183)(17,61,407,184)(18,62,408,185)(19,63,409,186)(20,64,410,187)(21,65,411,188)(22,66,412,189)(23,67,413,190)(24,68,414,191)(25,69,415,192)(26,70,416,193)(27,356,133,256)(28,357,134,257)(29,358,135,258)(30,359,136,259)(31,360,137,260)(32,361,138,235)(33,362,139,236)(34,363,140,237)(35,364,141,238)(36,339,142,239)(37,340,143,240)(38,341,144,241)(39,342,145,242)(40,343,146,243)(41,344,147,244)(42,345,148,245)(43,346,149,246)(44,347,150,247)(45,348,151,248)(46,349,152,249)(47,350,153,250)(48,351,154,251)(49,352,155,252)(50,353,156,253)(51,354,131,254)(52,355,132,255)(79,381,173,127)(80,382,174,128)(81,383,175,129)(82,384,176,130)(83,385,177,105)(84,386,178,106)(85,387,179,107)(86,388,180,108)(87,389,181,109)(88,390,182,110)(89,365,157,111)(90,366,158,112)(91,367,159,113)(92,368,160,114)(93,369,161,115)(94,370,162,116)(95,371,163,117)(96,372,164,118)(97,373,165,119)(98,374,166,120)(99,375,167,121)(100,376,168,122)(101,377,169,123)(102,378,170,124)(103,379,171,125)(104,380,172,126)(209,329,308,277)(210,330,309,278)(211,331,310,279)(212,332,311,280)(213,333,312,281)(214,334,287,282)(215,335,288,283)(216,336,289,284)(217,337,290,285)(218,338,291,286)(219,313,292,261)(220,314,293,262)(221,315,294,263)(222,316,295,264)(223,317,296,265)(224,318,297,266)(225,319,298,267)(226,320,299,268)(227,321,300,269)(228,322,301,270)(229,323,302,271)(230,324,303,272)(231,325,304,273)(232,326,305,274)(233,327,306,275)(234,328,307,276), (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), (1,145,14,132)(2,144,15,131)(3,143,16,156)(4,142,17,155)(5,141,18,154)(6,140,19,153)(7,139,20,152)(8,138,21,151)(9,137,22,150)(10,136,23,149)(11,135,24,148)(12,134,25,147)(13,133,26,146)(27,416,40,403)(28,415,41,402)(29,414,42,401)(30,413,43,400)(31,412,44,399)(32,411,45,398)(33,410,46,397)(34,409,47,396)(35,408,48,395)(36,407,49,394)(37,406,50,393)(38,405,51,392)(39,404,52,391)(53,260,66,247)(54,259,67,246)(55,258,68,245)(56,257,69,244)(57,256,70,243)(58,255,71,242)(59,254,72,241)(60,253,73,240)(61,252,74,239)(62,251,75,238)(63,250,76,237)(64,249,77,236)(65,248,78,235)(79,209,92,222)(80,234,93,221)(81,233,94,220)(82,232,95,219)(83,231,96,218)(84,230,97,217)(85,229,98,216)(86,228,99,215)(87,227,100,214)(88,226,101,213)(89,225,102,212)(90,224,103,211)(91,223,104,210)(105,273,118,286)(106,272,119,285)(107,271,120,284)(108,270,121,283)(109,269,122,282)(110,268,123,281)(111,267,124,280)(112,266,125,279)(113,265,126,278)(114,264,127,277)(115,263,128,276)(116,262,129,275)(117,261,130,274)(157,298,170,311)(158,297,171,310)(159,296,172,309)(160,295,173,308)(161,294,174,307)(162,293,175,306)(163,292,176,305)(164,291,177,304)(165,290,178,303)(166,289,179,302)(167,288,180,301)(168,287,181,300)(169,312,182,299)(183,353,196,340)(184,352,197,339)(185,351,198,364)(186,350,199,363)(187,349,200,362)(188,348,201,361)(189,347,202,360)(190,346,203,359)(191,345,204,358)(192,344,205,357)(193,343,206,356)(194,342,207,355)(195,341,208,354)(313,384,326,371)(314,383,327,370)(315,382,328,369)(316,381,329,368)(317,380,330,367)(318,379,331,366)(319,378,332,365)(320,377,333,390)(321,376,334,389)(322,375,335,388)(323,374,336,387)(324,373,337,386)(325,372,338,385)>;
G:=Group( (1,371)(2,372)(3,373)(4,374)(5,375)(6,376)(7,377)(8,378)(9,379)(10,380)(11,381)(12,382)(13,383)(14,384)(15,385)(16,386)(17,387)(18,388)(19,389)(20,390)(21,365)(22,366)(23,367)(24,368)(25,369)(26,370)(27,275)(28,276)(29,277)(30,278)(31,279)(32,280)(33,281)(34,282)(35,283)(36,284)(37,285)(38,286)(39,261)(40,262)(41,263)(42,264)(43,265)(44,266)(45,267)(46,268)(47,269)(48,270)(49,271)(50,272)(51,273)(52,274)(53,171)(54,172)(55,173)(56,174)(57,175)(58,176)(59,177)(60,178)(61,179)(62,180)(63,181)(64,182)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,204)(80,205)(81,206)(82,207)(83,208)(84,183)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,191)(93,192)(94,193)(95,194)(96,195)(97,196)(98,197)(99,198)(100,199)(101,200)(102,201)(103,202)(104,203)(105,405)(106,406)(107,407)(108,408)(109,409)(110,410)(111,411)(112,412)(113,413)(114,414)(115,415)(116,416)(117,391)(118,392)(119,393)(120,394)(121,395)(122,396)(123,397)(124,398)(125,399)(126,400)(127,401)(128,402)(129,403)(130,404)(131,325)(132,326)(133,327)(134,328)(135,329)(136,330)(137,331)(138,332)(139,333)(140,334)(141,335)(142,336)(143,337)(144,338)(145,313)(146,314)(147,315)(148,316)(149,317)(150,318)(151,319)(152,320)(153,321)(154,322)(155,323)(156,324)(209,358)(210,359)(211,360)(212,361)(213,362)(214,363)(215,364)(216,339)(217,340)(218,341)(219,342)(220,343)(221,344)(222,345)(223,346)(224,347)(225,348)(226,349)(227,350)(228,351)(229,352)(230,353)(231,354)(232,355)(233,356)(234,357)(235,311)(236,312)(237,287)(238,288)(239,289)(240,290)(241,291)(242,292)(243,293)(244,294)(245,295)(246,296)(247,297)(248,298)(249,299)(250,300)(251,301)(252,302)(253,303)(254,304)(255,305)(256,306)(257,307)(258,308)(259,309)(260,310), (1,71,391,194)(2,72,392,195)(3,73,393,196)(4,74,394,197)(5,75,395,198)(6,76,396,199)(7,77,397,200)(8,78,398,201)(9,53,399,202)(10,54,400,203)(11,55,401,204)(12,56,402,205)(13,57,403,206)(14,58,404,207)(15,59,405,208)(16,60,406,183)(17,61,407,184)(18,62,408,185)(19,63,409,186)(20,64,410,187)(21,65,411,188)(22,66,412,189)(23,67,413,190)(24,68,414,191)(25,69,415,192)(26,70,416,193)(27,356,133,256)(28,357,134,257)(29,358,135,258)(30,359,136,259)(31,360,137,260)(32,361,138,235)(33,362,139,236)(34,363,140,237)(35,364,141,238)(36,339,142,239)(37,340,143,240)(38,341,144,241)(39,342,145,242)(40,343,146,243)(41,344,147,244)(42,345,148,245)(43,346,149,246)(44,347,150,247)(45,348,151,248)(46,349,152,249)(47,350,153,250)(48,351,154,251)(49,352,155,252)(50,353,156,253)(51,354,131,254)(52,355,132,255)(79,381,173,127)(80,382,174,128)(81,383,175,129)(82,384,176,130)(83,385,177,105)(84,386,178,106)(85,387,179,107)(86,388,180,108)(87,389,181,109)(88,390,182,110)(89,365,157,111)(90,366,158,112)(91,367,159,113)(92,368,160,114)(93,369,161,115)(94,370,162,116)(95,371,163,117)(96,372,164,118)(97,373,165,119)(98,374,166,120)(99,375,167,121)(100,376,168,122)(101,377,169,123)(102,378,170,124)(103,379,171,125)(104,380,172,126)(209,329,308,277)(210,330,309,278)(211,331,310,279)(212,332,311,280)(213,333,312,281)(214,334,287,282)(215,335,288,283)(216,336,289,284)(217,337,290,285)(218,338,291,286)(219,313,292,261)(220,314,293,262)(221,315,294,263)(222,316,295,264)(223,317,296,265)(224,318,297,266)(225,319,298,267)(226,320,299,268)(227,321,300,269)(228,322,301,270)(229,323,302,271)(230,324,303,272)(231,325,304,273)(232,326,305,274)(233,327,306,275)(234,328,307,276), (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), (1,145,14,132)(2,144,15,131)(3,143,16,156)(4,142,17,155)(5,141,18,154)(6,140,19,153)(7,139,20,152)(8,138,21,151)(9,137,22,150)(10,136,23,149)(11,135,24,148)(12,134,25,147)(13,133,26,146)(27,416,40,403)(28,415,41,402)(29,414,42,401)(30,413,43,400)(31,412,44,399)(32,411,45,398)(33,410,46,397)(34,409,47,396)(35,408,48,395)(36,407,49,394)(37,406,50,393)(38,405,51,392)(39,404,52,391)(53,260,66,247)(54,259,67,246)(55,258,68,245)(56,257,69,244)(57,256,70,243)(58,255,71,242)(59,254,72,241)(60,253,73,240)(61,252,74,239)(62,251,75,238)(63,250,76,237)(64,249,77,236)(65,248,78,235)(79,209,92,222)(80,234,93,221)(81,233,94,220)(82,232,95,219)(83,231,96,218)(84,230,97,217)(85,229,98,216)(86,228,99,215)(87,227,100,214)(88,226,101,213)(89,225,102,212)(90,224,103,211)(91,223,104,210)(105,273,118,286)(106,272,119,285)(107,271,120,284)(108,270,121,283)(109,269,122,282)(110,268,123,281)(111,267,124,280)(112,266,125,279)(113,265,126,278)(114,264,127,277)(115,263,128,276)(116,262,129,275)(117,261,130,274)(157,298,170,311)(158,297,171,310)(159,296,172,309)(160,295,173,308)(161,294,174,307)(162,293,175,306)(163,292,176,305)(164,291,177,304)(165,290,178,303)(166,289,179,302)(167,288,180,301)(168,287,181,300)(169,312,182,299)(183,353,196,340)(184,352,197,339)(185,351,198,364)(186,350,199,363)(187,349,200,362)(188,348,201,361)(189,347,202,360)(190,346,203,359)(191,345,204,358)(192,344,205,357)(193,343,206,356)(194,342,207,355)(195,341,208,354)(313,384,326,371)(314,383,327,370)(315,382,328,369)(316,381,329,368)(317,380,330,367)(318,379,331,366)(319,378,332,365)(320,377,333,390)(321,376,334,389)(322,375,335,388)(323,374,336,387)(324,373,337,386)(325,372,338,385) );
G=PermutationGroup([[(1,371),(2,372),(3,373),(4,374),(5,375),(6,376),(7,377),(8,378),(9,379),(10,380),(11,381),(12,382),(13,383),(14,384),(15,385),(16,386),(17,387),(18,388),(19,389),(20,390),(21,365),(22,366),(23,367),(24,368),(25,369),(26,370),(27,275),(28,276),(29,277),(30,278),(31,279),(32,280),(33,281),(34,282),(35,283),(36,284),(37,285),(38,286),(39,261),(40,262),(41,263),(42,264),(43,265),(44,266),(45,267),(46,268),(47,269),(48,270),(49,271),(50,272),(51,273),(52,274),(53,171),(54,172),(55,173),(56,174),(57,175),(58,176),(59,177),(60,178),(61,179),(62,180),(63,181),(64,182),(65,157),(66,158),(67,159),(68,160),(69,161),(70,162),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,169),(78,170),(79,204),(80,205),(81,206),(82,207),(83,208),(84,183),(85,184),(86,185),(87,186),(88,187),(89,188),(90,189),(91,190),(92,191),(93,192),(94,193),(95,194),(96,195),(97,196),(98,197),(99,198),(100,199),(101,200),(102,201),(103,202),(104,203),(105,405),(106,406),(107,407),(108,408),(109,409),(110,410),(111,411),(112,412),(113,413),(114,414),(115,415),(116,416),(117,391),(118,392),(119,393),(120,394),(121,395),(122,396),(123,397),(124,398),(125,399),(126,400),(127,401),(128,402),(129,403),(130,404),(131,325),(132,326),(133,327),(134,328),(135,329),(136,330),(137,331),(138,332),(139,333),(140,334),(141,335),(142,336),(143,337),(144,338),(145,313),(146,314),(147,315),(148,316),(149,317),(150,318),(151,319),(152,320),(153,321),(154,322),(155,323),(156,324),(209,358),(210,359),(211,360),(212,361),(213,362),(214,363),(215,364),(216,339),(217,340),(218,341),(219,342),(220,343),(221,344),(222,345),(223,346),(224,347),(225,348),(226,349),(227,350),(228,351),(229,352),(230,353),(231,354),(232,355),(233,356),(234,357),(235,311),(236,312),(237,287),(238,288),(239,289),(240,290),(241,291),(242,292),(243,293),(244,294),(245,295),(246,296),(247,297),(248,298),(249,299),(250,300),(251,301),(252,302),(253,303),(254,304),(255,305),(256,306),(257,307),(258,308),(259,309),(260,310)], [(1,71,391,194),(2,72,392,195),(3,73,393,196),(4,74,394,197),(5,75,395,198),(6,76,396,199),(7,77,397,200),(8,78,398,201),(9,53,399,202),(10,54,400,203),(11,55,401,204),(12,56,402,205),(13,57,403,206),(14,58,404,207),(15,59,405,208),(16,60,406,183),(17,61,407,184),(18,62,408,185),(19,63,409,186),(20,64,410,187),(21,65,411,188),(22,66,412,189),(23,67,413,190),(24,68,414,191),(25,69,415,192),(26,70,416,193),(27,356,133,256),(28,357,134,257),(29,358,135,258),(30,359,136,259),(31,360,137,260),(32,361,138,235),(33,362,139,236),(34,363,140,237),(35,364,141,238),(36,339,142,239),(37,340,143,240),(38,341,144,241),(39,342,145,242),(40,343,146,243),(41,344,147,244),(42,345,148,245),(43,346,149,246),(44,347,150,247),(45,348,151,248),(46,349,152,249),(47,350,153,250),(48,351,154,251),(49,352,155,252),(50,353,156,253),(51,354,131,254),(52,355,132,255),(79,381,173,127),(80,382,174,128),(81,383,175,129),(82,384,176,130),(83,385,177,105),(84,386,178,106),(85,387,179,107),(86,388,180,108),(87,389,181,109),(88,390,182,110),(89,365,157,111),(90,366,158,112),(91,367,159,113),(92,368,160,114),(93,369,161,115),(94,370,162,116),(95,371,163,117),(96,372,164,118),(97,373,165,119),(98,374,166,120),(99,375,167,121),(100,376,168,122),(101,377,169,123),(102,378,170,124),(103,379,171,125),(104,380,172,126),(209,329,308,277),(210,330,309,278),(211,331,310,279),(212,332,311,280),(213,333,312,281),(214,334,287,282),(215,335,288,283),(216,336,289,284),(217,337,290,285),(218,338,291,286),(219,313,292,261),(220,314,293,262),(221,315,294,263),(222,316,295,264),(223,317,296,265),(224,318,297,266),(225,319,298,267),(226,320,299,268),(227,321,300,269),(228,322,301,270),(229,323,302,271),(230,324,303,272),(231,325,304,273),(232,326,305,274),(233,327,306,275),(234,328,307,276)], [(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)], [(1,145,14,132),(2,144,15,131),(3,143,16,156),(4,142,17,155),(5,141,18,154),(6,140,19,153),(7,139,20,152),(8,138,21,151),(9,137,22,150),(10,136,23,149),(11,135,24,148),(12,134,25,147),(13,133,26,146),(27,416,40,403),(28,415,41,402),(29,414,42,401),(30,413,43,400),(31,412,44,399),(32,411,45,398),(33,410,46,397),(34,409,47,396),(35,408,48,395),(36,407,49,394),(37,406,50,393),(38,405,51,392),(39,404,52,391),(53,260,66,247),(54,259,67,246),(55,258,68,245),(56,257,69,244),(57,256,70,243),(58,255,71,242),(59,254,72,241),(60,253,73,240),(61,252,74,239),(62,251,75,238),(63,250,76,237),(64,249,77,236),(65,248,78,235),(79,209,92,222),(80,234,93,221),(81,233,94,220),(82,232,95,219),(83,231,96,218),(84,230,97,217),(85,229,98,216),(86,228,99,215),(87,227,100,214),(88,226,101,213),(89,225,102,212),(90,224,103,211),(91,223,104,210),(105,273,118,286),(106,272,119,285),(107,271,120,284),(108,270,121,283),(109,269,122,282),(110,268,123,281),(111,267,124,280),(112,266,125,279),(113,265,126,278),(114,264,127,277),(115,263,128,276),(116,262,129,275),(117,261,130,274),(157,298,170,311),(158,297,171,310),(159,296,172,309),(160,295,173,308),(161,294,174,307),(162,293,175,306),(163,292,176,305),(164,291,177,304),(165,290,178,303),(166,289,179,302),(167,288,180,301),(168,287,181,300),(169,312,182,299),(183,353,196,340),(184,352,197,339),(185,351,198,364),(186,350,199,363),(187,349,200,362),(188,348,201,361),(189,347,202,360),(190,346,203,359),(191,345,204,358),(192,344,205,357),(193,343,206,356),(194,342,207,355),(195,341,208,354),(313,384,326,371),(314,383,327,370),(315,382,328,369),(316,381,329,368),(317,380,330,367),(318,379,331,366),(319,378,332,365),(320,377,333,390),(321,376,334,389),(322,375,335,388),(323,374,336,387),(324,373,337,386),(325,372,338,385)]])
128 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4X | 13A | ··· | 13F | 26A | ··· | 26AP | 52A | ··· | 52AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D13 | Dic13 | D26 | D26 | C4×D13 |
| kernel | C2×C4×Dic13 | C4×Dic13 | C22×Dic13 | C22×C52 | C2×Dic13 | C2×C52 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 16 | 8 | 6 | 24 | 12 | 6 | 48 |
Matrix representation of C2×C4×Dic13 ►in GL4(𝔽53) generated by
| 1 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 52 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 52 | 11 |
| 30 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 16 | 14 |
| 0 | 0 | 31 | 37 |
G:=sub<GL(4,GF(53))| [1,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,52,0,0,0,0,23,0,0,0,0,23],[52,0,0,0,0,1,0,0,0,0,0,52,0,0,1,11],[30,0,0,0,0,52,0,0,0,0,16,31,0,0,14,37] >;
C2×C4×Dic13 in GAP, Magma, Sage, TeX
C_2\times C_4\times {\rm Dic}_{13} % in TeX
G:=Group("C2xC4xDic13"); // GroupNames label
G:=SmallGroup(416,143);
// by ID
G=gap.SmallGroup(416,143);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,86,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^26=1,d^2=c^13,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations