direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Dic26, C52⋊3Q8, C42.3D13, C13⋊2(C4×Q8), (C4×C52).5C2, C4.9(C4×D13), C26.1(C2×Q8), C52.40(C2×C4), (C2×C4).72D26, C26.1(C4○D4), (C2×C26).9C23, C52⋊3C4.13C2, C2.1(C2×Dic26), (C2×C52).84C22, C26.14(C22×C4), Dic13.3(C2×C4), (C4×Dic13).9C2, C26.D4.7C2, C2.1(D52⋊5C2), (C2×Dic26).11C2, C22.8(C22×D13), (C2×Dic13).26C22, C2.4(C2×C4×D13), SmallGroup(416,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Dic26
G = < a,b,c | a4=b52=1, c2=b26, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 352 in 70 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C13, C42, C42, C4⋊C4, C2×Q8, C26, C4×Q8, Dic13, Dic13, C52, C52, C2×C26, Dic26, C2×Dic13, C2×C52, C4×Dic13, C26.D4, C52⋊3C4, C4×C52, C2×Dic26, C4×Dic26
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, D13, C4×Q8, D26, Dic26, C4×D13, C22×D13, C2×Dic26, C2×C4×D13, D52⋊5C2, C4×Dic26
►Regular action on 416 points
Generators in S416
(1 219 308 116)(2 220 309 117)(3 221 310 118)(4 222 311 119)(5 223 312 120)(6 224 261 121)(7 225 262 122)(8 226 263 123)(9 227 264 124)(10 228 265 125)(11 229 266 126)(12 230 267 127)(13 231 268 128)(14 232 269 129)(15 233 270 130)(16 234 271 131)(17 235 272 132)(18 236 273 133)(19 237 274 134)(20 238 275 135)(21 239 276 136)(22 240 277 137)(23 241 278 138)(24 242 279 139)(25 243 280 140)(26 244 281 141)(27 245 282 142)(28 246 283 143)(29 247 284 144)(30 248 285 145)(31 249 286 146)(32 250 287 147)(33 251 288 148)(34 252 289 149)(35 253 290 150)(36 254 291 151)(37 255 292 152)(38 256 293 153)(39 257 294 154)(40 258 295 155)(41 259 296 156)(42 260 297 105)(43 209 298 106)(44 210 299 107)(45 211 300 108)(46 212 301 109)(47 213 302 110)(48 214 303 111)(49 215 304 112)(50 216 305 113)(51 217 306 114)(52 218 307 115)(53 188 349 373)(54 189 350 374)(55 190 351 375)(56 191 352 376)(57 192 353 377)(58 193 354 378)(59 194 355 379)(60 195 356 380)(61 196 357 381)(62 197 358 382)(63 198 359 383)(64 199 360 384)(65 200 361 385)(66 201 362 386)(67 202 363 387)(68 203 364 388)(69 204 313 389)(70 205 314 390)(71 206 315 391)(72 207 316 392)(73 208 317 393)(74 157 318 394)(75 158 319 395)(76 159 320 396)(77 160 321 397)(78 161 322 398)(79 162 323 399)(80 163 324 400)(81 164 325 401)(82 165 326 402)(83 166 327 403)(84 167 328 404)(85 168 329 405)(86 169 330 406)(87 170 331 407)(88 171 332 408)(89 172 333 409)(90 173 334 410)(91 174 335 411)(92 175 336 412)(93 176 337 413)(94 177 338 414)(95 178 339 415)(96 179 340 416)(97 180 341 365)(98 181 342 366)(99 182 343 367)(100 183 344 368)(101 184 345 369)(102 185 346 370)(103 186 347 371)(104 187 348 372)
(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)
(1 168 27 194)(2 167 28 193)(3 166 29 192)(4 165 30 191)(5 164 31 190)(6 163 32 189)(7 162 33 188)(8 161 34 187)(9 160 35 186)(10 159 36 185)(11 158 37 184)(12 157 38 183)(13 208 39 182)(14 207 40 181)(15 206 41 180)(16 205 42 179)(17 204 43 178)(18 203 44 177)(19 202 45 176)(20 201 46 175)(21 200 47 174)(22 199 48 173)(23 198 49 172)(24 197 50 171)(25 196 51 170)(26 195 52 169)(53 122 79 148)(54 121 80 147)(55 120 81 146)(56 119 82 145)(57 118 83 144)(58 117 84 143)(59 116 85 142)(60 115 86 141)(61 114 87 140)(62 113 88 139)(63 112 89 138)(64 111 90 137)(65 110 91 136)(66 109 92 135)(67 108 93 134)(68 107 94 133)(69 106 95 132)(70 105 96 131)(71 156 97 130)(72 155 98 129)(73 154 99 128)(74 153 100 127)(75 152 101 126)(76 151 102 125)(77 150 103 124)(78 149 104 123)(209 339 235 313)(210 338 236 364)(211 337 237 363)(212 336 238 362)(213 335 239 361)(214 334 240 360)(215 333 241 359)(216 332 242 358)(217 331 243 357)(218 330 244 356)(219 329 245 355)(220 328 246 354)(221 327 247 353)(222 326 248 352)(223 325 249 351)(224 324 250 350)(225 323 251 349)(226 322 252 348)(227 321 253 347)(228 320 254 346)(229 319 255 345)(230 318 256 344)(231 317 257 343)(232 316 258 342)(233 315 259 341)(234 314 260 340)(261 400 287 374)(262 399 288 373)(263 398 289 372)(264 397 290 371)(265 396 291 370)(266 395 292 369)(267 394 293 368)(268 393 294 367)(269 392 295 366)(270 391 296 365)(271 390 297 416)(272 389 298 415)(273 388 299 414)(274 387 300 413)(275 386 301 412)(276 385 302 411)(277 384 303 410)(278 383 304 409)(279 382 305 408)(280 381 306 407)(281 380 307 406)(282 379 308 405)(283 378 309 404)(284 377 310 403)(285 376 311 402)(286 375 312 401)
G:=sub<Sym(416)| (1,219,308,116)(2,220,309,117)(3,221,310,118)(4,222,311,119)(5,223,312,120)(6,224,261,121)(7,225,262,122)(8,226,263,123)(9,227,264,124)(10,228,265,125)(11,229,266,126)(12,230,267,127)(13,231,268,128)(14,232,269,129)(15,233,270,130)(16,234,271,131)(17,235,272,132)(18,236,273,133)(19,237,274,134)(20,238,275,135)(21,239,276,136)(22,240,277,137)(23,241,278,138)(24,242,279,139)(25,243,280,140)(26,244,281,141)(27,245,282,142)(28,246,283,143)(29,247,284,144)(30,248,285,145)(31,249,286,146)(32,250,287,147)(33,251,288,148)(34,252,289,149)(35,253,290,150)(36,254,291,151)(37,255,292,152)(38,256,293,153)(39,257,294,154)(40,258,295,155)(41,259,296,156)(42,260,297,105)(43,209,298,106)(44,210,299,107)(45,211,300,108)(46,212,301,109)(47,213,302,110)(48,214,303,111)(49,215,304,112)(50,216,305,113)(51,217,306,114)(52,218,307,115)(53,188,349,373)(54,189,350,374)(55,190,351,375)(56,191,352,376)(57,192,353,377)(58,193,354,378)(59,194,355,379)(60,195,356,380)(61,196,357,381)(62,197,358,382)(63,198,359,383)(64,199,360,384)(65,200,361,385)(66,201,362,386)(67,202,363,387)(68,203,364,388)(69,204,313,389)(70,205,314,390)(71,206,315,391)(72,207,316,392)(73,208,317,393)(74,157,318,394)(75,158,319,395)(76,159,320,396)(77,160,321,397)(78,161,322,398)(79,162,323,399)(80,163,324,400)(81,164,325,401)(82,165,326,402)(83,166,327,403)(84,167,328,404)(85,168,329,405)(86,169,330,406)(87,170,331,407)(88,171,332,408)(89,172,333,409)(90,173,334,410)(91,174,335,411)(92,175,336,412)(93,176,337,413)(94,177,338,414)(95,178,339,415)(96,179,340,416)(97,180,341,365)(98,181,342,366)(99,182,343,367)(100,183,344,368)(101,184,345,369)(102,185,346,370)(103,186,347,371)(104,187,348,372), (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), (1,168,27,194)(2,167,28,193)(3,166,29,192)(4,165,30,191)(5,164,31,190)(6,163,32,189)(7,162,33,188)(8,161,34,187)(9,160,35,186)(10,159,36,185)(11,158,37,184)(12,157,38,183)(13,208,39,182)(14,207,40,181)(15,206,41,180)(16,205,42,179)(17,204,43,178)(18,203,44,177)(19,202,45,176)(20,201,46,175)(21,200,47,174)(22,199,48,173)(23,198,49,172)(24,197,50,171)(25,196,51,170)(26,195,52,169)(53,122,79,148)(54,121,80,147)(55,120,81,146)(56,119,82,145)(57,118,83,144)(58,117,84,143)(59,116,85,142)(60,115,86,141)(61,114,87,140)(62,113,88,139)(63,112,89,138)(64,111,90,137)(65,110,91,136)(66,109,92,135)(67,108,93,134)(68,107,94,133)(69,106,95,132)(70,105,96,131)(71,156,97,130)(72,155,98,129)(73,154,99,128)(74,153,100,127)(75,152,101,126)(76,151,102,125)(77,150,103,124)(78,149,104,123)(209,339,235,313)(210,338,236,364)(211,337,237,363)(212,336,238,362)(213,335,239,361)(214,334,240,360)(215,333,241,359)(216,332,242,358)(217,331,243,357)(218,330,244,356)(219,329,245,355)(220,328,246,354)(221,327,247,353)(222,326,248,352)(223,325,249,351)(224,324,250,350)(225,323,251,349)(226,322,252,348)(227,321,253,347)(228,320,254,346)(229,319,255,345)(230,318,256,344)(231,317,257,343)(232,316,258,342)(233,315,259,341)(234,314,260,340)(261,400,287,374)(262,399,288,373)(263,398,289,372)(264,397,290,371)(265,396,291,370)(266,395,292,369)(267,394,293,368)(268,393,294,367)(269,392,295,366)(270,391,296,365)(271,390,297,416)(272,389,298,415)(273,388,299,414)(274,387,300,413)(275,386,301,412)(276,385,302,411)(277,384,303,410)(278,383,304,409)(279,382,305,408)(280,381,306,407)(281,380,307,406)(282,379,308,405)(283,378,309,404)(284,377,310,403)(285,376,311,402)(286,375,312,401)>;
G:=Group( (1,219,308,116)(2,220,309,117)(3,221,310,118)(4,222,311,119)(5,223,312,120)(6,224,261,121)(7,225,262,122)(8,226,263,123)(9,227,264,124)(10,228,265,125)(11,229,266,126)(12,230,267,127)(13,231,268,128)(14,232,269,129)(15,233,270,130)(16,234,271,131)(17,235,272,132)(18,236,273,133)(19,237,274,134)(20,238,275,135)(21,239,276,136)(22,240,277,137)(23,241,278,138)(24,242,279,139)(25,243,280,140)(26,244,281,141)(27,245,282,142)(28,246,283,143)(29,247,284,144)(30,248,285,145)(31,249,286,146)(32,250,287,147)(33,251,288,148)(34,252,289,149)(35,253,290,150)(36,254,291,151)(37,255,292,152)(38,256,293,153)(39,257,294,154)(40,258,295,155)(41,259,296,156)(42,260,297,105)(43,209,298,106)(44,210,299,107)(45,211,300,108)(46,212,301,109)(47,213,302,110)(48,214,303,111)(49,215,304,112)(50,216,305,113)(51,217,306,114)(52,218,307,115)(53,188,349,373)(54,189,350,374)(55,190,351,375)(56,191,352,376)(57,192,353,377)(58,193,354,378)(59,194,355,379)(60,195,356,380)(61,196,357,381)(62,197,358,382)(63,198,359,383)(64,199,360,384)(65,200,361,385)(66,201,362,386)(67,202,363,387)(68,203,364,388)(69,204,313,389)(70,205,314,390)(71,206,315,391)(72,207,316,392)(73,208,317,393)(74,157,318,394)(75,158,319,395)(76,159,320,396)(77,160,321,397)(78,161,322,398)(79,162,323,399)(80,163,324,400)(81,164,325,401)(82,165,326,402)(83,166,327,403)(84,167,328,404)(85,168,329,405)(86,169,330,406)(87,170,331,407)(88,171,332,408)(89,172,333,409)(90,173,334,410)(91,174,335,411)(92,175,336,412)(93,176,337,413)(94,177,338,414)(95,178,339,415)(96,179,340,416)(97,180,341,365)(98,181,342,366)(99,182,343,367)(100,183,344,368)(101,184,345,369)(102,185,346,370)(103,186,347,371)(104,187,348,372), (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), (1,168,27,194)(2,167,28,193)(3,166,29,192)(4,165,30,191)(5,164,31,190)(6,163,32,189)(7,162,33,188)(8,161,34,187)(9,160,35,186)(10,159,36,185)(11,158,37,184)(12,157,38,183)(13,208,39,182)(14,207,40,181)(15,206,41,180)(16,205,42,179)(17,204,43,178)(18,203,44,177)(19,202,45,176)(20,201,46,175)(21,200,47,174)(22,199,48,173)(23,198,49,172)(24,197,50,171)(25,196,51,170)(26,195,52,169)(53,122,79,148)(54,121,80,147)(55,120,81,146)(56,119,82,145)(57,118,83,144)(58,117,84,143)(59,116,85,142)(60,115,86,141)(61,114,87,140)(62,113,88,139)(63,112,89,138)(64,111,90,137)(65,110,91,136)(66,109,92,135)(67,108,93,134)(68,107,94,133)(69,106,95,132)(70,105,96,131)(71,156,97,130)(72,155,98,129)(73,154,99,128)(74,153,100,127)(75,152,101,126)(76,151,102,125)(77,150,103,124)(78,149,104,123)(209,339,235,313)(210,338,236,364)(211,337,237,363)(212,336,238,362)(213,335,239,361)(214,334,240,360)(215,333,241,359)(216,332,242,358)(217,331,243,357)(218,330,244,356)(219,329,245,355)(220,328,246,354)(221,327,247,353)(222,326,248,352)(223,325,249,351)(224,324,250,350)(225,323,251,349)(226,322,252,348)(227,321,253,347)(228,320,254,346)(229,319,255,345)(230,318,256,344)(231,317,257,343)(232,316,258,342)(233,315,259,341)(234,314,260,340)(261,400,287,374)(262,399,288,373)(263,398,289,372)(264,397,290,371)(265,396,291,370)(266,395,292,369)(267,394,293,368)(268,393,294,367)(269,392,295,366)(270,391,296,365)(271,390,297,416)(272,389,298,415)(273,388,299,414)(274,387,300,413)(275,386,301,412)(276,385,302,411)(277,384,303,410)(278,383,304,409)(279,382,305,408)(280,381,306,407)(281,380,307,406)(282,379,308,405)(283,378,309,404)(284,377,310,403)(285,376,311,402)(286,375,312,401) );
G=PermutationGroup([[(1,219,308,116),(2,220,309,117),(3,221,310,118),(4,222,311,119),(5,223,312,120),(6,224,261,121),(7,225,262,122),(8,226,263,123),(9,227,264,124),(10,228,265,125),(11,229,266,126),(12,230,267,127),(13,231,268,128),(14,232,269,129),(15,233,270,130),(16,234,271,131),(17,235,272,132),(18,236,273,133),(19,237,274,134),(20,238,275,135),(21,239,276,136),(22,240,277,137),(23,241,278,138),(24,242,279,139),(25,243,280,140),(26,244,281,141),(27,245,282,142),(28,246,283,143),(29,247,284,144),(30,248,285,145),(31,249,286,146),(32,250,287,147),(33,251,288,148),(34,252,289,149),(35,253,290,150),(36,254,291,151),(37,255,292,152),(38,256,293,153),(39,257,294,154),(40,258,295,155),(41,259,296,156),(42,260,297,105),(43,209,298,106),(44,210,299,107),(45,211,300,108),(46,212,301,109),(47,213,302,110),(48,214,303,111),(49,215,304,112),(50,216,305,113),(51,217,306,114),(52,218,307,115),(53,188,349,373),(54,189,350,374),(55,190,351,375),(56,191,352,376),(57,192,353,377),(58,193,354,378),(59,194,355,379),(60,195,356,380),(61,196,357,381),(62,197,358,382),(63,198,359,383),(64,199,360,384),(65,200,361,385),(66,201,362,386),(67,202,363,387),(68,203,364,388),(69,204,313,389),(70,205,314,390),(71,206,315,391),(72,207,316,392),(73,208,317,393),(74,157,318,394),(75,158,319,395),(76,159,320,396),(77,160,321,397),(78,161,322,398),(79,162,323,399),(80,163,324,400),(81,164,325,401),(82,165,326,402),(83,166,327,403),(84,167,328,404),(85,168,329,405),(86,169,330,406),(87,170,331,407),(88,171,332,408),(89,172,333,409),(90,173,334,410),(91,174,335,411),(92,175,336,412),(93,176,337,413),(94,177,338,414),(95,178,339,415),(96,179,340,416),(97,180,341,365),(98,181,342,366),(99,182,343,367),(100,183,344,368),(101,184,345,369),(102,185,346,370),(103,186,347,371),(104,187,348,372)], [(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)], [(1,168,27,194),(2,167,28,193),(3,166,29,192),(4,165,30,191),(5,164,31,190),(6,163,32,189),(7,162,33,188),(8,161,34,187),(9,160,35,186),(10,159,36,185),(11,158,37,184),(12,157,38,183),(13,208,39,182),(14,207,40,181),(15,206,41,180),(16,205,42,179),(17,204,43,178),(18,203,44,177),(19,202,45,176),(20,201,46,175),(21,200,47,174),(22,199,48,173),(23,198,49,172),(24,197,50,171),(25,196,51,170),(26,195,52,169),(53,122,79,148),(54,121,80,147),(55,120,81,146),(56,119,82,145),(57,118,83,144),(58,117,84,143),(59,116,85,142),(60,115,86,141),(61,114,87,140),(62,113,88,139),(63,112,89,138),(64,111,90,137),(65,110,91,136),(66,109,92,135),(67,108,93,134),(68,107,94,133),(69,106,95,132),(70,105,96,131),(71,156,97,130),(72,155,98,129),(73,154,99,128),(74,153,100,127),(75,152,101,126),(76,151,102,125),(77,150,103,124),(78,149,104,123),(209,339,235,313),(210,338,236,364),(211,337,237,363),(212,336,238,362),(213,335,239,361),(214,334,240,360),(215,333,241,359),(216,332,242,358),(217,331,243,357),(218,330,244,356),(219,329,245,355),(220,328,246,354),(221,327,247,353),(222,326,248,352),(223,325,249,351),(224,324,250,350),(225,323,251,349),(226,322,252,348),(227,321,253,347),(228,320,254,346),(229,319,255,345),(230,318,256,344),(231,317,257,343),(232,316,258,342),(233,315,259,341),(234,314,260,340),(261,400,287,374),(262,399,288,373),(263,398,289,372),(264,397,290,371),(265,396,291,370),(266,395,292,369),(267,394,293,368),(268,393,294,367),(269,392,295,366),(270,391,296,365),(271,390,297,416),(272,389,298,415),(273,388,299,414),(274,387,300,413),(275,386,301,412),(276,385,302,411),(277,384,303,410),(278,383,304,409),(279,382,305,408),(280,381,306,407),(281,380,307,406),(282,379,308,405),(283,378,309,404),(284,377,310,403),(285,376,311,402),(286,375,312,401)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | C4○D4 | D13 | D26 | Dic26 | C4×D13 | D52⋊5C2 |
| kernel | C4×Dic26 | C4×Dic13 | C26.D4 | C52⋊3C4 | C4×C52 | C2×Dic26 | Dic26 | C52 | C26 | C42 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 2 | 2 | 6 | 18 | 24 | 24 | 24 |
Matrix representation of C4×Dic26 ►in GL3(𝔽53) generated by
| 30 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 30 |
| 1 | 0 | 0 |
| 0 | 37 | 19 |
| 0 | 34 | 49 |
| 1 | 0 | 0 |
| 0 | 48 | 30 |
| 0 | 38 | 5 |
G:=sub<GL(3,GF(53))| [30,0,0,0,30,0,0,0,30],[1,0,0,0,37,34,0,19,49],[1,0,0,0,48,38,0,30,5] >;
C4×Dic26 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{26} % in TeX
G:=Group("C4xDic26"); // GroupNames label
G:=SmallGroup(416,89);
// by ID
G=gap.SmallGroup(416,89);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,217,103,50,13829]);
// Polycyclic
G:=Group<a,b,c|a^4=b^52=1,c^2=b^26,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations