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## G = C4.Dic26order 416 = 25·13

### 3rd non-split extension by C4 of Dic26 acting via Dic26/Dic13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C4.Dic26
 Chief series C1 — C13 — C26 — C2×C26 — C2×Dic13 — C4×Dic13 — C4.Dic26
 Lower central C13 — C2×C26 — C4.Dic26
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4.Dic26
G = < a,b,c | a4=b52=1, c2=b26, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Subgroups: 296 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, C2×C4, C13, C42, C4⋊C4, C4⋊C4, C26, C42.C2, Dic13, C52, C52, C2×C26, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C4×Dic13, C26.D4, C523C4, C523C4, C13×C4⋊C4, C4.Dic26
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, D13, C42.C2, D26, Dic26, C22×D13, C2×Dic26, D42D13, D52⋊C2, C4.Dic26

Smallest permutation representation of C4.Dic26
Regular action on 416 points
Generators in S416
(1 202 128 88)(2 89 129 203)(3 204 130 90)(4 91 131 205)(5 206 132 92)(6 93 133 207)(7 208 134 94)(8 95 135 157)(9 158 136 96)(10 97 137 159)(11 160 138 98)(12 99 139 161)(13 162 140 100)(14 101 141 163)(15 164 142 102)(16 103 143 165)(17 166 144 104)(18 53 145 167)(19 168 146 54)(20 55 147 169)(21 170 148 56)(22 57 149 171)(23 172 150 58)(24 59 151 173)(25 174 152 60)(26 61 153 175)(27 176 154 62)(28 63 155 177)(29 178 156 64)(30 65 105 179)(31 180 106 66)(32 67 107 181)(33 182 108 68)(34 69 109 183)(35 184 110 70)(36 71 111 185)(37 186 112 72)(38 73 113 187)(39 188 114 74)(40 75 115 189)(41 190 116 76)(42 77 117 191)(43 192 118 78)(44 79 119 193)(45 194 120 80)(46 81 121 195)(47 196 122 82)(48 83 123 197)(49 198 124 84)(50 85 125 199)(51 200 126 86)(52 87 127 201)(209 361 270 389)(210 390 271 362)(211 363 272 391)(212 392 273 364)(213 313 274 393)(214 394 275 314)(215 315 276 395)(216 396 277 316)(217 317 278 397)(218 398 279 318)(219 319 280 399)(220 400 281 320)(221 321 282 401)(222 402 283 322)(223 323 284 403)(224 404 285 324)(225 325 286 405)(226 406 287 326)(227 327 288 407)(228 408 289 328)(229 329 290 409)(230 410 291 330)(231 331 292 411)(232 412 293 332)(233 333 294 413)(234 414 295 334)(235 335 296 415)(236 416 297 336)(237 337 298 365)(238 366 299 338)(239 339 300 367)(240 368 301 340)(241 341 302 369)(242 370 303 342)(243 343 304 371)(244 372 305 344)(245 345 306 373)(246 374 307 346)(247 347 308 375)(248 376 309 348)(249 349 310 377)(250 378 311 350)(251 351 312 379)(252 380 261 352)(253 353 262 381)(254 382 263 354)(255 355 264 383)(256 384 265 356)(257 357 266 385)(258 386 267 358)(259 359 268 387)(260 388 269 360)
(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)
(1 252 27 226)(2 312 28 286)(3 250 29 224)(4 310 30 284)(5 248 31 222)(6 308 32 282)(7 246 33 220)(8 306 34 280)(9 244 35 218)(10 304 36 278)(11 242 37 216)(12 302 38 276)(13 240 39 214)(14 300 40 274)(15 238 41 212)(16 298 42 272)(17 236 43 210)(18 296 44 270)(19 234 45 260)(20 294 46 268)(21 232 47 258)(22 292 48 266)(23 230 49 256)(24 290 50 264)(25 228 51 254)(26 288 52 262)(53 415 79 389)(54 334 80 360)(55 413 81 387)(56 332 82 358)(57 411 83 385)(58 330 84 356)(59 409 85 383)(60 328 86 354)(61 407 87 381)(62 326 88 352)(63 405 89 379)(64 324 90 350)(65 403 91 377)(66 322 92 348)(67 401 93 375)(68 320 94 346)(69 399 95 373)(70 318 96 344)(71 397 97 371)(72 316 98 342)(73 395 99 369)(74 314 100 340)(75 393 101 367)(76 364 102 338)(77 391 103 365)(78 362 104 336)(105 223 131 249)(106 283 132 309)(107 221 133 247)(108 281 134 307)(109 219 135 245)(110 279 136 305)(111 217 137 243)(112 277 138 303)(113 215 139 241)(114 275 140 301)(115 213 141 239)(116 273 142 299)(117 211 143 237)(118 271 144 297)(119 209 145 235)(120 269 146 295)(121 259 147 233)(122 267 148 293)(123 257 149 231)(124 265 150 291)(125 255 151 229)(126 263 152 289)(127 253 153 227)(128 261 154 287)(129 251 155 225)(130 311 156 285)(157 345 183 319)(158 372 184 398)(159 343 185 317)(160 370 186 396)(161 341 187 315)(162 368 188 394)(163 339 189 313)(164 366 190 392)(165 337 191 363)(166 416 192 390)(167 335 193 361)(168 414 194 388)(169 333 195 359)(170 412 196 386)(171 331 197 357)(172 410 198 384)(173 329 199 355)(174 408 200 382)(175 327 201 353)(176 406 202 380)(177 325 203 351)(178 404 204 378)(179 323 205 349)(180 402 206 376)(181 321 207 347)(182 400 208 374)

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 13A ··· 13F 26A ··· 26R 52A ··· 52AJ 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 2 2 4 4 26 26 26 26 52 52 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + + - - + image C1 C2 C2 C2 C2 Q8 C4○D4 D13 D26 Dic26 D4⋊2D13 D52⋊C2 kernel C4.Dic26 C4×Dic13 C26.D4 C52⋊3C4 C13×C4⋊C4 C52 C26 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 1 2 3 1 2 4 6 18 24 6 6

Matrix representation of C4.Dic26 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 19 45 0 0 32 34
,
 21 49 0 0 4 22 0 0 0 0 37 36 0 0 15 16
,
 47 39 0 0 14 6 0 0 0 0 13 28 0 0 47 40
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,19,32,0,0,45,34],[21,4,0,0,49,22,0,0,0,0,37,15,0,0,36,16],[47,14,0,0,39,6,0,0,0,0,13,47,0,0,28,40] >;

C4.Dic26 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{26}
% in TeX

G:=Group("C4.Dic26");
// GroupNames label

G:=SmallGroup(416,111);
// by ID

G=gap.SmallGroup(416,111);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,506,188,50,13829]);
// Polycyclic

G:=Group<a,b,c|a^4=b^52=1,c^2=b^26,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;
// generators/relations

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