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## G = C4×Dic26order 416 = 25·13

### Direct product of C4 and Dic26

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C4×Dic26
 Chief series C1 — C13 — C26 — C2×C26 — C2×Dic13 — C2×Dic26 — C4×Dic26
 Lower central C13 — C26 — C4×Dic26
 Upper central C1 — C2×C4 — C42

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 [×3], C4 [×4], C4 [×7], C22, C2×C4 [×3], C2×C4 [×4], Q8 [×4], C13, C42, C42 [×2], C4⋊C4 [×3], C2×Q8, C26 [×3], C4×Q8, Dic13 [×4], Dic13 [×2], C52 [×4], C52, C2×C26, Dic26 [×4], C2×Dic13 [×4], C2×C52 [×3], C4×Dic13 [×2], C26.D4 [×2], C523C4, C4×C52, C2×Dic26, C4×Dic26
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, C22×C4, C2×Q8, C4○D4, D13, C4×Q8, D26 [×3], Dic26 [×2], C4×D13 [×2], C22×D13, C2×Dic26, C2×C4×D13, D525C2, C4×Dic26

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

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

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

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

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

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