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## G = C2×C26.D4order 416 = 25·13

### Direct product of C2 and C26.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×C26.D4
 Chief series C1 — C13 — C26 — C2×C26 — C2×Dic13 — C22×Dic13 — C2×C26.D4
 Lower central C13 — C26 — C2×C26.D4
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×C26.D4
G = < a,b,c,d | a2=b26=c4=1, d2=b13, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 432 in 92 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C13, C4⋊C4, C22×C4, C22×C4, C26, C26, C2×C4⋊C4, Dic13, Dic13, C52, C2×C26, C2×C26, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C22×C26, C26.D4, C22×Dic13, C22×C52, C2×C26.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, D13, C2×C4⋊C4, D26, Dic26, C4×D13, C13⋊D4, C22×D13, C26.D4, C2×Dic26, C2×C4×D13, C2×C13⋊D4, C2×C26.D4

Smallest permutation representation of C2×C26.D4
Regular action on 416 points
Generators in S416
(1 207)(2 208)(3 183)(4 184)(5 185)(6 186)(7 187)(8 188)(9 189)(10 190)(11 191)(12 192)(13 193)(14 194)(15 195)(16 196)(17 197)(18 198)(19 199)(20 200)(21 201)(22 202)(23 203)(24 204)(25 205)(26 206)(27 264)(28 265)(29 266)(30 267)(31 268)(32 269)(33 270)(34 271)(35 272)(36 273)(37 274)(38 275)(39 276)(40 277)(41 278)(42 279)(43 280)(44 281)(45 282)(46 283)(47 284)(48 285)(49 286)(50 261)(51 262)(52 263)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 213)(80 214)(81 215)(82 216)(83 217)(84 218)(85 219)(86 220)(87 221)(88 222)(89 223)(90 224)(91 225)(92 226)(93 227)(94 228)(95 229)(96 230)(97 231)(98 232)(99 233)(100 234)(101 209)(102 210)(103 211)(104 212)(131 393)(132 394)(133 395)(134 396)(135 397)(136 398)(137 399)(138 400)(139 401)(140 402)(141 403)(142 404)(143 405)(144 406)(145 407)(146 408)(147 409)(148 410)(149 411)(150 412)(151 413)(152 414)(153 415)(154 416)(155 391)(156 392)(157 332)(158 333)(159 334)(160 335)(161 336)(162 337)(163 338)(164 313)(165 314)(166 315)(167 316)(168 317)(169 318)(170 319)(171 320)(172 321)(173 322)(174 323)(175 324)(176 325)(177 326)(178 327)(179 328)(180 329)(181 330)(182 331)(235 356)(236 357)(237 358)(238 359)(239 360)(240 361)(241 362)(242 363)(243 364)(244 339)(245 340)(246 341)(247 342)(248 343)(249 344)(250 345)(251 346)(252 347)(253 348)(254 349)(255 350)(256 351)(257 352)(258 353)(259 354)(260 355)(287 381)(288 382)(289 383)(290 384)(291 385)(292 386)(293 387)(294 388)(295 389)(296 390)(297 365)(298 366)(299 367)(300 368)(301 369)(302 370)(303 371)(304 372)(305 373)(306 374)(307 375)(308 376)(309 377)(310 378)(311 379)(312 380)
(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 30 380 341)(2 29 381 340)(3 28 382 339)(4 27 383 364)(5 52 384 363)(6 51 385 362)(7 50 386 361)(8 49 387 360)(9 48 388 359)(10 47 389 358)(11 46 390 357)(12 45 365 356)(13 44 366 355)(14 43 367 354)(15 42 368 353)(16 41 369 352)(17 40 370 351)(18 39 371 350)(19 38 372 349)(20 37 373 348)(21 36 374 347)(22 35 375 346)(23 34 376 345)(24 33 377 344)(25 32 378 343)(26 31 379 342)(53 215 329 415)(54 214 330 414)(55 213 331 413)(56 212 332 412)(57 211 333 411)(58 210 334 410)(59 209 335 409)(60 234 336 408)(61 233 337 407)(62 232 338 406)(63 231 313 405)(64 230 314 404)(65 229 315 403)(66 228 316 402)(67 227 317 401)(68 226 318 400)(69 225 319 399)(70 224 320 398)(71 223 321 397)(72 222 322 396)(73 221 323 395)(74 220 324 394)(75 219 325 393)(76 218 326 392)(77 217 327 391)(78 216 328 416)(79 182 151 115)(80 181 152 114)(81 180 153 113)(82 179 154 112)(83 178 155 111)(84 177 156 110)(85 176 131 109)(86 175 132 108)(87 174 133 107)(88 173 134 106)(89 172 135 105)(90 171 136 130)(91 170 137 129)(92 169 138 128)(93 168 139 127)(94 167 140 126)(95 166 141 125)(96 165 142 124)(97 164 143 123)(98 163 144 122)(99 162 145 121)(100 161 146 120)(101 160 147 119)(102 159 148 118)(103 158 149 117)(104 157 150 116)(183 265 288 244)(184 264 289 243)(185 263 290 242)(186 262 291 241)(187 261 292 240)(188 286 293 239)(189 285 294 238)(190 284 295 237)(191 283 296 236)(192 282 297 235)(193 281 298 260)(194 280 299 259)(195 279 300 258)(196 278 301 257)(197 277 302 256)(198 276 303 255)(199 275 304 254)(200 274 305 253)(201 273 306 252)(202 272 307 251)(203 271 308 250)(204 270 309 249)(205 269 310 248)(206 268 311 247)(207 267 312 246)(208 266 287 245)
(1 234 14 221)(2 233 15 220)(3 232 16 219)(4 231 17 218)(5 230 18 217)(6 229 19 216)(7 228 20 215)(8 227 21 214)(9 226 22 213)(10 225 23 212)(11 224 24 211)(12 223 25 210)(13 222 26 209)(27 63 40 76)(28 62 41 75)(29 61 42 74)(30 60 43 73)(31 59 44 72)(32 58 45 71)(33 57 46 70)(34 56 47 69)(35 55 48 68)(36 54 49 67)(37 53 50 66)(38 78 51 65)(39 77 52 64)(79 189 92 202)(80 188 93 201)(81 187 94 200)(82 186 95 199)(83 185 96 198)(84 184 97 197)(85 183 98 196)(86 208 99 195)(87 207 100 194)(88 206 101 193)(89 205 102 192)(90 204 103 191)(91 203 104 190)(105 269 118 282)(106 268 119 281)(107 267 120 280)(108 266 121 279)(109 265 122 278)(110 264 123 277)(111 263 124 276)(112 262 125 275)(113 261 126 274)(114 286 127 273)(115 285 128 272)(116 284 129 271)(117 283 130 270)(131 288 144 301)(132 287 145 300)(133 312 146 299)(134 311 147 298)(135 310 148 297)(136 309 149 296)(137 308 150 295)(138 307 151 294)(139 306 152 293)(140 305 153 292)(141 304 154 291)(142 303 155 290)(143 302 156 289)(157 237 170 250)(158 236 171 249)(159 235 172 248)(160 260 173 247)(161 259 174 246)(162 258 175 245)(163 257 176 244)(164 256 177 243)(165 255 178 242)(166 254 179 241)(167 253 180 240)(168 252 181 239)(169 251 182 238)(313 351 326 364)(314 350 327 363)(315 349 328 362)(316 348 329 361)(317 347 330 360)(318 346 331 359)(319 345 332 358)(320 344 333 357)(321 343 334 356)(322 342 335 355)(323 341 336 354)(324 340 337 353)(325 339 338 352)(365 397 378 410)(366 396 379 409)(367 395 380 408)(368 394 381 407)(369 393 382 406)(370 392 383 405)(371 391 384 404)(372 416 385 403)(373 415 386 402)(374 414 387 401)(375 413 388 400)(376 412 389 399)(377 411 390 398)

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

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

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

116 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 13A ··· 13F 26A ··· 26AP 52A ··· 52AV order 1 2 ··· 2 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 2 2 2 2 26 ··· 26 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + + + - image C1 C2 C2 C2 C4 D4 Q8 D13 D26 D26 Dic26 C4×D13 C13⋊D4 kernel C2×C26.D4 C26.D4 C22×Dic13 C22×C52 C2×Dic13 C2×C26 C2×C26 C22×C4 C2×C4 C23 C22 C22 C22 # reps 1 4 2 1 8 2 2 6 12 6 24 24 24

Matrix representation of C2×C26.D4 in GL5(𝔽53)

 52 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 52 0 0 0 0 0 52
,
 1 0 0 0 0 0 18 29 0 0 0 24 24 0 0 0 0 0 35 2 0 0 0 1 44
,
 1 0 0 0 0 0 18 2 0 0 0 24 35 0 0 0 0 0 49 8 0 0 0 31 4
,
 52 0 0 0 0 0 43 46 0 0 0 22 10 0 0 0 0 0 9 26 0 0 0 1 44

G:=sub<GL(5,GF(53))| [52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,18,24,0,0,0,29,24,0,0,0,0,0,35,1,0,0,0,2,44],[1,0,0,0,0,0,18,24,0,0,0,2,35,0,0,0,0,0,49,31,0,0,0,8,4],[52,0,0,0,0,0,43,22,0,0,0,46,10,0,0,0,0,0,9,1,0,0,0,26,44] >;

C2×C26.D4 in GAP, Magma, Sage, TeX

C_2\times C_{26}.D_4
% in TeX

G:=Group("C2xC26.D4");
// GroupNames label

G:=SmallGroup(416,144);
// by ID

G=gap.SmallGroup(416,144);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,362,50,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^26=c^4=1,d^2=b^13,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽