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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 [×3], C2 [×4], C4 [×8], C22, C22 [×6], C2×C4 [×2], C2×C4 [×12], C23, C13, C4⋊C4 [×4], C22×C4, C22×C4 [×2], C26 [×3], C26 [×4], C2×C4⋊C4, Dic13 [×4], Dic13 [×2], C52 [×2], C2×C26, C2×C26 [×6], C2×Dic13 [×8], C2×Dic13 [×2], C2×C52 [×2], C2×C52 [×2], C22×C26, C26.D4 [×4], C22×Dic13 [×2], C22×C52, C2×C26.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D13, C2×C4⋊C4, D26 [×3], Dic26 [×2], C4×D13 [×2], C13⋊D4 [×2], C22×D13, C26.D4 [×4], 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
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 384)(28 385)(29 386)(30 387)(31 388)(32 389)(33 390)(34 365)(35 366)(36 367)(37 368)(38 369)(39 370)(40 371)(41 372)(42 373)(43 374)(44 375)(45 376)(46 377)(47 378)(48 379)(49 380)(50 381)(51 382)(52 383)(53 398)(54 399)(55 400)(56 401)(57 402)(58 403)(59 404)(60 405)(61 406)(62 407)(63 408)(64 409)(65 410)(66 411)(67 412)(68 413)(69 414)(70 415)(71 416)(72 391)(73 392)(74 393)(75 394)(76 395)(77 396)(78 397)(79 310)(80 311)(81 312)(82 287)(83 288)(84 289)(85 290)(86 291)(87 292)(88 293)(89 294)(90 295)(91 296)(92 297)(93 298)(94 299)(95 300)(96 301)(97 302)(98 303)(99 304)(100 305)(101 306)(102 307)(103 308)(104 309)(105 246)(106 247)(107 248)(108 249)(109 250)(110 251)(111 252)(112 253)(113 254)(114 255)(115 256)(116 257)(117 258)(118 259)(119 260)(120 235)(121 236)(122 237)(123 238)(124 239)(125 240)(126 241)(127 242)(128 243)(129 244)(130 245)(157 219)(158 220)(159 221)(160 222)(161 223)(162 224)(163 225)(164 226)(165 227)(166 228)(167 229)(168 230)(169 231)(170 232)(171 233)(172 234)(173 209)(174 210)(175 211)(176 212)(177 213)(178 214)(179 215)(180 216)(181 217)(182 218)(183 280)(184 281)(185 282)(186 283)(187 284)(188 285)(189 286)(190 261)(191 262)(192 263)(193 264)(194 265)(195 266)(196 267)(197 268)(198 269)(199 270)(200 271)(201 272)(202 273)(203 274)(204 275)(205 276)(206 277)(207 278)(208 279)(313 353)(314 354)(315 355)(316 356)(317 357)(318 358)(319 359)(320 360)(321 361)(322 362)(323 363)(324 364)(325 339)(326 340)(327 341)(328 342)(329 343)(330 344)(331 345)(332 346)(333 347)(334 348)(335 349)(336 350)(337 351)(338 352)
(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)
(1 404 171 250)(2 403 172 249)(3 402 173 248)(4 401 174 247)(5 400 175 246)(6 399 176 245)(7 398 177 244)(8 397 178 243)(9 396 179 242)(10 395 180 241)(11 394 181 240)(12 393 182 239)(13 392 157 238)(14 391 158 237)(15 416 159 236)(16 415 160 235)(17 414 161 260)(18 413 162 259)(19 412 163 258)(20 411 164 257)(21 410 165 256)(22 409 166 255)(23 408 167 254)(24 407 168 253)(25 406 169 252)(26 405 170 251)(27 295 363 202)(28 294 364 201)(29 293 339 200)(30 292 340 199)(31 291 341 198)(32 290 342 197)(33 289 343 196)(34 288 344 195)(35 287 345 194)(36 312 346 193)(37 311 347 192)(38 310 348 191)(39 309 349 190)(40 308 350 189)(41 307 351 188)(42 306 352 187)(43 305 353 186)(44 304 354 185)(45 303 355 184)(46 302 356 183)(47 301 357 208)(48 300 358 207)(49 299 359 206)(50 298 360 205)(51 297 361 204)(52 296 362 203)(53 213 129 147)(54 212 130 146)(55 211 105 145)(56 210 106 144)(57 209 107 143)(58 234 108 142)(59 233 109 141)(60 232 110 140)(61 231 111 139)(62 230 112 138)(63 229 113 137)(64 228 114 136)(65 227 115 135)(66 226 116 134)(67 225 117 133)(68 224 118 132)(69 223 119 131)(70 222 120 156)(71 221 121 155)(72 220 122 154)(73 219 123 153)(74 218 124 152)(75 217 125 151)(76 216 126 150)(77 215 127 149)(78 214 128 148)(79 334 262 369)(80 333 263 368)(81 332 264 367)(82 331 265 366)(83 330 266 365)(84 329 267 390)(85 328 268 389)(86 327 269 388)(87 326 270 387)(88 325 271 386)(89 324 272 385)(90 323 273 384)(91 322 274 383)(92 321 275 382)(93 320 276 381)(94 319 277 380)(95 318 278 379)(96 317 279 378)(97 316 280 377)(98 315 281 376)(99 314 282 375)(100 313 283 374)(101 338 284 373)(102 337 285 372)(103 336 286 371)(104 335 261 370)
(1 199 14 186)(2 198 15 185)(3 197 16 184)(4 196 17 183)(5 195 18 208)(6 194 19 207)(7 193 20 206)(8 192 21 205)(9 191 22 204)(10 190 23 203)(11 189 24 202)(12 188 25 201)(13 187 26 200)(27 240 40 253)(28 239 41 252)(29 238 42 251)(30 237 43 250)(31 236 44 249)(32 235 45 248)(33 260 46 247)(34 259 47 246)(35 258 48 245)(36 257 49 244)(37 256 50 243)(38 255 51 242)(39 254 52 241)(53 332 66 319)(54 331 67 318)(55 330 68 317)(56 329 69 316)(57 328 70 315)(58 327 71 314)(59 326 72 313)(60 325 73 338)(61 324 74 337)(62 323 75 336)(63 322 76 335)(64 321 77 334)(65 320 78 333)(79 228 92 215)(80 227 93 214)(81 226 94 213)(82 225 95 212)(83 224 96 211)(84 223 97 210)(85 222 98 209)(86 221 99 234)(87 220 100 233)(88 219 101 232)(89 218 102 231)(90 217 103 230)(91 216 104 229)(105 365 118 378)(106 390 119 377)(107 389 120 376)(108 388 121 375)(109 387 122 374)(110 386 123 373)(111 385 124 372)(112 384 125 371)(113 383 126 370)(114 382 127 369)(115 381 128 368)(116 380 129 367)(117 379 130 366)(131 280 144 267)(132 279 145 266)(133 278 146 265)(134 277 147 264)(135 276 148 263)(136 275 149 262)(137 274 150 261)(138 273 151 286)(139 272 152 285)(140 271 153 284)(141 270 154 283)(142 269 155 282)(143 268 156 281)(157 306 170 293)(158 305 171 292)(159 304 172 291)(160 303 173 290)(161 302 174 289)(162 301 175 288)(163 300 176 287)(164 299 177 312)(165 298 178 311)(166 297 179 310)(167 296 180 309)(168 295 181 308)(169 294 182 307)(339 392 352 405)(340 391 353 404)(341 416 354 403)(342 415 355 402)(343 414 356 401)(344 413 357 400)(345 412 358 399)(346 411 359 398)(347 410 360 397)(348 409 361 396)(349 408 362 395)(350 407 363 394)(351 406 364 393)

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

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

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

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