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## G = Dic13⋊C8order 416 = 25·13

### 3rd semidirect product of Dic13 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — Dic13⋊C8
 Chief series C1 — C13 — C26 — Dic13 — C2×Dic13 — C2×C13⋊C8 — Dic13⋊C8
 Lower central C13 — C26 — Dic13⋊C8
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic13⋊C8
G = < a,b,c | a26=c8=1, b2=a13, bab-1=a-1, cac-1=a5, cbc-1=a13b >

Smallest permutation representation of Dic13⋊C8
Regular action on 416 points
Generators in S416
(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)
(1 301 14 288)(2 300 15 287)(3 299 16 312)(4 298 17 311)(5 297 18 310)(6 296 19 309)(7 295 20 308)(8 294 21 307)(9 293 22 306)(10 292 23 305)(11 291 24 304)(12 290 25 303)(13 289 26 302)(27 323 40 336)(28 322 41 335)(29 321 42 334)(30 320 43 333)(31 319 44 332)(32 318 45 331)(33 317 46 330)(34 316 47 329)(35 315 48 328)(36 314 49 327)(37 313 50 326)(38 338 51 325)(39 337 52 324)(53 346 66 359)(54 345 67 358)(55 344 68 357)(56 343 69 356)(57 342 70 355)(58 341 71 354)(59 340 72 353)(60 339 73 352)(61 364 74 351)(62 363 75 350)(63 362 76 349)(64 361 77 348)(65 360 78 347)(79 284 92 271)(80 283 93 270)(81 282 94 269)(82 281 95 268)(83 280 96 267)(84 279 97 266)(85 278 98 265)(86 277 99 264)(87 276 100 263)(88 275 101 262)(89 274 102 261)(90 273 103 286)(91 272 104 285)(105 197 118 184)(106 196 119 183)(107 195 120 208)(108 194 121 207)(109 193 122 206)(110 192 123 205)(111 191 124 204)(112 190 125 203)(113 189 126 202)(114 188 127 201)(115 187 128 200)(116 186 129 199)(117 185 130 198)(131 399 144 412)(132 398 145 411)(133 397 146 410)(134 396 147 409)(135 395 148 408)(136 394 149 407)(137 393 150 406)(138 392 151 405)(139 391 152 404)(140 416 153 403)(141 415 154 402)(142 414 155 401)(143 413 156 400)(157 232 170 219)(158 231 171 218)(159 230 172 217)(160 229 173 216)(161 228 174 215)(162 227 175 214)(163 226 176 213)(164 225 177 212)(165 224 178 211)(166 223 179 210)(167 222 180 209)(168 221 181 234)(169 220 182 233)(235 368 248 381)(236 367 249 380)(237 366 250 379)(238 365 251 378)(239 390 252 377)(240 389 253 376)(241 388 254 375)(242 387 255 374)(243 386 256 373)(244 385 257 372)(245 384 258 371)(246 383 259 370)(247 382 260 369)
(1 232 59 391 51 246 199 282)(2 227 58 396 52 241 198 261)(3 222 57 401 27 236 197 266)(4 217 56 406 28 257 196 271)(5 212 55 411 29 252 195 276)(6 233 54 416 30 247 194 281)(7 228 53 395 31 242 193 286)(8 223 78 400 32 237 192 265)(9 218 77 405 33 258 191 270)(10 213 76 410 34 253 190 275)(11 234 75 415 35 248 189 280)(12 229 74 394 36 243 188 285)(13 224 73 399 37 238 187 264)(14 219 72 404 38 259 186 269)(15 214 71 409 39 254 185 274)(16 209 70 414 40 249 184 279)(17 230 69 393 41 244 183 284)(18 225 68 398 42 239 208 263)(19 220 67 403 43 260 207 268)(20 215 66 408 44 255 206 273)(21 210 65 413 45 250 205 278)(22 231 64 392 46 245 204 283)(23 226 63 397 47 240 203 262)(24 221 62 402 48 235 202 267)(25 216 61 407 49 256 201 272)(26 211 60 412 50 251 200 277)(79 311 159 356 137 335 372 106)(80 306 158 361 138 330 371 111)(81 301 157 340 139 325 370 116)(82 296 182 345 140 320 369 121)(83 291 181 350 141 315 368 126)(84 312 180 355 142 336 367 105)(85 307 179 360 143 331 366 110)(86 302 178 339 144 326 365 115)(87 297 177 344 145 321 390 120)(88 292 176 349 146 316 389 125)(89 287 175 354 147 337 388 130)(90 308 174 359 148 332 387 109)(91 303 173 364 149 327 386 114)(92 298 172 343 150 322 385 119)(93 293 171 348 151 317 384 124)(94 288 170 353 152 338 383 129)(95 309 169 358 153 333 382 108)(96 304 168 363 154 328 381 113)(97 299 167 342 155 323 380 118)(98 294 166 347 156 318 379 123)(99 289 165 352 131 313 378 128)(100 310 164 357 132 334 377 107)(101 305 163 362 133 329 376 112)(102 300 162 341 134 324 375 117)(103 295 161 346 135 319 374 122)(104 290 160 351 136 314 373 127)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 13A 13B 13C 26A ··· 26I 52A ··· 52L order 1 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 13 13 13 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 13 13 13 13 26 26 26 ··· 26 4 4 4 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 4 4 4 type + + + + - + + - image C1 C2 C2 C4 C4 C8 D4 Q8 M4(2) C13⋊C4 C2×C13⋊C4 D13⋊C8 C52⋊C4 C13⋊M4(2) kernel Dic13⋊C8 C4×Dic13 C2×C13⋊C8 C2×Dic13 C2×C52 Dic13 Dic13 Dic13 C26 C2×C4 C22 C2 C2 C2 # reps 1 1 2 2 2 8 1 1 2 3 3 6 6 6

Matrix representation of Dic13⋊C8 in GL8(𝔽313)

 312 0 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 0 312 1 0 0 0 0 0 0 312 0 1 0 0 0 0 0 312 0 0 1 0 0 0 0 70 31 282 242
,
 312 311 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 258 35 0 0 0 0 0 0 155 55 0 0 0 0 0 0 0 0 129 133 72 1 0 0 0 0 5 149 76 292 0 0 0 0 184 303 236 105 0 0 0 0 121 54 233 112
,
 244 131 0 0 0 0 0 0 291 69 0 0 0 0 0 0 0 0 224 245 0 0 0 0 0 0 245 89 0 0 0 0 0 0 0 0 307 35 61 117 0 0 0 0 4 189 17 185 0 0 0 0 249 19 204 232 0 0 0 0 210 76 287 239

G:=sub<GL(8,GF(313))| [312,0,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,0,312,312,312,70,0,0,0,0,1,0,0,31,0,0,0,0,0,1,0,282,0,0,0,0,0,0,1,242],[312,1,0,0,0,0,0,0,311,1,0,0,0,0,0,0,0,0,258,155,0,0,0,0,0,0,35,55,0,0,0,0,0,0,0,0,129,5,184,121,0,0,0,0,133,149,303,54,0,0,0,0,72,76,236,233,0,0,0,0,1,292,105,112],[244,291,0,0,0,0,0,0,131,69,0,0,0,0,0,0,0,0,224,245,0,0,0,0,0,0,245,89,0,0,0,0,0,0,0,0,307,4,249,210,0,0,0,0,35,189,19,76,0,0,0,0,61,17,204,287,0,0,0,0,117,185,232,239] >;

Dic13⋊C8 in GAP, Magma, Sage, TeX

{\rm Dic}_{13}\rtimes C_8
% in TeX

G:=Group("Dic13:C8");
// GroupNames label

G:=SmallGroup(416,79);
// by ID

G=gap.SmallGroup(416,79);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,151,86,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^26=c^8=1,b^2=a^13,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^13*b>;
// generators/relations

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