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## G = Q8⋊Dic13order 416 = 25·13

### 1st semidirect product of Q8 and Dic13 acting via Dic13/C26=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — Q8⋊Dic13
 Chief series C1 — C13 — C26 — C2×C26 — C2×C52 — C52⋊3C4 — Q8⋊Dic13
 Lower central C13 — C26 — C52 — Q8⋊Dic13
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for Q8⋊Dic13
G = < a,b,c,d | a4=c26=1, b2=a2, d2=c13, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

Smallest permutation representation of Q8⋊Dic13
Regular action on 416 points
Generators in S416
(1 68 83 269)(2 69 84 270)(3 70 85 271)(4 71 86 272)(5 72 87 273)(6 73 88 274)(7 74 89 275)(8 75 90 276)(9 76 91 277)(10 77 92 278)(11 78 93 279)(12 53 94 280)(13 54 95 281)(14 55 96 282)(15 56 97 283)(16 57 98 284)(17 58 99 285)(18 59 100 286)(19 60 101 261)(20 61 102 262)(21 62 103 263)(22 63 104 264)(23 64 79 265)(24 65 80 266)(25 66 81 267)(26 67 82 268)(27 208 128 257)(28 183 129 258)(29 184 130 259)(30 185 105 260)(31 186 106 235)(32 187 107 236)(33 188 108 237)(34 189 109 238)(35 190 110 239)(36 191 111 240)(37 192 112 241)(38 193 113 242)(39 194 114 243)(40 195 115 244)(41 196 116 245)(42 197 117 246)(43 198 118 247)(44 199 119 248)(45 200 120 249)(46 201 121 250)(47 202 122 251)(48 203 123 252)(49 204 124 253)(50 205 125 254)(51 206 126 255)(52 207 127 256)(131 408 317 174)(132 409 318 175)(133 410 319 176)(134 411 320 177)(135 412 321 178)(136 413 322 179)(137 414 323 180)(138 415 324 181)(139 416 325 182)(140 391 326 157)(141 392 327 158)(142 393 328 159)(143 394 329 160)(144 395 330 161)(145 396 331 162)(146 397 332 163)(147 398 333 164)(148 399 334 165)(149 400 335 166)(150 401 336 167)(151 402 337 168)(152 403 338 169)(153 404 313 170)(154 405 314 171)(155 406 315 172)(156 407 316 173)(209 294 343 372)(210 295 344 373)(211 296 345 374)(212 297 346 375)(213 298 347 376)(214 299 348 377)(215 300 349 378)(216 301 350 379)(217 302 351 380)(218 303 352 381)(219 304 353 382)(220 305 354 383)(221 306 355 384)(222 307 356 385)(223 308 357 386)(224 309 358 387)(225 310 359 388)(226 311 360 389)(227 312 361 390)(228 287 362 365)(229 288 363 366)(230 289 364 367)(231 290 339 368)(232 291 340 369)(233 292 341 370)(234 293 342 371)
(1 45 83 120)(2 46 84 121)(3 47 85 122)(4 48 86 123)(5 49 87 124)(6 50 88 125)(7 51 89 126)(8 52 90 127)(9 27 91 128)(10 28 92 129)(11 29 93 130)(12 30 94 105)(13 31 95 106)(14 32 96 107)(15 33 97 108)(16 34 98 109)(17 35 99 110)(18 36 100 111)(19 37 101 112)(20 38 102 113)(21 39 103 114)(22 40 104 115)(23 41 79 116)(24 42 80 117)(25 43 81 118)(26 44 82 119)(53 260 280 185)(54 235 281 186)(55 236 282 187)(56 237 283 188)(57 238 284 189)(58 239 285 190)(59 240 286 191)(60 241 261 192)(61 242 262 193)(62 243 263 194)(63 244 264 195)(64 245 265 196)(65 246 266 197)(66 247 267 198)(67 248 268 199)(68 249 269 200)(69 250 270 201)(70 251 271 202)(71 252 272 203)(72 253 273 204)(73 254 274 205)(74 255 275 206)(75 256 276 207)(76 257 277 208)(77 258 278 183)(78 259 279 184)(131 306 317 384)(132 307 318 385)(133 308 319 386)(134 309 320 387)(135 310 321 388)(136 311 322 389)(137 312 323 390)(138 287 324 365)(139 288 325 366)(140 289 326 367)(141 290 327 368)(142 291 328 369)(143 292 329 370)(144 293 330 371)(145 294 331 372)(146 295 332 373)(147 296 333 374)(148 297 334 375)(149 298 335 376)(150 299 336 377)(151 300 337 378)(152 301 338 379)(153 302 313 380)(154 303 314 381)(155 304 315 382)(156 305 316 383)(157 364 391 230)(158 339 392 231)(159 340 393 232)(160 341 394 233)(161 342 395 234)(162 343 396 209)(163 344 397 210)(164 345 398 211)(165 346 399 212)(166 347 400 213)(167 348 401 214)(168 349 402 215)(169 350 403 216)(170 351 404 217)(171 352 405 218)(172 353 406 219)(173 354 407 220)(174 355 408 221)(175 356 409 222)(176 357 410 223)(177 358 411 224)(178 359 412 225)(179 360 413 226)(180 361 414 227)(181 362 415 228)(182 363 416 229)
(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)
(1 402 14 415)(2 401 15 414)(3 400 16 413)(4 399 17 412)(5 398 18 411)(6 397 19 410)(7 396 20 409)(8 395 21 408)(9 394 22 407)(10 393 23 406)(11 392 24 405)(12 391 25 404)(13 416 26 403)(27 292 40 305)(28 291 41 304)(29 290 42 303)(30 289 43 302)(31 288 44 301)(32 287 45 300)(33 312 46 299)(34 311 47 298)(35 310 48 297)(36 309 49 296)(37 308 50 295)(38 307 51 294)(39 306 52 293)(53 140 66 153)(54 139 67 152)(55 138 68 151)(56 137 69 150)(57 136 70 149)(58 135 71 148)(59 134 72 147)(60 133 73 146)(61 132 74 145)(62 131 75 144)(63 156 76 143)(64 155 77 142)(65 154 78 141)(79 172 92 159)(80 171 93 158)(81 170 94 157)(82 169 95 182)(83 168 96 181)(84 167 97 180)(85 166 98 179)(86 165 99 178)(87 164 100 177)(88 163 101 176)(89 162 102 175)(90 161 103 174)(91 160 104 173)(105 367 118 380)(106 366 119 379)(107 365 120 378)(108 390 121 377)(109 389 122 376)(110 388 123 375)(111 387 124 374)(112 386 125 373)(113 385 126 372)(114 384 127 371)(115 383 128 370)(116 382 129 369)(117 381 130 368)(183 232 196 219)(184 231 197 218)(185 230 198 217)(186 229 199 216)(187 228 200 215)(188 227 201 214)(189 226 202 213)(190 225 203 212)(191 224 204 211)(192 223 205 210)(193 222 206 209)(194 221 207 234)(195 220 208 233)(235 363 248 350)(236 362 249 349)(237 361 250 348)(238 360 251 347)(239 359 252 346)(240 358 253 345)(241 357 254 344)(242 356 255 343)(243 355 256 342)(244 354 257 341)(245 353 258 340)(246 352 259 339)(247 351 260 364)(261 319 274 332)(262 318 275 331)(263 317 276 330)(264 316 277 329)(265 315 278 328)(266 314 279 327)(267 313 280 326)(268 338 281 325)(269 337 282 324)(270 336 283 323)(271 335 284 322)(272 334 285 321)(273 333 286 320)

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 4 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 4 4 52 52 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + + - + - image C1 C2 C2 C2 C4 D4 D4 SD16 Q16 D13 D26 Dic13 C13⋊D4 C13⋊D4 Q8⋊D13 C13⋊Q16 kernel Q8⋊Dic13 C2×C13⋊2C8 C52⋊3C4 Q8×C26 Q8×C13 C52 C2×C26 C26 C26 C2×Q8 C2×C4 Q8 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 6 6 12 12 12 6 6

Matrix representation of Q8⋊Dic13 in GL5(𝔽313)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 192 0 0 0 238 312
,
 312 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 131 233 0 0 0 238 182
,
 312 0 0 0 0 0 0 1 0 0 0 312 293 0 0 0 0 0 312 0 0 0 0 0 312
,
 288 0 0 0 0 0 170 29 0 0 0 72 143 0 0 0 0 0 50 44 0 0 0 121 263

G:=sub<GL(5,GF(313))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,238,0,0,0,192,312],[312,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,131,238,0,0,0,233,182],[312,0,0,0,0,0,0,312,0,0,0,1,293,0,0,0,0,0,312,0,0,0,0,0,312],[288,0,0,0,0,0,170,72,0,0,0,29,143,0,0,0,0,0,50,121,0,0,0,44,263] >;

Q8⋊Dic13 in GAP, Magma, Sage, TeX

Q_8\rtimes {\rm Dic}_{13}
% in TeX

G:=Group("Q8:Dic13");
// GroupNames label

G:=SmallGroup(416,42);
// by ID

G=gap.SmallGroup(416,42);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,579,297,69,13829]);
// Polycyclic

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

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