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

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

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

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

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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