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## G = Dic13⋊3Q8order 416 = 25·13

### The semidirect product of Dic13 and Q8 acting through Inn(Dic13)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic133Q8
G = < a,b,c | a52=c4=1, b2=a26, bab-1=a-1, cac-1=a27, bc=cb >

Subgroups: 352 in 70 conjugacy classes, 43 normal (19 characteristic)
C1, C2 [×3], C4 [×2], C4 [×9], C22, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C13, C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×Q8, C26 [×3], C4×Q8, Dic13 [×6], Dic13, C52 [×2], C52 [×2], C2×C26, Dic26 [×4], C2×Dic13 [×2], C2×Dic13 [×2], C2×C52, C2×C52 [×2], C4×Dic13, C4×Dic13 [×2], C26.D4 [×2], C13×C4⋊C4, C2×Dic26, Dic133Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, C22×C4, C2×Q8, C4○D4, D13, C4×Q8, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D42D13, Q8×D13, Dic133Q8

Smallest permutation representation of Dic133Q8
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)
(1 237 27 211)(2 236 28 210)(3 235 29 209)(4 234 30 260)(5 233 31 259)(6 232 32 258)(7 231 33 257)(8 230 34 256)(9 229 35 255)(10 228 36 254)(11 227 37 253)(12 226 38 252)(13 225 39 251)(14 224 40 250)(15 223 41 249)(16 222 42 248)(17 221 43 247)(18 220 44 246)(19 219 45 245)(20 218 46 244)(21 217 47 243)(22 216 48 242)(23 215 49 241)(24 214 50 240)(25 213 51 239)(26 212 52 238)(53 352 79 326)(54 351 80 325)(55 350 81 324)(56 349 82 323)(57 348 83 322)(58 347 84 321)(59 346 85 320)(60 345 86 319)(61 344 87 318)(62 343 88 317)(63 342 89 316)(64 341 90 315)(65 340 91 314)(66 339 92 313)(67 338 93 364)(68 337 94 363)(69 336 95 362)(70 335 96 361)(71 334 97 360)(72 333 98 359)(73 332 99 358)(74 331 100 357)(75 330 101 356)(76 329 102 355)(77 328 103 354)(78 327 104 353)(105 378 131 404)(106 377 132 403)(107 376 133 402)(108 375 134 401)(109 374 135 400)(110 373 136 399)(111 372 137 398)(112 371 138 397)(113 370 139 396)(114 369 140 395)(115 368 141 394)(116 367 142 393)(117 366 143 392)(118 365 144 391)(119 416 145 390)(120 415 146 389)(121 414 147 388)(122 413 148 387)(123 412 149 386)(124 411 150 385)(125 410 151 384)(126 409 152 383)(127 408 153 382)(128 407 154 381)(129 406 155 380)(130 405 156 379)(157 291 183 265)(158 290 184 264)(159 289 185 263)(160 288 186 262)(161 287 187 261)(162 286 188 312)(163 285 189 311)(164 284 190 310)(165 283 191 309)(166 282 192 308)(167 281 193 307)(168 280 194 306)(169 279 195 305)(170 278 196 304)(171 277 197 303)(172 276 198 302)(173 275 199 301)(174 274 200 300)(175 273 201 299)(176 272 202 298)(177 271 203 297)(178 270 204 296)(179 269 205 295)(180 268 206 294)(181 267 207 293)(182 266 208 292)
(1 330 286 114)(2 357 287 141)(3 332 288 116)(4 359 289 143)(5 334 290 118)(6 361 291 145)(7 336 292 120)(8 363 293 147)(9 338 294 122)(10 313 295 149)(11 340 296 124)(12 315 297 151)(13 342 298 126)(14 317 299 153)(15 344 300 128)(16 319 301 155)(17 346 302 130)(18 321 303 105)(19 348 304 132)(20 323 305 107)(21 350 306 134)(22 325 307 109)(23 352 308 136)(24 327 309 111)(25 354 310 138)(26 329 311 113)(27 356 312 140)(28 331 261 115)(29 358 262 142)(30 333 263 117)(31 360 264 144)(32 335 265 119)(33 362 266 146)(34 337 267 121)(35 364 268 148)(36 339 269 123)(37 314 270 150)(38 341 271 125)(39 316 272 152)(40 343 273 127)(41 318 274 154)(42 345 275 129)(43 320 276 156)(44 347 277 131)(45 322 278 106)(46 349 279 133)(47 324 280 108)(48 351 281 135)(49 326 282 110)(50 353 283 137)(51 328 284 112)(52 355 285 139)(53 192 373 241)(54 167 374 216)(55 194 375 243)(56 169 376 218)(57 196 377 245)(58 171 378 220)(59 198 379 247)(60 173 380 222)(61 200 381 249)(62 175 382 224)(63 202 383 251)(64 177 384 226)(65 204 385 253)(66 179 386 228)(67 206 387 255)(68 181 388 230)(69 208 389 257)(70 183 390 232)(71 158 391 259)(72 185 392 234)(73 160 393 209)(74 187 394 236)(75 162 395 211)(76 189 396 238)(77 164 397 213)(78 191 398 240)(79 166 399 215)(80 193 400 242)(81 168 401 217)(82 195 402 244)(83 170 403 219)(84 197 404 246)(85 172 405 221)(86 199 406 248)(87 174 407 223)(88 201 408 250)(89 176 409 225)(90 203 410 252)(91 178 411 227)(92 205 412 254)(93 180 413 229)(94 207 414 256)(95 182 415 231)(96 157 416 258)(97 184 365 233)(98 159 366 260)(99 186 367 235)(100 161 368 210)(101 188 369 237)(102 163 370 212)(103 190 371 239)(104 165 372 214)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 2 ··· 2 13 13 13 13 26 ··· 26 2 ··· 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + + - - image C1 C2 C2 C2 C2 C4 Q8 C4○D4 D13 D26 C4×D13 D4⋊2D13 Q8×D13 kernel Dic13⋊3Q8 C4×Dic13 C26.D4 C13×C4⋊C4 C2×Dic26 Dic26 Dic13 C26 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 3 2 1 1 8 2 2 6 18 24 6 6

Matrix representation of Dic133Q8 in GL4(𝔽53) generated by

 8 52 0 0 47 34 0 0 0 0 11 49 0 0 4 42
,
 49 42 0 0 11 4 0 0 0 0 0 23 0 0 23 0
,
 23 0 0 0 0 23 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(53))| [8,47,0,0,52,34,0,0,0,0,11,4,0,0,49,42],[49,11,0,0,42,4,0,0,0,0,0,23,0,0,23,0],[23,0,0,0,0,23,0,0,0,0,0,1,0,0,1,0] >;

Dic133Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{13}\rtimes_3Q_8
% in TeX

G:=Group("Dic13:3Q8");
// GroupNames label

G:=SmallGroup(416,108);
// by ID

G=gap.SmallGroup(416,108);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,55,116,122,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=c^4=1,b^2=a^26,b*a*b^-1=a^-1,c*a*c^-1=a^27,b*c=c*b>;
// generators/relations

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