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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, C4, C4, C22, C2×C4, C2×C4, C2×C4, Q8, C13, C42, C4⋊C4, C4⋊C4, C2×Q8, C26, C4×Q8, Dic13, Dic13, C52, C52, C2×C26, Dic26, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C4×Dic13, C4×Dic13, C26.D4, C13×C4⋊C4, C2×Dic26, Dic133Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, D13, C4×Q8, D26, C4×D13, 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)
(1 334 27 360)(2 333 28 359)(3 332 29 358)(4 331 30 357)(5 330 31 356)(6 329 32 355)(7 328 33 354)(8 327 34 353)(9 326 35 352)(10 325 36 351)(11 324 37 350)(12 323 38 349)(13 322 39 348)(14 321 40 347)(15 320 41 346)(16 319 42 345)(17 318 43 344)(18 317 44 343)(19 316 45 342)(20 315 46 341)(21 314 47 340)(22 313 48 339)(23 364 49 338)(24 363 50 337)(25 362 51 336)(26 361 52 335)(53 294 79 268)(54 293 80 267)(55 292 81 266)(56 291 82 265)(57 290 83 264)(58 289 84 263)(59 288 85 262)(60 287 86 261)(61 286 87 312)(62 285 88 311)(63 284 89 310)(64 283 90 309)(65 282 91 308)(66 281 92 307)(67 280 93 306)(68 279 94 305)(69 278 95 304)(70 277 96 303)(71 276 97 302)(72 275 98 301)(73 274 99 300)(74 273 100 299)(75 272 101 298)(76 271 102 297)(77 270 103 296)(78 269 104 295)(105 409 131 383)(106 408 132 382)(107 407 133 381)(108 406 134 380)(109 405 135 379)(110 404 136 378)(111 403 137 377)(112 402 138 376)(113 401 139 375)(114 400 140 374)(115 399 141 373)(116 398 142 372)(117 397 143 371)(118 396 144 370)(119 395 145 369)(120 394 146 368)(121 393 147 367)(122 392 148 366)(123 391 149 365)(124 390 150 416)(125 389 151 415)(126 388 152 414)(127 387 153 413)(128 386 154 412)(129 385 155 411)(130 384 156 410)(157 226 183 252)(158 225 184 251)(159 224 185 250)(160 223 186 249)(161 222 187 248)(162 221 188 247)(163 220 189 246)(164 219 190 245)(165 218 191 244)(166 217 192 243)(167 216 193 242)(168 215 194 241)(169 214 195 240)(170 213 196 239)(171 212 197 238)(172 211 198 237)(173 210 199 236)(174 209 200 235)(175 260 201 234)(176 259 202 233)(177 258 203 232)(178 257 204 231)(179 256 205 230)(180 255 206 229)(181 254 207 228)(182 253 208 227)
(1 241 288 408)(2 216 289 383)(3 243 290 410)(4 218 291 385)(5 245 292 412)(6 220 293 387)(7 247 294 414)(8 222 295 389)(9 249 296 416)(10 224 297 391)(11 251 298 366)(12 226 299 393)(13 253 300 368)(14 228 301 395)(15 255 302 370)(16 230 303 397)(17 257 304 372)(18 232 305 399)(19 259 306 374)(20 234 307 401)(21 209 308 376)(22 236 309 403)(23 211 310 378)(24 238 311 405)(25 213 312 380)(26 240 261 407)(27 215 262 382)(28 242 263 409)(29 217 264 384)(30 244 265 411)(31 219 266 386)(32 246 267 413)(33 221 268 388)(34 248 269 415)(35 223 270 390)(36 250 271 365)(37 225 272 392)(38 252 273 367)(39 227 274 394)(40 254 275 369)(41 229 276 396)(42 256 277 371)(43 231 278 398)(44 258 279 373)(45 233 280 400)(46 260 281 375)(47 235 282 402)(48 210 283 377)(49 237 284 404)(50 212 285 379)(51 239 286 406)(52 214 287 381)(53 152 354 188)(54 127 355 163)(55 154 356 190)(56 129 357 165)(57 156 358 192)(58 131 359 167)(59 106 360 194)(60 133 361 169)(61 108 362 196)(62 135 363 171)(63 110 364 198)(64 137 313 173)(65 112 314 200)(66 139 315 175)(67 114 316 202)(68 141 317 177)(69 116 318 204)(70 143 319 179)(71 118 320 206)(72 145 321 181)(73 120 322 208)(74 147 323 183)(75 122 324 158)(76 149 325 185)(77 124 326 160)(78 151 327 187)(79 126 328 162)(80 153 329 189)(81 128 330 164)(82 155 331 191)(83 130 332 166)(84 105 333 193)(85 132 334 168)(86 107 335 195)(87 134 336 170)(88 109 337 197)(89 136 338 172)(90 111 339 199)(91 138 340 174)(92 113 341 201)(93 140 342 176)(94 115 343 203)(95 142 344 178)(96 117 345 205)(97 144 346 180)(98 119 347 207)(99 146 348 182)(100 121 349 157)(101 148 350 184)(102 123 351 159)(103 150 352 186)(104 125 353 161)

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

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

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

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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