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G = Dic13⋊Q8order 416 = 25·13

2nd semidirect product of Dic13 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.21D4, Dic132Q8, C133(C4⋊Q8), C2.8(Q8×D13), C26.56(C2×D4), (C2×C4).55D26, (C2×Q8).4D13, (Q8×C26).4C2, C26.15(C2×Q8), C4.10(C13⋊D4), (C2×C26).56C23, (C2×C52).63C22, (C4×Dic13).3C2, C26.D4.6C2, (C2×Dic26).10C2, C22.63(C22×D13), (C2×Dic13).20C22, C2.20(C2×C13⋊D4), SmallGroup(416,165)

Series: Derived Chief Lower central Upper central

C1C2×C26 — Dic13⋊Q8
C1C13C26C2×C26C2×Dic13C4×Dic13 — Dic13⋊Q8
C13C2×C26 — Dic13⋊Q8
C1C22C2×Q8

Generators and relations for Dic13⋊Q8
 G = < a,b,c,d | a26=c4=1, b2=a13, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a13b, bd=db, dcd-1=c-1 >

Subgroups: 360 in 68 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×8], C22, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C13, C42, C4⋊C4 [×4], C2×Q8, C2×Q8, C26, C26 [×2], C4⋊Q8, Dic13 [×4], Dic13 [×2], C52 [×2], C52 [×2], C2×C26, Dic26 [×2], C2×Dic13 [×4], C2×C52, C2×C52 [×2], Q8×C13 [×2], C4×Dic13, C26.D4 [×4], C2×Dic26, Q8×C26, Dic13⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, C2×D4, C2×Q8 [×2], D13, C4⋊Q8, D26 [×3], C13⋊D4 [×2], C22×D13, Q8×D13 [×2], C2×C13⋊D4, Dic13⋊Q8

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

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

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

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

74 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J13A···13F26A···26R52A···52AJ
order1222444444444413···1326···2652···52
size111122442626262652522···22···24···4

74 irreducible representations

dim11111222224
type+++++-+++-
imageC1C2C2C2C2Q8D4D13D26C13⋊D4Q8×D13
kernelDic13⋊Q8C4×Dic13C26.D4C2×Dic26Q8×C26Dic13C52C2×Q8C2×C4C4C2
# reps11411426182412

Matrix representation of Dic13⋊Q8 in GL6(𝔽53)

5200000
0520000
0052000
0005200
0000521
00002527
,
9360000
36440000
0092000
00334400
00003334
00002120
,
010000
5200000
000100
001000
000010
000001
,
44170000
1790000
0052000
0005200
000010
000001

G:=sub<GL(6,GF(53))| [52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,25,0,0,0,0,1,27],[9,36,0,0,0,0,36,44,0,0,0,0,0,0,9,33,0,0,0,0,20,44,0,0,0,0,0,0,33,21,0,0,0,0,34,20],[0,52,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[44,17,0,0,0,0,17,9,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic13⋊Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(416,165);
// by ID

G=gap.SmallGroup(416,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,55,362,116,50,13829]);
// Polycyclic

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

׿
×
𝔽