Dic13:Q8 - GroupNames

G = Dic₁₃⋊Q₈ order 416 = 2⁵·13

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₅₂.₂₁D₄, Dic₁₃⋊₂Q₈, C₁₃⋊₃(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

Derived series

C₁ — C₂×C₂₆ — Dic₁₃⋊Q₈

Chief series

C₁ — C₁₃ — C₂₆ — C₂×C₂₆ — C₂×Dic₁₃ — C₄×Dic₁₃ — Dic₁₃⋊Q₈

Lower central

C₁₃ — C₂×C₂₆ — Dic₁₃⋊Q₈

Upper central

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

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

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

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

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

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

74 conjugacy classes

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

74 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	4
type	+	+	+	+	+	-	+	+	+		-
image	C₁	C₂	C₂	C₂	C₂	Q₈	D₄	D₁₃	D₂₆	C₁₃⋊D₄	Q₈×D₁₃
kernel	Dic₁₃⋊Q₈	C₄×Dic₁₃	C₂₆.D₄	C₂×Dic₂₆	Q₈×C₂₆	Dic₁₃	C₅₂	C₂×Q₈	C₂×C₄	C₄	C₂
# reps	1	1	4	1	1	4	2	6	18	24	12

Matrix representation of Dic₁₃⋊Q₈ ►in GL₆(𝔽₅₃)

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	1
0	0	0	0	25	27

9	36	0	0	0	0
36	44	0	0	0	0
0	0	9	20	0	0
0	0	33	44	0	0
0	0	0	0	33	34
0	0	0	0	21	20

0	1	0	0	0	0
52	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

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

G = Dic13⋊Q8 order 416 = 25·13

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

G = Dic₁₃⋊Q₈ order 416 = 2⁵·13

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