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## G = Dic13.Q8order 416 = 25·13

### 1st non-split extension by Dic13 of Q8 acting via Q8/C4=C2

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 — C4⋊C4

Generators and relations for Dic13.Q8
G = < a,b,c,d | a26=c4=1, b2=a13, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a13b, bd=db, dcd-1=a13c-1 >

Subgroups: 296 in 56 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, C13, C42, C4⋊C4, C4⋊C4, C26, C42.C2, Dic13, Dic13, C52, C2×C26, C2×Dic13, C2×C52, C4×Dic13, C26.D4, C523C4, C13×C4⋊C4, Dic13.Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, D13, C42.C2, D26, C22×D13, D525C2, D42D13, Q8×D13, 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 98 14 85)(2 97 15 84)(3 96 16 83)(4 95 17 82)(5 94 18 81)(6 93 19 80)(7 92 20 79)(8 91 21 104)(9 90 22 103)(10 89 23 102)(11 88 24 101)(12 87 25 100)(13 86 26 99)(27 105 40 118)(28 130 41 117)(29 129 42 116)(30 128 43 115)(31 127 44 114)(32 126 45 113)(33 125 46 112)(34 124 47 111)(35 123 48 110)(36 122 49 109)(37 121 50 108)(38 120 51 107)(39 119 52 106)(53 264 66 277)(54 263 67 276)(55 262 68 275)(56 261 69 274)(57 286 70 273)(58 285 71 272)(59 284 72 271)(60 283 73 270)(61 282 74 269)(62 281 75 268)(63 280 76 267)(64 279 77 266)(65 278 78 265)(131 221 144 234)(132 220 145 233)(133 219 146 232)(134 218 147 231)(135 217 148 230)(136 216 149 229)(137 215 150 228)(138 214 151 227)(139 213 152 226)(140 212 153 225)(141 211 154 224)(142 210 155 223)(143 209 156 222)(157 336 170 323)(158 335 171 322)(159 334 172 321)(160 333 173 320)(161 332 174 319)(162 331 175 318)(163 330 176 317)(164 329 177 316)(165 328 178 315)(166 327 179 314)(167 326 180 313)(168 325 181 338)(169 324 182 337)(183 259 196 246)(184 258 197 245)(185 257 198 244)(186 256 199 243)(187 255 200 242)(188 254 201 241)(189 253 202 240)(190 252 203 239)(191 251 204 238)(192 250 205 237)(193 249 206 236)(194 248 207 235)(195 247 208 260)(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)(339 379 352 366)(340 378 353 365)(341 377 354 390)(342 376 355 389)(343 375 356 388)(344 374 357 387)(345 373 358 386)(346 372 359 385)(347 371 360 384)(348 370 361 383)(349 369 362 382)(350 368 363 381)(351 367 364 380)
(1 289 137 59)(2 290 138 60)(3 291 139 61)(4 292 140 62)(5 293 141 63)(6 294 142 64)(7 295 143 65)(8 296 144 66)(9 297 145 67)(10 298 146 68)(11 299 147 69)(12 300 148 70)(13 301 149 71)(14 302 150 72)(15 303 151 73)(16 304 152 74)(17 305 153 75)(18 306 154 76)(19 307 155 77)(20 308 156 78)(21 309 131 53)(22 310 132 54)(23 311 133 55)(24 312 134 56)(25 287 135 57)(26 288 136 58)(27 389 188 337)(28 390 189 338)(29 365 190 313)(30 366 191 314)(31 367 192 315)(32 368 193 316)(33 369 194 317)(34 370 195 318)(35 371 196 319)(36 372 197 320)(37 373 198 321)(38 374 199 322)(39 375 200 323)(40 376 201 324)(41 377 202 325)(42 378 203 326)(43 379 204 327)(44 380 205 328)(45 381 206 329)(46 382 207 330)(47 383 208 331)(48 384 183 332)(49 385 184 333)(50 386 185 334)(51 387 186 335)(52 388 187 336)(79 415 222 278)(80 416 223 279)(81 391 224 280)(82 392 225 281)(83 393 226 282)(84 394 227 283)(85 395 228 284)(86 396 229 285)(87 397 230 286)(88 398 231 261)(89 399 232 262)(90 400 233 263)(91 401 234 264)(92 402 209 265)(93 403 210 266)(94 404 211 267)(95 405 212 268)(96 406 213 269)(97 407 214 270)(98 408 215 271)(99 409 216 272)(100 410 217 273)(101 411 218 274)(102 412 219 275)(103 413 220 276)(104 414 221 277)(105 355 254 182)(106 356 255 157)(107 357 256 158)(108 358 257 159)(109 359 258 160)(110 360 259 161)(111 361 260 162)(112 362 235 163)(113 363 236 164)(114 364 237 165)(115 339 238 166)(116 340 239 167)(117 341 240 168)(118 342 241 169)(119 343 242 170)(120 344 243 171)(121 345 244 172)(122 346 245 173)(123 347 246 174)(124 348 247 175)(125 349 248 176)(126 350 249 177)(127 351 250 178)(128 352 251 179)(129 353 252 180)(130 354 253 181)
(1 241 137 118)(2 240 138 117)(3 239 139 116)(4 238 140 115)(5 237 141 114)(6 236 142 113)(7 235 143 112)(8 260 144 111)(9 259 145 110)(10 258 146 109)(11 257 147 108)(12 256 148 107)(13 255 149 106)(14 254 150 105)(15 253 151 130)(16 252 152 129)(17 251 153 128)(18 250 154 127)(19 249 155 126)(20 248 156 125)(21 247 131 124)(22 246 132 123)(23 245 133 122)(24 244 134 121)(25 243 135 120)(26 242 136 119)(27 98 188 215)(28 97 189 214)(29 96 190 213)(30 95 191 212)(31 94 192 211)(32 93 193 210)(33 92 194 209)(34 91 195 234)(35 90 196 233)(36 89 197 232)(37 88 198 231)(38 87 199 230)(39 86 200 229)(40 85 201 228)(41 84 202 227)(42 83 203 226)(43 82 204 225)(44 81 205 224)(45 80 206 223)(46 79 207 222)(47 104 208 221)(48 103 183 220)(49 102 184 219)(50 101 185 218)(51 100 186 217)(52 99 187 216)(53 162 309 361)(54 161 310 360)(55 160 311 359)(56 159 312 358)(57 158 287 357)(58 157 288 356)(59 182 289 355)(60 181 290 354)(61 180 291 353)(62 179 292 352)(63 178 293 351)(64 177 294 350)(65 176 295 349)(66 175 296 348)(67 174 297 347)(68 173 298 346)(69 172 299 345)(70 171 300 344)(71 170 301 343)(72 169 302 342)(73 168 303 341)(74 167 304 340)(75 166 305 339)(76 165 306 364)(77 164 307 363)(78 163 308 362)(261 334 398 386)(262 333 399 385)(263 332 400 384)(264 331 401 383)(265 330 402 382)(266 329 403 381)(267 328 404 380)(268 327 405 379)(269 326 406 378)(270 325 407 377)(271 324 408 376)(272 323 409 375)(273 322 410 374)(274 321 411 373)(275 320 412 372)(276 319 413 371)(277 318 414 370)(278 317 415 369)(279 316 416 368)(280 315 391 367)(281 314 392 366)(282 313 393 365)(283 338 394 390)(284 337 395 389)(285 336 396 388)(286 335 397 387)

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

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

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

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 4 type + + + + + - + + - - image C1 C2 C2 C2 C2 Q8 C4○D4 D13 D26 D52⋊5C2 D4⋊2D13 Q8×D13 kernel Dic13.Q8 C4×Dic13 C26.D4 C52⋊3C4 C13×C4⋊C4 Dic13 C26 C4⋊C4 C2×C4 C2 C2 C2 # reps 1 1 4 1 1 2 4 6 18 24 6 6

Matrix representation of Dic13.Q8 in GL4(𝔽53) generated by

 0 52 0 0 1 42 0 0 0 0 52 0 0 0 0 52
,
 18 35 0 0 21 35 0 0 0 0 6 50 0 0 30 47
,
 45 40 0 0 13 8 0 0 0 0 52 44 0 0 12 1
,
 43 10 0 0 6 10 0 0 0 0 30 0 0 0 0 30
G:=sub<GL(4,GF(53))| [0,1,0,0,52,42,0,0,0,0,52,0,0,0,0,52],[18,21,0,0,35,35,0,0,0,0,6,30,0,0,50,47],[45,13,0,0,40,8,0,0,0,0,52,12,0,0,44,1],[43,6,0,0,10,10,0,0,0,0,30,0,0,0,0,30] >;

Dic13.Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(416,110);
// by ID

G=gap.SmallGroup(416,110);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,55,218,188,86,13829]);
// Polycyclic

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

׿
×
𝔽