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## G = Q8×Dic13order 416 = 25·13

### Direct product of Q8 and Dic13

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×Dic13
G = < a,b,c,d | a4=c26=1, b2=a2, d2=c13, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 304 in 70 conjugacy classes, 51 normal (14 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C13, C42, C4⋊C4, C2×Q8, C26, C4×Q8, Dic13, Dic13, C52, C2×C26, C2×Dic13, C2×Dic13, C2×C52, Q8×C13, C4×Dic13, C523C4, Q8×C26, Q8×Dic13
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, D13, C4×Q8, Dic13, D26, C2×Dic13, C22×D13, Q8×D13, D52⋊C2, C22×Dic13, Q8×Dic13

Smallest permutation representation of Q8×Dic13
Regular action on 416 points
Generators in S416
(1 342 121 313)(2 343 122 314)(3 344 123 315)(4 345 124 316)(5 346 125 317)(6 347 126 318)(7 348 127 319)(8 349 128 320)(9 350 129 321)(10 351 130 322)(11 352 105 323)(12 353 106 324)(13 354 107 325)(14 355 108 326)(15 356 109 327)(16 357 110 328)(17 358 111 329)(18 359 112 330)(19 360 113 331)(20 361 114 332)(21 362 115 333)(22 363 116 334)(23 364 117 335)(24 339 118 336)(25 340 119 337)(26 341 120 338)(27 304 251 204)(28 305 252 205)(29 306 253 206)(30 307 254 207)(31 308 255 208)(32 309 256 183)(33 310 257 184)(34 311 258 185)(35 312 259 186)(36 287 260 187)(37 288 235 188)(38 289 236 189)(39 290 237 190)(40 291 238 191)(41 292 239 192)(42 293 240 193)(43 294 241 194)(44 295 242 195)(45 296 243 196)(46 297 244 197)(47 298 245 198)(48 299 246 199)(49 300 247 200)(50 301 248 201)(51 302 249 202)(52 303 250 203)(53 99 405 285)(54 100 406 286)(55 101 407 261)(56 102 408 262)(57 103 409 263)(58 104 410 264)(59 79 411 265)(60 80 412 266)(61 81 413 267)(62 82 414 268)(63 83 415 269)(64 84 416 270)(65 85 391 271)(66 86 392 272)(67 87 393 273)(68 88 394 274)(69 89 395 275)(70 90 396 276)(71 91 397 277)(72 92 398 278)(73 93 399 279)(74 94 400 280)(75 95 401 281)(76 96 402 282)(77 97 403 283)(78 98 404 284)(131 162 231 389)(132 163 232 390)(133 164 233 365)(134 165 234 366)(135 166 209 367)(136 167 210 368)(137 168 211 369)(138 169 212 370)(139 170 213 371)(140 171 214 372)(141 172 215 373)(142 173 216 374)(143 174 217 375)(144 175 218 376)(145 176 219 377)(146 177 220 378)(147 178 221 379)(148 179 222 380)(149 180 223 381)(150 181 224 382)(151 182 225 383)(152 157 226 384)(153 158 227 385)(154 159 228 386)(155 160 229 387)(156 161 230 388)
(1 179 121 380)(2 180 122 381)(3 181 123 382)(4 182 124 383)(5 157 125 384)(6 158 126 385)(7 159 127 386)(8 160 128 387)(9 161 129 388)(10 162 130 389)(11 163 105 390)(12 164 106 365)(13 165 107 366)(14 166 108 367)(15 167 109 368)(16 168 110 369)(17 169 111 370)(18 170 112 371)(19 171 113 372)(20 172 114 373)(21 173 115 374)(22 174 116 375)(23 175 117 376)(24 176 118 377)(25 177 119 378)(26 178 120 379)(27 412 251 60)(28 413 252 61)(29 414 253 62)(30 415 254 63)(31 416 255 64)(32 391 256 65)(33 392 257 66)(34 393 258 67)(35 394 259 68)(36 395 260 69)(37 396 235 70)(38 397 236 71)(39 398 237 72)(40 399 238 73)(41 400 239 74)(42 401 240 75)(43 402 241 76)(44 403 242 77)(45 404 243 78)(46 405 244 53)(47 406 245 54)(48 407 246 55)(49 408 247 56)(50 409 248 57)(51 410 249 58)(52 411 250 59)(79 203 265 303)(80 204 266 304)(81 205 267 305)(82 206 268 306)(83 207 269 307)(84 208 270 308)(85 183 271 309)(86 184 272 310)(87 185 273 311)(88 186 274 312)(89 187 275 287)(90 188 276 288)(91 189 277 289)(92 190 278 290)(93 191 279 291)(94 192 280 292)(95 193 281 293)(96 194 282 294)(97 195 283 295)(98 196 284 296)(99 197 285 297)(100 198 286 298)(101 199 261 299)(102 200 262 300)(103 201 263 301)(104 202 264 302)(131 322 231 351)(132 323 232 352)(133 324 233 353)(134 325 234 354)(135 326 209 355)(136 327 210 356)(137 328 211 357)(138 329 212 358)(139 330 213 359)(140 331 214 360)(141 332 215 361)(142 333 216 362)(143 334 217 363)(144 335 218 364)(145 336 219 339)(146 337 220 340)(147 338 221 341)(148 313 222 342)(149 314 223 343)(150 315 224 344)(151 316 225 345)(152 317 226 346)(153 318 227 347)(154 319 228 348)(155 320 229 349)(156 321 230 350)
(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)
(1 190 14 203)(2 189 15 202)(3 188 16 201)(4 187 17 200)(5 186 18 199)(6 185 19 198)(7 184 20 197)(8 183 21 196)(9 208 22 195)(10 207 23 194)(11 206 24 193)(12 205 25 192)(13 204 26 191)(27 341 40 354)(28 340 41 353)(29 339 42 352)(30 364 43 351)(31 363 44 350)(32 362 45 349)(33 361 46 348)(34 360 47 347)(35 359 48 346)(36 358 49 345)(37 357 50 344)(38 356 51 343)(39 355 52 342)(53 228 66 215)(54 227 67 214)(55 226 68 213)(56 225 69 212)(57 224 70 211)(58 223 71 210)(59 222 72 209)(60 221 73 234)(61 220 74 233)(62 219 75 232)(63 218 76 231)(64 217 77 230)(65 216 78 229)(79 380 92 367)(80 379 93 366)(81 378 94 365)(82 377 95 390)(83 376 96 389)(84 375 97 388)(85 374 98 387)(86 373 99 386)(87 372 100 385)(88 371 101 384)(89 370 102 383)(90 369 103 382)(91 368 104 381)(105 306 118 293)(106 305 119 292)(107 304 120 291)(108 303 121 290)(109 302 122 289)(110 301 123 288)(111 300 124 287)(112 299 125 312)(113 298 126 311)(114 297 127 310)(115 296 128 309)(116 295 129 308)(117 294 130 307)(131 415 144 402)(132 414 145 401)(133 413 146 400)(134 412 147 399)(135 411 148 398)(136 410 149 397)(137 409 150 396)(138 408 151 395)(139 407 152 394)(140 406 153 393)(141 405 154 392)(142 404 155 391)(143 403 156 416)(157 274 170 261)(158 273 171 286)(159 272 172 285)(160 271 173 284)(161 270 174 283)(162 269 175 282)(163 268 176 281)(164 267 177 280)(165 266 178 279)(166 265 179 278)(167 264 180 277)(168 263 181 276)(169 262 182 275)(235 328 248 315)(236 327 249 314)(237 326 250 313)(238 325 251 338)(239 324 252 337)(240 323 253 336)(241 322 254 335)(242 321 255 334)(243 320 256 333)(244 319 257 332)(245 318 258 331)(246 317 259 330)(247 316 260 329)

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

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

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

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 2 2 2 2 2 4 4 type + + + + - + + - - + image C1 C2 C2 C2 C4 Q8 C4○D4 D13 D26 Dic13 Q8×D13 D52⋊C2 kernel Q8×Dic13 C4×Dic13 C52⋊3C4 Q8×C26 Q8×C13 Dic13 C26 C2×Q8 C2×C4 Q8 C2 C2 # reps 1 3 3 1 8 2 2 6 18 24 6 6

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

 1 0 0 0 0 1 0 0 0 0 52 51 0 0 1 1
,
 52 0 0 0 0 52 0 0 0 0 47 36 0 0 24 6
,
 49 52 0 0 38 49 0 0 0 0 1 0 0 0 0 1
,
 25 21 0 0 3 28 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(53))| [1,0,0,0,0,1,0,0,0,0,52,1,0,0,51,1],[52,0,0,0,0,52,0,0,0,0,47,24,0,0,36,6],[49,38,0,0,52,49,0,0,0,0,1,0,0,0,0,1],[25,3,0,0,21,28,0,0,0,0,1,0,0,0,0,1] >;

Q8×Dic13 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xDic13");
// GroupNames label

G:=SmallGroup(416,166);
// by ID

G=gap.SmallGroup(416,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,188,86,13829]);
// Polycyclic

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

׿
×
𝔽