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G = Q8×Dic13order 416 = 25·13

Direct product of Q8 and Dic13

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8×Dic13, C134(C4×Q8), (Q8×C13)⋊6C4, C2.3(Q8×D13), C52.35(C2×C4), (C2×C4).56D26, (C2×Q8).5D13, (Q8×C26).5C2, C26.16(C2×Q8), C523C4.12C2, C4.4(C2×Dic13), C26.35(C4○D4), C26.39(C22×C4), (C2×C26).57C23, (C2×C52).39C22, (C4×Dic13).4C2, C2.3(D52⋊C2), C2.7(C22×Dic13), C22.26(C22×D13), (C2×Dic13).43C22, SmallGroup(416,166)

Series: Derived Chief Lower central Upper central

C1C26 — Q8×Dic13
C1C13C26C2×C26C2×Dic13C4×Dic13 — Q8×Dic13
C13C26 — Q8×Dic13
C1C22C2×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 [×3], C4 [×6], C4 [×5], C22, C2×C4 [×3], C2×C4 [×4], Q8 [×4], C13, C42 [×3], C4⋊C4 [×3], C2×Q8, C26 [×3], C4×Q8, Dic13 [×2], Dic13 [×3], C52 [×6], C2×C26, C2×Dic13, C2×Dic13 [×3], C2×C52 [×3], Q8×C13 [×4], C4×Dic13 [×3], C523C4 [×3], Q8×C26, Q8×Dic13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, C22×C4, C2×Q8, C4○D4, D13, C4×Q8, Dic13 [×4], D26 [×3], C2×Dic13 [×6], C22×D13, Q8×D13, D52⋊C2, C22×Dic13, Q8×Dic13

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

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

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

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

80 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J4K···4P13A···13F26A···26R52A···52AJ
order12224···444444···413···1326···2652···52
size11112···21313131326···262···22···24···4

80 irreducible representations

dim111112222244
type++++-++--+
imageC1C2C2C2C4Q8C4○D4D13D26Dic13Q8×D13D52⋊C2
kernelQ8×Dic13C4×Dic13C523C4Q8×C26Q8×C13Dic13C26C2×Q8C2×C4Q8C2C2
# reps13318226182466

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

1000
0100
005251
0011
,
52000
05200
004736
00246
,
495200
384900
0010
0001
,
252100
32800
0010
0001
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