direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×Q8⋊C4, Q8⋊1C52, C52.61D4, C26.6Q16, C26.10SD16, C4⋊C4.1C26, C4.2(C2×C52), (C2×C8).1C26, (Q8×C13)⋊7C4, (C2×C104).3C2, C52.50(C2×C4), (C2×C26).47D4, C4.12(D4×C13), (Q8×C26).7C2, (C2×Q8).2C26, C2.1(C13×Q16), C2.2(C13×SD16), C22.9(D4×C13), C26.36(C22⋊C4), (C2×C52).115C22, (C13×C4⋊C4).8C2, (C2×C4).18(C2×C26), C2.7(C13×C22⋊C4), SmallGroup(416,53)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×Q8⋊C4
G = < a,b,c,d | a13=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Smallest permutation representation of C13×Q8⋊C4
►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)
(1 326 229 49)(2 327 230 50)(3 328 231 51)(4 329 232 52)(5 330 233 40)(6 331 234 41)(7 332 222 42)(8 333 223 43)(9 334 224 44)(10 335 225 45)(11 336 226 46)(12 337 227 47)(13 338 228 48)(14 219 366 243)(15 220 367 244)(16 221 368 245)(17 209 369 246)(18 210 370 247)(19 211 371 235)(20 212 372 236)(21 213 373 237)(22 214 374 238)(23 215 375 239)(24 216 376 240)(25 217 377 241)(26 218 365 242)(27 172 386 90)(28 173 387 91)(29 174 388 79)(30 175 389 80)(31 176 390 81)(32 177 378 82)(33 178 379 83)(34 179 380 84)(35 180 381 85)(36 181 382 86)(37 182 383 87)(38 170 384 88)(39 171 385 89)(53 351 248 276)(54 339 249 277)(55 340 250 278)(56 341 251 279)(57 342 252 280)(58 343 253 281)(59 344 254 282)(60 345 255 283)(61 346 256 284)(62 347 257 285)(63 348 258 286)(64 349 259 274)(65 350 260 275)(66 415 109 185)(67 416 110 186)(68 404 111 187)(69 405 112 188)(70 406 113 189)(71 407 114 190)(72 408 115 191)(73 409 116 192)(74 410 117 193)(75 411 105 194)(76 412 106 195)(77 413 107 183)(78 414 108 184)(92 358 159 197)(93 359 160 198)(94 360 161 199)(95 361 162 200)(96 362 163 201)(97 363 164 202)(98 364 165 203)(99 352 166 204)(100 353 167 205)(101 354 168 206)(102 355 169 207)(103 356 157 208)(104 357 158 196)(118 149 319 270)(119 150 320 271)(120 151 321 272)(121 152 322 273)(122 153 323 261)(123 154 324 262)(124 155 325 263)(125 156 313 264)(126 144 314 265)(127 145 315 266)(128 146 316 267)(129 147 317 268)(130 148 318 269)(131 288 305 394)(132 289 306 395)(133 290 307 396)(134 291 308 397)(135 292 309 398)(136 293 310 399)(137 294 311 400)(138 295 312 401)(139 296 300 402)(140 297 301 403)(141 298 302 391)(142 299 303 392)(143 287 304 393)
(1 381 229 35)(2 382 230 36)(3 383 231 37)(4 384 232 38)(5 385 233 39)(6 386 234 27)(7 387 222 28)(8 388 223 29)(9 389 224 30)(10 390 225 31)(11 378 226 32)(12 379 227 33)(13 380 228 34)(14 299 366 392)(15 287 367 393)(16 288 368 394)(17 289 369 395)(18 290 370 396)(19 291 371 397)(20 292 372 398)(21 293 373 399)(22 294 374 400)(23 295 375 401)(24 296 376 402)(25 297 377 403)(26 298 365 391)(40 89 330 171)(41 90 331 172)(42 91 332 173)(43 79 333 174)(44 80 334 175)(45 81 335 176)(46 82 336 177)(47 83 337 178)(48 84 338 179)(49 85 326 180)(50 86 327 181)(51 87 328 182)(52 88 329 170)(53 404 248 187)(54 405 249 188)(55 406 250 189)(56 407 251 190)(57 408 252 191)(58 409 253 192)(59 410 254 193)(60 411 255 194)(61 412 256 195)(62 413 257 183)(63 414 258 184)(64 415 259 185)(65 416 260 186)(66 274 109 349)(67 275 110 350)(68 276 111 351)(69 277 112 339)(70 278 113 340)(71 279 114 341)(72 280 115 342)(73 281 116 343)(74 282 117 344)(75 283 105 345)(76 284 106 346)(77 285 107 347)(78 286 108 348)(92 322 159 121)(93 323 160 122)(94 324 161 123)(95 325 162 124)(96 313 163 125)(97 314 164 126)(98 315 165 127)(99 316 166 128)(100 317 167 129)(101 318 168 130)(102 319 169 118)(103 320 157 119)(104 321 158 120)(131 245 305 221)(132 246 306 209)(133 247 307 210)(134 235 308 211)(135 236 309 212)(136 237 310 213)(137 238 311 214)(138 239 312 215)(139 240 300 216)(140 241 301 217)(141 242 302 218)(142 243 303 219)(143 244 304 220)(144 202 265 363)(145 203 266 364)(146 204 267 352)(147 205 268 353)(148 206 269 354)(149 207 270 355)(150 208 271 356)(151 196 272 357)(152 197 273 358)(153 198 261 359)(154 199 262 360)(155 200 263 361)(156 201 264 362)
(1 123 309 54)(2 124 310 55)(3 125 311 56)(4 126 312 57)(5 127 300 58)(6 128 301 59)(7 129 302 60)(8 130 303 61)(9 118 304 62)(10 119 305 63)(11 120 306 64)(12 121 307 65)(13 122 308 53)(14 195 174 168)(15 183 175 169)(16 184 176 157)(17 185 177 158)(18 186 178 159)(19 187 179 160)(20 188 180 161)(21 189 181 162)(22 190 182 163)(23 191 170 164)(24 192 171 165)(25 193 172 166)(26 194 173 167)(27 204 241 74)(28 205 242 75)(29 206 243 76)(30 207 244 77)(31 208 245 78)(32 196 246 66)(33 197 247 67)(34 198 235 68)(35 199 236 69)(36 200 237 70)(37 201 238 71)(38 202 239 72)(39 203 240 73)(40 145 296 343)(41 146 297 344)(42 147 298 345)(43 148 299 346)(44 149 287 347)(45 150 288 348)(46 151 289 349)(47 152 290 350)(48 153 291 351)(49 154 292 339)(50 155 293 340)(51 156 294 341)(52 144 295 342)(79 101 366 412)(80 102 367 413)(81 103 368 414)(82 104 369 415)(83 92 370 416)(84 93 371 404)(85 94 372 405)(86 95 373 406)(87 96 374 407)(88 97 375 408)(89 98 376 409)(90 99 377 410)(91 100 365 411)(105 387 353 218)(106 388 354 219)(107 389 355 220)(108 390 356 221)(109 378 357 209)(110 379 358 210)(111 380 359 211)(112 381 360 212)(113 382 361 213)(114 383 362 214)(115 384 363 215)(116 385 364 216)(117 386 352 217)(131 258 225 320)(132 259 226 321)(133 260 227 322)(134 248 228 323)(135 249 229 324)(136 250 230 325)(137 251 231 313)(138 252 232 314)(139 253 233 315)(140 254 234 316)(141 255 222 317)(142 256 223 318)(143 257 224 319)(261 397 276 338)(262 398 277 326)(263 399 278 327)(264 400 279 328)(265 401 280 329)(266 402 281 330)(267 403 282 331)(268 391 283 332)(269 392 284 333)(270 393 285 334)(271 394 286 335)(272 395 274 336)(273 396 275 337)
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), (1,326,229,49)(2,327,230,50)(3,328,231,51)(4,329,232,52)(5,330,233,40)(6,331,234,41)(7,332,222,42)(8,333,223,43)(9,334,224,44)(10,335,225,45)(11,336,226,46)(12,337,227,47)(13,338,228,48)(14,219,366,243)(15,220,367,244)(16,221,368,245)(17,209,369,246)(18,210,370,247)(19,211,371,235)(20,212,372,236)(21,213,373,237)(22,214,374,238)(23,215,375,239)(24,216,376,240)(25,217,377,241)(26,218,365,242)(27,172,386,90)(28,173,387,91)(29,174,388,79)(30,175,389,80)(31,176,390,81)(32,177,378,82)(33,178,379,83)(34,179,380,84)(35,180,381,85)(36,181,382,86)(37,182,383,87)(38,170,384,88)(39,171,385,89)(53,351,248,276)(54,339,249,277)(55,340,250,278)(56,341,251,279)(57,342,252,280)(58,343,253,281)(59,344,254,282)(60,345,255,283)(61,346,256,284)(62,347,257,285)(63,348,258,286)(64,349,259,274)(65,350,260,275)(66,415,109,185)(67,416,110,186)(68,404,111,187)(69,405,112,188)(70,406,113,189)(71,407,114,190)(72,408,115,191)(73,409,116,192)(74,410,117,193)(75,411,105,194)(76,412,106,195)(77,413,107,183)(78,414,108,184)(92,358,159,197)(93,359,160,198)(94,360,161,199)(95,361,162,200)(96,362,163,201)(97,363,164,202)(98,364,165,203)(99,352,166,204)(100,353,167,205)(101,354,168,206)(102,355,169,207)(103,356,157,208)(104,357,158,196)(118,149,319,270)(119,150,320,271)(120,151,321,272)(121,152,322,273)(122,153,323,261)(123,154,324,262)(124,155,325,263)(125,156,313,264)(126,144,314,265)(127,145,315,266)(128,146,316,267)(129,147,317,268)(130,148,318,269)(131,288,305,394)(132,289,306,395)(133,290,307,396)(134,291,308,397)(135,292,309,398)(136,293,310,399)(137,294,311,400)(138,295,312,401)(139,296,300,402)(140,297,301,403)(141,298,302,391)(142,299,303,392)(143,287,304,393), (1,381,229,35)(2,382,230,36)(3,383,231,37)(4,384,232,38)(5,385,233,39)(6,386,234,27)(7,387,222,28)(8,388,223,29)(9,389,224,30)(10,390,225,31)(11,378,226,32)(12,379,227,33)(13,380,228,34)(14,299,366,392)(15,287,367,393)(16,288,368,394)(17,289,369,395)(18,290,370,396)(19,291,371,397)(20,292,372,398)(21,293,373,399)(22,294,374,400)(23,295,375,401)(24,296,376,402)(25,297,377,403)(26,298,365,391)(40,89,330,171)(41,90,331,172)(42,91,332,173)(43,79,333,174)(44,80,334,175)(45,81,335,176)(46,82,336,177)(47,83,337,178)(48,84,338,179)(49,85,326,180)(50,86,327,181)(51,87,328,182)(52,88,329,170)(53,404,248,187)(54,405,249,188)(55,406,250,189)(56,407,251,190)(57,408,252,191)(58,409,253,192)(59,410,254,193)(60,411,255,194)(61,412,256,195)(62,413,257,183)(63,414,258,184)(64,415,259,185)(65,416,260,186)(66,274,109,349)(67,275,110,350)(68,276,111,351)(69,277,112,339)(70,278,113,340)(71,279,114,341)(72,280,115,342)(73,281,116,343)(74,282,117,344)(75,283,105,345)(76,284,106,346)(77,285,107,347)(78,286,108,348)(92,322,159,121)(93,323,160,122)(94,324,161,123)(95,325,162,124)(96,313,163,125)(97,314,164,126)(98,315,165,127)(99,316,166,128)(100,317,167,129)(101,318,168,130)(102,319,169,118)(103,320,157,119)(104,321,158,120)(131,245,305,221)(132,246,306,209)(133,247,307,210)(134,235,308,211)(135,236,309,212)(136,237,310,213)(137,238,311,214)(138,239,312,215)(139,240,300,216)(140,241,301,217)(141,242,302,218)(142,243,303,219)(143,244,304,220)(144,202,265,363)(145,203,266,364)(146,204,267,352)(147,205,268,353)(148,206,269,354)(149,207,270,355)(150,208,271,356)(151,196,272,357)(152,197,273,358)(153,198,261,359)(154,199,262,360)(155,200,263,361)(156,201,264,362), (1,123,309,54)(2,124,310,55)(3,125,311,56)(4,126,312,57)(5,127,300,58)(6,128,301,59)(7,129,302,60)(8,130,303,61)(9,118,304,62)(10,119,305,63)(11,120,306,64)(12,121,307,65)(13,122,308,53)(14,195,174,168)(15,183,175,169)(16,184,176,157)(17,185,177,158)(18,186,178,159)(19,187,179,160)(20,188,180,161)(21,189,181,162)(22,190,182,163)(23,191,170,164)(24,192,171,165)(25,193,172,166)(26,194,173,167)(27,204,241,74)(28,205,242,75)(29,206,243,76)(30,207,244,77)(31,208,245,78)(32,196,246,66)(33,197,247,67)(34,198,235,68)(35,199,236,69)(36,200,237,70)(37,201,238,71)(38,202,239,72)(39,203,240,73)(40,145,296,343)(41,146,297,344)(42,147,298,345)(43,148,299,346)(44,149,287,347)(45,150,288,348)(46,151,289,349)(47,152,290,350)(48,153,291,351)(49,154,292,339)(50,155,293,340)(51,156,294,341)(52,144,295,342)(79,101,366,412)(80,102,367,413)(81,103,368,414)(82,104,369,415)(83,92,370,416)(84,93,371,404)(85,94,372,405)(86,95,373,406)(87,96,374,407)(88,97,375,408)(89,98,376,409)(90,99,377,410)(91,100,365,411)(105,387,353,218)(106,388,354,219)(107,389,355,220)(108,390,356,221)(109,378,357,209)(110,379,358,210)(111,380,359,211)(112,381,360,212)(113,382,361,213)(114,383,362,214)(115,384,363,215)(116,385,364,216)(117,386,352,217)(131,258,225,320)(132,259,226,321)(133,260,227,322)(134,248,228,323)(135,249,229,324)(136,250,230,325)(137,251,231,313)(138,252,232,314)(139,253,233,315)(140,254,234,316)(141,255,222,317)(142,256,223,318)(143,257,224,319)(261,397,276,338)(262,398,277,326)(263,399,278,327)(264,400,279,328)(265,401,280,329)(266,402,281,330)(267,403,282,331)(268,391,283,332)(269,392,284,333)(270,393,285,334)(271,394,286,335)(272,395,274,336)(273,396,275,337)>;
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), (1,326,229,49)(2,327,230,50)(3,328,231,51)(4,329,232,52)(5,330,233,40)(6,331,234,41)(7,332,222,42)(8,333,223,43)(9,334,224,44)(10,335,225,45)(11,336,226,46)(12,337,227,47)(13,338,228,48)(14,219,366,243)(15,220,367,244)(16,221,368,245)(17,209,369,246)(18,210,370,247)(19,211,371,235)(20,212,372,236)(21,213,373,237)(22,214,374,238)(23,215,375,239)(24,216,376,240)(25,217,377,241)(26,218,365,242)(27,172,386,90)(28,173,387,91)(29,174,388,79)(30,175,389,80)(31,176,390,81)(32,177,378,82)(33,178,379,83)(34,179,380,84)(35,180,381,85)(36,181,382,86)(37,182,383,87)(38,170,384,88)(39,171,385,89)(53,351,248,276)(54,339,249,277)(55,340,250,278)(56,341,251,279)(57,342,252,280)(58,343,253,281)(59,344,254,282)(60,345,255,283)(61,346,256,284)(62,347,257,285)(63,348,258,286)(64,349,259,274)(65,350,260,275)(66,415,109,185)(67,416,110,186)(68,404,111,187)(69,405,112,188)(70,406,113,189)(71,407,114,190)(72,408,115,191)(73,409,116,192)(74,410,117,193)(75,411,105,194)(76,412,106,195)(77,413,107,183)(78,414,108,184)(92,358,159,197)(93,359,160,198)(94,360,161,199)(95,361,162,200)(96,362,163,201)(97,363,164,202)(98,364,165,203)(99,352,166,204)(100,353,167,205)(101,354,168,206)(102,355,169,207)(103,356,157,208)(104,357,158,196)(118,149,319,270)(119,150,320,271)(120,151,321,272)(121,152,322,273)(122,153,323,261)(123,154,324,262)(124,155,325,263)(125,156,313,264)(126,144,314,265)(127,145,315,266)(128,146,316,267)(129,147,317,268)(130,148,318,269)(131,288,305,394)(132,289,306,395)(133,290,307,396)(134,291,308,397)(135,292,309,398)(136,293,310,399)(137,294,311,400)(138,295,312,401)(139,296,300,402)(140,297,301,403)(141,298,302,391)(142,299,303,392)(143,287,304,393), (1,381,229,35)(2,382,230,36)(3,383,231,37)(4,384,232,38)(5,385,233,39)(6,386,234,27)(7,387,222,28)(8,388,223,29)(9,389,224,30)(10,390,225,31)(11,378,226,32)(12,379,227,33)(13,380,228,34)(14,299,366,392)(15,287,367,393)(16,288,368,394)(17,289,369,395)(18,290,370,396)(19,291,371,397)(20,292,372,398)(21,293,373,399)(22,294,374,400)(23,295,375,401)(24,296,376,402)(25,297,377,403)(26,298,365,391)(40,89,330,171)(41,90,331,172)(42,91,332,173)(43,79,333,174)(44,80,334,175)(45,81,335,176)(46,82,336,177)(47,83,337,178)(48,84,338,179)(49,85,326,180)(50,86,327,181)(51,87,328,182)(52,88,329,170)(53,404,248,187)(54,405,249,188)(55,406,250,189)(56,407,251,190)(57,408,252,191)(58,409,253,192)(59,410,254,193)(60,411,255,194)(61,412,256,195)(62,413,257,183)(63,414,258,184)(64,415,259,185)(65,416,260,186)(66,274,109,349)(67,275,110,350)(68,276,111,351)(69,277,112,339)(70,278,113,340)(71,279,114,341)(72,280,115,342)(73,281,116,343)(74,282,117,344)(75,283,105,345)(76,284,106,346)(77,285,107,347)(78,286,108,348)(92,322,159,121)(93,323,160,122)(94,324,161,123)(95,325,162,124)(96,313,163,125)(97,314,164,126)(98,315,165,127)(99,316,166,128)(100,317,167,129)(101,318,168,130)(102,319,169,118)(103,320,157,119)(104,321,158,120)(131,245,305,221)(132,246,306,209)(133,247,307,210)(134,235,308,211)(135,236,309,212)(136,237,310,213)(137,238,311,214)(138,239,312,215)(139,240,300,216)(140,241,301,217)(141,242,302,218)(142,243,303,219)(143,244,304,220)(144,202,265,363)(145,203,266,364)(146,204,267,352)(147,205,268,353)(148,206,269,354)(149,207,270,355)(150,208,271,356)(151,196,272,357)(152,197,273,358)(153,198,261,359)(154,199,262,360)(155,200,263,361)(156,201,264,362), (1,123,309,54)(2,124,310,55)(3,125,311,56)(4,126,312,57)(5,127,300,58)(6,128,301,59)(7,129,302,60)(8,130,303,61)(9,118,304,62)(10,119,305,63)(11,120,306,64)(12,121,307,65)(13,122,308,53)(14,195,174,168)(15,183,175,169)(16,184,176,157)(17,185,177,158)(18,186,178,159)(19,187,179,160)(20,188,180,161)(21,189,181,162)(22,190,182,163)(23,191,170,164)(24,192,171,165)(25,193,172,166)(26,194,173,167)(27,204,241,74)(28,205,242,75)(29,206,243,76)(30,207,244,77)(31,208,245,78)(32,196,246,66)(33,197,247,67)(34,198,235,68)(35,199,236,69)(36,200,237,70)(37,201,238,71)(38,202,239,72)(39,203,240,73)(40,145,296,343)(41,146,297,344)(42,147,298,345)(43,148,299,346)(44,149,287,347)(45,150,288,348)(46,151,289,349)(47,152,290,350)(48,153,291,351)(49,154,292,339)(50,155,293,340)(51,156,294,341)(52,144,295,342)(79,101,366,412)(80,102,367,413)(81,103,368,414)(82,104,369,415)(83,92,370,416)(84,93,371,404)(85,94,372,405)(86,95,373,406)(87,96,374,407)(88,97,375,408)(89,98,376,409)(90,99,377,410)(91,100,365,411)(105,387,353,218)(106,388,354,219)(107,389,355,220)(108,390,356,221)(109,378,357,209)(110,379,358,210)(111,380,359,211)(112,381,360,212)(113,382,361,213)(114,383,362,214)(115,384,363,215)(116,385,364,216)(117,386,352,217)(131,258,225,320)(132,259,226,321)(133,260,227,322)(134,248,228,323)(135,249,229,324)(136,250,230,325)(137,251,231,313)(138,252,232,314)(139,253,233,315)(140,254,234,316)(141,255,222,317)(142,256,223,318)(143,257,224,319)(261,397,276,338)(262,398,277,326)(263,399,278,327)(264,400,279,328)(265,401,280,329)(266,402,281,330)(267,403,282,331)(268,391,283,332)(269,392,284,333)(270,393,285,334)(271,394,286,335)(272,395,274,336)(273,396,275,337) );
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)], [(1,326,229,49),(2,327,230,50),(3,328,231,51),(4,329,232,52),(5,330,233,40),(6,331,234,41),(7,332,222,42),(8,333,223,43),(9,334,224,44),(10,335,225,45),(11,336,226,46),(12,337,227,47),(13,338,228,48),(14,219,366,243),(15,220,367,244),(16,221,368,245),(17,209,369,246),(18,210,370,247),(19,211,371,235),(20,212,372,236),(21,213,373,237),(22,214,374,238),(23,215,375,239),(24,216,376,240),(25,217,377,241),(26,218,365,242),(27,172,386,90),(28,173,387,91),(29,174,388,79),(30,175,389,80),(31,176,390,81),(32,177,378,82),(33,178,379,83),(34,179,380,84),(35,180,381,85),(36,181,382,86),(37,182,383,87),(38,170,384,88),(39,171,385,89),(53,351,248,276),(54,339,249,277),(55,340,250,278),(56,341,251,279),(57,342,252,280),(58,343,253,281),(59,344,254,282),(60,345,255,283),(61,346,256,284),(62,347,257,285),(63,348,258,286),(64,349,259,274),(65,350,260,275),(66,415,109,185),(67,416,110,186),(68,404,111,187),(69,405,112,188),(70,406,113,189),(71,407,114,190),(72,408,115,191),(73,409,116,192),(74,410,117,193),(75,411,105,194),(76,412,106,195),(77,413,107,183),(78,414,108,184),(92,358,159,197),(93,359,160,198),(94,360,161,199),(95,361,162,200),(96,362,163,201),(97,363,164,202),(98,364,165,203),(99,352,166,204),(100,353,167,205),(101,354,168,206),(102,355,169,207),(103,356,157,208),(104,357,158,196),(118,149,319,270),(119,150,320,271),(120,151,321,272),(121,152,322,273),(122,153,323,261),(123,154,324,262),(124,155,325,263),(125,156,313,264),(126,144,314,265),(127,145,315,266),(128,146,316,267),(129,147,317,268),(130,148,318,269),(131,288,305,394),(132,289,306,395),(133,290,307,396),(134,291,308,397),(135,292,309,398),(136,293,310,399),(137,294,311,400),(138,295,312,401),(139,296,300,402),(140,297,301,403),(141,298,302,391),(142,299,303,392),(143,287,304,393)], [(1,381,229,35),(2,382,230,36),(3,383,231,37),(4,384,232,38),(5,385,233,39),(6,386,234,27),(7,387,222,28),(8,388,223,29),(9,389,224,30),(10,390,225,31),(11,378,226,32),(12,379,227,33),(13,380,228,34),(14,299,366,392),(15,287,367,393),(16,288,368,394),(17,289,369,395),(18,290,370,396),(19,291,371,397),(20,292,372,398),(21,293,373,399),(22,294,374,400),(23,295,375,401),(24,296,376,402),(25,297,377,403),(26,298,365,391),(40,89,330,171),(41,90,331,172),(42,91,332,173),(43,79,333,174),(44,80,334,175),(45,81,335,176),(46,82,336,177),(47,83,337,178),(48,84,338,179),(49,85,326,180),(50,86,327,181),(51,87,328,182),(52,88,329,170),(53,404,248,187),(54,405,249,188),(55,406,250,189),(56,407,251,190),(57,408,252,191),(58,409,253,192),(59,410,254,193),(60,411,255,194),(61,412,256,195),(62,413,257,183),(63,414,258,184),(64,415,259,185),(65,416,260,186),(66,274,109,349),(67,275,110,350),(68,276,111,351),(69,277,112,339),(70,278,113,340),(71,279,114,341),(72,280,115,342),(73,281,116,343),(74,282,117,344),(75,283,105,345),(76,284,106,346),(77,285,107,347),(78,286,108,348),(92,322,159,121),(93,323,160,122),(94,324,161,123),(95,325,162,124),(96,313,163,125),(97,314,164,126),(98,315,165,127),(99,316,166,128),(100,317,167,129),(101,318,168,130),(102,319,169,118),(103,320,157,119),(104,321,158,120),(131,245,305,221),(132,246,306,209),(133,247,307,210),(134,235,308,211),(135,236,309,212),(136,237,310,213),(137,238,311,214),(138,239,312,215),(139,240,300,216),(140,241,301,217),(141,242,302,218),(142,243,303,219),(143,244,304,220),(144,202,265,363),(145,203,266,364),(146,204,267,352),(147,205,268,353),(148,206,269,354),(149,207,270,355),(150,208,271,356),(151,196,272,357),(152,197,273,358),(153,198,261,359),(154,199,262,360),(155,200,263,361),(156,201,264,362)], [(1,123,309,54),(2,124,310,55),(3,125,311,56),(4,126,312,57),(5,127,300,58),(6,128,301,59),(7,129,302,60),(8,130,303,61),(9,118,304,62),(10,119,305,63),(11,120,306,64),(12,121,307,65),(13,122,308,53),(14,195,174,168),(15,183,175,169),(16,184,176,157),(17,185,177,158),(18,186,178,159),(19,187,179,160),(20,188,180,161),(21,189,181,162),(22,190,182,163),(23,191,170,164),(24,192,171,165),(25,193,172,166),(26,194,173,167),(27,204,241,74),(28,205,242,75),(29,206,243,76),(30,207,244,77),(31,208,245,78),(32,196,246,66),(33,197,247,67),(34,198,235,68),(35,199,236,69),(36,200,237,70),(37,201,238,71),(38,202,239,72),(39,203,240,73),(40,145,296,343),(41,146,297,344),(42,147,298,345),(43,148,299,346),(44,149,287,347),(45,150,288,348),(46,151,289,349),(47,152,290,350),(48,153,291,351),(49,154,292,339),(50,155,293,340),(51,156,294,341),(52,144,295,342),(79,101,366,412),(80,102,367,413),(81,103,368,414),(82,104,369,415),(83,92,370,416),(84,93,371,404),(85,94,372,405),(86,95,373,406),(87,96,374,407),(88,97,375,408),(89,98,376,409),(90,99,377,410),(91,100,365,411),(105,387,353,218),(106,388,354,219),(107,389,355,220),(108,390,356,221),(109,378,357,209),(110,379,358,210),(111,380,359,211),(112,381,360,212),(113,382,361,213),(114,383,362,214),(115,384,363,215),(116,385,364,216),(117,386,352,217),(131,258,225,320),(132,259,226,321),(133,260,227,322),(134,248,228,323),(135,249,229,324),(136,250,230,325),(137,251,231,313),(138,252,232,314),(139,253,233,315),(140,254,234,316),(141,255,222,317),(142,256,223,318),(143,257,224,319),(261,397,276,338),(262,398,277,326),(263,399,278,327),(264,400,279,328),(265,401,280,329),(266,402,281,330),(267,403,282,331),(268,391,283,332),(269,392,284,333),(270,393,285,334),(271,394,286,335),(272,395,274,336),(273,396,275,337)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | ··· | 13L | 26A | ··· | 26AJ | 52A | ··· | 52X | 52Y | ··· | 52BT | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C13 | C26 | C26 | C26 | C52 | D4 | D4 | SD16 | Q16 | D4×C13 | D4×C13 | C13×SD16 | C13×Q16 |
| kernel | C13×Q8⋊C4 | C13×C4⋊C4 | C2×C104 | Q8×C26 | Q8×C13 | Q8⋊C4 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C52 | C2×C26 | C26 | C26 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 12 | 12 | 48 | 1 | 1 | 2 | 2 | 12 | 12 | 24 | 24 |
Matrix representation of C13×Q8⋊C4 ►in GL4(𝔽313) generated by
| 103 | 0 | 0 | 0 |
| 0 | 103 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 312 | 0 |
| 159 | 2 | 0 | 0 |
| 193 | 154 | 0 | 0 |
| 0 | 0 | 230 | 263 |
| 0 | 0 | 263 | 83 |
| 155 | 14 | 0 | 0 |
| 296 | 158 | 0 | 0 |
| 0 | 0 | 119 | 267 |
| 0 | 0 | 267 | 194 |
G:=sub<GL(4,GF(313))| [103,0,0,0,0,103,0,0,0,0,27,0,0,0,0,27],[312,0,0,0,0,312,0,0,0,0,0,312,0,0,1,0],[159,193,0,0,2,154,0,0,0,0,230,263,0,0,263,83],[155,296,0,0,14,158,0,0,0,0,119,267,0,0,267,194] >;
C13×Q8⋊C4 in GAP, Magma, Sage, TeX
C_{13}\times Q_8\rtimes C_4 % in TeX
G:=Group("C13xQ8:C4"); // GroupNames label
G:=SmallGroup(416,53);
// by ID
G=gap.SmallGroup(416,53);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,1255,6243,3129,117]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations
Export