direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×C4.Q8, C8⋊2C52, C104⋊10C4, C52.9Q8, C26.11SD16, C4⋊C4.2C26, (C2×C8).6C26, C4.6(C2×C52), C4.1(Q8×C13), C52.64(C2×C4), (C2×C26).48D4, C26.19(C4⋊C4), (C2×C104).16C2, C2.3(C13×SD16), C22.10(D4×C13), (C2×C52).117C22, C2.3(C13×C4⋊C4), (C13×C4⋊C4).9C2, (C2×C4).20(C2×C26), SmallGroup(416,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4.Q8
G = < a,b,c,d | a13=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C13×C4.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)
(1 208 77 155)(2 196 78 156)(3 197 66 144)(4 198 67 145)(5 199 68 146)(6 200 69 147)(7 201 70 148)(8 202 71 149)(9 203 72 150)(10 204 73 151)(11 205 74 152)(12 206 75 153)(13 207 76 154)(14 331 373 315)(15 332 374 316)(16 333 375 317)(17 334 376 318)(18 335 377 319)(19 336 365 320)(20 337 366 321)(21 338 367 322)(22 326 368 323)(23 327 369 324)(24 328 370 325)(25 329 371 313)(26 330 372 314)(27 224 45 254)(28 225 46 255)(29 226 47 256)(30 227 48 257)(31 228 49 258)(32 229 50 259)(33 230 51 260)(34 231 52 248)(35 232 40 249)(36 233 41 250)(37 234 42 251)(38 222 43 252)(39 223 44 253)(53 304 216 410)(54 305 217 411)(55 306 218 412)(56 307 219 413)(57 308 220 414)(58 309 221 415)(59 310 209 416)(60 311 210 404)(61 312 211 405)(62 300 212 406)(63 301 213 407)(64 302 214 408)(65 303 215 409)(79 360 278 102)(80 361 279 103)(81 362 280 104)(82 363 281 92)(83 364 282 93)(84 352 283 94)(85 353 284 95)(86 354 285 96)(87 355 286 97)(88 356 274 98)(89 357 275 99)(90 358 276 100)(91 359 277 101)(105 383 125 394)(106 384 126 395)(107 385 127 396)(108 386 128 397)(109 387 129 398)(110 388 130 399)(111 389 118 400)(112 390 119 401)(113 378 120 402)(114 379 121 403)(115 380 122 391)(116 381 123 392)(117 382 124 393)(131 238 267 190)(132 239 268 191)(133 240 269 192)(134 241 270 193)(135 242 271 194)(136 243 272 195)(137 244 273 183)(138 245 261 184)(139 246 262 185)(140 247 263 186)(141 235 264 187)(142 236 265 188)(143 237 266 189)(157 296 179 342)(158 297 180 343)(159 298 181 344)(160 299 182 345)(161 287 170 346)(162 288 171 347)(163 289 172 348)(164 290 173 349)(165 291 174 350)(166 292 175 351)(167 293 176 339)(168 294 177 340)(169 295 178 341)
(1 171 19 48 77 162 365 30)(2 172 20 49 78 163 366 31)(3 173 21 50 66 164 367 32)(4 174 22 51 67 165 368 33)(5 175 23 52 68 166 369 34)(6 176 24 40 69 167 370 35)(7 177 25 41 70 168 371 36)(8 178 26 42 71 169 372 37)(9 179 14 43 72 157 373 38)(10 180 15 44 73 158 374 39)(11 181 16 45 74 159 375 27)(12 182 17 46 75 160 376 28)(13 170 18 47 76 161 377 29)(53 268 364 119 216 132 93 112)(54 269 352 120 217 133 94 113)(55 270 353 121 218 134 95 114)(56 271 354 122 219 135 96 115)(57 272 355 123 220 136 97 116)(58 273 356 124 221 137 98 117)(59 261 357 125 209 138 99 105)(60 262 358 126 210 139 100 106)(61 263 359 127 211 140 101 107)(62 264 360 128 212 141 102 108)(63 265 361 129 213 142 103 109)(64 266 362 130 214 143 104 110)(65 267 363 118 215 131 92 111)(79 386 300 187 278 397 406 235)(80 387 301 188 279 398 407 236)(81 388 302 189 280 399 408 237)(82 389 303 190 281 400 409 238)(83 390 304 191 282 401 410 239)(84 378 305 192 283 402 411 240)(85 379 306 193 284 403 412 241)(86 380 307 194 285 391 413 242)(87 381 308 195 286 392 414 243)(88 382 309 183 274 393 415 244)(89 383 310 184 275 394 416 245)(90 384 311 185 276 395 404 246)(91 385 312 186 277 396 405 247)(144 290 322 229 197 349 338 259)(145 291 323 230 198 350 326 260)(146 292 324 231 199 351 327 248)(147 293 325 232 200 339 328 249)(148 294 313 233 201 340 329 250)(149 295 314 234 202 341 330 251)(150 296 315 222 203 342 331 252)(151 297 316 223 204 343 332 253)(152 298 317 224 205 344 333 254)(153 299 318 225 206 345 334 255)(154 287 319 226 207 346 335 256)(155 288 320 227 208 347 336 257)(156 289 321 228 196 348 337 258)
(1 113 320 240)(2 114 321 241)(3 115 322 242)(4 116 323 243)(5 117 324 244)(6 105 325 245)(7 106 313 246)(8 107 314 247)(9 108 315 235)(10 109 316 236)(11 110 317 237)(12 111 318 238)(13 112 319 239)(14 141 203 397)(15 142 204 398)(16 143 205 399)(17 131 206 400)(18 132 207 401)(19 133 208 402)(20 134 196 403)(21 135 197 391)(22 136 198 392)(23 137 199 393)(24 138 200 394)(25 139 201 395)(26 140 202 396)(27 214 298 280)(28 215 299 281)(29 216 287 282)(30 217 288 283)(31 218 289 284)(32 219 290 285)(33 220 291 286)(34 221 292 274)(35 209 293 275)(36 210 294 276)(37 211 295 277)(38 212 296 278)(39 213 297 279)(40 59 339 89)(41 60 340 90)(42 61 341 91)(43 62 342 79)(44 63 343 80)(45 64 344 81)(46 65 345 82)(47 53 346 83)(48 54 347 84)(49 55 348 85)(50 56 349 86)(51 57 350 87)(52 58 351 88)(66 122 338 194)(67 123 326 195)(68 124 327 183)(69 125 328 184)(70 126 329 185)(71 127 330 186)(72 128 331 187)(73 129 332 188)(74 130 333 189)(75 118 334 190)(76 119 335 191)(77 120 336 192)(78 121 337 193)(92 255 409 160)(93 256 410 161)(94 257 411 162)(95 258 412 163)(96 259 413 164)(97 260 414 165)(98 248 415 166)(99 249 416 167)(100 250 404 168)(101 251 405 169)(102 252 406 157)(103 253 407 158)(104 254 408 159)(144 380 367 271)(145 381 368 272)(146 382 369 273)(147 383 370 261)(148 384 371 262)(149 385 372 263)(150 386 373 264)(151 387 374 265)(152 388 375 266)(153 389 376 267)(154 390 377 268)(155 378 365 269)(156 379 366 270)(170 364 226 304)(171 352 227 305)(172 353 228 306)(173 354 229 307)(174 355 230 308)(175 356 231 309)(176 357 232 310)(177 358 233 311)(178 359 234 312)(179 360 222 300)(180 361 223 301)(181 362 224 302)(182 363 225 303)
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), (1,208,77,155)(2,196,78,156)(3,197,66,144)(4,198,67,145)(5,199,68,146)(6,200,69,147)(7,201,70,148)(8,202,71,149)(9,203,72,150)(10,204,73,151)(11,205,74,152)(12,206,75,153)(13,207,76,154)(14,331,373,315)(15,332,374,316)(16,333,375,317)(17,334,376,318)(18,335,377,319)(19,336,365,320)(20,337,366,321)(21,338,367,322)(22,326,368,323)(23,327,369,324)(24,328,370,325)(25,329,371,313)(26,330,372,314)(27,224,45,254)(28,225,46,255)(29,226,47,256)(30,227,48,257)(31,228,49,258)(32,229,50,259)(33,230,51,260)(34,231,52,248)(35,232,40,249)(36,233,41,250)(37,234,42,251)(38,222,43,252)(39,223,44,253)(53,304,216,410)(54,305,217,411)(55,306,218,412)(56,307,219,413)(57,308,220,414)(58,309,221,415)(59,310,209,416)(60,311,210,404)(61,312,211,405)(62,300,212,406)(63,301,213,407)(64,302,214,408)(65,303,215,409)(79,360,278,102)(80,361,279,103)(81,362,280,104)(82,363,281,92)(83,364,282,93)(84,352,283,94)(85,353,284,95)(86,354,285,96)(87,355,286,97)(88,356,274,98)(89,357,275,99)(90,358,276,100)(91,359,277,101)(105,383,125,394)(106,384,126,395)(107,385,127,396)(108,386,128,397)(109,387,129,398)(110,388,130,399)(111,389,118,400)(112,390,119,401)(113,378,120,402)(114,379,121,403)(115,380,122,391)(116,381,123,392)(117,382,124,393)(131,238,267,190)(132,239,268,191)(133,240,269,192)(134,241,270,193)(135,242,271,194)(136,243,272,195)(137,244,273,183)(138,245,261,184)(139,246,262,185)(140,247,263,186)(141,235,264,187)(142,236,265,188)(143,237,266,189)(157,296,179,342)(158,297,180,343)(159,298,181,344)(160,299,182,345)(161,287,170,346)(162,288,171,347)(163,289,172,348)(164,290,173,349)(165,291,174,350)(166,292,175,351)(167,293,176,339)(168,294,177,340)(169,295,178,341), (1,171,19,48,77,162,365,30)(2,172,20,49,78,163,366,31)(3,173,21,50,66,164,367,32)(4,174,22,51,67,165,368,33)(5,175,23,52,68,166,369,34)(6,176,24,40,69,167,370,35)(7,177,25,41,70,168,371,36)(8,178,26,42,71,169,372,37)(9,179,14,43,72,157,373,38)(10,180,15,44,73,158,374,39)(11,181,16,45,74,159,375,27)(12,182,17,46,75,160,376,28)(13,170,18,47,76,161,377,29)(53,268,364,119,216,132,93,112)(54,269,352,120,217,133,94,113)(55,270,353,121,218,134,95,114)(56,271,354,122,219,135,96,115)(57,272,355,123,220,136,97,116)(58,273,356,124,221,137,98,117)(59,261,357,125,209,138,99,105)(60,262,358,126,210,139,100,106)(61,263,359,127,211,140,101,107)(62,264,360,128,212,141,102,108)(63,265,361,129,213,142,103,109)(64,266,362,130,214,143,104,110)(65,267,363,118,215,131,92,111)(79,386,300,187,278,397,406,235)(80,387,301,188,279,398,407,236)(81,388,302,189,280,399,408,237)(82,389,303,190,281,400,409,238)(83,390,304,191,282,401,410,239)(84,378,305,192,283,402,411,240)(85,379,306,193,284,403,412,241)(86,380,307,194,285,391,413,242)(87,381,308,195,286,392,414,243)(88,382,309,183,274,393,415,244)(89,383,310,184,275,394,416,245)(90,384,311,185,276,395,404,246)(91,385,312,186,277,396,405,247)(144,290,322,229,197,349,338,259)(145,291,323,230,198,350,326,260)(146,292,324,231,199,351,327,248)(147,293,325,232,200,339,328,249)(148,294,313,233,201,340,329,250)(149,295,314,234,202,341,330,251)(150,296,315,222,203,342,331,252)(151,297,316,223,204,343,332,253)(152,298,317,224,205,344,333,254)(153,299,318,225,206,345,334,255)(154,287,319,226,207,346,335,256)(155,288,320,227,208,347,336,257)(156,289,321,228,196,348,337,258), (1,113,320,240)(2,114,321,241)(3,115,322,242)(4,116,323,243)(5,117,324,244)(6,105,325,245)(7,106,313,246)(8,107,314,247)(9,108,315,235)(10,109,316,236)(11,110,317,237)(12,111,318,238)(13,112,319,239)(14,141,203,397)(15,142,204,398)(16,143,205,399)(17,131,206,400)(18,132,207,401)(19,133,208,402)(20,134,196,403)(21,135,197,391)(22,136,198,392)(23,137,199,393)(24,138,200,394)(25,139,201,395)(26,140,202,396)(27,214,298,280)(28,215,299,281)(29,216,287,282)(30,217,288,283)(31,218,289,284)(32,219,290,285)(33,220,291,286)(34,221,292,274)(35,209,293,275)(36,210,294,276)(37,211,295,277)(38,212,296,278)(39,213,297,279)(40,59,339,89)(41,60,340,90)(42,61,341,91)(43,62,342,79)(44,63,343,80)(45,64,344,81)(46,65,345,82)(47,53,346,83)(48,54,347,84)(49,55,348,85)(50,56,349,86)(51,57,350,87)(52,58,351,88)(66,122,338,194)(67,123,326,195)(68,124,327,183)(69,125,328,184)(70,126,329,185)(71,127,330,186)(72,128,331,187)(73,129,332,188)(74,130,333,189)(75,118,334,190)(76,119,335,191)(77,120,336,192)(78,121,337,193)(92,255,409,160)(93,256,410,161)(94,257,411,162)(95,258,412,163)(96,259,413,164)(97,260,414,165)(98,248,415,166)(99,249,416,167)(100,250,404,168)(101,251,405,169)(102,252,406,157)(103,253,407,158)(104,254,408,159)(144,380,367,271)(145,381,368,272)(146,382,369,273)(147,383,370,261)(148,384,371,262)(149,385,372,263)(150,386,373,264)(151,387,374,265)(152,388,375,266)(153,389,376,267)(154,390,377,268)(155,378,365,269)(156,379,366,270)(170,364,226,304)(171,352,227,305)(172,353,228,306)(173,354,229,307)(174,355,230,308)(175,356,231,309)(176,357,232,310)(177,358,233,311)(178,359,234,312)(179,360,222,300)(180,361,223,301)(181,362,224,302)(182,363,225,303)>;
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), (1,208,77,155)(2,196,78,156)(3,197,66,144)(4,198,67,145)(5,199,68,146)(6,200,69,147)(7,201,70,148)(8,202,71,149)(9,203,72,150)(10,204,73,151)(11,205,74,152)(12,206,75,153)(13,207,76,154)(14,331,373,315)(15,332,374,316)(16,333,375,317)(17,334,376,318)(18,335,377,319)(19,336,365,320)(20,337,366,321)(21,338,367,322)(22,326,368,323)(23,327,369,324)(24,328,370,325)(25,329,371,313)(26,330,372,314)(27,224,45,254)(28,225,46,255)(29,226,47,256)(30,227,48,257)(31,228,49,258)(32,229,50,259)(33,230,51,260)(34,231,52,248)(35,232,40,249)(36,233,41,250)(37,234,42,251)(38,222,43,252)(39,223,44,253)(53,304,216,410)(54,305,217,411)(55,306,218,412)(56,307,219,413)(57,308,220,414)(58,309,221,415)(59,310,209,416)(60,311,210,404)(61,312,211,405)(62,300,212,406)(63,301,213,407)(64,302,214,408)(65,303,215,409)(79,360,278,102)(80,361,279,103)(81,362,280,104)(82,363,281,92)(83,364,282,93)(84,352,283,94)(85,353,284,95)(86,354,285,96)(87,355,286,97)(88,356,274,98)(89,357,275,99)(90,358,276,100)(91,359,277,101)(105,383,125,394)(106,384,126,395)(107,385,127,396)(108,386,128,397)(109,387,129,398)(110,388,130,399)(111,389,118,400)(112,390,119,401)(113,378,120,402)(114,379,121,403)(115,380,122,391)(116,381,123,392)(117,382,124,393)(131,238,267,190)(132,239,268,191)(133,240,269,192)(134,241,270,193)(135,242,271,194)(136,243,272,195)(137,244,273,183)(138,245,261,184)(139,246,262,185)(140,247,263,186)(141,235,264,187)(142,236,265,188)(143,237,266,189)(157,296,179,342)(158,297,180,343)(159,298,181,344)(160,299,182,345)(161,287,170,346)(162,288,171,347)(163,289,172,348)(164,290,173,349)(165,291,174,350)(166,292,175,351)(167,293,176,339)(168,294,177,340)(169,295,178,341), (1,171,19,48,77,162,365,30)(2,172,20,49,78,163,366,31)(3,173,21,50,66,164,367,32)(4,174,22,51,67,165,368,33)(5,175,23,52,68,166,369,34)(6,176,24,40,69,167,370,35)(7,177,25,41,70,168,371,36)(8,178,26,42,71,169,372,37)(9,179,14,43,72,157,373,38)(10,180,15,44,73,158,374,39)(11,181,16,45,74,159,375,27)(12,182,17,46,75,160,376,28)(13,170,18,47,76,161,377,29)(53,268,364,119,216,132,93,112)(54,269,352,120,217,133,94,113)(55,270,353,121,218,134,95,114)(56,271,354,122,219,135,96,115)(57,272,355,123,220,136,97,116)(58,273,356,124,221,137,98,117)(59,261,357,125,209,138,99,105)(60,262,358,126,210,139,100,106)(61,263,359,127,211,140,101,107)(62,264,360,128,212,141,102,108)(63,265,361,129,213,142,103,109)(64,266,362,130,214,143,104,110)(65,267,363,118,215,131,92,111)(79,386,300,187,278,397,406,235)(80,387,301,188,279,398,407,236)(81,388,302,189,280,399,408,237)(82,389,303,190,281,400,409,238)(83,390,304,191,282,401,410,239)(84,378,305,192,283,402,411,240)(85,379,306,193,284,403,412,241)(86,380,307,194,285,391,413,242)(87,381,308,195,286,392,414,243)(88,382,309,183,274,393,415,244)(89,383,310,184,275,394,416,245)(90,384,311,185,276,395,404,246)(91,385,312,186,277,396,405,247)(144,290,322,229,197,349,338,259)(145,291,323,230,198,350,326,260)(146,292,324,231,199,351,327,248)(147,293,325,232,200,339,328,249)(148,294,313,233,201,340,329,250)(149,295,314,234,202,341,330,251)(150,296,315,222,203,342,331,252)(151,297,316,223,204,343,332,253)(152,298,317,224,205,344,333,254)(153,299,318,225,206,345,334,255)(154,287,319,226,207,346,335,256)(155,288,320,227,208,347,336,257)(156,289,321,228,196,348,337,258), (1,113,320,240)(2,114,321,241)(3,115,322,242)(4,116,323,243)(5,117,324,244)(6,105,325,245)(7,106,313,246)(8,107,314,247)(9,108,315,235)(10,109,316,236)(11,110,317,237)(12,111,318,238)(13,112,319,239)(14,141,203,397)(15,142,204,398)(16,143,205,399)(17,131,206,400)(18,132,207,401)(19,133,208,402)(20,134,196,403)(21,135,197,391)(22,136,198,392)(23,137,199,393)(24,138,200,394)(25,139,201,395)(26,140,202,396)(27,214,298,280)(28,215,299,281)(29,216,287,282)(30,217,288,283)(31,218,289,284)(32,219,290,285)(33,220,291,286)(34,221,292,274)(35,209,293,275)(36,210,294,276)(37,211,295,277)(38,212,296,278)(39,213,297,279)(40,59,339,89)(41,60,340,90)(42,61,341,91)(43,62,342,79)(44,63,343,80)(45,64,344,81)(46,65,345,82)(47,53,346,83)(48,54,347,84)(49,55,348,85)(50,56,349,86)(51,57,350,87)(52,58,351,88)(66,122,338,194)(67,123,326,195)(68,124,327,183)(69,125,328,184)(70,126,329,185)(71,127,330,186)(72,128,331,187)(73,129,332,188)(74,130,333,189)(75,118,334,190)(76,119,335,191)(77,120,336,192)(78,121,337,193)(92,255,409,160)(93,256,410,161)(94,257,411,162)(95,258,412,163)(96,259,413,164)(97,260,414,165)(98,248,415,166)(99,249,416,167)(100,250,404,168)(101,251,405,169)(102,252,406,157)(103,253,407,158)(104,254,408,159)(144,380,367,271)(145,381,368,272)(146,382,369,273)(147,383,370,261)(148,384,371,262)(149,385,372,263)(150,386,373,264)(151,387,374,265)(152,388,375,266)(153,389,376,267)(154,390,377,268)(155,378,365,269)(156,379,366,270)(170,364,226,304)(171,352,227,305)(172,353,228,306)(173,354,229,307)(174,355,230,308)(175,356,231,309)(176,357,232,310)(177,358,233,311)(178,359,234,312)(179,360,222,300)(180,361,223,301)(181,362,224,302)(182,363,225,303) );
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)], [(1,208,77,155),(2,196,78,156),(3,197,66,144),(4,198,67,145),(5,199,68,146),(6,200,69,147),(7,201,70,148),(8,202,71,149),(9,203,72,150),(10,204,73,151),(11,205,74,152),(12,206,75,153),(13,207,76,154),(14,331,373,315),(15,332,374,316),(16,333,375,317),(17,334,376,318),(18,335,377,319),(19,336,365,320),(20,337,366,321),(21,338,367,322),(22,326,368,323),(23,327,369,324),(24,328,370,325),(25,329,371,313),(26,330,372,314),(27,224,45,254),(28,225,46,255),(29,226,47,256),(30,227,48,257),(31,228,49,258),(32,229,50,259),(33,230,51,260),(34,231,52,248),(35,232,40,249),(36,233,41,250),(37,234,42,251),(38,222,43,252),(39,223,44,253),(53,304,216,410),(54,305,217,411),(55,306,218,412),(56,307,219,413),(57,308,220,414),(58,309,221,415),(59,310,209,416),(60,311,210,404),(61,312,211,405),(62,300,212,406),(63,301,213,407),(64,302,214,408),(65,303,215,409),(79,360,278,102),(80,361,279,103),(81,362,280,104),(82,363,281,92),(83,364,282,93),(84,352,283,94),(85,353,284,95),(86,354,285,96),(87,355,286,97),(88,356,274,98),(89,357,275,99),(90,358,276,100),(91,359,277,101),(105,383,125,394),(106,384,126,395),(107,385,127,396),(108,386,128,397),(109,387,129,398),(110,388,130,399),(111,389,118,400),(112,390,119,401),(113,378,120,402),(114,379,121,403),(115,380,122,391),(116,381,123,392),(117,382,124,393),(131,238,267,190),(132,239,268,191),(133,240,269,192),(134,241,270,193),(135,242,271,194),(136,243,272,195),(137,244,273,183),(138,245,261,184),(139,246,262,185),(140,247,263,186),(141,235,264,187),(142,236,265,188),(143,237,266,189),(157,296,179,342),(158,297,180,343),(159,298,181,344),(160,299,182,345),(161,287,170,346),(162,288,171,347),(163,289,172,348),(164,290,173,349),(165,291,174,350),(166,292,175,351),(167,293,176,339),(168,294,177,340),(169,295,178,341)], [(1,171,19,48,77,162,365,30),(2,172,20,49,78,163,366,31),(3,173,21,50,66,164,367,32),(4,174,22,51,67,165,368,33),(5,175,23,52,68,166,369,34),(6,176,24,40,69,167,370,35),(7,177,25,41,70,168,371,36),(8,178,26,42,71,169,372,37),(9,179,14,43,72,157,373,38),(10,180,15,44,73,158,374,39),(11,181,16,45,74,159,375,27),(12,182,17,46,75,160,376,28),(13,170,18,47,76,161,377,29),(53,268,364,119,216,132,93,112),(54,269,352,120,217,133,94,113),(55,270,353,121,218,134,95,114),(56,271,354,122,219,135,96,115),(57,272,355,123,220,136,97,116),(58,273,356,124,221,137,98,117),(59,261,357,125,209,138,99,105),(60,262,358,126,210,139,100,106),(61,263,359,127,211,140,101,107),(62,264,360,128,212,141,102,108),(63,265,361,129,213,142,103,109),(64,266,362,130,214,143,104,110),(65,267,363,118,215,131,92,111),(79,386,300,187,278,397,406,235),(80,387,301,188,279,398,407,236),(81,388,302,189,280,399,408,237),(82,389,303,190,281,400,409,238),(83,390,304,191,282,401,410,239),(84,378,305,192,283,402,411,240),(85,379,306,193,284,403,412,241),(86,380,307,194,285,391,413,242),(87,381,308,195,286,392,414,243),(88,382,309,183,274,393,415,244),(89,383,310,184,275,394,416,245),(90,384,311,185,276,395,404,246),(91,385,312,186,277,396,405,247),(144,290,322,229,197,349,338,259),(145,291,323,230,198,350,326,260),(146,292,324,231,199,351,327,248),(147,293,325,232,200,339,328,249),(148,294,313,233,201,340,329,250),(149,295,314,234,202,341,330,251),(150,296,315,222,203,342,331,252),(151,297,316,223,204,343,332,253),(152,298,317,224,205,344,333,254),(153,299,318,225,206,345,334,255),(154,287,319,226,207,346,335,256),(155,288,320,227,208,347,336,257),(156,289,321,228,196,348,337,258)], [(1,113,320,240),(2,114,321,241),(3,115,322,242),(4,116,323,243),(5,117,324,244),(6,105,325,245),(7,106,313,246),(8,107,314,247),(9,108,315,235),(10,109,316,236),(11,110,317,237),(12,111,318,238),(13,112,319,239),(14,141,203,397),(15,142,204,398),(16,143,205,399),(17,131,206,400),(18,132,207,401),(19,133,208,402),(20,134,196,403),(21,135,197,391),(22,136,198,392),(23,137,199,393),(24,138,200,394),(25,139,201,395),(26,140,202,396),(27,214,298,280),(28,215,299,281),(29,216,287,282),(30,217,288,283),(31,218,289,284),(32,219,290,285),(33,220,291,286),(34,221,292,274),(35,209,293,275),(36,210,294,276),(37,211,295,277),(38,212,296,278),(39,213,297,279),(40,59,339,89),(41,60,340,90),(42,61,341,91),(43,62,342,79),(44,63,343,80),(45,64,344,81),(46,65,345,82),(47,53,346,83),(48,54,347,84),(49,55,348,85),(50,56,349,86),(51,57,350,87),(52,58,351,88),(66,122,338,194),(67,123,326,195),(68,124,327,183),(69,125,328,184),(70,126,329,185),(71,127,330,186),(72,128,331,187),(73,129,332,188),(74,130,333,189),(75,118,334,190),(76,119,335,191),(77,120,336,192),(78,121,337,193),(92,255,409,160),(93,256,410,161),(94,257,411,162),(95,258,412,163),(96,259,413,164),(97,260,414,165),(98,248,415,166),(99,249,416,167),(100,250,404,168),(101,251,405,169),(102,252,406,157),(103,253,407,158),(104,254,408,159),(144,380,367,271),(145,381,368,272),(146,382,369,273),(147,383,370,261),(148,384,371,262),(149,385,372,263),(150,386,373,264),(151,387,374,265),(152,388,375,266),(153,389,376,267),(154,390,377,268),(155,378,365,269),(156,379,366,270),(170,364,226,304),(171,352,227,305),(172,353,228,306),(173,354,229,307),(174,355,230,308),(175,356,231,309),(176,357,232,310),(177,358,233,311),(178,359,234,312),(179,360,222,300),(180,361,223,301),(181,362,224,302),(182,363,225,303)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C13 | C26 | C26 | C52 | Q8 | D4 | SD16 | Q8×C13 | D4×C13 | C13×SD16 |
| kernel | C13×C4.Q8 | C13×C4⋊C4 | C2×C104 | C104 | C4.Q8 | C4⋊C4 | C2×C8 | C8 | C52 | C2×C26 | C26 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 12 | 24 | 12 | 48 | 1 | 1 | 4 | 12 | 12 | 48 |
Matrix representation of C13×C4.Q8 ►in GL4(𝔽313) generated by
| 150 | 0 | 0 | 0 |
| 0 | 150 | 0 | 0 |
| 0 | 0 | 113 | 0 |
| 0 | 0 | 0 | 113 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 312 | 0 |
| 1 | 311 | 0 | 0 |
| 1 | 312 | 0 | 0 |
| 0 | 0 | 248 | 65 |
| 0 | 0 | 248 | 248 |
| 37 | 103 | 0 | 0 |
| 245 | 276 | 0 | 0 |
| 0 | 0 | 221 | 232 |
| 0 | 0 | 232 | 92 |
G:=sub<GL(4,GF(313))| [150,0,0,0,0,150,0,0,0,0,113,0,0,0,0,113],[1,0,0,0,0,1,0,0,0,0,0,312,0,0,1,0],[1,1,0,0,311,312,0,0,0,0,248,248,0,0,65,248],[37,245,0,0,103,276,0,0,0,0,221,232,0,0,232,92] >;
C13×C4.Q8 in GAP, Magma, Sage, TeX
C_{13}\times C_4.Q_8 % in TeX
G:=Group("C13xC4.Q8"); // GroupNames label
G:=SmallGroup(416,56);
// by ID
G=gap.SmallGroup(416,56);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,319,6243,117]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
Export