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## G = C4×C13⋊C8order 416 = 25·13

### Direct product of C4 and C13⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C4×C13⋊C8
 Chief series C1 — C13 — C26 — Dic13 — C2×Dic13 — C2×C13⋊C8 — C4×C13⋊C8
 Lower central C13 — C4×C13⋊C8
 Upper central C1 — C2×C4

Generators and relations for C4×C13⋊C8
G = < a,b,c | a4=b13=c8=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C4×C13⋊C8
Regular action on 416 points
Generators in S416
(1 131 27 105)(2 132 28 106)(3 133 29 107)(4 134 30 108)(5 135 31 109)(6 136 32 110)(7 137 33 111)(8 138 34 112)(9 139 35 113)(10 140 36 114)(11 141 37 115)(12 142 38 116)(13 143 39 117)(14 144 40 118)(15 145 41 119)(16 146 42 120)(17 147 43 121)(18 148 44 122)(19 149 45 123)(20 150 46 124)(21 151 47 125)(22 152 48 126)(23 153 49 127)(24 154 50 128)(25 155 51 129)(26 156 52 130)(53 188 87 157)(54 189 88 158)(55 190 89 159)(56 191 90 160)(57 192 91 161)(58 193 79 162)(59 194 80 163)(60 195 81 164)(61 183 82 165)(62 184 83 166)(63 185 84 167)(64 186 85 168)(65 187 86 169)(66 208 97 180)(67 196 98 181)(68 197 99 182)(69 198 100 170)(70 199 101 171)(71 200 102 172)(72 201 103 173)(73 202 104 174)(74 203 92 175)(75 204 93 176)(76 205 94 177)(77 206 95 178)(78 207 96 179)(209 344 241 320)(210 345 242 321)(211 346 243 322)(212 347 244 323)(213 348 245 324)(214 349 246 325)(215 350 247 313)(216 351 235 314)(217 339 236 315)(218 340 237 316)(219 341 238 317)(220 342 239 318)(221 343 240 319)(222 360 248 335)(223 361 249 336)(224 362 250 337)(225 363 251 338)(226 364 252 326)(227 352 253 327)(228 353 254 328)(229 354 255 329)(230 355 256 330)(231 356 257 331)(232 357 258 332)(233 358 259 333)(234 359 260 334)(261 392 289 372)(262 393 290 373)(263 394 291 374)(264 395 292 375)(265 396 293 376)(266 397 294 377)(267 398 295 365)(268 399 296 366)(269 400 297 367)(270 401 298 368)(271 402 299 369)(272 403 287 370)(273 391 288 371)(274 412 303 382)(275 413 304 383)(276 414 305 384)(277 415 306 385)(278 416 307 386)(279 404 308 387)(280 405 309 388)(281 406 310 389)(282 407 311 390)(283 408 312 378)(284 409 300 379)(285 410 301 380)(286 411 302 381)
(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)
(1 279 70 228 14 264 62 219)(2 274 69 233 15 272 61 211)(3 282 68 225 16 267 60 216)(4 277 67 230 17 262 59 221)(5 285 66 222 18 270 58 213)(6 280 78 227 19 265 57 218)(7 275 77 232 20 273 56 210)(8 283 76 224 21 268 55 215)(9 278 75 229 22 263 54 220)(10 286 74 234 23 271 53 212)(11 281 73 226 24 266 65 217)(12 276 72 231 25 261 64 209)(13 284 71 223 26 269 63 214)(27 308 101 254 40 292 83 238)(28 303 100 259 41 287 82 243)(29 311 99 251 42 295 81 235)(30 306 98 256 43 290 80 240)(31 301 97 248 44 298 79 245)(32 309 96 253 45 293 91 237)(33 304 95 258 46 288 90 242)(34 312 94 250 47 296 89 247)(35 307 93 255 48 291 88 239)(36 302 92 260 49 299 87 244)(37 310 104 252 50 294 86 236)(38 305 103 257 51 289 85 241)(39 300 102 249 52 297 84 246)(105 387 171 328 118 375 166 317)(106 382 170 333 119 370 165 322)(107 390 182 338 120 365 164 314)(108 385 181 330 121 373 163 319)(109 380 180 335 122 368 162 324)(110 388 179 327 123 376 161 316)(111 383 178 332 124 371 160 321)(112 378 177 337 125 366 159 313)(113 386 176 329 126 374 158 318)(114 381 175 334 127 369 157 323)(115 389 174 326 128 377 169 315)(116 384 173 331 129 372 168 320)(117 379 172 336 130 367 167 325)(131 404 199 353 144 395 184 341)(132 412 198 358 145 403 183 346)(133 407 197 363 146 398 195 351)(134 415 196 355 147 393 194 343)(135 410 208 360 148 401 193 348)(136 405 207 352 149 396 192 340)(137 413 206 357 150 391 191 345)(138 408 205 362 151 399 190 350)(139 416 204 354 152 394 189 342)(140 411 203 359 153 402 188 347)(141 406 202 364 154 397 187 339)(142 414 201 356 155 392 186 344)(143 409 200 361 156 400 185 349)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4L 8A ··· 8P 13A 13B 13C 26A ··· 26I 52A ··· 52L order 1 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 13 13 13 26 ··· 26 52 ··· 52 size 1 1 1 1 1 1 1 1 13 ··· 13 13 ··· 13 4 4 4 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C4 C8 C8 C13⋊C4 C13⋊C8 C2×C13⋊C4 D13⋊C8 C4×C13⋊C4 kernel C4×C13⋊C8 C4×Dic13 C2×C13⋊C8 C13⋊C8 C2×Dic13 C2×C52 Dic13 C52 C2×C4 C4 C22 C2 C2 # reps 1 1 2 8 2 2 8 8 3 6 3 6 6

Matrix representation of C4×C13⋊C8 in GL5(𝔽313)

 288 0 0 0 0 0 288 0 0 0 0 0 288 0 0 0 0 0 288 0 0 0 0 0 288
,
 1 0 0 0 0 0 31 282 242 312 0 32 282 242 312 0 31 283 242 312 0 31 282 243 312
,
 288 0 0 0 0 0 232 244 17 125 0 61 291 168 132 0 248 24 307 196 0 272 46 129 109

G:=sub<GL(5,GF(313))| [288,0,0,0,0,0,288,0,0,0,0,0,288,0,0,0,0,0,288,0,0,0,0,0,288],[1,0,0,0,0,0,31,32,31,31,0,282,282,283,282,0,242,242,242,243,0,312,312,312,312],[288,0,0,0,0,0,232,61,248,272,0,244,291,24,46,0,17,168,307,129,0,125,132,196,109] >;

C4×C13⋊C8 in GAP, Magma, Sage, TeX

C_4\times C_{13}\rtimes C_8
% in TeX

G:=Group("C4xC13:C8");
// GroupNames label

G:=SmallGroup(416,75);
// by ID

G=gap.SmallGroup(416,75);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,55,86,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^4=b^13=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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