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## G = C4×C13⋊2C8order 416 = 25·13

### Direct product of C4 and C13⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C4×C13⋊2C8
 Chief series C1 — C13 — C26 — C52 — C2×C52 — C2×C13⋊2C8 — C4×C13⋊2C8
 Lower central C13 — C4×C13⋊2C8
 Upper central C1 — C42

Generators and relations for C4×C132C8
G = < a,b,c | a4=b13=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C4×C132C8
Regular action on 416 points
Generators in S416
(1 196 40 157)(2 197 41 158)(3 198 42 159)(4 199 43 160)(5 200 44 161)(6 201 45 162)(7 202 46 163)(8 203 47 164)(9 204 48 165)(10 205 49 166)(11 206 50 167)(12 207 51 168)(13 208 52 169)(14 183 27 170)(15 184 28 171)(16 185 29 172)(17 186 30 173)(18 187 31 174)(19 188 32 175)(20 189 33 176)(21 190 34 177)(22 191 35 178)(23 192 36 179)(24 193 37 180)(25 194 38 181)(26 195 39 182)(53 131 92 118)(54 132 93 119)(55 133 94 120)(56 134 95 121)(57 135 96 122)(58 136 97 123)(59 137 98 124)(60 138 99 125)(61 139 100 126)(62 140 101 127)(63 141 102 128)(64 142 103 129)(65 143 104 130)(66 144 79 105)(67 145 80 106)(68 146 81 107)(69 147 82 108)(70 148 83 109)(71 149 84 110)(72 150 85 111)(73 151 86 112)(74 152 87 113)(75 153 88 114)(76 154 89 115)(77 155 90 116)(78 156 91 117)(209 414 260 365)(210 415 248 366)(211 416 249 367)(212 404 250 368)(213 405 251 369)(214 406 252 370)(215 407 253 371)(216 408 254 372)(217 409 255 373)(218 410 256 374)(219 411 257 375)(220 412 258 376)(221 413 259 377)(222 400 239 381)(223 401 240 382)(224 402 241 383)(225 403 242 384)(226 391 243 385)(227 392 244 386)(228 393 245 387)(229 394 246 388)(230 395 247 389)(231 396 235 390)(232 397 236 378)(233 398 237 379)(234 399 238 380)(261 343 304 332)(262 344 305 333)(263 345 306 334)(264 346 307 335)(265 347 308 336)(266 348 309 337)(267 349 310 338)(268 350 311 326)(269 351 312 327)(270 339 300 328)(271 340 301 329)(272 341 302 330)(273 342 303 331)(274 364 292 317)(275 352 293 318)(276 353 294 319)(277 354 295 320)(278 355 296 321)(279 356 297 322)(280 357 298 323)(281 358 299 324)(282 359 287 325)(283 360 288 313)(284 361 289 314)(285 362 290 315)(286 363 291 316)
(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)
(1 281 66 228 14 261 53 218)(2 280 67 227 15 273 54 217)(3 279 68 226 16 272 55 216)(4 278 69 225 17 271 56 215)(5 277 70 224 18 270 57 214)(6 276 71 223 19 269 58 213)(7 275 72 222 20 268 59 212)(8 274 73 234 21 267 60 211)(9 286 74 233 22 266 61 210)(10 285 75 232 23 265 62 209)(11 284 76 231 24 264 63 221)(12 283 77 230 25 263 64 220)(13 282 78 229 26 262 65 219)(27 304 92 256 40 299 79 245)(28 303 93 255 41 298 80 244)(29 302 94 254 42 297 81 243)(30 301 95 253 43 296 82 242)(31 300 96 252 44 295 83 241)(32 312 97 251 45 294 84 240)(33 311 98 250 46 293 85 239)(34 310 99 249 47 292 86 238)(35 309 100 248 48 291 87 237)(36 308 101 260 49 290 88 236)(37 307 102 259 50 289 89 235)(38 306 103 258 51 288 90 247)(39 305 104 257 52 287 91 246)(105 387 170 332 118 374 157 324)(106 386 171 331 119 373 158 323)(107 385 172 330 120 372 159 322)(108 384 173 329 121 371 160 321)(109 383 174 328 122 370 161 320)(110 382 175 327 123 369 162 319)(111 381 176 326 124 368 163 318)(112 380 177 338 125 367 164 317)(113 379 178 337 126 366 165 316)(114 378 179 336 127 365 166 315)(115 390 180 335 128 377 167 314)(116 389 181 334 129 376 168 313)(117 388 182 333 130 375 169 325)(131 410 196 358 144 393 183 343)(132 409 197 357 145 392 184 342)(133 408 198 356 146 391 185 341)(134 407 199 355 147 403 186 340)(135 406 200 354 148 402 187 339)(136 405 201 353 149 401 188 351)(137 404 202 352 150 400 189 350)(138 416 203 364 151 399 190 349)(139 415 204 363 152 398 191 348)(140 414 205 362 153 397 192 347)(141 413 206 361 154 396 193 346)(142 412 207 360 155 395 194 345)(143 411 208 359 156 394 195 344)

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

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

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

128 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 8A ··· 8P 13A ··· 13F 26A ··· 26R 52A ··· 52BT order 1 2 2 2 4 ··· 4 8 ··· 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 1 ··· 1 13 ··· 13 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 C8 D13 Dic13 D26 C13⋊2C8 C4×D13 kernel C4×C13⋊2C8 C2×C13⋊2C8 C4×C52 C13⋊2C8 C2×C52 C52 C42 C2×C4 C2×C4 C4 C4 # reps 1 2 1 8 4 16 6 12 6 48 24

Matrix representation of C4×C132C8 in GL3(𝔽313) generated by

 288 0 0 0 288 0 0 0 288
,
 1 0 0 0 27 0 0 256 58
,
 25 0 0 0 188 57 0 0 125
G:=sub<GL(3,GF(313))| [288,0,0,0,288,0,0,0,288],[1,0,0,0,27,256,0,0,58],[25,0,0,0,188,0,0,57,125] >;

C4×C132C8 in GAP, Magma, Sage, TeX

C_4\times C_{13}\rtimes_2C_8
% in TeX

G:=Group("C4xC13:2C8");
// GroupNames label

G:=SmallGroup(416,9);
// by ID

G=gap.SmallGroup(416,9);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,55,86,13829]);
// 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^-1>;
// generators/relations

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