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G = C13×C8⋊C4order 416 = 25·13

Direct product of C13 and C8⋊C4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C13×C8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C52 — C2×C104 — C13×C8⋊C4
 Lower central C1 — C2 — C13×C8⋊C4
 Upper central C1 — C2×C52 — C13×C8⋊C4

Generators and relations for C13×C8⋊C4
G = < a,b,c | a13=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >

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

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

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

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

260 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 13A ··· 13L 26A ··· 26AJ 52A ··· 52AV 52AW ··· 52CR 104A ··· 104CR order 1 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 1 1 1 1 1 1 2 2 2 2 2 ··· 2 1 ··· 1 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C13 C26 C26 C52 C52 M4(2) C13×M4(2) kernel C13×C8⋊C4 C4×C52 C2×C104 C104 C2×C52 C8⋊C4 C42 C2×C8 C8 C2×C4 C26 C2 # reps 1 1 2 8 4 12 12 24 96 48 4 48

Matrix representation of C13×C8⋊C4 in GL3(𝔽313) generated by

 1 0 0 0 294 0 0 0 294
,
 1 0 0 0 218 249 0 249 95
,
 288 0 0 0 0 1 0 312 0
G:=sub<GL(3,GF(313))| [1,0,0,0,294,0,0,0,294],[1,0,0,0,218,249,0,249,95],[288,0,0,0,0,312,0,1,0] >;

C13×C8⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C13xC8:C4");
// GroupNames label

G:=SmallGroup(416,47);
// by ID

G=gap.SmallGroup(416,47);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,312,2521,631,117]);
// Polycyclic

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

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