direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C8⋊C4, C8⋊3C52, C104⋊11C4, C42.1C26, C26.12C42, C26.12M4(2), (C2×C8).7C26, (C2×C4).2C52, C2.2(C4×C52), (C4×C52).1C2, C4.11(C2×C52), C52.69(C2×C4), (C2×C52).16C4, (C2×C104).17C2, C22.8(C2×C52), C2.1(C13×M4(2)), (C2×C52).134C22, (C2×C4).30(C2×C26), (C2×C26).57(C2×C4), SmallGroup(416,47)
Series: Derived ►Chief ►Lower central ►Upper central
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)
(1 228 243 272 52 379 62 274)(2 229 244 273 40 380 63 275)(3 230 245 261 41 381 64 276)(4 231 246 262 42 382 65 277)(5 232 247 263 43 383 53 278)(6 233 235 264 44 384 54 279)(7 234 236 265 45 385 55 280)(8 222 237 266 46 386 56 281)(9 223 238 267 47 387 57 282)(10 224 239 268 48 388 58 283)(11 225 240 269 49 389 59 284)(12 226 241 270 50 390 60 285)(13 227 242 271 51 378 61 286)(14 191 136 156 392 213 127 318)(15 192 137 144 393 214 128 319)(16 193 138 145 394 215 129 320)(17 194 139 146 395 216 130 321)(18 195 140 147 396 217 118 322)(19 183 141 148 397 218 119 323)(20 184 142 149 398 219 120 324)(21 185 143 150 399 220 121 325)(22 186 131 151 400 221 122 313)(23 187 132 152 401 209 123 314)(24 188 133 153 402 210 124 315)(25 189 134 154 403 211 125 316)(26 190 135 155 391 212 126 317)(27 79 359 405 77 204 305 101)(28 80 360 406 78 205 306 102)(29 81 361 407 66 206 307 103)(30 82 362 408 67 207 308 104)(31 83 363 409 68 208 309 92)(32 84 364 410 69 196 310 93)(33 85 352 411 70 197 311 94)(34 86 353 412 71 198 312 95)(35 87 354 413 72 199 300 96)(36 88 355 414 73 200 301 97)(37 89 356 415 74 201 302 98)(38 90 357 416 75 202 303 99)(39 91 358 404 76 203 304 100)(105 253 295 161 171 371 347 326)(106 254 296 162 172 372 348 327)(107 255 297 163 173 373 349 328)(108 256 298 164 174 374 350 329)(109 257 299 165 175 375 351 330)(110 258 287 166 176 376 339 331)(111 259 288 167 177 377 340 332)(112 260 289 168 178 365 341 333)(113 248 290 169 179 366 342 334)(114 249 291 157 180 367 343 335)(115 250 292 158 181 368 344 336)(116 251 293 159 182 369 345 337)(117 252 294 160 170 370 346 338)
(1 30 257 20)(2 31 258 21)(3 32 259 22)(4 33 260 23)(5 34 248 24)(6 35 249 25)(7 36 250 26)(8 37 251 14)(9 38 252 15)(10 39 253 16)(11 27 254 17)(12 28 255 18)(13 29 256 19)(40 68 376 399)(41 69 377 400)(42 70 365 401)(43 71 366 402)(44 72 367 403)(45 73 368 391)(46 74 369 392)(47 75 370 393)(48 76 371 394)(49 77 372 395)(50 78 373 396)(51 66 374 397)(52 67 375 398)(53 312 334 124)(54 300 335 125)(55 301 336 126)(56 302 337 127)(57 303 338 128)(58 304 326 129)(59 305 327 130)(60 306 328 118)(61 307 329 119)(62 308 330 120)(63 309 331 121)(64 310 332 122)(65 311 333 123)(79 348 194 389)(80 349 195 390)(81 350 183 378)(82 351 184 379)(83 339 185 380)(84 340 186 381)(85 341 187 382)(86 342 188 383)(87 343 189 384)(88 344 190 385)(89 345 191 386)(90 346 192 387)(91 347 193 388)(92 176 325 273)(93 177 313 261)(94 178 314 262)(95 179 315 263)(96 180 316 264)(97 181 317 265)(98 182 318 266)(99 170 319 267)(100 171 320 268)(101 172 321 269)(102 173 322 270)(103 174 323 271)(104 175 324 272)(105 145 283 404)(106 146 284 405)(107 147 285 406)(108 148 286 407)(109 149 274 408)(110 150 275 409)(111 151 276 410)(112 152 277 411)(113 153 278 412)(114 154 279 413)(115 155 280 414)(116 156 281 415)(117 144 282 416)(131 245 364 167)(132 246 352 168)(133 247 353 169)(134 235 354 157)(135 236 355 158)(136 237 356 159)(137 238 357 160)(138 239 358 161)(139 240 359 162)(140 241 360 163)(141 242 361 164)(142 243 362 165)(143 244 363 166)(196 288 221 230)(197 289 209 231)(198 290 210 232)(199 291 211 233)(200 292 212 234)(201 293 213 222)(202 294 214 223)(203 295 215 224)(204 296 216 225)(205 297 217 226)(206 298 218 227)(207 299 219 228)(208 287 220 229)
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), (1,228,243,272,52,379,62,274)(2,229,244,273,40,380,63,275)(3,230,245,261,41,381,64,276)(4,231,246,262,42,382,65,277)(5,232,247,263,43,383,53,278)(6,233,235,264,44,384,54,279)(7,234,236,265,45,385,55,280)(8,222,237,266,46,386,56,281)(9,223,238,267,47,387,57,282)(10,224,239,268,48,388,58,283)(11,225,240,269,49,389,59,284)(12,226,241,270,50,390,60,285)(13,227,242,271,51,378,61,286)(14,191,136,156,392,213,127,318)(15,192,137,144,393,214,128,319)(16,193,138,145,394,215,129,320)(17,194,139,146,395,216,130,321)(18,195,140,147,396,217,118,322)(19,183,141,148,397,218,119,323)(20,184,142,149,398,219,120,324)(21,185,143,150,399,220,121,325)(22,186,131,151,400,221,122,313)(23,187,132,152,401,209,123,314)(24,188,133,153,402,210,124,315)(25,189,134,154,403,211,125,316)(26,190,135,155,391,212,126,317)(27,79,359,405,77,204,305,101)(28,80,360,406,78,205,306,102)(29,81,361,407,66,206,307,103)(30,82,362,408,67,207,308,104)(31,83,363,409,68,208,309,92)(32,84,364,410,69,196,310,93)(33,85,352,411,70,197,311,94)(34,86,353,412,71,198,312,95)(35,87,354,413,72,199,300,96)(36,88,355,414,73,200,301,97)(37,89,356,415,74,201,302,98)(38,90,357,416,75,202,303,99)(39,91,358,404,76,203,304,100)(105,253,295,161,171,371,347,326)(106,254,296,162,172,372,348,327)(107,255,297,163,173,373,349,328)(108,256,298,164,174,374,350,329)(109,257,299,165,175,375,351,330)(110,258,287,166,176,376,339,331)(111,259,288,167,177,377,340,332)(112,260,289,168,178,365,341,333)(113,248,290,169,179,366,342,334)(114,249,291,157,180,367,343,335)(115,250,292,158,181,368,344,336)(116,251,293,159,182,369,345,337)(117,252,294,160,170,370,346,338), (1,30,257,20)(2,31,258,21)(3,32,259,22)(4,33,260,23)(5,34,248,24)(6,35,249,25)(7,36,250,26)(8,37,251,14)(9,38,252,15)(10,39,253,16)(11,27,254,17)(12,28,255,18)(13,29,256,19)(40,68,376,399)(41,69,377,400)(42,70,365,401)(43,71,366,402)(44,72,367,403)(45,73,368,391)(46,74,369,392)(47,75,370,393)(48,76,371,394)(49,77,372,395)(50,78,373,396)(51,66,374,397)(52,67,375,398)(53,312,334,124)(54,300,335,125)(55,301,336,126)(56,302,337,127)(57,303,338,128)(58,304,326,129)(59,305,327,130)(60,306,328,118)(61,307,329,119)(62,308,330,120)(63,309,331,121)(64,310,332,122)(65,311,333,123)(79,348,194,389)(80,349,195,390)(81,350,183,378)(82,351,184,379)(83,339,185,380)(84,340,186,381)(85,341,187,382)(86,342,188,383)(87,343,189,384)(88,344,190,385)(89,345,191,386)(90,346,192,387)(91,347,193,388)(92,176,325,273)(93,177,313,261)(94,178,314,262)(95,179,315,263)(96,180,316,264)(97,181,317,265)(98,182,318,266)(99,170,319,267)(100,171,320,268)(101,172,321,269)(102,173,322,270)(103,174,323,271)(104,175,324,272)(105,145,283,404)(106,146,284,405)(107,147,285,406)(108,148,286,407)(109,149,274,408)(110,150,275,409)(111,151,276,410)(112,152,277,411)(113,153,278,412)(114,154,279,413)(115,155,280,414)(116,156,281,415)(117,144,282,416)(131,245,364,167)(132,246,352,168)(133,247,353,169)(134,235,354,157)(135,236,355,158)(136,237,356,159)(137,238,357,160)(138,239,358,161)(139,240,359,162)(140,241,360,163)(141,242,361,164)(142,243,362,165)(143,244,363,166)(196,288,221,230)(197,289,209,231)(198,290,210,232)(199,291,211,233)(200,292,212,234)(201,293,213,222)(202,294,214,223)(203,295,215,224)(204,296,216,225)(205,297,217,226)(206,298,218,227)(207,299,219,228)(208,287,220,229)>;
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), (1,228,243,272,52,379,62,274)(2,229,244,273,40,380,63,275)(3,230,245,261,41,381,64,276)(4,231,246,262,42,382,65,277)(5,232,247,263,43,383,53,278)(6,233,235,264,44,384,54,279)(7,234,236,265,45,385,55,280)(8,222,237,266,46,386,56,281)(9,223,238,267,47,387,57,282)(10,224,239,268,48,388,58,283)(11,225,240,269,49,389,59,284)(12,226,241,270,50,390,60,285)(13,227,242,271,51,378,61,286)(14,191,136,156,392,213,127,318)(15,192,137,144,393,214,128,319)(16,193,138,145,394,215,129,320)(17,194,139,146,395,216,130,321)(18,195,140,147,396,217,118,322)(19,183,141,148,397,218,119,323)(20,184,142,149,398,219,120,324)(21,185,143,150,399,220,121,325)(22,186,131,151,400,221,122,313)(23,187,132,152,401,209,123,314)(24,188,133,153,402,210,124,315)(25,189,134,154,403,211,125,316)(26,190,135,155,391,212,126,317)(27,79,359,405,77,204,305,101)(28,80,360,406,78,205,306,102)(29,81,361,407,66,206,307,103)(30,82,362,408,67,207,308,104)(31,83,363,409,68,208,309,92)(32,84,364,410,69,196,310,93)(33,85,352,411,70,197,311,94)(34,86,353,412,71,198,312,95)(35,87,354,413,72,199,300,96)(36,88,355,414,73,200,301,97)(37,89,356,415,74,201,302,98)(38,90,357,416,75,202,303,99)(39,91,358,404,76,203,304,100)(105,253,295,161,171,371,347,326)(106,254,296,162,172,372,348,327)(107,255,297,163,173,373,349,328)(108,256,298,164,174,374,350,329)(109,257,299,165,175,375,351,330)(110,258,287,166,176,376,339,331)(111,259,288,167,177,377,340,332)(112,260,289,168,178,365,341,333)(113,248,290,169,179,366,342,334)(114,249,291,157,180,367,343,335)(115,250,292,158,181,368,344,336)(116,251,293,159,182,369,345,337)(117,252,294,160,170,370,346,338), (1,30,257,20)(2,31,258,21)(3,32,259,22)(4,33,260,23)(5,34,248,24)(6,35,249,25)(7,36,250,26)(8,37,251,14)(9,38,252,15)(10,39,253,16)(11,27,254,17)(12,28,255,18)(13,29,256,19)(40,68,376,399)(41,69,377,400)(42,70,365,401)(43,71,366,402)(44,72,367,403)(45,73,368,391)(46,74,369,392)(47,75,370,393)(48,76,371,394)(49,77,372,395)(50,78,373,396)(51,66,374,397)(52,67,375,398)(53,312,334,124)(54,300,335,125)(55,301,336,126)(56,302,337,127)(57,303,338,128)(58,304,326,129)(59,305,327,130)(60,306,328,118)(61,307,329,119)(62,308,330,120)(63,309,331,121)(64,310,332,122)(65,311,333,123)(79,348,194,389)(80,349,195,390)(81,350,183,378)(82,351,184,379)(83,339,185,380)(84,340,186,381)(85,341,187,382)(86,342,188,383)(87,343,189,384)(88,344,190,385)(89,345,191,386)(90,346,192,387)(91,347,193,388)(92,176,325,273)(93,177,313,261)(94,178,314,262)(95,179,315,263)(96,180,316,264)(97,181,317,265)(98,182,318,266)(99,170,319,267)(100,171,320,268)(101,172,321,269)(102,173,322,270)(103,174,323,271)(104,175,324,272)(105,145,283,404)(106,146,284,405)(107,147,285,406)(108,148,286,407)(109,149,274,408)(110,150,275,409)(111,151,276,410)(112,152,277,411)(113,153,278,412)(114,154,279,413)(115,155,280,414)(116,156,281,415)(117,144,282,416)(131,245,364,167)(132,246,352,168)(133,247,353,169)(134,235,354,157)(135,236,355,158)(136,237,356,159)(137,238,357,160)(138,239,358,161)(139,240,359,162)(140,241,360,163)(141,242,361,164)(142,243,362,165)(143,244,363,166)(196,288,221,230)(197,289,209,231)(198,290,210,232)(199,291,211,233)(200,292,212,234)(201,293,213,222)(202,294,214,223)(203,295,215,224)(204,296,216,225)(205,297,217,226)(206,298,218,227)(207,299,219,228)(208,287,220,229) );
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)], [(1,228,243,272,52,379,62,274),(2,229,244,273,40,380,63,275),(3,230,245,261,41,381,64,276),(4,231,246,262,42,382,65,277),(5,232,247,263,43,383,53,278),(6,233,235,264,44,384,54,279),(7,234,236,265,45,385,55,280),(8,222,237,266,46,386,56,281),(9,223,238,267,47,387,57,282),(10,224,239,268,48,388,58,283),(11,225,240,269,49,389,59,284),(12,226,241,270,50,390,60,285),(13,227,242,271,51,378,61,286),(14,191,136,156,392,213,127,318),(15,192,137,144,393,214,128,319),(16,193,138,145,394,215,129,320),(17,194,139,146,395,216,130,321),(18,195,140,147,396,217,118,322),(19,183,141,148,397,218,119,323),(20,184,142,149,398,219,120,324),(21,185,143,150,399,220,121,325),(22,186,131,151,400,221,122,313),(23,187,132,152,401,209,123,314),(24,188,133,153,402,210,124,315),(25,189,134,154,403,211,125,316),(26,190,135,155,391,212,126,317),(27,79,359,405,77,204,305,101),(28,80,360,406,78,205,306,102),(29,81,361,407,66,206,307,103),(30,82,362,408,67,207,308,104),(31,83,363,409,68,208,309,92),(32,84,364,410,69,196,310,93),(33,85,352,411,70,197,311,94),(34,86,353,412,71,198,312,95),(35,87,354,413,72,199,300,96),(36,88,355,414,73,200,301,97),(37,89,356,415,74,201,302,98),(38,90,357,416,75,202,303,99),(39,91,358,404,76,203,304,100),(105,253,295,161,171,371,347,326),(106,254,296,162,172,372,348,327),(107,255,297,163,173,373,349,328),(108,256,298,164,174,374,350,329),(109,257,299,165,175,375,351,330),(110,258,287,166,176,376,339,331),(111,259,288,167,177,377,340,332),(112,260,289,168,178,365,341,333),(113,248,290,169,179,366,342,334),(114,249,291,157,180,367,343,335),(115,250,292,158,181,368,344,336),(116,251,293,159,182,369,345,337),(117,252,294,160,170,370,346,338)], [(1,30,257,20),(2,31,258,21),(3,32,259,22),(4,33,260,23),(5,34,248,24),(6,35,249,25),(7,36,250,26),(8,37,251,14),(9,38,252,15),(10,39,253,16),(11,27,254,17),(12,28,255,18),(13,29,256,19),(40,68,376,399),(41,69,377,400),(42,70,365,401),(43,71,366,402),(44,72,367,403),(45,73,368,391),(46,74,369,392),(47,75,370,393),(48,76,371,394),(49,77,372,395),(50,78,373,396),(51,66,374,397),(52,67,375,398),(53,312,334,124),(54,300,335,125),(55,301,336,126),(56,302,337,127),(57,303,338,128),(58,304,326,129),(59,305,327,130),(60,306,328,118),(61,307,329,119),(62,308,330,120),(63,309,331,121),(64,310,332,122),(65,311,333,123),(79,348,194,389),(80,349,195,390),(81,350,183,378),(82,351,184,379),(83,339,185,380),(84,340,186,381),(85,341,187,382),(86,342,188,383),(87,343,189,384),(88,344,190,385),(89,345,191,386),(90,346,192,387),(91,347,193,388),(92,176,325,273),(93,177,313,261),(94,178,314,262),(95,179,315,263),(96,180,316,264),(97,181,317,265),(98,182,318,266),(99,170,319,267),(100,171,320,268),(101,172,321,269),(102,173,322,270),(103,174,323,271),(104,175,324,272),(105,145,283,404),(106,146,284,405),(107,147,285,406),(108,148,286,407),(109,149,274,408),(110,150,275,409),(111,151,276,410),(112,152,277,411),(113,153,278,412),(114,154,279,413),(115,155,280,414),(116,156,281,415),(117,144,282,416),(131,245,364,167),(132,246,352,168),(133,247,353,169),(134,235,354,157),(135,236,355,158),(136,237,356,159),(137,238,357,160),(138,239,358,161),(139,240,359,162),(140,241,360,163),(141,242,361,164),(142,243,362,165),(143,244,363,166),(196,288,221,230),(197,289,209,231),(198,290,210,232),(199,291,211,233),(200,292,212,234),(201,293,213,222),(202,294,214,223),(203,295,215,224),(204,296,216,225),(205,297,217,226),(206,298,218,227),(207,299,219,228),(208,287,220,229)]])
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
Export