direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C4⋊C8, C4⋊C104, C52⋊5C8, C52.67D4, C52.12Q8, C42.2C26, C26.14M4(2), (C2×C4).4C52, (C4×C52).8C2, (C2×C8).2C26, C4.4(Q8×C13), C26.21(C2×C8), (C2×C52).24C4, (C2×C104).4C2, C2.2(C2×C104), C4.18(D4×C13), C26.18(C4⋊C4), C22.10(C2×C52), C2.3(C13×M4(2)), (C2×C52).136C22, C2.2(C13×C4⋊C4), (C2×C4).32(C2×C26), (C2×C26).59(C2×C4), SmallGroup(416,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4⋊C8
G = < a,b,c | a13=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C13×C4⋊C8
►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)
(1 274 178 85)(2 275 179 86)(3 276 180 87)(4 277 181 88)(5 278 182 89)(6 279 170 90)(7 280 171 91)(8 281 172 79)(9 282 173 80)(10 283 174 81)(11 284 175 82)(12 285 176 83)(13 286 177 84)(14 154 327 222)(15 155 328 223)(16 156 329 224)(17 144 330 225)(18 145 331 226)(19 146 332 227)(20 147 333 228)(21 148 334 229)(22 149 335 230)(23 150 336 231)(24 151 337 232)(25 152 338 233)(26 153 326 234)(27 199 250 212)(28 200 251 213)(29 201 252 214)(30 202 253 215)(31 203 254 216)(32 204 255 217)(33 205 256 218)(34 206 257 219)(35 207 258 220)(36 208 259 221)(37 196 260 209)(38 197 248 210)(39 198 249 211)(40 241 267 143)(41 242 268 131)(42 243 269 132)(43 244 270 133)(44 245 271 134)(45 246 272 135)(46 247 273 136)(47 235 261 137)(48 236 262 138)(49 237 263 139)(50 238 264 140)(51 239 265 141)(52 240 266 142)(53 169 392 366)(54 157 393 367)(55 158 394 368)(56 159 395 369)(57 160 396 370)(58 161 397 371)(59 162 398 372)(60 163 399 373)(61 164 400 374)(62 165 401 375)(63 166 402 376)(64 167 403 377)(65 168 391 365)(66 363 320 106)(67 364 321 107)(68 352 322 108)(69 353 323 109)(70 354 324 110)(71 355 325 111)(72 356 313 112)(73 357 314 113)(74 358 315 114)(75 359 316 115)(76 360 317 116)(77 361 318 117)(78 362 319 105)(92 390 194 119)(93 378 195 120)(94 379 183 121)(95 380 184 122)(96 381 185 123)(97 382 186 124)(98 383 187 125)(99 384 188 126)(100 385 189 127)(101 386 190 128)(102 387 191 129)(103 388 192 130)(104 389 193 118)(287 341 312 409)(288 342 300 410)(289 343 301 411)(290 344 302 412)(291 345 303 413)(292 346 304 414)(293 347 305 415)(294 348 306 416)(295 349 307 404)(296 350 308 405)(297 351 309 406)(298 339 310 407)(299 340 311 408)
(1 123 401 17 114 50 203 346)(2 124 402 18 115 51 204 347)(3 125 403 19 116 52 205 348)(4 126 391 20 117 40 206 349)(5 127 392 21 105 41 207 350)(6 128 393 22 106 42 208 351)(7 129 394 23 107 43 196 339)(8 130 395 24 108 44 197 340)(9 118 396 25 109 45 198 341)(10 119 397 26 110 46 199 342)(11 120 398 14 111 47 200 343)(12 121 399 15 112 48 201 344)(13 122 400 16 113 49 202 345)(27 300 81 92 161 153 324 247)(28 301 82 93 162 154 325 235)(29 302 83 94 163 155 313 236)(30 303 84 95 164 156 314 237)(31 304 85 96 165 144 315 238)(32 305 86 97 166 145 316 239)(33 306 87 98 167 146 317 240)(34 307 88 99 168 147 318 241)(35 308 89 100 169 148 319 242)(36 309 90 101 157 149 320 243)(37 310 91 102 158 150 321 244)(38 311 79 103 159 151 322 245)(39 312 80 104 160 152 323 246)(53 334 362 268 220 405 182 385)(54 335 363 269 221 406 170 386)(55 336 364 270 209 407 171 387)(56 337 352 271 210 408 172 388)(57 338 353 272 211 409 173 389)(58 326 354 273 212 410 174 390)(59 327 355 261 213 411 175 378)(60 328 356 262 214 412 176 379)(61 329 357 263 215 413 177 380)(62 330 358 264 216 414 178 381)(63 331 359 265 217 415 179 382)(64 332 360 266 218 416 180 383)(65 333 361 267 219 404 181 384)(66 132 259 297 279 190 367 230)(67 133 260 298 280 191 368 231)(68 134 248 299 281 192 369 232)(69 135 249 287 282 193 370 233)(70 136 250 288 283 194 371 234)(71 137 251 289 284 195 372 222)(72 138 252 290 285 183 373 223)(73 139 253 291 286 184 374 224)(74 140 254 292 274 185 375 225)(75 141 255 293 275 186 376 226)(76 142 256 294 276 187 377 227)(77 143 257 295 277 188 365 228)(78 131 258 296 278 189 366 229)
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), (1,274,178,85)(2,275,179,86)(3,276,180,87)(4,277,181,88)(5,278,182,89)(6,279,170,90)(7,280,171,91)(8,281,172,79)(9,282,173,80)(10,283,174,81)(11,284,175,82)(12,285,176,83)(13,286,177,84)(14,154,327,222)(15,155,328,223)(16,156,329,224)(17,144,330,225)(18,145,331,226)(19,146,332,227)(20,147,333,228)(21,148,334,229)(22,149,335,230)(23,150,336,231)(24,151,337,232)(25,152,338,233)(26,153,326,234)(27,199,250,212)(28,200,251,213)(29,201,252,214)(30,202,253,215)(31,203,254,216)(32,204,255,217)(33,205,256,218)(34,206,257,219)(35,207,258,220)(36,208,259,221)(37,196,260,209)(38,197,248,210)(39,198,249,211)(40,241,267,143)(41,242,268,131)(42,243,269,132)(43,244,270,133)(44,245,271,134)(45,246,272,135)(46,247,273,136)(47,235,261,137)(48,236,262,138)(49,237,263,139)(50,238,264,140)(51,239,265,141)(52,240,266,142)(53,169,392,366)(54,157,393,367)(55,158,394,368)(56,159,395,369)(57,160,396,370)(58,161,397,371)(59,162,398,372)(60,163,399,373)(61,164,400,374)(62,165,401,375)(63,166,402,376)(64,167,403,377)(65,168,391,365)(66,363,320,106)(67,364,321,107)(68,352,322,108)(69,353,323,109)(70,354,324,110)(71,355,325,111)(72,356,313,112)(73,357,314,113)(74,358,315,114)(75,359,316,115)(76,360,317,116)(77,361,318,117)(78,362,319,105)(92,390,194,119)(93,378,195,120)(94,379,183,121)(95,380,184,122)(96,381,185,123)(97,382,186,124)(98,383,187,125)(99,384,188,126)(100,385,189,127)(101,386,190,128)(102,387,191,129)(103,388,192,130)(104,389,193,118)(287,341,312,409)(288,342,300,410)(289,343,301,411)(290,344,302,412)(291,345,303,413)(292,346,304,414)(293,347,305,415)(294,348,306,416)(295,349,307,404)(296,350,308,405)(297,351,309,406)(298,339,310,407)(299,340,311,408), (1,123,401,17,114,50,203,346)(2,124,402,18,115,51,204,347)(3,125,403,19,116,52,205,348)(4,126,391,20,117,40,206,349)(5,127,392,21,105,41,207,350)(6,128,393,22,106,42,208,351)(7,129,394,23,107,43,196,339)(8,130,395,24,108,44,197,340)(9,118,396,25,109,45,198,341)(10,119,397,26,110,46,199,342)(11,120,398,14,111,47,200,343)(12,121,399,15,112,48,201,344)(13,122,400,16,113,49,202,345)(27,300,81,92,161,153,324,247)(28,301,82,93,162,154,325,235)(29,302,83,94,163,155,313,236)(30,303,84,95,164,156,314,237)(31,304,85,96,165,144,315,238)(32,305,86,97,166,145,316,239)(33,306,87,98,167,146,317,240)(34,307,88,99,168,147,318,241)(35,308,89,100,169,148,319,242)(36,309,90,101,157,149,320,243)(37,310,91,102,158,150,321,244)(38,311,79,103,159,151,322,245)(39,312,80,104,160,152,323,246)(53,334,362,268,220,405,182,385)(54,335,363,269,221,406,170,386)(55,336,364,270,209,407,171,387)(56,337,352,271,210,408,172,388)(57,338,353,272,211,409,173,389)(58,326,354,273,212,410,174,390)(59,327,355,261,213,411,175,378)(60,328,356,262,214,412,176,379)(61,329,357,263,215,413,177,380)(62,330,358,264,216,414,178,381)(63,331,359,265,217,415,179,382)(64,332,360,266,218,416,180,383)(65,333,361,267,219,404,181,384)(66,132,259,297,279,190,367,230)(67,133,260,298,280,191,368,231)(68,134,248,299,281,192,369,232)(69,135,249,287,282,193,370,233)(70,136,250,288,283,194,371,234)(71,137,251,289,284,195,372,222)(72,138,252,290,285,183,373,223)(73,139,253,291,286,184,374,224)(74,140,254,292,274,185,375,225)(75,141,255,293,275,186,376,226)(76,142,256,294,276,187,377,227)(77,143,257,295,277,188,365,228)(78,131,258,296,278,189,366,229)>;
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), (1,274,178,85)(2,275,179,86)(3,276,180,87)(4,277,181,88)(5,278,182,89)(6,279,170,90)(7,280,171,91)(8,281,172,79)(9,282,173,80)(10,283,174,81)(11,284,175,82)(12,285,176,83)(13,286,177,84)(14,154,327,222)(15,155,328,223)(16,156,329,224)(17,144,330,225)(18,145,331,226)(19,146,332,227)(20,147,333,228)(21,148,334,229)(22,149,335,230)(23,150,336,231)(24,151,337,232)(25,152,338,233)(26,153,326,234)(27,199,250,212)(28,200,251,213)(29,201,252,214)(30,202,253,215)(31,203,254,216)(32,204,255,217)(33,205,256,218)(34,206,257,219)(35,207,258,220)(36,208,259,221)(37,196,260,209)(38,197,248,210)(39,198,249,211)(40,241,267,143)(41,242,268,131)(42,243,269,132)(43,244,270,133)(44,245,271,134)(45,246,272,135)(46,247,273,136)(47,235,261,137)(48,236,262,138)(49,237,263,139)(50,238,264,140)(51,239,265,141)(52,240,266,142)(53,169,392,366)(54,157,393,367)(55,158,394,368)(56,159,395,369)(57,160,396,370)(58,161,397,371)(59,162,398,372)(60,163,399,373)(61,164,400,374)(62,165,401,375)(63,166,402,376)(64,167,403,377)(65,168,391,365)(66,363,320,106)(67,364,321,107)(68,352,322,108)(69,353,323,109)(70,354,324,110)(71,355,325,111)(72,356,313,112)(73,357,314,113)(74,358,315,114)(75,359,316,115)(76,360,317,116)(77,361,318,117)(78,362,319,105)(92,390,194,119)(93,378,195,120)(94,379,183,121)(95,380,184,122)(96,381,185,123)(97,382,186,124)(98,383,187,125)(99,384,188,126)(100,385,189,127)(101,386,190,128)(102,387,191,129)(103,388,192,130)(104,389,193,118)(287,341,312,409)(288,342,300,410)(289,343,301,411)(290,344,302,412)(291,345,303,413)(292,346,304,414)(293,347,305,415)(294,348,306,416)(295,349,307,404)(296,350,308,405)(297,351,309,406)(298,339,310,407)(299,340,311,408), (1,123,401,17,114,50,203,346)(2,124,402,18,115,51,204,347)(3,125,403,19,116,52,205,348)(4,126,391,20,117,40,206,349)(5,127,392,21,105,41,207,350)(6,128,393,22,106,42,208,351)(7,129,394,23,107,43,196,339)(8,130,395,24,108,44,197,340)(9,118,396,25,109,45,198,341)(10,119,397,26,110,46,199,342)(11,120,398,14,111,47,200,343)(12,121,399,15,112,48,201,344)(13,122,400,16,113,49,202,345)(27,300,81,92,161,153,324,247)(28,301,82,93,162,154,325,235)(29,302,83,94,163,155,313,236)(30,303,84,95,164,156,314,237)(31,304,85,96,165,144,315,238)(32,305,86,97,166,145,316,239)(33,306,87,98,167,146,317,240)(34,307,88,99,168,147,318,241)(35,308,89,100,169,148,319,242)(36,309,90,101,157,149,320,243)(37,310,91,102,158,150,321,244)(38,311,79,103,159,151,322,245)(39,312,80,104,160,152,323,246)(53,334,362,268,220,405,182,385)(54,335,363,269,221,406,170,386)(55,336,364,270,209,407,171,387)(56,337,352,271,210,408,172,388)(57,338,353,272,211,409,173,389)(58,326,354,273,212,410,174,390)(59,327,355,261,213,411,175,378)(60,328,356,262,214,412,176,379)(61,329,357,263,215,413,177,380)(62,330,358,264,216,414,178,381)(63,331,359,265,217,415,179,382)(64,332,360,266,218,416,180,383)(65,333,361,267,219,404,181,384)(66,132,259,297,279,190,367,230)(67,133,260,298,280,191,368,231)(68,134,248,299,281,192,369,232)(69,135,249,287,282,193,370,233)(70,136,250,288,283,194,371,234)(71,137,251,289,284,195,372,222)(72,138,252,290,285,183,373,223)(73,139,253,291,286,184,374,224)(74,140,254,292,274,185,375,225)(75,141,255,293,275,186,376,226)(76,142,256,294,276,187,377,227)(77,143,257,295,277,188,365,228)(78,131,258,296,278,189,366,229) );
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)], [(1,274,178,85),(2,275,179,86),(3,276,180,87),(4,277,181,88),(5,278,182,89),(6,279,170,90),(7,280,171,91),(8,281,172,79),(9,282,173,80),(10,283,174,81),(11,284,175,82),(12,285,176,83),(13,286,177,84),(14,154,327,222),(15,155,328,223),(16,156,329,224),(17,144,330,225),(18,145,331,226),(19,146,332,227),(20,147,333,228),(21,148,334,229),(22,149,335,230),(23,150,336,231),(24,151,337,232),(25,152,338,233),(26,153,326,234),(27,199,250,212),(28,200,251,213),(29,201,252,214),(30,202,253,215),(31,203,254,216),(32,204,255,217),(33,205,256,218),(34,206,257,219),(35,207,258,220),(36,208,259,221),(37,196,260,209),(38,197,248,210),(39,198,249,211),(40,241,267,143),(41,242,268,131),(42,243,269,132),(43,244,270,133),(44,245,271,134),(45,246,272,135),(46,247,273,136),(47,235,261,137),(48,236,262,138),(49,237,263,139),(50,238,264,140),(51,239,265,141),(52,240,266,142),(53,169,392,366),(54,157,393,367),(55,158,394,368),(56,159,395,369),(57,160,396,370),(58,161,397,371),(59,162,398,372),(60,163,399,373),(61,164,400,374),(62,165,401,375),(63,166,402,376),(64,167,403,377),(65,168,391,365),(66,363,320,106),(67,364,321,107),(68,352,322,108),(69,353,323,109),(70,354,324,110),(71,355,325,111),(72,356,313,112),(73,357,314,113),(74,358,315,114),(75,359,316,115),(76,360,317,116),(77,361,318,117),(78,362,319,105),(92,390,194,119),(93,378,195,120),(94,379,183,121),(95,380,184,122),(96,381,185,123),(97,382,186,124),(98,383,187,125),(99,384,188,126),(100,385,189,127),(101,386,190,128),(102,387,191,129),(103,388,192,130),(104,389,193,118),(287,341,312,409),(288,342,300,410),(289,343,301,411),(290,344,302,412),(291,345,303,413),(292,346,304,414),(293,347,305,415),(294,348,306,416),(295,349,307,404),(296,350,308,405),(297,351,309,406),(298,339,310,407),(299,340,311,408)], [(1,123,401,17,114,50,203,346),(2,124,402,18,115,51,204,347),(3,125,403,19,116,52,205,348),(4,126,391,20,117,40,206,349),(5,127,392,21,105,41,207,350),(6,128,393,22,106,42,208,351),(7,129,394,23,107,43,196,339),(8,130,395,24,108,44,197,340),(9,118,396,25,109,45,198,341),(10,119,397,26,110,46,199,342),(11,120,398,14,111,47,200,343),(12,121,399,15,112,48,201,344),(13,122,400,16,113,49,202,345),(27,300,81,92,161,153,324,247),(28,301,82,93,162,154,325,235),(29,302,83,94,163,155,313,236),(30,303,84,95,164,156,314,237),(31,304,85,96,165,144,315,238),(32,305,86,97,166,145,316,239),(33,306,87,98,167,146,317,240),(34,307,88,99,168,147,318,241),(35,308,89,100,169,148,319,242),(36,309,90,101,157,149,320,243),(37,310,91,102,158,150,321,244),(38,311,79,103,159,151,322,245),(39,312,80,104,160,152,323,246),(53,334,362,268,220,405,182,385),(54,335,363,269,221,406,170,386),(55,336,364,270,209,407,171,387),(56,337,352,271,210,408,172,388),(57,338,353,272,211,409,173,389),(58,326,354,273,212,410,174,390),(59,327,355,261,213,411,175,378),(60,328,356,262,214,412,176,379),(61,329,357,263,215,413,177,380),(62,330,358,264,216,414,178,381),(63,331,359,265,217,415,179,382),(64,332,360,266,218,416,180,383),(65,333,361,267,219,404,181,384),(66,132,259,297,279,190,367,230),(67,133,260,298,280,191,368,231),(68,134,248,299,281,192,369,232),(69,135,249,287,282,193,370,233),(70,136,250,288,283,194,371,234),(71,137,251,289,284,195,372,222),(72,138,252,290,285,183,373,223),(73,139,253,291,286,184,374,224),(74,140,254,292,274,185,375,225),(75,141,255,293,275,186,376,226),(76,142,256,294,276,187,377,227),(77,143,257,295,277,188,365,228),(78,131,258,296,278,189,366,229)]])
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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C8 | C13 | C26 | C26 | C52 | C104 | D4 | Q8 | M4(2) | D4×C13 | Q8×C13 | C13×M4(2) |
| kernel | C13×C4⋊C8 | C4×C52 | C2×C104 | C2×C52 | C52 | C4⋊C8 | C42 | C2×C8 | C2×C4 | C4 | C52 | C52 | C26 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 8 | 12 | 12 | 24 | 48 | 96 | 1 | 1 | 2 | 12 | 12 | 24 |
Matrix representation of C13×C4⋊C8 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 312 | 0 | 0 |
| 0 | 197 | 2 |
| 0 | 1 | 116 |
| 125 | 0 | 0 |
| 0 | 185 | 102 |
| 0 | 125 | 128 |
G:=sub<GL(3,GF(313))| [1,0,0,0,27,0,0,0,27],[312,0,0,0,197,1,0,2,116],[125,0,0,0,185,125,0,102,128] >;
C13×C4⋊C8 in GAP, Magma, Sage, TeX
C_{13}\times C_4\rtimes C_8 % in TeX
G:=Group("C13xC4:C8"); // GroupNames label
G:=SmallGroup(416,55);
// by ID
G=gap.SmallGroup(416,55);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,319,88]);
// Polycyclic
G:=Group<a,b,c|a^13=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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