direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×Dic13, C104⋊7C4, C26.8C42, C13⋊4(C4×C8), C13⋊2C8⋊9C4, C2.2(C8×D13), C52.61(C2×C4), C26.12(C2×C8), (C2×C8).10D13, C4.20(C4×D13), (C2×C4).90D26, (C2×C104).11C2, C2.2(C4×Dic13), C22.8(C4×D13), C4.12(C2×Dic13), (C2×C52).104C22, (C4×Dic13).14C2, (C2×Dic13).15C4, (C2×C26).29(C2×C4), (C2×C13⋊2C8).14C2, SmallGroup(416,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C8×Dic13 |
Generators and relations for C8×Dic13
G = < a,b,c | a8=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C8×Dic13
►Regular action on 416 points
Generators in S416
(1 332 142 251 270 199 213 81)(2 333 143 252 271 200 214 82)(3 334 144 253 272 201 215 83)(4 335 145 254 273 202 216 84)(5 336 146 255 274 203 217 85)(6 337 147 256 275 204 218 86)(7 338 148 257 276 205 219 87)(8 313 149 258 277 206 220 88)(9 314 150 259 278 207 221 89)(10 315 151 260 279 208 222 90)(11 316 152 235 280 183 223 91)(12 317 153 236 281 184 224 92)(13 318 154 237 282 185 225 93)(14 319 155 238 283 186 226 94)(15 320 156 239 284 187 227 95)(16 321 131 240 285 188 228 96)(17 322 132 241 286 189 229 97)(18 323 133 242 261 190 230 98)(19 324 134 243 262 191 231 99)(20 325 135 244 263 192 232 100)(21 326 136 245 264 193 233 101)(22 327 137 246 265 194 234 102)(23 328 138 247 266 195 209 103)(24 329 139 248 267 196 210 104)(25 330 140 249 268 197 211 79)(26 331 141 250 269 198 212 80)(27 343 64 292 396 373 110 167)(28 344 65 293 397 374 111 168)(29 345 66 294 398 375 112 169)(30 346 67 295 399 376 113 170)(31 347 68 296 400 377 114 171)(32 348 69 297 401 378 115 172)(33 349 70 298 402 379 116 173)(34 350 71 299 403 380 117 174)(35 351 72 300 404 381 118 175)(36 352 73 301 405 382 119 176)(37 353 74 302 406 383 120 177)(38 354 75 303 407 384 121 178)(39 355 76 304 408 385 122 179)(40 356 77 305 409 386 123 180)(41 357 78 306 410 387 124 181)(42 358 53 307 411 388 125 182)(43 359 54 308 412 389 126 157)(44 360 55 309 413 390 127 158)(45 361 56 310 414 365 128 159)(46 362 57 311 415 366 129 160)(47 363 58 312 416 367 130 161)(48 364 59 287 391 368 105 162)(49 339 60 288 392 369 106 163)(50 340 61 289 393 370 107 164)(51 341 62 290 394 371 108 165)(52 342 63 291 395 372 109 166)
(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)
(1 403 14 416)(2 402 15 415)(3 401 16 414)(4 400 17 413)(5 399 18 412)(6 398 19 411)(7 397 20 410)(8 396 21 409)(9 395 22 408)(10 394 23 407)(11 393 24 406)(12 392 25 405)(13 391 26 404)(27 264 40 277)(28 263 41 276)(29 262 42 275)(30 261 43 274)(31 286 44 273)(32 285 45 272)(33 284 46 271)(34 283 47 270)(35 282 48 269)(36 281 49 268)(37 280 50 267)(38 279 51 266)(39 278 52 265)(53 218 66 231)(54 217 67 230)(55 216 68 229)(56 215 69 228)(57 214 70 227)(58 213 71 226)(59 212 72 225)(60 211 73 224)(61 210 74 223)(62 209 75 222)(63 234 76 221)(64 233 77 220)(65 232 78 219)(79 301 92 288)(80 300 93 287)(81 299 94 312)(82 298 95 311)(83 297 96 310)(84 296 97 309)(85 295 98 308)(86 294 99 307)(87 293 100 306)(88 292 101 305)(89 291 102 304)(90 290 103 303)(91 289 104 302)(105 141 118 154)(106 140 119 153)(107 139 120 152)(108 138 121 151)(109 137 122 150)(110 136 123 149)(111 135 124 148)(112 134 125 147)(113 133 126 146)(114 132 127 145)(115 131 128 144)(116 156 129 143)(117 155 130 142)(157 255 170 242)(158 254 171 241)(159 253 172 240)(160 252 173 239)(161 251 174 238)(162 250 175 237)(163 249 176 236)(164 248 177 235)(165 247 178 260)(166 246 179 259)(167 245 180 258)(168 244 181 257)(169 243 182 256)(183 340 196 353)(184 339 197 352)(185 364 198 351)(186 363 199 350)(187 362 200 349)(188 361 201 348)(189 360 202 347)(190 359 203 346)(191 358 204 345)(192 357 205 344)(193 356 206 343)(194 355 207 342)(195 354 208 341)(313 373 326 386)(314 372 327 385)(315 371 328 384)(316 370 329 383)(317 369 330 382)(318 368 331 381)(319 367 332 380)(320 366 333 379)(321 365 334 378)(322 390 335 377)(323 389 336 376)(324 388 337 375)(325 387 338 374)
G:=sub<Sym(416)| (1,332,142,251,270,199,213,81)(2,333,143,252,271,200,214,82)(3,334,144,253,272,201,215,83)(4,335,145,254,273,202,216,84)(5,336,146,255,274,203,217,85)(6,337,147,256,275,204,218,86)(7,338,148,257,276,205,219,87)(8,313,149,258,277,206,220,88)(9,314,150,259,278,207,221,89)(10,315,151,260,279,208,222,90)(11,316,152,235,280,183,223,91)(12,317,153,236,281,184,224,92)(13,318,154,237,282,185,225,93)(14,319,155,238,283,186,226,94)(15,320,156,239,284,187,227,95)(16,321,131,240,285,188,228,96)(17,322,132,241,286,189,229,97)(18,323,133,242,261,190,230,98)(19,324,134,243,262,191,231,99)(20,325,135,244,263,192,232,100)(21,326,136,245,264,193,233,101)(22,327,137,246,265,194,234,102)(23,328,138,247,266,195,209,103)(24,329,139,248,267,196,210,104)(25,330,140,249,268,197,211,79)(26,331,141,250,269,198,212,80)(27,343,64,292,396,373,110,167)(28,344,65,293,397,374,111,168)(29,345,66,294,398,375,112,169)(30,346,67,295,399,376,113,170)(31,347,68,296,400,377,114,171)(32,348,69,297,401,378,115,172)(33,349,70,298,402,379,116,173)(34,350,71,299,403,380,117,174)(35,351,72,300,404,381,118,175)(36,352,73,301,405,382,119,176)(37,353,74,302,406,383,120,177)(38,354,75,303,407,384,121,178)(39,355,76,304,408,385,122,179)(40,356,77,305,409,386,123,180)(41,357,78,306,410,387,124,181)(42,358,53,307,411,388,125,182)(43,359,54,308,412,389,126,157)(44,360,55,309,413,390,127,158)(45,361,56,310,414,365,128,159)(46,362,57,311,415,366,129,160)(47,363,58,312,416,367,130,161)(48,364,59,287,391,368,105,162)(49,339,60,288,392,369,106,163)(50,340,61,289,393,370,107,164)(51,341,62,290,394,371,108,165)(52,342,63,291,395,372,109,166), (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), (1,403,14,416)(2,402,15,415)(3,401,16,414)(4,400,17,413)(5,399,18,412)(6,398,19,411)(7,397,20,410)(8,396,21,409)(9,395,22,408)(10,394,23,407)(11,393,24,406)(12,392,25,405)(13,391,26,404)(27,264,40,277)(28,263,41,276)(29,262,42,275)(30,261,43,274)(31,286,44,273)(32,285,45,272)(33,284,46,271)(34,283,47,270)(35,282,48,269)(36,281,49,268)(37,280,50,267)(38,279,51,266)(39,278,52,265)(53,218,66,231)(54,217,67,230)(55,216,68,229)(56,215,69,228)(57,214,70,227)(58,213,71,226)(59,212,72,225)(60,211,73,224)(61,210,74,223)(62,209,75,222)(63,234,76,221)(64,233,77,220)(65,232,78,219)(79,301,92,288)(80,300,93,287)(81,299,94,312)(82,298,95,311)(83,297,96,310)(84,296,97,309)(85,295,98,308)(86,294,99,307)(87,293,100,306)(88,292,101,305)(89,291,102,304)(90,290,103,303)(91,289,104,302)(105,141,118,154)(106,140,119,153)(107,139,120,152)(108,138,121,151)(109,137,122,150)(110,136,123,149)(111,135,124,148)(112,134,125,147)(113,133,126,146)(114,132,127,145)(115,131,128,144)(116,156,129,143)(117,155,130,142)(157,255,170,242)(158,254,171,241)(159,253,172,240)(160,252,173,239)(161,251,174,238)(162,250,175,237)(163,249,176,236)(164,248,177,235)(165,247,178,260)(166,246,179,259)(167,245,180,258)(168,244,181,257)(169,243,182,256)(183,340,196,353)(184,339,197,352)(185,364,198,351)(186,363,199,350)(187,362,200,349)(188,361,201,348)(189,360,202,347)(190,359,203,346)(191,358,204,345)(192,357,205,344)(193,356,206,343)(194,355,207,342)(195,354,208,341)(313,373,326,386)(314,372,327,385)(315,371,328,384)(316,370,329,383)(317,369,330,382)(318,368,331,381)(319,367,332,380)(320,366,333,379)(321,365,334,378)(322,390,335,377)(323,389,336,376)(324,388,337,375)(325,387,338,374)>;
G:=Group( (1,332,142,251,270,199,213,81)(2,333,143,252,271,200,214,82)(3,334,144,253,272,201,215,83)(4,335,145,254,273,202,216,84)(5,336,146,255,274,203,217,85)(6,337,147,256,275,204,218,86)(7,338,148,257,276,205,219,87)(8,313,149,258,277,206,220,88)(9,314,150,259,278,207,221,89)(10,315,151,260,279,208,222,90)(11,316,152,235,280,183,223,91)(12,317,153,236,281,184,224,92)(13,318,154,237,282,185,225,93)(14,319,155,238,283,186,226,94)(15,320,156,239,284,187,227,95)(16,321,131,240,285,188,228,96)(17,322,132,241,286,189,229,97)(18,323,133,242,261,190,230,98)(19,324,134,243,262,191,231,99)(20,325,135,244,263,192,232,100)(21,326,136,245,264,193,233,101)(22,327,137,246,265,194,234,102)(23,328,138,247,266,195,209,103)(24,329,139,248,267,196,210,104)(25,330,140,249,268,197,211,79)(26,331,141,250,269,198,212,80)(27,343,64,292,396,373,110,167)(28,344,65,293,397,374,111,168)(29,345,66,294,398,375,112,169)(30,346,67,295,399,376,113,170)(31,347,68,296,400,377,114,171)(32,348,69,297,401,378,115,172)(33,349,70,298,402,379,116,173)(34,350,71,299,403,380,117,174)(35,351,72,300,404,381,118,175)(36,352,73,301,405,382,119,176)(37,353,74,302,406,383,120,177)(38,354,75,303,407,384,121,178)(39,355,76,304,408,385,122,179)(40,356,77,305,409,386,123,180)(41,357,78,306,410,387,124,181)(42,358,53,307,411,388,125,182)(43,359,54,308,412,389,126,157)(44,360,55,309,413,390,127,158)(45,361,56,310,414,365,128,159)(46,362,57,311,415,366,129,160)(47,363,58,312,416,367,130,161)(48,364,59,287,391,368,105,162)(49,339,60,288,392,369,106,163)(50,340,61,289,393,370,107,164)(51,341,62,290,394,371,108,165)(52,342,63,291,395,372,109,166), (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), (1,403,14,416)(2,402,15,415)(3,401,16,414)(4,400,17,413)(5,399,18,412)(6,398,19,411)(7,397,20,410)(8,396,21,409)(9,395,22,408)(10,394,23,407)(11,393,24,406)(12,392,25,405)(13,391,26,404)(27,264,40,277)(28,263,41,276)(29,262,42,275)(30,261,43,274)(31,286,44,273)(32,285,45,272)(33,284,46,271)(34,283,47,270)(35,282,48,269)(36,281,49,268)(37,280,50,267)(38,279,51,266)(39,278,52,265)(53,218,66,231)(54,217,67,230)(55,216,68,229)(56,215,69,228)(57,214,70,227)(58,213,71,226)(59,212,72,225)(60,211,73,224)(61,210,74,223)(62,209,75,222)(63,234,76,221)(64,233,77,220)(65,232,78,219)(79,301,92,288)(80,300,93,287)(81,299,94,312)(82,298,95,311)(83,297,96,310)(84,296,97,309)(85,295,98,308)(86,294,99,307)(87,293,100,306)(88,292,101,305)(89,291,102,304)(90,290,103,303)(91,289,104,302)(105,141,118,154)(106,140,119,153)(107,139,120,152)(108,138,121,151)(109,137,122,150)(110,136,123,149)(111,135,124,148)(112,134,125,147)(113,133,126,146)(114,132,127,145)(115,131,128,144)(116,156,129,143)(117,155,130,142)(157,255,170,242)(158,254,171,241)(159,253,172,240)(160,252,173,239)(161,251,174,238)(162,250,175,237)(163,249,176,236)(164,248,177,235)(165,247,178,260)(166,246,179,259)(167,245,180,258)(168,244,181,257)(169,243,182,256)(183,340,196,353)(184,339,197,352)(185,364,198,351)(186,363,199,350)(187,362,200,349)(188,361,201,348)(189,360,202,347)(190,359,203,346)(191,358,204,345)(192,357,205,344)(193,356,206,343)(194,355,207,342)(195,354,208,341)(313,373,326,386)(314,372,327,385)(315,371,328,384)(316,370,329,383)(317,369,330,382)(318,368,331,381)(319,367,332,380)(320,366,333,379)(321,365,334,378)(322,390,335,377)(323,389,336,376)(324,388,337,375)(325,387,338,374) );
G=PermutationGroup([[(1,332,142,251,270,199,213,81),(2,333,143,252,271,200,214,82),(3,334,144,253,272,201,215,83),(4,335,145,254,273,202,216,84),(5,336,146,255,274,203,217,85),(6,337,147,256,275,204,218,86),(7,338,148,257,276,205,219,87),(8,313,149,258,277,206,220,88),(9,314,150,259,278,207,221,89),(10,315,151,260,279,208,222,90),(11,316,152,235,280,183,223,91),(12,317,153,236,281,184,224,92),(13,318,154,237,282,185,225,93),(14,319,155,238,283,186,226,94),(15,320,156,239,284,187,227,95),(16,321,131,240,285,188,228,96),(17,322,132,241,286,189,229,97),(18,323,133,242,261,190,230,98),(19,324,134,243,262,191,231,99),(20,325,135,244,263,192,232,100),(21,326,136,245,264,193,233,101),(22,327,137,246,265,194,234,102),(23,328,138,247,266,195,209,103),(24,329,139,248,267,196,210,104),(25,330,140,249,268,197,211,79),(26,331,141,250,269,198,212,80),(27,343,64,292,396,373,110,167),(28,344,65,293,397,374,111,168),(29,345,66,294,398,375,112,169),(30,346,67,295,399,376,113,170),(31,347,68,296,400,377,114,171),(32,348,69,297,401,378,115,172),(33,349,70,298,402,379,116,173),(34,350,71,299,403,380,117,174),(35,351,72,300,404,381,118,175),(36,352,73,301,405,382,119,176),(37,353,74,302,406,383,120,177),(38,354,75,303,407,384,121,178),(39,355,76,304,408,385,122,179),(40,356,77,305,409,386,123,180),(41,357,78,306,410,387,124,181),(42,358,53,307,411,388,125,182),(43,359,54,308,412,389,126,157),(44,360,55,309,413,390,127,158),(45,361,56,310,414,365,128,159),(46,362,57,311,415,366,129,160),(47,363,58,312,416,367,130,161),(48,364,59,287,391,368,105,162),(49,339,60,288,392,369,106,163),(50,340,61,289,393,370,107,164),(51,341,62,290,394,371,108,165),(52,342,63,291,395,372,109,166)], [(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)], [(1,403,14,416),(2,402,15,415),(3,401,16,414),(4,400,17,413),(5,399,18,412),(6,398,19,411),(7,397,20,410),(8,396,21,409),(9,395,22,408),(10,394,23,407),(11,393,24,406),(12,392,25,405),(13,391,26,404),(27,264,40,277),(28,263,41,276),(29,262,42,275),(30,261,43,274),(31,286,44,273),(32,285,45,272),(33,284,46,271),(34,283,47,270),(35,282,48,269),(36,281,49,268),(37,280,50,267),(38,279,51,266),(39,278,52,265),(53,218,66,231),(54,217,67,230),(55,216,68,229),(56,215,69,228),(57,214,70,227),(58,213,71,226),(59,212,72,225),(60,211,73,224),(61,210,74,223),(62,209,75,222),(63,234,76,221),(64,233,77,220),(65,232,78,219),(79,301,92,288),(80,300,93,287),(81,299,94,312),(82,298,95,311),(83,297,96,310),(84,296,97,309),(85,295,98,308),(86,294,99,307),(87,293,100,306),(88,292,101,305),(89,291,102,304),(90,290,103,303),(91,289,104,302),(105,141,118,154),(106,140,119,153),(107,139,120,152),(108,138,121,151),(109,137,122,150),(110,136,123,149),(111,135,124,148),(112,134,125,147),(113,133,126,146),(114,132,127,145),(115,131,128,144),(116,156,129,143),(117,155,130,142),(157,255,170,242),(158,254,171,241),(159,253,172,240),(160,252,173,239),(161,251,174,238),(162,250,175,237),(163,249,176,236),(164,248,177,235),(165,247,178,260),(166,246,179,259),(167,245,180,258),(168,244,181,257),(169,243,182,256),(183,340,196,353),(184,339,197,352),(185,364,198,351),(186,363,199,350),(187,362,200,349),(188,361,201,348),(189,360,202,347),(190,359,203,346),(191,358,204,345),(192,357,205,344),(193,356,206,343),(194,355,207,342),(195,354,208,341),(313,373,326,386),(314,372,327,385),(315,371,328,384),(316,370,329,383),(317,369,330,382),(318,368,331,381),(319,367,332,380),(320,366,333,379),(321,365,334,378),(322,390,335,377),(323,389,336,376),(324,388,337,375),(325,387,338,374)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 8A | ··· | 8H | 8I | ··· | 8P | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D13 | Dic13 | D26 | C4×D13 | C4×D13 | C8×D13 |
| kernel | C8×Dic13 | C2×C13⋊2C8 | C4×Dic13 | C2×C104 | C13⋊2C8 | C104 | C2×Dic13 | Dic13 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 6 | 12 | 6 | 12 | 12 | 48 |
Matrix representation of C8×Dic13 ►in GL3(𝔽313) generated by
| 312 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 312 | 0 | 0 |
| 0 | 0 | 312 |
| 0 | 1 | 289 |
| 25 | 0 | 0 |
| 0 | 169 | 70 |
| 0 | 57 | 144 |
G:=sub<GL(3,GF(313))| [312,0,0,0,5,0,0,0,5],[312,0,0,0,0,1,0,312,289],[25,0,0,0,169,57,0,70,144] >;
C8×Dic13 in GAP, Magma, Sage, TeX
C_8\times {\rm Dic}_{13} % in TeX
G:=Group("C8xDic13"); // GroupNames label
G:=SmallGroup(416,20);
// by ID
G=gap.SmallGroup(416,20);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,55,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^8=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export