direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×C2.D8, C8⋊1C52, C104⋊9C4, C26.14D8, C52.10Q8, C26.7Q16, C4⋊C4.3C26, C4.7(C2×C52), (C2×C8).3C26, C2.2(C13×D8), C4.2(Q8×C13), C52.65(C2×C4), (C2×C26).49D4, C26.20(C4⋊C4), C2.2(C13×Q16), (C2×C104).13C2, C22.11(D4×C13), (C2×C52).118C22, C2.4(C13×C4⋊C4), (C13×C4⋊C4).10C2, (C2×C4).21(C2×C26), SmallGroup(416,57)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C2.D8
G = < a,b,c,d | a13=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of C13×C2.D8
►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)
(1 175)(2 176)(3 177)(4 178)(5 179)(6 180)(7 181)(8 182)(9 170)(10 171)(11 172)(12 173)(13 174)(14 116)(15 117)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 65)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 207)(41 208)(42 196)(43 197)(44 198)(45 199)(46 200)(47 201)(48 202)(49 203)(50 204)(51 205)(52 206)(66 317)(67 318)(68 319)(69 320)(70 321)(71 322)(72 323)(73 324)(74 325)(75 313)(76 314)(77 315)(78 316)(79 167)(80 168)(81 169)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 237)(93 238)(94 239)(95 240)(96 241)(97 242)(98 243)(99 244)(100 245)(101 246)(102 247)(103 235)(104 236)(118 220)(119 221)(120 209)(121 210)(122 211)(123 212)(124 213)(125 214)(126 215)(127 216)(128 217)(129 218)(130 219)(131 396)(132 397)(133 398)(134 399)(135 400)(136 401)(137 402)(138 403)(139 391)(140 392)(141 393)(142 394)(143 395)(144 386)(145 387)(146 388)(147 389)(148 390)(149 378)(150 379)(151 380)(152 381)(153 382)(154 383)(155 384)(156 385)(183 336)(184 337)(185 338)(186 326)(187 327)(188 328)(189 329)(190 330)(191 331)(192 332)(193 333)(194 334)(195 335)(222 278)(223 279)(224 280)(225 281)(226 282)(227 283)(228 284)(229 285)(230 286)(231 274)(232 275)(233 276)(234 277)(248 266)(249 267)(250 268)(251 269)(252 270)(253 271)(254 272)(255 273)(256 261)(257 262)(258 263)(259 264)(260 265)(287 411)(288 412)(289 413)(290 414)(291 415)(292 416)(293 404)(294 405)(295 406)(296 407)(297 408)(298 409)(299 410)(300 377)(301 365)(302 366)(303 367)(304 368)(305 369)(306 370)(307 371)(308 372)(309 373)(310 374)(311 375)(312 376)(339 357)(340 358)(341 359)(342 360)(343 361)(344 362)(345 363)(346 364)(347 352)(348 353)(349 354)(350 355)(351 356)
(1 100 316 273 394 379 194 115)(2 101 317 261 395 380 195 116)(3 102 318 262 396 381 183 117)(4 103 319 263 397 382 184 105)(5 104 320 264 398 383 185 106)(6 92 321 265 399 384 186 107)(7 93 322 266 400 385 187 108)(8 94 323 267 401 386 188 109)(9 95 324 268 402 387 189 110)(10 96 325 269 403 388 190 111)(11 97 313 270 391 389 191 112)(12 98 314 271 392 390 192 113)(13 99 315 272 393 378 193 114)(14 176 246 66 256 143 151 335)(15 177 247 67 257 131 152 336)(16 178 235 68 258 132 153 337)(17 179 236 69 259 133 154 338)(18 180 237 70 260 134 155 326)(19 181 238 71 248 135 156 327)(20 182 239 72 249 136 144 328)(21 170 240 73 250 137 145 329)(22 171 241 74 251 138 146 330)(23 172 242 75 252 139 147 331)(24 173 243 76 253 140 148 332)(25 174 244 77 254 141 149 333)(26 175 245 78 255 142 150 334)(27 79 304 282 340 221 197 406)(28 80 305 283 341 209 198 407)(29 81 306 284 342 210 199 408)(30 82 307 285 343 211 200 409)(31 83 308 286 344 212 201 410)(32 84 309 274 345 213 202 411)(33 85 310 275 346 214 203 412)(34 86 311 276 347 215 204 413)(35 87 312 277 348 216 205 414)(36 88 300 278 349 217 206 415)(37 89 301 279 350 218 207 416)(38 90 302 280 351 219 208 404)(39 91 303 281 339 220 196 405)(40 292 62 164 365 223 355 129)(41 293 63 165 366 224 356 130)(42 294 64 166 367 225 357 118)(43 295 65 167 368 226 358 119)(44 296 53 168 369 227 359 120)(45 297 54 169 370 228 360 121)(46 298 55 157 371 229 361 122)(47 299 56 158 372 230 362 123)(48 287 57 159 373 231 363 124)(49 288 58 160 374 232 364 125)(50 289 59 161 375 233 352 126)(51 290 60 162 376 234 353 127)(52 291 61 163 377 222 354 128)
(1 220 175 118)(2 221 176 119)(3 209 177 120)(4 210 178 121)(5 211 179 122)(6 212 180 123)(7 213 181 124)(8 214 182 125)(9 215 170 126)(10 216 171 127)(11 217 172 128)(12 218 173 129)(13 219 174 130)(14 43 116 197)(15 44 117 198)(16 45 105 199)(17 46 106 200)(18 47 107 201)(19 48 108 202)(20 49 109 203)(21 50 110 204)(22 51 111 205)(23 52 112 206)(24 40 113 207)(25 41 114 208)(26 42 115 196)(27 151 65 380)(28 152 53 381)(29 153 54 382)(30 154 55 383)(31 155 56 384)(32 156 57 385)(33 144 58 386)(34 145 59 387)(35 146 60 388)(36 147 61 389)(37 148 62 390)(38 149 63 378)(39 150 64 379)(66 226 317 282)(67 227 318 283)(68 228 319 284)(69 229 320 285)(70 230 321 286)(71 231 322 274)(72 232 323 275)(73 233 324 276)(74 234 325 277)(75 222 313 278)(76 223 314 279)(77 224 315 280)(78 225 316 281)(79 143 167 395)(80 131 168 396)(81 132 169 397)(82 133 157 398)(83 134 158 399)(84 135 159 400)(85 136 160 401)(86 137 161 402)(87 138 162 403)(88 139 163 391)(89 140 164 392)(90 141 165 393)(91 142 166 394)(92 344 237 362)(93 345 238 363)(94 346 239 364)(95 347 240 352)(96 348 241 353)(97 349 242 354)(98 350 243 355)(99 351 244 356)(100 339 245 357)(101 340 246 358)(102 341 247 359)(103 342 235 360)(104 343 236 361)(183 407 336 296)(184 408 337 297)(185 409 338 298)(186 410 326 299)(187 411 327 287)(188 412 328 288)(189 413 329 289)(190 414 330 290)(191 415 331 291)(192 416 332 292)(193 404 333 293)(194 405 334 294)(195 406 335 295)(248 373 266 309)(249 374 267 310)(250 375 268 311)(251 376 269 312)(252 377 270 300)(253 365 271 301)(254 366 272 302)(255 367 273 303)(256 368 261 304)(257 369 262 305)(258 370 263 306)(259 371 264 307)(260 372 265 308)
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), (1,175)(2,176)(3,177)(4,178)(5,179)(6,180)(7,181)(8,182)(9,170)(10,171)(11,172)(12,173)(13,174)(14,116)(15,117)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,65)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,207)(41,208)(42,196)(43,197)(44,198)(45,199)(46,200)(47,201)(48,202)(49,203)(50,204)(51,205)(52,206)(66,317)(67,318)(68,319)(69,320)(70,321)(71,322)(72,323)(73,324)(74,325)(75,313)(76,314)(77,315)(78,316)(79,167)(80,168)(81,169)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,237)(93,238)(94,239)(95,240)(96,241)(97,242)(98,243)(99,244)(100,245)(101,246)(102,247)(103,235)(104,236)(118,220)(119,221)(120,209)(121,210)(122,211)(123,212)(124,213)(125,214)(126,215)(127,216)(128,217)(129,218)(130,219)(131,396)(132,397)(133,398)(134,399)(135,400)(136,401)(137,402)(138,403)(139,391)(140,392)(141,393)(142,394)(143,395)(144,386)(145,387)(146,388)(147,389)(148,390)(149,378)(150,379)(151,380)(152,381)(153,382)(154,383)(155,384)(156,385)(183,336)(184,337)(185,338)(186,326)(187,327)(188,328)(189,329)(190,330)(191,331)(192,332)(193,333)(194,334)(195,335)(222,278)(223,279)(224,280)(225,281)(226,282)(227,283)(228,284)(229,285)(230,286)(231,274)(232,275)(233,276)(234,277)(248,266)(249,267)(250,268)(251,269)(252,270)(253,271)(254,272)(255,273)(256,261)(257,262)(258,263)(259,264)(260,265)(287,411)(288,412)(289,413)(290,414)(291,415)(292,416)(293,404)(294,405)(295,406)(296,407)(297,408)(298,409)(299,410)(300,377)(301,365)(302,366)(303,367)(304,368)(305,369)(306,370)(307,371)(308,372)(309,373)(310,374)(311,375)(312,376)(339,357)(340,358)(341,359)(342,360)(343,361)(344,362)(345,363)(346,364)(347,352)(348,353)(349,354)(350,355)(351,356), (1,100,316,273,394,379,194,115)(2,101,317,261,395,380,195,116)(3,102,318,262,396,381,183,117)(4,103,319,263,397,382,184,105)(5,104,320,264,398,383,185,106)(6,92,321,265,399,384,186,107)(7,93,322,266,400,385,187,108)(8,94,323,267,401,386,188,109)(9,95,324,268,402,387,189,110)(10,96,325,269,403,388,190,111)(11,97,313,270,391,389,191,112)(12,98,314,271,392,390,192,113)(13,99,315,272,393,378,193,114)(14,176,246,66,256,143,151,335)(15,177,247,67,257,131,152,336)(16,178,235,68,258,132,153,337)(17,179,236,69,259,133,154,338)(18,180,237,70,260,134,155,326)(19,181,238,71,248,135,156,327)(20,182,239,72,249,136,144,328)(21,170,240,73,250,137,145,329)(22,171,241,74,251,138,146,330)(23,172,242,75,252,139,147,331)(24,173,243,76,253,140,148,332)(25,174,244,77,254,141,149,333)(26,175,245,78,255,142,150,334)(27,79,304,282,340,221,197,406)(28,80,305,283,341,209,198,407)(29,81,306,284,342,210,199,408)(30,82,307,285,343,211,200,409)(31,83,308,286,344,212,201,410)(32,84,309,274,345,213,202,411)(33,85,310,275,346,214,203,412)(34,86,311,276,347,215,204,413)(35,87,312,277,348,216,205,414)(36,88,300,278,349,217,206,415)(37,89,301,279,350,218,207,416)(38,90,302,280,351,219,208,404)(39,91,303,281,339,220,196,405)(40,292,62,164,365,223,355,129)(41,293,63,165,366,224,356,130)(42,294,64,166,367,225,357,118)(43,295,65,167,368,226,358,119)(44,296,53,168,369,227,359,120)(45,297,54,169,370,228,360,121)(46,298,55,157,371,229,361,122)(47,299,56,158,372,230,362,123)(48,287,57,159,373,231,363,124)(49,288,58,160,374,232,364,125)(50,289,59,161,375,233,352,126)(51,290,60,162,376,234,353,127)(52,291,61,163,377,222,354,128), (1,220,175,118)(2,221,176,119)(3,209,177,120)(4,210,178,121)(5,211,179,122)(6,212,180,123)(7,213,181,124)(8,214,182,125)(9,215,170,126)(10,216,171,127)(11,217,172,128)(12,218,173,129)(13,219,174,130)(14,43,116,197)(15,44,117,198)(16,45,105,199)(17,46,106,200)(18,47,107,201)(19,48,108,202)(20,49,109,203)(21,50,110,204)(22,51,111,205)(23,52,112,206)(24,40,113,207)(25,41,114,208)(26,42,115,196)(27,151,65,380)(28,152,53,381)(29,153,54,382)(30,154,55,383)(31,155,56,384)(32,156,57,385)(33,144,58,386)(34,145,59,387)(35,146,60,388)(36,147,61,389)(37,148,62,390)(38,149,63,378)(39,150,64,379)(66,226,317,282)(67,227,318,283)(68,228,319,284)(69,229,320,285)(70,230,321,286)(71,231,322,274)(72,232,323,275)(73,233,324,276)(74,234,325,277)(75,222,313,278)(76,223,314,279)(77,224,315,280)(78,225,316,281)(79,143,167,395)(80,131,168,396)(81,132,169,397)(82,133,157,398)(83,134,158,399)(84,135,159,400)(85,136,160,401)(86,137,161,402)(87,138,162,403)(88,139,163,391)(89,140,164,392)(90,141,165,393)(91,142,166,394)(92,344,237,362)(93,345,238,363)(94,346,239,364)(95,347,240,352)(96,348,241,353)(97,349,242,354)(98,350,243,355)(99,351,244,356)(100,339,245,357)(101,340,246,358)(102,341,247,359)(103,342,235,360)(104,343,236,361)(183,407,336,296)(184,408,337,297)(185,409,338,298)(186,410,326,299)(187,411,327,287)(188,412,328,288)(189,413,329,289)(190,414,330,290)(191,415,331,291)(192,416,332,292)(193,404,333,293)(194,405,334,294)(195,406,335,295)(248,373,266,309)(249,374,267,310)(250,375,268,311)(251,376,269,312)(252,377,270,300)(253,365,271,301)(254,366,272,302)(255,367,273,303)(256,368,261,304)(257,369,262,305)(258,370,263,306)(259,371,264,307)(260,372,265,308)>;
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), (1,175)(2,176)(3,177)(4,178)(5,179)(6,180)(7,181)(8,182)(9,170)(10,171)(11,172)(12,173)(13,174)(14,116)(15,117)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,65)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,207)(41,208)(42,196)(43,197)(44,198)(45,199)(46,200)(47,201)(48,202)(49,203)(50,204)(51,205)(52,206)(66,317)(67,318)(68,319)(69,320)(70,321)(71,322)(72,323)(73,324)(74,325)(75,313)(76,314)(77,315)(78,316)(79,167)(80,168)(81,169)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,237)(93,238)(94,239)(95,240)(96,241)(97,242)(98,243)(99,244)(100,245)(101,246)(102,247)(103,235)(104,236)(118,220)(119,221)(120,209)(121,210)(122,211)(123,212)(124,213)(125,214)(126,215)(127,216)(128,217)(129,218)(130,219)(131,396)(132,397)(133,398)(134,399)(135,400)(136,401)(137,402)(138,403)(139,391)(140,392)(141,393)(142,394)(143,395)(144,386)(145,387)(146,388)(147,389)(148,390)(149,378)(150,379)(151,380)(152,381)(153,382)(154,383)(155,384)(156,385)(183,336)(184,337)(185,338)(186,326)(187,327)(188,328)(189,329)(190,330)(191,331)(192,332)(193,333)(194,334)(195,335)(222,278)(223,279)(224,280)(225,281)(226,282)(227,283)(228,284)(229,285)(230,286)(231,274)(232,275)(233,276)(234,277)(248,266)(249,267)(250,268)(251,269)(252,270)(253,271)(254,272)(255,273)(256,261)(257,262)(258,263)(259,264)(260,265)(287,411)(288,412)(289,413)(290,414)(291,415)(292,416)(293,404)(294,405)(295,406)(296,407)(297,408)(298,409)(299,410)(300,377)(301,365)(302,366)(303,367)(304,368)(305,369)(306,370)(307,371)(308,372)(309,373)(310,374)(311,375)(312,376)(339,357)(340,358)(341,359)(342,360)(343,361)(344,362)(345,363)(346,364)(347,352)(348,353)(349,354)(350,355)(351,356), (1,100,316,273,394,379,194,115)(2,101,317,261,395,380,195,116)(3,102,318,262,396,381,183,117)(4,103,319,263,397,382,184,105)(5,104,320,264,398,383,185,106)(6,92,321,265,399,384,186,107)(7,93,322,266,400,385,187,108)(8,94,323,267,401,386,188,109)(9,95,324,268,402,387,189,110)(10,96,325,269,403,388,190,111)(11,97,313,270,391,389,191,112)(12,98,314,271,392,390,192,113)(13,99,315,272,393,378,193,114)(14,176,246,66,256,143,151,335)(15,177,247,67,257,131,152,336)(16,178,235,68,258,132,153,337)(17,179,236,69,259,133,154,338)(18,180,237,70,260,134,155,326)(19,181,238,71,248,135,156,327)(20,182,239,72,249,136,144,328)(21,170,240,73,250,137,145,329)(22,171,241,74,251,138,146,330)(23,172,242,75,252,139,147,331)(24,173,243,76,253,140,148,332)(25,174,244,77,254,141,149,333)(26,175,245,78,255,142,150,334)(27,79,304,282,340,221,197,406)(28,80,305,283,341,209,198,407)(29,81,306,284,342,210,199,408)(30,82,307,285,343,211,200,409)(31,83,308,286,344,212,201,410)(32,84,309,274,345,213,202,411)(33,85,310,275,346,214,203,412)(34,86,311,276,347,215,204,413)(35,87,312,277,348,216,205,414)(36,88,300,278,349,217,206,415)(37,89,301,279,350,218,207,416)(38,90,302,280,351,219,208,404)(39,91,303,281,339,220,196,405)(40,292,62,164,365,223,355,129)(41,293,63,165,366,224,356,130)(42,294,64,166,367,225,357,118)(43,295,65,167,368,226,358,119)(44,296,53,168,369,227,359,120)(45,297,54,169,370,228,360,121)(46,298,55,157,371,229,361,122)(47,299,56,158,372,230,362,123)(48,287,57,159,373,231,363,124)(49,288,58,160,374,232,364,125)(50,289,59,161,375,233,352,126)(51,290,60,162,376,234,353,127)(52,291,61,163,377,222,354,128), (1,220,175,118)(2,221,176,119)(3,209,177,120)(4,210,178,121)(5,211,179,122)(6,212,180,123)(7,213,181,124)(8,214,182,125)(9,215,170,126)(10,216,171,127)(11,217,172,128)(12,218,173,129)(13,219,174,130)(14,43,116,197)(15,44,117,198)(16,45,105,199)(17,46,106,200)(18,47,107,201)(19,48,108,202)(20,49,109,203)(21,50,110,204)(22,51,111,205)(23,52,112,206)(24,40,113,207)(25,41,114,208)(26,42,115,196)(27,151,65,380)(28,152,53,381)(29,153,54,382)(30,154,55,383)(31,155,56,384)(32,156,57,385)(33,144,58,386)(34,145,59,387)(35,146,60,388)(36,147,61,389)(37,148,62,390)(38,149,63,378)(39,150,64,379)(66,226,317,282)(67,227,318,283)(68,228,319,284)(69,229,320,285)(70,230,321,286)(71,231,322,274)(72,232,323,275)(73,233,324,276)(74,234,325,277)(75,222,313,278)(76,223,314,279)(77,224,315,280)(78,225,316,281)(79,143,167,395)(80,131,168,396)(81,132,169,397)(82,133,157,398)(83,134,158,399)(84,135,159,400)(85,136,160,401)(86,137,161,402)(87,138,162,403)(88,139,163,391)(89,140,164,392)(90,141,165,393)(91,142,166,394)(92,344,237,362)(93,345,238,363)(94,346,239,364)(95,347,240,352)(96,348,241,353)(97,349,242,354)(98,350,243,355)(99,351,244,356)(100,339,245,357)(101,340,246,358)(102,341,247,359)(103,342,235,360)(104,343,236,361)(183,407,336,296)(184,408,337,297)(185,409,338,298)(186,410,326,299)(187,411,327,287)(188,412,328,288)(189,413,329,289)(190,414,330,290)(191,415,331,291)(192,416,332,292)(193,404,333,293)(194,405,334,294)(195,406,335,295)(248,373,266,309)(249,374,267,310)(250,375,268,311)(251,376,269,312)(252,377,270,300)(253,365,271,301)(254,366,272,302)(255,367,273,303)(256,368,261,304)(257,369,262,305)(258,370,263,306)(259,371,264,307)(260,372,265,308) );
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)], [(1,175),(2,176),(3,177),(4,178),(5,179),(6,180),(7,181),(8,182),(9,170),(10,171),(11,172),(12,173),(13,174),(14,116),(15,117),(16,105),(17,106),(18,107),(19,108),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,65),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,207),(41,208),(42,196),(43,197),(44,198),(45,199),(46,200),(47,201),(48,202),(49,203),(50,204),(51,205),(52,206),(66,317),(67,318),(68,319),(69,320),(70,321),(71,322),(72,323),(73,324),(74,325),(75,313),(76,314),(77,315),(78,316),(79,167),(80,168),(81,169),(82,157),(83,158),(84,159),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(91,166),(92,237),(93,238),(94,239),(95,240),(96,241),(97,242),(98,243),(99,244),(100,245),(101,246),(102,247),(103,235),(104,236),(118,220),(119,221),(120,209),(121,210),(122,211),(123,212),(124,213),(125,214),(126,215),(127,216),(128,217),(129,218),(130,219),(131,396),(132,397),(133,398),(134,399),(135,400),(136,401),(137,402),(138,403),(139,391),(140,392),(141,393),(142,394),(143,395),(144,386),(145,387),(146,388),(147,389),(148,390),(149,378),(150,379),(151,380),(152,381),(153,382),(154,383),(155,384),(156,385),(183,336),(184,337),(185,338),(186,326),(187,327),(188,328),(189,329),(190,330),(191,331),(192,332),(193,333),(194,334),(195,335),(222,278),(223,279),(224,280),(225,281),(226,282),(227,283),(228,284),(229,285),(230,286),(231,274),(232,275),(233,276),(234,277),(248,266),(249,267),(250,268),(251,269),(252,270),(253,271),(254,272),(255,273),(256,261),(257,262),(258,263),(259,264),(260,265),(287,411),(288,412),(289,413),(290,414),(291,415),(292,416),(293,404),(294,405),(295,406),(296,407),(297,408),(298,409),(299,410),(300,377),(301,365),(302,366),(303,367),(304,368),(305,369),(306,370),(307,371),(308,372),(309,373),(310,374),(311,375),(312,376),(339,357),(340,358),(341,359),(342,360),(343,361),(344,362),(345,363),(346,364),(347,352),(348,353),(349,354),(350,355),(351,356)], [(1,100,316,273,394,379,194,115),(2,101,317,261,395,380,195,116),(3,102,318,262,396,381,183,117),(4,103,319,263,397,382,184,105),(5,104,320,264,398,383,185,106),(6,92,321,265,399,384,186,107),(7,93,322,266,400,385,187,108),(8,94,323,267,401,386,188,109),(9,95,324,268,402,387,189,110),(10,96,325,269,403,388,190,111),(11,97,313,270,391,389,191,112),(12,98,314,271,392,390,192,113),(13,99,315,272,393,378,193,114),(14,176,246,66,256,143,151,335),(15,177,247,67,257,131,152,336),(16,178,235,68,258,132,153,337),(17,179,236,69,259,133,154,338),(18,180,237,70,260,134,155,326),(19,181,238,71,248,135,156,327),(20,182,239,72,249,136,144,328),(21,170,240,73,250,137,145,329),(22,171,241,74,251,138,146,330),(23,172,242,75,252,139,147,331),(24,173,243,76,253,140,148,332),(25,174,244,77,254,141,149,333),(26,175,245,78,255,142,150,334),(27,79,304,282,340,221,197,406),(28,80,305,283,341,209,198,407),(29,81,306,284,342,210,199,408),(30,82,307,285,343,211,200,409),(31,83,308,286,344,212,201,410),(32,84,309,274,345,213,202,411),(33,85,310,275,346,214,203,412),(34,86,311,276,347,215,204,413),(35,87,312,277,348,216,205,414),(36,88,300,278,349,217,206,415),(37,89,301,279,350,218,207,416),(38,90,302,280,351,219,208,404),(39,91,303,281,339,220,196,405),(40,292,62,164,365,223,355,129),(41,293,63,165,366,224,356,130),(42,294,64,166,367,225,357,118),(43,295,65,167,368,226,358,119),(44,296,53,168,369,227,359,120),(45,297,54,169,370,228,360,121),(46,298,55,157,371,229,361,122),(47,299,56,158,372,230,362,123),(48,287,57,159,373,231,363,124),(49,288,58,160,374,232,364,125),(50,289,59,161,375,233,352,126),(51,290,60,162,376,234,353,127),(52,291,61,163,377,222,354,128)], [(1,220,175,118),(2,221,176,119),(3,209,177,120),(4,210,178,121),(5,211,179,122),(6,212,180,123),(7,213,181,124),(8,214,182,125),(9,215,170,126),(10,216,171,127),(11,217,172,128),(12,218,173,129),(13,219,174,130),(14,43,116,197),(15,44,117,198),(16,45,105,199),(17,46,106,200),(18,47,107,201),(19,48,108,202),(20,49,109,203),(21,50,110,204),(22,51,111,205),(23,52,112,206),(24,40,113,207),(25,41,114,208),(26,42,115,196),(27,151,65,380),(28,152,53,381),(29,153,54,382),(30,154,55,383),(31,155,56,384),(32,156,57,385),(33,144,58,386),(34,145,59,387),(35,146,60,388),(36,147,61,389),(37,148,62,390),(38,149,63,378),(39,150,64,379),(66,226,317,282),(67,227,318,283),(68,228,319,284),(69,229,320,285),(70,230,321,286),(71,231,322,274),(72,232,323,275),(73,233,324,276),(74,234,325,277),(75,222,313,278),(76,223,314,279),(77,224,315,280),(78,225,316,281),(79,143,167,395),(80,131,168,396),(81,132,169,397),(82,133,157,398),(83,134,158,399),(84,135,159,400),(85,136,160,401),(86,137,161,402),(87,138,162,403),(88,139,163,391),(89,140,164,392),(90,141,165,393),(91,142,166,394),(92,344,237,362),(93,345,238,363),(94,346,239,364),(95,347,240,352),(96,348,241,353),(97,349,242,354),(98,350,243,355),(99,351,244,356),(100,339,245,357),(101,340,246,358),(102,341,247,359),(103,342,235,360),(104,343,236,361),(183,407,336,296),(184,408,337,297),(185,409,338,298),(186,410,326,299),(187,411,327,287),(188,412,328,288),(189,413,329,289),(190,414,330,290),(191,415,331,291),(192,416,332,292),(193,404,333,293),(194,405,334,294),(195,406,335,295),(248,373,266,309),(249,374,267,310),(250,375,268,311),(251,376,269,312),(252,377,270,300),(253,365,271,301),(254,366,272,302),(255,367,273,303),(256,368,261,304),(257,369,262,305),(258,370,263,306),(259,371,264,307),(260,372,265,308)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | ··· | 13L | 26A | ··· | 26AJ | 52A | ··· | 52X | 52Y | ··· | 52BT | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C13 | C26 | C26 | C52 | Q8 | D4 | D8 | Q16 | Q8×C13 | D4×C13 | C13×D8 | C13×Q16 |
| kernel | C13×C2.D8 | C13×C4⋊C4 | C2×C104 | C104 | C2.D8 | C4⋊C4 | C2×C8 | C8 | C52 | C2×C26 | C26 | C26 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 12 | 24 | 12 | 48 | 1 | 1 | 2 | 2 | 12 | 12 | 24 | 24 |
Matrix representation of C13×C2.D8 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 103 | 0 |
| 0 | 0 | 103 |
| 312 | 0 | 0 |
| 0 | 312 | 0 |
| 0 | 0 | 312 |
| 312 | 0 | 0 |
| 0 | 60 | 253 |
| 0 | 60 | 60 |
| 288 | 0 | 0 |
| 0 | 147 | 109 |
| 0 | 109 | 166 |
G:=sub<GL(3,GF(313))| [1,0,0,0,103,0,0,0,103],[312,0,0,0,312,0,0,0,312],[312,0,0,0,60,60,0,253,60],[288,0,0,0,147,109,0,109,166] >;
C13×C2.D8 in GAP, Magma, Sage, TeX
C_{13}\times C_2.D_8 % in TeX
G:=Group("C13xC2.D8"); // GroupNames label
G:=SmallGroup(416,57);
// by ID
G=gap.SmallGroup(416,57);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,1567,6243,117]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export