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G = C13×C2.D8order 416 = 25·13

Direct product of C13 and C2.D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C13×C2.D8, C81C52, C1049C4, 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

C1C4 — C13×C2.D8
C1C2C22C2×C4C2×C52C13×C4⋊C4 — C13×C2.D8
C1C2C4 — C13×C2.D8
C1C2×C26C2×C52 — C13×C2.D8

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 >

4C4
4C4
2C2×C4
2C2×C4
4C52
4C52
2C2×C52
2C2×C52

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

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

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

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

182 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F8A8B8C8D13A···13L26A···26AJ52A···52X52Y···52BT104A···104AV
order1222444444888813···1326···2652···5252···52104···104
size111122444422221···11···12···24···42···2

182 irreducible representations

dim1111111122222222
type+++-++-
imageC1C2C2C4C13C26C26C52Q8D4D8Q16Q8×C13D4×C13C13×D8C13×Q16
kernelC13×C2.D8C13×C4⋊C4C2×C104C104C2.D8C4⋊C4C2×C8C8C52C2×C26C26C26C4C22C2C2
# reps121412241248112212122424

Matrix representation of C13×C2.D8 in GL3(𝔽313) generated by

100
01030
00103
,
31200
03120
00312
,
31200
060253
06060
,
28800
0147109
0109166
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

Subgroup lattice of C13×C2.D8 in TeX

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