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## G = C13×C2.C42order 416 = 25·13

### Direct product of C13 and C2.C42

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C13×C2.C42
 Chief series C1 — C2 — C22 — C23 — C22×C26 — C22×C52 — C13×C2.C42
 Lower central C1 — C2 — C13×C2.C42
 Upper central C1 — C22×C26 — C13×C2.C42

Generators and relations for C13×C2.C42
G = < a,b,c,d | a13=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 100 in 76 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C13, C22×C4, C26, C26, C2.C42, C52, C2×C26, C2×C26, C2×C52, C2×C52, C22×C26, C22×C52, C13×C2.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C13, C42, C22⋊C4, C4⋊C4, C26, C2.C42, C52, C2×C26, C2×C52, D4×C13, Q8×C13, C4×C52, C13×C22⋊C4, C13×C4⋊C4, C13×C2.C42

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

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

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

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

260 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 13A ··· 13L 26A ··· 26CF 52A ··· 52EN order 1 2 ··· 2 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C13 C26 C52 D4 Q8 D4×C13 Q8×C13 kernel C13×C2.C42 C22×C52 C2×C52 C2.C42 C22×C4 C2×C4 C2×C26 C2×C26 C22 C22 # reps 1 3 12 12 36 144 3 1 36 12

Matrix representation of C13×C2.C42 in GL4(𝔽53) generated by

 1 0 0 0 0 1 0 0 0 0 24 0 0 0 0 24
,
 1 0 0 0 0 1 0 0 0 0 52 0 0 0 0 52
,
 52 0 0 0 0 23 0 0 0 0 4 21 0 0 32 49
,
 23 0 0 0 0 30 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(53))| [1,0,0,0,0,1,0,0,0,0,24,0,0,0,0,24],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,23,0,0,0,0,4,32,0,0,21,49],[23,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0] >;

C13×C2.C42 in GAP, Magma, Sage, TeX

C_{13}\times C_2.C_4^2
% in TeX

G:=Group("C13xC2.C4^2");
// GroupNames label

G:=SmallGroup(416,45);
// by ID

G=gap.SmallGroup(416,45);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,1255]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations

׿
×
𝔽