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## G = C13×C42.C2order 416 = 25·13

### Direct product of C13 and C42.C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C13×C42.C2
 Chief series C1 — C2 — C22 — C2×C26 — C2×C52 — C13×C4⋊C4 — C13×C42.C2
 Lower central C1 — C22 — C13×C42.C2
 Upper central C1 — C2×C26 — C13×C42.C2

Generators and relations for C13×C42.C2
G = < a,b,c,d | a13=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 68 in 56 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, C13, C42, C4⋊C4, C26, C26, C42.C2, C52, C52, C2×C26, C2×C52, C2×C52, C4×C52, C13×C4⋊C4, C13×C42.C2
Quotients: C1, C2, C22, Q8, C23, C13, C2×Q8, C4○D4, C26, C42.C2, C2×C26, Q8×C13, C22×C26, Q8×C26, C13×C4○D4, C13×C42.C2

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

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 13A ··· 13L 26A ··· 26AJ 52A ··· 52BT 52BU ··· 52DP order 1 2 2 2 4 ··· 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4

182 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C2 C13 C26 C26 Q8 C4○D4 Q8×C13 C13×C4○D4 kernel C13×C42.C2 C4×C52 C13×C4⋊C4 C42.C2 C42 C4⋊C4 C52 C26 C4 C2 # reps 1 1 6 12 12 72 2 4 24 48

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

 15 0 0 0 0 15 0 0 0 0 10 0 0 0 0 10
,
 52 51 0 0 0 1 0 0 0 0 23 0 0 0 0 23
,
 23 0 0 0 0 23 0 0 0 0 25 51 0 0 47 28
,
 49 9 0 0 4 4 0 0 0 0 46 35 0 0 38 7
G:=sub<GL(4,GF(53))| [15,0,0,0,0,15,0,0,0,0,10,0,0,0,0,10],[52,0,0,0,51,1,0,0,0,0,23,0,0,0,0,23],[23,0,0,0,0,23,0,0,0,0,25,47,0,0,51,28],[49,4,0,0,9,4,0,0,0,0,46,38,0,0,35,7] >;

C13×C42.C2 in GAP, Magma, Sage, TeX

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

G:=Group("C13xC4^2.C2");
// GroupNames label

G:=SmallGroup(416,186);
// by ID

G=gap.SmallGroup(416,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,1273,1255,3818,482]);
// Polycyclic

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

׿
×
𝔽