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## G = C2×C13⋊C16order 416 = 25·13

### Direct product of C2 and C13⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C2×C13⋊C16
 Chief series C1 — C13 — C26 — C52 — C13⋊2C8 — C13⋊C16 — C2×C13⋊C16
 Lower central C13 — C2×C13⋊C16
 Upper central C1 — C2×C4

Generators and relations for C2×C13⋊C16
G = < a,b,c | a2=b13=c16=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C2×C13⋊C16
Regular action on 416 points
Generators in S416
(1 210)(2 211)(3 212)(4 213)(5 214)(6 215)(7 216)(8 217)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 224)(16 209)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 403)(34 404)(35 405)(36 406)(37 407)(38 408)(39 409)(40 410)(41 411)(42 412)(43 413)(44 414)(45 415)(46 416)(47 401)(48 402)(49 395)(50 396)(51 397)(52 398)(53 399)(54 400)(55 385)(56 386)(57 387)(58 388)(59 389)(60 390)(61 391)(62 392)(63 393)(64 394)(65 148)(66 149)(67 150)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 159)(77 160)(78 145)(79 146)(80 147)(81 340)(82 341)(83 342)(84 343)(85 344)(86 345)(87 346)(88 347)(89 348)(90 349)(91 350)(92 351)(93 352)(94 337)(95 338)(96 339)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(161 298)(162 299)(163 300)(164 301)(165 302)(166 303)(167 304)(168 289)(169 290)(170 291)(171 292)(172 293)(173 294)(174 295)(175 296)(176 297)(177 236)(178 237)(179 238)(180 239)(181 240)(182 225)(183 226)(184 227)(185 228)(186 229)(187 230)(188 231)(189 232)(190 233)(191 234)(192 235)(193 314)(194 315)(195 316)(196 317)(197 318)(198 319)(199 320)(200 305)(201 306)(202 307)(203 308)(204 309)(205 310)(206 311)(207 312)(208 313)(241 361)(242 362)(243 363)(244 364)(245 365)(246 366)(247 367)(248 368)(249 353)(250 354)(251 355)(252 356)(253 357)(254 358)(255 359)(256 360)(257 273)(258 274)(259 275)(260 276)(261 277)(262 278)(263 279)(264 280)(265 281)(266 282)(267 283)(268 284)(269 285)(270 286)(271 287)(272 288)(321 372)(322 373)(323 374)(324 375)(325 376)(326 377)(327 378)(328 379)(329 380)(330 381)(331 382)(332 383)(333 384)(334 369)(335 370)(336 371)
(1 143 279 158 353 18 49 182 404 206 92 164 332)(2 19 93 280 183 333 354 207 144 50 165 159 405)(3 334 166 94 208 406 184 51 20 355 160 281 129)(4 407 145 167 52 130 193 356 335 185 282 95 21)(5 131 283 146 357 22 53 186 408 194 96 168 336)(6 23 81 284 187 321 358 195 132 54 169 147 409)(7 322 170 82 196 410 188 55 24 359 148 285 133)(8 411 149 171 56 134 197 360 323 189 286 83 25)(9 135 287 150 361 26 57 190 412 198 84 172 324)(10 27 85 288 191 325 362 199 136 58 173 151 413)(11 326 174 86 200 414 192 59 28 363 152 273 137)(12 415 153 175 60 138 201 364 327 177 274 87 29)(13 139 275 154 365 30 61 178 416 202 88 176 328)(14 31 89 276 179 329 366 203 140 62 161 155 401)(15 330 162 90 204 402 180 63 32 367 156 277 141)(16 403 157 163 64 142 205 368 331 181 278 91 17)(33 74 300 394 102 310 248 382 240 262 350 113 209)(34 311 351 301 383 210 103 263 75 249 114 395 225)(35 211 115 352 264 226 384 250 312 104 396 302 76)(36 227 397 116 251 77 265 105 212 369 303 337 313)(37 78 304 398 106 314 252 370 228 266 338 117 213)(38 315 339 289 371 214 107 267 79 253 118 399 229)(39 215 119 340 268 230 372 254 316 108 400 290 80)(40 231 385 120 255 65 269 109 216 373 291 341 317)(41 66 292 386 110 318 256 374 232 270 342 121 217)(42 319 343 293 375 218 111 271 67 241 122 387 233)(43 219 123 344 272 234 376 242 320 112 388 294 68)(44 235 389 124 243 69 257 97 220 377 295 345 305)(45 70 296 390 98 306 244 378 236 258 346 125 221)(46 307 347 297 379 222 99 259 71 245 126 391 237)(47 223 127 348 260 238 380 246 308 100 392 298 72)(48 239 393 128 247 73 261 101 224 381 299 349 309)
(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)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 8A ··· 8H 13A 13B 13C 16A ··· 16P 26A ··· 26I 52A ··· 52L order 1 2 2 2 4 4 4 4 8 ··· 8 13 13 13 16 ··· 16 26 ··· 26 52 ··· 52 size 1 1 1 1 1 1 1 1 13 ··· 13 4 4 4 13 ··· 13 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 4 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 C13⋊C4 C13⋊C8 C2×C13⋊C4 C13⋊C8 C13⋊C16 kernel C2×C13⋊C16 C13⋊C16 C2×C13⋊2C8 C13⋊2C8 C2×C52 C52 C2×C26 C26 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 4 4 16 3 3 3 3 12

Matrix representation of C2×C13⋊C16 in GL5(𝔽1249)

 1248 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 685 564 44 1248 0 686 564 44 1248 0 685 565 44 1248 0 685 564 45 1248
,
 1248 0 0 0 0 0 36 771 889 964 0 434 870 554 1190 0 524 355 271 723 0 793 418 1000 72

G:=sub<GL(5,GF(1249))| [1248,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,685,686,685,685,0,564,564,565,564,0,44,44,44,45,0,1248,1248,1248,1248],[1248,0,0,0,0,0,36,434,524,793,0,771,870,355,418,0,889,554,271,1000,0,964,1190,723,72] >;

C2×C13⋊C16 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes C_{16}
% in TeX

G:=Group("C2xC13:C16");
// GroupNames label

G:=SmallGroup(416,72);
// by ID

G=gap.SmallGroup(416,72);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,50,69,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^2=b^13=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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