direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C13⋊C16, C26⋊C16, C52.3C8, C13⋊2(C2×C16), C4.3(C13⋊C8), (C2×C52).9C4, C26.5(C2×C8), (C2×C26).1C8, C13⋊2C8.5C4, C52.17(C2×C4), C22.2(C13⋊C8), C13⋊2C8.16C22, C2.1(C2×C13⋊C8), C4.18(C2×C13⋊C4), (C2×C4).9(C13⋊C4), (C2×C13⋊2C8).12C2, SmallGroup(416,72)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C13⋊C16 |
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
(1 292)(2 293)(3 294)(4 295)(5 296)(6 297)(7 298)(8 299)(9 300)(10 301)(11 302)(12 303)(13 304)(14 289)(15 290)(16 291)(17 402)(18 403)(19 404)(20 405)(21 406)(22 407)(23 408)(24 409)(25 410)(26 411)(27 412)(28 413)(29 414)(30 415)(31 416)(32 401)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 93)(50 94)(51 95)(52 96)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(97 346)(98 347)(99 348)(100 349)(101 350)(102 351)(103 352)(104 337)(105 338)(106 339)(107 340)(108 341)(109 342)(110 343)(111 344)(112 345)(113 143)(114 144)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(127 141)(128 142)(145 356)(146 357)(147 358)(148 359)(149 360)(150 361)(151 362)(152 363)(153 364)(154 365)(155 366)(156 367)(157 368)(158 353)(159 354)(160 355)(161 254)(162 255)(163 256)(164 241)(165 242)(166 243)(167 244)(168 245)(169 246)(170 247)(171 248)(172 249)(173 250)(174 251)(175 252)(176 253)(177 214)(178 215)(179 216)(180 217)(181 218)(182 219)(183 220)(184 221)(185 222)(186 223)(187 224)(188 209)(189 210)(190 211)(191 212)(192 213)(193 284)(194 285)(195 286)(196 287)(197 288)(198 273)(199 274)(200 275)(201 276)(202 277)(203 278)(204 279)(205 280)(206 281)(207 282)(208 283)(225 374)(226 375)(227 376)(228 377)(229 378)(230 379)(231 380)(232 381)(233 382)(234 383)(235 384)(236 369)(237 370)(238 371)(239 372)(240 373)(257 389)(258 390)(259 391)(260 392)(261 393)(262 394)(263 395)(264 396)(265 397)(266 398)(267 399)(268 400)(269 385)(270 386)(271 387)(272 388)(305 334)(306 335)(307 336)(308 321)(309 322)(310 323)(311 324)(312 325)(313 326)(314 327)(315 328)(316 329)(317 330)(318 331)(319 332)(320 333)
(1 416 168 67 324 128 257 376 342 219 59 149 284)(2 113 60 169 377 285 325 220 401 258 150 68 343)(3 286 151 61 221 344 378 259 114 326 69 170 402)(4 345 70 152 260 403 222 327 287 379 171 62 115)(5 404 172 71 328 116 261 380 346 223 63 153 288)(6 117 64 173 381 273 329 224 405 262 154 72 347)(7 274 155 49 209 348 382 263 118 330 73 174 406)(8 349 74 156 264 407 210 331 275 383 175 50 119)(9 408 176 75 332 120 265 384 350 211 51 157 276)(10 121 52 161 369 277 333 212 409 266 158 76 351)(11 278 159 53 213 352 370 267 122 334 77 162 410)(12 337 78 160 268 411 214 335 279 371 163 54 123)(13 412 164 79 336 124 269 372 338 215 55 145 280)(14 125 56 165 373 281 321 216 413 270 146 80 339)(15 282 147 57 217 340 374 271 126 322 65 166 414)(16 341 66 148 272 415 218 323 283 375 167 58 127)(17 294 195 362 89 184 111 229 391 144 313 46 247)(18 185 314 196 230 248 90 129 295 112 47 363 392)(19 249 48 315 130 393 231 97 186 91 364 197 296)(20 394 365 33 98 297 131 92 250 232 198 316 187)(21 298 199 366 93 188 99 233 395 132 317 34 251)(22 189 318 200 234 252 94 133 299 100 35 367 396)(23 253 36 319 134 397 235 101 190 95 368 201 300)(24 398 353 37 102 301 135 96 254 236 202 320 191)(25 302 203 354 81 192 103 237 399 136 305 38 255)(26 177 306 204 238 256 82 137 303 104 39 355 400)(27 241 40 307 138 385 239 105 178 83 356 205 304)(28 386 357 41 106 289 139 84 242 240 206 308 179)(29 290 207 358 85 180 107 225 387 140 309 42 243)(30 181 310 208 226 244 86 141 291 108 43 359 388)(31 245 44 311 142 389 227 109 182 87 360 193 292)(32 390 361 45 110 293 143 88 246 228 194 312 183)
(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)| (1,292)(2,293)(3,294)(4,295)(5,296)(6,297)(7,298)(8,299)(9,300)(10,301)(11,302)(12,303)(13,304)(14,289)(15,290)(16,291)(17,402)(18,403)(19,404)(20,405)(21,406)(22,407)(23,408)(24,409)(25,410)(26,411)(27,412)(28,413)(29,414)(30,415)(31,416)(32,401)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,93)(50,94)(51,95)(52,96)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(97,346)(98,347)(99,348)(100,349)(101,350)(102,351)(103,352)(104,337)(105,338)(106,339)(107,340)(108,341)(109,342)(110,343)(111,344)(112,345)(113,143)(114,144)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(145,356)(146,357)(147,358)(148,359)(149,360)(150,361)(151,362)(152,363)(153,364)(154,365)(155,366)(156,367)(157,368)(158,353)(159,354)(160,355)(161,254)(162,255)(163,256)(164,241)(165,242)(166,243)(167,244)(168,245)(169,246)(170,247)(171,248)(172,249)(173,250)(174,251)(175,252)(176,253)(177,214)(178,215)(179,216)(180,217)(181,218)(182,219)(183,220)(184,221)(185,222)(186,223)(187,224)(188,209)(189,210)(190,211)(191,212)(192,213)(193,284)(194,285)(195,286)(196,287)(197,288)(198,273)(199,274)(200,275)(201,276)(202,277)(203,278)(204,279)(205,280)(206,281)(207,282)(208,283)(225,374)(226,375)(227,376)(228,377)(229,378)(230,379)(231,380)(232,381)(233,382)(234,383)(235,384)(236,369)(237,370)(238,371)(239,372)(240,373)(257,389)(258,390)(259,391)(260,392)(261,393)(262,394)(263,395)(264,396)(265,397)(266,398)(267,399)(268,400)(269,385)(270,386)(271,387)(272,388)(305,334)(306,335)(307,336)(308,321)(309,322)(310,323)(311,324)(312,325)(313,326)(314,327)(315,328)(316,329)(317,330)(318,331)(319,332)(320,333), (1,416,168,67,324,128,257,376,342,219,59,149,284)(2,113,60,169,377,285,325,220,401,258,150,68,343)(3,286,151,61,221,344,378,259,114,326,69,170,402)(4,345,70,152,260,403,222,327,287,379,171,62,115)(5,404,172,71,328,116,261,380,346,223,63,153,288)(6,117,64,173,381,273,329,224,405,262,154,72,347)(7,274,155,49,209,348,382,263,118,330,73,174,406)(8,349,74,156,264,407,210,331,275,383,175,50,119)(9,408,176,75,332,120,265,384,350,211,51,157,276)(10,121,52,161,369,277,333,212,409,266,158,76,351)(11,278,159,53,213,352,370,267,122,334,77,162,410)(12,337,78,160,268,411,214,335,279,371,163,54,123)(13,412,164,79,336,124,269,372,338,215,55,145,280)(14,125,56,165,373,281,321,216,413,270,146,80,339)(15,282,147,57,217,340,374,271,126,322,65,166,414)(16,341,66,148,272,415,218,323,283,375,167,58,127)(17,294,195,362,89,184,111,229,391,144,313,46,247)(18,185,314,196,230,248,90,129,295,112,47,363,392)(19,249,48,315,130,393,231,97,186,91,364,197,296)(20,394,365,33,98,297,131,92,250,232,198,316,187)(21,298,199,366,93,188,99,233,395,132,317,34,251)(22,189,318,200,234,252,94,133,299,100,35,367,396)(23,253,36,319,134,397,235,101,190,95,368,201,300)(24,398,353,37,102,301,135,96,254,236,202,320,191)(25,302,203,354,81,192,103,237,399,136,305,38,255)(26,177,306,204,238,256,82,137,303,104,39,355,400)(27,241,40,307,138,385,239,105,178,83,356,205,304)(28,386,357,41,106,289,139,84,242,240,206,308,179)(29,290,207,358,85,180,107,225,387,140,309,42,243)(30,181,310,208,226,244,86,141,291,108,43,359,388)(31,245,44,311,142,389,227,109,182,87,360,193,292)(32,390,361,45,110,293,143,88,246,228,194,312,183), (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( (1,292)(2,293)(3,294)(4,295)(5,296)(6,297)(7,298)(8,299)(9,300)(10,301)(11,302)(12,303)(13,304)(14,289)(15,290)(16,291)(17,402)(18,403)(19,404)(20,405)(21,406)(22,407)(23,408)(24,409)(25,410)(26,411)(27,412)(28,413)(29,414)(30,415)(31,416)(32,401)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,93)(50,94)(51,95)(52,96)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(97,346)(98,347)(99,348)(100,349)(101,350)(102,351)(103,352)(104,337)(105,338)(106,339)(107,340)(108,341)(109,342)(110,343)(111,344)(112,345)(113,143)(114,144)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(145,356)(146,357)(147,358)(148,359)(149,360)(150,361)(151,362)(152,363)(153,364)(154,365)(155,366)(156,367)(157,368)(158,353)(159,354)(160,355)(161,254)(162,255)(163,256)(164,241)(165,242)(166,243)(167,244)(168,245)(169,246)(170,247)(171,248)(172,249)(173,250)(174,251)(175,252)(176,253)(177,214)(178,215)(179,216)(180,217)(181,218)(182,219)(183,220)(184,221)(185,222)(186,223)(187,224)(188,209)(189,210)(190,211)(191,212)(192,213)(193,284)(194,285)(195,286)(196,287)(197,288)(198,273)(199,274)(200,275)(201,276)(202,277)(203,278)(204,279)(205,280)(206,281)(207,282)(208,283)(225,374)(226,375)(227,376)(228,377)(229,378)(230,379)(231,380)(232,381)(233,382)(234,383)(235,384)(236,369)(237,370)(238,371)(239,372)(240,373)(257,389)(258,390)(259,391)(260,392)(261,393)(262,394)(263,395)(264,396)(265,397)(266,398)(267,399)(268,400)(269,385)(270,386)(271,387)(272,388)(305,334)(306,335)(307,336)(308,321)(309,322)(310,323)(311,324)(312,325)(313,326)(314,327)(315,328)(316,329)(317,330)(318,331)(319,332)(320,333), (1,416,168,67,324,128,257,376,342,219,59,149,284)(2,113,60,169,377,285,325,220,401,258,150,68,343)(3,286,151,61,221,344,378,259,114,326,69,170,402)(4,345,70,152,260,403,222,327,287,379,171,62,115)(5,404,172,71,328,116,261,380,346,223,63,153,288)(6,117,64,173,381,273,329,224,405,262,154,72,347)(7,274,155,49,209,348,382,263,118,330,73,174,406)(8,349,74,156,264,407,210,331,275,383,175,50,119)(9,408,176,75,332,120,265,384,350,211,51,157,276)(10,121,52,161,369,277,333,212,409,266,158,76,351)(11,278,159,53,213,352,370,267,122,334,77,162,410)(12,337,78,160,268,411,214,335,279,371,163,54,123)(13,412,164,79,336,124,269,372,338,215,55,145,280)(14,125,56,165,373,281,321,216,413,270,146,80,339)(15,282,147,57,217,340,374,271,126,322,65,166,414)(16,341,66,148,272,415,218,323,283,375,167,58,127)(17,294,195,362,89,184,111,229,391,144,313,46,247)(18,185,314,196,230,248,90,129,295,112,47,363,392)(19,249,48,315,130,393,231,97,186,91,364,197,296)(20,394,365,33,98,297,131,92,250,232,198,316,187)(21,298,199,366,93,188,99,233,395,132,317,34,251)(22,189,318,200,234,252,94,133,299,100,35,367,396)(23,253,36,319,134,397,235,101,190,95,368,201,300)(24,398,353,37,102,301,135,96,254,236,202,320,191)(25,302,203,354,81,192,103,237,399,136,305,38,255)(26,177,306,204,238,256,82,137,303,104,39,355,400)(27,241,40,307,138,385,239,105,178,83,356,205,304)(28,386,357,41,106,289,139,84,242,240,206,308,179)(29,290,207,358,85,180,107,225,387,140,309,42,243)(30,181,310,208,226,244,86,141,291,108,43,359,388)(31,245,44,311,142,389,227,109,182,87,360,193,292)(32,390,361,45,110,293,143,88,246,228,194,312,183), (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([[(1,292),(2,293),(3,294),(4,295),(5,296),(6,297),(7,298),(8,299),(9,300),(10,301),(11,302),(12,303),(13,304),(14,289),(15,290),(16,291),(17,402),(18,403),(19,404),(20,405),(21,406),(22,407),(23,408),(24,409),(25,410),(26,411),(27,412),(28,413),(29,414),(30,415),(31,416),(32,401),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,93),(50,94),(51,95),(52,96),(53,81),(54,82),(55,83),(56,84),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(97,346),(98,347),(99,348),(100,349),(101,350),(102,351),(103,352),(104,337),(105,338),(106,339),(107,340),(108,341),(109,342),(110,343),(111,344),(112,345),(113,143),(114,144),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(127,141),(128,142),(145,356),(146,357),(147,358),(148,359),(149,360),(150,361),(151,362),(152,363),(153,364),(154,365),(155,366),(156,367),(157,368),(158,353),(159,354),(160,355),(161,254),(162,255),(163,256),(164,241),(165,242),(166,243),(167,244),(168,245),(169,246),(170,247),(171,248),(172,249),(173,250),(174,251),(175,252),(176,253),(177,214),(178,215),(179,216),(180,217),(181,218),(182,219),(183,220),(184,221),(185,222),(186,223),(187,224),(188,209),(189,210),(190,211),(191,212),(192,213),(193,284),(194,285),(195,286),(196,287),(197,288),(198,273),(199,274),(200,275),(201,276),(202,277),(203,278),(204,279),(205,280),(206,281),(207,282),(208,283),(225,374),(226,375),(227,376),(228,377),(229,378),(230,379),(231,380),(232,381),(233,382),(234,383),(235,384),(236,369),(237,370),(238,371),(239,372),(240,373),(257,389),(258,390),(259,391),(260,392),(261,393),(262,394),(263,395),(264,396),(265,397),(266,398),(267,399),(268,400),(269,385),(270,386),(271,387),(272,388),(305,334),(306,335),(307,336),(308,321),(309,322),(310,323),(311,324),(312,325),(313,326),(314,327),(315,328),(316,329),(317,330),(318,331),(319,332),(320,333)], [(1,416,168,67,324,128,257,376,342,219,59,149,284),(2,113,60,169,377,285,325,220,401,258,150,68,343),(3,286,151,61,221,344,378,259,114,326,69,170,402),(4,345,70,152,260,403,222,327,287,379,171,62,115),(5,404,172,71,328,116,261,380,346,223,63,153,288),(6,117,64,173,381,273,329,224,405,262,154,72,347),(7,274,155,49,209,348,382,263,118,330,73,174,406),(8,349,74,156,264,407,210,331,275,383,175,50,119),(9,408,176,75,332,120,265,384,350,211,51,157,276),(10,121,52,161,369,277,333,212,409,266,158,76,351),(11,278,159,53,213,352,370,267,122,334,77,162,410),(12,337,78,160,268,411,214,335,279,371,163,54,123),(13,412,164,79,336,124,269,372,338,215,55,145,280),(14,125,56,165,373,281,321,216,413,270,146,80,339),(15,282,147,57,217,340,374,271,126,322,65,166,414),(16,341,66,148,272,415,218,323,283,375,167,58,127),(17,294,195,362,89,184,111,229,391,144,313,46,247),(18,185,314,196,230,248,90,129,295,112,47,363,392),(19,249,48,315,130,393,231,97,186,91,364,197,296),(20,394,365,33,98,297,131,92,250,232,198,316,187),(21,298,199,366,93,188,99,233,395,132,317,34,251),(22,189,318,200,234,252,94,133,299,100,35,367,396),(23,253,36,319,134,397,235,101,190,95,368,201,300),(24,398,353,37,102,301,135,96,254,236,202,320,191),(25,302,203,354,81,192,103,237,399,136,305,38,255),(26,177,306,204,238,256,82,137,303,104,39,355,400),(27,241,40,307,138,385,239,105,178,83,356,205,304),(28,386,357,41,106,289,139,84,242,240,206,308,179),(29,290,207,358,85,180,107,225,387,140,309,42,243),(30,181,310,208,226,244,86,141,291,108,43,359,388),(31,245,44,311,142,389,227,109,182,87,360,193,292),(32,390,361,45,110,293,143,88,246,228,194,312,183)], [(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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