direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C13×Q32, C16.C26, Q16.C26, C208.3C2, C26.17D8, C52.38D4, C104.26C22, C8.4(C2×C26), C4.3(D4×C13), C2.5(C13×D8), (C13×Q16).2C2, SmallGroup(416,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×Q32
G = < a,b,c | a13=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C13×Q32
►Regular action on 416 points
Generators in S416
(1 17 149 382 391 298 273 351 142 80 38 237 250)(2 18 150 383 392 299 274 352 143 65 39 238 251)(3 19 151 384 393 300 275 337 144 66 40 239 252)(4 20 152 369 394 301 276 338 129 67 41 240 253)(5 21 153 370 395 302 277 339 130 68 42 225 254)(6 22 154 371 396 303 278 340 131 69 43 226 255)(7 23 155 372 397 304 279 341 132 70 44 227 256)(8 24 156 373 398 289 280 342 133 71 45 228 241)(9 25 157 374 399 290 281 343 134 72 46 229 242)(10 26 158 375 400 291 282 344 135 73 47 230 243)(11 27 159 376 385 292 283 345 136 74 48 231 244)(12 28 160 377 386 293 284 346 137 75 33 232 245)(13 29 145 378 387 294 285 347 138 76 34 233 246)(14 30 146 379 388 295 286 348 139 77 35 234 247)(15 31 147 380 389 296 287 349 140 78 36 235 248)(16 32 148 381 390 297 288 350 141 79 37 236 249)(49 264 331 167 180 313 415 110 197 360 124 91 220)(50 265 332 168 181 314 416 111 198 361 125 92 221)(51 266 333 169 182 315 401 112 199 362 126 93 222)(52 267 334 170 183 316 402 97 200 363 127 94 223)(53 268 335 171 184 317 403 98 201 364 128 95 224)(54 269 336 172 185 318 404 99 202 365 113 96 209)(55 270 321 173 186 319 405 100 203 366 114 81 210)(56 271 322 174 187 320 406 101 204 367 115 82 211)(57 272 323 175 188 305 407 102 205 368 116 83 212)(58 257 324 176 189 306 408 103 206 353 117 84 213)(59 258 325 161 190 307 409 104 207 354 118 85 214)(60 259 326 162 191 308 410 105 208 355 119 86 215)(61 260 327 163 192 309 411 106 193 356 120 87 216)(62 261 328 164 177 310 412 107 194 357 121 88 217)(63 262 329 165 178 311 413 108 195 358 122 89 218)(64 263 330 166 179 312 414 109 196 359 123 90 219)
(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)
(1 114 9 122)(2 113 10 121)(3 128 11 120)(4 127 12 119)(5 126 13 118)(6 125 14 117)(7 124 15 116)(8 123 16 115)(17 81 25 89)(18 96 26 88)(19 95 27 87)(20 94 28 86)(21 93 29 85)(22 92 30 84)(23 91 31 83)(24 90 32 82)(33 105 41 97)(34 104 42 112)(35 103 43 111)(36 102 44 110)(37 101 45 109)(38 100 46 108)(39 99 47 107)(40 98 48 106)(49 380 57 372)(50 379 58 371)(51 378 59 370)(52 377 60 369)(53 376 61 384)(54 375 62 383)(55 374 63 382)(56 373 64 381)(65 404 73 412)(66 403 74 411)(67 402 75 410)(68 401 76 409)(69 416 77 408)(70 415 78 407)(71 414 79 406)(72 413 80 405)(129 316 137 308)(130 315 138 307)(131 314 139 306)(132 313 140 305)(133 312 141 320)(134 311 142 319)(135 310 143 318)(136 309 144 317)(145 214 153 222)(146 213 154 221)(147 212 155 220)(148 211 156 219)(149 210 157 218)(150 209 158 217)(151 224 159 216)(152 223 160 215)(161 277 169 285)(162 276 170 284)(163 275 171 283)(164 274 172 282)(165 273 173 281)(166 288 174 280)(167 287 175 279)(168 286 176 278)(177 352 185 344)(178 351 186 343)(179 350 187 342)(180 349 188 341)(181 348 189 340)(182 347 190 339)(183 346 191 338)(184 345 192 337)(193 239 201 231)(194 238 202 230)(195 237 203 229)(196 236 204 228)(197 235 205 227)(198 234 206 226)(199 233 207 225)(200 232 208 240)(241 359 249 367)(242 358 250 366)(243 357 251 365)(244 356 252 364)(245 355 253 363)(246 354 254 362)(247 353 255 361)(248 368 256 360)(257 396 265 388)(258 395 266 387)(259 394 267 386)(260 393 268 385)(261 392 269 400)(262 391 270 399)(263 390 271 398)(264 389 272 397)(289 330 297 322)(290 329 298 321)(291 328 299 336)(292 327 300 335)(293 326 301 334)(294 325 302 333)(295 324 303 332)(296 323 304 331)
G:=sub<Sym(416)| (1,17,149,382,391,298,273,351,142,80,38,237,250)(2,18,150,383,392,299,274,352,143,65,39,238,251)(3,19,151,384,393,300,275,337,144,66,40,239,252)(4,20,152,369,394,301,276,338,129,67,41,240,253)(5,21,153,370,395,302,277,339,130,68,42,225,254)(6,22,154,371,396,303,278,340,131,69,43,226,255)(7,23,155,372,397,304,279,341,132,70,44,227,256)(8,24,156,373,398,289,280,342,133,71,45,228,241)(9,25,157,374,399,290,281,343,134,72,46,229,242)(10,26,158,375,400,291,282,344,135,73,47,230,243)(11,27,159,376,385,292,283,345,136,74,48,231,244)(12,28,160,377,386,293,284,346,137,75,33,232,245)(13,29,145,378,387,294,285,347,138,76,34,233,246)(14,30,146,379,388,295,286,348,139,77,35,234,247)(15,31,147,380,389,296,287,349,140,78,36,235,248)(16,32,148,381,390,297,288,350,141,79,37,236,249)(49,264,331,167,180,313,415,110,197,360,124,91,220)(50,265,332,168,181,314,416,111,198,361,125,92,221)(51,266,333,169,182,315,401,112,199,362,126,93,222)(52,267,334,170,183,316,402,97,200,363,127,94,223)(53,268,335,171,184,317,403,98,201,364,128,95,224)(54,269,336,172,185,318,404,99,202,365,113,96,209)(55,270,321,173,186,319,405,100,203,366,114,81,210)(56,271,322,174,187,320,406,101,204,367,115,82,211)(57,272,323,175,188,305,407,102,205,368,116,83,212)(58,257,324,176,189,306,408,103,206,353,117,84,213)(59,258,325,161,190,307,409,104,207,354,118,85,214)(60,259,326,162,191,308,410,105,208,355,119,86,215)(61,260,327,163,192,309,411,106,193,356,120,87,216)(62,261,328,164,177,310,412,107,194,357,121,88,217)(63,262,329,165,178,311,413,108,195,358,122,89,218)(64,263,330,166,179,312,414,109,196,359,123,90,219), (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), (1,114,9,122)(2,113,10,121)(3,128,11,120)(4,127,12,119)(5,126,13,118)(6,125,14,117)(7,124,15,116)(8,123,16,115)(17,81,25,89)(18,96,26,88)(19,95,27,87)(20,94,28,86)(21,93,29,85)(22,92,30,84)(23,91,31,83)(24,90,32,82)(33,105,41,97)(34,104,42,112)(35,103,43,111)(36,102,44,110)(37,101,45,109)(38,100,46,108)(39,99,47,107)(40,98,48,106)(49,380,57,372)(50,379,58,371)(51,378,59,370)(52,377,60,369)(53,376,61,384)(54,375,62,383)(55,374,63,382)(56,373,64,381)(65,404,73,412)(66,403,74,411)(67,402,75,410)(68,401,76,409)(69,416,77,408)(70,415,78,407)(71,414,79,406)(72,413,80,405)(129,316,137,308)(130,315,138,307)(131,314,139,306)(132,313,140,305)(133,312,141,320)(134,311,142,319)(135,310,143,318)(136,309,144,317)(145,214,153,222)(146,213,154,221)(147,212,155,220)(148,211,156,219)(149,210,157,218)(150,209,158,217)(151,224,159,216)(152,223,160,215)(161,277,169,285)(162,276,170,284)(163,275,171,283)(164,274,172,282)(165,273,173,281)(166,288,174,280)(167,287,175,279)(168,286,176,278)(177,352,185,344)(178,351,186,343)(179,350,187,342)(180,349,188,341)(181,348,189,340)(182,347,190,339)(183,346,191,338)(184,345,192,337)(193,239,201,231)(194,238,202,230)(195,237,203,229)(196,236,204,228)(197,235,205,227)(198,234,206,226)(199,233,207,225)(200,232,208,240)(241,359,249,367)(242,358,250,366)(243,357,251,365)(244,356,252,364)(245,355,253,363)(246,354,254,362)(247,353,255,361)(248,368,256,360)(257,396,265,388)(258,395,266,387)(259,394,267,386)(260,393,268,385)(261,392,269,400)(262,391,270,399)(263,390,271,398)(264,389,272,397)(289,330,297,322)(290,329,298,321)(291,328,299,336)(292,327,300,335)(293,326,301,334)(294,325,302,333)(295,324,303,332)(296,323,304,331)>;
G:=Group( (1,17,149,382,391,298,273,351,142,80,38,237,250)(2,18,150,383,392,299,274,352,143,65,39,238,251)(3,19,151,384,393,300,275,337,144,66,40,239,252)(4,20,152,369,394,301,276,338,129,67,41,240,253)(5,21,153,370,395,302,277,339,130,68,42,225,254)(6,22,154,371,396,303,278,340,131,69,43,226,255)(7,23,155,372,397,304,279,341,132,70,44,227,256)(8,24,156,373,398,289,280,342,133,71,45,228,241)(9,25,157,374,399,290,281,343,134,72,46,229,242)(10,26,158,375,400,291,282,344,135,73,47,230,243)(11,27,159,376,385,292,283,345,136,74,48,231,244)(12,28,160,377,386,293,284,346,137,75,33,232,245)(13,29,145,378,387,294,285,347,138,76,34,233,246)(14,30,146,379,388,295,286,348,139,77,35,234,247)(15,31,147,380,389,296,287,349,140,78,36,235,248)(16,32,148,381,390,297,288,350,141,79,37,236,249)(49,264,331,167,180,313,415,110,197,360,124,91,220)(50,265,332,168,181,314,416,111,198,361,125,92,221)(51,266,333,169,182,315,401,112,199,362,126,93,222)(52,267,334,170,183,316,402,97,200,363,127,94,223)(53,268,335,171,184,317,403,98,201,364,128,95,224)(54,269,336,172,185,318,404,99,202,365,113,96,209)(55,270,321,173,186,319,405,100,203,366,114,81,210)(56,271,322,174,187,320,406,101,204,367,115,82,211)(57,272,323,175,188,305,407,102,205,368,116,83,212)(58,257,324,176,189,306,408,103,206,353,117,84,213)(59,258,325,161,190,307,409,104,207,354,118,85,214)(60,259,326,162,191,308,410,105,208,355,119,86,215)(61,260,327,163,192,309,411,106,193,356,120,87,216)(62,261,328,164,177,310,412,107,194,357,121,88,217)(63,262,329,165,178,311,413,108,195,358,122,89,218)(64,263,330,166,179,312,414,109,196,359,123,90,219), (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), (1,114,9,122)(2,113,10,121)(3,128,11,120)(4,127,12,119)(5,126,13,118)(6,125,14,117)(7,124,15,116)(8,123,16,115)(17,81,25,89)(18,96,26,88)(19,95,27,87)(20,94,28,86)(21,93,29,85)(22,92,30,84)(23,91,31,83)(24,90,32,82)(33,105,41,97)(34,104,42,112)(35,103,43,111)(36,102,44,110)(37,101,45,109)(38,100,46,108)(39,99,47,107)(40,98,48,106)(49,380,57,372)(50,379,58,371)(51,378,59,370)(52,377,60,369)(53,376,61,384)(54,375,62,383)(55,374,63,382)(56,373,64,381)(65,404,73,412)(66,403,74,411)(67,402,75,410)(68,401,76,409)(69,416,77,408)(70,415,78,407)(71,414,79,406)(72,413,80,405)(129,316,137,308)(130,315,138,307)(131,314,139,306)(132,313,140,305)(133,312,141,320)(134,311,142,319)(135,310,143,318)(136,309,144,317)(145,214,153,222)(146,213,154,221)(147,212,155,220)(148,211,156,219)(149,210,157,218)(150,209,158,217)(151,224,159,216)(152,223,160,215)(161,277,169,285)(162,276,170,284)(163,275,171,283)(164,274,172,282)(165,273,173,281)(166,288,174,280)(167,287,175,279)(168,286,176,278)(177,352,185,344)(178,351,186,343)(179,350,187,342)(180,349,188,341)(181,348,189,340)(182,347,190,339)(183,346,191,338)(184,345,192,337)(193,239,201,231)(194,238,202,230)(195,237,203,229)(196,236,204,228)(197,235,205,227)(198,234,206,226)(199,233,207,225)(200,232,208,240)(241,359,249,367)(242,358,250,366)(243,357,251,365)(244,356,252,364)(245,355,253,363)(246,354,254,362)(247,353,255,361)(248,368,256,360)(257,396,265,388)(258,395,266,387)(259,394,267,386)(260,393,268,385)(261,392,269,400)(262,391,270,399)(263,390,271,398)(264,389,272,397)(289,330,297,322)(290,329,298,321)(291,328,299,336)(292,327,300,335)(293,326,301,334)(294,325,302,333)(295,324,303,332)(296,323,304,331) );
G=PermutationGroup([[(1,17,149,382,391,298,273,351,142,80,38,237,250),(2,18,150,383,392,299,274,352,143,65,39,238,251),(3,19,151,384,393,300,275,337,144,66,40,239,252),(4,20,152,369,394,301,276,338,129,67,41,240,253),(5,21,153,370,395,302,277,339,130,68,42,225,254),(6,22,154,371,396,303,278,340,131,69,43,226,255),(7,23,155,372,397,304,279,341,132,70,44,227,256),(8,24,156,373,398,289,280,342,133,71,45,228,241),(9,25,157,374,399,290,281,343,134,72,46,229,242),(10,26,158,375,400,291,282,344,135,73,47,230,243),(11,27,159,376,385,292,283,345,136,74,48,231,244),(12,28,160,377,386,293,284,346,137,75,33,232,245),(13,29,145,378,387,294,285,347,138,76,34,233,246),(14,30,146,379,388,295,286,348,139,77,35,234,247),(15,31,147,380,389,296,287,349,140,78,36,235,248),(16,32,148,381,390,297,288,350,141,79,37,236,249),(49,264,331,167,180,313,415,110,197,360,124,91,220),(50,265,332,168,181,314,416,111,198,361,125,92,221),(51,266,333,169,182,315,401,112,199,362,126,93,222),(52,267,334,170,183,316,402,97,200,363,127,94,223),(53,268,335,171,184,317,403,98,201,364,128,95,224),(54,269,336,172,185,318,404,99,202,365,113,96,209),(55,270,321,173,186,319,405,100,203,366,114,81,210),(56,271,322,174,187,320,406,101,204,367,115,82,211),(57,272,323,175,188,305,407,102,205,368,116,83,212),(58,257,324,176,189,306,408,103,206,353,117,84,213),(59,258,325,161,190,307,409,104,207,354,118,85,214),(60,259,326,162,191,308,410,105,208,355,119,86,215),(61,260,327,163,192,309,411,106,193,356,120,87,216),(62,261,328,164,177,310,412,107,194,357,121,88,217),(63,262,329,165,178,311,413,108,195,358,122,89,218),(64,263,330,166,179,312,414,109,196,359,123,90,219)], [(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)], [(1,114,9,122),(2,113,10,121),(3,128,11,120),(4,127,12,119),(5,126,13,118),(6,125,14,117),(7,124,15,116),(8,123,16,115),(17,81,25,89),(18,96,26,88),(19,95,27,87),(20,94,28,86),(21,93,29,85),(22,92,30,84),(23,91,31,83),(24,90,32,82),(33,105,41,97),(34,104,42,112),(35,103,43,111),(36,102,44,110),(37,101,45,109),(38,100,46,108),(39,99,47,107),(40,98,48,106),(49,380,57,372),(50,379,58,371),(51,378,59,370),(52,377,60,369),(53,376,61,384),(54,375,62,383),(55,374,63,382),(56,373,64,381),(65,404,73,412),(66,403,74,411),(67,402,75,410),(68,401,76,409),(69,416,77,408),(70,415,78,407),(71,414,79,406),(72,413,80,405),(129,316,137,308),(130,315,138,307),(131,314,139,306),(132,313,140,305),(133,312,141,320),(134,311,142,319),(135,310,143,318),(136,309,144,317),(145,214,153,222),(146,213,154,221),(147,212,155,220),(148,211,156,219),(149,210,157,218),(150,209,158,217),(151,224,159,216),(152,223,160,215),(161,277,169,285),(162,276,170,284),(163,275,171,283),(164,274,172,282),(165,273,173,281),(166,288,174,280),(167,287,175,279),(168,286,176,278),(177,352,185,344),(178,351,186,343),(179,350,187,342),(180,349,188,341),(181,348,189,340),(182,347,190,339),(183,346,191,338),(184,345,192,337),(193,239,201,231),(194,238,202,230),(195,237,203,229),(196,236,204,228),(197,235,205,227),(198,234,206,226),(199,233,207,225),(200,232,208,240),(241,359,249,367),(242,358,250,366),(243,357,251,365),(244,356,252,364),(245,355,253,363),(246,354,254,362),(247,353,255,361),(248,368,256,360),(257,396,265,388),(258,395,266,387),(259,394,267,386),(260,393,268,385),(261,392,269,400),(262,391,270,399),(263,390,271,398),(264,389,272,397),(289,330,297,322),(290,329,298,321),(291,328,299,336),(292,327,300,335),(293,326,301,334),(294,325,302,333),(295,324,303,332),(296,323,304,331)]])
143 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13L | 16A | 16B | 16C | 16D | 26A | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52AJ | 104A | ··· | 104X | 208A | ··· | 208AV |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 16 | 16 | 16 | 16 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | D4 | D8 | Q32 | D4×C13 | C13×D8 | C13×Q32 |
| kernel | C13×Q32 | C208 | C13×Q16 | Q32 | C16 | Q16 | C52 | C26 | C13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 | 1 | 2 | 4 | 12 | 24 | 48 |
Matrix representation of C13×Q32 ►in GL2(𝔽1249) generated by
| 349 | 0 |
| 0 | 349 |
| 35 | 678 |
| 910 | 713 |
| 473 | 743 |
| 220 | 776 |
G:=sub<GL(2,GF(1249))| [349,0,0,349],[35,910,678,713],[473,220,743,776] >;
C13×Q32 in GAP, Magma, Sage, TeX
C_{13}\times Q_{32} % in TeX
G:=Group("C13xQ32"); // GroupNames label
G:=SmallGroup(416,63);
// by ID
G=gap.SmallGroup(416,63);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,1248,649,1255,3747,1881,165,9364,4690,88]);
// Polycyclic
G:=Group<a,b,c|a^13=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export