metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13⋊2C32, C52.5C8, C208.2C2, C104.7C4, C26.2C16, C16.2D13, C8.3Dic13, C2.(C13⋊2C16), C4.2(C13⋊2C8), SmallGroup(416,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊2C32 |
Generators and relations for C13⋊2C32
G = < a,b | a13=b32=1, bab-1=a-1 >
Smallest permutation representation of C13⋊2C32
►Regular action on 416 points
Generators in S416
(1 186 376 311 75 409 322 220 44 160 277 122 230)(2 231 123 278 129 45 221 323 410 76 312 377 187)(3 188 378 313 77 411 324 222 46 130 279 124 232)(4 233 125 280 131 47 223 325 412 78 314 379 189)(5 190 380 315 79 413 326 224 48 132 281 126 234)(6 235 127 282 133 49 193 327 414 80 316 381 191)(7 192 382 317 81 415 328 194 50 134 283 128 236)(8 237 97 284 135 51 195 329 416 82 318 383 161)(9 162 384 319 83 385 330 196 52 136 285 98 238)(10 239 99 286 137 53 197 331 386 84 320 353 163)(11 164 354 289 85 387 332 198 54 138 287 100 240)(12 241 101 288 139 55 199 333 388 86 290 355 165)(13 166 356 291 87 389 334 200 56 140 257 102 242)(14 243 103 258 141 57 201 335 390 88 292 357 167)(15 168 358 293 89 391 336 202 58 142 259 104 244)(16 245 105 260 143 59 203 337 392 90 294 359 169)(17 170 360 295 91 393 338 204 60 144 261 106 246)(18 247 107 262 145 61 205 339 394 92 296 361 171)(19 172 362 297 93 395 340 206 62 146 263 108 248)(20 249 109 264 147 63 207 341 396 94 298 363 173)(21 174 364 299 95 397 342 208 64 148 265 110 250)(22 251 111 266 149 33 209 343 398 96 300 365 175)(23 176 366 301 65 399 344 210 34 150 267 112 252)(24 253 113 268 151 35 211 345 400 66 302 367 177)(25 178 368 303 67 401 346 212 36 152 269 114 254)(26 255 115 270 153 37 213 347 402 68 304 369 179)(27 180 370 305 69 403 348 214 38 154 271 116 256)(28 225 117 272 155 39 215 349 404 70 306 371 181)(29 182 372 307 71 405 350 216 40 156 273 118 226)(30 227 119 274 157 41 217 351 406 72 308 373 183)(31 184 374 309 73 407 352 218 42 158 275 120 228)(32 229 121 276 159 43 219 321 408 74 310 375 185)
(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)| (1,186,376,311,75,409,322,220,44,160,277,122,230)(2,231,123,278,129,45,221,323,410,76,312,377,187)(3,188,378,313,77,411,324,222,46,130,279,124,232)(4,233,125,280,131,47,223,325,412,78,314,379,189)(5,190,380,315,79,413,326,224,48,132,281,126,234)(6,235,127,282,133,49,193,327,414,80,316,381,191)(7,192,382,317,81,415,328,194,50,134,283,128,236)(8,237,97,284,135,51,195,329,416,82,318,383,161)(9,162,384,319,83,385,330,196,52,136,285,98,238)(10,239,99,286,137,53,197,331,386,84,320,353,163)(11,164,354,289,85,387,332,198,54,138,287,100,240)(12,241,101,288,139,55,199,333,388,86,290,355,165)(13,166,356,291,87,389,334,200,56,140,257,102,242)(14,243,103,258,141,57,201,335,390,88,292,357,167)(15,168,358,293,89,391,336,202,58,142,259,104,244)(16,245,105,260,143,59,203,337,392,90,294,359,169)(17,170,360,295,91,393,338,204,60,144,261,106,246)(18,247,107,262,145,61,205,339,394,92,296,361,171)(19,172,362,297,93,395,340,206,62,146,263,108,248)(20,249,109,264,147,63,207,341,396,94,298,363,173)(21,174,364,299,95,397,342,208,64,148,265,110,250)(22,251,111,266,149,33,209,343,398,96,300,365,175)(23,176,366,301,65,399,344,210,34,150,267,112,252)(24,253,113,268,151,35,211,345,400,66,302,367,177)(25,178,368,303,67,401,346,212,36,152,269,114,254)(26,255,115,270,153,37,213,347,402,68,304,369,179)(27,180,370,305,69,403,348,214,38,154,271,116,256)(28,225,117,272,155,39,215,349,404,70,306,371,181)(29,182,372,307,71,405,350,216,40,156,273,118,226)(30,227,119,274,157,41,217,351,406,72,308,373,183)(31,184,374,309,73,407,352,218,42,158,275,120,228)(32,229,121,276,159,43,219,321,408,74,310,375,185), (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( (1,186,376,311,75,409,322,220,44,160,277,122,230)(2,231,123,278,129,45,221,323,410,76,312,377,187)(3,188,378,313,77,411,324,222,46,130,279,124,232)(4,233,125,280,131,47,223,325,412,78,314,379,189)(5,190,380,315,79,413,326,224,48,132,281,126,234)(6,235,127,282,133,49,193,327,414,80,316,381,191)(7,192,382,317,81,415,328,194,50,134,283,128,236)(8,237,97,284,135,51,195,329,416,82,318,383,161)(9,162,384,319,83,385,330,196,52,136,285,98,238)(10,239,99,286,137,53,197,331,386,84,320,353,163)(11,164,354,289,85,387,332,198,54,138,287,100,240)(12,241,101,288,139,55,199,333,388,86,290,355,165)(13,166,356,291,87,389,334,200,56,140,257,102,242)(14,243,103,258,141,57,201,335,390,88,292,357,167)(15,168,358,293,89,391,336,202,58,142,259,104,244)(16,245,105,260,143,59,203,337,392,90,294,359,169)(17,170,360,295,91,393,338,204,60,144,261,106,246)(18,247,107,262,145,61,205,339,394,92,296,361,171)(19,172,362,297,93,395,340,206,62,146,263,108,248)(20,249,109,264,147,63,207,341,396,94,298,363,173)(21,174,364,299,95,397,342,208,64,148,265,110,250)(22,251,111,266,149,33,209,343,398,96,300,365,175)(23,176,366,301,65,399,344,210,34,150,267,112,252)(24,253,113,268,151,35,211,345,400,66,302,367,177)(25,178,368,303,67,401,346,212,36,152,269,114,254)(26,255,115,270,153,37,213,347,402,68,304,369,179)(27,180,370,305,69,403,348,214,38,154,271,116,256)(28,225,117,272,155,39,215,349,404,70,306,371,181)(29,182,372,307,71,405,350,216,40,156,273,118,226)(30,227,119,274,157,41,217,351,406,72,308,373,183)(31,184,374,309,73,407,352,218,42,158,275,120,228)(32,229,121,276,159,43,219,321,408,74,310,375,185), (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([[(1,186,376,311,75,409,322,220,44,160,277,122,230),(2,231,123,278,129,45,221,323,410,76,312,377,187),(3,188,378,313,77,411,324,222,46,130,279,124,232),(4,233,125,280,131,47,223,325,412,78,314,379,189),(5,190,380,315,79,413,326,224,48,132,281,126,234),(6,235,127,282,133,49,193,327,414,80,316,381,191),(7,192,382,317,81,415,328,194,50,134,283,128,236),(8,237,97,284,135,51,195,329,416,82,318,383,161),(9,162,384,319,83,385,330,196,52,136,285,98,238),(10,239,99,286,137,53,197,331,386,84,320,353,163),(11,164,354,289,85,387,332,198,54,138,287,100,240),(12,241,101,288,139,55,199,333,388,86,290,355,165),(13,166,356,291,87,389,334,200,56,140,257,102,242),(14,243,103,258,141,57,201,335,390,88,292,357,167),(15,168,358,293,89,391,336,202,58,142,259,104,244),(16,245,105,260,143,59,203,337,392,90,294,359,169),(17,170,360,295,91,393,338,204,60,144,261,106,246),(18,247,107,262,145,61,205,339,394,92,296,361,171),(19,172,362,297,93,395,340,206,62,146,263,108,248),(20,249,109,264,147,63,207,341,396,94,298,363,173),(21,174,364,299,95,397,342,208,64,148,265,110,250),(22,251,111,266,149,33,209,343,398,96,300,365,175),(23,176,366,301,65,399,344,210,34,150,267,112,252),(24,253,113,268,151,35,211,345,400,66,302,367,177),(25,178,368,303,67,401,346,212,36,152,269,114,254),(26,255,115,270,153,37,213,347,402,68,304,369,179),(27,180,370,305,69,403,348,214,38,154,271,116,256),(28,225,117,272,155,39,215,349,404,70,306,371,181),(29,182,372,307,71,405,350,216,40,156,273,118,226),(30,227,119,274,157,41,217,351,406,72,308,373,183),(31,184,374,309,73,407,352,218,42,158,275,120,228),(32,229,121,276,159,43,219,321,408,74,310,375,185)], [(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)]])
128 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 16A | ··· | 16H | 26A | ··· | 26F | 32A | ··· | 32P | 52A | ··· | 52L | 104A | ··· | 104X | 208A | ··· | 208AV |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 16 | ··· | 16 | 26 | ··· | 26 | 32 | ··· | 32 | 52 | ··· | 52 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | D13 | Dic13 | C13⋊2C8 | C13⋊2C16 | C13⋊2C32 |
| kernel | C13⋊2C32 | C208 | C104 | C52 | C26 | C13 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 6 | 6 | 12 | 24 | 48 |
Matrix representation of C13⋊2C32 ►in GL2(𝔽1249) generated by
| 1248 | 1 |
| 1024 | 224 |
| 725 | 277 |
| 630 | 524 |
G:=sub<GL(2,GF(1249))| [1248,1024,1,224],[725,630,277,524] >;
C13⋊2C32 in GAP, Magma, Sage, TeX
C_{13}\rtimes_2C_{32} % in TeX
G:=Group("C13:2C32"); // GroupNames label
G:=SmallGroup(416,1);
// by ID
G=gap.SmallGroup(416,1);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,12,31,50,69,13829]);
// Polycyclic
G:=Group<a,b|a^13=b^32=1,b*a*b^-1=a^-1>;
// generators/relations
Export