metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13⋊C32, C26.C16, C52.2C8, C104.3C4, C2.(C13⋊C16), C4.2(C13⋊C8), C8.4(C13⋊C4), C13⋊2C16.2C2, SmallGroup(416,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊C32 |
Generators and relations for C13⋊C32
G = < a,b | a13=b32=1, bab-1=a5 >
Smallest permutation representation of C13⋊C32
►Regular action on 416 points
Generators in S416
(1 94 169 116 224 393 62 158 325 231 367 297 273)(2 394 368 170 159 274 193 232 95 63 298 117 326)(3 275 299 369 233 327 160 64 395 194 118 171 96)(4 328 119 300 33 65 234 195 276 129 172 370 396)(5 66 173 120 196 397 34 130 329 235 371 301 277)(6 398 372 174 131 278 197 236 67 35 302 121 330)(7 279 303 373 237 331 132 36 399 198 122 175 68)(8 332 123 304 37 69 238 199 280 133 176 374 400)(9 70 177 124 200 401 38 134 333 239 375 305 281)(10 402 376 178 135 282 201 240 71 39 306 125 334)(11 283 307 377 241 335 136 40 403 202 126 179 72)(12 336 127 308 41 73 242 203 284 137 180 378 404)(13 74 181 128 204 405 42 138 337 243 379 309 285)(14 406 380 182 139 286 205 244 75 43 310 97 338)(15 287 311 381 245 339 140 44 407 206 98 183 76)(16 340 99 312 45 77 246 207 288 141 184 382 408)(17 78 185 100 208 409 46 142 341 247 383 313 257)(18 410 384 186 143 258 209 248 79 47 314 101 342)(19 259 315 353 249 343 144 48 411 210 102 187 80)(20 344 103 316 49 81 250 211 260 145 188 354 412)(21 82 189 104 212 413 50 146 345 251 355 317 261)(22 414 356 190 147 262 213 252 83 51 318 105 346)(23 263 319 357 253 347 148 52 415 214 106 191 84)(24 348 107 320 53 85 254 215 264 149 192 358 416)(25 86 161 108 216 385 54 150 349 255 359 289 265)(26 386 360 162 151 266 217 256 87 55 290 109 350)(27 267 291 361 225 351 152 56 387 218 110 163 88)(28 352 111 292 57 89 226 219 268 153 164 362 388)(29 90 165 112 220 389 58 154 321 227 363 293 269)(30 390 364 166 155 270 221 228 91 59 294 113 322)(31 271 295 365 229 323 156 60 391 222 114 167 92)(32 324 115 296 61 93 230 223 272 157 168 366 392)
(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)| (1,94,169,116,224,393,62,158,325,231,367,297,273)(2,394,368,170,159,274,193,232,95,63,298,117,326)(3,275,299,369,233,327,160,64,395,194,118,171,96)(4,328,119,300,33,65,234,195,276,129,172,370,396)(5,66,173,120,196,397,34,130,329,235,371,301,277)(6,398,372,174,131,278,197,236,67,35,302,121,330)(7,279,303,373,237,331,132,36,399,198,122,175,68)(8,332,123,304,37,69,238,199,280,133,176,374,400)(9,70,177,124,200,401,38,134,333,239,375,305,281)(10,402,376,178,135,282,201,240,71,39,306,125,334)(11,283,307,377,241,335,136,40,403,202,126,179,72)(12,336,127,308,41,73,242,203,284,137,180,378,404)(13,74,181,128,204,405,42,138,337,243,379,309,285)(14,406,380,182,139,286,205,244,75,43,310,97,338)(15,287,311,381,245,339,140,44,407,206,98,183,76)(16,340,99,312,45,77,246,207,288,141,184,382,408)(17,78,185,100,208,409,46,142,341,247,383,313,257)(18,410,384,186,143,258,209,248,79,47,314,101,342)(19,259,315,353,249,343,144,48,411,210,102,187,80)(20,344,103,316,49,81,250,211,260,145,188,354,412)(21,82,189,104,212,413,50,146,345,251,355,317,261)(22,414,356,190,147,262,213,252,83,51,318,105,346)(23,263,319,357,253,347,148,52,415,214,106,191,84)(24,348,107,320,53,85,254,215,264,149,192,358,416)(25,86,161,108,216,385,54,150,349,255,359,289,265)(26,386,360,162,151,266,217,256,87,55,290,109,350)(27,267,291,361,225,351,152,56,387,218,110,163,88)(28,352,111,292,57,89,226,219,268,153,164,362,388)(29,90,165,112,220,389,58,154,321,227,363,293,269)(30,390,364,166,155,270,221,228,91,59,294,113,322)(31,271,295,365,229,323,156,60,391,222,114,167,92)(32,324,115,296,61,93,230,223,272,157,168,366,392), (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( (1,94,169,116,224,393,62,158,325,231,367,297,273)(2,394,368,170,159,274,193,232,95,63,298,117,326)(3,275,299,369,233,327,160,64,395,194,118,171,96)(4,328,119,300,33,65,234,195,276,129,172,370,396)(5,66,173,120,196,397,34,130,329,235,371,301,277)(6,398,372,174,131,278,197,236,67,35,302,121,330)(7,279,303,373,237,331,132,36,399,198,122,175,68)(8,332,123,304,37,69,238,199,280,133,176,374,400)(9,70,177,124,200,401,38,134,333,239,375,305,281)(10,402,376,178,135,282,201,240,71,39,306,125,334)(11,283,307,377,241,335,136,40,403,202,126,179,72)(12,336,127,308,41,73,242,203,284,137,180,378,404)(13,74,181,128,204,405,42,138,337,243,379,309,285)(14,406,380,182,139,286,205,244,75,43,310,97,338)(15,287,311,381,245,339,140,44,407,206,98,183,76)(16,340,99,312,45,77,246,207,288,141,184,382,408)(17,78,185,100,208,409,46,142,341,247,383,313,257)(18,410,384,186,143,258,209,248,79,47,314,101,342)(19,259,315,353,249,343,144,48,411,210,102,187,80)(20,344,103,316,49,81,250,211,260,145,188,354,412)(21,82,189,104,212,413,50,146,345,251,355,317,261)(22,414,356,190,147,262,213,252,83,51,318,105,346)(23,263,319,357,253,347,148,52,415,214,106,191,84)(24,348,107,320,53,85,254,215,264,149,192,358,416)(25,86,161,108,216,385,54,150,349,255,359,289,265)(26,386,360,162,151,266,217,256,87,55,290,109,350)(27,267,291,361,225,351,152,56,387,218,110,163,88)(28,352,111,292,57,89,226,219,268,153,164,362,388)(29,90,165,112,220,389,58,154,321,227,363,293,269)(30,390,364,166,155,270,221,228,91,59,294,113,322)(31,271,295,365,229,323,156,60,391,222,114,167,92)(32,324,115,296,61,93,230,223,272,157,168,366,392), (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([[(1,94,169,116,224,393,62,158,325,231,367,297,273),(2,394,368,170,159,274,193,232,95,63,298,117,326),(3,275,299,369,233,327,160,64,395,194,118,171,96),(4,328,119,300,33,65,234,195,276,129,172,370,396),(5,66,173,120,196,397,34,130,329,235,371,301,277),(6,398,372,174,131,278,197,236,67,35,302,121,330),(7,279,303,373,237,331,132,36,399,198,122,175,68),(8,332,123,304,37,69,238,199,280,133,176,374,400),(9,70,177,124,200,401,38,134,333,239,375,305,281),(10,402,376,178,135,282,201,240,71,39,306,125,334),(11,283,307,377,241,335,136,40,403,202,126,179,72),(12,336,127,308,41,73,242,203,284,137,180,378,404),(13,74,181,128,204,405,42,138,337,243,379,309,285),(14,406,380,182,139,286,205,244,75,43,310,97,338),(15,287,311,381,245,339,140,44,407,206,98,183,76),(16,340,99,312,45,77,246,207,288,141,184,382,408),(17,78,185,100,208,409,46,142,341,247,383,313,257),(18,410,384,186,143,258,209,248,79,47,314,101,342),(19,259,315,353,249,343,144,48,411,210,102,187,80),(20,344,103,316,49,81,250,211,260,145,188,354,412),(21,82,189,104,212,413,50,146,345,251,355,317,261),(22,414,356,190,147,262,213,252,83,51,318,105,346),(23,263,319,357,253,347,148,52,415,214,106,191,84),(24,348,107,320,53,85,254,215,264,149,192,358,416),(25,86,161,108,216,385,54,150,349,255,359,289,265),(26,386,360,162,151,266,217,256,87,55,290,109,350),(27,267,291,361,225,351,152,56,387,218,110,163,88),(28,352,111,292,57,89,226,219,268,153,164,362,388),(29,90,165,112,220,389,58,154,321,227,363,293,269),(30,390,364,166,155,270,221,228,91,59,294,113,322),(31,271,295,365,229,323,156,60,391,222,114,167,92),(32,324,115,296,61,93,230,223,272,157,168,366,392)], [(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 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 13A | 13B | 13C | 16A | ··· | 16H | 26A | 26B | 26C | 32A | ··· | 32P | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | 13 | 13 | 16 | ··· | 16 | 26 | 26 | 26 | 32 | ··· | 32 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 13 | ··· | 13 | 4 | 4 | 4 | 13 | ··· | 13 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | C13⋊C4 | C13⋊C8 | C13⋊C16 | C13⋊C32 |
| kernel | C13⋊C32 | C13⋊2C16 | C104 | C52 | C26 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 3 | 3 | 6 | 12 |
Matrix representation of C13⋊C32 ►in GL5(𝔽1249)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1248 | 1 | 0 | 0 |
| 0 | 1248 | 0 | 1 | 0 |
| 0 | 1248 | 0 | 0 | 1 |
| 0 | 682 | 610 | 639 | 566 |
| 908 | 0 | 0 | 0 | 0 |
| 0 | 319 | 681 | 209 | 383 |
| 0 | 1242 | 834 | 1075 | 16 |
| 0 | 1101 | 445 | 799 | 709 |
| 0 | 1012 | 1194 | 641 | 546 |
G:=sub<GL(5,GF(1249))| [1,0,0,0,0,0,1248,1248,1248,682,0,1,0,0,610,0,0,1,0,639,0,0,0,1,566],[908,0,0,0,0,0,319,1242,1101,1012,0,681,834,445,1194,0,209,1075,799,641,0,383,16,709,546] >;
C13⋊C32 in GAP, Magma, Sage, TeX
C_{13}\rtimes C_{32} % in TeX
G:=Group("C13:C32"); // GroupNames label
G:=SmallGroup(416,3);
// by ID
G=gap.SmallGroup(416,3);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,12,31,50,69,9221,6923]);
// Polycyclic
G:=Group<a,b|a^13=b^32=1,b*a*b^-1=a^5>;
// generators/relations
Export