metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C13⋊C27, C39.C9, C117.C3, C3.(C13⋊C9), C9.(C13⋊C3), SmallGroup(351,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊C27 |
Generators and relations for C13⋊C27
G = < a,b | a13=b27=1, bab-1=a9 >
Smallest permutation representation of C13⋊C27
►Regular action on 351 points
Generators in S351
(1 163 221 306 117 326 153 75 30 83 204 275 257)(2 84 327 164 205 154 222 276 76 307 258 31 118)(3 308 155 85 259 223 328 32 277 165 119 77 206)(4 166 224 309 120 329 156 78 33 86 207 278 260)(5 87 330 167 208 157 225 279 79 310 261 34 121)(6 311 158 88 262 226 331 35 280 168 122 80 209)(7 169 227 312 123 332 159 81 36 89 210 281 263)(8 90 333 170 211 160 228 282 55 313 264 37 124)(9 314 161 91 265 229 334 38 283 171 125 56 212)(10 172 230 315 126 335 162 57 39 92 213 284 266)(11 93 336 173 214 136 231 285 58 316 267 40 127)(12 317 137 94 268 232 337 41 286 174 128 59 215)(13 175 233 318 129 338 138 60 42 95 216 287 269)(14 96 339 176 190 139 234 288 61 319 270 43 130)(15 320 140 97 244 235 340 44 289 177 131 62 191)(16 178 236 321 132 341 141 63 45 98 192 290 245)(17 99 342 179 193 142 237 291 64 322 246 46 133)(18 323 143 100 247 238 343 47 292 180 134 65 194)(19 181 239 324 135 344 144 66 48 101 195 293 248)(20 102 345 182 196 145 240 294 67 298 249 49 109)(21 299 146 103 250 241 346 50 295 183 110 68 197)(22 184 242 300 111 347 147 69 51 104 198 296 251)(23 105 348 185 199 148 243 297 70 301 252 52 112)(24 302 149 106 253 217 349 53 271 186 113 71 200)(25 187 218 303 114 350 150 72 54 107 201 272 254)(26 108 351 188 202 151 219 273 73 304 255 28 115)(27 305 152 82 256 220 325 29 274 189 116 74 203)
(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)
G:=sub<Sym(351)| (1,163,221,306,117,326,153,75,30,83,204,275,257)(2,84,327,164,205,154,222,276,76,307,258,31,118)(3,308,155,85,259,223,328,32,277,165,119,77,206)(4,166,224,309,120,329,156,78,33,86,207,278,260)(5,87,330,167,208,157,225,279,79,310,261,34,121)(6,311,158,88,262,226,331,35,280,168,122,80,209)(7,169,227,312,123,332,159,81,36,89,210,281,263)(8,90,333,170,211,160,228,282,55,313,264,37,124)(9,314,161,91,265,229,334,38,283,171,125,56,212)(10,172,230,315,126,335,162,57,39,92,213,284,266)(11,93,336,173,214,136,231,285,58,316,267,40,127)(12,317,137,94,268,232,337,41,286,174,128,59,215)(13,175,233,318,129,338,138,60,42,95,216,287,269)(14,96,339,176,190,139,234,288,61,319,270,43,130)(15,320,140,97,244,235,340,44,289,177,131,62,191)(16,178,236,321,132,341,141,63,45,98,192,290,245)(17,99,342,179,193,142,237,291,64,322,246,46,133)(18,323,143,100,247,238,343,47,292,180,134,65,194)(19,181,239,324,135,344,144,66,48,101,195,293,248)(20,102,345,182,196,145,240,294,67,298,249,49,109)(21,299,146,103,250,241,346,50,295,183,110,68,197)(22,184,242,300,111,347,147,69,51,104,198,296,251)(23,105,348,185,199,148,243,297,70,301,252,52,112)(24,302,149,106,253,217,349,53,271,186,113,71,200)(25,187,218,303,114,350,150,72,54,107,201,272,254)(26,108,351,188,202,151,219,273,73,304,255,28,115)(27,305,152,82,256,220,325,29,274,189,116,74,203), (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)>;
G:=Group( (1,163,221,306,117,326,153,75,30,83,204,275,257)(2,84,327,164,205,154,222,276,76,307,258,31,118)(3,308,155,85,259,223,328,32,277,165,119,77,206)(4,166,224,309,120,329,156,78,33,86,207,278,260)(5,87,330,167,208,157,225,279,79,310,261,34,121)(6,311,158,88,262,226,331,35,280,168,122,80,209)(7,169,227,312,123,332,159,81,36,89,210,281,263)(8,90,333,170,211,160,228,282,55,313,264,37,124)(9,314,161,91,265,229,334,38,283,171,125,56,212)(10,172,230,315,126,335,162,57,39,92,213,284,266)(11,93,336,173,214,136,231,285,58,316,267,40,127)(12,317,137,94,268,232,337,41,286,174,128,59,215)(13,175,233,318,129,338,138,60,42,95,216,287,269)(14,96,339,176,190,139,234,288,61,319,270,43,130)(15,320,140,97,244,235,340,44,289,177,131,62,191)(16,178,236,321,132,341,141,63,45,98,192,290,245)(17,99,342,179,193,142,237,291,64,322,246,46,133)(18,323,143,100,247,238,343,47,292,180,134,65,194)(19,181,239,324,135,344,144,66,48,101,195,293,248)(20,102,345,182,196,145,240,294,67,298,249,49,109)(21,299,146,103,250,241,346,50,295,183,110,68,197)(22,184,242,300,111,347,147,69,51,104,198,296,251)(23,105,348,185,199,148,243,297,70,301,252,52,112)(24,302,149,106,253,217,349,53,271,186,113,71,200)(25,187,218,303,114,350,150,72,54,107,201,272,254)(26,108,351,188,202,151,219,273,73,304,255,28,115)(27,305,152,82,256,220,325,29,274,189,116,74,203), (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) );
G=PermutationGroup([[(1,163,221,306,117,326,153,75,30,83,204,275,257),(2,84,327,164,205,154,222,276,76,307,258,31,118),(3,308,155,85,259,223,328,32,277,165,119,77,206),(4,166,224,309,120,329,156,78,33,86,207,278,260),(5,87,330,167,208,157,225,279,79,310,261,34,121),(6,311,158,88,262,226,331,35,280,168,122,80,209),(7,169,227,312,123,332,159,81,36,89,210,281,263),(8,90,333,170,211,160,228,282,55,313,264,37,124),(9,314,161,91,265,229,334,38,283,171,125,56,212),(10,172,230,315,126,335,162,57,39,92,213,284,266),(11,93,336,173,214,136,231,285,58,316,267,40,127),(12,317,137,94,268,232,337,41,286,174,128,59,215),(13,175,233,318,129,338,138,60,42,95,216,287,269),(14,96,339,176,190,139,234,288,61,319,270,43,130),(15,320,140,97,244,235,340,44,289,177,131,62,191),(16,178,236,321,132,341,141,63,45,98,192,290,245),(17,99,342,179,193,142,237,291,64,322,246,46,133),(18,323,143,100,247,238,343,47,292,180,134,65,194),(19,181,239,324,135,344,144,66,48,101,195,293,248),(20,102,345,182,196,145,240,294,67,298,249,49,109),(21,299,146,103,250,241,346,50,295,183,110,68,197),(22,184,242,300,111,347,147,69,51,104,198,296,251),(23,105,348,185,199,148,243,297,70,301,252,52,112),(24,302,149,106,253,217,349,53,271,186,113,71,200),(25,187,218,303,114,350,150,72,54,107,201,272,254),(26,108,351,188,202,151,219,273,73,304,255,28,115),(27,305,152,82,256,220,325,29,274,189,116,74,203)], [(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)]])
63 conjugacy classes
| class | 1 | 3A | 3B | 9A | ··· | 9F | 13A | 13B | 13C | 13D | 27A | ··· | 27R | 39A | ··· | 39H | 117A | ··· | 117X |
| order | 1 | 3 | 3 | 9 | ··· | 9 | 13 | 13 | 13 | 13 | 27 | ··· | 27 | 39 | ··· | 39 | 117 | ··· | 117 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 3 | ··· | 3 | 3 | ··· | 3 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C9 | C27 | C13⋊C3 | C13⋊C9 | C13⋊C27 |
| kernel | C13⋊C27 | C117 | C39 | C13 | C9 | C3 | C1 |
| # reps | 1 | 2 | 6 | 18 | 4 | 8 | 24 |
Matrix representation of C13⋊C27 ►in GL3(𝔽3511) generated by
| 3510 | 1 | 0 |
| 3510 | 0 | 1 |
| 270 | 2361 | 881 |
| 2943 | 1456 | 2072 |
| 1172 | 2864 | 2626 |
| 2820 | 2916 | 1215 |
G:=sub<GL(3,GF(3511))| [3510,3510,270,1,0,2361,0,1,881],[2943,1172,2820,1456,2864,2916,2072,2626,1215] >;
C13⋊C27 in GAP, Magma, Sage, TeX
C_{13}\rtimes C_{27} % in TeX
G:=Group("C13:C27"); // GroupNames label
G:=SmallGroup(351,1);
// by ID
G=gap.SmallGroup(351,1);
# by ID
G:=PCGroup([4,-3,-3,-3,-13,12,29,1299]);
// Polycyclic
G:=Group<a,b|a^13=b^27=1,b*a*b^-1=a^9>;
// generators/relations
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