metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C43⋊C9, C129.C3, C3.(C43⋊C3), SmallGroup(387,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C43⋊C9 |
Generators and relations for C43⋊C9
G = < a,b | a43=b9=1, bab-1=a6 >
Smallest permutation representation of C43⋊C9
►Regular action on 387 points
Generators in S387
(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)
(1 347 217 128 310 195 85 293 136)(2 383 223 129 303 201 86 286 142)(3 376 229 87 339 207 44 279 148)(4 369 235 88 332 213 45 272 154)(5 362 241 89 325 176 46 265 160)(6 355 247 90 318 182 47 301 166)(7 348 253 91 311 188 48 294 172)(8 384 216 92 304 194 49 287 135)(9 377 222 93 340 200 50 280 141)(10 370 228 94 333 206 51 273 147)(11 363 234 95 326 212 52 266 153)(12 356 240 96 319 175 53 259 159)(13 349 246 97 312 181 54 295 165)(14 385 252 98 305 187 55 288 171)(15 378 258 99 341 193 56 281 134)(16 371 221 100 334 199 57 274 140)(17 364 227 101 327 205 58 267 146)(18 357 233 102 320 211 59 260 152)(19 350 239 103 313 174 60 296 158)(20 386 245 104 306 180 61 289 164)(21 379 251 105 342 186 62 282 170)(22 372 257 106 335 192 63 275 133)(23 365 220 107 328 198 64 268 139)(24 358 226 108 321 204 65 261 145)(25 351 232 109 314 210 66 297 151)(26 387 238 110 307 173 67 290 157)(27 380 244 111 343 179 68 283 163)(28 373 250 112 336 185 69 276 169)(29 366 256 113 329 191 70 269 132)(30 359 219 114 322 197 71 262 138)(31 352 225 115 315 203 72 298 144)(32 345 231 116 308 209 73 291 150)(33 381 237 117 344 215 74 284 156)(34 374 243 118 337 178 75 277 162)(35 367 249 119 330 184 76 270 168)(36 360 255 120 323 190 77 263 131)(37 353 218 121 316 196 78 299 137)(38 346 224 122 309 202 79 292 143)(39 382 230 123 302 208 80 285 149)(40 375 236 124 338 214 81 278 155)(41 368 242 125 331 177 82 271 161)(42 361 248 126 324 183 83 264 167)(43 354 254 127 317 189 84 300 130)
G:=sub<Sym(387)| (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), (1,347,217,128,310,195,85,293,136)(2,383,223,129,303,201,86,286,142)(3,376,229,87,339,207,44,279,148)(4,369,235,88,332,213,45,272,154)(5,362,241,89,325,176,46,265,160)(6,355,247,90,318,182,47,301,166)(7,348,253,91,311,188,48,294,172)(8,384,216,92,304,194,49,287,135)(9,377,222,93,340,200,50,280,141)(10,370,228,94,333,206,51,273,147)(11,363,234,95,326,212,52,266,153)(12,356,240,96,319,175,53,259,159)(13,349,246,97,312,181,54,295,165)(14,385,252,98,305,187,55,288,171)(15,378,258,99,341,193,56,281,134)(16,371,221,100,334,199,57,274,140)(17,364,227,101,327,205,58,267,146)(18,357,233,102,320,211,59,260,152)(19,350,239,103,313,174,60,296,158)(20,386,245,104,306,180,61,289,164)(21,379,251,105,342,186,62,282,170)(22,372,257,106,335,192,63,275,133)(23,365,220,107,328,198,64,268,139)(24,358,226,108,321,204,65,261,145)(25,351,232,109,314,210,66,297,151)(26,387,238,110,307,173,67,290,157)(27,380,244,111,343,179,68,283,163)(28,373,250,112,336,185,69,276,169)(29,366,256,113,329,191,70,269,132)(30,359,219,114,322,197,71,262,138)(31,352,225,115,315,203,72,298,144)(32,345,231,116,308,209,73,291,150)(33,381,237,117,344,215,74,284,156)(34,374,243,118,337,178,75,277,162)(35,367,249,119,330,184,76,270,168)(36,360,255,120,323,190,77,263,131)(37,353,218,121,316,196,78,299,137)(38,346,224,122,309,202,79,292,143)(39,382,230,123,302,208,80,285,149)(40,375,236,124,338,214,81,278,155)(41,368,242,125,331,177,82,271,161)(42,361,248,126,324,183,83,264,167)(43,354,254,127,317,189,84,300,130)>;
G:=Group( (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), (1,347,217,128,310,195,85,293,136)(2,383,223,129,303,201,86,286,142)(3,376,229,87,339,207,44,279,148)(4,369,235,88,332,213,45,272,154)(5,362,241,89,325,176,46,265,160)(6,355,247,90,318,182,47,301,166)(7,348,253,91,311,188,48,294,172)(8,384,216,92,304,194,49,287,135)(9,377,222,93,340,200,50,280,141)(10,370,228,94,333,206,51,273,147)(11,363,234,95,326,212,52,266,153)(12,356,240,96,319,175,53,259,159)(13,349,246,97,312,181,54,295,165)(14,385,252,98,305,187,55,288,171)(15,378,258,99,341,193,56,281,134)(16,371,221,100,334,199,57,274,140)(17,364,227,101,327,205,58,267,146)(18,357,233,102,320,211,59,260,152)(19,350,239,103,313,174,60,296,158)(20,386,245,104,306,180,61,289,164)(21,379,251,105,342,186,62,282,170)(22,372,257,106,335,192,63,275,133)(23,365,220,107,328,198,64,268,139)(24,358,226,108,321,204,65,261,145)(25,351,232,109,314,210,66,297,151)(26,387,238,110,307,173,67,290,157)(27,380,244,111,343,179,68,283,163)(28,373,250,112,336,185,69,276,169)(29,366,256,113,329,191,70,269,132)(30,359,219,114,322,197,71,262,138)(31,352,225,115,315,203,72,298,144)(32,345,231,116,308,209,73,291,150)(33,381,237,117,344,215,74,284,156)(34,374,243,118,337,178,75,277,162)(35,367,249,119,330,184,76,270,168)(36,360,255,120,323,190,77,263,131)(37,353,218,121,316,196,78,299,137)(38,346,224,122,309,202,79,292,143)(39,382,230,123,302,208,80,285,149)(40,375,236,124,338,214,81,278,155)(41,368,242,125,331,177,82,271,161)(42,361,248,126,324,183,83,264,167)(43,354,254,127,317,189,84,300,130) );
G=PermutationGroup([[(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)], [(1,347,217,128,310,195,85,293,136),(2,383,223,129,303,201,86,286,142),(3,376,229,87,339,207,44,279,148),(4,369,235,88,332,213,45,272,154),(5,362,241,89,325,176,46,265,160),(6,355,247,90,318,182,47,301,166),(7,348,253,91,311,188,48,294,172),(8,384,216,92,304,194,49,287,135),(9,377,222,93,340,200,50,280,141),(10,370,228,94,333,206,51,273,147),(11,363,234,95,326,212,52,266,153),(12,356,240,96,319,175,53,259,159),(13,349,246,97,312,181,54,295,165),(14,385,252,98,305,187,55,288,171),(15,378,258,99,341,193,56,281,134),(16,371,221,100,334,199,57,274,140),(17,364,227,101,327,205,58,267,146),(18,357,233,102,320,211,59,260,152),(19,350,239,103,313,174,60,296,158),(20,386,245,104,306,180,61,289,164),(21,379,251,105,342,186,62,282,170),(22,372,257,106,335,192,63,275,133),(23,365,220,107,328,198,64,268,139),(24,358,226,108,321,204,65,261,145),(25,351,232,109,314,210,66,297,151),(26,387,238,110,307,173,67,290,157),(27,380,244,111,343,179,68,283,163),(28,373,250,112,336,185,69,276,169),(29,366,256,113,329,191,70,269,132),(30,359,219,114,322,197,71,262,138),(31,352,225,115,315,203,72,298,144),(32,345,231,116,308,209,73,291,150),(33,381,237,117,344,215,74,284,156),(34,374,243,118,337,178,75,277,162),(35,367,249,119,330,184,76,270,168),(36,360,255,120,323,190,77,263,131),(37,353,218,121,316,196,78,299,137),(38,346,224,122,309,202,79,292,143),(39,382,230,123,302,208,80,285,149),(40,375,236,124,338,214,81,278,155),(41,368,242,125,331,177,82,271,161),(42,361,248,126,324,183,83,264,167),(43,354,254,127,317,189,84,300,130)]])
51 conjugacy classes
| class | 1 | 3A | 3B | 9A | ··· | 9F | 43A | ··· | 43N | 129A | ··· | 129AB |
| order | 1 | 3 | 3 | 9 | ··· | 9 | 43 | ··· | 43 | 129 | ··· | 129 |
| size | 1 | 1 | 1 | 43 | ··· | 43 | 3 | ··· | 3 | 3 | ··· | 3 |
51 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C9 | C43⋊C3 | C43⋊C9 |
| kernel | C43⋊C9 | C129 | C43 | C3 | C1 |
| # reps | 1 | 2 | 6 | 14 | 28 |
Matrix representation of C43⋊C9 ►in GL3(𝔽1549) generated by
| 1548 | 1 | 0 |
| 1548 | 0 | 1 |
| 971 | 944 | 1184 |
| 957 | 1056 | 1243 |
| 528 | 1132 | 154 |
| 1133 | 1056 | 1009 |
G:=sub<GL(3,GF(1549))| [1548,1548,971,1,0,944,0,1,1184],[957,528,1133,1056,1132,1056,1243,154,1009] >;
C43⋊C9 in GAP, Magma, Sage, TeX
C_{43}\rtimes C_9 % in TeX
G:=Group("C43:C9"); // GroupNames label
G:=SmallGroup(387,1);
// by ID
G=gap.SmallGroup(387,1);
# by ID
G:=PCGroup([3,-3,-3,-43,9,2918]);
// Polycyclic
G:=Group<a,b|a^43=b^9=1,b*a*b^-1=a^6>;
// generators/relations
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