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G = C43⋊C9order 387 = 32·43

The semidirect product of C43 and C9 acting via C9/C3=C3

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

C1C43 — C43⋊C9
C1C43C129 — C43⋊C9
C43 — C43⋊C9
C1C3

Generators and relations for C43⋊C9
 G = < a,b | a43=b9=1, bab-1=a6 >

43C9

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)
(1 370 218 112 306 183 53 272 168)(2 363 224 113 342 189 54 265 131)(3 356 230 114 335 195 55 301 137)(4 349 236 115 328 201 56 294 143)(5 385 242 116 321 207 57 287 149)(6 378 248 117 314 213 58 280 155)(7 371 254 118 307 176 59 273 161)(8 364 217 119 343 182 60 266 167)(9 357 223 120 336 188 61 259 130)(10 350 229 121 329 194 62 295 136)(11 386 235 122 322 200 63 288 142)(12 379 241 123 315 206 64 281 148)(13 372 247 124 308 212 65 274 154)(14 365 253 125 344 175 66 267 160)(15 358 216 126 337 181 67 260 166)(16 351 222 127 330 187 68 296 172)(17 387 228 128 323 193 69 289 135)(18 380 234 129 316 199 70 282 141)(19 373 240 87 309 205 71 275 147)(20 366 246 88 302 211 72 268 153)(21 359 252 89 338 174 73 261 159)(22 352 258 90 331 180 74 297 165)(23 345 221 91 324 186 75 290 171)(24 381 227 92 317 192 76 283 134)(25 374 233 93 310 198 77 276 140)(26 367 239 94 303 204 78 269 146)(27 360 245 95 339 210 79 262 152)(28 353 251 96 332 173 80 298 158)(29 346 257 97 325 179 81 291 164)(30 382 220 98 318 185 82 284 170)(31 375 226 99 311 191 83 277 133)(32 368 232 100 304 197 84 270 139)(33 361 238 101 340 203 85 263 145)(34 354 244 102 333 209 86 299 151)(35 347 250 103 326 215 44 292 157)(36 383 256 104 319 178 45 285 163)(37 376 219 105 312 184 46 278 169)(38 369 225 106 305 190 47 271 132)(39 362 231 107 341 196 48 264 138)(40 355 237 108 334 202 49 300 144)(41 348 243 109 327 208 50 293 150)(42 384 249 110 320 214 51 286 156)(43 377 255 111 313 177 52 279 162)

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), (1,370,218,112,306,183,53,272,168)(2,363,224,113,342,189,54,265,131)(3,356,230,114,335,195,55,301,137)(4,349,236,115,328,201,56,294,143)(5,385,242,116,321,207,57,287,149)(6,378,248,117,314,213,58,280,155)(7,371,254,118,307,176,59,273,161)(8,364,217,119,343,182,60,266,167)(9,357,223,120,336,188,61,259,130)(10,350,229,121,329,194,62,295,136)(11,386,235,122,322,200,63,288,142)(12,379,241,123,315,206,64,281,148)(13,372,247,124,308,212,65,274,154)(14,365,253,125,344,175,66,267,160)(15,358,216,126,337,181,67,260,166)(16,351,222,127,330,187,68,296,172)(17,387,228,128,323,193,69,289,135)(18,380,234,129,316,199,70,282,141)(19,373,240,87,309,205,71,275,147)(20,366,246,88,302,211,72,268,153)(21,359,252,89,338,174,73,261,159)(22,352,258,90,331,180,74,297,165)(23,345,221,91,324,186,75,290,171)(24,381,227,92,317,192,76,283,134)(25,374,233,93,310,198,77,276,140)(26,367,239,94,303,204,78,269,146)(27,360,245,95,339,210,79,262,152)(28,353,251,96,332,173,80,298,158)(29,346,257,97,325,179,81,291,164)(30,382,220,98,318,185,82,284,170)(31,375,226,99,311,191,83,277,133)(32,368,232,100,304,197,84,270,139)(33,361,238,101,340,203,85,263,145)(34,354,244,102,333,209,86,299,151)(35,347,250,103,326,215,44,292,157)(36,383,256,104,319,178,45,285,163)(37,376,219,105,312,184,46,278,169)(38,369,225,106,305,190,47,271,132)(39,362,231,107,341,196,48,264,138)(40,355,237,108,334,202,49,300,144)(41,348,243,109,327,208,50,293,150)(42,384,249,110,320,214,51,286,156)(43,377,255,111,313,177,52,279,162)>;

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), (1,370,218,112,306,183,53,272,168)(2,363,224,113,342,189,54,265,131)(3,356,230,114,335,195,55,301,137)(4,349,236,115,328,201,56,294,143)(5,385,242,116,321,207,57,287,149)(6,378,248,117,314,213,58,280,155)(7,371,254,118,307,176,59,273,161)(8,364,217,119,343,182,60,266,167)(9,357,223,120,336,188,61,259,130)(10,350,229,121,329,194,62,295,136)(11,386,235,122,322,200,63,288,142)(12,379,241,123,315,206,64,281,148)(13,372,247,124,308,212,65,274,154)(14,365,253,125,344,175,66,267,160)(15,358,216,126,337,181,67,260,166)(16,351,222,127,330,187,68,296,172)(17,387,228,128,323,193,69,289,135)(18,380,234,129,316,199,70,282,141)(19,373,240,87,309,205,71,275,147)(20,366,246,88,302,211,72,268,153)(21,359,252,89,338,174,73,261,159)(22,352,258,90,331,180,74,297,165)(23,345,221,91,324,186,75,290,171)(24,381,227,92,317,192,76,283,134)(25,374,233,93,310,198,77,276,140)(26,367,239,94,303,204,78,269,146)(27,360,245,95,339,210,79,262,152)(28,353,251,96,332,173,80,298,158)(29,346,257,97,325,179,81,291,164)(30,382,220,98,318,185,82,284,170)(31,375,226,99,311,191,83,277,133)(32,368,232,100,304,197,84,270,139)(33,361,238,101,340,203,85,263,145)(34,354,244,102,333,209,86,299,151)(35,347,250,103,326,215,44,292,157)(36,383,256,104,319,178,45,285,163)(37,376,219,105,312,184,46,278,169)(38,369,225,106,305,190,47,271,132)(39,362,231,107,341,196,48,264,138)(40,355,237,108,334,202,49,300,144)(41,348,243,109,327,208,50,293,150)(42,384,249,110,320,214,51,286,156)(43,377,255,111,313,177,52,279,162) );

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)], [(1,370,218,112,306,183,53,272,168),(2,363,224,113,342,189,54,265,131),(3,356,230,114,335,195,55,301,137),(4,349,236,115,328,201,56,294,143),(5,385,242,116,321,207,57,287,149),(6,378,248,117,314,213,58,280,155),(7,371,254,118,307,176,59,273,161),(8,364,217,119,343,182,60,266,167),(9,357,223,120,336,188,61,259,130),(10,350,229,121,329,194,62,295,136),(11,386,235,122,322,200,63,288,142),(12,379,241,123,315,206,64,281,148),(13,372,247,124,308,212,65,274,154),(14,365,253,125,344,175,66,267,160),(15,358,216,126,337,181,67,260,166),(16,351,222,127,330,187,68,296,172),(17,387,228,128,323,193,69,289,135),(18,380,234,129,316,199,70,282,141),(19,373,240,87,309,205,71,275,147),(20,366,246,88,302,211,72,268,153),(21,359,252,89,338,174,73,261,159),(22,352,258,90,331,180,74,297,165),(23,345,221,91,324,186,75,290,171),(24,381,227,92,317,192,76,283,134),(25,374,233,93,310,198,77,276,140),(26,367,239,94,303,204,78,269,146),(27,360,245,95,339,210,79,262,152),(28,353,251,96,332,173,80,298,158),(29,346,257,97,325,179,81,291,164),(30,382,220,98,318,185,82,284,170),(31,375,226,99,311,191,83,277,133),(32,368,232,100,304,197,84,270,139),(33,361,238,101,340,203,85,263,145),(34,354,244,102,333,209,86,299,151),(35,347,250,103,326,215,44,292,157),(36,383,256,104,319,178,45,285,163),(37,376,219,105,312,184,46,278,169),(38,369,225,106,305,190,47,271,132),(39,362,231,107,341,196,48,264,138),(40,355,237,108,334,202,49,300,144),(41,348,243,109,327,208,50,293,150),(42,384,249,110,320,214,51,286,156),(43,377,255,111,313,177,52,279,162)])

51 conjugacy classes

class 1 3A3B9A···9F43A···43N129A···129AB
order1339···943···43129···129
size11143···433···33···3

51 irreducible representations

dim11133
type+
imageC1C3C9C43⋊C3C43⋊C9
kernelC43⋊C9C129C43C3C1
# reps1261428

Matrix representation of C43⋊C9 in GL3(𝔽1549) generated by

154810
154801
9719441184
,
95710561243
5281132154
113310561009
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

Export

Subgroup lattice of C43⋊C9 in TeX

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