metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C27⋊C16, C54.C8, C72.5S3, C24.6D9, C8.2D27, C108.2C4, C216.2C2, C36.4Dic3, C12.4Dic9, C4.2Dic27, C3.(C9⋊C16), C2.(C27⋊C8), C9.(C3⋊C16), C6.1(C9⋊C8), C18.1(C3⋊C8), SmallGroup(432,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C27⋊C16 |
Generators and relations for C27⋊C16
G = < a,b | a27=b16=1, bab-1=a-1 >
Smallest permutation representation of C27⋊C16
►Regular action on 432 points
Generators in S432
(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)(406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)
(1 416 208 320 91 374 145 258 46 405 164 290 72 328 113 239)(2 415 209 319 92 373 146 257 47 404 165 289 73 327 114 238)(3 414 210 318 93 372 147 256 48 403 166 288 74 326 115 237)(4 413 211 317 94 371 148 255 49 402 167 287 75 325 116 236)(5 412 212 316 95 370 149 254 50 401 168 286 76 351 117 235)(6 411 213 315 96 369 150 253 51 400 169 285 77 350 118 234)(7 410 214 314 97 368 151 252 52 399 170 284 78 349 119 233)(8 409 215 313 98 367 152 251 53 398 171 283 79 348 120 232)(9 408 216 312 99 366 153 250 54 397 172 282 80 347 121 231)(10 407 190 311 100 365 154 249 28 396 173 281 81 346 122 230)(11 406 191 310 101 364 155 248 29 395 174 280 55 345 123 229)(12 432 192 309 102 363 156 247 30 394 175 279 56 344 124 228)(13 431 193 308 103 362 157 246 31 393 176 278 57 343 125 227)(14 430 194 307 104 361 158 245 32 392 177 277 58 342 126 226)(15 429 195 306 105 360 159 244 33 391 178 276 59 341 127 225)(16 428 196 305 106 359 160 270 34 390 179 275 60 340 128 224)(17 427 197 304 107 358 161 269 35 389 180 274 61 339 129 223)(18 426 198 303 108 357 162 268 36 388 181 273 62 338 130 222)(19 425 199 302 82 356 136 267 37 387 182 272 63 337 131 221)(20 424 200 301 83 355 137 266 38 386 183 271 64 336 132 220)(21 423 201 300 84 354 138 265 39 385 184 297 65 335 133 219)(22 422 202 299 85 353 139 264 40 384 185 296 66 334 134 218)(23 421 203 298 86 352 140 263 41 383 186 295 67 333 135 217)(24 420 204 324 87 378 141 262 42 382 187 294 68 332 109 243)(25 419 205 323 88 377 142 261 43 381 188 293 69 331 110 242)(26 418 206 322 89 376 143 260 44 380 189 292 70 330 111 241)(27 417 207 321 90 375 144 259 45 379 163 291 71 329 112 240)
G:=sub<Sym(432)| (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)(406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432), (1,416,208,320,91,374,145,258,46,405,164,290,72,328,113,239)(2,415,209,319,92,373,146,257,47,404,165,289,73,327,114,238)(3,414,210,318,93,372,147,256,48,403,166,288,74,326,115,237)(4,413,211,317,94,371,148,255,49,402,167,287,75,325,116,236)(5,412,212,316,95,370,149,254,50,401,168,286,76,351,117,235)(6,411,213,315,96,369,150,253,51,400,169,285,77,350,118,234)(7,410,214,314,97,368,151,252,52,399,170,284,78,349,119,233)(8,409,215,313,98,367,152,251,53,398,171,283,79,348,120,232)(9,408,216,312,99,366,153,250,54,397,172,282,80,347,121,231)(10,407,190,311,100,365,154,249,28,396,173,281,81,346,122,230)(11,406,191,310,101,364,155,248,29,395,174,280,55,345,123,229)(12,432,192,309,102,363,156,247,30,394,175,279,56,344,124,228)(13,431,193,308,103,362,157,246,31,393,176,278,57,343,125,227)(14,430,194,307,104,361,158,245,32,392,177,277,58,342,126,226)(15,429,195,306,105,360,159,244,33,391,178,276,59,341,127,225)(16,428,196,305,106,359,160,270,34,390,179,275,60,340,128,224)(17,427,197,304,107,358,161,269,35,389,180,274,61,339,129,223)(18,426,198,303,108,357,162,268,36,388,181,273,62,338,130,222)(19,425,199,302,82,356,136,267,37,387,182,272,63,337,131,221)(20,424,200,301,83,355,137,266,38,386,183,271,64,336,132,220)(21,423,201,300,84,354,138,265,39,385,184,297,65,335,133,219)(22,422,202,299,85,353,139,264,40,384,185,296,66,334,134,218)(23,421,203,298,86,352,140,263,41,383,186,295,67,333,135,217)(24,420,204,324,87,378,141,262,42,382,187,294,68,332,109,243)(25,419,205,323,88,377,142,261,43,381,188,293,69,331,110,242)(26,418,206,322,89,376,143,260,44,380,189,292,70,330,111,241)(27,417,207,321,90,375,144,259,45,379,163,291,71,329,112,240)>;
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)(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405)(406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432), (1,416,208,320,91,374,145,258,46,405,164,290,72,328,113,239)(2,415,209,319,92,373,146,257,47,404,165,289,73,327,114,238)(3,414,210,318,93,372,147,256,48,403,166,288,74,326,115,237)(4,413,211,317,94,371,148,255,49,402,167,287,75,325,116,236)(5,412,212,316,95,370,149,254,50,401,168,286,76,351,117,235)(6,411,213,315,96,369,150,253,51,400,169,285,77,350,118,234)(7,410,214,314,97,368,151,252,52,399,170,284,78,349,119,233)(8,409,215,313,98,367,152,251,53,398,171,283,79,348,120,232)(9,408,216,312,99,366,153,250,54,397,172,282,80,347,121,231)(10,407,190,311,100,365,154,249,28,396,173,281,81,346,122,230)(11,406,191,310,101,364,155,248,29,395,174,280,55,345,123,229)(12,432,192,309,102,363,156,247,30,394,175,279,56,344,124,228)(13,431,193,308,103,362,157,246,31,393,176,278,57,343,125,227)(14,430,194,307,104,361,158,245,32,392,177,277,58,342,126,226)(15,429,195,306,105,360,159,244,33,391,178,276,59,341,127,225)(16,428,196,305,106,359,160,270,34,390,179,275,60,340,128,224)(17,427,197,304,107,358,161,269,35,389,180,274,61,339,129,223)(18,426,198,303,108,357,162,268,36,388,181,273,62,338,130,222)(19,425,199,302,82,356,136,267,37,387,182,272,63,337,131,221)(20,424,200,301,83,355,137,266,38,386,183,271,64,336,132,220)(21,423,201,300,84,354,138,265,39,385,184,297,65,335,133,219)(22,422,202,299,85,353,139,264,40,384,185,296,66,334,134,218)(23,421,203,298,86,352,140,263,41,383,186,295,67,333,135,217)(24,420,204,324,87,378,141,262,42,382,187,294,68,332,109,243)(25,419,205,323,88,377,142,261,43,381,188,293,69,331,110,242)(26,418,206,322,89,376,143,260,44,380,189,292,70,330,111,241)(27,417,207,321,90,375,144,259,45,379,163,291,71,329,112,240) );
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),(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405),(406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)], [(1,416,208,320,91,374,145,258,46,405,164,290,72,328,113,239),(2,415,209,319,92,373,146,257,47,404,165,289,73,327,114,238),(3,414,210,318,93,372,147,256,48,403,166,288,74,326,115,237),(4,413,211,317,94,371,148,255,49,402,167,287,75,325,116,236),(5,412,212,316,95,370,149,254,50,401,168,286,76,351,117,235),(6,411,213,315,96,369,150,253,51,400,169,285,77,350,118,234),(7,410,214,314,97,368,151,252,52,399,170,284,78,349,119,233),(8,409,215,313,98,367,152,251,53,398,171,283,79,348,120,232),(9,408,216,312,99,366,153,250,54,397,172,282,80,347,121,231),(10,407,190,311,100,365,154,249,28,396,173,281,81,346,122,230),(11,406,191,310,101,364,155,248,29,395,174,280,55,345,123,229),(12,432,192,309,102,363,156,247,30,394,175,279,56,344,124,228),(13,431,193,308,103,362,157,246,31,393,176,278,57,343,125,227),(14,430,194,307,104,361,158,245,32,392,177,277,58,342,126,226),(15,429,195,306,105,360,159,244,33,391,178,276,59,341,127,225),(16,428,196,305,106,359,160,270,34,390,179,275,60,340,128,224),(17,427,197,304,107,358,161,269,35,389,180,274,61,339,129,223),(18,426,198,303,108,357,162,268,36,388,181,273,62,338,130,222),(19,425,199,302,82,356,136,267,37,387,182,272,63,337,131,221),(20,424,200,301,83,355,137,266,38,386,183,271,64,336,132,220),(21,423,201,300,84,354,138,265,39,385,184,297,65,335,133,219),(22,422,202,299,85,353,139,264,40,384,185,296,66,334,134,218),(23,421,203,298,86,352,140,263,41,383,186,295,67,333,135,217),(24,420,204,324,87,378,141,262,42,382,187,294,68,332,109,243),(25,419,205,323,88,377,142,261,43,381,188,293,69,331,110,242),(26,418,206,322,89,376,143,260,44,380,189,292,70,330,111,241),(27,417,207,321,90,375,144,259,45,379,163,291,71,329,112,240)]])
120 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 16A | ··· | 16H | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 27A | ··· | 27I | 36A | ··· | 36F | 54A | ··· | 54I | 72A | ··· | 72L | 108A | ··· | 108R | 216A | ··· | 216AJ |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 16 | ··· | 16 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 72 | ··· | 72 | 108 | ··· | 108 | 216 | ··· | 216 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 27 | ··· | 27 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | S3 | Dic3 | D9 | C3⋊C8 | Dic9 | C3⋊C16 | D27 | C9⋊C8 | Dic27 | C9⋊C16 | C27⋊C8 | C27⋊C16 |
| kernel | C27⋊C16 | C216 | C108 | C54 | C27 | C72 | C36 | C24 | C18 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 3 | 2 | 3 | 4 | 9 | 6 | 9 | 12 | 18 | 36 |
Matrix representation of C27⋊C16 ►in GL2(𝔽433) generated by
| 263 | 25 |
| 408 | 238 |
| 47 | 380 |
| 333 | 386 |
G:=sub<GL(2,GF(433))| [263,408,25,238],[47,333,380,386] >;
C27⋊C16 in GAP, Magma, Sage, TeX
C_{27}\rtimes C_{16} % in TeX
G:=Group("C27:C16"); // GroupNames label
G:=SmallGroup(432,1);
// by ID
G=gap.SmallGroup(432,1);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,14,36,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b|a^27=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
Export