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G = C27⋊C16order 432 = 24·33

The semidirect product of C27 and C16 acting via C16/C8=C2

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

C1C27 — C27⋊C16
C1C3C9C27C54C108C216 — C27⋊C16
C27 — C27⋊C16
C1C8

Generators and relations for C27⋊C16
 G = < a,b | a27=b16=1, bab-1=a-1 >

27C16
9C3⋊C16
3C9⋊C16

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)
(1 432 203 298 100 363 136 255 45 382 163 285 65 331 113 224)(2 431 204 324 101 362 137 254 46 381 164 284 66 330 114 223)(3 430 205 323 102 361 138 253 47 380 165 283 67 329 115 222)(4 429 206 322 103 360 139 252 48 379 166 282 68 328 116 221)(5 428 207 321 104 359 140 251 49 405 167 281 69 327 117 220)(6 427 208 320 105 358 141 250 50 404 168 280 70 326 118 219)(7 426 209 319 106 357 142 249 51 403 169 279 71 325 119 218)(8 425 210 318 107 356 143 248 52 402 170 278 72 351 120 217)(9 424 211 317 108 355 144 247 53 401 171 277 73 350 121 243)(10 423 212 316 82 354 145 246 54 400 172 276 74 349 122 242)(11 422 213 315 83 353 146 245 28 399 173 275 75 348 123 241)(12 421 214 314 84 352 147 244 29 398 174 274 76 347 124 240)(13 420 215 313 85 378 148 270 30 397 175 273 77 346 125 239)(14 419 216 312 86 377 149 269 31 396 176 272 78 345 126 238)(15 418 190 311 87 376 150 268 32 395 177 271 79 344 127 237)(16 417 191 310 88 375 151 267 33 394 178 297 80 343 128 236)(17 416 192 309 89 374 152 266 34 393 179 296 81 342 129 235)(18 415 193 308 90 373 153 265 35 392 180 295 55 341 130 234)(19 414 194 307 91 372 154 264 36 391 181 294 56 340 131 233)(20 413 195 306 92 371 155 263 37 390 182 293 57 339 132 232)(21 412 196 305 93 370 156 262 38 389 183 292 58 338 133 231)(22 411 197 304 94 369 157 261 39 388 184 291 59 337 134 230)(23 410 198 303 95 368 158 260 40 387 185 290 60 336 135 229)(24 409 199 302 96 367 159 259 41 386 186 289 61 335 109 228)(25 408 200 301 97 366 160 258 42 385 187 288 62 334 110 227)(26 407 201 300 98 365 161 257 43 384 188 287 63 333 111 226)(27 406 202 299 99 364 162 256 44 383 189 286 64 332 112 225)

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), (1,432,203,298,100,363,136,255,45,382,163,285,65,331,113,224)(2,431,204,324,101,362,137,254,46,381,164,284,66,330,114,223)(3,430,205,323,102,361,138,253,47,380,165,283,67,329,115,222)(4,429,206,322,103,360,139,252,48,379,166,282,68,328,116,221)(5,428,207,321,104,359,140,251,49,405,167,281,69,327,117,220)(6,427,208,320,105,358,141,250,50,404,168,280,70,326,118,219)(7,426,209,319,106,357,142,249,51,403,169,279,71,325,119,218)(8,425,210,318,107,356,143,248,52,402,170,278,72,351,120,217)(9,424,211,317,108,355,144,247,53,401,171,277,73,350,121,243)(10,423,212,316,82,354,145,246,54,400,172,276,74,349,122,242)(11,422,213,315,83,353,146,245,28,399,173,275,75,348,123,241)(12,421,214,314,84,352,147,244,29,398,174,274,76,347,124,240)(13,420,215,313,85,378,148,270,30,397,175,273,77,346,125,239)(14,419,216,312,86,377,149,269,31,396,176,272,78,345,126,238)(15,418,190,311,87,376,150,268,32,395,177,271,79,344,127,237)(16,417,191,310,88,375,151,267,33,394,178,297,80,343,128,236)(17,416,192,309,89,374,152,266,34,393,179,296,81,342,129,235)(18,415,193,308,90,373,153,265,35,392,180,295,55,341,130,234)(19,414,194,307,91,372,154,264,36,391,181,294,56,340,131,233)(20,413,195,306,92,371,155,263,37,390,182,293,57,339,132,232)(21,412,196,305,93,370,156,262,38,389,183,292,58,338,133,231)(22,411,197,304,94,369,157,261,39,388,184,291,59,337,134,230)(23,410,198,303,95,368,158,260,40,387,185,290,60,336,135,229)(24,409,199,302,96,367,159,259,41,386,186,289,61,335,109,228)(25,408,200,301,97,366,160,258,42,385,187,288,62,334,110,227)(26,407,201,300,98,365,161,257,43,384,188,287,63,333,111,226)(27,406,202,299,99,364,162,256,44,383,189,286,64,332,112,225)>;

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), (1,432,203,298,100,363,136,255,45,382,163,285,65,331,113,224)(2,431,204,324,101,362,137,254,46,381,164,284,66,330,114,223)(3,430,205,323,102,361,138,253,47,380,165,283,67,329,115,222)(4,429,206,322,103,360,139,252,48,379,166,282,68,328,116,221)(5,428,207,321,104,359,140,251,49,405,167,281,69,327,117,220)(6,427,208,320,105,358,141,250,50,404,168,280,70,326,118,219)(7,426,209,319,106,357,142,249,51,403,169,279,71,325,119,218)(8,425,210,318,107,356,143,248,52,402,170,278,72,351,120,217)(9,424,211,317,108,355,144,247,53,401,171,277,73,350,121,243)(10,423,212,316,82,354,145,246,54,400,172,276,74,349,122,242)(11,422,213,315,83,353,146,245,28,399,173,275,75,348,123,241)(12,421,214,314,84,352,147,244,29,398,174,274,76,347,124,240)(13,420,215,313,85,378,148,270,30,397,175,273,77,346,125,239)(14,419,216,312,86,377,149,269,31,396,176,272,78,345,126,238)(15,418,190,311,87,376,150,268,32,395,177,271,79,344,127,237)(16,417,191,310,88,375,151,267,33,394,178,297,80,343,128,236)(17,416,192,309,89,374,152,266,34,393,179,296,81,342,129,235)(18,415,193,308,90,373,153,265,35,392,180,295,55,341,130,234)(19,414,194,307,91,372,154,264,36,391,181,294,56,340,131,233)(20,413,195,306,92,371,155,263,37,390,182,293,57,339,132,232)(21,412,196,305,93,370,156,262,38,389,183,292,58,338,133,231)(22,411,197,304,94,369,157,261,39,388,184,291,59,337,134,230)(23,410,198,303,95,368,158,260,40,387,185,290,60,336,135,229)(24,409,199,302,96,367,159,259,41,386,186,289,61,335,109,228)(25,408,200,301,97,366,160,258,42,385,187,288,62,334,110,227)(26,407,201,300,98,365,161,257,43,384,188,287,63,333,111,226)(27,406,202,299,99,364,162,256,44,383,189,286,64,332,112,225) );

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)], [(1,432,203,298,100,363,136,255,45,382,163,285,65,331,113,224),(2,431,204,324,101,362,137,254,46,381,164,284,66,330,114,223),(3,430,205,323,102,361,138,253,47,380,165,283,67,329,115,222),(4,429,206,322,103,360,139,252,48,379,166,282,68,328,116,221),(5,428,207,321,104,359,140,251,49,405,167,281,69,327,117,220),(6,427,208,320,105,358,141,250,50,404,168,280,70,326,118,219),(7,426,209,319,106,357,142,249,51,403,169,279,71,325,119,218),(8,425,210,318,107,356,143,248,52,402,170,278,72,351,120,217),(9,424,211,317,108,355,144,247,53,401,171,277,73,350,121,243),(10,423,212,316,82,354,145,246,54,400,172,276,74,349,122,242),(11,422,213,315,83,353,146,245,28,399,173,275,75,348,123,241),(12,421,214,314,84,352,147,244,29,398,174,274,76,347,124,240),(13,420,215,313,85,378,148,270,30,397,175,273,77,346,125,239),(14,419,216,312,86,377,149,269,31,396,176,272,78,345,126,238),(15,418,190,311,87,376,150,268,32,395,177,271,79,344,127,237),(16,417,191,310,88,375,151,267,33,394,178,297,80,343,128,236),(17,416,192,309,89,374,152,266,34,393,179,296,81,342,129,235),(18,415,193,308,90,373,153,265,35,392,180,295,55,341,130,234),(19,414,194,307,91,372,154,264,36,391,181,294,56,340,131,233),(20,413,195,306,92,371,155,263,37,390,182,293,57,339,132,232),(21,412,196,305,93,370,156,262,38,389,183,292,58,338,133,231),(22,411,197,304,94,369,157,261,39,388,184,291,59,337,134,230),(23,410,198,303,95,368,158,260,40,387,185,290,60,336,135,229),(24,409,199,302,96,367,159,259,41,386,186,289,61,335,109,228),(25,408,200,301,97,366,160,258,42,385,187,288,62,334,110,227),(26,407,201,300,98,365,161,257,43,384,188,287,63,333,111,226),(27,406,202,299,99,364,162,256,44,383,189,286,64,332,112,225)])

120 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D9A9B9C12A12B16A···16H18A18B18C24A24B24C24D27A···27I36A···36F54A···54I72A···72L108A···108R216A···216AJ
order1234468888999121216···161818182424242427···2736···3654···5472···72108···108216···216
size11211211112222227···2722222222···22···22···22···22···22···2

120 irreducible representations

dim11111222222222222
type+++-+-+-
imageC1C2C4C8C16S3Dic3D9C3⋊C8Dic9C3⋊C16D27C9⋊C8Dic27C9⋊C16C27⋊C8C27⋊C16
kernelC27⋊C16C216C108C54C27C72C36C24C18C12C9C8C6C4C3C2C1
# reps11248113234969121836

Matrix representation of C27⋊C16 in GL2(𝔽433) generated by

26325
408238
,
47380
333386
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

Subgroup lattice of C27⋊C16 in TeX

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