metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C27⋊2Q16, C54.9D4, C4.3D54, C36.3D6, C12.3D18, Q8.2D27, C108.3C22, Dic54.2C2, C27⋊C8.C2, C9.(C3⋊Q16), C3.(C9⋊Q16), (C3×Q8).5D9, (Q8×C9).5S3, (Q8×C27).1C2, C6.18(C9⋊D4), C2.6(C27⋊D4), C18.18(C3⋊D4), SmallGroup(432,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C27⋊Q16
G = < a,b,c | a27=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C27⋊Q16
►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)
(1 402 78 362 38 428 106 336)(2 401 79 361 39 427 107 335)(3 400 80 360 40 426 108 334)(4 399 81 359 41 425 82 333)(5 398 55 358 42 424 83 332)(6 397 56 357 43 423 84 331)(7 396 57 356 44 422 85 330)(8 395 58 355 45 421 86 329)(9 394 59 354 46 420 87 328)(10 393 60 353 47 419 88 327)(11 392 61 352 48 418 89 326)(12 391 62 378 49 417 90 325)(13 390 63 377 50 416 91 351)(14 389 64 376 51 415 92 350)(15 388 65 375 52 414 93 349)(16 387 66 374 53 413 94 348)(17 386 67 373 54 412 95 347)(18 385 68 372 28 411 96 346)(19 384 69 371 29 410 97 345)(20 383 70 370 30 409 98 344)(21 382 71 369 31 408 99 343)(22 381 72 368 32 407 100 342)(23 380 73 367 33 406 101 341)(24 379 74 366 34 432 102 340)(25 405 75 365 35 431 103 339)(26 404 76 364 36 430 104 338)(27 403 77 363 37 429 105 337)(109 294 205 220 142 320 167 254)(110 293 206 219 143 319 168 253)(111 292 207 218 144 318 169 252)(112 291 208 217 145 317 170 251)(113 290 209 243 146 316 171 250)(114 289 210 242 147 315 172 249)(115 288 211 241 148 314 173 248)(116 287 212 240 149 313 174 247)(117 286 213 239 150 312 175 246)(118 285 214 238 151 311 176 245)(119 284 215 237 152 310 177 244)(120 283 216 236 153 309 178 270)(121 282 190 235 154 308 179 269)(122 281 191 234 155 307 180 268)(123 280 192 233 156 306 181 267)(124 279 193 232 157 305 182 266)(125 278 194 231 158 304 183 265)(126 277 195 230 159 303 184 264)(127 276 196 229 160 302 185 263)(128 275 197 228 161 301 186 262)(129 274 198 227 162 300 187 261)(130 273 199 226 136 299 188 260)(131 272 200 225 137 298 189 259)(132 271 201 224 138 324 163 258)(133 297 202 223 139 323 164 257)(134 296 203 222 140 322 165 256)(135 295 204 221 141 321 166 255)
(1 146 38 113)(2 147 39 114)(3 148 40 115)(4 149 41 116)(5 150 42 117)(6 151 43 118)(7 152 44 119)(8 153 45 120)(9 154 46 121)(10 155 47 122)(11 156 48 123)(12 157 49 124)(13 158 50 125)(14 159 51 126)(15 160 52 127)(16 161 53 128)(17 162 54 129)(18 136 28 130)(19 137 29 131)(20 138 30 132)(21 139 31 133)(22 140 32 134)(23 141 33 135)(24 142 34 109)(25 143 35 110)(26 144 36 111)(27 145 37 112)(55 213 83 175)(56 214 84 176)(57 215 85 177)(58 216 86 178)(59 190 87 179)(60 191 88 180)(61 192 89 181)(62 193 90 182)(63 194 91 183)(64 195 92 184)(65 196 93 185)(66 197 94 186)(67 198 95 187)(68 199 96 188)(69 200 97 189)(70 201 98 163)(71 202 99 164)(72 203 100 165)(73 204 101 166)(74 205 102 167)(75 206 103 168)(76 207 104 169)(77 208 105 170)(78 209 106 171)(79 210 107 172)(80 211 108 173)(81 212 82 174)(217 429 251 403)(218 430 252 404)(219 431 253 405)(220 432 254 379)(221 406 255 380)(222 407 256 381)(223 408 257 382)(224 409 258 383)(225 410 259 384)(226 411 260 385)(227 412 261 386)(228 413 262 387)(229 414 263 388)(230 415 264 389)(231 416 265 390)(232 417 266 391)(233 418 267 392)(234 419 268 393)(235 420 269 394)(236 421 270 395)(237 422 244 396)(238 423 245 397)(239 424 246 398)(240 425 247 399)(241 426 248 400)(242 427 249 401)(243 428 250 402)(271 344 324 370)(272 345 298 371)(273 346 299 372)(274 347 300 373)(275 348 301 374)(276 349 302 375)(277 350 303 376)(278 351 304 377)(279 325 305 378)(280 326 306 352)(281 327 307 353)(282 328 308 354)(283 329 309 355)(284 330 310 356)(285 331 311 357)(286 332 312 358)(287 333 313 359)(288 334 314 360)(289 335 315 361)(290 336 316 362)(291 337 317 363)(292 338 318 364)(293 339 319 365)(294 340 320 366)(295 341 321 367)(296 342 322 368)(297 343 323 369)
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), (1,402,78,362,38,428,106,336)(2,401,79,361,39,427,107,335)(3,400,80,360,40,426,108,334)(4,399,81,359,41,425,82,333)(5,398,55,358,42,424,83,332)(6,397,56,357,43,423,84,331)(7,396,57,356,44,422,85,330)(8,395,58,355,45,421,86,329)(9,394,59,354,46,420,87,328)(10,393,60,353,47,419,88,327)(11,392,61,352,48,418,89,326)(12,391,62,378,49,417,90,325)(13,390,63,377,50,416,91,351)(14,389,64,376,51,415,92,350)(15,388,65,375,52,414,93,349)(16,387,66,374,53,413,94,348)(17,386,67,373,54,412,95,347)(18,385,68,372,28,411,96,346)(19,384,69,371,29,410,97,345)(20,383,70,370,30,409,98,344)(21,382,71,369,31,408,99,343)(22,381,72,368,32,407,100,342)(23,380,73,367,33,406,101,341)(24,379,74,366,34,432,102,340)(25,405,75,365,35,431,103,339)(26,404,76,364,36,430,104,338)(27,403,77,363,37,429,105,337)(109,294,205,220,142,320,167,254)(110,293,206,219,143,319,168,253)(111,292,207,218,144,318,169,252)(112,291,208,217,145,317,170,251)(113,290,209,243,146,316,171,250)(114,289,210,242,147,315,172,249)(115,288,211,241,148,314,173,248)(116,287,212,240,149,313,174,247)(117,286,213,239,150,312,175,246)(118,285,214,238,151,311,176,245)(119,284,215,237,152,310,177,244)(120,283,216,236,153,309,178,270)(121,282,190,235,154,308,179,269)(122,281,191,234,155,307,180,268)(123,280,192,233,156,306,181,267)(124,279,193,232,157,305,182,266)(125,278,194,231,158,304,183,265)(126,277,195,230,159,303,184,264)(127,276,196,229,160,302,185,263)(128,275,197,228,161,301,186,262)(129,274,198,227,162,300,187,261)(130,273,199,226,136,299,188,260)(131,272,200,225,137,298,189,259)(132,271,201,224,138,324,163,258)(133,297,202,223,139,323,164,257)(134,296,203,222,140,322,165,256)(135,295,204,221,141,321,166,255), (1,146,38,113)(2,147,39,114)(3,148,40,115)(4,149,41,116)(5,150,42,117)(6,151,43,118)(7,152,44,119)(8,153,45,120)(9,154,46,121)(10,155,47,122)(11,156,48,123)(12,157,49,124)(13,158,50,125)(14,159,51,126)(15,160,52,127)(16,161,53,128)(17,162,54,129)(18,136,28,130)(19,137,29,131)(20,138,30,132)(21,139,31,133)(22,140,32,134)(23,141,33,135)(24,142,34,109)(25,143,35,110)(26,144,36,111)(27,145,37,112)(55,213,83,175)(56,214,84,176)(57,215,85,177)(58,216,86,178)(59,190,87,179)(60,191,88,180)(61,192,89,181)(62,193,90,182)(63,194,91,183)(64,195,92,184)(65,196,93,185)(66,197,94,186)(67,198,95,187)(68,199,96,188)(69,200,97,189)(70,201,98,163)(71,202,99,164)(72,203,100,165)(73,204,101,166)(74,205,102,167)(75,206,103,168)(76,207,104,169)(77,208,105,170)(78,209,106,171)(79,210,107,172)(80,211,108,173)(81,212,82,174)(217,429,251,403)(218,430,252,404)(219,431,253,405)(220,432,254,379)(221,406,255,380)(222,407,256,381)(223,408,257,382)(224,409,258,383)(225,410,259,384)(226,411,260,385)(227,412,261,386)(228,413,262,387)(229,414,263,388)(230,415,264,389)(231,416,265,390)(232,417,266,391)(233,418,267,392)(234,419,268,393)(235,420,269,394)(236,421,270,395)(237,422,244,396)(238,423,245,397)(239,424,246,398)(240,425,247,399)(241,426,248,400)(242,427,249,401)(243,428,250,402)(271,344,324,370)(272,345,298,371)(273,346,299,372)(274,347,300,373)(275,348,301,374)(276,349,302,375)(277,350,303,376)(278,351,304,377)(279,325,305,378)(280,326,306,352)(281,327,307,353)(282,328,308,354)(283,329,309,355)(284,330,310,356)(285,331,311,357)(286,332,312,358)(287,333,313,359)(288,334,314,360)(289,335,315,361)(290,336,316,362)(291,337,317,363)(292,338,318,364)(293,339,319,365)(294,340,320,366)(295,341,321,367)(296,342,322,368)(297,343,323,369)>;
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), (1,402,78,362,38,428,106,336)(2,401,79,361,39,427,107,335)(3,400,80,360,40,426,108,334)(4,399,81,359,41,425,82,333)(5,398,55,358,42,424,83,332)(6,397,56,357,43,423,84,331)(7,396,57,356,44,422,85,330)(8,395,58,355,45,421,86,329)(9,394,59,354,46,420,87,328)(10,393,60,353,47,419,88,327)(11,392,61,352,48,418,89,326)(12,391,62,378,49,417,90,325)(13,390,63,377,50,416,91,351)(14,389,64,376,51,415,92,350)(15,388,65,375,52,414,93,349)(16,387,66,374,53,413,94,348)(17,386,67,373,54,412,95,347)(18,385,68,372,28,411,96,346)(19,384,69,371,29,410,97,345)(20,383,70,370,30,409,98,344)(21,382,71,369,31,408,99,343)(22,381,72,368,32,407,100,342)(23,380,73,367,33,406,101,341)(24,379,74,366,34,432,102,340)(25,405,75,365,35,431,103,339)(26,404,76,364,36,430,104,338)(27,403,77,363,37,429,105,337)(109,294,205,220,142,320,167,254)(110,293,206,219,143,319,168,253)(111,292,207,218,144,318,169,252)(112,291,208,217,145,317,170,251)(113,290,209,243,146,316,171,250)(114,289,210,242,147,315,172,249)(115,288,211,241,148,314,173,248)(116,287,212,240,149,313,174,247)(117,286,213,239,150,312,175,246)(118,285,214,238,151,311,176,245)(119,284,215,237,152,310,177,244)(120,283,216,236,153,309,178,270)(121,282,190,235,154,308,179,269)(122,281,191,234,155,307,180,268)(123,280,192,233,156,306,181,267)(124,279,193,232,157,305,182,266)(125,278,194,231,158,304,183,265)(126,277,195,230,159,303,184,264)(127,276,196,229,160,302,185,263)(128,275,197,228,161,301,186,262)(129,274,198,227,162,300,187,261)(130,273,199,226,136,299,188,260)(131,272,200,225,137,298,189,259)(132,271,201,224,138,324,163,258)(133,297,202,223,139,323,164,257)(134,296,203,222,140,322,165,256)(135,295,204,221,141,321,166,255), (1,146,38,113)(2,147,39,114)(3,148,40,115)(4,149,41,116)(5,150,42,117)(6,151,43,118)(7,152,44,119)(8,153,45,120)(9,154,46,121)(10,155,47,122)(11,156,48,123)(12,157,49,124)(13,158,50,125)(14,159,51,126)(15,160,52,127)(16,161,53,128)(17,162,54,129)(18,136,28,130)(19,137,29,131)(20,138,30,132)(21,139,31,133)(22,140,32,134)(23,141,33,135)(24,142,34,109)(25,143,35,110)(26,144,36,111)(27,145,37,112)(55,213,83,175)(56,214,84,176)(57,215,85,177)(58,216,86,178)(59,190,87,179)(60,191,88,180)(61,192,89,181)(62,193,90,182)(63,194,91,183)(64,195,92,184)(65,196,93,185)(66,197,94,186)(67,198,95,187)(68,199,96,188)(69,200,97,189)(70,201,98,163)(71,202,99,164)(72,203,100,165)(73,204,101,166)(74,205,102,167)(75,206,103,168)(76,207,104,169)(77,208,105,170)(78,209,106,171)(79,210,107,172)(80,211,108,173)(81,212,82,174)(217,429,251,403)(218,430,252,404)(219,431,253,405)(220,432,254,379)(221,406,255,380)(222,407,256,381)(223,408,257,382)(224,409,258,383)(225,410,259,384)(226,411,260,385)(227,412,261,386)(228,413,262,387)(229,414,263,388)(230,415,264,389)(231,416,265,390)(232,417,266,391)(233,418,267,392)(234,419,268,393)(235,420,269,394)(236,421,270,395)(237,422,244,396)(238,423,245,397)(239,424,246,398)(240,425,247,399)(241,426,248,400)(242,427,249,401)(243,428,250,402)(271,344,324,370)(272,345,298,371)(273,346,299,372)(274,347,300,373)(275,348,301,374)(276,349,302,375)(277,350,303,376)(278,351,304,377)(279,325,305,378)(280,326,306,352)(281,327,307,353)(282,328,308,354)(283,329,309,355)(284,330,310,356)(285,331,311,357)(286,332,312,358)(287,333,313,359)(288,334,314,360)(289,335,315,361)(290,336,316,362)(291,337,317,363)(292,338,318,364)(293,339,319,365)(294,340,320,366)(295,341,321,367)(296,342,322,368)(297,343,323,369) );
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)], [(1,402,78,362,38,428,106,336),(2,401,79,361,39,427,107,335),(3,400,80,360,40,426,108,334),(4,399,81,359,41,425,82,333),(5,398,55,358,42,424,83,332),(6,397,56,357,43,423,84,331),(7,396,57,356,44,422,85,330),(8,395,58,355,45,421,86,329),(9,394,59,354,46,420,87,328),(10,393,60,353,47,419,88,327),(11,392,61,352,48,418,89,326),(12,391,62,378,49,417,90,325),(13,390,63,377,50,416,91,351),(14,389,64,376,51,415,92,350),(15,388,65,375,52,414,93,349),(16,387,66,374,53,413,94,348),(17,386,67,373,54,412,95,347),(18,385,68,372,28,411,96,346),(19,384,69,371,29,410,97,345),(20,383,70,370,30,409,98,344),(21,382,71,369,31,408,99,343),(22,381,72,368,32,407,100,342),(23,380,73,367,33,406,101,341),(24,379,74,366,34,432,102,340),(25,405,75,365,35,431,103,339),(26,404,76,364,36,430,104,338),(27,403,77,363,37,429,105,337),(109,294,205,220,142,320,167,254),(110,293,206,219,143,319,168,253),(111,292,207,218,144,318,169,252),(112,291,208,217,145,317,170,251),(113,290,209,243,146,316,171,250),(114,289,210,242,147,315,172,249),(115,288,211,241,148,314,173,248),(116,287,212,240,149,313,174,247),(117,286,213,239,150,312,175,246),(118,285,214,238,151,311,176,245),(119,284,215,237,152,310,177,244),(120,283,216,236,153,309,178,270),(121,282,190,235,154,308,179,269),(122,281,191,234,155,307,180,268),(123,280,192,233,156,306,181,267),(124,279,193,232,157,305,182,266),(125,278,194,231,158,304,183,265),(126,277,195,230,159,303,184,264),(127,276,196,229,160,302,185,263),(128,275,197,228,161,301,186,262),(129,274,198,227,162,300,187,261),(130,273,199,226,136,299,188,260),(131,272,200,225,137,298,189,259),(132,271,201,224,138,324,163,258),(133,297,202,223,139,323,164,257),(134,296,203,222,140,322,165,256),(135,295,204,221,141,321,166,255)], [(1,146,38,113),(2,147,39,114),(3,148,40,115),(4,149,41,116),(5,150,42,117),(6,151,43,118),(7,152,44,119),(8,153,45,120),(9,154,46,121),(10,155,47,122),(11,156,48,123),(12,157,49,124),(13,158,50,125),(14,159,51,126),(15,160,52,127),(16,161,53,128),(17,162,54,129),(18,136,28,130),(19,137,29,131),(20,138,30,132),(21,139,31,133),(22,140,32,134),(23,141,33,135),(24,142,34,109),(25,143,35,110),(26,144,36,111),(27,145,37,112),(55,213,83,175),(56,214,84,176),(57,215,85,177),(58,216,86,178),(59,190,87,179),(60,191,88,180),(61,192,89,181),(62,193,90,182),(63,194,91,183),(64,195,92,184),(65,196,93,185),(66,197,94,186),(67,198,95,187),(68,199,96,188),(69,200,97,189),(70,201,98,163),(71,202,99,164),(72,203,100,165),(73,204,101,166),(74,205,102,167),(75,206,103,168),(76,207,104,169),(77,208,105,170),(78,209,106,171),(79,210,107,172),(80,211,108,173),(81,212,82,174),(217,429,251,403),(218,430,252,404),(219,431,253,405),(220,432,254,379),(221,406,255,380),(222,407,256,381),(223,408,257,382),(224,409,258,383),(225,410,259,384),(226,411,260,385),(227,412,261,386),(228,413,262,387),(229,414,263,388),(230,415,264,389),(231,416,265,390),(232,417,266,391),(233,418,267,392),(234,419,268,393),(235,420,269,394),(236,421,270,395),(237,422,244,396),(238,423,245,397),(239,424,246,398),(240,425,247,399),(241,426,248,400),(242,427,249,401),(243,428,250,402),(271,344,324,370),(272,345,298,371),(273,346,299,372),(274,347,300,373),(275,348,301,374),(276,349,302,375),(277,350,303,376),(278,351,304,377),(279,325,305,378),(280,326,306,352),(281,327,307,353),(282,328,308,354),(283,329,309,355),(284,330,310,356),(285,331,311,357),(286,332,312,358),(287,333,313,359),(288,334,314,360),(289,335,315,361),(290,336,316,362),(291,337,317,363),(292,338,318,364),(293,339,319,365),(294,340,320,366),(295,341,321,367),(296,342,322,368),(297,343,323,369)]])
72 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 27A | ··· | 27I | 36A | ··· | 36I | 54A | ··· | 54I | 108A | ··· | 108AA |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 18 | 18 | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 2 | 2 | 4 | 108 | 2 | 54 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | Q16 | D9 | C3⋊D4 | D18 | D27 | C9⋊D4 | D54 | C27⋊D4 | C3⋊Q16 | C9⋊Q16 | C27⋊Q16 |
| kernel | C27⋊Q16 | C27⋊C8 | Dic54 | Q8×C27 | Q8×C9 | C54 | C36 | C27 | C3×Q8 | C18 | C12 | Q8 | C6 | C4 | C2 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 9 | 6 | 9 | 18 | 1 | 3 | 9 |
Matrix representation of C27⋊Q16 ►in GL4(𝔽433) generated by
| 401 | 130 | 0 | 0 |
| 303 | 271 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 47 | 83 | 0 | 0 |
| 36 | 386 | 0 | 0 |
| 0 | 0 | 103 | 330 |
| 0 | 0 | 103 | 103 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 268 | 273 |
| 0 | 0 | 273 | 165 |
G:=sub<GL(4,GF(433))| [401,303,0,0,130,271,0,0,0,0,1,0,0,0,0,1],[47,36,0,0,83,386,0,0,0,0,103,103,0,0,330,103],[1,0,0,0,0,1,0,0,0,0,268,273,0,0,273,165] >;
C27⋊Q16 in GAP, Magma, Sage, TeX
C_{27}\rtimes Q_{16} % in TeX
G:=Group("C27:Q16"); // GroupNames label
G:=SmallGroup(432,17);
// by ID
G=gap.SmallGroup(432,17);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,64,254,135,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^27=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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