direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C27, C8.C54, C72.8C6, C216.3C2, C24.3C18, C54.16D4, Q8.2C54, C108.19C22, C3.(C9×Q16), C9.(C3×Q16), (C3×Q16).C9, (C9×Q16).C3, C4.3(C2×C54), C6.16(D4×C9), C2.5(D4×C27), C36.42(C2×C6), C18.32(C3×D4), (C3×Q8).7C18, (Q8×C9).11C6, (Q8×C27).2C2, C12.19(C2×C18), SmallGroup(432,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C27
G = < a,b,c | a27=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q16×C27
►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)
(1 256 173 160 407 283 323 380)(2 257 174 161 408 284 324 381)(3 258 175 162 409 285 298 382)(4 259 176 136 410 286 299 383)(5 260 177 137 411 287 300 384)(6 261 178 138 412 288 301 385)(7 262 179 139 413 289 302 386)(8 263 180 140 414 290 303 387)(9 264 181 141 415 291 304 388)(10 265 182 142 416 292 305 389)(11 266 183 143 417 293 306 390)(12 267 184 144 418 294 307 391)(13 268 185 145 419 295 308 392)(14 269 186 146 420 296 309 393)(15 270 187 147 421 297 310 394)(16 244 188 148 422 271 311 395)(17 245 189 149 423 272 312 396)(18 246 163 150 424 273 313 397)(19 247 164 151 425 274 314 398)(20 248 165 152 426 275 315 399)(21 249 166 153 427 276 316 400)(22 250 167 154 428 277 317 401)(23 251 168 155 429 278 318 402)(24 252 169 156 430 279 319 403)(25 253 170 157 431 280 320 404)(26 254 171 158 432 281 321 405)(27 255 172 159 406 282 322 379)(28 209 96 131 357 231 59 342)(29 210 97 132 358 232 60 343)(30 211 98 133 359 233 61 344)(31 212 99 134 360 234 62 345)(32 213 100 135 361 235 63 346)(33 214 101 109 362 236 64 347)(34 215 102 110 363 237 65 348)(35 216 103 111 364 238 66 349)(36 190 104 112 365 239 67 350)(37 191 105 113 366 240 68 351)(38 192 106 114 367 241 69 325)(39 193 107 115 368 242 70 326)(40 194 108 116 369 243 71 327)(41 195 82 117 370 217 72 328)(42 196 83 118 371 218 73 329)(43 197 84 119 372 219 74 330)(44 198 85 120 373 220 75 331)(45 199 86 121 374 221 76 332)(46 200 87 122 375 222 77 333)(47 201 88 123 376 223 78 334)(48 202 89 124 377 224 79 335)(49 203 90 125 378 225 80 336)(50 204 91 126 352 226 81 337)(51 205 92 127 353 227 55 338)(52 206 93 128 354 228 56 339)(53 207 94 129 355 229 57 340)(54 208 95 130 356 230 58 341)
(1 87 407 77)(2 88 408 78)(3 89 409 79)(4 90 410 80)(5 91 411 81)(6 92 412 55)(7 93 413 56)(8 94 414 57)(9 95 415 58)(10 96 416 59)(11 97 417 60)(12 98 418 61)(13 99 419 62)(14 100 420 63)(15 101 421 64)(16 102 422 65)(17 103 423 66)(18 104 424 67)(19 105 425 68)(20 106 426 69)(21 107 427 70)(22 108 428 71)(23 82 429 72)(24 83 430 73)(25 84 431 74)(26 85 432 75)(27 86 406 76)(28 305 357 182)(29 306 358 183)(30 307 359 184)(31 308 360 185)(32 309 361 186)(33 310 362 187)(34 311 363 188)(35 312 364 189)(36 313 365 163)(37 314 366 164)(38 315 367 165)(39 316 368 166)(40 317 369 167)(41 318 370 168)(42 319 371 169)(43 320 372 170)(44 321 373 171)(45 322 374 172)(46 323 375 173)(47 324 376 174)(48 298 377 175)(49 299 378 176)(50 300 352 177)(51 301 353 178)(52 302 354 179)(53 303 355 180)(54 304 356 181)(109 147 347 394)(110 148 348 395)(111 149 349 396)(112 150 350 397)(113 151 351 398)(114 152 325 399)(115 153 326 400)(116 154 327 401)(117 155 328 402)(118 156 329 403)(119 157 330 404)(120 158 331 405)(121 159 332 379)(122 160 333 380)(123 161 334 381)(124 162 335 382)(125 136 336 383)(126 137 337 384)(127 138 338 385)(128 139 339 386)(129 140 340 387)(130 141 341 388)(131 142 342 389)(132 143 343 390)(133 144 344 391)(134 145 345 392)(135 146 346 393)(190 273 239 246)(191 274 240 247)(192 275 241 248)(193 276 242 249)(194 277 243 250)(195 278 217 251)(196 279 218 252)(197 280 219 253)(198 281 220 254)(199 282 221 255)(200 283 222 256)(201 284 223 257)(202 285 224 258)(203 286 225 259)(204 287 226 260)(205 288 227 261)(206 289 228 262)(207 290 229 263)(208 291 230 264)(209 292 231 265)(210 293 232 266)(211 294 233 267)(212 295 234 268)(213 296 235 269)(214 297 236 270)(215 271 237 244)(216 272 238 245)
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), (1,256,173,160,407,283,323,380)(2,257,174,161,408,284,324,381)(3,258,175,162,409,285,298,382)(4,259,176,136,410,286,299,383)(5,260,177,137,411,287,300,384)(6,261,178,138,412,288,301,385)(7,262,179,139,413,289,302,386)(8,263,180,140,414,290,303,387)(9,264,181,141,415,291,304,388)(10,265,182,142,416,292,305,389)(11,266,183,143,417,293,306,390)(12,267,184,144,418,294,307,391)(13,268,185,145,419,295,308,392)(14,269,186,146,420,296,309,393)(15,270,187,147,421,297,310,394)(16,244,188,148,422,271,311,395)(17,245,189,149,423,272,312,396)(18,246,163,150,424,273,313,397)(19,247,164,151,425,274,314,398)(20,248,165,152,426,275,315,399)(21,249,166,153,427,276,316,400)(22,250,167,154,428,277,317,401)(23,251,168,155,429,278,318,402)(24,252,169,156,430,279,319,403)(25,253,170,157,431,280,320,404)(26,254,171,158,432,281,321,405)(27,255,172,159,406,282,322,379)(28,209,96,131,357,231,59,342)(29,210,97,132,358,232,60,343)(30,211,98,133,359,233,61,344)(31,212,99,134,360,234,62,345)(32,213,100,135,361,235,63,346)(33,214,101,109,362,236,64,347)(34,215,102,110,363,237,65,348)(35,216,103,111,364,238,66,349)(36,190,104,112,365,239,67,350)(37,191,105,113,366,240,68,351)(38,192,106,114,367,241,69,325)(39,193,107,115,368,242,70,326)(40,194,108,116,369,243,71,327)(41,195,82,117,370,217,72,328)(42,196,83,118,371,218,73,329)(43,197,84,119,372,219,74,330)(44,198,85,120,373,220,75,331)(45,199,86,121,374,221,76,332)(46,200,87,122,375,222,77,333)(47,201,88,123,376,223,78,334)(48,202,89,124,377,224,79,335)(49,203,90,125,378,225,80,336)(50,204,91,126,352,226,81,337)(51,205,92,127,353,227,55,338)(52,206,93,128,354,228,56,339)(53,207,94,129,355,229,57,340)(54,208,95,130,356,230,58,341), (1,87,407,77)(2,88,408,78)(3,89,409,79)(4,90,410,80)(5,91,411,81)(6,92,412,55)(7,93,413,56)(8,94,414,57)(9,95,415,58)(10,96,416,59)(11,97,417,60)(12,98,418,61)(13,99,419,62)(14,100,420,63)(15,101,421,64)(16,102,422,65)(17,103,423,66)(18,104,424,67)(19,105,425,68)(20,106,426,69)(21,107,427,70)(22,108,428,71)(23,82,429,72)(24,83,430,73)(25,84,431,74)(26,85,432,75)(27,86,406,76)(28,305,357,182)(29,306,358,183)(30,307,359,184)(31,308,360,185)(32,309,361,186)(33,310,362,187)(34,311,363,188)(35,312,364,189)(36,313,365,163)(37,314,366,164)(38,315,367,165)(39,316,368,166)(40,317,369,167)(41,318,370,168)(42,319,371,169)(43,320,372,170)(44,321,373,171)(45,322,374,172)(46,323,375,173)(47,324,376,174)(48,298,377,175)(49,299,378,176)(50,300,352,177)(51,301,353,178)(52,302,354,179)(53,303,355,180)(54,304,356,181)(109,147,347,394)(110,148,348,395)(111,149,349,396)(112,150,350,397)(113,151,351,398)(114,152,325,399)(115,153,326,400)(116,154,327,401)(117,155,328,402)(118,156,329,403)(119,157,330,404)(120,158,331,405)(121,159,332,379)(122,160,333,380)(123,161,334,381)(124,162,335,382)(125,136,336,383)(126,137,337,384)(127,138,338,385)(128,139,339,386)(129,140,340,387)(130,141,341,388)(131,142,342,389)(132,143,343,390)(133,144,344,391)(134,145,345,392)(135,146,346,393)(190,273,239,246)(191,274,240,247)(192,275,241,248)(193,276,242,249)(194,277,243,250)(195,278,217,251)(196,279,218,252)(197,280,219,253)(198,281,220,254)(199,282,221,255)(200,283,222,256)(201,284,223,257)(202,285,224,258)(203,286,225,259)(204,287,226,260)(205,288,227,261)(206,289,228,262)(207,290,229,263)(208,291,230,264)(209,292,231,265)(210,293,232,266)(211,294,233,267)(212,295,234,268)(213,296,235,269)(214,297,236,270)(215,271,237,244)(216,272,238,245)>;
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), (1,256,173,160,407,283,323,380)(2,257,174,161,408,284,324,381)(3,258,175,162,409,285,298,382)(4,259,176,136,410,286,299,383)(5,260,177,137,411,287,300,384)(6,261,178,138,412,288,301,385)(7,262,179,139,413,289,302,386)(8,263,180,140,414,290,303,387)(9,264,181,141,415,291,304,388)(10,265,182,142,416,292,305,389)(11,266,183,143,417,293,306,390)(12,267,184,144,418,294,307,391)(13,268,185,145,419,295,308,392)(14,269,186,146,420,296,309,393)(15,270,187,147,421,297,310,394)(16,244,188,148,422,271,311,395)(17,245,189,149,423,272,312,396)(18,246,163,150,424,273,313,397)(19,247,164,151,425,274,314,398)(20,248,165,152,426,275,315,399)(21,249,166,153,427,276,316,400)(22,250,167,154,428,277,317,401)(23,251,168,155,429,278,318,402)(24,252,169,156,430,279,319,403)(25,253,170,157,431,280,320,404)(26,254,171,158,432,281,321,405)(27,255,172,159,406,282,322,379)(28,209,96,131,357,231,59,342)(29,210,97,132,358,232,60,343)(30,211,98,133,359,233,61,344)(31,212,99,134,360,234,62,345)(32,213,100,135,361,235,63,346)(33,214,101,109,362,236,64,347)(34,215,102,110,363,237,65,348)(35,216,103,111,364,238,66,349)(36,190,104,112,365,239,67,350)(37,191,105,113,366,240,68,351)(38,192,106,114,367,241,69,325)(39,193,107,115,368,242,70,326)(40,194,108,116,369,243,71,327)(41,195,82,117,370,217,72,328)(42,196,83,118,371,218,73,329)(43,197,84,119,372,219,74,330)(44,198,85,120,373,220,75,331)(45,199,86,121,374,221,76,332)(46,200,87,122,375,222,77,333)(47,201,88,123,376,223,78,334)(48,202,89,124,377,224,79,335)(49,203,90,125,378,225,80,336)(50,204,91,126,352,226,81,337)(51,205,92,127,353,227,55,338)(52,206,93,128,354,228,56,339)(53,207,94,129,355,229,57,340)(54,208,95,130,356,230,58,341), (1,87,407,77)(2,88,408,78)(3,89,409,79)(4,90,410,80)(5,91,411,81)(6,92,412,55)(7,93,413,56)(8,94,414,57)(9,95,415,58)(10,96,416,59)(11,97,417,60)(12,98,418,61)(13,99,419,62)(14,100,420,63)(15,101,421,64)(16,102,422,65)(17,103,423,66)(18,104,424,67)(19,105,425,68)(20,106,426,69)(21,107,427,70)(22,108,428,71)(23,82,429,72)(24,83,430,73)(25,84,431,74)(26,85,432,75)(27,86,406,76)(28,305,357,182)(29,306,358,183)(30,307,359,184)(31,308,360,185)(32,309,361,186)(33,310,362,187)(34,311,363,188)(35,312,364,189)(36,313,365,163)(37,314,366,164)(38,315,367,165)(39,316,368,166)(40,317,369,167)(41,318,370,168)(42,319,371,169)(43,320,372,170)(44,321,373,171)(45,322,374,172)(46,323,375,173)(47,324,376,174)(48,298,377,175)(49,299,378,176)(50,300,352,177)(51,301,353,178)(52,302,354,179)(53,303,355,180)(54,304,356,181)(109,147,347,394)(110,148,348,395)(111,149,349,396)(112,150,350,397)(113,151,351,398)(114,152,325,399)(115,153,326,400)(116,154,327,401)(117,155,328,402)(118,156,329,403)(119,157,330,404)(120,158,331,405)(121,159,332,379)(122,160,333,380)(123,161,334,381)(124,162,335,382)(125,136,336,383)(126,137,337,384)(127,138,338,385)(128,139,339,386)(129,140,340,387)(130,141,341,388)(131,142,342,389)(132,143,343,390)(133,144,344,391)(134,145,345,392)(135,146,346,393)(190,273,239,246)(191,274,240,247)(192,275,241,248)(193,276,242,249)(194,277,243,250)(195,278,217,251)(196,279,218,252)(197,280,219,253)(198,281,220,254)(199,282,221,255)(200,283,222,256)(201,284,223,257)(202,285,224,258)(203,286,225,259)(204,287,226,260)(205,288,227,261)(206,289,228,262)(207,290,229,263)(208,291,230,264)(209,292,231,265)(210,293,232,266)(211,294,233,267)(212,295,234,268)(213,296,235,269)(214,297,236,270)(215,271,237,244)(216,272,238,245) );
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)], [(1,256,173,160,407,283,323,380),(2,257,174,161,408,284,324,381),(3,258,175,162,409,285,298,382),(4,259,176,136,410,286,299,383),(5,260,177,137,411,287,300,384),(6,261,178,138,412,288,301,385),(7,262,179,139,413,289,302,386),(8,263,180,140,414,290,303,387),(9,264,181,141,415,291,304,388),(10,265,182,142,416,292,305,389),(11,266,183,143,417,293,306,390),(12,267,184,144,418,294,307,391),(13,268,185,145,419,295,308,392),(14,269,186,146,420,296,309,393),(15,270,187,147,421,297,310,394),(16,244,188,148,422,271,311,395),(17,245,189,149,423,272,312,396),(18,246,163,150,424,273,313,397),(19,247,164,151,425,274,314,398),(20,248,165,152,426,275,315,399),(21,249,166,153,427,276,316,400),(22,250,167,154,428,277,317,401),(23,251,168,155,429,278,318,402),(24,252,169,156,430,279,319,403),(25,253,170,157,431,280,320,404),(26,254,171,158,432,281,321,405),(27,255,172,159,406,282,322,379),(28,209,96,131,357,231,59,342),(29,210,97,132,358,232,60,343),(30,211,98,133,359,233,61,344),(31,212,99,134,360,234,62,345),(32,213,100,135,361,235,63,346),(33,214,101,109,362,236,64,347),(34,215,102,110,363,237,65,348),(35,216,103,111,364,238,66,349),(36,190,104,112,365,239,67,350),(37,191,105,113,366,240,68,351),(38,192,106,114,367,241,69,325),(39,193,107,115,368,242,70,326),(40,194,108,116,369,243,71,327),(41,195,82,117,370,217,72,328),(42,196,83,118,371,218,73,329),(43,197,84,119,372,219,74,330),(44,198,85,120,373,220,75,331),(45,199,86,121,374,221,76,332),(46,200,87,122,375,222,77,333),(47,201,88,123,376,223,78,334),(48,202,89,124,377,224,79,335),(49,203,90,125,378,225,80,336),(50,204,91,126,352,226,81,337),(51,205,92,127,353,227,55,338),(52,206,93,128,354,228,56,339),(53,207,94,129,355,229,57,340),(54,208,95,130,356,230,58,341)], [(1,87,407,77),(2,88,408,78),(3,89,409,79),(4,90,410,80),(5,91,411,81),(6,92,412,55),(7,93,413,56),(8,94,414,57),(9,95,415,58),(10,96,416,59),(11,97,417,60),(12,98,418,61),(13,99,419,62),(14,100,420,63),(15,101,421,64),(16,102,422,65),(17,103,423,66),(18,104,424,67),(19,105,425,68),(20,106,426,69),(21,107,427,70),(22,108,428,71),(23,82,429,72),(24,83,430,73),(25,84,431,74),(26,85,432,75),(27,86,406,76),(28,305,357,182),(29,306,358,183),(30,307,359,184),(31,308,360,185),(32,309,361,186),(33,310,362,187),(34,311,363,188),(35,312,364,189),(36,313,365,163),(37,314,366,164),(38,315,367,165),(39,316,368,166),(40,317,369,167),(41,318,370,168),(42,319,371,169),(43,320,372,170),(44,321,373,171),(45,322,374,172),(46,323,375,173),(47,324,376,174),(48,298,377,175),(49,299,378,176),(50,300,352,177),(51,301,353,178),(52,302,354,179),(53,303,355,180),(54,304,356,181),(109,147,347,394),(110,148,348,395),(111,149,349,396),(112,150,350,397),(113,151,351,398),(114,152,325,399),(115,153,326,400),(116,154,327,401),(117,155,328,402),(118,156,329,403),(119,157,330,404),(120,158,331,405),(121,159,332,379),(122,160,333,380),(123,161,334,381),(124,162,335,382),(125,136,336,383),(126,137,337,384),(127,138,338,385),(128,139,339,386),(129,140,340,387),(130,141,341,388),(131,142,342,389),(132,143,343,390),(133,144,344,391),(134,145,345,392),(135,146,346,393),(190,273,239,246),(191,274,240,247),(192,275,241,248),(193,276,242,249),(194,277,243,250),(195,278,217,251),(196,279,218,252),(197,280,219,253),(198,281,220,254),(199,282,221,255),(200,283,222,256),(201,284,223,257),(202,285,224,258),(203,286,225,259),(204,287,226,260),(205,288,227,261),(206,289,228,262),(207,290,229,263),(208,291,230,264),(209,292,231,265),(210,293,232,266),(211,294,233,267),(212,295,234,268),(213,296,235,269),(214,297,236,270),(215,271,237,244),(216,272,238,245)]])
189 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | ··· | 18F | 24A | 24B | 24C | 24D | 27A | ··· | 27R | 36A | ··· | 36F | 36G | ··· | 36R | 54A | ··· | 54R | 72A | ··· | 72L | 108A | ··· | 108R | 108S | ··· | 108BB | 216A | ··· | 216AJ |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 27 | ··· | 27 | 36 | ··· | 36 | 36 | ··· | 36 | 54 | ··· | 54 | 72 | ··· | 72 | 108 | ··· | 108 | 108 | ··· | 108 | 216 | ··· | 216 |
| size | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
189 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | C27 | C54 | C54 | D4 | Q16 | C3×D4 | C3×Q16 | D4×C9 | C9×Q16 | D4×C27 | Q16×C27 |
| kernel | Q16×C27 | C216 | Q8×C27 | C9×Q16 | C72 | Q8×C9 | C3×Q16 | C24 | C3×Q8 | Q16 | C8 | Q8 | C54 | C27 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 18 | 18 | 36 | 1 | 2 | 2 | 4 | 6 | 12 | 18 | 36 |
Matrix representation of Q16×C27 ►in GL2(𝔽433) generated by
| 66 | 0 |
| 0 | 66 |
| 420 | 244 |
| 351 | 240 |
| 402 | 137 |
| 328 | 31 |
G:=sub<GL(2,GF(433))| [66,0,0,66],[420,351,244,240],[402,328,137,31] >;
Q16×C27 in GAP, Magma, Sage, TeX
Q_{16}\times C_{27} % in TeX
G:=Group("Q16xC27"); // GroupNames label
G:=SmallGroup(432,27);
// by ID
G=gap.SmallGroup(432,27);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,504,197,512,142,2355,1186,192,242]);
// Polycyclic
G:=Group<a,b,c|a^27=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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