Copied to
clipboard

G = Q16×C27order 432 = 24·33

Direct product of C27 and Q16

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

C1C4 — Q16×C27
C1C3C6C18C36C108Q8×C27 — Q16×C27
C1C2C4 — Q16×C27
C1C54C108 — Q16×C27

Generators and relations for Q16×C27
 G = < a,b,c | a27=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C12
2C12
2C36
2C36
2C108
2C108

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

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

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

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

189 conjugacy classes

class 1  2 3A3B4A4B4C6A6B8A8B9A···9F12A12B12C12D12E12F18A···18F24A24B24C24D27A···27R36A···36F36G···36R54A···54R72A···72L108A···108R108S···108BB216A···216AJ
order123344466889···912121212121218···182424242427···2736···3636···3654···5472···72108···108108···108216···216
size111124411221···12244441···122221···12···24···41···12···22···24···42···2

189 irreducible representations

dim11111111111122222222
type++++-
imageC1C2C2C3C6C6C9C18C18C27C54C54D4Q16C3×D4C3×Q16D4×C9C9×Q16D4×C27Q16×C27
kernelQ16×C27C216Q8×C27C9×Q16C72Q8×C9C3×Q16C24C3×Q8Q16C8Q8C54C27C18C9C6C3C2C1
# reps112224661218183612246121836

Matrix representation of Q16×C27 in GL2(𝔽433) generated by

660
066
,
420244
351240
,
402137
32831
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

Export

Subgroup lattice of Q16×C27 in TeX

׿
×
𝔽