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G = C27⋊Q16order 432 = 24·33

The semidirect product of C27 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C272Q16, 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

C1C108 — C27⋊Q16
C1C3C9C27C54C108Dic54 — C27⋊Q16
C27C54C108 — C27⋊Q16
C1C2C4Q8

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 >

2C4
54C4
27Q8
27C8
2C12
18Dic3
27Q16
9C3⋊C8
9Dic6
2C36
6Dic9
9C3⋊Q16
3C9⋊C8
3Dic18
2Dic27
2C108
3C9⋊Q16

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

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

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

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

72 conjugacy classes

class 1  2  3 4A4B4C 6 8A8B9A9B9C12A12B12C18A18B18C27A···27I36A···36I54A···54I108A···108AA
order12344468899912121218181827···2736···3654···54108···108
size11224108254542224442222···24···42···24···4

72 irreducible representations

dim111122222222222444
type+++++++-++++---
imageC1C2C2C2S3D4D6Q16D9C3⋊D4D18D27C9⋊D4D54C27⋊D4C3⋊Q16C9⋊Q16C27⋊Q16
kernelC27⋊Q16C27⋊C8Dic54Q8×C27Q8×C9C54C36C27C3×Q8C18C12Q8C6C4C2C9C3C1
# reps1111111232396918139

Matrix representation of C27⋊Q16 in GL4(𝔽433) generated by

40113000
30327100
0010
0001
,
478300
3638600
00103330
00103103
,
1000
0100
00268273
00273165
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

Export

Subgroup lattice of C27⋊Q16 in TeX

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