Copied to
clipboard

## G = Q8.D27order 432 = 24·33

### The non-split extension by Q8 of D27 acting via D27/C9=S3

Aliases: Q8.D27, C18.1S4, C9.CSU2(𝔽3), Q8⋊C27.C2, (C3×Q8).1D9, (Q8×C9).1S3, C3.(Q8.D9), C2.2(C9.S4), C6.1(C3.S4), SmallGroup(432,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8⋊C27 — Q8.D27
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C9 — Q8⋊C27 — Q8.D27
 Lower central Q8⋊C27 — Q8.D27
 Upper central C1 — C2

Generators and relations for Q8.D27
G = < a,b,c,d | a4=c27=1, b2=d2=a2, bab-1=dbd-1=a-1, cac-1=b, dad-1=a2b, cbc-1=ab, dcd-1=c-1 >

Smallest permutation representation of Q8.D27
Regular action on 432 points
Generators in S432
```(1 153 189 289)(2 115 163 341)(3 241 164 309)(4 156 165 292)(5 118 166 344)(6 217 167 312)(7 159 168 295)(8 121 169 347)(9 220 170 315)(10 162 171 271)(11 124 172 350)(12 223 173 318)(13 138 174 274)(14 127 175 326)(15 226 176 321)(16 141 177 277)(17 130 178 329)(18 229 179 324)(19 144 180 280)(20 133 181 332)(21 232 182 300)(22 147 183 283)(23 109 184 335)(24 235 185 303)(25 150 186 286)(26 112 187 338)(27 238 188 306)(28 72 253 363)(29 399 254 93)(30 420 255 205)(31 75 256 366)(32 402 257 96)(33 423 258 208)(34 78 259 369)(35 405 260 99)(36 426 261 211)(37 81 262 372)(38 381 263 102)(39 429 264 214)(40 57 265 375)(41 384 266 105)(42 432 267 190)(43 60 268 378)(44 387 269 108)(45 408 270 193)(46 63 244 354)(47 390 245 84)(48 411 246 196)(49 66 247 357)(50 393 248 87)(51 414 249 199)(52 69 250 360)(53 396 251 90)(54 417 252 202)(55 428 373 213)(56 103 374 382)(58 431 376 216)(59 106 377 385)(61 407 352 192)(62 82 353 388)(64 410 355 195)(65 85 356 391)(67 413 358 198)(68 88 359 394)(70 416 361 201)(71 91 362 397)(73 419 364 204)(74 94 365 400)(76 422 367 207)(77 97 368 403)(79 425 370 210)(80 100 371 379)(83 409 389 194)(86 412 392 197)(89 415 395 200)(92 418 398 203)(95 421 401 206)(98 424 404 209)(101 427 380 212)(104 430 383 215)(107 406 386 191)(110 149 336 285)(111 304 337 236)(113 152 339 288)(114 307 340 239)(116 155 342 291)(117 310 343 242)(119 158 345 294)(120 313 346 218)(122 161 348 297)(123 316 349 221)(125 137 351 273)(126 319 325 224)(128 140 327 276)(129 322 328 227)(131 143 330 279)(132 298 331 230)(134 146 333 282)(135 301 334 233)(136 222 272 317)(139 225 275 320)(142 228 278 323)(145 231 281 299)(148 234 284 302)(151 237 287 305)(154 240 290 308)(157 243 293 311)(160 219 296 314)
(1 114 189 340)(2 240 163 308)(3 155 164 291)(4 117 165 343)(5 243 166 311)(6 158 167 294)(7 120 168 346)(8 219 169 314)(9 161 170 297)(10 123 171 349)(11 222 172 317)(12 137 173 273)(13 126 174 325)(14 225 175 320)(15 140 176 276)(16 129 177 328)(17 228 178 323)(18 143 179 279)(19 132 180 331)(20 231 181 299)(21 146 182 282)(22 135 183 334)(23 234 184 302)(24 149 185 285)(25 111 186 337)(26 237 187 305)(27 152 188 288)(28 398 253 92)(29 419 254 204)(30 74 255 365)(31 401 256 95)(32 422 257 207)(33 77 258 368)(34 404 259 98)(35 425 260 210)(36 80 261 371)(37 380 262 101)(38 428 263 213)(39 56 264 374)(40 383 265 104)(41 431 266 216)(42 59 267 377)(43 386 268 107)(44 407 269 192)(45 62 270 353)(46 389 244 83)(47 410 245 195)(48 65 246 356)(49 392 247 86)(50 413 248 198)(51 68 249 359)(52 395 250 89)(53 416 251 201)(54 71 252 362)(55 102 373 381)(57 430 375 215)(58 105 376 384)(60 406 378 191)(61 108 352 387)(63 409 354 194)(64 84 355 390)(66 412 357 197)(67 87 358 393)(69 415 360 200)(70 90 361 396)(72 418 363 203)(73 93 364 399)(75 421 366 206)(76 96 367 402)(78 424 369 209)(79 99 370 405)(81 427 372 212)(82 408 388 193)(85 411 391 196)(88 414 394 199)(91 417 397 202)(94 420 400 205)(97 423 403 208)(100 426 379 211)(103 429 382 214)(106 432 385 190)(109 148 335 284)(110 303 336 235)(112 151 338 287)(113 306 339 238)(115 154 341 290)(116 309 342 241)(118 157 344 293)(119 312 345 217)(121 160 347 296)(122 315 348 220)(124 136 350 272)(125 318 351 223)(127 139 326 275)(128 321 327 226)(130 142 329 278)(131 324 330 229)(133 145 332 281)(134 300 333 232)(138 224 274 319)(141 227 277 322)(144 230 280 298)(147 233 283 301)(150 236 286 304)(153 239 289 307)(156 242 292 310)(159 218 295 313)(162 221 271 316)
(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)
(1 270 189 45)(2 269 163 44)(3 268 164 43)(4 267 165 42)(5 266 166 41)(6 265 167 40)(7 264 168 39)(8 263 169 38)(9 262 170 37)(10 261 171 36)(11 260 172 35)(12 259 173 34)(13 258 174 33)(14 257 175 32)(15 256 176 31)(16 255 177 30)(17 254 178 29)(18 253 179 28)(19 252 180 54)(20 251 181 53)(21 250 182 52)(22 249 183 51)(23 248 184 50)(24 247 185 49)(25 246 186 48)(26 245 187 47)(27 244 188 46)(55 296 373 160)(56 295 374 159)(57 294 375 158)(58 293 376 157)(59 292 377 156)(60 291 378 155)(61 290 352 154)(62 289 353 153)(63 288 354 152)(64 287 355 151)(65 286 356 150)(66 285 357 149)(67 284 358 148)(68 283 359 147)(69 282 360 146)(70 281 361 145)(71 280 362 144)(72 279 363 143)(73 278 364 142)(74 277 365 141)(75 276 366 140)(76 275 367 139)(77 274 368 138)(78 273 369 137)(79 272 370 136)(80 271 371 162)(81 297 372 161)(82 239 388 307)(83 238 389 306)(84 237 390 305)(85 236 391 304)(86 235 392 303)(87 234 393 302)(88 233 394 301)(89 232 395 300)(90 231 396 299)(91 230 397 298)(92 229 398 324)(93 228 399 323)(94 227 400 322)(95 226 401 321)(96 225 402 320)(97 224 403 319)(98 223 404 318)(99 222 405 317)(100 221 379 316)(101 220 380 315)(102 219 381 314)(103 218 382 313)(104 217 383 312)(105 243 384 311)(106 242 385 310)(107 241 386 309)(108 240 387 308)(109 413 335 198)(110 412 336 197)(111 411 337 196)(112 410 338 195)(113 409 339 194)(114 408 340 193)(115 407 341 192)(116 406 342 191)(117 432 343 190)(118 431 344 216)(119 430 345 215)(120 429 346 214)(121 428 347 213)(122 427 348 212)(123 426 349 211)(124 425 350 210)(125 424 351 209)(126 423 325 208)(127 422 326 207)(128 421 327 206)(129 420 328 205)(130 419 329 204)(131 418 330 203)(132 417 331 202)(133 416 332 201)(134 415 333 200)(135 414 334 199)```

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

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

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

36 conjugacy classes

 class 1 2 3 4A 4B 6 8A 8B 9A 9B 9C 12 18A 18B 18C 27A ··· 27I 36A 36B 36C 54A ··· 54I order 1 2 3 4 4 6 8 8 9 9 9 12 18 18 18 27 ··· 27 36 36 36 54 ··· 54 size 1 1 2 6 108 2 54 54 2 2 2 12 2 2 2 8 ··· 8 12 12 12 8 ··· 8

36 irreducible representations

 dim 1 1 2 2 2 2 3 4 4 4 6 6 type + + + + - + + - - - + + image C1 C2 S3 D9 CSU2(𝔽3) D27 S4 CSU2(𝔽3) Q8.D9 Q8.D27 C3.S4 C9.S4 kernel Q8.D27 Q8⋊C27 Q8×C9 C3×Q8 C9 Q8 C18 C9 C3 C1 C6 C2 # reps 1 1 1 3 2 9 2 1 3 9 1 3

Matrix representation of Q8.D27 in GL4(𝔽433) generated by

 1 0 0 0 0 1 0 0 0 0 267 281 0 0 281 166
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 432 0
,
 65 64 0 0 217 227 0 0 0 0 375 209 0 0 210 57
,
 228 260 0 0 318 205 0 0 0 0 222 262 0 0 35 211
`G:=sub<GL(4,GF(433))| [1,0,0,0,0,1,0,0,0,0,267,281,0,0,281,166],[1,0,0,0,0,1,0,0,0,0,0,432,0,0,1,0],[65,217,0,0,64,227,0,0,0,0,375,210,0,0,209,57],[228,318,0,0,260,205,0,0,0,0,222,35,0,0,262,211] >;`

Q8.D27 in GAP, Magma, Sage, TeX

`Q_8.D_{27}`
`% in TeX`

`G:=Group("Q8.D27");`
`// GroupNames label`

`G:=SmallGroup(432,37);`
`// by ID`

`G=gap.SmallGroup(432,37);`
`# by ID`

`G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,141,218,632,142,1011,3784,5681,172,2273,3414,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^27=1,b^2=d^2=a^2,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^2*b,c*b*c^-1=a*b,d*c*d^-1=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽