metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic27⋊C4, C54.5D4, C54.1Q8, C6.4Dic18, C2.1Dic54, C18.4Dic6, C22.4D54, C27⋊1(C4⋊C4), C6.8(C4×D9), C18.9(C4×S3), (C2×C36).1S3, C54.4(C2×C4), C2.4(C4×D27), (C2×C4).1D27, (C2×C12).1D9, (C2×C108).1C2, (C2×C18).25D6, (C2×C6).25D18, C3.(Dic9⋊C4), C9.(Dic3⋊C4), C2.1(C27⋊D4), C6.14(C9⋊D4), (C2×C54).4C22, C18.14(C3⋊D4), (C2×Dic27).1C2, SmallGroup(432,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic27⋊C4
G = < a,b,c | a54=c4=1, b2=a27, bab-1=a-1, ac=ca, cbc-1=a27b >
Smallest permutation representation of Dic27⋊C4
►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)
(1 278 28 305)(2 277 29 304)(3 276 30 303)(4 275 31 302)(5 274 32 301)(6 273 33 300)(7 272 34 299)(8 271 35 298)(9 324 36 297)(10 323 37 296)(11 322 38 295)(12 321 39 294)(13 320 40 293)(14 319 41 292)(15 318 42 291)(16 317 43 290)(17 316 44 289)(18 315 45 288)(19 314 46 287)(20 313 47 286)(21 312 48 285)(22 311 49 284)(23 310 50 283)(24 309 51 282)(25 308 52 281)(26 307 53 280)(27 306 54 279)(55 217 82 244)(56 270 83 243)(57 269 84 242)(58 268 85 241)(59 267 86 240)(60 266 87 239)(61 265 88 238)(62 264 89 237)(63 263 90 236)(64 262 91 235)(65 261 92 234)(66 260 93 233)(67 259 94 232)(68 258 95 231)(69 257 96 230)(70 256 97 229)(71 255 98 228)(72 254 99 227)(73 253 100 226)(74 252 101 225)(75 251 102 224)(76 250 103 223)(77 249 104 222)(78 248 105 221)(79 247 106 220)(80 246 107 219)(81 245 108 218)(109 401 136 428)(110 400 137 427)(111 399 138 426)(112 398 139 425)(113 397 140 424)(114 396 141 423)(115 395 142 422)(116 394 143 421)(117 393 144 420)(118 392 145 419)(119 391 146 418)(120 390 147 417)(121 389 148 416)(122 388 149 415)(123 387 150 414)(124 386 151 413)(125 385 152 412)(126 384 153 411)(127 383 154 410)(128 382 155 409)(129 381 156 408)(130 380 157 407)(131 379 158 406)(132 432 159 405)(133 431 160 404)(134 430 161 403)(135 429 162 402)(163 372 190 345)(164 371 191 344)(165 370 192 343)(166 369 193 342)(167 368 194 341)(168 367 195 340)(169 366 196 339)(170 365 197 338)(171 364 198 337)(172 363 199 336)(173 362 200 335)(174 361 201 334)(175 360 202 333)(176 359 203 332)(177 358 204 331)(178 357 205 330)(179 356 206 329)(180 355 207 328)(181 354 208 327)(182 353 209 326)(183 352 210 325)(184 351 211 378)(185 350 212 377)(186 349 213 376)(187 348 214 375)(188 347 215 374)(189 346 216 373)
(1 185 82 129)(2 186 83 130)(3 187 84 131)(4 188 85 132)(5 189 86 133)(6 190 87 134)(7 191 88 135)(8 192 89 136)(9 193 90 137)(10 194 91 138)(11 195 92 139)(12 196 93 140)(13 197 94 141)(14 198 95 142)(15 199 96 143)(16 200 97 144)(17 201 98 145)(18 202 99 146)(19 203 100 147)(20 204 101 148)(21 205 102 149)(22 206 103 150)(23 207 104 151)(24 208 105 152)(25 209 106 153)(26 210 107 154)(27 211 108 155)(28 212 55 156)(29 213 56 157)(30 214 57 158)(31 215 58 159)(32 216 59 160)(33 163 60 161)(34 164 61 162)(35 165 62 109)(36 166 63 110)(37 167 64 111)(38 168 65 112)(39 169 66 113)(40 170 67 114)(41 171 68 115)(42 172 69 116)(43 173 70 117)(44 174 71 118)(45 175 72 119)(46 176 73 120)(47 177 74 121)(48 178 75 122)(49 179 76 123)(50 180 77 124)(51 181 78 125)(52 182 79 126)(53 183 80 127)(54 184 81 128)(217 381 305 350)(218 382 306 351)(219 383 307 352)(220 384 308 353)(221 385 309 354)(222 386 310 355)(223 387 311 356)(224 388 312 357)(225 389 313 358)(226 390 314 359)(227 391 315 360)(228 392 316 361)(229 393 317 362)(230 394 318 363)(231 395 319 364)(232 396 320 365)(233 397 321 366)(234 398 322 367)(235 399 323 368)(236 400 324 369)(237 401 271 370)(238 402 272 371)(239 403 273 372)(240 404 274 373)(241 405 275 374)(242 406 276 375)(243 407 277 376)(244 408 278 377)(245 409 279 378)(246 410 280 325)(247 411 281 326)(248 412 282 327)(249 413 283 328)(250 414 284 329)(251 415 285 330)(252 416 286 331)(253 417 287 332)(254 418 288 333)(255 419 289 334)(256 420 290 335)(257 421 291 336)(258 422 292 337)(259 423 293 338)(260 424 294 339)(261 425 295 340)(262 426 296 341)(263 427 297 342)(264 428 298 343)(265 429 299 344)(266 430 300 345)(267 431 301 346)(268 432 302 347)(269 379 303 348)(270 380 304 349)
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), (1,278,28,305)(2,277,29,304)(3,276,30,303)(4,275,31,302)(5,274,32,301)(6,273,33,300)(7,272,34,299)(8,271,35,298)(9,324,36,297)(10,323,37,296)(11,322,38,295)(12,321,39,294)(13,320,40,293)(14,319,41,292)(15,318,42,291)(16,317,43,290)(17,316,44,289)(18,315,45,288)(19,314,46,287)(20,313,47,286)(21,312,48,285)(22,311,49,284)(23,310,50,283)(24,309,51,282)(25,308,52,281)(26,307,53,280)(27,306,54,279)(55,217,82,244)(56,270,83,243)(57,269,84,242)(58,268,85,241)(59,267,86,240)(60,266,87,239)(61,265,88,238)(62,264,89,237)(63,263,90,236)(64,262,91,235)(65,261,92,234)(66,260,93,233)(67,259,94,232)(68,258,95,231)(69,257,96,230)(70,256,97,229)(71,255,98,228)(72,254,99,227)(73,253,100,226)(74,252,101,225)(75,251,102,224)(76,250,103,223)(77,249,104,222)(78,248,105,221)(79,247,106,220)(80,246,107,219)(81,245,108,218)(109,401,136,428)(110,400,137,427)(111,399,138,426)(112,398,139,425)(113,397,140,424)(114,396,141,423)(115,395,142,422)(116,394,143,421)(117,393,144,420)(118,392,145,419)(119,391,146,418)(120,390,147,417)(121,389,148,416)(122,388,149,415)(123,387,150,414)(124,386,151,413)(125,385,152,412)(126,384,153,411)(127,383,154,410)(128,382,155,409)(129,381,156,408)(130,380,157,407)(131,379,158,406)(132,432,159,405)(133,431,160,404)(134,430,161,403)(135,429,162,402)(163,372,190,345)(164,371,191,344)(165,370,192,343)(166,369,193,342)(167,368,194,341)(168,367,195,340)(169,366,196,339)(170,365,197,338)(171,364,198,337)(172,363,199,336)(173,362,200,335)(174,361,201,334)(175,360,202,333)(176,359,203,332)(177,358,204,331)(178,357,205,330)(179,356,206,329)(180,355,207,328)(181,354,208,327)(182,353,209,326)(183,352,210,325)(184,351,211,378)(185,350,212,377)(186,349,213,376)(187,348,214,375)(188,347,215,374)(189,346,216,373), (1,185,82,129)(2,186,83,130)(3,187,84,131)(4,188,85,132)(5,189,86,133)(6,190,87,134)(7,191,88,135)(8,192,89,136)(9,193,90,137)(10,194,91,138)(11,195,92,139)(12,196,93,140)(13,197,94,141)(14,198,95,142)(15,199,96,143)(16,200,97,144)(17,201,98,145)(18,202,99,146)(19,203,100,147)(20,204,101,148)(21,205,102,149)(22,206,103,150)(23,207,104,151)(24,208,105,152)(25,209,106,153)(26,210,107,154)(27,211,108,155)(28,212,55,156)(29,213,56,157)(30,214,57,158)(31,215,58,159)(32,216,59,160)(33,163,60,161)(34,164,61,162)(35,165,62,109)(36,166,63,110)(37,167,64,111)(38,168,65,112)(39,169,66,113)(40,170,67,114)(41,171,68,115)(42,172,69,116)(43,173,70,117)(44,174,71,118)(45,175,72,119)(46,176,73,120)(47,177,74,121)(48,178,75,122)(49,179,76,123)(50,180,77,124)(51,181,78,125)(52,182,79,126)(53,183,80,127)(54,184,81,128)(217,381,305,350)(218,382,306,351)(219,383,307,352)(220,384,308,353)(221,385,309,354)(222,386,310,355)(223,387,311,356)(224,388,312,357)(225,389,313,358)(226,390,314,359)(227,391,315,360)(228,392,316,361)(229,393,317,362)(230,394,318,363)(231,395,319,364)(232,396,320,365)(233,397,321,366)(234,398,322,367)(235,399,323,368)(236,400,324,369)(237,401,271,370)(238,402,272,371)(239,403,273,372)(240,404,274,373)(241,405,275,374)(242,406,276,375)(243,407,277,376)(244,408,278,377)(245,409,279,378)(246,410,280,325)(247,411,281,326)(248,412,282,327)(249,413,283,328)(250,414,284,329)(251,415,285,330)(252,416,286,331)(253,417,287,332)(254,418,288,333)(255,419,289,334)(256,420,290,335)(257,421,291,336)(258,422,292,337)(259,423,293,338)(260,424,294,339)(261,425,295,340)(262,426,296,341)(263,427,297,342)(264,428,298,343)(265,429,299,344)(266,430,300,345)(267,431,301,346)(268,432,302,347)(269,379,303,348)(270,380,304,349)>;
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), (1,278,28,305)(2,277,29,304)(3,276,30,303)(4,275,31,302)(5,274,32,301)(6,273,33,300)(7,272,34,299)(8,271,35,298)(9,324,36,297)(10,323,37,296)(11,322,38,295)(12,321,39,294)(13,320,40,293)(14,319,41,292)(15,318,42,291)(16,317,43,290)(17,316,44,289)(18,315,45,288)(19,314,46,287)(20,313,47,286)(21,312,48,285)(22,311,49,284)(23,310,50,283)(24,309,51,282)(25,308,52,281)(26,307,53,280)(27,306,54,279)(55,217,82,244)(56,270,83,243)(57,269,84,242)(58,268,85,241)(59,267,86,240)(60,266,87,239)(61,265,88,238)(62,264,89,237)(63,263,90,236)(64,262,91,235)(65,261,92,234)(66,260,93,233)(67,259,94,232)(68,258,95,231)(69,257,96,230)(70,256,97,229)(71,255,98,228)(72,254,99,227)(73,253,100,226)(74,252,101,225)(75,251,102,224)(76,250,103,223)(77,249,104,222)(78,248,105,221)(79,247,106,220)(80,246,107,219)(81,245,108,218)(109,401,136,428)(110,400,137,427)(111,399,138,426)(112,398,139,425)(113,397,140,424)(114,396,141,423)(115,395,142,422)(116,394,143,421)(117,393,144,420)(118,392,145,419)(119,391,146,418)(120,390,147,417)(121,389,148,416)(122,388,149,415)(123,387,150,414)(124,386,151,413)(125,385,152,412)(126,384,153,411)(127,383,154,410)(128,382,155,409)(129,381,156,408)(130,380,157,407)(131,379,158,406)(132,432,159,405)(133,431,160,404)(134,430,161,403)(135,429,162,402)(163,372,190,345)(164,371,191,344)(165,370,192,343)(166,369,193,342)(167,368,194,341)(168,367,195,340)(169,366,196,339)(170,365,197,338)(171,364,198,337)(172,363,199,336)(173,362,200,335)(174,361,201,334)(175,360,202,333)(176,359,203,332)(177,358,204,331)(178,357,205,330)(179,356,206,329)(180,355,207,328)(181,354,208,327)(182,353,209,326)(183,352,210,325)(184,351,211,378)(185,350,212,377)(186,349,213,376)(187,348,214,375)(188,347,215,374)(189,346,216,373), (1,185,82,129)(2,186,83,130)(3,187,84,131)(4,188,85,132)(5,189,86,133)(6,190,87,134)(7,191,88,135)(8,192,89,136)(9,193,90,137)(10,194,91,138)(11,195,92,139)(12,196,93,140)(13,197,94,141)(14,198,95,142)(15,199,96,143)(16,200,97,144)(17,201,98,145)(18,202,99,146)(19,203,100,147)(20,204,101,148)(21,205,102,149)(22,206,103,150)(23,207,104,151)(24,208,105,152)(25,209,106,153)(26,210,107,154)(27,211,108,155)(28,212,55,156)(29,213,56,157)(30,214,57,158)(31,215,58,159)(32,216,59,160)(33,163,60,161)(34,164,61,162)(35,165,62,109)(36,166,63,110)(37,167,64,111)(38,168,65,112)(39,169,66,113)(40,170,67,114)(41,171,68,115)(42,172,69,116)(43,173,70,117)(44,174,71,118)(45,175,72,119)(46,176,73,120)(47,177,74,121)(48,178,75,122)(49,179,76,123)(50,180,77,124)(51,181,78,125)(52,182,79,126)(53,183,80,127)(54,184,81,128)(217,381,305,350)(218,382,306,351)(219,383,307,352)(220,384,308,353)(221,385,309,354)(222,386,310,355)(223,387,311,356)(224,388,312,357)(225,389,313,358)(226,390,314,359)(227,391,315,360)(228,392,316,361)(229,393,317,362)(230,394,318,363)(231,395,319,364)(232,396,320,365)(233,397,321,366)(234,398,322,367)(235,399,323,368)(236,400,324,369)(237,401,271,370)(238,402,272,371)(239,403,273,372)(240,404,274,373)(241,405,275,374)(242,406,276,375)(243,407,277,376)(244,408,278,377)(245,409,279,378)(246,410,280,325)(247,411,281,326)(248,412,282,327)(249,413,283,328)(250,414,284,329)(251,415,285,330)(252,416,286,331)(253,417,287,332)(254,418,288,333)(255,419,289,334)(256,420,290,335)(257,421,291,336)(258,422,292,337)(259,423,293,338)(260,424,294,339)(261,425,295,340)(262,426,296,341)(263,427,297,342)(264,428,298,343)(265,429,299,344)(266,430,300,345)(267,431,301,346)(268,432,302,347)(269,379,303,348)(270,380,304,349) );
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)], [(1,278,28,305),(2,277,29,304),(3,276,30,303),(4,275,31,302),(5,274,32,301),(6,273,33,300),(7,272,34,299),(8,271,35,298),(9,324,36,297),(10,323,37,296),(11,322,38,295),(12,321,39,294),(13,320,40,293),(14,319,41,292),(15,318,42,291),(16,317,43,290),(17,316,44,289),(18,315,45,288),(19,314,46,287),(20,313,47,286),(21,312,48,285),(22,311,49,284),(23,310,50,283),(24,309,51,282),(25,308,52,281),(26,307,53,280),(27,306,54,279),(55,217,82,244),(56,270,83,243),(57,269,84,242),(58,268,85,241),(59,267,86,240),(60,266,87,239),(61,265,88,238),(62,264,89,237),(63,263,90,236),(64,262,91,235),(65,261,92,234),(66,260,93,233),(67,259,94,232),(68,258,95,231),(69,257,96,230),(70,256,97,229),(71,255,98,228),(72,254,99,227),(73,253,100,226),(74,252,101,225),(75,251,102,224),(76,250,103,223),(77,249,104,222),(78,248,105,221),(79,247,106,220),(80,246,107,219),(81,245,108,218),(109,401,136,428),(110,400,137,427),(111,399,138,426),(112,398,139,425),(113,397,140,424),(114,396,141,423),(115,395,142,422),(116,394,143,421),(117,393,144,420),(118,392,145,419),(119,391,146,418),(120,390,147,417),(121,389,148,416),(122,388,149,415),(123,387,150,414),(124,386,151,413),(125,385,152,412),(126,384,153,411),(127,383,154,410),(128,382,155,409),(129,381,156,408),(130,380,157,407),(131,379,158,406),(132,432,159,405),(133,431,160,404),(134,430,161,403),(135,429,162,402),(163,372,190,345),(164,371,191,344),(165,370,192,343),(166,369,193,342),(167,368,194,341),(168,367,195,340),(169,366,196,339),(170,365,197,338),(171,364,198,337),(172,363,199,336),(173,362,200,335),(174,361,201,334),(175,360,202,333),(176,359,203,332),(177,358,204,331),(178,357,205,330),(179,356,206,329),(180,355,207,328),(181,354,208,327),(182,353,209,326),(183,352,210,325),(184,351,211,378),(185,350,212,377),(186,349,213,376),(187,348,214,375),(188,347,215,374),(189,346,216,373)], [(1,185,82,129),(2,186,83,130),(3,187,84,131),(4,188,85,132),(5,189,86,133),(6,190,87,134),(7,191,88,135),(8,192,89,136),(9,193,90,137),(10,194,91,138),(11,195,92,139),(12,196,93,140),(13,197,94,141),(14,198,95,142),(15,199,96,143),(16,200,97,144),(17,201,98,145),(18,202,99,146),(19,203,100,147),(20,204,101,148),(21,205,102,149),(22,206,103,150),(23,207,104,151),(24,208,105,152),(25,209,106,153),(26,210,107,154),(27,211,108,155),(28,212,55,156),(29,213,56,157),(30,214,57,158),(31,215,58,159),(32,216,59,160),(33,163,60,161),(34,164,61,162),(35,165,62,109),(36,166,63,110),(37,167,64,111),(38,168,65,112),(39,169,66,113),(40,170,67,114),(41,171,68,115),(42,172,69,116),(43,173,70,117),(44,174,71,118),(45,175,72,119),(46,176,73,120),(47,177,74,121),(48,178,75,122),(49,179,76,123),(50,180,77,124),(51,181,78,125),(52,182,79,126),(53,183,80,127),(54,184,81,128),(217,381,305,350),(218,382,306,351),(219,383,307,352),(220,384,308,353),(221,385,309,354),(222,386,310,355),(223,387,311,356),(224,388,312,357),(225,389,313,358),(226,390,314,359),(227,391,315,360),(228,392,316,361),(229,393,317,362),(230,394,318,363),(231,395,319,364),(232,396,320,365),(233,397,321,366),(234,398,322,367),(235,399,323,368),(236,400,324,369),(237,401,271,370),(238,402,272,371),(239,403,273,372),(240,404,274,373),(241,405,275,374),(242,406,276,375),(243,407,277,376),(244,408,278,377),(245,409,279,378),(246,410,280,325),(247,411,281,326),(248,412,282,327),(249,413,283,328),(250,414,284,329),(251,415,285,330),(252,416,286,331),(253,417,287,332),(254,418,288,333),(255,419,289,334),(256,420,290,335),(257,421,291,336),(258,422,292,337),(259,423,293,338),(260,424,294,339),(261,425,295,340),(262,426,296,341),(263,427,297,342),(264,428,298,343),(265,429,299,344),(266,430,300,345),(267,431,301,346),(268,432,302,347),(269,379,303,348),(270,380,304,349)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 27A | ··· | 27I | 36A | ··· | 36L | 54A | ··· | 54AA | 108A | ··· | 108AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 54 | 54 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | D9 | Dic6 | C4×S3 | C3⋊D4 | D18 | D27 | Dic18 | C4×D9 | C9⋊D4 | D54 | Dic54 | C4×D27 | C27⋊D4 |
| kernel | Dic27⋊C4 | C2×Dic27 | C2×C108 | Dic27 | C2×C36 | C54 | C54 | C2×C18 | C2×C12 | C18 | C18 | C18 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 2 | 3 | 9 | 6 | 6 | 6 | 9 | 18 | 18 | 18 |
Matrix representation of Dic27⋊C4 ►in GL3(𝔽109) generated by
| 1 | 0 | 0 |
| 0 | 92 | 7 |
| 0 | 102 | 99 |
| 1 | 0 | 0 |
| 0 | 37 | 48 |
| 0 | 85 | 72 |
| 76 | 0 | 0 |
| 0 | 89 | 40 |
| 0 | 69 | 20 |
G:=sub<GL(3,GF(109))| [1,0,0,0,92,102,0,7,99],[1,0,0,0,37,85,0,48,72],[76,0,0,0,89,69,0,40,20] >;
Dic27⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{27}\rtimes C_4 % in TeX
G:=Group("Dic27:C4"); // GroupNames label
G:=SmallGroup(432,12);
// by ID
G=gap.SmallGroup(432,12);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^54=c^4=1,b^2=a^27,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^27*b>;
// generators/relations
Export