metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic27, C108⋊1C4, C6.4D36, C54.4D4, C54.2Q8, C2.1D108, C18.4D12, C12.2Dic9, C18.5Dic6, C36.2Dic3, C6.5Dic18, C2.2Dic54, C22.5D54, C27⋊2(C4⋊C4), (C2×C36).5S3, C54.8(C2×C4), (C2×C4).3D27, (C2×C12).5D9, C3.(C4⋊Dic9), C9.(C4⋊Dic3), (C2×C108).3C2, (C2×C6).26D18, (C2×C18).26D6, C6.9(C2×Dic9), (C2×C54).5C22, C18.9(C2×Dic3), C2.4(C2×Dic27), (C2×Dic27).2C2, SmallGroup(432,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic27
G = < a,b,c | a4=b54=1, c2=b27, ab=ba, cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C4⋊Dic27
►Regular action on 432 points
Generators in S432
(1 305 139 362)(2 306 140 363)(3 307 141 364)(4 308 142 365)(5 309 143 366)(6 310 144 367)(7 311 145 368)(8 312 146 369)(9 313 147 370)(10 314 148 371)(11 315 149 372)(12 316 150 373)(13 317 151 374)(14 318 152 375)(15 319 153 376)(16 320 154 377)(17 321 155 378)(18 322 156 325)(19 323 157 326)(20 324 158 327)(21 271 159 328)(22 272 160 329)(23 273 161 330)(24 274 162 331)(25 275 109 332)(26 276 110 333)(27 277 111 334)(28 278 112 335)(29 279 113 336)(30 280 114 337)(31 281 115 338)(32 282 116 339)(33 283 117 340)(34 284 118 341)(35 285 119 342)(36 286 120 343)(37 287 121 344)(38 288 122 345)(39 289 123 346)(40 290 124 347)(41 291 125 348)(42 292 126 349)(43 293 127 350)(44 294 128 351)(45 295 129 352)(46 296 130 353)(47 297 131 354)(48 298 132 355)(49 299 133 356)(50 300 134 357)(51 301 135 358)(52 302 136 359)(53 303 137 360)(54 304 138 361)(55 388 176 220)(56 389 177 221)(57 390 178 222)(58 391 179 223)(59 392 180 224)(60 393 181 225)(61 394 182 226)(62 395 183 227)(63 396 184 228)(64 397 185 229)(65 398 186 230)(66 399 187 231)(67 400 188 232)(68 401 189 233)(69 402 190 234)(70 403 191 235)(71 404 192 236)(72 405 193 237)(73 406 194 238)(74 407 195 239)(75 408 196 240)(76 409 197 241)(77 410 198 242)(78 411 199 243)(79 412 200 244)(80 413 201 245)(81 414 202 246)(82 415 203 247)(83 416 204 248)(84 417 205 249)(85 418 206 250)(86 419 207 251)(87 420 208 252)(88 421 209 253)(89 422 210 254)(90 423 211 255)(91 424 212 256)(92 425 213 257)(93 426 214 258)(94 427 215 259)(95 428 216 260)(96 429 163 261)(97 430 164 262)(98 431 165 263)(99 432 166 264)(100 379 167 265)(101 380 168 266)(102 381 169 267)(103 382 170 268)(104 383 171 269)(105 384 172 270)(106 385 173 217)(107 386 174 218)(108 387 175 219)
(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)
(1 217 28 244)(2 270 29 243)(3 269 30 242)(4 268 31 241)(5 267 32 240)(6 266 33 239)(7 265 34 238)(8 264 35 237)(9 263 36 236)(10 262 37 235)(11 261 38 234)(12 260 39 233)(13 259 40 232)(14 258 41 231)(15 257 42 230)(16 256 43 229)(17 255 44 228)(18 254 45 227)(19 253 46 226)(20 252 47 225)(21 251 48 224)(22 250 49 223)(23 249 50 222)(24 248 51 221)(25 247 52 220)(26 246 53 219)(27 245 54 218)(55 332 82 359)(56 331 83 358)(57 330 84 357)(58 329 85 356)(59 328 86 355)(60 327 87 354)(61 326 88 353)(62 325 89 352)(63 378 90 351)(64 377 91 350)(65 376 92 349)(66 375 93 348)(67 374 94 347)(68 373 95 346)(69 372 96 345)(70 371 97 344)(71 370 98 343)(72 369 99 342)(73 368 100 341)(74 367 101 340)(75 366 102 339)(76 365 103 338)(77 364 104 337)(78 363 105 336)(79 362 106 335)(80 361 107 334)(81 360 108 333)(109 415 136 388)(110 414 137 387)(111 413 138 386)(112 412 139 385)(113 411 140 384)(114 410 141 383)(115 409 142 382)(116 408 143 381)(117 407 144 380)(118 406 145 379)(119 405 146 432)(120 404 147 431)(121 403 148 430)(122 402 149 429)(123 401 150 428)(124 400 151 427)(125 399 152 426)(126 398 153 425)(127 397 154 424)(128 396 155 423)(129 395 156 422)(130 394 157 421)(131 393 158 420)(132 392 159 419)(133 391 160 418)(134 390 161 417)(135 389 162 416)(163 288 190 315)(164 287 191 314)(165 286 192 313)(166 285 193 312)(167 284 194 311)(168 283 195 310)(169 282 196 309)(170 281 197 308)(171 280 198 307)(172 279 199 306)(173 278 200 305)(174 277 201 304)(175 276 202 303)(176 275 203 302)(177 274 204 301)(178 273 205 300)(179 272 206 299)(180 271 207 298)(181 324 208 297)(182 323 209 296)(183 322 210 295)(184 321 211 294)(185 320 212 293)(186 319 213 292)(187 318 214 291)(188 317 215 290)(189 316 216 289)
G:=sub<Sym(432)| (1,305,139,362)(2,306,140,363)(3,307,141,364)(4,308,142,365)(5,309,143,366)(6,310,144,367)(7,311,145,368)(8,312,146,369)(9,313,147,370)(10,314,148,371)(11,315,149,372)(12,316,150,373)(13,317,151,374)(14,318,152,375)(15,319,153,376)(16,320,154,377)(17,321,155,378)(18,322,156,325)(19,323,157,326)(20,324,158,327)(21,271,159,328)(22,272,160,329)(23,273,161,330)(24,274,162,331)(25,275,109,332)(26,276,110,333)(27,277,111,334)(28,278,112,335)(29,279,113,336)(30,280,114,337)(31,281,115,338)(32,282,116,339)(33,283,117,340)(34,284,118,341)(35,285,119,342)(36,286,120,343)(37,287,121,344)(38,288,122,345)(39,289,123,346)(40,290,124,347)(41,291,125,348)(42,292,126,349)(43,293,127,350)(44,294,128,351)(45,295,129,352)(46,296,130,353)(47,297,131,354)(48,298,132,355)(49,299,133,356)(50,300,134,357)(51,301,135,358)(52,302,136,359)(53,303,137,360)(54,304,138,361)(55,388,176,220)(56,389,177,221)(57,390,178,222)(58,391,179,223)(59,392,180,224)(60,393,181,225)(61,394,182,226)(62,395,183,227)(63,396,184,228)(64,397,185,229)(65,398,186,230)(66,399,187,231)(67,400,188,232)(68,401,189,233)(69,402,190,234)(70,403,191,235)(71,404,192,236)(72,405,193,237)(73,406,194,238)(74,407,195,239)(75,408,196,240)(76,409,197,241)(77,410,198,242)(78,411,199,243)(79,412,200,244)(80,413,201,245)(81,414,202,246)(82,415,203,247)(83,416,204,248)(84,417,205,249)(85,418,206,250)(86,419,207,251)(87,420,208,252)(88,421,209,253)(89,422,210,254)(90,423,211,255)(91,424,212,256)(92,425,213,257)(93,426,214,258)(94,427,215,259)(95,428,216,260)(96,429,163,261)(97,430,164,262)(98,431,165,263)(99,432,166,264)(100,379,167,265)(101,380,168,266)(102,381,169,267)(103,382,170,268)(104,383,171,269)(105,384,172,270)(106,385,173,217)(107,386,174,218)(108,387,175,219), (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), (1,217,28,244)(2,270,29,243)(3,269,30,242)(4,268,31,241)(5,267,32,240)(6,266,33,239)(7,265,34,238)(8,264,35,237)(9,263,36,236)(10,262,37,235)(11,261,38,234)(12,260,39,233)(13,259,40,232)(14,258,41,231)(15,257,42,230)(16,256,43,229)(17,255,44,228)(18,254,45,227)(19,253,46,226)(20,252,47,225)(21,251,48,224)(22,250,49,223)(23,249,50,222)(24,248,51,221)(25,247,52,220)(26,246,53,219)(27,245,54,218)(55,332,82,359)(56,331,83,358)(57,330,84,357)(58,329,85,356)(59,328,86,355)(60,327,87,354)(61,326,88,353)(62,325,89,352)(63,378,90,351)(64,377,91,350)(65,376,92,349)(66,375,93,348)(67,374,94,347)(68,373,95,346)(69,372,96,345)(70,371,97,344)(71,370,98,343)(72,369,99,342)(73,368,100,341)(74,367,101,340)(75,366,102,339)(76,365,103,338)(77,364,104,337)(78,363,105,336)(79,362,106,335)(80,361,107,334)(81,360,108,333)(109,415,136,388)(110,414,137,387)(111,413,138,386)(112,412,139,385)(113,411,140,384)(114,410,141,383)(115,409,142,382)(116,408,143,381)(117,407,144,380)(118,406,145,379)(119,405,146,432)(120,404,147,431)(121,403,148,430)(122,402,149,429)(123,401,150,428)(124,400,151,427)(125,399,152,426)(126,398,153,425)(127,397,154,424)(128,396,155,423)(129,395,156,422)(130,394,157,421)(131,393,158,420)(132,392,159,419)(133,391,160,418)(134,390,161,417)(135,389,162,416)(163,288,190,315)(164,287,191,314)(165,286,192,313)(166,285,193,312)(167,284,194,311)(168,283,195,310)(169,282,196,309)(170,281,197,308)(171,280,198,307)(172,279,199,306)(173,278,200,305)(174,277,201,304)(175,276,202,303)(176,275,203,302)(177,274,204,301)(178,273,205,300)(179,272,206,299)(180,271,207,298)(181,324,208,297)(182,323,209,296)(183,322,210,295)(184,321,211,294)(185,320,212,293)(186,319,213,292)(187,318,214,291)(188,317,215,290)(189,316,216,289)>;
G:=Group( (1,305,139,362)(2,306,140,363)(3,307,141,364)(4,308,142,365)(5,309,143,366)(6,310,144,367)(7,311,145,368)(8,312,146,369)(9,313,147,370)(10,314,148,371)(11,315,149,372)(12,316,150,373)(13,317,151,374)(14,318,152,375)(15,319,153,376)(16,320,154,377)(17,321,155,378)(18,322,156,325)(19,323,157,326)(20,324,158,327)(21,271,159,328)(22,272,160,329)(23,273,161,330)(24,274,162,331)(25,275,109,332)(26,276,110,333)(27,277,111,334)(28,278,112,335)(29,279,113,336)(30,280,114,337)(31,281,115,338)(32,282,116,339)(33,283,117,340)(34,284,118,341)(35,285,119,342)(36,286,120,343)(37,287,121,344)(38,288,122,345)(39,289,123,346)(40,290,124,347)(41,291,125,348)(42,292,126,349)(43,293,127,350)(44,294,128,351)(45,295,129,352)(46,296,130,353)(47,297,131,354)(48,298,132,355)(49,299,133,356)(50,300,134,357)(51,301,135,358)(52,302,136,359)(53,303,137,360)(54,304,138,361)(55,388,176,220)(56,389,177,221)(57,390,178,222)(58,391,179,223)(59,392,180,224)(60,393,181,225)(61,394,182,226)(62,395,183,227)(63,396,184,228)(64,397,185,229)(65,398,186,230)(66,399,187,231)(67,400,188,232)(68,401,189,233)(69,402,190,234)(70,403,191,235)(71,404,192,236)(72,405,193,237)(73,406,194,238)(74,407,195,239)(75,408,196,240)(76,409,197,241)(77,410,198,242)(78,411,199,243)(79,412,200,244)(80,413,201,245)(81,414,202,246)(82,415,203,247)(83,416,204,248)(84,417,205,249)(85,418,206,250)(86,419,207,251)(87,420,208,252)(88,421,209,253)(89,422,210,254)(90,423,211,255)(91,424,212,256)(92,425,213,257)(93,426,214,258)(94,427,215,259)(95,428,216,260)(96,429,163,261)(97,430,164,262)(98,431,165,263)(99,432,166,264)(100,379,167,265)(101,380,168,266)(102,381,169,267)(103,382,170,268)(104,383,171,269)(105,384,172,270)(106,385,173,217)(107,386,174,218)(108,387,175,219), (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), (1,217,28,244)(2,270,29,243)(3,269,30,242)(4,268,31,241)(5,267,32,240)(6,266,33,239)(7,265,34,238)(8,264,35,237)(9,263,36,236)(10,262,37,235)(11,261,38,234)(12,260,39,233)(13,259,40,232)(14,258,41,231)(15,257,42,230)(16,256,43,229)(17,255,44,228)(18,254,45,227)(19,253,46,226)(20,252,47,225)(21,251,48,224)(22,250,49,223)(23,249,50,222)(24,248,51,221)(25,247,52,220)(26,246,53,219)(27,245,54,218)(55,332,82,359)(56,331,83,358)(57,330,84,357)(58,329,85,356)(59,328,86,355)(60,327,87,354)(61,326,88,353)(62,325,89,352)(63,378,90,351)(64,377,91,350)(65,376,92,349)(66,375,93,348)(67,374,94,347)(68,373,95,346)(69,372,96,345)(70,371,97,344)(71,370,98,343)(72,369,99,342)(73,368,100,341)(74,367,101,340)(75,366,102,339)(76,365,103,338)(77,364,104,337)(78,363,105,336)(79,362,106,335)(80,361,107,334)(81,360,108,333)(109,415,136,388)(110,414,137,387)(111,413,138,386)(112,412,139,385)(113,411,140,384)(114,410,141,383)(115,409,142,382)(116,408,143,381)(117,407,144,380)(118,406,145,379)(119,405,146,432)(120,404,147,431)(121,403,148,430)(122,402,149,429)(123,401,150,428)(124,400,151,427)(125,399,152,426)(126,398,153,425)(127,397,154,424)(128,396,155,423)(129,395,156,422)(130,394,157,421)(131,393,158,420)(132,392,159,419)(133,391,160,418)(134,390,161,417)(135,389,162,416)(163,288,190,315)(164,287,191,314)(165,286,192,313)(166,285,193,312)(167,284,194,311)(168,283,195,310)(169,282,196,309)(170,281,197,308)(171,280,198,307)(172,279,199,306)(173,278,200,305)(174,277,201,304)(175,276,202,303)(176,275,203,302)(177,274,204,301)(178,273,205,300)(179,272,206,299)(180,271,207,298)(181,324,208,297)(182,323,209,296)(183,322,210,295)(184,321,211,294)(185,320,212,293)(186,319,213,292)(187,318,214,291)(188,317,215,290)(189,316,216,289) );
G=PermutationGroup([[(1,305,139,362),(2,306,140,363),(3,307,141,364),(4,308,142,365),(5,309,143,366),(6,310,144,367),(7,311,145,368),(8,312,146,369),(9,313,147,370),(10,314,148,371),(11,315,149,372),(12,316,150,373),(13,317,151,374),(14,318,152,375),(15,319,153,376),(16,320,154,377),(17,321,155,378),(18,322,156,325),(19,323,157,326),(20,324,158,327),(21,271,159,328),(22,272,160,329),(23,273,161,330),(24,274,162,331),(25,275,109,332),(26,276,110,333),(27,277,111,334),(28,278,112,335),(29,279,113,336),(30,280,114,337),(31,281,115,338),(32,282,116,339),(33,283,117,340),(34,284,118,341),(35,285,119,342),(36,286,120,343),(37,287,121,344),(38,288,122,345),(39,289,123,346),(40,290,124,347),(41,291,125,348),(42,292,126,349),(43,293,127,350),(44,294,128,351),(45,295,129,352),(46,296,130,353),(47,297,131,354),(48,298,132,355),(49,299,133,356),(50,300,134,357),(51,301,135,358),(52,302,136,359),(53,303,137,360),(54,304,138,361),(55,388,176,220),(56,389,177,221),(57,390,178,222),(58,391,179,223),(59,392,180,224),(60,393,181,225),(61,394,182,226),(62,395,183,227),(63,396,184,228),(64,397,185,229),(65,398,186,230),(66,399,187,231),(67,400,188,232),(68,401,189,233),(69,402,190,234),(70,403,191,235),(71,404,192,236),(72,405,193,237),(73,406,194,238),(74,407,195,239),(75,408,196,240),(76,409,197,241),(77,410,198,242),(78,411,199,243),(79,412,200,244),(80,413,201,245),(81,414,202,246),(82,415,203,247),(83,416,204,248),(84,417,205,249),(85,418,206,250),(86,419,207,251),(87,420,208,252),(88,421,209,253),(89,422,210,254),(90,423,211,255),(91,424,212,256),(92,425,213,257),(93,426,214,258),(94,427,215,259),(95,428,216,260),(96,429,163,261),(97,430,164,262),(98,431,165,263),(99,432,166,264),(100,379,167,265),(101,380,168,266),(102,381,169,267),(103,382,170,268),(104,383,171,269),(105,384,172,270),(106,385,173,217),(107,386,174,218),(108,387,175,219)], [(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)], [(1,217,28,244),(2,270,29,243),(3,269,30,242),(4,268,31,241),(5,267,32,240),(6,266,33,239),(7,265,34,238),(8,264,35,237),(9,263,36,236),(10,262,37,235),(11,261,38,234),(12,260,39,233),(13,259,40,232),(14,258,41,231),(15,257,42,230),(16,256,43,229),(17,255,44,228),(18,254,45,227),(19,253,46,226),(20,252,47,225),(21,251,48,224),(22,250,49,223),(23,249,50,222),(24,248,51,221),(25,247,52,220),(26,246,53,219),(27,245,54,218),(55,332,82,359),(56,331,83,358),(57,330,84,357),(58,329,85,356),(59,328,86,355),(60,327,87,354),(61,326,88,353),(62,325,89,352),(63,378,90,351),(64,377,91,350),(65,376,92,349),(66,375,93,348),(67,374,94,347),(68,373,95,346),(69,372,96,345),(70,371,97,344),(71,370,98,343),(72,369,99,342),(73,368,100,341),(74,367,101,340),(75,366,102,339),(76,365,103,338),(77,364,104,337),(78,363,105,336),(79,362,106,335),(80,361,107,334),(81,360,108,333),(109,415,136,388),(110,414,137,387),(111,413,138,386),(112,412,139,385),(113,411,140,384),(114,410,141,383),(115,409,142,382),(116,408,143,381),(117,407,144,380),(118,406,145,379),(119,405,146,432),(120,404,147,431),(121,403,148,430),(122,402,149,429),(123,401,150,428),(124,400,151,427),(125,399,152,426),(126,398,153,425),(127,397,154,424),(128,396,155,423),(129,395,156,422),(130,394,157,421),(131,393,158,420),(132,392,159,419),(133,391,160,418),(134,390,161,417),(135,389,162,416),(163,288,190,315),(164,287,191,314),(165,286,192,313),(166,285,193,312),(167,284,194,311),(168,283,195,310),(169,282,196,309),(170,281,197,308),(171,280,198,307),(172,279,199,306),(173,278,200,305),(174,277,201,304),(175,276,202,303),(176,275,203,302),(177,274,204,301),(178,273,205,300),(179,272,206,299),(180,271,207,298),(181,324,208,297),(182,323,209,296),(183,322,210,295),(184,321,211,294),(185,320,212,293),(186,319,213,292),(187,318,214,291),(188,317,215,290),(189,316,216,289)]])
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 | Dic3 | D6 | D9 | Dic6 | D12 | Dic9 | D18 | D27 | Dic18 | D36 | Dic27 | D54 | Dic54 | D108 |
| kernel | C4⋊Dic27 | C2×Dic27 | C2×C108 | C108 | C2×C36 | C54 | C54 | C36 | C2×C18 | C2×C12 | C18 | C18 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 6 | 3 | 9 | 6 | 6 | 18 | 9 | 18 | 18 |
Matrix representation of C4⋊Dic27 ►in GL4(𝔽109) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 89 | 40 |
| 0 | 0 | 69 | 20 |
| 59 | 32 | 0 | 0 |
| 77 | 27 | 0 | 0 |
| 0 | 0 | 92 | 7 |
| 0 | 0 | 102 | 99 |
| 16 | 13 | 0 | 0 |
| 106 | 93 | 0 | 0 |
| 0 | 0 | 94 | 90 |
| 0 | 0 | 75 | 15 |
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,89,69,0,0,40,20],[59,77,0,0,32,27,0,0,0,0,92,102,0,0,7,99],[16,106,0,0,13,93,0,0,0,0,94,75,0,0,90,15] >;
C4⋊Dic27 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_{27} % in TeX
G:=Group("C4:Dic27"); // GroupNames label
G:=SmallGroup(432,13);
// by ID
G=gap.SmallGroup(432,13);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,64,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^4=b^54=1,c^2=b^27,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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