direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic27, C27⋊C42, C108⋊2C4, C36.6Dic3, C12.6Dic9, C22.3D54, C6.7(C4×D9), C3.(C4×Dic9), C9.(C4×Dic3), C18.8(C4×S3), C54.3(C2×C4), C2.2(C4×D27), (C2×C4).6D27, (C2×C108).7C2, (C2×C36).14S3, (C2×C12).14D9, (C2×C6).24D18, (C2×C18).24D6, C6.8(C2×Dic9), (C2×C54).3C22, C2.2(C2×Dic27), C18.8(C2×Dic3), (C2×Dic27).4C2, SmallGroup(432,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C4×Dic27 |
Generators and relations for C4×Dic27
G = < a,b,c | a4=b54=1, c2=b27, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 312 in 60 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, Dic3, C12, C2×C6, C42, C18, C18, C2×Dic3, C2×C12, C27, Dic9, C36, C2×C18, C4×Dic3, C54, C54, C2×Dic9, C2×C36, Dic27, C108, C2×C54, C4×Dic9, C2×Dic27, C2×C108, C4×Dic27
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, D9, C4×S3, C2×Dic3, Dic9, D18, C4×Dic3, D27, C4×D9, C2×Dic9, Dic27, D54, C4×Dic9, C4×D27, C2×Dic27, C4×Dic27
►Regular action on 432 points
Generators in S432
(1 313 203 350)(2 314 204 351)(3 315 205 352)(4 316 206 353)(5 317 207 354)(6 318 208 355)(7 319 209 356)(8 320 210 357)(9 321 211 358)(10 322 212 359)(11 323 213 360)(12 324 214 361)(13 271 215 362)(14 272 216 363)(15 273 163 364)(16 274 164 365)(17 275 165 366)(18 276 166 367)(19 277 167 368)(20 278 168 369)(21 279 169 370)(22 280 170 371)(23 281 171 372)(24 282 172 373)(25 283 173 374)(26 284 174 375)(27 285 175 376)(28 286 176 377)(29 287 177 378)(30 288 178 325)(31 289 179 326)(32 290 180 327)(33 291 181 328)(34 292 182 329)(35 293 183 330)(36 294 184 331)(37 295 185 332)(38 296 186 333)(39 297 187 334)(40 298 188 335)(41 299 189 336)(42 300 190 337)(43 301 191 338)(44 302 192 339)(45 303 193 340)(46 304 194 341)(47 305 195 342)(48 306 196 343)(49 307 197 344)(50 308 198 345)(51 309 199 346)(52 310 200 347)(53 311 201 348)(54 312 202 349)(55 250 424 162)(56 251 425 109)(57 252 426 110)(58 253 427 111)(59 254 428 112)(60 255 429 113)(61 256 430 114)(62 257 431 115)(63 258 432 116)(64 259 379 117)(65 260 380 118)(66 261 381 119)(67 262 382 120)(68 263 383 121)(69 264 384 122)(70 265 385 123)(71 266 386 124)(72 267 387 125)(73 268 388 126)(74 269 389 127)(75 270 390 128)(76 217 391 129)(77 218 392 130)(78 219 393 131)(79 220 394 132)(80 221 395 133)(81 222 396 134)(82 223 397 135)(83 224 398 136)(84 225 399 137)(85 226 400 138)(86 227 401 139)(87 228 402 140)(88 229 403 141)(89 230 404 142)(90 231 405 143)(91 232 406 144)(92 233 407 145)(93 234 408 146)(94 235 409 147)(95 236 410 148)(96 237 411 149)(97 238 412 150)(98 239 413 151)(99 240 414 152)(100 241 415 153)(101 242 416 154)(102 243 417 155)(103 244 418 156)(104 245 419 157)(105 246 420 158)(106 247 421 159)(107 248 422 160)(108 249 423 161)
(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)
(1 151 28 124)(2 150 29 123)(3 149 30 122)(4 148 31 121)(5 147 32 120)(6 146 33 119)(7 145 34 118)(8 144 35 117)(9 143 36 116)(10 142 37 115)(11 141 38 114)(12 140 39 113)(13 139 40 112)(14 138 41 111)(15 137 42 110)(16 136 43 109)(17 135 44 162)(18 134 45 161)(19 133 46 160)(20 132 47 159)(21 131 48 158)(22 130 49 157)(23 129 50 156)(24 128 51 155)(25 127 52 154)(26 126 53 153)(27 125 54 152)(55 275 82 302)(56 274 83 301)(57 273 84 300)(58 272 85 299)(59 271 86 298)(60 324 87 297)(61 323 88 296)(62 322 89 295)(63 321 90 294)(64 320 91 293)(65 319 92 292)(66 318 93 291)(67 317 94 290)(68 316 95 289)(69 315 96 288)(70 314 97 287)(71 313 98 286)(72 312 99 285)(73 311 100 284)(74 310 101 283)(75 309 102 282)(76 308 103 281)(77 307 104 280)(78 306 105 279)(79 305 106 278)(80 304 107 277)(81 303 108 276)(163 225 190 252)(164 224 191 251)(165 223 192 250)(166 222 193 249)(167 221 194 248)(168 220 195 247)(169 219 196 246)(170 218 197 245)(171 217 198 244)(172 270 199 243)(173 269 200 242)(174 268 201 241)(175 267 202 240)(176 266 203 239)(177 265 204 238)(178 264 205 237)(179 263 206 236)(180 262 207 235)(181 261 208 234)(182 260 209 233)(183 259 210 232)(184 258 211 231)(185 257 212 230)(186 256 213 229)(187 255 214 228)(188 254 215 227)(189 253 216 226)(325 384 352 411)(326 383 353 410)(327 382 354 409)(328 381 355 408)(329 380 356 407)(330 379 357 406)(331 432 358 405)(332 431 359 404)(333 430 360 403)(334 429 361 402)(335 428 362 401)(336 427 363 400)(337 426 364 399)(338 425 365 398)(339 424 366 397)(340 423 367 396)(341 422 368 395)(342 421 369 394)(343 420 370 393)(344 419 371 392)(345 418 372 391)(346 417 373 390)(347 416 374 389)(348 415 375 388)(349 414 376 387)(350 413 377 386)(351 412 378 385)
G:=sub<Sym(432)| (1,313,203,350)(2,314,204,351)(3,315,205,352)(4,316,206,353)(5,317,207,354)(6,318,208,355)(7,319,209,356)(8,320,210,357)(9,321,211,358)(10,322,212,359)(11,323,213,360)(12,324,214,361)(13,271,215,362)(14,272,216,363)(15,273,163,364)(16,274,164,365)(17,275,165,366)(18,276,166,367)(19,277,167,368)(20,278,168,369)(21,279,169,370)(22,280,170,371)(23,281,171,372)(24,282,172,373)(25,283,173,374)(26,284,174,375)(27,285,175,376)(28,286,176,377)(29,287,177,378)(30,288,178,325)(31,289,179,326)(32,290,180,327)(33,291,181,328)(34,292,182,329)(35,293,183,330)(36,294,184,331)(37,295,185,332)(38,296,186,333)(39,297,187,334)(40,298,188,335)(41,299,189,336)(42,300,190,337)(43,301,191,338)(44,302,192,339)(45,303,193,340)(46,304,194,341)(47,305,195,342)(48,306,196,343)(49,307,197,344)(50,308,198,345)(51,309,199,346)(52,310,200,347)(53,311,201,348)(54,312,202,349)(55,250,424,162)(56,251,425,109)(57,252,426,110)(58,253,427,111)(59,254,428,112)(60,255,429,113)(61,256,430,114)(62,257,431,115)(63,258,432,116)(64,259,379,117)(65,260,380,118)(66,261,381,119)(67,262,382,120)(68,263,383,121)(69,264,384,122)(70,265,385,123)(71,266,386,124)(72,267,387,125)(73,268,388,126)(74,269,389,127)(75,270,390,128)(76,217,391,129)(77,218,392,130)(78,219,393,131)(79,220,394,132)(80,221,395,133)(81,222,396,134)(82,223,397,135)(83,224,398,136)(84,225,399,137)(85,226,400,138)(86,227,401,139)(87,228,402,140)(88,229,403,141)(89,230,404,142)(90,231,405,143)(91,232,406,144)(92,233,407,145)(93,234,408,146)(94,235,409,147)(95,236,410,148)(96,237,411,149)(97,238,412,150)(98,239,413,151)(99,240,414,152)(100,241,415,153)(101,242,416,154)(102,243,417,155)(103,244,418,156)(104,245,419,157)(105,246,420,158)(106,247,421,159)(107,248,422,160)(108,249,423,161), (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), (1,151,28,124)(2,150,29,123)(3,149,30,122)(4,148,31,121)(5,147,32,120)(6,146,33,119)(7,145,34,118)(8,144,35,117)(9,143,36,116)(10,142,37,115)(11,141,38,114)(12,140,39,113)(13,139,40,112)(14,138,41,111)(15,137,42,110)(16,136,43,109)(17,135,44,162)(18,134,45,161)(19,133,46,160)(20,132,47,159)(21,131,48,158)(22,130,49,157)(23,129,50,156)(24,128,51,155)(25,127,52,154)(26,126,53,153)(27,125,54,152)(55,275,82,302)(56,274,83,301)(57,273,84,300)(58,272,85,299)(59,271,86,298)(60,324,87,297)(61,323,88,296)(62,322,89,295)(63,321,90,294)(64,320,91,293)(65,319,92,292)(66,318,93,291)(67,317,94,290)(68,316,95,289)(69,315,96,288)(70,314,97,287)(71,313,98,286)(72,312,99,285)(73,311,100,284)(74,310,101,283)(75,309,102,282)(76,308,103,281)(77,307,104,280)(78,306,105,279)(79,305,106,278)(80,304,107,277)(81,303,108,276)(163,225,190,252)(164,224,191,251)(165,223,192,250)(166,222,193,249)(167,221,194,248)(168,220,195,247)(169,219,196,246)(170,218,197,245)(171,217,198,244)(172,270,199,243)(173,269,200,242)(174,268,201,241)(175,267,202,240)(176,266,203,239)(177,265,204,238)(178,264,205,237)(179,263,206,236)(180,262,207,235)(181,261,208,234)(182,260,209,233)(183,259,210,232)(184,258,211,231)(185,257,212,230)(186,256,213,229)(187,255,214,228)(188,254,215,227)(189,253,216,226)(325,384,352,411)(326,383,353,410)(327,382,354,409)(328,381,355,408)(329,380,356,407)(330,379,357,406)(331,432,358,405)(332,431,359,404)(333,430,360,403)(334,429,361,402)(335,428,362,401)(336,427,363,400)(337,426,364,399)(338,425,365,398)(339,424,366,397)(340,423,367,396)(341,422,368,395)(342,421,369,394)(343,420,370,393)(344,419,371,392)(345,418,372,391)(346,417,373,390)(347,416,374,389)(348,415,375,388)(349,414,376,387)(350,413,377,386)(351,412,378,385)>;
G:=Group( (1,313,203,350)(2,314,204,351)(3,315,205,352)(4,316,206,353)(5,317,207,354)(6,318,208,355)(7,319,209,356)(8,320,210,357)(9,321,211,358)(10,322,212,359)(11,323,213,360)(12,324,214,361)(13,271,215,362)(14,272,216,363)(15,273,163,364)(16,274,164,365)(17,275,165,366)(18,276,166,367)(19,277,167,368)(20,278,168,369)(21,279,169,370)(22,280,170,371)(23,281,171,372)(24,282,172,373)(25,283,173,374)(26,284,174,375)(27,285,175,376)(28,286,176,377)(29,287,177,378)(30,288,178,325)(31,289,179,326)(32,290,180,327)(33,291,181,328)(34,292,182,329)(35,293,183,330)(36,294,184,331)(37,295,185,332)(38,296,186,333)(39,297,187,334)(40,298,188,335)(41,299,189,336)(42,300,190,337)(43,301,191,338)(44,302,192,339)(45,303,193,340)(46,304,194,341)(47,305,195,342)(48,306,196,343)(49,307,197,344)(50,308,198,345)(51,309,199,346)(52,310,200,347)(53,311,201,348)(54,312,202,349)(55,250,424,162)(56,251,425,109)(57,252,426,110)(58,253,427,111)(59,254,428,112)(60,255,429,113)(61,256,430,114)(62,257,431,115)(63,258,432,116)(64,259,379,117)(65,260,380,118)(66,261,381,119)(67,262,382,120)(68,263,383,121)(69,264,384,122)(70,265,385,123)(71,266,386,124)(72,267,387,125)(73,268,388,126)(74,269,389,127)(75,270,390,128)(76,217,391,129)(77,218,392,130)(78,219,393,131)(79,220,394,132)(80,221,395,133)(81,222,396,134)(82,223,397,135)(83,224,398,136)(84,225,399,137)(85,226,400,138)(86,227,401,139)(87,228,402,140)(88,229,403,141)(89,230,404,142)(90,231,405,143)(91,232,406,144)(92,233,407,145)(93,234,408,146)(94,235,409,147)(95,236,410,148)(96,237,411,149)(97,238,412,150)(98,239,413,151)(99,240,414,152)(100,241,415,153)(101,242,416,154)(102,243,417,155)(103,244,418,156)(104,245,419,157)(105,246,420,158)(106,247,421,159)(107,248,422,160)(108,249,423,161), (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), (1,151,28,124)(2,150,29,123)(3,149,30,122)(4,148,31,121)(5,147,32,120)(6,146,33,119)(7,145,34,118)(8,144,35,117)(9,143,36,116)(10,142,37,115)(11,141,38,114)(12,140,39,113)(13,139,40,112)(14,138,41,111)(15,137,42,110)(16,136,43,109)(17,135,44,162)(18,134,45,161)(19,133,46,160)(20,132,47,159)(21,131,48,158)(22,130,49,157)(23,129,50,156)(24,128,51,155)(25,127,52,154)(26,126,53,153)(27,125,54,152)(55,275,82,302)(56,274,83,301)(57,273,84,300)(58,272,85,299)(59,271,86,298)(60,324,87,297)(61,323,88,296)(62,322,89,295)(63,321,90,294)(64,320,91,293)(65,319,92,292)(66,318,93,291)(67,317,94,290)(68,316,95,289)(69,315,96,288)(70,314,97,287)(71,313,98,286)(72,312,99,285)(73,311,100,284)(74,310,101,283)(75,309,102,282)(76,308,103,281)(77,307,104,280)(78,306,105,279)(79,305,106,278)(80,304,107,277)(81,303,108,276)(163,225,190,252)(164,224,191,251)(165,223,192,250)(166,222,193,249)(167,221,194,248)(168,220,195,247)(169,219,196,246)(170,218,197,245)(171,217,198,244)(172,270,199,243)(173,269,200,242)(174,268,201,241)(175,267,202,240)(176,266,203,239)(177,265,204,238)(178,264,205,237)(179,263,206,236)(180,262,207,235)(181,261,208,234)(182,260,209,233)(183,259,210,232)(184,258,211,231)(185,257,212,230)(186,256,213,229)(187,255,214,228)(188,254,215,227)(189,253,216,226)(325,384,352,411)(326,383,353,410)(327,382,354,409)(328,381,355,408)(329,380,356,407)(330,379,357,406)(331,432,358,405)(332,431,359,404)(333,430,360,403)(334,429,361,402)(335,428,362,401)(336,427,363,400)(337,426,364,399)(338,425,365,398)(339,424,366,397)(340,423,367,396)(341,422,368,395)(342,421,369,394)(343,420,370,393)(344,419,371,392)(345,418,372,391)(346,417,373,390)(347,416,374,389)(348,415,375,388)(349,414,376,387)(350,413,377,386)(351,412,378,385) );
G=PermutationGroup([[(1,313,203,350),(2,314,204,351),(3,315,205,352),(4,316,206,353),(5,317,207,354),(6,318,208,355),(7,319,209,356),(8,320,210,357),(9,321,211,358),(10,322,212,359),(11,323,213,360),(12,324,214,361),(13,271,215,362),(14,272,216,363),(15,273,163,364),(16,274,164,365),(17,275,165,366),(18,276,166,367),(19,277,167,368),(20,278,168,369),(21,279,169,370),(22,280,170,371),(23,281,171,372),(24,282,172,373),(25,283,173,374),(26,284,174,375),(27,285,175,376),(28,286,176,377),(29,287,177,378),(30,288,178,325),(31,289,179,326),(32,290,180,327),(33,291,181,328),(34,292,182,329),(35,293,183,330),(36,294,184,331),(37,295,185,332),(38,296,186,333),(39,297,187,334),(40,298,188,335),(41,299,189,336),(42,300,190,337),(43,301,191,338),(44,302,192,339),(45,303,193,340),(46,304,194,341),(47,305,195,342),(48,306,196,343),(49,307,197,344),(50,308,198,345),(51,309,199,346),(52,310,200,347),(53,311,201,348),(54,312,202,349),(55,250,424,162),(56,251,425,109),(57,252,426,110),(58,253,427,111),(59,254,428,112),(60,255,429,113),(61,256,430,114),(62,257,431,115),(63,258,432,116),(64,259,379,117),(65,260,380,118),(66,261,381,119),(67,262,382,120),(68,263,383,121),(69,264,384,122),(70,265,385,123),(71,266,386,124),(72,267,387,125),(73,268,388,126),(74,269,389,127),(75,270,390,128),(76,217,391,129),(77,218,392,130),(78,219,393,131),(79,220,394,132),(80,221,395,133),(81,222,396,134),(82,223,397,135),(83,224,398,136),(84,225,399,137),(85,226,400,138),(86,227,401,139),(87,228,402,140),(88,229,403,141),(89,230,404,142),(90,231,405,143),(91,232,406,144),(92,233,407,145),(93,234,408,146),(94,235,409,147),(95,236,410,148),(96,237,411,149),(97,238,412,150),(98,239,413,151),(99,240,414,152),(100,241,415,153),(101,242,416,154),(102,243,417,155),(103,244,418,156),(104,245,419,157),(105,246,420,158),(106,247,421,159),(107,248,422,160),(108,249,423,161)], [(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)], [(1,151,28,124),(2,150,29,123),(3,149,30,122),(4,148,31,121),(5,147,32,120),(6,146,33,119),(7,145,34,118),(8,144,35,117),(9,143,36,116),(10,142,37,115),(11,141,38,114),(12,140,39,113),(13,139,40,112),(14,138,41,111),(15,137,42,110),(16,136,43,109),(17,135,44,162),(18,134,45,161),(19,133,46,160),(20,132,47,159),(21,131,48,158),(22,130,49,157),(23,129,50,156),(24,128,51,155),(25,127,52,154),(26,126,53,153),(27,125,54,152),(55,275,82,302),(56,274,83,301),(57,273,84,300),(58,272,85,299),(59,271,86,298),(60,324,87,297),(61,323,88,296),(62,322,89,295),(63,321,90,294),(64,320,91,293),(65,319,92,292),(66,318,93,291),(67,317,94,290),(68,316,95,289),(69,315,96,288),(70,314,97,287),(71,313,98,286),(72,312,99,285),(73,311,100,284),(74,310,101,283),(75,309,102,282),(76,308,103,281),(77,307,104,280),(78,306,105,279),(79,305,106,278),(80,304,107,277),(81,303,108,276),(163,225,190,252),(164,224,191,251),(165,223,192,250),(166,222,193,249),(167,221,194,248),(168,220,195,247),(169,219,196,246),(170,218,197,245),(171,217,198,244),(172,270,199,243),(173,269,200,242),(174,268,201,241),(175,267,202,240),(176,266,203,239),(177,265,204,238),(178,264,205,237),(179,263,206,236),(180,262,207,235),(181,261,208,234),(182,260,209,233),(183,259,210,232),(184,258,211,231),(185,257,212,230),(186,256,213,229),(187,255,214,228),(188,254,215,227),(189,253,216,226),(325,384,352,411),(326,383,353,410),(327,382,354,409),(328,381,355,408),(329,380,356,407),(330,379,357,406),(331,432,358,405),(332,431,359,404),(333,430,360,403),(334,429,361,402),(335,428,362,401),(336,427,363,400),(337,426,364,399),(338,425,365,398),(339,424,366,397),(340,423,367,396),(341,422,368,395),(342,421,369,394),(343,420,370,393),(344,419,371,392),(345,418,372,391),(346,417,373,390),(347,416,374,389),(348,415,375,388),(349,414,376,387),(350,413,377,386),(351,412,378,385)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 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 | 1 | 1 | 1 | 1 | 27 | ··· | 27 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | D9 | C4×S3 | Dic9 | D18 | D27 | C4×D9 | Dic27 | D54 | C4×D27 |
| kernel | C4×Dic27 | C2×Dic27 | C2×C108 | Dic27 | C108 | C2×C36 | C36 | C2×C18 | C2×C12 | C18 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 3 | 4 | 6 | 3 | 9 | 12 | 18 | 9 | 36 |
Matrix representation of C4×Dic27 ►in GL4(𝔽109) generated by
| 76 | 0 | 0 | 0 |
| 0 | 76 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 59 | 0 | 0 |
| 50 | 77 | 0 | 0 |
| 0 | 0 | 10 | 92 |
| 0 | 0 | 17 | 102 |
| 77 | 50 | 0 | 0 |
| 82 | 32 | 0 | 0 |
| 0 | 0 | 106 | 96 |
| 0 | 0 | 93 | 3 |
G:=sub<GL(4,GF(109))| [76,0,0,0,0,76,0,0,0,0,1,0,0,0,0,1],[27,50,0,0,59,77,0,0,0,0,10,17,0,0,92,102],[77,82,0,0,50,32,0,0,0,0,106,93,0,0,96,3] >;
C4×Dic27 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{27} % in TeX
G:=Group("C4xDic27"); // GroupNames label
G:=SmallGroup(432,11);
// by ID
G=gap.SmallGroup(432,11);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,64,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^4=b^54=1,c^2=b^27,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations