direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C27⋊C8, C54⋊C8, C108.3C4, C36.62D6, C4.15D54, C12.62D18, C4.3Dic27, C12.5Dic9, C36.5Dic3, C108.15C22, C22.2Dic27, C27⋊2(C2×C8), C6.2(C9⋊C8), C18.2(C3⋊C8), C54.6(C2×C4), (C2×C54).2C4, (C2×C4).5D27, (C2×C108).6C2, (C2×C36).13S3, (C2×C12).13D9, (C2×C6).3Dic9, C6.6(C2×Dic9), C2.1(C2×Dic27), C18.6(C2×Dic3), (C2×C18).3Dic3, C3.(C2×C9⋊C8), C9.(C2×C3⋊C8), SmallGroup(432,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C2×C27⋊C8 |
Generators and relations for C2×C27⋊C8
G = < a,b,c | a2=b27=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C27⋊C8
►Regular action on 432 points
Generators in S432
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 109)(10 110)(11 111)(12 112)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 159)(42 160)(43 161)(44 162)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 144)(54 145)(55 187)(56 188)(57 189)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 176)(72 177)(73 178)(74 179)(75 180)(76 181)(77 182)(78 183)(79 184)(80 185)(81 186)(82 198)(83 199)(84 200)(85 201)(86 202)(87 203)(88 204)(89 205)(90 206)(91 207)(92 208)(93 209)(94 210)(95 211)(96 212)(97 213)(98 214)(99 215)(100 216)(101 190)(102 191)(103 192)(104 193)(105 194)(106 195)(107 196)(108 197)(217 330)(218 331)(219 332)(220 333)(221 334)(222 335)(223 336)(224 337)(225 338)(226 339)(227 340)(228 341)(229 342)(230 343)(231 344)(232 345)(233 346)(234 347)(235 348)(236 349)(237 350)(238 351)(239 325)(240 326)(241 327)(242 328)(243 329)(244 364)(245 365)(246 366)(247 367)(248 368)(249 369)(250 370)(251 371)(252 372)(253 373)(254 374)(255 375)(256 376)(257 377)(258 378)(259 352)(260 353)(261 354)(262 355)(263 356)(264 357)(265 358)(266 359)(267 360)(268 361)(269 362)(270 363)(271 389)(272 390)(273 391)(274 392)(275 393)(276 394)(277 395)(278 396)(279 397)(280 398)(281 399)(282 400)(283 401)(284 402)(285 403)(286 404)(287 405)(288 379)(289 380)(290 381)(291 382)(292 383)(293 384)(294 385)(295 386)(296 387)(297 388)(298 428)(299 429)(300 430)(301 431)(302 432)(303 406)(304 407)(305 408)(306 409)(307 410)(308 411)(309 412)(310 413)(311 414)(312 415)(313 416)(314 417)(315 418)(316 419)(317 420)(318 421)(319 422)(320 423)(321 424)(322 425)(323 426)(324 427)
(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)
(1 416 106 356 35 393 67 348)(2 415 107 355 36 392 68 347)(3 414 108 354 37 391 69 346)(4 413 82 353 38 390 70 345)(5 412 83 352 39 389 71 344)(6 411 84 378 40 388 72 343)(7 410 85 377 41 387 73 342)(8 409 86 376 42 386 74 341)(9 408 87 375 43 385 75 340)(10 407 88 374 44 384 76 339)(11 406 89 373 45 383 77 338)(12 432 90 372 46 382 78 337)(13 431 91 371 47 381 79 336)(14 430 92 370 48 380 80 335)(15 429 93 369 49 379 81 334)(16 428 94 368 50 405 55 333)(17 427 95 367 51 404 56 332)(18 426 96 366 52 403 57 331)(19 425 97 365 53 402 58 330)(20 424 98 364 54 401 59 329)(21 423 99 363 28 400 60 328)(22 422 100 362 29 399 61 327)(23 421 101 361 30 398 62 326)(24 420 102 360 31 397 63 325)(25 419 103 359 32 396 64 351)(26 418 104 358 33 395 65 350)(27 417 105 357 34 394 66 349)(109 305 203 255 161 294 180 227)(110 304 204 254 162 293 181 226)(111 303 205 253 136 292 182 225)(112 302 206 252 137 291 183 224)(113 301 207 251 138 290 184 223)(114 300 208 250 139 289 185 222)(115 299 209 249 140 288 186 221)(116 298 210 248 141 287 187 220)(117 324 211 247 142 286 188 219)(118 323 212 246 143 285 189 218)(119 322 213 245 144 284 163 217)(120 321 214 244 145 283 164 243)(121 320 215 270 146 282 165 242)(122 319 216 269 147 281 166 241)(123 318 190 268 148 280 167 240)(124 317 191 267 149 279 168 239)(125 316 192 266 150 278 169 238)(126 315 193 265 151 277 170 237)(127 314 194 264 152 276 171 236)(128 313 195 263 153 275 172 235)(129 312 196 262 154 274 173 234)(130 311 197 261 155 273 174 233)(131 310 198 260 156 272 175 232)(132 309 199 259 157 271 176 231)(133 308 200 258 158 297 177 230)(134 307 201 257 159 296 178 229)(135 306 202 256 160 295 179 228)
G:=sub<Sym(432)| (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,109)(10,110)(11,111)(12,112)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,144)(54,145)(55,187)(56,188)(57,189)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,183)(79,184)(80,185)(81,186)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,208)(93,209)(94,210)(95,211)(96,212)(97,213)(98,214)(99,215)(100,216)(101,190)(102,191)(103,192)(104,193)(105,194)(106,195)(107,196)(108,197)(217,330)(218,331)(219,332)(220,333)(221,334)(222,335)(223,336)(224,337)(225,338)(226,339)(227,340)(228,341)(229,342)(230,343)(231,344)(232,345)(233,346)(234,347)(235,348)(236,349)(237,350)(238,351)(239,325)(240,326)(241,327)(242,328)(243,329)(244,364)(245,365)(246,366)(247,367)(248,368)(249,369)(250,370)(251,371)(252,372)(253,373)(254,374)(255,375)(256,376)(257,377)(258,378)(259,352)(260,353)(261,354)(262,355)(263,356)(264,357)(265,358)(266,359)(267,360)(268,361)(269,362)(270,363)(271,389)(272,390)(273,391)(274,392)(275,393)(276,394)(277,395)(278,396)(279,397)(280,398)(281,399)(282,400)(283,401)(284,402)(285,403)(286,404)(287,405)(288,379)(289,380)(290,381)(291,382)(292,383)(293,384)(294,385)(295,386)(296,387)(297,388)(298,428)(299,429)(300,430)(301,431)(302,432)(303,406)(304,407)(305,408)(306,409)(307,410)(308,411)(309,412)(310,413)(311,414)(312,415)(313,416)(314,417)(315,418)(316,419)(317,420)(318,421)(319,422)(320,423)(321,424)(322,425)(323,426)(324,427), (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), (1,416,106,356,35,393,67,348)(2,415,107,355,36,392,68,347)(3,414,108,354,37,391,69,346)(4,413,82,353,38,390,70,345)(5,412,83,352,39,389,71,344)(6,411,84,378,40,388,72,343)(7,410,85,377,41,387,73,342)(8,409,86,376,42,386,74,341)(9,408,87,375,43,385,75,340)(10,407,88,374,44,384,76,339)(11,406,89,373,45,383,77,338)(12,432,90,372,46,382,78,337)(13,431,91,371,47,381,79,336)(14,430,92,370,48,380,80,335)(15,429,93,369,49,379,81,334)(16,428,94,368,50,405,55,333)(17,427,95,367,51,404,56,332)(18,426,96,366,52,403,57,331)(19,425,97,365,53,402,58,330)(20,424,98,364,54,401,59,329)(21,423,99,363,28,400,60,328)(22,422,100,362,29,399,61,327)(23,421,101,361,30,398,62,326)(24,420,102,360,31,397,63,325)(25,419,103,359,32,396,64,351)(26,418,104,358,33,395,65,350)(27,417,105,357,34,394,66,349)(109,305,203,255,161,294,180,227)(110,304,204,254,162,293,181,226)(111,303,205,253,136,292,182,225)(112,302,206,252,137,291,183,224)(113,301,207,251,138,290,184,223)(114,300,208,250,139,289,185,222)(115,299,209,249,140,288,186,221)(116,298,210,248,141,287,187,220)(117,324,211,247,142,286,188,219)(118,323,212,246,143,285,189,218)(119,322,213,245,144,284,163,217)(120,321,214,244,145,283,164,243)(121,320,215,270,146,282,165,242)(122,319,216,269,147,281,166,241)(123,318,190,268,148,280,167,240)(124,317,191,267,149,279,168,239)(125,316,192,266,150,278,169,238)(126,315,193,265,151,277,170,237)(127,314,194,264,152,276,171,236)(128,313,195,263,153,275,172,235)(129,312,196,262,154,274,173,234)(130,311,197,261,155,273,174,233)(131,310,198,260,156,272,175,232)(132,309,199,259,157,271,176,231)(133,308,200,258,158,297,177,230)(134,307,201,257,159,296,178,229)(135,306,202,256,160,295,179,228)>;
G:=Group( (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,109)(10,110)(11,111)(12,112)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,144)(54,145)(55,187)(56,188)(57,189)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,183)(79,184)(80,185)(81,186)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,208)(93,209)(94,210)(95,211)(96,212)(97,213)(98,214)(99,215)(100,216)(101,190)(102,191)(103,192)(104,193)(105,194)(106,195)(107,196)(108,197)(217,330)(218,331)(219,332)(220,333)(221,334)(222,335)(223,336)(224,337)(225,338)(226,339)(227,340)(228,341)(229,342)(230,343)(231,344)(232,345)(233,346)(234,347)(235,348)(236,349)(237,350)(238,351)(239,325)(240,326)(241,327)(242,328)(243,329)(244,364)(245,365)(246,366)(247,367)(248,368)(249,369)(250,370)(251,371)(252,372)(253,373)(254,374)(255,375)(256,376)(257,377)(258,378)(259,352)(260,353)(261,354)(262,355)(263,356)(264,357)(265,358)(266,359)(267,360)(268,361)(269,362)(270,363)(271,389)(272,390)(273,391)(274,392)(275,393)(276,394)(277,395)(278,396)(279,397)(280,398)(281,399)(282,400)(283,401)(284,402)(285,403)(286,404)(287,405)(288,379)(289,380)(290,381)(291,382)(292,383)(293,384)(294,385)(295,386)(296,387)(297,388)(298,428)(299,429)(300,430)(301,431)(302,432)(303,406)(304,407)(305,408)(306,409)(307,410)(308,411)(309,412)(310,413)(311,414)(312,415)(313,416)(314,417)(315,418)(316,419)(317,420)(318,421)(319,422)(320,423)(321,424)(322,425)(323,426)(324,427), (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), (1,416,106,356,35,393,67,348)(2,415,107,355,36,392,68,347)(3,414,108,354,37,391,69,346)(4,413,82,353,38,390,70,345)(5,412,83,352,39,389,71,344)(6,411,84,378,40,388,72,343)(7,410,85,377,41,387,73,342)(8,409,86,376,42,386,74,341)(9,408,87,375,43,385,75,340)(10,407,88,374,44,384,76,339)(11,406,89,373,45,383,77,338)(12,432,90,372,46,382,78,337)(13,431,91,371,47,381,79,336)(14,430,92,370,48,380,80,335)(15,429,93,369,49,379,81,334)(16,428,94,368,50,405,55,333)(17,427,95,367,51,404,56,332)(18,426,96,366,52,403,57,331)(19,425,97,365,53,402,58,330)(20,424,98,364,54,401,59,329)(21,423,99,363,28,400,60,328)(22,422,100,362,29,399,61,327)(23,421,101,361,30,398,62,326)(24,420,102,360,31,397,63,325)(25,419,103,359,32,396,64,351)(26,418,104,358,33,395,65,350)(27,417,105,357,34,394,66,349)(109,305,203,255,161,294,180,227)(110,304,204,254,162,293,181,226)(111,303,205,253,136,292,182,225)(112,302,206,252,137,291,183,224)(113,301,207,251,138,290,184,223)(114,300,208,250,139,289,185,222)(115,299,209,249,140,288,186,221)(116,298,210,248,141,287,187,220)(117,324,211,247,142,286,188,219)(118,323,212,246,143,285,189,218)(119,322,213,245,144,284,163,217)(120,321,214,244,145,283,164,243)(121,320,215,270,146,282,165,242)(122,319,216,269,147,281,166,241)(123,318,190,268,148,280,167,240)(124,317,191,267,149,279,168,239)(125,316,192,266,150,278,169,238)(126,315,193,265,151,277,170,237)(127,314,194,264,152,276,171,236)(128,313,195,263,153,275,172,235)(129,312,196,262,154,274,173,234)(130,311,197,261,155,273,174,233)(131,310,198,260,156,272,175,232)(132,309,199,259,157,271,176,231)(133,308,200,258,158,297,177,230)(134,307,201,257,159,296,178,229)(135,306,202,256,160,295,179,228) );
G=PermutationGroup([[(1,128),(2,129),(3,130),(4,131),(5,132),(6,133),(7,134),(8,135),(9,109),(10,110),(11,111),(12,112),(13,113),(14,114),(15,115),(16,116),(17,117),(18,118),(19,119),(20,120),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,127),(28,146),(29,147),(30,148),(31,149),(32,150),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,157),(40,158),(41,159),(42,160),(43,161),(44,162),(45,136),(46,137),(47,138),(48,139),(49,140),(50,141),(51,142),(52,143),(53,144),(54,145),(55,187),(56,188),(57,189),(58,163),(59,164),(60,165),(61,166),(62,167),(63,168),(64,169),(65,170),(66,171),(67,172),(68,173),(69,174),(70,175),(71,176),(72,177),(73,178),(74,179),(75,180),(76,181),(77,182),(78,183),(79,184),(80,185),(81,186),(82,198),(83,199),(84,200),(85,201),(86,202),(87,203),(88,204),(89,205),(90,206),(91,207),(92,208),(93,209),(94,210),(95,211),(96,212),(97,213),(98,214),(99,215),(100,216),(101,190),(102,191),(103,192),(104,193),(105,194),(106,195),(107,196),(108,197),(217,330),(218,331),(219,332),(220,333),(221,334),(222,335),(223,336),(224,337),(225,338),(226,339),(227,340),(228,341),(229,342),(230,343),(231,344),(232,345),(233,346),(234,347),(235,348),(236,349),(237,350),(238,351),(239,325),(240,326),(241,327),(242,328),(243,329),(244,364),(245,365),(246,366),(247,367),(248,368),(249,369),(250,370),(251,371),(252,372),(253,373),(254,374),(255,375),(256,376),(257,377),(258,378),(259,352),(260,353),(261,354),(262,355),(263,356),(264,357),(265,358),(266,359),(267,360),(268,361),(269,362),(270,363),(271,389),(272,390),(273,391),(274,392),(275,393),(276,394),(277,395),(278,396),(279,397),(280,398),(281,399),(282,400),(283,401),(284,402),(285,403),(286,404),(287,405),(288,379),(289,380),(290,381),(291,382),(292,383),(293,384),(294,385),(295,386),(296,387),(297,388),(298,428),(299,429),(300,430),(301,431),(302,432),(303,406),(304,407),(305,408),(306,409),(307,410),(308,411),(309,412),(310,413),(311,414),(312,415),(313,416),(314,417),(315,418),(316,419),(317,420),(318,421),(319,422),(320,423),(321,424),(322,425),(323,426),(324,427)], [(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)], [(1,416,106,356,35,393,67,348),(2,415,107,355,36,392,68,347),(3,414,108,354,37,391,69,346),(4,413,82,353,38,390,70,345),(5,412,83,352,39,389,71,344),(6,411,84,378,40,388,72,343),(7,410,85,377,41,387,73,342),(8,409,86,376,42,386,74,341),(9,408,87,375,43,385,75,340),(10,407,88,374,44,384,76,339),(11,406,89,373,45,383,77,338),(12,432,90,372,46,382,78,337),(13,431,91,371,47,381,79,336),(14,430,92,370,48,380,80,335),(15,429,93,369,49,379,81,334),(16,428,94,368,50,405,55,333),(17,427,95,367,51,404,56,332),(18,426,96,366,52,403,57,331),(19,425,97,365,53,402,58,330),(20,424,98,364,54,401,59,329),(21,423,99,363,28,400,60,328),(22,422,100,362,29,399,61,327),(23,421,101,361,30,398,62,326),(24,420,102,360,31,397,63,325),(25,419,103,359,32,396,64,351),(26,418,104,358,33,395,65,350),(27,417,105,357,34,394,66,349),(109,305,203,255,161,294,180,227),(110,304,204,254,162,293,181,226),(111,303,205,253,136,292,182,225),(112,302,206,252,137,291,183,224),(113,301,207,251,138,290,184,223),(114,300,208,250,139,289,185,222),(115,299,209,249,140,288,186,221),(116,298,210,248,141,287,187,220),(117,324,211,247,142,286,188,219),(118,323,212,246,143,285,189,218),(119,322,213,245,144,284,163,217),(120,321,214,244,145,283,164,243),(121,320,215,270,146,282,165,242),(122,319,216,269,147,281,166,241),(123,318,190,268,148,280,167,240),(124,317,191,267,149,279,168,239),(125,316,192,266,150,278,169,238),(126,315,193,265,151,277,170,237),(127,314,194,264,152,276,171,236),(128,313,195,263,153,275,172,235),(129,312,196,262,154,274,173,234),(130,311,197,261,155,273,174,233),(131,310,198,260,156,272,175,232),(132,309,199,259,157,271,176,231),(133,308,200,258,158,297,177,230),(134,307,201,257,159,296,178,229),(135,306,202,256,160,295,179,228)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | ··· | 8H | 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 | 6 | 6 | 6 | 8 | ··· | 8 | 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 | 2 | 2 | 2 | 27 | ··· | 27 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | Dic3 | D6 | Dic3 | D9 | C3⋊C8 | Dic9 | D18 | Dic9 | D27 | C9⋊C8 | Dic27 | D54 | Dic27 | C27⋊C8 |
| kernel | C2×C27⋊C8 | C27⋊C8 | C2×C108 | C108 | C2×C54 | C54 | C2×C36 | C36 | C36 | C2×C18 | C2×C12 | C18 | C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 9 | 12 | 9 | 9 | 9 | 36 |
Matrix representation of C2×C27⋊C8 ►in GL3(𝔽433) generated by
| 432 | 0 | 0 |
| 0 | 432 | 0 |
| 0 | 0 | 432 |
| 1 | 0 | 0 |
| 0 | 238 | 408 |
| 0 | 25 | 263 |
| 432 | 0 | 0 |
| 0 | 2 | 95 |
| 0 | 93 | 431 |
G:=sub<GL(3,GF(433))| [432,0,0,0,432,0,0,0,432],[1,0,0,0,238,25,0,408,263],[432,0,0,0,2,93,0,95,431] >;
C2×C27⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_{27}\rtimes C_8 % in TeX
G:=Group("C2xC27:C8"); // GroupNames label
G:=SmallGroup(432,9);
// by ID
G=gap.SmallGroup(432,9);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^2=b^27=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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