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## G = C2×C27⋊C8order 432 = 24·33

### Direct product of C2 and C27⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C2×C27⋊C8
 Chief series C1 — C3 — C9 — C27 — C54 — C108 — C27⋊C8 — C2×C27⋊C8
 Lower central C27 — C2×C27⋊C8
 Upper central C1 — C2×C4

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

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

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

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

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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