direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C27, C4⋊C108, C108⋊3C4, C54.3Q8, C12.4C36, C54.13D4, C36.10C12, C2.(Q8×C27), C6.3(Q8×C9), (C2×C4).1C54, C2.2(D4×C27), C6.13(D4×C9), C18.6(C3×Q8), (C2×C12).8C18, C54.11(C2×C4), C2.2(C2×C108), C6.11(C2×C36), (C2×C108).2C2, (C2×C36).13C6, C18.29(C3×D4), C18.25(C2×C12), C22.3(C2×C54), (C2×C54).14C22, C9.(C3×C4⋊C4), C3.(C9×C4⋊C4), (C3×C4⋊C4).C9, (C9×C4⋊C4).C3, (C2×C6).17(C2×C18), (C2×C18).34(C2×C6), SmallGroup(432,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×C27
G = < a,b,c | a27=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4⋊C4×C27
►Regular action on 432 points
Generators in S432
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)(163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189)(190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243)(244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297)(298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324)(325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351)(352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378)(379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405)(406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)
(1 157 340 417)(2 158 341 418)(3 159 342 419)(4 160 343 420)(5 161 344 421)(6 162 345 422)(7 136 346 423)(8 137 347 424)(9 138 348 425)(10 139 349 426)(11 140 350 427)(12 141 351 428)(13 142 325 429)(14 143 326 430)(15 144 327 431)(16 145 328 432)(17 146 329 406)(18 147 330 407)(19 148 331 408)(20 149 332 409)(21 150 333 410)(22 151 334 411)(23 152 335 412)(24 153 336 413)(25 154 337 414)(26 155 338 415)(27 156 339 416)(28 122 237 91)(29 123 238 92)(30 124 239 93)(31 125 240 94)(32 126 241 95)(33 127 242 96)(34 128 243 97)(35 129 217 98)(36 130 218 99)(37 131 219 100)(38 132 220 101)(39 133 221 102)(40 134 222 103)(41 135 223 104)(42 109 224 105)(43 110 225 106)(44 111 226 107)(45 112 227 108)(46 113 228 82)(47 114 229 83)(48 115 230 84)(49 116 231 85)(50 117 232 86)(51 118 233 87)(52 119 234 88)(53 120 235 89)(54 121 236 90)(55 247 170 274)(56 248 171 275)(57 249 172 276)(58 250 173 277)(59 251 174 278)(60 252 175 279)(61 253 176 280)(62 254 177 281)(63 255 178 282)(64 256 179 283)(65 257 180 284)(66 258 181 285)(67 259 182 286)(68 260 183 287)(69 261 184 288)(70 262 185 289)(71 263 186 290)(72 264 187 291)(73 265 188 292)(74 266 189 293)(75 267 163 294)(76 268 164 295)(77 269 165 296)(78 270 166 297)(79 244 167 271)(80 245 168 272)(81 246 169 273)(190 317 369 386)(191 318 370 387)(192 319 371 388)(193 320 372 389)(194 321 373 390)(195 322 374 391)(196 323 375 392)(197 324 376 393)(198 298 377 394)(199 299 378 395)(200 300 352 396)(201 301 353 397)(202 302 354 398)(203 303 355 399)(204 304 356 400)(205 305 357 401)(206 306 358 402)(207 307 359 403)(208 308 360 404)(209 309 361 405)(210 310 362 379)(211 311 363 380)(212 312 364 381)(213 313 365 382)(214 314 366 383)(215 315 367 384)(216 316 368 385)
(1 93 274 364)(2 94 275 365)(3 95 276 366)(4 96 277 367)(5 97 278 368)(6 98 279 369)(7 99 280 370)(8 100 281 371)(9 101 282 372)(10 102 283 373)(11 103 284 374)(12 104 285 375)(13 105 286 376)(14 106 287 377)(15 107 288 378)(16 108 289 352)(17 82 290 353)(18 83 291 354)(19 84 292 355)(20 85 293 356)(21 86 294 357)(22 87 295 358)(23 88 296 359)(24 89 297 360)(25 90 271 361)(26 91 272 362)(27 92 273 363)(28 168 379 415)(29 169 380 416)(30 170 381 417)(31 171 382 418)(32 172 383 419)(33 173 384 420)(34 174 385 421)(35 175 386 422)(36 176 387 423)(37 177 388 424)(38 178 389 425)(39 179 390 426)(40 180 391 427)(41 181 392 428)(42 182 393 429)(43 183 394 430)(44 184 395 431)(45 185 396 432)(46 186 397 406)(47 187 398 407)(48 188 399 408)(49 189 400 409)(50 163 401 410)(51 164 402 411)(52 165 403 412)(53 166 404 413)(54 167 405 414)(55 312 157 239)(56 313 158 240)(57 314 159 241)(58 315 160 242)(59 316 161 243)(60 317 162 217)(61 318 136 218)(62 319 137 219)(63 320 138 220)(64 321 139 221)(65 322 140 222)(66 323 141 223)(67 324 142 224)(68 298 143 225)(69 299 144 226)(70 300 145 227)(71 301 146 228)(72 302 147 229)(73 303 148 230)(74 304 149 231)(75 305 150 232)(76 306 151 233)(77 307 152 234)(78 308 153 235)(79 309 154 236)(80 310 155 237)(81 311 156 238)(109 259 197 325)(110 260 198 326)(111 261 199 327)(112 262 200 328)(113 263 201 329)(114 264 202 330)(115 265 203 331)(116 266 204 332)(117 267 205 333)(118 268 206 334)(119 269 207 335)(120 270 208 336)(121 244 209 337)(122 245 210 338)(123 246 211 339)(124 247 212 340)(125 248 213 341)(126 249 214 342)(127 250 215 343)(128 251 216 344)(129 252 190 345)(130 253 191 346)(131 254 192 347)(132 255 193 348)(133 256 194 349)(134 257 195 350)(135 258 196 351)
G:=sub<Sym(432)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189)(190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243)(244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297)(298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324)(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351)(352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405)(406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432), (1,157,340,417)(2,158,341,418)(3,159,342,419)(4,160,343,420)(5,161,344,421)(6,162,345,422)(7,136,346,423)(8,137,347,424)(9,138,348,425)(10,139,349,426)(11,140,350,427)(12,141,351,428)(13,142,325,429)(14,143,326,430)(15,144,327,431)(16,145,328,432)(17,146,329,406)(18,147,330,407)(19,148,331,408)(20,149,332,409)(21,150,333,410)(22,151,334,411)(23,152,335,412)(24,153,336,413)(25,154,337,414)(26,155,338,415)(27,156,339,416)(28,122,237,91)(29,123,238,92)(30,124,239,93)(31,125,240,94)(32,126,241,95)(33,127,242,96)(34,128,243,97)(35,129,217,98)(36,130,218,99)(37,131,219,100)(38,132,220,101)(39,133,221,102)(40,134,222,103)(41,135,223,104)(42,109,224,105)(43,110,225,106)(44,111,226,107)(45,112,227,108)(46,113,228,82)(47,114,229,83)(48,115,230,84)(49,116,231,85)(50,117,232,86)(51,118,233,87)(52,119,234,88)(53,120,235,89)(54,121,236,90)(55,247,170,274)(56,248,171,275)(57,249,172,276)(58,250,173,277)(59,251,174,278)(60,252,175,279)(61,253,176,280)(62,254,177,281)(63,255,178,282)(64,256,179,283)(65,257,180,284)(66,258,181,285)(67,259,182,286)(68,260,183,287)(69,261,184,288)(70,262,185,289)(71,263,186,290)(72,264,187,291)(73,265,188,292)(74,266,189,293)(75,267,163,294)(76,268,164,295)(77,269,165,296)(78,270,166,297)(79,244,167,271)(80,245,168,272)(81,246,169,273)(190,317,369,386)(191,318,370,387)(192,319,371,388)(193,320,372,389)(194,321,373,390)(195,322,374,391)(196,323,375,392)(197,324,376,393)(198,298,377,394)(199,299,378,395)(200,300,352,396)(201,301,353,397)(202,302,354,398)(203,303,355,399)(204,304,356,400)(205,305,357,401)(206,306,358,402)(207,307,359,403)(208,308,360,404)(209,309,361,405)(210,310,362,379)(211,311,363,380)(212,312,364,381)(213,313,365,382)(214,314,366,383)(215,315,367,384)(216,316,368,385), (1,93,274,364)(2,94,275,365)(3,95,276,366)(4,96,277,367)(5,97,278,368)(6,98,279,369)(7,99,280,370)(8,100,281,371)(9,101,282,372)(10,102,283,373)(11,103,284,374)(12,104,285,375)(13,105,286,376)(14,106,287,377)(15,107,288,378)(16,108,289,352)(17,82,290,353)(18,83,291,354)(19,84,292,355)(20,85,293,356)(21,86,294,357)(22,87,295,358)(23,88,296,359)(24,89,297,360)(25,90,271,361)(26,91,272,362)(27,92,273,363)(28,168,379,415)(29,169,380,416)(30,170,381,417)(31,171,382,418)(32,172,383,419)(33,173,384,420)(34,174,385,421)(35,175,386,422)(36,176,387,423)(37,177,388,424)(38,178,389,425)(39,179,390,426)(40,180,391,427)(41,181,392,428)(42,182,393,429)(43,183,394,430)(44,184,395,431)(45,185,396,432)(46,186,397,406)(47,187,398,407)(48,188,399,408)(49,189,400,409)(50,163,401,410)(51,164,402,411)(52,165,403,412)(53,166,404,413)(54,167,405,414)(55,312,157,239)(56,313,158,240)(57,314,159,241)(58,315,160,242)(59,316,161,243)(60,317,162,217)(61,318,136,218)(62,319,137,219)(63,320,138,220)(64,321,139,221)(65,322,140,222)(66,323,141,223)(67,324,142,224)(68,298,143,225)(69,299,144,226)(70,300,145,227)(71,301,146,228)(72,302,147,229)(73,303,148,230)(74,304,149,231)(75,305,150,232)(76,306,151,233)(77,307,152,234)(78,308,153,235)(79,309,154,236)(80,310,155,237)(81,311,156,238)(109,259,197,325)(110,260,198,326)(111,261,199,327)(112,262,200,328)(113,263,201,329)(114,264,202,330)(115,265,203,331)(116,266,204,332)(117,267,205,333)(118,268,206,334)(119,269,207,335)(120,270,208,336)(121,244,209,337)(122,245,210,338)(123,246,211,339)(124,247,212,340)(125,248,213,341)(126,249,214,342)(127,250,215,343)(128,251,216,344)(129,252,190,345)(130,253,191,346)(131,254,192,347)(132,255,193,348)(133,256,194,349)(134,257,195,350)(135,258,196,351)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189)(190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243)(244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297)(298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324)(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351)(352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405)(406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432), (1,157,340,417)(2,158,341,418)(3,159,342,419)(4,160,343,420)(5,161,344,421)(6,162,345,422)(7,136,346,423)(8,137,347,424)(9,138,348,425)(10,139,349,426)(11,140,350,427)(12,141,351,428)(13,142,325,429)(14,143,326,430)(15,144,327,431)(16,145,328,432)(17,146,329,406)(18,147,330,407)(19,148,331,408)(20,149,332,409)(21,150,333,410)(22,151,334,411)(23,152,335,412)(24,153,336,413)(25,154,337,414)(26,155,338,415)(27,156,339,416)(28,122,237,91)(29,123,238,92)(30,124,239,93)(31,125,240,94)(32,126,241,95)(33,127,242,96)(34,128,243,97)(35,129,217,98)(36,130,218,99)(37,131,219,100)(38,132,220,101)(39,133,221,102)(40,134,222,103)(41,135,223,104)(42,109,224,105)(43,110,225,106)(44,111,226,107)(45,112,227,108)(46,113,228,82)(47,114,229,83)(48,115,230,84)(49,116,231,85)(50,117,232,86)(51,118,233,87)(52,119,234,88)(53,120,235,89)(54,121,236,90)(55,247,170,274)(56,248,171,275)(57,249,172,276)(58,250,173,277)(59,251,174,278)(60,252,175,279)(61,253,176,280)(62,254,177,281)(63,255,178,282)(64,256,179,283)(65,257,180,284)(66,258,181,285)(67,259,182,286)(68,260,183,287)(69,261,184,288)(70,262,185,289)(71,263,186,290)(72,264,187,291)(73,265,188,292)(74,266,189,293)(75,267,163,294)(76,268,164,295)(77,269,165,296)(78,270,166,297)(79,244,167,271)(80,245,168,272)(81,246,169,273)(190,317,369,386)(191,318,370,387)(192,319,371,388)(193,320,372,389)(194,321,373,390)(195,322,374,391)(196,323,375,392)(197,324,376,393)(198,298,377,394)(199,299,378,395)(200,300,352,396)(201,301,353,397)(202,302,354,398)(203,303,355,399)(204,304,356,400)(205,305,357,401)(206,306,358,402)(207,307,359,403)(208,308,360,404)(209,309,361,405)(210,310,362,379)(211,311,363,380)(212,312,364,381)(213,313,365,382)(214,314,366,383)(215,315,367,384)(216,316,368,385), (1,93,274,364)(2,94,275,365)(3,95,276,366)(4,96,277,367)(5,97,278,368)(6,98,279,369)(7,99,280,370)(8,100,281,371)(9,101,282,372)(10,102,283,373)(11,103,284,374)(12,104,285,375)(13,105,286,376)(14,106,287,377)(15,107,288,378)(16,108,289,352)(17,82,290,353)(18,83,291,354)(19,84,292,355)(20,85,293,356)(21,86,294,357)(22,87,295,358)(23,88,296,359)(24,89,297,360)(25,90,271,361)(26,91,272,362)(27,92,273,363)(28,168,379,415)(29,169,380,416)(30,170,381,417)(31,171,382,418)(32,172,383,419)(33,173,384,420)(34,174,385,421)(35,175,386,422)(36,176,387,423)(37,177,388,424)(38,178,389,425)(39,179,390,426)(40,180,391,427)(41,181,392,428)(42,182,393,429)(43,183,394,430)(44,184,395,431)(45,185,396,432)(46,186,397,406)(47,187,398,407)(48,188,399,408)(49,189,400,409)(50,163,401,410)(51,164,402,411)(52,165,403,412)(53,166,404,413)(54,167,405,414)(55,312,157,239)(56,313,158,240)(57,314,159,241)(58,315,160,242)(59,316,161,243)(60,317,162,217)(61,318,136,218)(62,319,137,219)(63,320,138,220)(64,321,139,221)(65,322,140,222)(66,323,141,223)(67,324,142,224)(68,298,143,225)(69,299,144,226)(70,300,145,227)(71,301,146,228)(72,302,147,229)(73,303,148,230)(74,304,149,231)(75,305,150,232)(76,306,151,233)(77,307,152,234)(78,308,153,235)(79,309,154,236)(80,310,155,237)(81,311,156,238)(109,259,197,325)(110,260,198,326)(111,261,199,327)(112,262,200,328)(113,263,201,329)(114,264,202,330)(115,265,203,331)(116,266,204,332)(117,267,205,333)(118,268,206,334)(119,269,207,335)(120,270,208,336)(121,244,209,337)(122,245,210,338)(123,246,211,339)(124,247,212,340)(125,248,213,341)(126,249,214,342)(127,250,215,343)(128,251,216,344)(129,252,190,345)(130,253,191,346)(131,254,192,347)(132,255,193,348)(133,256,194,349)(134,257,195,350)(135,258,196,351) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162),(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189),(190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243),(244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270),(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297),(298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324),(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351),(352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378),(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405),(406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)], [(1,157,340,417),(2,158,341,418),(3,159,342,419),(4,160,343,420),(5,161,344,421),(6,162,345,422),(7,136,346,423),(8,137,347,424),(9,138,348,425),(10,139,349,426),(11,140,350,427),(12,141,351,428),(13,142,325,429),(14,143,326,430),(15,144,327,431),(16,145,328,432),(17,146,329,406),(18,147,330,407),(19,148,331,408),(20,149,332,409),(21,150,333,410),(22,151,334,411),(23,152,335,412),(24,153,336,413),(25,154,337,414),(26,155,338,415),(27,156,339,416),(28,122,237,91),(29,123,238,92),(30,124,239,93),(31,125,240,94),(32,126,241,95),(33,127,242,96),(34,128,243,97),(35,129,217,98),(36,130,218,99),(37,131,219,100),(38,132,220,101),(39,133,221,102),(40,134,222,103),(41,135,223,104),(42,109,224,105),(43,110,225,106),(44,111,226,107),(45,112,227,108),(46,113,228,82),(47,114,229,83),(48,115,230,84),(49,116,231,85),(50,117,232,86),(51,118,233,87),(52,119,234,88),(53,120,235,89),(54,121,236,90),(55,247,170,274),(56,248,171,275),(57,249,172,276),(58,250,173,277),(59,251,174,278),(60,252,175,279),(61,253,176,280),(62,254,177,281),(63,255,178,282),(64,256,179,283),(65,257,180,284),(66,258,181,285),(67,259,182,286),(68,260,183,287),(69,261,184,288),(70,262,185,289),(71,263,186,290),(72,264,187,291),(73,265,188,292),(74,266,189,293),(75,267,163,294),(76,268,164,295),(77,269,165,296),(78,270,166,297),(79,244,167,271),(80,245,168,272),(81,246,169,273),(190,317,369,386),(191,318,370,387),(192,319,371,388),(193,320,372,389),(194,321,373,390),(195,322,374,391),(196,323,375,392),(197,324,376,393),(198,298,377,394),(199,299,378,395),(200,300,352,396),(201,301,353,397),(202,302,354,398),(203,303,355,399),(204,304,356,400),(205,305,357,401),(206,306,358,402),(207,307,359,403),(208,308,360,404),(209,309,361,405),(210,310,362,379),(211,311,363,380),(212,312,364,381),(213,313,365,382),(214,314,366,383),(215,315,367,384),(216,316,368,385)], [(1,93,274,364),(2,94,275,365),(3,95,276,366),(4,96,277,367),(5,97,278,368),(6,98,279,369),(7,99,280,370),(8,100,281,371),(9,101,282,372),(10,102,283,373),(11,103,284,374),(12,104,285,375),(13,105,286,376),(14,106,287,377),(15,107,288,378),(16,108,289,352),(17,82,290,353),(18,83,291,354),(19,84,292,355),(20,85,293,356),(21,86,294,357),(22,87,295,358),(23,88,296,359),(24,89,297,360),(25,90,271,361),(26,91,272,362),(27,92,273,363),(28,168,379,415),(29,169,380,416),(30,170,381,417),(31,171,382,418),(32,172,383,419),(33,173,384,420),(34,174,385,421),(35,175,386,422),(36,176,387,423),(37,177,388,424),(38,178,389,425),(39,179,390,426),(40,180,391,427),(41,181,392,428),(42,182,393,429),(43,183,394,430),(44,184,395,431),(45,185,396,432),(46,186,397,406),(47,187,398,407),(48,188,399,408),(49,189,400,409),(50,163,401,410),(51,164,402,411),(52,165,403,412),(53,166,404,413),(54,167,405,414),(55,312,157,239),(56,313,158,240),(57,314,159,241),(58,315,160,242),(59,316,161,243),(60,317,162,217),(61,318,136,218),(62,319,137,219),(63,320,138,220),(64,321,139,221),(65,322,140,222),(66,323,141,223),(67,324,142,224),(68,298,143,225),(69,299,144,226),(70,300,145,227),(71,301,146,228),(72,302,147,229),(73,303,148,230),(74,304,149,231),(75,305,150,232),(76,306,151,233),(77,307,152,234),(78,308,153,235),(79,309,154,236),(80,310,155,237),(81,311,156,238),(109,259,197,325),(110,260,198,326),(111,261,199,327),(112,262,200,328),(113,263,201,329),(114,264,202,330),(115,265,203,331),(116,266,204,332),(117,267,205,333),(118,268,206,334),(119,269,207,335),(120,270,208,336),(121,244,209,337),(122,245,210,338),(123,246,211,339),(124,247,212,340),(125,248,213,341),(126,249,214,342),(127,250,215,343),(128,251,216,344),(129,252,190,345),(130,253,191,346),(131,254,192,347),(132,255,193,348),(133,256,194,349),(134,257,195,350),(135,258,196,351)]])
270 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 6A | ··· | 6F | 9A | ··· | 9F | 12A | ··· | 12L | 18A | ··· | 18R | 27A | ··· | 27R | 36A | ··· | 36AJ | 54A | ··· | 54BB | 108A | ··· | 108DD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
270 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C27 | C36 | C54 | C108 | D4 | Q8 | C3×D4 | C3×Q8 | D4×C9 | Q8×C9 | D4×C27 | Q8×C27 |
| kernel | C4⋊C4×C27 | C2×C108 | C9×C4⋊C4 | C108 | C2×C36 | C3×C4⋊C4 | C36 | C2×C12 | C4⋊C4 | C12 | C2×C4 | C4 | C54 | C54 | C18 | C18 | C6 | C6 | C2 | C2 |
| # reps | 1 | 3 | 2 | 4 | 6 | 6 | 8 | 18 | 18 | 24 | 54 | 72 | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 |
Matrix representation of C4⋊C4×C27 ►in GL4(𝔽109) generated by
| 80 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 75 | 0 |
| 0 | 0 | 0 | 75 |
| 108 | 0 | 0 | 0 |
| 0 | 108 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 108 | 108 |
| 108 | 0 | 0 | 0 |
| 0 | 33 | 0 | 0 |
| 0 | 0 | 38 | 13 |
| 0 | 0 | 23 | 71 |
G:=sub<GL(4,GF(109))| [80,0,0,0,0,1,0,0,0,0,75,0,0,0,0,75],[108,0,0,0,0,108,0,0,0,0,1,108,0,0,2,108],[108,0,0,0,0,33,0,0,0,0,38,23,0,0,13,71] >;
C4⋊C4×C27 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_{27} % in TeX
G:=Group("C4:C4xC27"); // GroupNames label
G:=SmallGroup(432,22);
// by ID
G=gap.SmallGroup(432,22);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,92,268,166]);
// Polycyclic
G:=Group<a,b,c|a^27=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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