Copied to
clipboard

## G = C4×Dic27order 432 = 24·33

### Direct product of C4 and Dic27

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C4×Dic27
 Chief series C1 — C3 — C9 — C27 — C54 — C2×C54 — C2×Dic27 — C4×Dic27
 Lower central C27 — C4×Dic27
 Upper central C1 — C2×C4

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 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], C9, Dic3 [×4], C12 [×2], C2×C6, C42, C18, C18 [×2], C2×Dic3 [×2], C2×C12, C27, Dic9 [×4], C36 [×2], C2×C18, C4×Dic3, C54, C54 [×2], C2×Dic9 [×2], C2×C36, Dic27 [×4], C108 [×2], C2×C54, C4×Dic9, C2×Dic27 [×2], C2×C108, C4×Dic27
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], Dic3 [×2], D6, C42, D9, C4×S3 [×2], C2×Dic3, Dic9 [×2], D18, C4×Dic3, D27, C4×D9 [×2], C2×Dic9, Dic27 [×2], D54, C4×Dic9, C4×D27 [×2], C2×Dic27, C4×Dic27

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

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

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

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

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

׿
×
𝔽