Copied to
clipboard

## G = C22×Dic27order 432 = 24·33

### Direct product of C22 and Dic27

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C22×Dic27
 Chief series C1 — C3 — C9 — C27 — C54 — Dic27 — C2×Dic27 — C22×Dic27
 Lower central C27 — C22×Dic27
 Upper central C1 — C23

Generators and relations for C22×Dic27
G = < a,b,c,d | a2=b2=c54=1, d2=c27, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 504 in 108 conjugacy classes, 75 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, C23, C9, Dic3, C2×C6, C22×C4, C18, C18, C2×Dic3, C22×C6, C27, Dic9, C2×C18, C22×Dic3, C54, C54, C2×Dic9, C22×C18, Dic27, C2×C54, C22×Dic9, C2×Dic27, C22×C54, C22×Dic27
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, D9, C2×Dic3, C22×S3, Dic9, D18, C22×Dic3, D27, C2×Dic9, C22×D9, Dic27, D54, C22×Dic9, C2×Dic27, C22×D27, C22×Dic27

Smallest permutation representation of C22×Dic27
Regular action on 432 points
Generators in S432
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 201)(56 202)(57 203)(58 204)(59 205)(60 206)(61 207)(62 208)(63 209)(64 210)(65 211)(66 212)(67 213)(68 214)(69 215)(70 216)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 169)(78 170)(79 171)(80 172)(81 173)(82 174)(83 175)(84 176)(85 177)(86 178)(87 179)(88 180)(89 181)(90 182)(91 183)(92 184)(93 185)(94 186)(95 187)(96 188)(97 189)(98 190)(99 191)(100 192)(101 193)(102 194)(103 195)(104 196)(105 197)(106 198)(107 199)(108 200)(217 343)(218 344)(219 345)(220 346)(221 347)(222 348)(223 349)(224 350)(225 351)(226 352)(227 353)(228 354)(229 355)(230 356)(231 357)(232 358)(233 359)(234 360)(235 361)(236 362)(237 363)(238 364)(239 365)(240 366)(241 367)(242 368)(243 369)(244 370)(245 371)(246 372)(247 373)(248 374)(249 375)(250 376)(251 377)(252 378)(253 325)(254 326)(255 327)(256 328)(257 329)(258 330)(259 331)(260 332)(261 333)(262 334)(263 335)(264 336)(265 337)(266 338)(267 339)(268 340)(269 341)(270 342)(271 406)(272 407)(273 408)(274 409)(275 410)(276 411)(277 412)(278 413)(279 414)(280 415)(281 416)(282 417)(283 418)(284 419)(285 420)(286 421)(287 422)(288 423)(289 424)(290 425)(291 426)(292 427)(293 428)(294 429)(295 430)(296 431)(297 432)(298 379)(299 380)(300 381)(301 382)(302 383)(303 384)(304 385)(305 386)(306 387)(307 388)(308 389)(309 390)(310 391)(311 392)(312 393)(313 394)(314 395)(315 396)(316 397)(317 398)(318 399)(319 400)(320 401)(321 402)(322 403)(323 404)(324 405)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(109 210)(110 211)(111 212)(112 213)(113 214)(114 215)(115 216)(116 163)(117 164)(118 165)(119 166)(120 167)(121 168)(122 169)(123 170)(124 171)(125 172)(126 173)(127 174)(128 175)(129 176)(130 177)(131 178)(132 179)(133 180)(134 181)(135 182)(136 183)(137 184)(138 185)(139 186)(140 187)(141 188)(142 189)(143 190)(144 191)(145 192)(146 193)(147 194)(148 195)(149 196)(150 197)(151 198)(152 199)(153 200)(154 201)(155 202)(156 203)(157 204)(158 205)(159 206)(160 207)(161 208)(162 209)(217 275)(218 276)(219 277)(220 278)(221 279)(222 280)(223 281)(224 282)(225 283)(226 284)(227 285)(228 286)(229 287)(230 288)(231 289)(232 290)(233 291)(234 292)(235 293)(236 294)(237 295)(238 296)(239 297)(240 298)(241 299)(242 300)(243 301)(244 302)(245 303)(246 304)(247 305)(248 306)(249 307)(250 308)(251 309)(252 310)(253 311)(254 312)(255 313)(256 314)(257 315)(258 316)(259 317)(260 318)(261 319)(262 320)(263 321)(264 322)(265 323)(266 324)(267 271)(268 272)(269 273)(270 274)(325 392)(326 393)(327 394)(328 395)(329 396)(330 397)(331 398)(332 399)(333 400)(334 401)(335 402)(336 403)(337 404)(338 405)(339 406)(340 407)(341 408)(342 409)(343 410)(344 411)(345 412)(346 413)(347 414)(348 415)(349 416)(350 417)(351 418)(352 419)(353 420)(354 421)(355 422)(356 423)(357 424)(358 425)(359 426)(360 427)(361 428)(362 429)(363 430)(364 431)(365 432)(366 379)(367 380)(368 381)(369 382)(370 383)(371 384)(372 385)(373 386)(374 387)(375 388)(376 389)(377 390)(378 391)
(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)
(1 316 28 289)(2 315 29 288)(3 314 30 287)(4 313 31 286)(5 312 32 285)(6 311 33 284)(7 310 34 283)(8 309 35 282)(9 308 36 281)(10 307 37 280)(11 306 38 279)(12 305 39 278)(13 304 40 277)(14 303 41 276)(15 302 42 275)(16 301 43 274)(17 300 44 273)(18 299 45 272)(19 298 46 271)(20 297 47 324)(21 296 48 323)(22 295 49 322)(23 294 50 321)(24 293 51 320)(25 292 52 319)(26 291 53 318)(27 290 54 317)(55 245 82 218)(56 244 83 217)(57 243 84 270)(58 242 85 269)(59 241 86 268)(60 240 87 267)(61 239 88 266)(62 238 89 265)(63 237 90 264)(64 236 91 263)(65 235 92 262)(66 234 93 261)(67 233 94 260)(68 232 95 259)(69 231 96 258)(70 230 97 257)(71 229 98 256)(72 228 99 255)(73 227 100 254)(74 226 101 253)(75 225 102 252)(76 224 103 251)(77 223 104 250)(78 222 105 249)(79 221 106 248)(80 220 107 247)(81 219 108 246)(109 429 136 402)(110 428 137 401)(111 427 138 400)(112 426 139 399)(113 425 140 398)(114 424 141 397)(115 423 142 396)(116 422 143 395)(117 421 144 394)(118 420 145 393)(119 419 146 392)(120 418 147 391)(121 417 148 390)(122 416 149 389)(123 415 150 388)(124 414 151 387)(125 413 152 386)(126 412 153 385)(127 411 154 384)(128 410 155 383)(129 409 156 382)(130 408 157 381)(131 407 158 380)(132 406 159 379)(133 405 160 432)(134 404 161 431)(135 403 162 430)(163 355 190 328)(164 354 191 327)(165 353 192 326)(166 352 193 325)(167 351 194 378)(168 350 195 377)(169 349 196 376)(170 348 197 375)(171 347 198 374)(172 346 199 373)(173 345 200 372)(174 344 201 371)(175 343 202 370)(176 342 203 369)(177 341 204 368)(178 340 205 367)(179 339 206 366)(180 338 207 365)(181 337 208 364)(182 336 209 363)(183 335 210 362)(184 334 211 361)(185 333 212 360)(186 332 213 359)(187 331 214 358)(188 330 215 357)(189 329 216 356)

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

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

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

120 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 6A ··· 6G 9A 9B 9C 18A ··· 18U 27A ··· 27I 54A ··· 54BK order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 9 9 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 ··· 1 2 27 ··· 27 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + + - + + - + image C1 C2 C2 C4 S3 Dic3 D6 D9 Dic9 D18 D27 Dic27 D54 kernel C22×Dic27 C2×Dic27 C22×C54 C2×C54 C22×C18 C2×C18 C2×C18 C22×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 8 1 4 3 3 12 9 9 36 27

Matrix representation of C22×Dic27 in GL4(𝔽109) generated by

 1 0 0 0 0 108 0 0 0 0 108 0 0 0 0 108
,
 108 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 80 58 0 0 51 22
,
 108 0 0 0 0 1 0 0 0 0 29 108 0 0 79 80
G:=sub<GL(4,GF(109))| [1,0,0,0,0,108,0,0,0,0,108,0,0,0,0,108],[108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,80,51,0,0,58,22],[108,0,0,0,0,1,0,0,0,0,29,79,0,0,108,80] >;

C22×Dic27 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{27}
% in TeX

G:=Group("C2^2xDic27");
// GroupNames label

G:=SmallGroup(432,51);
// by ID

G=gap.SmallGroup(432,51);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^54=1,d^2=c^27,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽