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## G = C23×C54order 432 = 24·33

### Abelian group of type [2,2,2,54]

Aliases: C23×C54, SmallGroup(432,228)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C23×C54
G = < a,b,c,d | a2=b2=c2=d54=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 268, all normal (8 characteristic)
C1, C2, C3, C22, C6, C23, C9, C2×C6, C24, C18, C22×C6, C27, C2×C18, C23×C6, C54, C22×C18, C2×C54, C23×C18, C22×C54, C23×C54
Quotients: C1, C2, C3, C22, C6, C23, C9, C2×C6, C24, C18, C22×C6, C27, C2×C18, C23×C6, C54, C22×C18, C2×C54, C23×C18, C22×C54, C23×C54

Smallest permutation representation of C23×C54
Regular action on 432 points
Generators in S432
(1 232)(2 233)(3 234)(4 235)(5 236)(6 237)(7 238)(8 239)(9 240)(10 241)(11 242)(12 243)(13 244)(14 245)(15 246)(16 247)(17 248)(18 249)(19 250)(20 251)(21 252)(22 253)(23 254)(24 255)(25 256)(26 257)(27 258)(28 259)(29 260)(30 261)(31 262)(32 263)(33 264)(34 265)(35 266)(36 267)(37 268)(38 269)(39 270)(40 217)(41 218)(42 219)(43 220)(44 221)(45 222)(46 223)(47 224)(48 225)(49 226)(50 227)(51 228)(52 229)(53 230)(54 231)(55 275)(56 276)(57 277)(58 278)(59 279)(60 280)(61 281)(62 282)(63 283)(64 284)(65 285)(66 286)(67 287)(68 288)(69 289)(70 290)(71 291)(72 292)(73 293)(74 294)(75 295)(76 296)(77 297)(78 298)(79 299)(80 300)(81 301)(82 302)(83 303)(84 304)(85 305)(86 306)(87 307)(88 308)(89 309)(90 310)(91 311)(92 312)(93 313)(94 314)(95 315)(96 316)(97 317)(98 318)(99 319)(100 320)(101 321)(102 322)(103 323)(104 324)(105 271)(106 272)(107 273)(108 274)(109 345)(110 346)(111 347)(112 348)(113 349)(114 350)(115 351)(116 352)(117 353)(118 354)(119 355)(120 356)(121 357)(122 358)(123 359)(124 360)(125 361)(126 362)(127 363)(128 364)(129 365)(130 366)(131 367)(132 368)(133 369)(134 370)(135 371)(136 372)(137 373)(138 374)(139 375)(140 376)(141 377)(142 378)(143 325)(144 326)(145 327)(146 328)(147 329)(148 330)(149 331)(150 332)(151 333)(152 334)(153 335)(154 336)(155 337)(156 338)(157 339)(158 340)(159 341)(160 342)(161 343)(162 344)(163 403)(164 404)(165 405)(166 406)(167 407)(168 408)(169 409)(170 410)(171 411)(172 412)(173 413)(174 414)(175 415)(176 416)(177 417)(178 418)(179 419)(180 420)(181 421)(182 422)(183 423)(184 424)(185 425)(186 426)(187 427)(188 428)(189 429)(190 430)(191 431)(192 432)(193 379)(194 380)(195 381)(196 382)(197 383)(198 384)(199 385)(200 386)(201 387)(202 388)(203 389)(204 390)(205 391)(206 392)(207 393)(208 394)(209 395)(210 396)(211 397)(212 398)(213 399)(214 400)(215 401)(216 402)
(1 125)(2 126)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 134)(11 135)(12 136)(13 137)(14 138)(15 139)(16 140)(17 141)(18 142)(19 143)(20 144)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 151)(28 152)(29 153)(30 154)(31 155)(32 156)(33 157)(34 158)(35 159)(36 160)(37 161)(38 162)(39 109)(40 110)(41 111)(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 178)(71 179)(72 180)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 189)(82 190)(83 191)(84 192)(85 193)(86 194)(87 195)(88 196)(89 197)(90 198)(91 199)(92 200)(93 201)(94 202)(95 203)(96 204)(97 205)(98 206)(99 207)(100 208)(101 209)(102 210)(103 211)(104 212)(105 213)(106 214)(107 215)(108 216)(217 346)(218 347)(219 348)(220 349)(221 350)(222 351)(223 352)(224 353)(225 354)(226 355)(227 356)(228 357)(229 358)(230 359)(231 360)(232 361)(233 362)(234 363)(235 364)(236 365)(237 366)(238 367)(239 368)(240 369)(241 370)(242 371)(243 372)(244 373)(245 374)(246 375)(247 376)(248 377)(249 378)(250 325)(251 326)(252 327)(253 328)(254 329)(255 330)(256 331)(257 332)(258 333)(259 334)(260 335)(261 336)(262 337)(263 338)(264 339)(265 340)(266 341)(267 342)(268 343)(269 344)(270 345)(271 399)(272 400)(273 401)(274 402)(275 403)(276 404)(277 405)(278 406)(279 407)(280 408)(281 409)(282 410)(283 411)(284 412)(285 413)(286 414)(287 415)(288 416)(289 417)(290 418)(291 419)(292 420)(293 421)(294 422)(295 423)(296 424)(297 425)(298 426)(299 427)(300 428)(301 429)(302 430)(303 431)(304 432)(305 379)(306 380)(307 381)(308 382)(309 383)(310 384)(311 385)(312 386)(313 387)(314 388)(315 389)(316 390)(317 391)(318 392)(319 393)(320 394)(321 395)(322 396)(323 397)(324 398)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 181)(128 182)(129 183)(130 184)(131 185)(132 186)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 193)(140 194)(141 195)(142 196)(143 197)(144 198)(145 199)(146 200)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(153 207)(154 208)(155 209)(156 210)(157 211)(158 212)(159 213)(160 214)(161 215)(162 216)(217 276)(218 277)(219 278)(220 279)(221 280)(222 281)(223 282)(224 283)(225 284)(226 285)(227 286)(228 287)(229 288)(230 289)(231 290)(232 291)(233 292)(234 293)(235 294)(236 295)(237 296)(238 297)(239 298)(240 299)(241 300)(242 301)(243 302)(244 303)(245 304)(246 305)(247 306)(248 307)(249 308)(250 309)(251 310)(252 311)(253 312)(254 313)(255 314)(256 315)(257 316)(258 317)(259 318)(260 319)(261 320)(262 321)(263 322)(264 323)(265 324)(266 271)(267 272)(268 273)(269 274)(270 275)(325 383)(326 384)(327 385)(328 386)(329 387)(330 388)(331 389)(332 390)(333 391)(334 392)(335 393)(336 394)(337 395)(338 396)(339 397)(340 398)(341 399)(342 400)(343 401)(344 402)(345 403)(346 404)(347 405)(348 406)(349 407)(350 408)(351 409)(352 410)(353 411)(354 412)(355 413)(356 414)(357 415)(358 416)(359 417)(360 418)(361 419)(362 420)(363 421)(364 422)(365 423)(366 424)(367 425)(368 426)(369 427)(370 428)(371 429)(372 430)(373 431)(374 432)(375 379)(376 380)(377 381)(378 382)
(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)

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

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

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

432 conjugacy classes

 class 1 2A ··· 2O 3A 3B 6A ··· 6AD 9A ··· 9F 18A ··· 18CL 27A ··· 27R 54A ··· 54JJ order 1 2 ··· 2 3 3 6 ··· 6 9 ··· 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

432 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C9 C18 C27 C54 kernel C23×C54 C22×C54 C23×C18 C22×C18 C23×C6 C22×C6 C24 C23 # reps 1 15 2 30 6 90 18 270

Matrix representation of C23×C54 in GL4(𝔽109) generated by

 108 0 0 0 0 1 0 0 0 0 1 0 0 0 0 108
,
 1 0 0 0 0 108 0 0 0 0 1 0 0 0 0 108
,
 1 0 0 0 0 108 0 0 0 0 108 0 0 0 0 108
,
 66 0 0 0 0 66 0 0 0 0 102 0 0 0 0 88
G:=sub<GL(4,GF(109))| [108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,108],[1,0,0,0,0,108,0,0,0,0,1,0,0,0,0,108],[1,0,0,0,0,108,0,0,0,0,108,0,0,0,0,108],[66,0,0,0,0,66,0,0,0,0,102,0,0,0,0,88] >;

C23×C54 in GAP, Magma, Sage, TeX

C_2^3\times C_{54}
% in TeX

G:=Group("C2^3xC54");
// GroupNames label

G:=SmallGroup(432,228);
// by ID

G=gap.SmallGroup(432,228);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,137,166]);
// Polycyclic

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

׿
×
𝔽