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G = C23×C54order 432 = 24·33

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

direct product, abelian, monomial, 2-elementary

Aliases: C23×C54, SmallGroup(432,228)

Series: Derived Chief Lower central Upper central

C1 — C23×C54
C1C3C9C27C54C2×C54C22×C54 — C23×C54
C1 — C23×C54
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 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C9, C2×C6 [×35], C24, C18 [×15], C22×C6 [×15], C27, C2×C18 [×35], C23×C6, C54 [×15], C22×C18 [×15], C2×C54 [×35], C23×C18, C22×C54 [×15], C23×C54
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C9, C2×C6 [×35], C24, C18 [×15], C22×C6 [×15], C27, C2×C18 [×35], C23×C6, C54 [×15], C22×C18 [×15], C2×C54 [×35], C23×C18, C22×C54 [×15], C23×C54

Smallest permutation representation of C23×C54
Regular action on 432 points
Generators in S432
(1 253)(2 254)(3 255)(4 256)(5 257)(6 258)(7 259)(8 260)(9 261)(10 262)(11 263)(12 264)(13 265)(14 266)(15 267)(16 268)(17 269)(18 270)(19 217)(20 218)(21 219)(22 220)(23 221)(24 222)(25 223)(26 224)(27 225)(28 226)(29 227)(30 228)(31 229)(32 230)(33 231)(34 232)(35 233)(36 234)(37 235)(38 236)(39 237)(40 238)(41 239)(42 240)(43 241)(44 242)(45 243)(46 244)(47 245)(48 246)(49 247)(50 248)(51 249)(52 250)(53 251)(54 252)(55 304)(56 305)(57 306)(58 307)(59 308)(60 309)(61 310)(62 311)(63 312)(64 313)(65 314)(66 315)(67 316)(68 317)(69 318)(70 319)(71 320)(72 321)(73 322)(74 323)(75 324)(76 271)(77 272)(78 273)(79 274)(80 275)(81 276)(82 277)(83 278)(84 279)(85 280)(86 281)(87 282)(88 283)(89 284)(90 285)(91 286)(92 287)(93 288)(94 289)(95 290)(96 291)(97 292)(98 293)(99 294)(100 295)(101 296)(102 297)(103 298)(104 299)(105 300)(106 301)(107 302)(108 303)(109 354)(110 355)(111 356)(112 357)(113 358)(114 359)(115 360)(116 361)(117 362)(118 363)(119 364)(120 365)(121 366)(122 367)(123 368)(124 369)(125 370)(126 371)(127 372)(128 373)(129 374)(130 375)(131 376)(132 377)(133 378)(134 325)(135 326)(136 327)(137 328)(138 329)(139 330)(140 331)(141 332)(142 333)(143 334)(144 335)(145 336)(146 337)(147 338)(148 339)(149 340)(150 341)(151 342)(152 343)(153 344)(154 345)(155 346)(156 347)(157 348)(158 349)(159 350)(160 351)(161 352)(162 353)(163 412)(164 413)(165 414)(166 415)(167 416)(168 417)(169 418)(170 419)(171 420)(172 421)(173 422)(174 423)(175 424)(176 425)(177 426)(178 427)(179 428)(180 429)(181 430)(182 431)(183 432)(184 379)(185 380)(186 381)(187 382)(188 383)(189 384)(190 385)(191 386)(192 387)(193 388)(194 389)(195 390)(196 391)(197 392)(198 393)(199 394)(200 395)(201 396)(202 397)(203 398)(204 399)(205 400)(206 401)(207 402)(208 403)(209 404)(210 405)(211 406)(212 407)(213 408)(214 409)(215 410)(216 411)
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 141)(14 142)(15 143)(16 144)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 161)(34 162)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 184)(56 185)(57 186)(58 187)(59 188)(60 189)(61 190)(62 191)(63 192)(64 193)(65 194)(66 195)(67 196)(68 197)(69 198)(70 199)(71 200)(72 201)(73 202)(74 203)(75 204)(76 205)(77 206)(78 207)(79 208)(80 209)(81 210)(82 211)(83 212)(84 213)(85 214)(86 215)(87 216)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 169)(95 170)(96 171)(97 172)(98 173)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 183)(217 338)(218 339)(219 340)(220 341)(221 342)(222 343)(223 344)(224 345)(225 346)(226 347)(227 348)(228 349)(229 350)(230 351)(231 352)(232 353)(233 354)(234 355)(235 356)(236 357)(237 358)(238 359)(239 360)(240 361)(241 362)(242 363)(243 364)(244 365)(245 366)(246 367)(247 368)(248 369)(249 370)(250 371)(251 372)(252 373)(253 374)(254 375)(255 376)(256 377)(257 378)(258 325)(259 326)(260 327)(261 328)(262 329)(263 330)(264 331)(265 332)(266 333)(267 334)(268 335)(269 336)(270 337)(271 400)(272 401)(273 402)(274 403)(275 404)(276 405)(277 406)(278 407)(279 408)(280 409)(281 410)(282 411)(283 412)(284 413)(285 414)(286 415)(287 416)(288 417)(289 418)(290 419)(291 420)(292 421)(293 422)(294 423)(295 424)(296 425)(297 426)(298 427)(299 428)(300 429)(301 430)(302 431)(303 432)(304 379)(305 380)(306 381)(307 382)(308 383)(309 384)(310 385)(311 386)(312 387)(313 388)(314 389)(315 390)(316 391)(317 392)(318 393)(319 394)(320 395)(321 396)(322 397)(323 398)(324 399)
(1 108)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(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)(217 321)(218 322)(219 323)(220 324)(221 271)(222 272)(223 273)(224 274)(225 275)(226 276)(227 277)(228 278)(229 279)(230 280)(231 281)(232 282)(233 283)(234 284)(235 285)(236 286)(237 287)(238 288)(239 289)(240 290)(241 291)(242 292)(243 293)(244 294)(245 295)(246 296)(247 297)(248 298)(249 299)(250 300)(251 301)(252 302)(253 303)(254 304)(255 305)(256 306)(257 307)(258 308)(259 309)(260 310)(261 311)(262 312)(263 313)(264 314)(265 315)(266 316)(267 317)(268 318)(269 319)(270 320)(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)| (1,253)(2,254)(3,255)(4,256)(5,257)(6,258)(7,259)(8,260)(9,261)(10,262)(11,263)(12,264)(13,265)(14,266)(15,267)(16,268)(17,269)(18,270)(19,217)(20,218)(21,219)(22,220)(23,221)(24,222)(25,223)(26,224)(27,225)(28,226)(29,227)(30,228)(31,229)(32,230)(33,231)(34,232)(35,233)(36,234)(37,235)(38,236)(39,237)(40,238)(41,239)(42,240)(43,241)(44,242)(45,243)(46,244)(47,245)(48,246)(49,247)(50,248)(51,249)(52,250)(53,251)(54,252)(55,304)(56,305)(57,306)(58,307)(59,308)(60,309)(61,310)(62,311)(63,312)(64,313)(65,314)(66,315)(67,316)(68,317)(69,318)(70,319)(71,320)(72,321)(73,322)(74,323)(75,324)(76,271)(77,272)(78,273)(79,274)(80,275)(81,276)(82,277)(83,278)(84,279)(85,280)(86,281)(87,282)(88,283)(89,284)(90,285)(91,286)(92,287)(93,288)(94,289)(95,290)(96,291)(97,292)(98,293)(99,294)(100,295)(101,296)(102,297)(103,298)(104,299)(105,300)(106,301)(107,302)(108,303)(109,354)(110,355)(111,356)(112,357)(113,358)(114,359)(115,360)(116,361)(117,362)(118,363)(119,364)(120,365)(121,366)(122,367)(123,368)(124,369)(125,370)(126,371)(127,372)(128,373)(129,374)(130,375)(131,376)(132,377)(133,378)(134,325)(135,326)(136,327)(137,328)(138,329)(139,330)(140,331)(141,332)(142,333)(143,334)(144,335)(145,336)(146,337)(147,338)(148,339)(149,340)(150,341)(151,342)(152,343)(153,344)(154,345)(155,346)(156,347)(157,348)(158,349)(159,350)(160,351)(161,352)(162,353)(163,412)(164,413)(165,414)(166,415)(167,416)(168,417)(169,418)(170,419)(171,420)(172,421)(173,422)(174,423)(175,424)(176,425)(177,426)(178,427)(179,428)(180,429)(181,430)(182,431)(183,432)(184,379)(185,380)(186,381)(187,382)(188,383)(189,384)(190,385)(191,386)(192,387)(193,388)(194,389)(195,390)(196,391)(197,392)(198,393)(199,394)(200,395)(201,396)(202,397)(203,398)(204,399)(205,400)(206,401)(207,402)(208,403)(209,404)(210,405)(211,406)(212,407)(213,408)(214,409)(215,410)(216,411), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,141)(14,142)(15,143)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,161)(34,162)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,184)(56,185)(57,186)(58,187)(59,188)(60,189)(61,190)(62,191)(63,192)(64,193)(65,194)(66,195)(67,196)(68,197)(69,198)(70,199)(71,200)(72,201)(73,202)(74,203)(75,204)(76,205)(77,206)(78,207)(79,208)(80,209)(81,210)(82,211)(83,212)(84,213)(85,214)(86,215)(87,216)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,171)(97,172)(98,173)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(217,338)(218,339)(219,340)(220,341)(221,342)(222,343)(223,344)(224,345)(225,346)(226,347)(227,348)(228,349)(229,350)(230,351)(231,352)(232,353)(233,354)(234,355)(235,356)(236,357)(237,358)(238,359)(239,360)(240,361)(241,362)(242,363)(243,364)(244,365)(245,366)(246,367)(247,368)(248,369)(249,370)(250,371)(251,372)(252,373)(253,374)(254,375)(255,376)(256,377)(257,378)(258,325)(259,326)(260,327)(261,328)(262,329)(263,330)(264,331)(265,332)(266,333)(267,334)(268,335)(269,336)(270,337)(271,400)(272,401)(273,402)(274,403)(275,404)(276,405)(277,406)(278,407)(279,408)(280,409)(281,410)(282,411)(283,412)(284,413)(285,414)(286,415)(287,416)(288,417)(289,418)(290,419)(291,420)(292,421)(293,422)(294,423)(295,424)(296,425)(297,426)(298,427)(299,428)(300,429)(301,430)(302,431)(303,432)(304,379)(305,380)(306,381)(307,382)(308,383)(309,384)(310,385)(311,386)(312,387)(313,388)(314,389)(315,390)(316,391)(317,392)(318,393)(319,394)(320,395)(321,396)(322,397)(323,398)(324,399), (1,108)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(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)(217,321)(218,322)(219,323)(220,324)(221,271)(222,272)(223,273)(224,274)(225,275)(226,276)(227,277)(228,278)(229,279)(230,280)(231,281)(232,282)(233,283)(234,284)(235,285)(236,286)(237,287)(238,288)(239,289)(240,290)(241,291)(242,292)(243,293)(244,294)(245,295)(246,296)(247,297)(248,298)(249,299)(250,300)(251,301)(252,302)(253,303)(254,304)(255,305)(256,306)(257,307)(258,308)(259,309)(260,310)(261,311)(262,312)(263,313)(264,314)(265,315)(266,316)(267,317)(268,318)(269,319)(270,320)(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( (1,253)(2,254)(3,255)(4,256)(5,257)(6,258)(7,259)(8,260)(9,261)(10,262)(11,263)(12,264)(13,265)(14,266)(15,267)(16,268)(17,269)(18,270)(19,217)(20,218)(21,219)(22,220)(23,221)(24,222)(25,223)(26,224)(27,225)(28,226)(29,227)(30,228)(31,229)(32,230)(33,231)(34,232)(35,233)(36,234)(37,235)(38,236)(39,237)(40,238)(41,239)(42,240)(43,241)(44,242)(45,243)(46,244)(47,245)(48,246)(49,247)(50,248)(51,249)(52,250)(53,251)(54,252)(55,304)(56,305)(57,306)(58,307)(59,308)(60,309)(61,310)(62,311)(63,312)(64,313)(65,314)(66,315)(67,316)(68,317)(69,318)(70,319)(71,320)(72,321)(73,322)(74,323)(75,324)(76,271)(77,272)(78,273)(79,274)(80,275)(81,276)(82,277)(83,278)(84,279)(85,280)(86,281)(87,282)(88,283)(89,284)(90,285)(91,286)(92,287)(93,288)(94,289)(95,290)(96,291)(97,292)(98,293)(99,294)(100,295)(101,296)(102,297)(103,298)(104,299)(105,300)(106,301)(107,302)(108,303)(109,354)(110,355)(111,356)(112,357)(113,358)(114,359)(115,360)(116,361)(117,362)(118,363)(119,364)(120,365)(121,366)(122,367)(123,368)(124,369)(125,370)(126,371)(127,372)(128,373)(129,374)(130,375)(131,376)(132,377)(133,378)(134,325)(135,326)(136,327)(137,328)(138,329)(139,330)(140,331)(141,332)(142,333)(143,334)(144,335)(145,336)(146,337)(147,338)(148,339)(149,340)(150,341)(151,342)(152,343)(153,344)(154,345)(155,346)(156,347)(157,348)(158,349)(159,350)(160,351)(161,352)(162,353)(163,412)(164,413)(165,414)(166,415)(167,416)(168,417)(169,418)(170,419)(171,420)(172,421)(173,422)(174,423)(175,424)(176,425)(177,426)(178,427)(179,428)(180,429)(181,430)(182,431)(183,432)(184,379)(185,380)(186,381)(187,382)(188,383)(189,384)(190,385)(191,386)(192,387)(193,388)(194,389)(195,390)(196,391)(197,392)(198,393)(199,394)(200,395)(201,396)(202,397)(203,398)(204,399)(205,400)(206,401)(207,402)(208,403)(209,404)(210,405)(211,406)(212,407)(213,408)(214,409)(215,410)(216,411), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,141)(14,142)(15,143)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,161)(34,162)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,184)(56,185)(57,186)(58,187)(59,188)(60,189)(61,190)(62,191)(63,192)(64,193)(65,194)(66,195)(67,196)(68,197)(69,198)(70,199)(71,200)(72,201)(73,202)(74,203)(75,204)(76,205)(77,206)(78,207)(79,208)(80,209)(81,210)(82,211)(83,212)(84,213)(85,214)(86,215)(87,216)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,171)(97,172)(98,173)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(217,338)(218,339)(219,340)(220,341)(221,342)(222,343)(223,344)(224,345)(225,346)(226,347)(227,348)(228,349)(229,350)(230,351)(231,352)(232,353)(233,354)(234,355)(235,356)(236,357)(237,358)(238,359)(239,360)(240,361)(241,362)(242,363)(243,364)(244,365)(245,366)(246,367)(247,368)(248,369)(249,370)(250,371)(251,372)(252,373)(253,374)(254,375)(255,376)(256,377)(257,378)(258,325)(259,326)(260,327)(261,328)(262,329)(263,330)(264,331)(265,332)(266,333)(267,334)(268,335)(269,336)(270,337)(271,400)(272,401)(273,402)(274,403)(275,404)(276,405)(277,406)(278,407)(279,408)(280,409)(281,410)(282,411)(283,412)(284,413)(285,414)(286,415)(287,416)(288,417)(289,418)(290,419)(291,420)(292,421)(293,422)(294,423)(295,424)(296,425)(297,426)(298,427)(299,428)(300,429)(301,430)(302,431)(303,432)(304,379)(305,380)(306,381)(307,382)(308,383)(309,384)(310,385)(311,386)(312,387)(313,388)(314,389)(315,390)(316,391)(317,392)(318,393)(319,394)(320,395)(321,396)(322,397)(323,398)(324,399), (1,108)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(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)(217,321)(218,322)(219,323)(220,324)(221,271)(222,272)(223,273)(224,274)(225,275)(226,276)(227,277)(228,278)(229,279)(230,280)(231,281)(232,282)(233,283)(234,284)(235,285)(236,286)(237,287)(238,288)(239,289)(240,290)(241,291)(242,292)(243,293)(244,294)(245,295)(246,296)(247,297)(248,298)(249,299)(250,300)(251,301)(252,302)(253,303)(254,304)(255,305)(256,306)(257,307)(258,308)(259,309)(260,310)(261,311)(262,312)(263,313)(264,314)(265,315)(266,316)(267,317)(268,318)(269,319)(270,320)(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([(1,253),(2,254),(3,255),(4,256),(5,257),(6,258),(7,259),(8,260),(9,261),(10,262),(11,263),(12,264),(13,265),(14,266),(15,267),(16,268),(17,269),(18,270),(19,217),(20,218),(21,219),(22,220),(23,221),(24,222),(25,223),(26,224),(27,225),(28,226),(29,227),(30,228),(31,229),(32,230),(33,231),(34,232),(35,233),(36,234),(37,235),(38,236),(39,237),(40,238),(41,239),(42,240),(43,241),(44,242),(45,243),(46,244),(47,245),(48,246),(49,247),(50,248),(51,249),(52,250),(53,251),(54,252),(55,304),(56,305),(57,306),(58,307),(59,308),(60,309),(61,310),(62,311),(63,312),(64,313),(65,314),(66,315),(67,316),(68,317),(69,318),(70,319),(71,320),(72,321),(73,322),(74,323),(75,324),(76,271),(77,272),(78,273),(79,274),(80,275),(81,276),(82,277),(83,278),(84,279),(85,280),(86,281),(87,282),(88,283),(89,284),(90,285),(91,286),(92,287),(93,288),(94,289),(95,290),(96,291),(97,292),(98,293),(99,294),(100,295),(101,296),(102,297),(103,298),(104,299),(105,300),(106,301),(107,302),(108,303),(109,354),(110,355),(111,356),(112,357),(113,358),(114,359),(115,360),(116,361),(117,362),(118,363),(119,364),(120,365),(121,366),(122,367),(123,368),(124,369),(125,370),(126,371),(127,372),(128,373),(129,374),(130,375),(131,376),(132,377),(133,378),(134,325),(135,326),(136,327),(137,328),(138,329),(139,330),(140,331),(141,332),(142,333),(143,334),(144,335),(145,336),(146,337),(147,338),(148,339),(149,340),(150,341),(151,342),(152,343),(153,344),(154,345),(155,346),(156,347),(157,348),(158,349),(159,350),(160,351),(161,352),(162,353),(163,412),(164,413),(165,414),(166,415),(167,416),(168,417),(169,418),(170,419),(171,420),(172,421),(173,422),(174,423),(175,424),(176,425),(177,426),(178,427),(179,428),(180,429),(181,430),(182,431),(183,432),(184,379),(185,380),(186,381),(187,382),(188,383),(189,384),(190,385),(191,386),(192,387),(193,388),(194,389),(195,390),(196,391),(197,392),(198,393),(199,394),(200,395),(201,396),(202,397),(203,398),(204,399),(205,400),(206,401),(207,402),(208,403),(209,404),(210,405),(211,406),(212,407),(213,408),(214,409),(215,410),(216,411)], [(1,129),(2,130),(3,131),(4,132),(5,133),(6,134),(7,135),(8,136),(9,137),(10,138),(11,139),(12,140),(13,141),(14,142),(15,143),(16,144),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,152),(25,153),(26,154),(27,155),(28,156),(29,157),(30,158),(31,159),(32,160),(33,161),(34,162),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,184),(56,185),(57,186),(58,187),(59,188),(60,189),(61,190),(62,191),(63,192),(64,193),(65,194),(66,195),(67,196),(68,197),(69,198),(70,199),(71,200),(72,201),(73,202),(74,203),(75,204),(76,205),(77,206),(78,207),(79,208),(80,209),(81,210),(82,211),(83,212),(84,213),(85,214),(86,215),(87,216),(88,163),(89,164),(90,165),(91,166),(92,167),(93,168),(94,169),(95,170),(96,171),(97,172),(98,173),(99,174),(100,175),(101,176),(102,177),(103,178),(104,179),(105,180),(106,181),(107,182),(108,183),(217,338),(218,339),(219,340),(220,341),(221,342),(222,343),(223,344),(224,345),(225,346),(226,347),(227,348),(228,349),(229,350),(230,351),(231,352),(232,353),(233,354),(234,355),(235,356),(236,357),(237,358),(238,359),(239,360),(240,361),(241,362),(242,363),(243,364),(244,365),(245,366),(246,367),(247,368),(248,369),(249,370),(250,371),(251,372),(252,373),(253,374),(254,375),(255,376),(256,377),(257,378),(258,325),(259,326),(260,327),(261,328),(262,329),(263,330),(264,331),(265,332),(266,333),(267,334),(268,335),(269,336),(270,337),(271,400),(272,401),(273,402),(274,403),(275,404),(276,405),(277,406),(278,407),(279,408),(280,409),(281,410),(282,411),(283,412),(284,413),(285,414),(286,415),(287,416),(288,417),(289,418),(290,419),(291,420),(292,421),(293,422),(294,423),(295,424),(296,425),(297,426),(298,427),(299,428),(300,429),(301,430),(302,431),(303,432),(304,379),(305,380),(306,381),(307,382),(308,383),(309,384),(310,385),(311,386),(312,387),(313,388),(314,389),(315,390),(316,391),(317,392),(318,393),(319,394),(320,395),(321,396),(322,397),(323,398),(324,399)], [(1,108),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(46,99),(47,100),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(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),(217,321),(218,322),(219,323),(220,324),(221,271),(222,272),(223,273),(224,274),(225,275),(226,276),(227,277),(228,278),(229,279),(230,280),(231,281),(232,282),(233,283),(234,284),(235,285),(236,286),(237,287),(238,288),(239,289),(240,290),(241,291),(242,292),(243,293),(244,294),(245,295),(246,296),(247,297),(248,298),(249,299),(250,300),(251,301),(252,302),(253,303),(254,304),(255,305),(256,306),(257,307),(258,308),(259,309),(260,310),(261,311),(262,312),(263,313),(264,314),(265,315),(266,316),(267,317),(268,318),(269,319),(270,320),(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···2O3A3B6A···6AD9A···9F18A···18CL27A···27R54A···54JJ
order12···2336···69···918···1827···2754···54
size11···1111···11···11···11···11···1

432 irreducible representations

dim11111111
type++
imageC1C2C3C6C9C18C27C54
kernelC23×C54C22×C54C23×C18C22×C18C23×C6C22×C6C24C23
# reps11523069018270

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

108000
0100
0010
000108
,
1000
010800
0010
000108
,
1000
010800
001080
000108
,
66000
06600
001020
00088
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