direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic54, C54⋊Q8, C4.11D54, C36.54D6, C12.54D18, C54.1C23, C6.6Dic18, C18.6Dic6, C22.8D54, C108.11C22, Dic27.1C22, C27⋊1(C2×Q8), C9.(C2×Dic6), (C2×C36).6S3, (C2×C12).6D9, (C2×C4).4D27, C3.(C2×Dic18), (C2×C108).4C2, (C2×C6).29D18, (C2×C18).29D6, (C2×C54).8C22, C2.3(C22×D27), C6.28(C22×D9), C18.28(C22×S3), (C2×Dic27).3C2, SmallGroup(432,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic54
G = < a,b,c | a2=b108=1, c2=b54, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 472 in 76 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C9, Dic3, C12, C2×C6, C2×Q8, C18, C18, Dic6, C2×Dic3, C2×C12, C27, Dic9, C36, C2×C18, C2×Dic6, C54, C54, Dic18, C2×Dic9, C2×C36, Dic27, C108, C2×C54, C2×Dic18, Dic54, C2×Dic27, C2×C108, C2×Dic54
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, C2×Dic6, D27, Dic18, C22×D9, D54, C2×Dic18, Dic54, C22×D27, C2×Dic54
►Regular action on 432 points
Generators in S432
(1 196)(2 197)(3 198)(4 199)(5 200)(6 201)(7 202)(8 203)(9 204)(10 205)(11 206)(12 207)(13 208)(14 209)(15 210)(16 211)(17 212)(18 213)(19 214)(20 215)(21 216)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 120)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 128)(42 129)(43 130)(44 131)(45 132)(46 133)(47 134)(48 135)(49 136)(50 137)(51 138)(52 139)(53 140)(54 141)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 168)(82 169)(83 170)(84 171)(85 172)(86 173)(87 174)(88 175)(89 176)(90 177)(91 178)(92 179)(93 180)(94 181)(95 182)(96 183)(97 184)(98 185)(99 186)(100 187)(101 188)(102 189)(103 190)(104 191)(105 192)(106 193)(107 194)(108 195)(217 352)(218 353)(219 354)(220 355)(221 356)(222 357)(223 358)(224 359)(225 360)(226 361)(227 362)(228 363)(229 364)(230 365)(231 366)(232 367)(233 368)(234 369)(235 370)(236 371)(237 372)(238 373)(239 374)(240 375)(241 376)(242 377)(243 378)(244 379)(245 380)(246 381)(247 382)(248 383)(249 384)(250 385)(251 386)(252 387)(253 388)(254 389)(255 390)(256 391)(257 392)(258 393)(259 394)(260 395)(261 396)(262 397)(263 398)(264 399)(265 400)(266 401)(267 402)(268 403)(269 404)(270 405)(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)(298 325)(299 326)(300 327)(301 328)(302 329)(303 330)(304 331)(305 332)(306 333)(307 334)(308 335)(309 336)(310 337)(311 338)(312 339)(313 340)(314 341)(315 342)(316 343)(317 344)(318 345)(319 346)(320 347)(321 348)(322 349)(323 350)(324 351)
(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)
(1 244 55 298)(2 243 56 297)(3 242 57 296)(4 241 58 295)(5 240 59 294)(6 239 60 293)(7 238 61 292)(8 237 62 291)(9 236 63 290)(10 235 64 289)(11 234 65 288)(12 233 66 287)(13 232 67 286)(14 231 68 285)(15 230 69 284)(16 229 70 283)(17 228 71 282)(18 227 72 281)(19 226 73 280)(20 225 74 279)(21 224 75 278)(22 223 76 277)(23 222 77 276)(24 221 78 275)(25 220 79 274)(26 219 80 273)(27 218 81 272)(28 217 82 271)(29 324 83 270)(30 323 84 269)(31 322 85 268)(32 321 86 267)(33 320 87 266)(34 319 88 265)(35 318 89 264)(36 317 90 263)(37 316 91 262)(38 315 92 261)(39 314 93 260)(40 313 94 259)(41 312 95 258)(42 311 96 257)(43 310 97 256)(44 309 98 255)(45 308 99 254)(46 307 100 253)(47 306 101 252)(48 305 102 251)(49 304 103 250)(50 303 104 249)(51 302 105 248)(52 301 106 247)(53 300 107 246)(54 299 108 245)(109 358 163 412)(110 357 164 411)(111 356 165 410)(112 355 166 409)(113 354 167 408)(114 353 168 407)(115 352 169 406)(116 351 170 405)(117 350 171 404)(118 349 172 403)(119 348 173 402)(120 347 174 401)(121 346 175 400)(122 345 176 399)(123 344 177 398)(124 343 178 397)(125 342 179 396)(126 341 180 395)(127 340 181 394)(128 339 182 393)(129 338 183 392)(130 337 184 391)(131 336 185 390)(132 335 186 389)(133 334 187 388)(134 333 188 387)(135 332 189 386)(136 331 190 385)(137 330 191 384)(138 329 192 383)(139 328 193 382)(140 327 194 381)(141 326 195 380)(142 325 196 379)(143 432 197 378)(144 431 198 377)(145 430 199 376)(146 429 200 375)(147 428 201 374)(148 427 202 373)(149 426 203 372)(150 425 204 371)(151 424 205 370)(152 423 206 369)(153 422 207 368)(154 421 208 367)(155 420 209 366)(156 419 210 365)(157 418 211 364)(158 417 212 363)(159 416 213 362)(160 415 214 361)(161 414 215 360)(162 413 216 359)
G:=sub<Sym(432)| (1,196)(2,197)(3,198)(4,199)(5,200)(6,201)(7,202)(8,203)(9,204)(10,205)(11,206)(12,207)(13,208)(14,209)(15,210)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,128)(42,129)(43,130)(44,131)(45,132)(46,133)(47,134)(48,135)(49,136)(50,137)(51,138)(52,139)(53,140)(54,141)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,169)(83,170)(84,171)(85,172)(86,173)(87,174)(88,175)(89,176)(90,177)(91,178)(92,179)(93,180)(94,181)(95,182)(96,183)(97,184)(98,185)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191)(105,192)(106,193)(107,194)(108,195)(217,352)(218,353)(219,354)(220,355)(221,356)(222,357)(223,358)(224,359)(225,360)(226,361)(227,362)(228,363)(229,364)(230,365)(231,366)(232,367)(233,368)(234,369)(235,370)(236,371)(237,372)(238,373)(239,374)(240,375)(241,376)(242,377)(243,378)(244,379)(245,380)(246,381)(247,382)(248,383)(249,384)(250,385)(251,386)(252,387)(253,388)(254,389)(255,390)(256,391)(257,392)(258,393)(259,394)(260,395)(261,396)(262,397)(263,398)(264,399)(265,400)(266,401)(267,402)(268,403)(269,404)(270,405)(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)(298,325)(299,326)(300,327)(301,328)(302,329)(303,330)(304,331)(305,332)(306,333)(307,334)(308,335)(309,336)(310,337)(311,338)(312,339)(313,340)(314,341)(315,342)(316,343)(317,344)(318,345)(319,346)(320,347)(321,348)(322,349)(323,350)(324,351), (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), (1,244,55,298)(2,243,56,297)(3,242,57,296)(4,241,58,295)(5,240,59,294)(6,239,60,293)(7,238,61,292)(8,237,62,291)(9,236,63,290)(10,235,64,289)(11,234,65,288)(12,233,66,287)(13,232,67,286)(14,231,68,285)(15,230,69,284)(16,229,70,283)(17,228,71,282)(18,227,72,281)(19,226,73,280)(20,225,74,279)(21,224,75,278)(22,223,76,277)(23,222,77,276)(24,221,78,275)(25,220,79,274)(26,219,80,273)(27,218,81,272)(28,217,82,271)(29,324,83,270)(30,323,84,269)(31,322,85,268)(32,321,86,267)(33,320,87,266)(34,319,88,265)(35,318,89,264)(36,317,90,263)(37,316,91,262)(38,315,92,261)(39,314,93,260)(40,313,94,259)(41,312,95,258)(42,311,96,257)(43,310,97,256)(44,309,98,255)(45,308,99,254)(46,307,100,253)(47,306,101,252)(48,305,102,251)(49,304,103,250)(50,303,104,249)(51,302,105,248)(52,301,106,247)(53,300,107,246)(54,299,108,245)(109,358,163,412)(110,357,164,411)(111,356,165,410)(112,355,166,409)(113,354,167,408)(114,353,168,407)(115,352,169,406)(116,351,170,405)(117,350,171,404)(118,349,172,403)(119,348,173,402)(120,347,174,401)(121,346,175,400)(122,345,176,399)(123,344,177,398)(124,343,178,397)(125,342,179,396)(126,341,180,395)(127,340,181,394)(128,339,182,393)(129,338,183,392)(130,337,184,391)(131,336,185,390)(132,335,186,389)(133,334,187,388)(134,333,188,387)(135,332,189,386)(136,331,190,385)(137,330,191,384)(138,329,192,383)(139,328,193,382)(140,327,194,381)(141,326,195,380)(142,325,196,379)(143,432,197,378)(144,431,198,377)(145,430,199,376)(146,429,200,375)(147,428,201,374)(148,427,202,373)(149,426,203,372)(150,425,204,371)(151,424,205,370)(152,423,206,369)(153,422,207,368)(154,421,208,367)(155,420,209,366)(156,419,210,365)(157,418,211,364)(158,417,212,363)(159,416,213,362)(160,415,214,361)(161,414,215,360)(162,413,216,359)>;
G:=Group( (1,196)(2,197)(3,198)(4,199)(5,200)(6,201)(7,202)(8,203)(9,204)(10,205)(11,206)(12,207)(13,208)(14,209)(15,210)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,128)(42,129)(43,130)(44,131)(45,132)(46,133)(47,134)(48,135)(49,136)(50,137)(51,138)(52,139)(53,140)(54,141)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,169)(83,170)(84,171)(85,172)(86,173)(87,174)(88,175)(89,176)(90,177)(91,178)(92,179)(93,180)(94,181)(95,182)(96,183)(97,184)(98,185)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191)(105,192)(106,193)(107,194)(108,195)(217,352)(218,353)(219,354)(220,355)(221,356)(222,357)(223,358)(224,359)(225,360)(226,361)(227,362)(228,363)(229,364)(230,365)(231,366)(232,367)(233,368)(234,369)(235,370)(236,371)(237,372)(238,373)(239,374)(240,375)(241,376)(242,377)(243,378)(244,379)(245,380)(246,381)(247,382)(248,383)(249,384)(250,385)(251,386)(252,387)(253,388)(254,389)(255,390)(256,391)(257,392)(258,393)(259,394)(260,395)(261,396)(262,397)(263,398)(264,399)(265,400)(266,401)(267,402)(268,403)(269,404)(270,405)(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)(298,325)(299,326)(300,327)(301,328)(302,329)(303,330)(304,331)(305,332)(306,333)(307,334)(308,335)(309,336)(310,337)(311,338)(312,339)(313,340)(314,341)(315,342)(316,343)(317,344)(318,345)(319,346)(320,347)(321,348)(322,349)(323,350)(324,351), (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), (1,244,55,298)(2,243,56,297)(3,242,57,296)(4,241,58,295)(5,240,59,294)(6,239,60,293)(7,238,61,292)(8,237,62,291)(9,236,63,290)(10,235,64,289)(11,234,65,288)(12,233,66,287)(13,232,67,286)(14,231,68,285)(15,230,69,284)(16,229,70,283)(17,228,71,282)(18,227,72,281)(19,226,73,280)(20,225,74,279)(21,224,75,278)(22,223,76,277)(23,222,77,276)(24,221,78,275)(25,220,79,274)(26,219,80,273)(27,218,81,272)(28,217,82,271)(29,324,83,270)(30,323,84,269)(31,322,85,268)(32,321,86,267)(33,320,87,266)(34,319,88,265)(35,318,89,264)(36,317,90,263)(37,316,91,262)(38,315,92,261)(39,314,93,260)(40,313,94,259)(41,312,95,258)(42,311,96,257)(43,310,97,256)(44,309,98,255)(45,308,99,254)(46,307,100,253)(47,306,101,252)(48,305,102,251)(49,304,103,250)(50,303,104,249)(51,302,105,248)(52,301,106,247)(53,300,107,246)(54,299,108,245)(109,358,163,412)(110,357,164,411)(111,356,165,410)(112,355,166,409)(113,354,167,408)(114,353,168,407)(115,352,169,406)(116,351,170,405)(117,350,171,404)(118,349,172,403)(119,348,173,402)(120,347,174,401)(121,346,175,400)(122,345,176,399)(123,344,177,398)(124,343,178,397)(125,342,179,396)(126,341,180,395)(127,340,181,394)(128,339,182,393)(129,338,183,392)(130,337,184,391)(131,336,185,390)(132,335,186,389)(133,334,187,388)(134,333,188,387)(135,332,189,386)(136,331,190,385)(137,330,191,384)(138,329,192,383)(139,328,193,382)(140,327,194,381)(141,326,195,380)(142,325,196,379)(143,432,197,378)(144,431,198,377)(145,430,199,376)(146,429,200,375)(147,428,201,374)(148,427,202,373)(149,426,203,372)(150,425,204,371)(151,424,205,370)(152,423,206,369)(153,422,207,368)(154,421,208,367)(155,420,209,366)(156,419,210,365)(157,418,211,364)(158,417,212,363)(159,416,213,362)(160,415,214,361)(161,414,215,360)(162,413,216,359) );
G=PermutationGroup([[(1,196),(2,197),(3,198),(4,199),(5,200),(6,201),(7,202),(8,203),(9,204),(10,205),(11,206),(12,207),(13,208),(14,209),(15,210),(16,211),(17,212),(18,213),(19,214),(20,215),(21,216),(22,109),(23,110),(24,111),(25,112),(26,113),(27,114),(28,115),(29,116),(30,117),(31,118),(32,119),(33,120),(34,121),(35,122),(36,123),(37,124),(38,125),(39,126),(40,127),(41,128),(42,129),(43,130),(44,131),(45,132),(46,133),(47,134),(48,135),(49,136),(50,137),(51,138),(52,139),(53,140),(54,141),(55,142),(56,143),(57,144),(58,145),(59,146),(60,147),(61,148),(62,149),(63,150),(64,151),(65,152),(66,153),(67,154),(68,155),(69,156),(70,157),(71,158),(72,159),(73,160),(74,161),(75,162),(76,163),(77,164),(78,165),(79,166),(80,167),(81,168),(82,169),(83,170),(84,171),(85,172),(86,173),(87,174),(88,175),(89,176),(90,177),(91,178),(92,179),(93,180),(94,181),(95,182),(96,183),(97,184),(98,185),(99,186),(100,187),(101,188),(102,189),(103,190),(104,191),(105,192),(106,193),(107,194),(108,195),(217,352),(218,353),(219,354),(220,355),(221,356),(222,357),(223,358),(224,359),(225,360),(226,361),(227,362),(228,363),(229,364),(230,365),(231,366),(232,367),(233,368),(234,369),(235,370),(236,371),(237,372),(238,373),(239,374),(240,375),(241,376),(242,377),(243,378),(244,379),(245,380),(246,381),(247,382),(248,383),(249,384),(250,385),(251,386),(252,387),(253,388),(254,389),(255,390),(256,391),(257,392),(258,393),(259,394),(260,395),(261,396),(262,397),(263,398),(264,399),(265,400),(266,401),(267,402),(268,403),(269,404),(270,405),(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),(298,325),(299,326),(300,327),(301,328),(302,329),(303,330),(304,331),(305,332),(306,333),(307,334),(308,335),(309,336),(310,337),(311,338),(312,339),(313,340),(314,341),(315,342),(316,343),(317,344),(318,345),(319,346),(320,347),(321,348),(322,349),(323,350),(324,351)], [(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)], [(1,244,55,298),(2,243,56,297),(3,242,57,296),(4,241,58,295),(5,240,59,294),(6,239,60,293),(7,238,61,292),(8,237,62,291),(9,236,63,290),(10,235,64,289),(11,234,65,288),(12,233,66,287),(13,232,67,286),(14,231,68,285),(15,230,69,284),(16,229,70,283),(17,228,71,282),(18,227,72,281),(19,226,73,280),(20,225,74,279),(21,224,75,278),(22,223,76,277),(23,222,77,276),(24,221,78,275),(25,220,79,274),(26,219,80,273),(27,218,81,272),(28,217,82,271),(29,324,83,270),(30,323,84,269),(31,322,85,268),(32,321,86,267),(33,320,87,266),(34,319,88,265),(35,318,89,264),(36,317,90,263),(37,316,91,262),(38,315,92,261),(39,314,93,260),(40,313,94,259),(41,312,95,258),(42,311,96,257),(43,310,97,256),(44,309,98,255),(45,308,99,254),(46,307,100,253),(47,306,101,252),(48,305,102,251),(49,304,103,250),(50,303,104,249),(51,302,105,248),(52,301,106,247),(53,300,107,246),(54,299,108,245),(109,358,163,412),(110,357,164,411),(111,356,165,410),(112,355,166,409),(113,354,167,408),(114,353,168,407),(115,352,169,406),(116,351,170,405),(117,350,171,404),(118,349,172,403),(119,348,173,402),(120,347,174,401),(121,346,175,400),(122,345,176,399),(123,344,177,398),(124,343,178,397),(125,342,179,396),(126,341,180,395),(127,340,181,394),(128,339,182,393),(129,338,183,392),(130,337,184,391),(131,336,185,390),(132,335,186,389),(133,334,187,388),(134,333,188,387),(135,332,189,386),(136,331,190,385),(137,330,191,384),(138,329,192,383),(139,328,193,382),(140,327,194,381),(141,326,195,380),(142,325,196,379),(143,432,197,378),(144,431,198,377),(145,430,199,376),(146,429,200,375),(147,428,201,374),(148,427,202,373),(149,426,203,372),(150,425,204,371),(151,424,205,370),(152,423,206,369),(153,422,207,368),(154,421,208,367),(155,420,209,366),(156,419,210,365),(157,418,211,364),(158,417,212,363),(159,416,213,362),(160,415,214,361),(161,414,215,360),(162,413,216,359)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 27A | ··· | 27I | 36A | ··· | 36L | 54A | ··· | 54AA | 108A | ··· | 108AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 54 | 54 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D9 | Dic6 | D18 | D18 | D27 | Dic18 | D54 | D54 | Dic54 |
| kernel | C2×Dic54 | Dic54 | C2×Dic27 | C2×C108 | C2×C36 | C54 | C36 | C2×C18 | C2×C12 | C18 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 1 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 9 | 12 | 18 | 9 | 36 |
Matrix representation of C2×Dic54 ►in GL3(𝔽109) generated by
| 108 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 6 | 47 |
| 0 | 62 | 68 |
| 1 | 0 | 0 |
| 0 | 61 | 24 |
| 0 | 72 | 48 |
G:=sub<GL(3,GF(109))| [108,0,0,0,1,0,0,0,1],[1,0,0,0,6,62,0,47,68],[1,0,0,0,61,72,0,24,48] >;
C2×Dic54 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{54} % in TeX
G:=Group("C2xDic54"); // GroupNames label
G:=SmallGroup(432,43);
// by ID
G=gap.SmallGroup(432,43);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^2=b^108=1,c^2=b^54,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations