direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic57, C38⋊Dic3, C6⋊Dic19, C114⋊1C4, C22.D57, C6.11D38, C38.11D6, C2.2D114, C114.11C22, C57⋊7(C2×C4), (C2×C38).S3, (C2×C6).D19, C19⋊2(C2×Dic3), C3⋊2(C2×Dic19), (C2×C114).1C2, SmallGroup(456,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — C2×Dic57 |
Generators and relations for C2×Dic57
G = < a,b,c | a2=b114=1, c2=b57, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic57
►Regular action on 456 points
Generators in S456
(1 190)(2 191)(3 192)(4 193)(5 194)(6 195)(7 196)(8 197)(9 198)(10 199)(11 200)(12 201)(13 202)(14 203)(15 204)(16 205)(17 206)(18 207)(19 208)(20 209)(21 210)(22 211)(23 212)(24 213)(25 214)(26 215)(27 216)(28 217)(29 218)(30 219)(31 220)(32 221)(33 222)(34 223)(35 224)(36 225)(37 226)(38 227)(39 228)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(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)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(229 343)(230 344)(231 345)(232 346)(233 347)(234 348)(235 349)(236 350)(237 351)(238 352)(239 353)(240 354)(241 355)(242 356)(243 357)(244 358)(245 359)(246 360)(247 361)(248 362)(249 363)(250 364)(251 365)(252 366)(253 367)(254 368)(255 369)(256 370)(257 371)(258 372)(259 373)(260 374)(261 375)(262 376)(263 377)(264 378)(265 379)(266 380)(267 381)(268 382)(269 383)(270 384)(271 385)(272 386)(273 387)(274 388)(275 389)(276 390)(277 391)(278 392)(279 393)(280 394)(281 395)(282 396)(283 397)(284 398)(285 399)(286 400)(287 401)(288 402)(289 403)(290 404)(291 405)(292 406)(293 407)(294 408)(295 409)(296 410)(297 411)(298 412)(299 413)(300 414)(301 415)(302 416)(303 417)(304 418)(305 419)(306 420)(307 421)(308 422)(309 423)(310 424)(311 425)(312 426)(313 427)(314 428)(315 429)(316 430)(317 431)(318 432)(319 433)(320 434)(321 435)(322 436)(323 437)(324 438)(325 439)(326 440)(327 441)(328 442)(329 443)(330 444)(331 445)(332 446)(333 447)(334 448)(335 449)(336 450)(337 451)(338 452)(339 453)(340 454)(341 455)(342 456)
(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)(343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456)
(1 343 58 400)(2 456 59 399)(3 455 60 398)(4 454 61 397)(5 453 62 396)(6 452 63 395)(7 451 64 394)(8 450 65 393)(9 449 66 392)(10 448 67 391)(11 447 68 390)(12 446 69 389)(13 445 70 388)(14 444 71 387)(15 443 72 386)(16 442 73 385)(17 441 74 384)(18 440 75 383)(19 439 76 382)(20 438 77 381)(21 437 78 380)(22 436 79 379)(23 435 80 378)(24 434 81 377)(25 433 82 376)(26 432 83 375)(27 431 84 374)(28 430 85 373)(29 429 86 372)(30 428 87 371)(31 427 88 370)(32 426 89 369)(33 425 90 368)(34 424 91 367)(35 423 92 366)(36 422 93 365)(37 421 94 364)(38 420 95 363)(39 419 96 362)(40 418 97 361)(41 417 98 360)(42 416 99 359)(43 415 100 358)(44 414 101 357)(45 413 102 356)(46 412 103 355)(47 411 104 354)(48 410 105 353)(49 409 106 352)(50 408 107 351)(51 407 108 350)(52 406 109 349)(53 405 110 348)(54 404 111 347)(55 403 112 346)(56 402 113 345)(57 401 114 344)(115 304 172 247)(116 303 173 246)(117 302 174 245)(118 301 175 244)(119 300 176 243)(120 299 177 242)(121 298 178 241)(122 297 179 240)(123 296 180 239)(124 295 181 238)(125 294 182 237)(126 293 183 236)(127 292 184 235)(128 291 185 234)(129 290 186 233)(130 289 187 232)(131 288 188 231)(132 287 189 230)(133 286 190 229)(134 285 191 342)(135 284 192 341)(136 283 193 340)(137 282 194 339)(138 281 195 338)(139 280 196 337)(140 279 197 336)(141 278 198 335)(142 277 199 334)(143 276 200 333)(144 275 201 332)(145 274 202 331)(146 273 203 330)(147 272 204 329)(148 271 205 328)(149 270 206 327)(150 269 207 326)(151 268 208 325)(152 267 209 324)(153 266 210 323)(154 265 211 322)(155 264 212 321)(156 263 213 320)(157 262 214 319)(158 261 215 318)(159 260 216 317)(160 259 217 316)(161 258 218 315)(162 257 219 314)(163 256 220 313)(164 255 221 312)(165 254 222 311)(166 253 223 310)(167 252 224 309)(168 251 225 308)(169 250 226 307)(170 249 227 306)(171 248 228 305)
G:=sub<Sym(456)| (1,190)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,197)(9,198)(10,199)(11,200)(12,201)(13,202)(14,203)(15,204)(16,205)(17,206)(18,207)(19,208)(20,209)(21,210)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,218)(30,219)(31,220)(32,221)(33,222)(34,223)(35,224)(36,225)(37,226)(38,227)(39,228)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(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)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(229,343)(230,344)(231,345)(232,346)(233,347)(234,348)(235,349)(236,350)(237,351)(238,352)(239,353)(240,354)(241,355)(242,356)(243,357)(244,358)(245,359)(246,360)(247,361)(248,362)(249,363)(250,364)(251,365)(252,366)(253,367)(254,368)(255,369)(256,370)(257,371)(258,372)(259,373)(260,374)(261,375)(262,376)(263,377)(264,378)(265,379)(266,380)(267,381)(268,382)(269,383)(270,384)(271,385)(272,386)(273,387)(274,388)(275,389)(276,390)(277,391)(278,392)(279,393)(280,394)(281,395)(282,396)(283,397)(284,398)(285,399)(286,400)(287,401)(288,402)(289,403)(290,404)(291,405)(292,406)(293,407)(294,408)(295,409)(296,410)(297,411)(298,412)(299,413)(300,414)(301,415)(302,416)(303,417)(304,418)(305,419)(306,420)(307,421)(308,422)(309,423)(310,424)(311,425)(312,426)(313,427)(314,428)(315,429)(316,430)(317,431)(318,432)(319,433)(320,434)(321,435)(322,436)(323,437)(324,438)(325,439)(326,440)(327,441)(328,442)(329,443)(330,444)(331,445)(332,446)(333,447)(334,448)(335,449)(336,450)(337,451)(338,452)(339,453)(340,454)(341,455)(342,456), (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)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456), (1,343,58,400)(2,456,59,399)(3,455,60,398)(4,454,61,397)(5,453,62,396)(6,452,63,395)(7,451,64,394)(8,450,65,393)(9,449,66,392)(10,448,67,391)(11,447,68,390)(12,446,69,389)(13,445,70,388)(14,444,71,387)(15,443,72,386)(16,442,73,385)(17,441,74,384)(18,440,75,383)(19,439,76,382)(20,438,77,381)(21,437,78,380)(22,436,79,379)(23,435,80,378)(24,434,81,377)(25,433,82,376)(26,432,83,375)(27,431,84,374)(28,430,85,373)(29,429,86,372)(30,428,87,371)(31,427,88,370)(32,426,89,369)(33,425,90,368)(34,424,91,367)(35,423,92,366)(36,422,93,365)(37,421,94,364)(38,420,95,363)(39,419,96,362)(40,418,97,361)(41,417,98,360)(42,416,99,359)(43,415,100,358)(44,414,101,357)(45,413,102,356)(46,412,103,355)(47,411,104,354)(48,410,105,353)(49,409,106,352)(50,408,107,351)(51,407,108,350)(52,406,109,349)(53,405,110,348)(54,404,111,347)(55,403,112,346)(56,402,113,345)(57,401,114,344)(115,304,172,247)(116,303,173,246)(117,302,174,245)(118,301,175,244)(119,300,176,243)(120,299,177,242)(121,298,178,241)(122,297,179,240)(123,296,180,239)(124,295,181,238)(125,294,182,237)(126,293,183,236)(127,292,184,235)(128,291,185,234)(129,290,186,233)(130,289,187,232)(131,288,188,231)(132,287,189,230)(133,286,190,229)(134,285,191,342)(135,284,192,341)(136,283,193,340)(137,282,194,339)(138,281,195,338)(139,280,196,337)(140,279,197,336)(141,278,198,335)(142,277,199,334)(143,276,200,333)(144,275,201,332)(145,274,202,331)(146,273,203,330)(147,272,204,329)(148,271,205,328)(149,270,206,327)(150,269,207,326)(151,268,208,325)(152,267,209,324)(153,266,210,323)(154,265,211,322)(155,264,212,321)(156,263,213,320)(157,262,214,319)(158,261,215,318)(159,260,216,317)(160,259,217,316)(161,258,218,315)(162,257,219,314)(163,256,220,313)(164,255,221,312)(165,254,222,311)(166,253,223,310)(167,252,224,309)(168,251,225,308)(169,250,226,307)(170,249,227,306)(171,248,228,305)>;
G:=Group( (1,190)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,197)(9,198)(10,199)(11,200)(12,201)(13,202)(14,203)(15,204)(16,205)(17,206)(18,207)(19,208)(20,209)(21,210)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,218)(30,219)(31,220)(32,221)(33,222)(34,223)(35,224)(36,225)(37,226)(38,227)(39,228)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(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)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(229,343)(230,344)(231,345)(232,346)(233,347)(234,348)(235,349)(236,350)(237,351)(238,352)(239,353)(240,354)(241,355)(242,356)(243,357)(244,358)(245,359)(246,360)(247,361)(248,362)(249,363)(250,364)(251,365)(252,366)(253,367)(254,368)(255,369)(256,370)(257,371)(258,372)(259,373)(260,374)(261,375)(262,376)(263,377)(264,378)(265,379)(266,380)(267,381)(268,382)(269,383)(270,384)(271,385)(272,386)(273,387)(274,388)(275,389)(276,390)(277,391)(278,392)(279,393)(280,394)(281,395)(282,396)(283,397)(284,398)(285,399)(286,400)(287,401)(288,402)(289,403)(290,404)(291,405)(292,406)(293,407)(294,408)(295,409)(296,410)(297,411)(298,412)(299,413)(300,414)(301,415)(302,416)(303,417)(304,418)(305,419)(306,420)(307,421)(308,422)(309,423)(310,424)(311,425)(312,426)(313,427)(314,428)(315,429)(316,430)(317,431)(318,432)(319,433)(320,434)(321,435)(322,436)(323,437)(324,438)(325,439)(326,440)(327,441)(328,442)(329,443)(330,444)(331,445)(332,446)(333,447)(334,448)(335,449)(336,450)(337,451)(338,452)(339,453)(340,454)(341,455)(342,456), (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)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456), (1,343,58,400)(2,456,59,399)(3,455,60,398)(4,454,61,397)(5,453,62,396)(6,452,63,395)(7,451,64,394)(8,450,65,393)(9,449,66,392)(10,448,67,391)(11,447,68,390)(12,446,69,389)(13,445,70,388)(14,444,71,387)(15,443,72,386)(16,442,73,385)(17,441,74,384)(18,440,75,383)(19,439,76,382)(20,438,77,381)(21,437,78,380)(22,436,79,379)(23,435,80,378)(24,434,81,377)(25,433,82,376)(26,432,83,375)(27,431,84,374)(28,430,85,373)(29,429,86,372)(30,428,87,371)(31,427,88,370)(32,426,89,369)(33,425,90,368)(34,424,91,367)(35,423,92,366)(36,422,93,365)(37,421,94,364)(38,420,95,363)(39,419,96,362)(40,418,97,361)(41,417,98,360)(42,416,99,359)(43,415,100,358)(44,414,101,357)(45,413,102,356)(46,412,103,355)(47,411,104,354)(48,410,105,353)(49,409,106,352)(50,408,107,351)(51,407,108,350)(52,406,109,349)(53,405,110,348)(54,404,111,347)(55,403,112,346)(56,402,113,345)(57,401,114,344)(115,304,172,247)(116,303,173,246)(117,302,174,245)(118,301,175,244)(119,300,176,243)(120,299,177,242)(121,298,178,241)(122,297,179,240)(123,296,180,239)(124,295,181,238)(125,294,182,237)(126,293,183,236)(127,292,184,235)(128,291,185,234)(129,290,186,233)(130,289,187,232)(131,288,188,231)(132,287,189,230)(133,286,190,229)(134,285,191,342)(135,284,192,341)(136,283,193,340)(137,282,194,339)(138,281,195,338)(139,280,196,337)(140,279,197,336)(141,278,198,335)(142,277,199,334)(143,276,200,333)(144,275,201,332)(145,274,202,331)(146,273,203,330)(147,272,204,329)(148,271,205,328)(149,270,206,327)(150,269,207,326)(151,268,208,325)(152,267,209,324)(153,266,210,323)(154,265,211,322)(155,264,212,321)(156,263,213,320)(157,262,214,319)(158,261,215,318)(159,260,216,317)(160,259,217,316)(161,258,218,315)(162,257,219,314)(163,256,220,313)(164,255,221,312)(165,254,222,311)(166,253,223,310)(167,252,224,309)(168,251,225,308)(169,250,226,307)(170,249,227,306)(171,248,228,305) );
G=PermutationGroup([[(1,190),(2,191),(3,192),(4,193),(5,194),(6,195),(7,196),(8,197),(9,198),(10,199),(11,200),(12,201),(13,202),(14,203),(15,204),(16,205),(17,206),(18,207),(19,208),(20,209),(21,210),(22,211),(23,212),(24,213),(25,214),(26,215),(27,216),(28,217),(29,218),(30,219),(31,220),(32,221),(33,222),(34,223),(35,224),(36,225),(37,226),(38,227),(39,228),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,140),(66,141),(67,142),(68,143),(69,144),(70,145),(71,146),(72,147),(73,148),(74,149),(75,150),(76,151),(77,152),(78,153),(79,154),(80,155),(81,156),(82,157),(83,158),(84,159),(85,160),(86,161),(87,162),(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),(109,184),(110,185),(111,186),(112,187),(113,188),(114,189),(229,343),(230,344),(231,345),(232,346),(233,347),(234,348),(235,349),(236,350),(237,351),(238,352),(239,353),(240,354),(241,355),(242,356),(243,357),(244,358),(245,359),(246,360),(247,361),(248,362),(249,363),(250,364),(251,365),(252,366),(253,367),(254,368),(255,369),(256,370),(257,371),(258,372),(259,373),(260,374),(261,375),(262,376),(263,377),(264,378),(265,379),(266,380),(267,381),(268,382),(269,383),(270,384),(271,385),(272,386),(273,387),(274,388),(275,389),(276,390),(277,391),(278,392),(279,393),(280,394),(281,395),(282,396),(283,397),(284,398),(285,399),(286,400),(287,401),(288,402),(289,403),(290,404),(291,405),(292,406),(293,407),(294,408),(295,409),(296,410),(297,411),(298,412),(299,413),(300,414),(301,415),(302,416),(303,417),(304,418),(305,419),(306,420),(307,421),(308,422),(309,423),(310,424),(311,425),(312,426),(313,427),(314,428),(315,429),(316,430),(317,431),(318,432),(319,433),(320,434),(321,435),(322,436),(323,437),(324,438),(325,439),(326,440),(327,441),(328,442),(329,443),(330,444),(331,445),(332,446),(333,447),(334,448),(335,449),(336,450),(337,451),(338,452),(339,453),(340,454),(341,455),(342,456)], [(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),(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456)], [(1,343,58,400),(2,456,59,399),(3,455,60,398),(4,454,61,397),(5,453,62,396),(6,452,63,395),(7,451,64,394),(8,450,65,393),(9,449,66,392),(10,448,67,391),(11,447,68,390),(12,446,69,389),(13,445,70,388),(14,444,71,387),(15,443,72,386),(16,442,73,385),(17,441,74,384),(18,440,75,383),(19,439,76,382),(20,438,77,381),(21,437,78,380),(22,436,79,379),(23,435,80,378),(24,434,81,377),(25,433,82,376),(26,432,83,375),(27,431,84,374),(28,430,85,373),(29,429,86,372),(30,428,87,371),(31,427,88,370),(32,426,89,369),(33,425,90,368),(34,424,91,367),(35,423,92,366),(36,422,93,365),(37,421,94,364),(38,420,95,363),(39,419,96,362),(40,418,97,361),(41,417,98,360),(42,416,99,359),(43,415,100,358),(44,414,101,357),(45,413,102,356),(46,412,103,355),(47,411,104,354),(48,410,105,353),(49,409,106,352),(50,408,107,351),(51,407,108,350),(52,406,109,349),(53,405,110,348),(54,404,111,347),(55,403,112,346),(56,402,113,345),(57,401,114,344),(115,304,172,247),(116,303,173,246),(117,302,174,245),(118,301,175,244),(119,300,176,243),(120,299,177,242),(121,298,178,241),(122,297,179,240),(123,296,180,239),(124,295,181,238),(125,294,182,237),(126,293,183,236),(127,292,184,235),(128,291,185,234),(129,290,186,233),(130,289,187,232),(131,288,188,231),(132,287,189,230),(133,286,190,229),(134,285,191,342),(135,284,192,341),(136,283,193,340),(137,282,194,339),(138,281,195,338),(139,280,196,337),(140,279,197,336),(141,278,198,335),(142,277,199,334),(143,276,200,333),(144,275,201,332),(145,274,202,331),(146,273,203,330),(147,272,204,329),(148,271,205,328),(149,270,206,327),(150,269,207,326),(151,268,208,325),(152,267,209,324),(153,266,210,323),(154,265,211,322),(155,264,212,321),(156,263,213,320),(157,262,214,319),(158,261,215,318),(159,260,216,317),(160,259,217,316),(161,258,218,315),(162,257,219,314),(163,256,220,313),(164,255,221,312),(165,254,222,311),(166,253,223,310),(167,252,224,309),(168,251,225,308),(169,250,226,307),(170,249,227,306),(171,248,228,305)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 19A | ··· | 19I | 38A | ··· | 38AA | 57A | ··· | 57R | 114A | ··· | 114BB |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 1 | 1 | 2 | 57 | 57 | 57 | 57 | 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 | D19 | Dic19 | D38 | D57 | Dic57 | D114 |
| kernel | C2×Dic57 | Dic57 | C2×C114 | C114 | C2×C38 | C38 | C38 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 9 | 18 | 9 | 18 | 36 | 18 |
Matrix representation of C2×Dic57 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 1 | 0 | 0 |
| 0 | 149 | 171 |
| 0 | 58 | 22 |
| 1 | 0 | 0 |
| 0 | 106 | 193 |
| 0 | 204 | 123 |
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[1,0,0,0,149,58,0,171,22],[1,0,0,0,106,204,0,193,123] >;
C2×Dic57 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{57} % in TeX
G:=Group("C2xDic57"); // GroupNames label
G:=SmallGroup(456,37);
// by ID
G=gap.SmallGroup(456,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,20,323,10804]);
// Polycyclic
G:=Group<a,b,c|a^2=b^114=1,c^2=b^57,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export