direct product, abelian, monomial, 2-elementary
Aliases: C22×C114, SmallGroup(456,54)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C114 |
| C1 — C22×C114 |
| C1 — C22×C114 |
Generators and relations for C22×C114
G = < a,b,c | a2=b2=c114=1, ab=ba, ac=ca, bc=cb >
Subgroups: 64, all normal (8 characteristic)
C1, C2, C3, C22, C6, C23, C2×C6, C19, C22×C6, C38, C57, C2×C38, C114, C22×C38, C2×C114, C22×C114
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C19, C22×C6, C38, C57, C2×C38, C114, C22×C38, C2×C114, C22×C114
►Regular action on 456 points
Generators in S456
(1 230)(2 231)(3 232)(4 233)(5 234)(6 235)(7 236)(8 237)(9 238)(10 239)(11 240)(12 241)(13 242)(14 243)(15 244)(16 245)(17 246)(18 247)(19 248)(20 249)(21 250)(22 251)(23 252)(24 253)(25 254)(26 255)(27 256)(28 257)(29 258)(30 259)(31 260)(32 261)(33 262)(34 263)(35 264)(36 265)(37 266)(38 267)(39 268)(40 269)(41 270)(42 271)(43 272)(44 273)(45 274)(46 275)(47 276)(48 277)(49 278)(50 279)(51 280)(52 281)(53 282)(54 283)(55 284)(56 285)(57 286)(58 287)(59 288)(60 289)(61 290)(62 291)(63 292)(64 293)(65 294)(66 295)(67 296)(68 297)(69 298)(70 299)(71 300)(72 301)(73 302)(74 303)(75 304)(76 305)(77 306)(78 307)(79 308)(80 309)(81 310)(82 311)(83 312)(84 313)(85 314)(86 315)(87 316)(88 317)(89 318)(90 319)(91 320)(92 321)(93 322)(94 323)(95 324)(96 325)(97 326)(98 327)(99 328)(100 329)(101 330)(102 331)(103 332)(104 333)(105 334)(106 335)(107 336)(108 337)(109 338)(110 339)(111 340)(112 341)(113 342)(114 229)(115 436)(116 437)(117 438)(118 439)(119 440)(120 441)(121 442)(122 443)(123 444)(124 445)(125 446)(126 447)(127 448)(128 449)(129 450)(130 451)(131 452)(132 453)(133 454)(134 455)(135 456)(136 343)(137 344)(138 345)(139 346)(140 347)(141 348)(142 349)(143 350)(144 351)(145 352)(146 353)(147 354)(148 355)(149 356)(150 357)(151 358)(152 359)(153 360)(154 361)(155 362)(156 363)(157 364)(158 365)(159 366)(160 367)(161 368)(162 369)(163 370)(164 371)(165 372)(166 373)(167 374)(168 375)(169 376)(170 377)(171 378)(172 379)(173 380)(174 381)(175 382)(176 383)(177 384)(178 385)(179 386)(180 387)(181 388)(182 389)(183 390)(184 391)(185 392)(186 393)(187 394)(188 395)(189 396)(190 397)(191 398)(192 399)(193 400)(194 401)(195 402)(196 403)(197 404)(198 405)(199 406)(200 407)(201 408)(202 409)(203 410)(204 411)(205 412)(206 413)(207 414)(208 415)(209 416)(210 417)(211 418)(212 419)(213 420)(214 421)(215 422)(216 423)(217 424)(218 425)(219 426)(220 427)(221 428)(222 429)(223 430)(224 431)(225 432)(226 433)(227 434)(228 435)
(1 146)(2 147)(3 148)(4 149)(5 150)(6 151)(7 152)(8 153)(9 154)(10 155)(11 156)(12 157)(13 158)(14 159)(15 160)(16 161)(17 162)(18 163)(19 164)(20 165)(21 166)(22 167)(23 168)(24 169)(25 170)(26 171)(27 172)(28 173)(29 174)(30 175)(31 176)(32 177)(33 178)(34 179)(35 180)(36 181)(37 182)(38 183)(39 184)(40 185)(41 186)(42 187)(43 188)(44 189)(45 190)(46 191)(47 192)(48 193)(49 194)(50 195)(51 196)(52 197)(53 198)(54 199)(55 200)(56 201)(57 202)(58 203)(59 204)(60 205)(61 206)(62 207)(63 208)(64 209)(65 210)(66 211)(67 212)(68 213)(69 214)(70 215)(71 216)(72 217)(73 218)(74 219)(75 220)(76 221)(77 222)(78 223)(79 224)(80 225)(81 226)(82 227)(83 228)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 131)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 141)(111 142)(112 143)(113 144)(114 145)(229 352)(230 353)(231 354)(232 355)(233 356)(234 357)(235 358)(236 359)(237 360)(238 361)(239 362)(240 363)(241 364)(242 365)(243 366)(244 367)(245 368)(246 369)(247 370)(248 371)(249 372)(250 373)(251 374)(252 375)(253 376)(254 377)(255 378)(256 379)(257 380)(258 381)(259 382)(260 383)(261 384)(262 385)(263 386)(264 387)(265 388)(266 389)(267 390)(268 391)(269 392)(270 393)(271 394)(272 395)(273 396)(274 397)(275 398)(276 399)(277 400)(278 401)(279 402)(280 403)(281 404)(282 405)(283 406)(284 407)(285 408)(286 409)(287 410)(288 411)(289 412)(290 413)(291 414)(292 415)(293 416)(294 417)(295 418)(296 419)(297 420)(298 421)(299 422)(300 423)(301 424)(302 425)(303 426)(304 427)(305 428)(306 429)(307 430)(308 431)(309 432)(310 433)(311 434)(312 435)(313 436)(314 437)(315 438)(316 439)(317 440)(318 441)(319 442)(320 443)(321 444)(322 445)(323 446)(324 447)(325 448)(326 449)(327 450)(328 451)(329 452)(330 453)(331 454)(332 455)(333 456)(334 343)(335 344)(336 345)(337 346)(338 347)(339 348)(340 349)(341 350)(342 351)
(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)(343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456)
G:=sub<Sym(456)| (1,230)(2,231)(3,232)(4,233)(5,234)(6,235)(7,236)(8,237)(9,238)(10,239)(11,240)(12,241)(13,242)(14,243)(15,244)(16,245)(17,246)(18,247)(19,248)(20,249)(21,250)(22,251)(23,252)(24,253)(25,254)(26,255)(27,256)(28,257)(29,258)(30,259)(31,260)(32,261)(33,262)(34,263)(35,264)(36,265)(37,266)(38,267)(39,268)(40,269)(41,270)(42,271)(43,272)(44,273)(45,274)(46,275)(47,276)(48,277)(49,278)(50,279)(51,280)(52,281)(53,282)(54,283)(55,284)(56,285)(57,286)(58,287)(59,288)(60,289)(61,290)(62,291)(63,292)(64,293)(65,294)(66,295)(67,296)(68,297)(69,298)(70,299)(71,300)(72,301)(73,302)(74,303)(75,304)(76,305)(77,306)(78,307)(79,308)(80,309)(81,310)(82,311)(83,312)(84,313)(85,314)(86,315)(87,316)(88,317)(89,318)(90,319)(91,320)(92,321)(93,322)(94,323)(95,324)(96,325)(97,326)(98,327)(99,328)(100,329)(101,330)(102,331)(103,332)(104,333)(105,334)(106,335)(107,336)(108,337)(109,338)(110,339)(111,340)(112,341)(113,342)(114,229)(115,436)(116,437)(117,438)(118,439)(119,440)(120,441)(121,442)(122,443)(123,444)(124,445)(125,446)(126,447)(127,448)(128,449)(129,450)(130,451)(131,452)(132,453)(133,454)(134,455)(135,456)(136,343)(137,344)(138,345)(139,346)(140,347)(141,348)(142,349)(143,350)(144,351)(145,352)(146,353)(147,354)(148,355)(149,356)(150,357)(151,358)(152,359)(153,360)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)(160,367)(161,368)(162,369)(163,370)(164,371)(165,372)(166,373)(167,374)(168,375)(169,376)(170,377)(171,378)(172,379)(173,380)(174,381)(175,382)(176,383)(177,384)(178,385)(179,386)(180,387)(181,388)(182,389)(183,390)(184,391)(185,392)(186,393)(187,394)(188,395)(189,396)(190,397)(191,398)(192,399)(193,400)(194,401)(195,402)(196,403)(197,404)(198,405)(199,406)(200,407)(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414)(208,415)(209,416)(210,417)(211,418)(212,419)(213,420)(214,421)(215,422)(216,423)(217,424)(218,425)(219,426)(220,427)(221,428)(222,429)(223,430)(224,431)(225,432)(226,433)(227,434)(228,435), (1,146)(2,147)(3,148)(4,149)(5,150)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,161)(17,162)(18,163)(19,164)(20,165)(21,166)(22,167)(23,168)(24,169)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,180)(36,181)(37,182)(38,183)(39,184)(40,185)(41,186)(42,187)(43,188)(44,189)(45,190)(46,191)(47,192)(48,193)(49,194)(50,195)(51,196)(52,197)(53,198)(54,199)(55,200)(56,201)(57,202)(58,203)(59,204)(60,205)(61,206)(62,207)(63,208)(64,209)(65,210)(66,211)(67,212)(68,213)(69,214)(70,215)(71,216)(72,217)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,225)(81,226)(82,227)(83,228)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,141)(111,142)(112,143)(113,144)(114,145)(229,352)(230,353)(231,354)(232,355)(233,356)(234,357)(235,358)(236,359)(237,360)(238,361)(239,362)(240,363)(241,364)(242,365)(243,366)(244,367)(245,368)(246,369)(247,370)(248,371)(249,372)(250,373)(251,374)(252,375)(253,376)(254,377)(255,378)(256,379)(257,380)(258,381)(259,382)(260,383)(261,384)(262,385)(263,386)(264,387)(265,388)(266,389)(267,390)(268,391)(269,392)(270,393)(271,394)(272,395)(273,396)(274,397)(275,398)(276,399)(277,400)(278,401)(279,402)(280,403)(281,404)(282,405)(283,406)(284,407)(285,408)(286,409)(287,410)(288,411)(289,412)(290,413)(291,414)(292,415)(293,416)(294,417)(295,418)(296,419)(297,420)(298,421)(299,422)(300,423)(301,424)(302,425)(303,426)(304,427)(305,428)(306,429)(307,430)(308,431)(309,432)(310,433)(311,434)(312,435)(313,436)(314,437)(315,438)(316,439)(317,440)(318,441)(319,442)(320,443)(321,444)(322,445)(323,446)(324,447)(325,448)(326,449)(327,450)(328,451)(329,452)(330,453)(331,454)(332,455)(333,456)(334,343)(335,344)(336,345)(337,346)(338,347)(339,348)(340,349)(341,350)(342,351), (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)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456)>;
G:=Group( (1,230)(2,231)(3,232)(4,233)(5,234)(6,235)(7,236)(8,237)(9,238)(10,239)(11,240)(12,241)(13,242)(14,243)(15,244)(16,245)(17,246)(18,247)(19,248)(20,249)(21,250)(22,251)(23,252)(24,253)(25,254)(26,255)(27,256)(28,257)(29,258)(30,259)(31,260)(32,261)(33,262)(34,263)(35,264)(36,265)(37,266)(38,267)(39,268)(40,269)(41,270)(42,271)(43,272)(44,273)(45,274)(46,275)(47,276)(48,277)(49,278)(50,279)(51,280)(52,281)(53,282)(54,283)(55,284)(56,285)(57,286)(58,287)(59,288)(60,289)(61,290)(62,291)(63,292)(64,293)(65,294)(66,295)(67,296)(68,297)(69,298)(70,299)(71,300)(72,301)(73,302)(74,303)(75,304)(76,305)(77,306)(78,307)(79,308)(80,309)(81,310)(82,311)(83,312)(84,313)(85,314)(86,315)(87,316)(88,317)(89,318)(90,319)(91,320)(92,321)(93,322)(94,323)(95,324)(96,325)(97,326)(98,327)(99,328)(100,329)(101,330)(102,331)(103,332)(104,333)(105,334)(106,335)(107,336)(108,337)(109,338)(110,339)(111,340)(112,341)(113,342)(114,229)(115,436)(116,437)(117,438)(118,439)(119,440)(120,441)(121,442)(122,443)(123,444)(124,445)(125,446)(126,447)(127,448)(128,449)(129,450)(130,451)(131,452)(132,453)(133,454)(134,455)(135,456)(136,343)(137,344)(138,345)(139,346)(140,347)(141,348)(142,349)(143,350)(144,351)(145,352)(146,353)(147,354)(148,355)(149,356)(150,357)(151,358)(152,359)(153,360)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)(160,367)(161,368)(162,369)(163,370)(164,371)(165,372)(166,373)(167,374)(168,375)(169,376)(170,377)(171,378)(172,379)(173,380)(174,381)(175,382)(176,383)(177,384)(178,385)(179,386)(180,387)(181,388)(182,389)(183,390)(184,391)(185,392)(186,393)(187,394)(188,395)(189,396)(190,397)(191,398)(192,399)(193,400)(194,401)(195,402)(196,403)(197,404)(198,405)(199,406)(200,407)(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414)(208,415)(209,416)(210,417)(211,418)(212,419)(213,420)(214,421)(215,422)(216,423)(217,424)(218,425)(219,426)(220,427)(221,428)(222,429)(223,430)(224,431)(225,432)(226,433)(227,434)(228,435), (1,146)(2,147)(3,148)(4,149)(5,150)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,161)(17,162)(18,163)(19,164)(20,165)(21,166)(22,167)(23,168)(24,169)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,180)(36,181)(37,182)(38,183)(39,184)(40,185)(41,186)(42,187)(43,188)(44,189)(45,190)(46,191)(47,192)(48,193)(49,194)(50,195)(51,196)(52,197)(53,198)(54,199)(55,200)(56,201)(57,202)(58,203)(59,204)(60,205)(61,206)(62,207)(63,208)(64,209)(65,210)(66,211)(67,212)(68,213)(69,214)(70,215)(71,216)(72,217)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,225)(81,226)(82,227)(83,228)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,141)(111,142)(112,143)(113,144)(114,145)(229,352)(230,353)(231,354)(232,355)(233,356)(234,357)(235,358)(236,359)(237,360)(238,361)(239,362)(240,363)(241,364)(242,365)(243,366)(244,367)(245,368)(246,369)(247,370)(248,371)(249,372)(250,373)(251,374)(252,375)(253,376)(254,377)(255,378)(256,379)(257,380)(258,381)(259,382)(260,383)(261,384)(262,385)(263,386)(264,387)(265,388)(266,389)(267,390)(268,391)(269,392)(270,393)(271,394)(272,395)(273,396)(274,397)(275,398)(276,399)(277,400)(278,401)(279,402)(280,403)(281,404)(282,405)(283,406)(284,407)(285,408)(286,409)(287,410)(288,411)(289,412)(290,413)(291,414)(292,415)(293,416)(294,417)(295,418)(296,419)(297,420)(298,421)(299,422)(300,423)(301,424)(302,425)(303,426)(304,427)(305,428)(306,429)(307,430)(308,431)(309,432)(310,433)(311,434)(312,435)(313,436)(314,437)(315,438)(316,439)(317,440)(318,441)(319,442)(320,443)(321,444)(322,445)(323,446)(324,447)(325,448)(326,449)(327,450)(328,451)(329,452)(330,453)(331,454)(332,455)(333,456)(334,343)(335,344)(336,345)(337,346)(338,347)(339,348)(340,349)(341,350)(342,351), (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)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456) );
G=PermutationGroup([[(1,230),(2,231),(3,232),(4,233),(5,234),(6,235),(7,236),(8,237),(9,238),(10,239),(11,240),(12,241),(13,242),(14,243),(15,244),(16,245),(17,246),(18,247),(19,248),(20,249),(21,250),(22,251),(23,252),(24,253),(25,254),(26,255),(27,256),(28,257),(29,258),(30,259),(31,260),(32,261),(33,262),(34,263),(35,264),(36,265),(37,266),(38,267),(39,268),(40,269),(41,270),(42,271),(43,272),(44,273),(45,274),(46,275),(47,276),(48,277),(49,278),(50,279),(51,280),(52,281),(53,282),(54,283),(55,284),(56,285),(57,286),(58,287),(59,288),(60,289),(61,290),(62,291),(63,292),(64,293),(65,294),(66,295),(67,296),(68,297),(69,298),(70,299),(71,300),(72,301),(73,302),(74,303),(75,304),(76,305),(77,306),(78,307),(79,308),(80,309),(81,310),(82,311),(83,312),(84,313),(85,314),(86,315),(87,316),(88,317),(89,318),(90,319),(91,320),(92,321),(93,322),(94,323),(95,324),(96,325),(97,326),(98,327),(99,328),(100,329),(101,330),(102,331),(103,332),(104,333),(105,334),(106,335),(107,336),(108,337),(109,338),(110,339),(111,340),(112,341),(113,342),(114,229),(115,436),(116,437),(117,438),(118,439),(119,440),(120,441),(121,442),(122,443),(123,444),(124,445),(125,446),(126,447),(127,448),(128,449),(129,450),(130,451),(131,452),(132,453),(133,454),(134,455),(135,456),(136,343),(137,344),(138,345),(139,346),(140,347),(141,348),(142,349),(143,350),(144,351),(145,352),(146,353),(147,354),(148,355),(149,356),(150,357),(151,358),(152,359),(153,360),(154,361),(155,362),(156,363),(157,364),(158,365),(159,366),(160,367),(161,368),(162,369),(163,370),(164,371),(165,372),(166,373),(167,374),(168,375),(169,376),(170,377),(171,378),(172,379),(173,380),(174,381),(175,382),(176,383),(177,384),(178,385),(179,386),(180,387),(181,388),(182,389),(183,390),(184,391),(185,392),(186,393),(187,394),(188,395),(189,396),(190,397),(191,398),(192,399),(193,400),(194,401),(195,402),(196,403),(197,404),(198,405),(199,406),(200,407),(201,408),(202,409),(203,410),(204,411),(205,412),(206,413),(207,414),(208,415),(209,416),(210,417),(211,418),(212,419),(213,420),(214,421),(215,422),(216,423),(217,424),(218,425),(219,426),(220,427),(221,428),(222,429),(223,430),(224,431),(225,432),(226,433),(227,434),(228,435)], [(1,146),(2,147),(3,148),(4,149),(5,150),(6,151),(7,152),(8,153),(9,154),(10,155),(11,156),(12,157),(13,158),(14,159),(15,160),(16,161),(17,162),(18,163),(19,164),(20,165),(21,166),(22,167),(23,168),(24,169),(25,170),(26,171),(27,172),(28,173),(29,174),(30,175),(31,176),(32,177),(33,178),(34,179),(35,180),(36,181),(37,182),(38,183),(39,184),(40,185),(41,186),(42,187),(43,188),(44,189),(45,190),(46,191),(47,192),(48,193),(49,194),(50,195),(51,196),(52,197),(53,198),(54,199),(55,200),(56,201),(57,202),(58,203),(59,204),(60,205),(61,206),(62,207),(63,208),(64,209),(65,210),(66,211),(67,212),(68,213),(69,214),(70,215),(71,216),(72,217),(73,218),(74,219),(75,220),(76,221),(77,222),(78,223),(79,224),(80,225),(81,226),(82,227),(83,228),(84,115),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127),(97,128),(98,129),(99,130),(100,131),(101,132),(102,133),(103,134),(104,135),(105,136),(106,137),(107,138),(108,139),(109,140),(110,141),(111,142),(112,143),(113,144),(114,145),(229,352),(230,353),(231,354),(232,355),(233,356),(234,357),(235,358),(236,359),(237,360),(238,361),(239,362),(240,363),(241,364),(242,365),(243,366),(244,367),(245,368),(246,369),(247,370),(248,371),(249,372),(250,373),(251,374),(252,375),(253,376),(254,377),(255,378),(256,379),(257,380),(258,381),(259,382),(260,383),(261,384),(262,385),(263,386),(264,387),(265,388),(266,389),(267,390),(268,391),(269,392),(270,393),(271,394),(272,395),(273,396),(274,397),(275,398),(276,399),(277,400),(278,401),(279,402),(280,403),(281,404),(282,405),(283,406),(284,407),(285,408),(286,409),(287,410),(288,411),(289,412),(290,413),(291,414),(292,415),(293,416),(294,417),(295,418),(296,419),(297,420),(298,421),(299,422),(300,423),(301,424),(302,425),(303,426),(304,427),(305,428),(306,429),(307,430),(308,431),(309,432),(310,433),(311,434),(312,435),(313,436),(314,437),(315,438),(316,439),(317,440),(318,441),(319,442),(320,443),(321,444),(322,445),(323,446),(324,447),(325,448),(326,449),(327,450),(328,451),(329,452),(330,453),(331,454),(332,455),(333,456),(334,343),(335,344),(336,345),(337,346),(338,347),(339,348),(340,349),(341,350),(342,351)], [(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),(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456)]])
456 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 6A | ··· | 6N | 19A | ··· | 19R | 38A | ··· | 38DV | 57A | ··· | 57AJ | 114A | ··· | 114IR |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
456 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C19 | C38 | C57 | C114 |
| kernel | C22×C114 | C2×C114 | C22×C38 | C2×C38 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 7 | 2 | 14 | 18 | 126 | 36 | 252 |
Matrix representation of C22×C114 ►in GL3(𝔽229) generated by
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 228 |
| 228 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 192 | 0 | 0 |
| 0 | 55 | 0 |
| 0 | 0 | 215 |
G:=sub<GL(3,GF(229))| [1,0,0,0,1,0,0,0,228],[228,0,0,0,228,0,0,0,228],[192,0,0,0,55,0,0,0,215] >;
C22×C114 in GAP, Magma, Sage, TeX
C_2^2\times C_{114} % in TeX
G:=Group("C2^2xC114"); // GroupNames label
G:=SmallGroup(456,54);
// by ID
G=gap.SmallGroup(456,54);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^114=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations