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## G = C22×C114order 456 = 23·3·19

### Abelian group of type [2,2,114]

Aliases: C22×C114, SmallGroup(456,54)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C114
 Chief series C1 — C19 — C57 — C114 — C2×C114 — C22×C114
 Lower central C1 — C22×C114
 Upper central 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 [×7], C3, C22 [×7], C6 [×7], C23, C2×C6 [×7], C19, C22×C6, C38 [×7], C57, C2×C38 [×7], C114 [×7], C22×C38, C2×C114 [×7], C22×C114
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C2×C6 [×7], C19, C22×C6, C38 [×7], C57, C2×C38 [×7], C114 [×7], C22×C38, C2×C114 [×7], C22×C114

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

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