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## G = C22×C116order 464 = 24·29

### Abelian group of type [2,2,116]

Aliases: C22×C116, SmallGroup(464,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C116
 Chief series C1 — C2 — C58 — C116 — C2×C116 — C22×C116
 Lower central C1 — C22×C116
 Upper central C1 — C22×C116

Generators and relations for C22×C116
G = < a,b,c | a2=b2=c116=1, ab=ba, ac=ca, bc=cb >

Subgroups: 54, all normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C29, C58, C58 [×6], C116 [×4], C2×C58 [×7], C2×C116 [×6], C22×C58, C22×C116
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C29, C58 [×7], C116 [×4], C2×C58 [×7], C2×C116 [×6], C22×C58, C22×C116

Smallest permutation representation of C22×C116
Regular action on 464 points
Generators in S464
(1 172)(2 173)(3 174)(4 175)(5 176)(6 177)(7 178)(8 179)(9 180)(10 181)(11 182)(12 183)(13 184)(14 185)(15 186)(16 187)(17 188)(18 189)(19 190)(20 191)(21 192)(22 193)(23 194)(24 195)(25 196)(26 197)(27 198)(28 199)(29 200)(30 201)(31 202)(32 203)(33 204)(34 205)(35 206)(36 207)(37 208)(38 209)(39 210)(40 211)(41 212)(42 213)(43 214)(44 215)(45 216)(46 217)(47 218)(48 219)(49 220)(50 221)(51 222)(52 223)(53 224)(54 225)(55 226)(56 227)(57 228)(58 229)(59 230)(60 231)(61 232)(62 117)(63 118)(64 119)(65 120)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)(101 156)(102 157)(103 158)(104 159)(105 160)(106 161)(107 162)(108 163)(109 164)(110 165)(111 166)(112 167)(113 168)(114 169)(115 170)(116 171)(233 370)(234 371)(235 372)(236 373)(237 374)(238 375)(239 376)(240 377)(241 378)(242 379)(243 380)(244 381)(245 382)(246 383)(247 384)(248 385)(249 386)(250 387)(251 388)(252 389)(253 390)(254 391)(255 392)(256 393)(257 394)(258 395)(259 396)(260 397)(261 398)(262 399)(263 400)(264 401)(265 402)(266 403)(267 404)(268 405)(269 406)(270 407)(271 408)(272 409)(273 410)(274 411)(275 412)(276 413)(277 414)(278 415)(279 416)(280 417)(281 418)(282 419)(283 420)(284 421)(285 422)(286 423)(287 424)(288 425)(289 426)(290 427)(291 428)(292 429)(293 430)(294 431)(295 432)(296 433)(297 434)(298 435)(299 436)(300 437)(301 438)(302 439)(303 440)(304 441)(305 442)(306 443)(307 444)(308 445)(309 446)(310 447)(311 448)(312 449)(313 450)(314 451)(315 452)(316 453)(317 454)(318 455)(319 456)(320 457)(321 458)(322 459)(323 460)(324 461)(325 462)(326 463)(327 464)(328 349)(329 350)(330 351)(331 352)(332 353)(333 354)(334 355)(335 356)(336 357)(337 358)(338 359)(339 360)(340 361)(341 362)(342 363)(343 364)(344 365)(345 366)(346 367)(347 368)(348 369)
(1 281)(2 282)(3 283)(4 284)(5 285)(6 286)(7 287)(8 288)(9 289)(10 290)(11 291)(12 292)(13 293)(14 294)(15 295)(16 296)(17 297)(18 298)(19 299)(20 300)(21 301)(22 302)(23 303)(24 304)(25 305)(26 306)(27 307)(28 308)(29 309)(30 310)(31 311)(32 312)(33 313)(34 314)(35 315)(36 316)(37 317)(38 318)(39 319)(40 320)(41 321)(42 322)(43 323)(44 324)(45 325)(46 326)(47 327)(48 328)(49 329)(50 330)(51 331)(52 332)(53 333)(54 334)(55 335)(56 336)(57 337)(58 338)(59 339)(60 340)(61 341)(62 342)(63 343)(64 344)(65 345)(66 346)(67 347)(68 348)(69 233)(70 234)(71 235)(72 236)(73 237)(74 238)(75 239)(76 240)(77 241)(78 242)(79 243)(80 244)(81 245)(82 246)(83 247)(84 248)(85 249)(86 250)(87 251)(88 252)(89 253)(90 254)(91 255)(92 256)(93 257)(94 258)(95 259)(96 260)(97 261)(98 262)(99 263)(100 264)(101 265)(102 266)(103 267)(104 268)(105 269)(106 270)(107 271)(108 272)(109 273)(110 274)(111 275)(112 276)(113 277)(114 278)(115 279)(116 280)(117 363)(118 364)(119 365)(120 366)(121 367)(122 368)(123 369)(124 370)(125 371)(126 372)(127 373)(128 374)(129 375)(130 376)(131 377)(132 378)(133 379)(134 380)(135 381)(136 382)(137 383)(138 384)(139 385)(140 386)(141 387)(142 388)(143 389)(144 390)(145 391)(146 392)(147 393)(148 394)(149 395)(150 396)(151 397)(152 398)(153 399)(154 400)(155 401)(156 402)(157 403)(158 404)(159 405)(160 406)(161 407)(162 408)(163 409)(164 410)(165 411)(166 412)(167 413)(168 414)(169 415)(170 416)(171 417)(172 418)(173 419)(174 420)(175 421)(176 422)(177 423)(178 424)(179 425)(180 426)(181 427)(182 428)(183 429)(184 430)(185 431)(186 432)(187 433)(188 434)(189 435)(190 436)(191 437)(192 438)(193 439)(194 440)(195 441)(196 442)(197 443)(198 444)(199 445)(200 446)(201 447)(202 448)(203 449)(204 450)(205 451)(206 452)(207 453)(208 454)(209 455)(210 456)(211 457)(212 458)(213 459)(214 460)(215 461)(216 462)(217 463)(218 464)(219 349)(220 350)(221 351)(222 352)(223 353)(224 354)(225 355)(226 356)(227 357)(228 358)(229 359)(230 360)(231 361)(232 362)
(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)(349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464)

G:=sub<Sym(464)| (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,181)(11,182)(12,183)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,197)(27,198)(28,199)(29,200)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,229)(59,230)(60,231)(61,232)(62,117)(63,118)(64,119)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)(101,156)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(233,370)(234,371)(235,372)(236,373)(237,374)(238,375)(239,376)(240,377)(241,378)(242,379)(243,380)(244,381)(245,382)(246,383)(247,384)(248,385)(249,386)(250,387)(251,388)(252,389)(253,390)(254,391)(255,392)(256,393)(257,394)(258,395)(259,396)(260,397)(261,398)(262,399)(263,400)(264,401)(265,402)(266,403)(267,404)(268,405)(269,406)(270,407)(271,408)(272,409)(273,410)(274,411)(275,412)(276,413)(277,414)(278,415)(279,416)(280,417)(281,418)(282,419)(283,420)(284,421)(285,422)(286,423)(287,424)(288,425)(289,426)(290,427)(291,428)(292,429)(293,430)(294,431)(295,432)(296,433)(297,434)(298,435)(299,436)(300,437)(301,438)(302,439)(303,440)(304,441)(305,442)(306,443)(307,444)(308,445)(309,446)(310,447)(311,448)(312,449)(313,450)(314,451)(315,452)(316,453)(317,454)(318,455)(319,456)(320,457)(321,458)(322,459)(323,460)(324,461)(325,462)(326,463)(327,464)(328,349)(329,350)(330,351)(331,352)(332,353)(333,354)(334,355)(335,356)(336,357)(337,358)(338,359)(339,360)(340,361)(341,362)(342,363)(343,364)(344,365)(345,366)(346,367)(347,368)(348,369), (1,281)(2,282)(3,283)(4,284)(5,285)(6,286)(7,287)(8,288)(9,289)(10,290)(11,291)(12,292)(13,293)(14,294)(15,295)(16,296)(17,297)(18,298)(19,299)(20,300)(21,301)(22,302)(23,303)(24,304)(25,305)(26,306)(27,307)(28,308)(29,309)(30,310)(31,311)(32,312)(33,313)(34,314)(35,315)(36,316)(37,317)(38,318)(39,319)(40,320)(41,321)(42,322)(43,323)(44,324)(45,325)(46,326)(47,327)(48,328)(49,329)(50,330)(51,331)(52,332)(53,333)(54,334)(55,335)(56,336)(57,337)(58,338)(59,339)(60,340)(61,341)(62,342)(63,343)(64,344)(65,345)(66,346)(67,347)(68,348)(69,233)(70,234)(71,235)(72,236)(73,237)(74,238)(75,239)(76,240)(77,241)(78,242)(79,243)(80,244)(81,245)(82,246)(83,247)(84,248)(85,249)(86,250)(87,251)(88,252)(89,253)(90,254)(91,255)(92,256)(93,257)(94,258)(95,259)(96,260)(97,261)(98,262)(99,263)(100,264)(101,265)(102,266)(103,267)(104,268)(105,269)(106,270)(107,271)(108,272)(109,273)(110,274)(111,275)(112,276)(113,277)(114,278)(115,279)(116,280)(117,363)(118,364)(119,365)(120,366)(121,367)(122,368)(123,369)(124,370)(125,371)(126,372)(127,373)(128,374)(129,375)(130,376)(131,377)(132,378)(133,379)(134,380)(135,381)(136,382)(137,383)(138,384)(139,385)(140,386)(141,387)(142,388)(143,389)(144,390)(145,391)(146,392)(147,393)(148,394)(149,395)(150,396)(151,397)(152,398)(153,399)(154,400)(155,401)(156,402)(157,403)(158,404)(159,405)(160,406)(161,407)(162,408)(163,409)(164,410)(165,411)(166,412)(167,413)(168,414)(169,415)(170,416)(171,417)(172,418)(173,419)(174,420)(175,421)(176,422)(177,423)(178,424)(179,425)(180,426)(181,427)(182,428)(183,429)(184,430)(185,431)(186,432)(187,433)(188,434)(189,435)(190,436)(191,437)(192,438)(193,439)(194,440)(195,441)(196,442)(197,443)(198,444)(199,445)(200,446)(201,447)(202,448)(203,449)(204,450)(205,451)(206,452)(207,453)(208,454)(209,455)(210,456)(211,457)(212,458)(213,459)(214,460)(215,461)(216,462)(217,463)(218,464)(219,349)(220,350)(221,351)(222,352)(223,353)(224,354)(225,355)(226,356)(227,357)(228,358)(229,359)(230,360)(231,361)(232,362), (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)(349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464)>;

G:=Group( (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,181)(11,182)(12,183)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,197)(27,198)(28,199)(29,200)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,229)(59,230)(60,231)(61,232)(62,117)(63,118)(64,119)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)(101,156)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(233,370)(234,371)(235,372)(236,373)(237,374)(238,375)(239,376)(240,377)(241,378)(242,379)(243,380)(244,381)(245,382)(246,383)(247,384)(248,385)(249,386)(250,387)(251,388)(252,389)(253,390)(254,391)(255,392)(256,393)(257,394)(258,395)(259,396)(260,397)(261,398)(262,399)(263,400)(264,401)(265,402)(266,403)(267,404)(268,405)(269,406)(270,407)(271,408)(272,409)(273,410)(274,411)(275,412)(276,413)(277,414)(278,415)(279,416)(280,417)(281,418)(282,419)(283,420)(284,421)(285,422)(286,423)(287,424)(288,425)(289,426)(290,427)(291,428)(292,429)(293,430)(294,431)(295,432)(296,433)(297,434)(298,435)(299,436)(300,437)(301,438)(302,439)(303,440)(304,441)(305,442)(306,443)(307,444)(308,445)(309,446)(310,447)(311,448)(312,449)(313,450)(314,451)(315,452)(316,453)(317,454)(318,455)(319,456)(320,457)(321,458)(322,459)(323,460)(324,461)(325,462)(326,463)(327,464)(328,349)(329,350)(330,351)(331,352)(332,353)(333,354)(334,355)(335,356)(336,357)(337,358)(338,359)(339,360)(340,361)(341,362)(342,363)(343,364)(344,365)(345,366)(346,367)(347,368)(348,369), (1,281)(2,282)(3,283)(4,284)(5,285)(6,286)(7,287)(8,288)(9,289)(10,290)(11,291)(12,292)(13,293)(14,294)(15,295)(16,296)(17,297)(18,298)(19,299)(20,300)(21,301)(22,302)(23,303)(24,304)(25,305)(26,306)(27,307)(28,308)(29,309)(30,310)(31,311)(32,312)(33,313)(34,314)(35,315)(36,316)(37,317)(38,318)(39,319)(40,320)(41,321)(42,322)(43,323)(44,324)(45,325)(46,326)(47,327)(48,328)(49,329)(50,330)(51,331)(52,332)(53,333)(54,334)(55,335)(56,336)(57,337)(58,338)(59,339)(60,340)(61,341)(62,342)(63,343)(64,344)(65,345)(66,346)(67,347)(68,348)(69,233)(70,234)(71,235)(72,236)(73,237)(74,238)(75,239)(76,240)(77,241)(78,242)(79,243)(80,244)(81,245)(82,246)(83,247)(84,248)(85,249)(86,250)(87,251)(88,252)(89,253)(90,254)(91,255)(92,256)(93,257)(94,258)(95,259)(96,260)(97,261)(98,262)(99,263)(100,264)(101,265)(102,266)(103,267)(104,268)(105,269)(106,270)(107,271)(108,272)(109,273)(110,274)(111,275)(112,276)(113,277)(114,278)(115,279)(116,280)(117,363)(118,364)(119,365)(120,366)(121,367)(122,368)(123,369)(124,370)(125,371)(126,372)(127,373)(128,374)(129,375)(130,376)(131,377)(132,378)(133,379)(134,380)(135,381)(136,382)(137,383)(138,384)(139,385)(140,386)(141,387)(142,388)(143,389)(144,390)(145,391)(146,392)(147,393)(148,394)(149,395)(150,396)(151,397)(152,398)(153,399)(154,400)(155,401)(156,402)(157,403)(158,404)(159,405)(160,406)(161,407)(162,408)(163,409)(164,410)(165,411)(166,412)(167,413)(168,414)(169,415)(170,416)(171,417)(172,418)(173,419)(174,420)(175,421)(176,422)(177,423)(178,424)(179,425)(180,426)(181,427)(182,428)(183,429)(184,430)(185,431)(186,432)(187,433)(188,434)(189,435)(190,436)(191,437)(192,438)(193,439)(194,440)(195,441)(196,442)(197,443)(198,444)(199,445)(200,446)(201,447)(202,448)(203,449)(204,450)(205,451)(206,452)(207,453)(208,454)(209,455)(210,456)(211,457)(212,458)(213,459)(214,460)(215,461)(216,462)(217,463)(218,464)(219,349)(220,350)(221,351)(222,352)(223,353)(224,354)(225,355)(226,356)(227,357)(228,358)(229,359)(230,360)(231,361)(232,362), (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)(349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464) );

G=PermutationGroup([(1,172),(2,173),(3,174),(4,175),(5,176),(6,177),(7,178),(8,179),(9,180),(10,181),(11,182),(12,183),(13,184),(14,185),(15,186),(16,187),(17,188),(18,189),(19,190),(20,191),(21,192),(22,193),(23,194),(24,195),(25,196),(26,197),(27,198),(28,199),(29,200),(30,201),(31,202),(32,203),(33,204),(34,205),(35,206),(36,207),(37,208),(38,209),(39,210),(40,211),(41,212),(42,213),(43,214),(44,215),(45,216),(46,217),(47,218),(48,219),(49,220),(50,221),(51,222),(52,223),(53,224),(54,225),(55,226),(56,227),(57,228),(58,229),(59,230),(60,231),(61,232),(62,117),(63,118),(64,119),(65,120),(66,121),(67,122),(68,123),(69,124),(70,125),(71,126),(72,127),(73,128),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,136),(82,137),(83,138),(84,139),(85,140),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,155),(101,156),(102,157),(103,158),(104,159),(105,160),(106,161),(107,162),(108,163),(109,164),(110,165),(111,166),(112,167),(113,168),(114,169),(115,170),(116,171),(233,370),(234,371),(235,372),(236,373),(237,374),(238,375),(239,376),(240,377),(241,378),(242,379),(243,380),(244,381),(245,382),(246,383),(247,384),(248,385),(249,386),(250,387),(251,388),(252,389),(253,390),(254,391),(255,392),(256,393),(257,394),(258,395),(259,396),(260,397),(261,398),(262,399),(263,400),(264,401),(265,402),(266,403),(267,404),(268,405),(269,406),(270,407),(271,408),(272,409),(273,410),(274,411),(275,412),(276,413),(277,414),(278,415),(279,416),(280,417),(281,418),(282,419),(283,420),(284,421),(285,422),(286,423),(287,424),(288,425),(289,426),(290,427),(291,428),(292,429),(293,430),(294,431),(295,432),(296,433),(297,434),(298,435),(299,436),(300,437),(301,438),(302,439),(303,440),(304,441),(305,442),(306,443),(307,444),(308,445),(309,446),(310,447),(311,448),(312,449),(313,450),(314,451),(315,452),(316,453),(317,454),(318,455),(319,456),(320,457),(321,458),(322,459),(323,460),(324,461),(325,462),(326,463),(327,464),(328,349),(329,350),(330,351),(331,352),(332,353),(333,354),(334,355),(335,356),(336,357),(337,358),(338,359),(339,360),(340,361),(341,362),(342,363),(343,364),(344,365),(345,366),(346,367),(347,368),(348,369)], [(1,281),(2,282),(3,283),(4,284),(5,285),(6,286),(7,287),(8,288),(9,289),(10,290),(11,291),(12,292),(13,293),(14,294),(15,295),(16,296),(17,297),(18,298),(19,299),(20,300),(21,301),(22,302),(23,303),(24,304),(25,305),(26,306),(27,307),(28,308),(29,309),(30,310),(31,311),(32,312),(33,313),(34,314),(35,315),(36,316),(37,317),(38,318),(39,319),(40,320),(41,321),(42,322),(43,323),(44,324),(45,325),(46,326),(47,327),(48,328),(49,329),(50,330),(51,331),(52,332),(53,333),(54,334),(55,335),(56,336),(57,337),(58,338),(59,339),(60,340),(61,341),(62,342),(63,343),(64,344),(65,345),(66,346),(67,347),(68,348),(69,233),(70,234),(71,235),(72,236),(73,237),(74,238),(75,239),(76,240),(77,241),(78,242),(79,243),(80,244),(81,245),(82,246),(83,247),(84,248),(85,249),(86,250),(87,251),(88,252),(89,253),(90,254),(91,255),(92,256),(93,257),(94,258),(95,259),(96,260),(97,261),(98,262),(99,263),(100,264),(101,265),(102,266),(103,267),(104,268),(105,269),(106,270),(107,271),(108,272),(109,273),(110,274),(111,275),(112,276),(113,277),(114,278),(115,279),(116,280),(117,363),(118,364),(119,365),(120,366),(121,367),(122,368),(123,369),(124,370),(125,371),(126,372),(127,373),(128,374),(129,375),(130,376),(131,377),(132,378),(133,379),(134,380),(135,381),(136,382),(137,383),(138,384),(139,385),(140,386),(141,387),(142,388),(143,389),(144,390),(145,391),(146,392),(147,393),(148,394),(149,395),(150,396),(151,397),(152,398),(153,399),(154,400),(155,401),(156,402),(157,403),(158,404),(159,405),(160,406),(161,407),(162,408),(163,409),(164,410),(165,411),(166,412),(167,413),(168,414),(169,415),(170,416),(171,417),(172,418),(173,419),(174,420),(175,421),(176,422),(177,423),(178,424),(179,425),(180,426),(181,427),(182,428),(183,429),(184,430),(185,431),(186,432),(187,433),(188,434),(189,435),(190,436),(191,437),(192,438),(193,439),(194,440),(195,441),(196,442),(197,443),(198,444),(199,445),(200,446),(201,447),(202,448),(203,449),(204,450),(205,451),(206,452),(207,453),(208,454),(209,455),(210,456),(211,457),(212,458),(213,459),(214,460),(215,461),(216,462),(217,463),(218,464),(219,349),(220,350),(221,351),(222,352),(223,353),(224,354),(225,355),(226,356),(227,357),(228,358),(229,359),(230,360),(231,361),(232,362)], [(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),(349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464)])

464 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 29A ··· 29AB 58A ··· 58GN 116A ··· 116HP order 1 2 ··· 2 4 ··· 4 29 ··· 29 58 ··· 58 116 ··· 116 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

464 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C29 C58 C58 C116 kernel C22×C116 C2×C116 C22×C58 C2×C58 C22×C4 C2×C4 C23 C22 # reps 1 6 1 8 28 168 28 224

Matrix representation of C22×C116 in GL3(𝔽233) generated by

 1 0 0 0 1 0 0 0 232
,
 232 0 0 0 1 0 0 0 232
,
 217 0 0 0 120 0 0 0 133
G:=sub<GL(3,GF(233))| [1,0,0,0,1,0,0,0,232],[232,0,0,0,1,0,0,0,232],[217,0,0,0,120,0,0,0,133] >;

C22×C116 in GAP, Magma, Sage, TeX

C_2^2\times C_{116}
% in TeX

G:=Group("C2^2xC116");
// GroupNames label

G:=SmallGroup(464,45);
// by ID

G=gap.SmallGroup(464,45);
# by ID

G:=PCGroup([5,-2,-2,-2,-29,-2,1160]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^116=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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