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G = C22×C110order 440 = 23·5·11

Abelian group of type [2,2,110]

Aliases: C22×C110, SmallGroup(440,51)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C110
 Chief series C1 — C11 — C55 — C110 — C2×C110 — C22×C110
 Lower central C1 — C22×C110
 Upper central C1 — C22×C110

Generators and relations for C22×C110
G = < a,b,c | a2=b2=c110=1, ab=ba, ac=ca, bc=cb >

Subgroups: 64, all normal (8 characteristic)
C1, C2 [×7], C22 [×7], C5, C23, C10 [×7], C11, C2×C10 [×7], C22 [×7], C22×C10, C2×C22 [×7], C55, C22×C22, C110 [×7], C2×C110 [×7], C22×C110
Quotients: C1, C2 [×7], C22 [×7], C5, C23, C10 [×7], C11, C2×C10 [×7], C22 [×7], C22×C10, C2×C22 [×7], C55, C22×C22, C110 [×7], C2×C110 [×7], C22×C110

Smallest permutation representation of C22×C110
Regular action on 440 points
Generators in S440
(1 318)(2 319)(3 320)(4 321)(5 322)(6 323)(7 324)(8 325)(9 326)(10 327)(11 328)(12 329)(13 330)(14 221)(15 222)(16 223)(17 224)(18 225)(19 226)(20 227)(21 228)(22 229)(23 230)(24 231)(25 232)(26 233)(27 234)(28 235)(29 236)(30 237)(31 238)(32 239)(33 240)(34 241)(35 242)(36 243)(37 244)(38 245)(39 246)(40 247)(41 248)(42 249)(43 250)(44 251)(45 252)(46 253)(47 254)(48 255)(49 256)(50 257)(51 258)(52 259)(53 260)(54 261)(55 262)(56 263)(57 264)(58 265)(59 266)(60 267)(61 268)(62 269)(63 270)(64 271)(65 272)(66 273)(67 274)(68 275)(69 276)(70 277)(71 278)(72 279)(73 280)(74 281)(75 282)(76 283)(77 284)(78 285)(79 286)(80 287)(81 288)(82 289)(83 290)(84 291)(85 292)(86 293)(87 294)(88 295)(89 296)(90 297)(91 298)(92 299)(93 300)(94 301)(95 302)(96 303)(97 304)(98 305)(99 306)(100 307)(101 308)(102 309)(103 310)(104 311)(105 312)(106 313)(107 314)(108 315)(109 316)(110 317)(111 421)(112 422)(113 423)(114 424)(115 425)(116 426)(117 427)(118 428)(119 429)(120 430)(121 431)(122 432)(123 433)(124 434)(125 435)(126 436)(127 437)(128 438)(129 439)(130 440)(131 331)(132 332)(133 333)(134 334)(135 335)(136 336)(137 337)(138 338)(139 339)(140 340)(141 341)(142 342)(143 343)(144 344)(145 345)(146 346)(147 347)(148 348)(149 349)(150 350)(151 351)(152 352)(153 353)(154 354)(155 355)(156 356)(157 357)(158 358)(159 359)(160 360)(161 361)(162 362)(163 363)(164 364)(165 365)(166 366)(167 367)(168 368)(169 369)(170 370)(171 371)(172 372)(173 373)(174 374)(175 375)(176 376)(177 377)(178 378)(179 379)(180 380)(181 381)(182 382)(183 383)(184 384)(185 385)(186 386)(187 387)(188 388)(189 389)(190 390)(191 391)(192 392)(193 393)(194 394)(195 395)(196 396)(197 397)(198 398)(199 399)(200 400)(201 401)(202 402)(203 403)(204 404)(205 405)(206 406)(207 407)(208 408)(209 409)(210 410)(211 411)(212 412)(213 413)(214 414)(215 415)(216 416)(217 417)(218 418)(219 419)(220 420)
(1 217)(2 218)(3 219)(4 220)(5 111)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 141)(36 142)(37 143)(38 144)(39 145)(40 146)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 161)(56 162)(57 163)(58 164)(59 165)(60 166)(61 167)(62 168)(63 169)(64 170)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 177)(72 178)(73 179)(74 180)(75 181)(76 182)(77 183)(78 184)(79 185)(80 186)(81 187)(82 188)(83 189)(84 190)(85 191)(86 192)(87 193)(88 194)(89 195)(90 196)(91 197)(92 198)(93 199)(94 200)(95 201)(96 202)(97 203)(98 204)(99 205)(100 206)(101 207)(102 208)(103 209)(104 210)(105 211)(106 212)(107 213)(108 214)(109 215)(110 216)(221 430)(222 431)(223 432)(224 433)(225 434)(226 435)(227 436)(228 437)(229 438)(230 439)(231 440)(232 331)(233 332)(234 333)(235 334)(236 335)(237 336)(238 337)(239 338)(240 339)(241 340)(242 341)(243 342)(244 343)(245 344)(246 345)(247 346)(248 347)(249 348)(250 349)(251 350)(252 351)(253 352)(254 353)(255 354)(256 355)(257 356)(258 357)(259 358)(260 359)(261 360)(262 361)(263 362)(264 363)(265 364)(266 365)(267 366)(268 367)(269 368)(270 369)(271 370)(272 371)(273 372)(274 373)(275 374)(276 375)(277 376)(278 377)(279 378)(280 379)(281 380)(282 381)(283 382)(284 383)(285 384)(286 385)(287 386)(288 387)(289 388)(290 389)(291 390)(292 391)(293 392)(294 393)(295 394)(296 395)(297 396)(298 397)(299 398)(300 399)(301 400)(302 401)(303 402)(304 403)(305 404)(306 405)(307 406)(308 407)(309 408)(310 409)(311 410)(312 411)(313 412)(314 413)(315 414)(316 415)(317 416)(318 417)(319 418)(320 419)(321 420)(322 421)(323 422)(324 423)(325 424)(326 425)(327 426)(328 427)(329 428)(330 429)
(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)

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

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

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

440 conjugacy classes

 class 1 2A ··· 2G 5A 5B 5C 5D 10A ··· 10AB 11A ··· 11J 22A ··· 22BR 55A ··· 55AN 110A ··· 110JT order 1 2 ··· 2 5 5 5 5 10 ··· 10 11 ··· 11 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

440 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C5 C10 C11 C22 C55 C110 kernel C22×C110 C2×C110 C22×C22 C2×C22 C22×C10 C2×C10 C23 C22 # reps 1 7 4 28 10 70 40 280

Matrix representation of C22×C110 in GL3(𝔽331) generated by

 330 0 0 0 1 0 0 0 330
,
 1 0 0 0 330 0 0 0 330
,
 246 0 0 0 215 0 0 0 1
G:=sub<GL(3,GF(331))| [330,0,0,0,1,0,0,0,330],[1,0,0,0,330,0,0,0,330],[246,0,0,0,215,0,0,0,1] >;

C22×C110 in GAP, Magma, Sage, TeX

C_2^2\times C_{110}
% in TeX

G:=Group("C2^2xC110");
// GroupNames label

G:=SmallGroup(440,51);
// by ID

G=gap.SmallGroup(440,51);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^110=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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