direct product, abelian, monomial, 2-elementary
Aliases: C22×C110, SmallGroup(440,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C110 |
| C1 — C22×C110 |
| 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, C22, C5, C23, C10, C11, C2×C10, C22, C22×C10, C2×C22, C55, C22×C22, C110, C2×C110, C22×C110
Quotients: C1, C2, C22, C5, C23, C10, C11, C2×C10, C22, C22×C10, C2×C22, C55, C22×C22, C110, C2×C110, C22×C110
►Regular action on 440 points
Generators in S440
(1 307)(2 308)(3 309)(4 310)(5 311)(6 312)(7 313)(8 314)(9 315)(10 316)(11 317)(12 318)(13 319)(14 320)(15 321)(16 322)(17 323)(18 324)(19 325)(20 326)(21 327)(22 328)(23 329)(24 330)(25 221)(26 222)(27 223)(28 224)(29 225)(30 226)(31 227)(32 228)(33 229)(34 230)(35 231)(36 232)(37 233)(38 234)(39 235)(40 236)(41 237)(42 238)(43 239)(44 240)(45 241)(46 242)(47 243)(48 244)(49 245)(50 246)(51 247)(52 248)(53 249)(54 250)(55 251)(56 252)(57 253)(58 254)(59 255)(60 256)(61 257)(62 258)(63 259)(64 260)(65 261)(66 262)(67 263)(68 264)(69 265)(70 266)(71 267)(72 268)(73 269)(74 270)(75 271)(76 272)(77 273)(78 274)(79 275)(80 276)(81 277)(82 278)(83 279)(84 280)(85 281)(86 282)(87 283)(88 284)(89 285)(90 286)(91 287)(92 288)(93 289)(94 290)(95 291)(96 292)(97 293)(98 294)(99 295)(100 296)(101 297)(102 298)(103 299)(104 300)(105 301)(106 302)(107 303)(108 304)(109 305)(110 306)(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)
(1 192)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 199)(9 200)(10 201)(11 202)(12 203)(13 204)(14 205)(15 206)(16 207)(17 208)(18 209)(19 210)(20 211)(21 212)(22 213)(23 214)(24 215)(25 216)(26 217)(27 218)(28 219)(29 220)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)(82 163)(83 164)(84 165)(85 166)(86 167)(87 168)(88 169)(89 170)(90 171)(91 172)(92 173)(93 174)(94 175)(95 176)(96 177)(97 178)(98 179)(99 180)(100 181)(101 182)(102 183)(103 184)(104 185)(105 186)(106 187)(107 188)(108 189)(109 190)(110 191)(221 416)(222 417)(223 418)(224 419)(225 420)(226 421)(227 422)(228 423)(229 424)(230 425)(231 426)(232 427)(233 428)(234 429)(235 430)(236 431)(237 432)(238 433)(239 434)(240 435)(241 436)(242 437)(243 438)(244 439)(245 440)(246 331)(247 332)(248 333)(249 334)(250 335)(251 336)(252 337)(253 338)(254 339)(255 340)(256 341)(257 342)(258 343)(259 344)(260 345)(261 346)(262 347)(263 348)(264 349)(265 350)(266 351)(267 352)(268 353)(269 354)(270 355)(271 356)(272 357)(273 358)(274 359)(275 360)(276 361)(277 362)(278 363)(279 364)(280 365)(281 366)(282 367)(283 368)(284 369)(285 370)(286 371)(287 372)(288 373)(289 374)(290 375)(291 376)(292 377)(293 378)(294 379)(295 380)(296 381)(297 382)(298 383)(299 384)(300 385)(301 386)(302 387)(303 388)(304 389)(305 390)(306 391)(307 392)(308 393)(309 394)(310 395)(311 396)(312 397)(313 398)(314 399)(315 400)(316 401)(317 402)(318 403)(319 404)(320 405)(321 406)(322 407)(323 408)(324 409)(325 410)(326 411)(327 412)(328 413)(329 414)(330 415)
(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)| (1,307)(2,308)(3,309)(4,310)(5,311)(6,312)(7,313)(8,314)(9,315)(10,316)(11,317)(12,318)(13,319)(14,320)(15,321)(16,322)(17,323)(18,324)(19,325)(20,326)(21,327)(22,328)(23,329)(24,330)(25,221)(26,222)(27,223)(28,224)(29,225)(30,226)(31,227)(32,228)(33,229)(34,230)(35,231)(36,232)(37,233)(38,234)(39,235)(40,236)(41,237)(42,238)(43,239)(44,240)(45,241)(46,242)(47,243)(48,244)(49,245)(50,246)(51,247)(52,248)(53,249)(54,250)(55,251)(56,252)(57,253)(58,254)(59,255)(60,256)(61,257)(62,258)(63,259)(64,260)(65,261)(66,262)(67,263)(68,264)(69,265)(70,266)(71,267)(72,268)(73,269)(74,270)(75,271)(76,272)(77,273)(78,274)(79,275)(80,276)(81,277)(82,278)(83,279)(84,280)(85,281)(86,282)(87,283)(88,284)(89,285)(90,286)(91,287)(92,288)(93,289)(94,290)(95,291)(96,292)(97,293)(98,294)(99,295)(100,296)(101,297)(102,298)(103,299)(104,300)(105,301)(106,302)(107,303)(108,304)(109,305)(110,306)(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), (1,192)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,200)(10,201)(11,202)(12,203)(13,204)(14,205)(15,206)(16,207)(17,208)(18,209)(19,210)(20,211)(21,212)(22,213)(23,214)(24,215)(25,216)(26,217)(27,218)(28,219)(29,220)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(109,190)(110,191)(221,416)(222,417)(223,418)(224,419)(225,420)(226,421)(227,422)(228,423)(229,424)(230,425)(231,426)(232,427)(233,428)(234,429)(235,430)(236,431)(237,432)(238,433)(239,434)(240,435)(241,436)(242,437)(243,438)(244,439)(245,440)(246,331)(247,332)(248,333)(249,334)(250,335)(251,336)(252,337)(253,338)(254,339)(255,340)(256,341)(257,342)(258,343)(259,344)(260,345)(261,346)(262,347)(263,348)(264,349)(265,350)(266,351)(267,352)(268,353)(269,354)(270,355)(271,356)(272,357)(273,358)(274,359)(275,360)(276,361)(277,362)(278,363)(279,364)(280,365)(281,366)(282,367)(283,368)(284,369)(285,370)(286,371)(287,372)(288,373)(289,374)(290,375)(291,376)(292,377)(293,378)(294,379)(295,380)(296,381)(297,382)(298,383)(299,384)(300,385)(301,386)(302,387)(303,388)(304,389)(305,390)(306,391)(307,392)(308,393)(309,394)(310,395)(311,396)(312,397)(313,398)(314,399)(315,400)(316,401)(317,402)(318,403)(319,404)(320,405)(321,406)(322,407)(323,408)(324,409)(325,410)(326,411)(327,412)(328,413)(329,414)(330,415), (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( (1,307)(2,308)(3,309)(4,310)(5,311)(6,312)(7,313)(8,314)(9,315)(10,316)(11,317)(12,318)(13,319)(14,320)(15,321)(16,322)(17,323)(18,324)(19,325)(20,326)(21,327)(22,328)(23,329)(24,330)(25,221)(26,222)(27,223)(28,224)(29,225)(30,226)(31,227)(32,228)(33,229)(34,230)(35,231)(36,232)(37,233)(38,234)(39,235)(40,236)(41,237)(42,238)(43,239)(44,240)(45,241)(46,242)(47,243)(48,244)(49,245)(50,246)(51,247)(52,248)(53,249)(54,250)(55,251)(56,252)(57,253)(58,254)(59,255)(60,256)(61,257)(62,258)(63,259)(64,260)(65,261)(66,262)(67,263)(68,264)(69,265)(70,266)(71,267)(72,268)(73,269)(74,270)(75,271)(76,272)(77,273)(78,274)(79,275)(80,276)(81,277)(82,278)(83,279)(84,280)(85,281)(86,282)(87,283)(88,284)(89,285)(90,286)(91,287)(92,288)(93,289)(94,290)(95,291)(96,292)(97,293)(98,294)(99,295)(100,296)(101,297)(102,298)(103,299)(104,300)(105,301)(106,302)(107,303)(108,304)(109,305)(110,306)(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), (1,192)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,200)(10,201)(11,202)(12,203)(13,204)(14,205)(15,206)(16,207)(17,208)(18,209)(19,210)(20,211)(21,212)(22,213)(23,214)(24,215)(25,216)(26,217)(27,218)(28,219)(29,220)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(109,190)(110,191)(221,416)(222,417)(223,418)(224,419)(225,420)(226,421)(227,422)(228,423)(229,424)(230,425)(231,426)(232,427)(233,428)(234,429)(235,430)(236,431)(237,432)(238,433)(239,434)(240,435)(241,436)(242,437)(243,438)(244,439)(245,440)(246,331)(247,332)(248,333)(249,334)(250,335)(251,336)(252,337)(253,338)(254,339)(255,340)(256,341)(257,342)(258,343)(259,344)(260,345)(261,346)(262,347)(263,348)(264,349)(265,350)(266,351)(267,352)(268,353)(269,354)(270,355)(271,356)(272,357)(273,358)(274,359)(275,360)(276,361)(277,362)(278,363)(279,364)(280,365)(281,366)(282,367)(283,368)(284,369)(285,370)(286,371)(287,372)(288,373)(289,374)(290,375)(291,376)(292,377)(293,378)(294,379)(295,380)(296,381)(297,382)(298,383)(299,384)(300,385)(301,386)(302,387)(303,388)(304,389)(305,390)(306,391)(307,392)(308,393)(309,394)(310,395)(311,396)(312,397)(313,398)(314,399)(315,400)(316,401)(317,402)(318,403)(319,404)(320,405)(321,406)(322,407)(323,408)(324,409)(325,410)(326,411)(327,412)(328,413)(329,414)(330,415), (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([[(1,307),(2,308),(3,309),(4,310),(5,311),(6,312),(7,313),(8,314),(9,315),(10,316),(11,317),(12,318),(13,319),(14,320),(15,321),(16,322),(17,323),(18,324),(19,325),(20,326),(21,327),(22,328),(23,329),(24,330),(25,221),(26,222),(27,223),(28,224),(29,225),(30,226),(31,227),(32,228),(33,229),(34,230),(35,231),(36,232),(37,233),(38,234),(39,235),(40,236),(41,237),(42,238),(43,239),(44,240),(45,241),(46,242),(47,243),(48,244),(49,245),(50,246),(51,247),(52,248),(53,249),(54,250),(55,251),(56,252),(57,253),(58,254),(59,255),(60,256),(61,257),(62,258),(63,259),(64,260),(65,261),(66,262),(67,263),(68,264),(69,265),(70,266),(71,267),(72,268),(73,269),(74,270),(75,271),(76,272),(77,273),(78,274),(79,275),(80,276),(81,277),(82,278),(83,279),(84,280),(85,281),(86,282),(87,283),(88,284),(89,285),(90,286),(91,287),(92,288),(93,289),(94,290),(95,291),(96,292),(97,293),(98,294),(99,295),(100,296),(101,297),(102,298),(103,299),(104,300),(105,301),(106,302),(107,303),(108,304),(109,305),(110,306),(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)], [(1,192),(2,193),(3,194),(4,195),(5,196),(6,197),(7,198),(8,199),(9,200),(10,201),(11,202),(12,203),(13,204),(14,205),(15,206),(16,207),(17,208),(18,209),(19,210),(20,211),(21,212),(22,213),(23,214),(24,215),(25,216),(26,217),(27,218),(28,219),(29,220),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,131),(51,132),(52,133),(53,134),(54,135),(55,136),(56,137),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,161),(81,162),(82,163),(83,164),(84,165),(85,166),(86,167),(87,168),(88,169),(89,170),(90,171),(91,172),(92,173),(93,174),(94,175),(95,176),(96,177),(97,178),(98,179),(99,180),(100,181),(101,182),(102,183),(103,184),(104,185),(105,186),(106,187),(107,188),(108,189),(109,190),(110,191),(221,416),(222,417),(223,418),(224,419),(225,420),(226,421),(227,422),(228,423),(229,424),(230,425),(231,426),(232,427),(233,428),(234,429),(235,430),(236,431),(237,432),(238,433),(239,434),(240,435),(241,436),(242,437),(243,438),(244,439),(245,440),(246,331),(247,332),(248,333),(249,334),(250,335),(251,336),(252,337),(253,338),(254,339),(255,340),(256,341),(257,342),(258,343),(259,344),(260,345),(261,346),(262,347),(263,348),(264,349),(265,350),(266,351),(267,352),(268,353),(269,354),(270,355),(271,356),(272,357),(273,358),(274,359),(275,360),(276,361),(277,362),(278,363),(279,364),(280,365),(281,366),(282,367),(283,368),(284,369),(285,370),(286,371),(287,372),(288,373),(289,374),(290,375),(291,376),(292,377),(293,378),(294,379),(295,380),(296,381),(297,382),(298,383),(299,384),(300,385),(301,386),(302,387),(303,388),(304,389),(305,390),(306,391),(307,392),(308,393),(309,394),(310,395),(311,396),(312,397),(313,398),(314,399),(315,400),(316,401),(317,402),(318,403),(319,404),(320,405),(321,406),(322,407),(323,408),(324,409),(325,410),(326,411),(327,412),(328,413),(329,414),(330,415)], [(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