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## G = C22×C112order 448 = 26·7

### Abelian group of type [2,2,112]

Aliases: C22×C112, SmallGroup(448,910)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C112
 Chief series C1 — C2 — C4 — C8 — C56 — C112 — C2×C112 — C22×C112
 Lower central C1 — C22×C112
 Upper central C1 — C22×C112

Generators and relations for C22×C112
G = < a,b,c | a2=b2=c112=1, ab=ba, ac=ca, bc=cb >

Subgroups: 98, all normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C23, C14, C14, C16, C2×C8, C22×C4, C28, C28, C2×C14, C2×C16, C22×C8, C56, C56, C2×C28, C22×C14, C22×C16, C112, C2×C56, C22×C28, C2×C112, C22×C56, C22×C112
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C23, C14, C16, C2×C8, C22×C4, C28, C2×C14, C2×C16, C22×C8, C56, C2×C28, C22×C14, C22×C16, C112, C2×C56, C22×C28, C2×C112, C22×C56, C22×C112

Smallest permutation representation of C22×C112
Regular action on 448 points
Generators in S448
(1 381)(2 382)(3 383)(4 384)(5 385)(6 386)(7 387)(8 388)(9 389)(10 390)(11 391)(12 392)(13 393)(14 394)(15 395)(16 396)(17 397)(18 398)(19 399)(20 400)(21 401)(22 402)(23 403)(24 404)(25 405)(26 406)(27 407)(28 408)(29 409)(30 410)(31 411)(32 412)(33 413)(34 414)(35 415)(36 416)(37 417)(38 418)(39 419)(40 420)(41 421)(42 422)(43 423)(44 424)(45 425)(46 426)(47 427)(48 428)(49 429)(50 430)(51 431)(52 432)(53 433)(54 434)(55 435)(56 436)(57 437)(58 438)(59 439)(60 440)(61 441)(62 442)(63 443)(64 444)(65 445)(66 446)(67 447)(68 448)(69 337)(70 338)(71 339)(72 340)(73 341)(74 342)(75 343)(76 344)(77 345)(78 346)(79 347)(80 348)(81 349)(82 350)(83 351)(84 352)(85 353)(86 354)(87 355)(88 356)(89 357)(90 358)(91 359)(92 360)(93 361)(94 362)(95 363)(96 364)(97 365)(98 366)(99 367)(100 368)(101 369)(102 370)(103 371)(104 372)(105 373)(106 374)(107 375)(108 376)(109 377)(110 378)(111 379)(112 380)(113 293)(114 294)(115 295)(116 296)(117 297)(118 298)(119 299)(120 300)(121 301)(122 302)(123 303)(124 304)(125 305)(126 306)(127 307)(128 308)(129 309)(130 310)(131 311)(132 312)(133 313)(134 314)(135 315)(136 316)(137 317)(138 318)(139 319)(140 320)(141 321)(142 322)(143 323)(144 324)(145 325)(146 326)(147 327)(148 328)(149 329)(150 330)(151 331)(152 332)(153 333)(154 334)(155 335)(156 336)(157 225)(158 226)(159 227)(160 228)(161 229)(162 230)(163 231)(164 232)(165 233)(166 234)(167 235)(168 236)(169 237)(170 238)(171 239)(172 240)(173 241)(174 242)(175 243)(176 244)(177 245)(178 246)(179 247)(180 248)(181 249)(182 250)(183 251)(184 252)(185 253)(186 254)(187 255)(188 256)(189 257)(190 258)(191 259)(192 260)(193 261)(194 262)(195 263)(196 264)(197 265)(198 266)(199 267)(200 268)(201 269)(202 270)(203 271)(204 272)(205 273)(206 274)(207 275)(208 276)(209 277)(210 278)(211 279)(212 280)(213 281)(214 282)(215 283)(216 284)(217 285)(218 286)(219 287)(220 288)(221 289)(222 290)(223 291)(224 292)
(1 179)(2 180)(3 181)(4 182)(5 183)(6 184)(7 185)(8 186)(9 187)(10 188)(11 189)(12 190)(13 191)(14 192)(15 193)(16 194)(17 195)(18 196)(19 197)(20 198)(21 199)(22 200)(23 201)(24 202)(25 203)(26 204)(27 205)(28 206)(29 207)(30 208)(31 209)(32 210)(33 211)(34 212)(35 213)(36 214)(37 215)(38 216)(39 217)(40 218)(41 219)(42 220)(43 221)(44 222)(45 223)(46 224)(47 113)(48 114)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 163)(98 164)(99 165)(100 166)(101 167)(102 168)(103 169)(104 170)(105 171)(106 172)(107 173)(108 174)(109 175)(110 176)(111 177)(112 178)(225 359)(226 360)(227 361)(228 362)(229 363)(230 364)(231 365)(232 366)(233 367)(234 368)(235 369)(236 370)(237 371)(238 372)(239 373)(240 374)(241 375)(242 376)(243 377)(244 378)(245 379)(246 380)(247 381)(248 382)(249 383)(250 384)(251 385)(252 386)(253 387)(254 388)(255 389)(256 390)(257 391)(258 392)(259 393)(260 394)(261 395)(262 396)(263 397)(264 398)(265 399)(266 400)(267 401)(268 402)(269 403)(270 404)(271 405)(272 406)(273 407)(274 408)(275 409)(276 410)(277 411)(278 412)(279 413)(280 414)(281 415)(282 416)(283 417)(284 418)(285 419)(286 420)(287 421)(288 422)(289 423)(290 424)(291 425)(292 426)(293 427)(294 428)(295 429)(296 430)(297 431)(298 432)(299 433)(300 434)(301 435)(302 436)(303 437)(304 438)(305 439)(306 440)(307 441)(308 442)(309 443)(310 444)(311 445)(312 446)(313 447)(314 448)(315 337)(316 338)(317 339)(318 340)(319 341)(320 342)(321 343)(322 344)(323 345)(324 346)(325 347)(326 348)(327 349)(328 350)(329 351)(330 352)(331 353)(332 354)(333 355)(334 356)(335 357)(336 358)
(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)

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

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

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

448 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 7A ··· 7F 8A ··· 8P 14A ··· 14AP 16A ··· 16AF 28A ··· 28AV 56A ··· 56CR 112A ··· 112GJ order 1 2 ··· 2 4 ··· 4 7 ··· 7 8 ··· 8 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

448 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C7 C8 C8 C14 C14 C16 C28 C28 C56 C56 C112 kernel C22×C112 C2×C112 C22×C56 C2×C56 C22×C28 C22×C16 C2×C28 C22×C14 C2×C16 C22×C8 C2×C14 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 6 12 4 36 6 32 36 12 72 24 192

Matrix representation of C22×C112 in GL3(𝔽113) generated by

 112 0 0 0 112 0 0 0 112
,
 1 0 0 0 112 0 0 0 112
,
 102 0 0 0 91 0 0 0 20
G:=sub<GL(3,GF(113))| [112,0,0,0,112,0,0,0,112],[1,0,0,0,112,0,0,0,112],[102,0,0,0,91,0,0,0,20] >;

C22×C112 in GAP, Magma, Sage, TeX

C_2^2\times C_{112}
% in TeX

G:=Group("C2^2xC112");
// GroupNames label

G:=SmallGroup(448,910);
// by ID

G=gap.SmallGroup(448,910);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,102,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^112=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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