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G = C22×C112order 448 = 26·7

Abelian group of type [2,2,112]

direct product, abelian, monomial, 2-elementary

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

Series: Derived Chief Lower central Upper central

C1 — C22×C112
C1C2C4C8C56C112C2×C112 — C22×C112
C1 — C22×C112
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 [×6], C4, C4 [×3], C22 [×7], C7, C8, C8 [×3], C2×C4 [×6], C23, C14, C14 [×6], C16 [×4], C2×C8 [×6], C22×C4, C28, C28 [×3], C2×C14 [×7], C2×C16 [×6], C22×C8, C56, C56 [×3], C2×C28 [×6], C22×C14, C22×C16, C112 [×4], C2×C56 [×6], C22×C28, C2×C112 [×6], C22×C56, C22×C112
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C7, C8 [×4], C2×C4 [×6], C23, C14 [×7], C16 [×4], C2×C8 [×6], C22×C4, C28 [×4], C2×C14 [×7], C2×C16 [×6], C22×C8, C56 [×4], C2×C28 [×6], C22×C14, C22×C16, C112 [×4], C2×C56 [×6], C22×C28, C2×C112 [×6], C22×C56, C22×C112

Smallest permutation representation of C22×C112
Regular action on 448 points
Generators in S448
(1 324)(2 325)(3 326)(4 327)(5 328)(6 329)(7 330)(8 331)(9 332)(10 333)(11 334)(12 335)(13 336)(14 225)(15 226)(16 227)(17 228)(18 229)(19 230)(20 231)(21 232)(22 233)(23 234)(24 235)(25 236)(26 237)(27 238)(28 239)(29 240)(30 241)(31 242)(32 243)(33 244)(34 245)(35 246)(36 247)(37 248)(38 249)(39 250)(40 251)(41 252)(42 253)(43 254)(44 255)(45 256)(46 257)(47 258)(48 259)(49 260)(50 261)(51 262)(52 263)(53 264)(54 265)(55 266)(56 267)(57 268)(58 269)(59 270)(60 271)(61 272)(62 273)(63 274)(64 275)(65 276)(66 277)(67 278)(68 279)(69 280)(70 281)(71 282)(72 283)(73 284)(74 285)(75 286)(76 287)(77 288)(78 289)(79 290)(80 291)(81 292)(82 293)(83 294)(84 295)(85 296)(86 297)(87 298)(88 299)(89 300)(90 301)(91 302)(92 303)(93 304)(94 305)(95 306)(96 307)(97 308)(98 309)(99 310)(100 311)(101 312)(102 313)(103 314)(104 315)(105 316)(106 317)(107 318)(108 319)(109 320)(110 321)(111 322)(112 323)(113 438)(114 439)(115 440)(116 441)(117 442)(118 443)(119 444)(120 445)(121 446)(122 447)(123 448)(124 337)(125 338)(126 339)(127 340)(128 341)(129 342)(130 343)(131 344)(132 345)(133 346)(134 347)(135 348)(136 349)(137 350)(138 351)(139 352)(140 353)(141 354)(142 355)(143 356)(144 357)(145 358)(146 359)(147 360)(148 361)(149 362)(150 363)(151 364)(152 365)(153 366)(154 367)(155 368)(156 369)(157 370)(158 371)(159 372)(160 373)(161 374)(162 375)(163 376)(164 377)(165 378)(166 379)(167 380)(168 381)(169 382)(170 383)(171 384)(172 385)(173 386)(174 387)(175 388)(176 389)(177 390)(178 391)(179 392)(180 393)(181 394)(182 395)(183 396)(184 397)(185 398)(186 399)(187 400)(188 401)(189 402)(190 403)(191 404)(192 405)(193 406)(194 407)(195 408)(196 409)(197 410)(198 411)(199 412)(200 413)(201 414)(202 415)(203 416)(204 417)(205 418)(206 419)(207 420)(208 421)(209 422)(210 423)(211 424)(212 425)(213 426)(214 427)(215 428)(216 429)(217 430)(218 431)(219 432)(220 433)(221 434)(222 435)(223 436)(224 437)
(1 448)(2 337)(3 338)(4 339)(5 340)(6 341)(7 342)(8 343)(9 344)(10 345)(11 346)(12 347)(13 348)(14 349)(15 350)(16 351)(17 352)(18 353)(19 354)(20 355)(21 356)(22 357)(23 358)(24 359)(25 360)(26 361)(27 362)(28 363)(29 364)(30 365)(31 366)(32 367)(33 368)(34 369)(35 370)(36 371)(37 372)(38 373)(39 374)(40 375)(41 376)(42 377)(43 378)(44 379)(45 380)(46 381)(47 382)(48 383)(49 384)(50 385)(51 386)(52 387)(53 388)(54 389)(55 390)(56 391)(57 392)(58 393)(59 394)(60 395)(61 396)(62 397)(63 398)(64 399)(65 400)(66 401)(67 402)(68 403)(69 404)(70 405)(71 406)(72 407)(73 408)(74 409)(75 410)(76 411)(77 412)(78 413)(79 414)(80 415)(81 416)(82 417)(83 418)(84 419)(85 420)(86 421)(87 422)(88 423)(89 424)(90 425)(91 426)(92 427)(93 428)(94 429)(95 430)(96 431)(97 432)(98 433)(99 434)(100 435)(101 436)(102 437)(103 438)(104 439)(105 440)(106 441)(107 442)(108 443)(109 444)(110 445)(111 446)(112 447)(113 314)(114 315)(115 316)(116 317)(117 318)(118 319)(119 320)(120 321)(121 322)(122 323)(123 324)(124 325)(125 326)(126 327)(127 328)(128 329)(129 330)(130 331)(131 332)(132 333)(133 334)(134 335)(135 336)(136 225)(137 226)(138 227)(139 228)(140 229)(141 230)(142 231)(143 232)(144 233)(145 234)(146 235)(147 236)(148 237)(149 238)(150 239)(151 240)(152 241)(153 242)(154 243)(155 244)(156 245)(157 246)(158 247)(159 248)(160 249)(161 250)(162 251)(163 252)(164 253)(165 254)(166 255)(167 256)(168 257)(169 258)(170 259)(171 260)(172 261)(173 262)(174 263)(175 264)(176 265)(177 266)(178 267)(179 268)(180 269)(181 270)(182 271)(183 272)(184 273)(185 274)(186 275)(187 276)(188 277)(189 278)(190 279)(191 280)(192 281)(193 282)(194 283)(195 284)(196 285)(197 286)(198 287)(199 288)(200 289)(201 290)(202 291)(203 292)(204 293)(205 294)(206 295)(207 296)(208 297)(209 298)(210 299)(211 300)(212 301)(213 302)(214 303)(215 304)(216 305)(217 306)(218 307)(219 308)(220 309)(221 310)(222 311)(223 312)(224 313)
(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)| (1,324)(2,325)(3,326)(4,327)(5,328)(6,329)(7,330)(8,331)(9,332)(10,333)(11,334)(12,335)(13,336)(14,225)(15,226)(16,227)(17,228)(18,229)(19,230)(20,231)(21,232)(22,233)(23,234)(24,235)(25,236)(26,237)(27,238)(28,239)(29,240)(30,241)(31,242)(32,243)(33,244)(34,245)(35,246)(36,247)(37,248)(38,249)(39,250)(40,251)(41,252)(42,253)(43,254)(44,255)(45,256)(46,257)(47,258)(48,259)(49,260)(50,261)(51,262)(52,263)(53,264)(54,265)(55,266)(56,267)(57,268)(58,269)(59,270)(60,271)(61,272)(62,273)(63,274)(64,275)(65,276)(66,277)(67,278)(68,279)(69,280)(70,281)(71,282)(72,283)(73,284)(74,285)(75,286)(76,287)(77,288)(78,289)(79,290)(80,291)(81,292)(82,293)(83,294)(84,295)(85,296)(86,297)(87,298)(88,299)(89,300)(90,301)(91,302)(92,303)(93,304)(94,305)(95,306)(96,307)(97,308)(98,309)(99,310)(100,311)(101,312)(102,313)(103,314)(104,315)(105,316)(106,317)(107,318)(108,319)(109,320)(110,321)(111,322)(112,323)(113,438)(114,439)(115,440)(116,441)(117,442)(118,443)(119,444)(120,445)(121,446)(122,447)(123,448)(124,337)(125,338)(126,339)(127,340)(128,341)(129,342)(130,343)(131,344)(132,345)(133,346)(134,347)(135,348)(136,349)(137,350)(138,351)(139,352)(140,353)(141,354)(142,355)(143,356)(144,357)(145,358)(146,359)(147,360)(148,361)(149,362)(150,363)(151,364)(152,365)(153,366)(154,367)(155,368)(156,369)(157,370)(158,371)(159,372)(160,373)(161,374)(162,375)(163,376)(164,377)(165,378)(166,379)(167,380)(168,381)(169,382)(170,383)(171,384)(172,385)(173,386)(174,387)(175,388)(176,389)(177,390)(178,391)(179,392)(180,393)(181,394)(182,395)(183,396)(184,397)(185,398)(186,399)(187,400)(188,401)(189,402)(190,403)(191,404)(192,405)(193,406)(194,407)(195,408)(196,409)(197,410)(198,411)(199,412)(200,413)(201,414)(202,415)(203,416)(204,417)(205,418)(206,419)(207,420)(208,421)(209,422)(210,423)(211,424)(212,425)(213,426)(214,427)(215,428)(216,429)(217,430)(218,431)(219,432)(220,433)(221,434)(222,435)(223,436)(224,437), (1,448)(2,337)(3,338)(4,339)(5,340)(6,341)(7,342)(8,343)(9,344)(10,345)(11,346)(12,347)(13,348)(14,349)(15,350)(16,351)(17,352)(18,353)(19,354)(20,355)(21,356)(22,357)(23,358)(24,359)(25,360)(26,361)(27,362)(28,363)(29,364)(30,365)(31,366)(32,367)(33,368)(34,369)(35,370)(36,371)(37,372)(38,373)(39,374)(40,375)(41,376)(42,377)(43,378)(44,379)(45,380)(46,381)(47,382)(48,383)(49,384)(50,385)(51,386)(52,387)(53,388)(54,389)(55,390)(56,391)(57,392)(58,393)(59,394)(60,395)(61,396)(62,397)(63,398)(64,399)(65,400)(66,401)(67,402)(68,403)(69,404)(70,405)(71,406)(72,407)(73,408)(74,409)(75,410)(76,411)(77,412)(78,413)(79,414)(80,415)(81,416)(82,417)(83,418)(84,419)(85,420)(86,421)(87,422)(88,423)(89,424)(90,425)(91,426)(92,427)(93,428)(94,429)(95,430)(96,431)(97,432)(98,433)(99,434)(100,435)(101,436)(102,437)(103,438)(104,439)(105,440)(106,441)(107,442)(108,443)(109,444)(110,445)(111,446)(112,447)(113,314)(114,315)(115,316)(116,317)(117,318)(118,319)(119,320)(120,321)(121,322)(122,323)(123,324)(124,325)(125,326)(126,327)(127,328)(128,329)(129,330)(130,331)(131,332)(132,333)(133,334)(134,335)(135,336)(136,225)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,238)(150,239)(151,240)(152,241)(153,242)(154,243)(155,244)(156,245)(157,246)(158,247)(159,248)(160,249)(161,250)(162,251)(163,252)(164,253)(165,254)(166,255)(167,256)(168,257)(169,258)(170,259)(171,260)(172,261)(173,262)(174,263)(175,264)(176,265)(177,266)(178,267)(179,268)(180,269)(181,270)(182,271)(183,272)(184,273)(185,274)(186,275)(187,276)(188,277)(189,278)(190,279)(191,280)(192,281)(193,282)(194,283)(195,284)(196,285)(197,286)(198,287)(199,288)(200,289)(201,290)(202,291)(203,292)(204,293)(205,294)(206,295)(207,296)(208,297)(209,298)(210,299)(211,300)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309)(221,310)(222,311)(223,312)(224,313), (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( (1,324)(2,325)(3,326)(4,327)(5,328)(6,329)(7,330)(8,331)(9,332)(10,333)(11,334)(12,335)(13,336)(14,225)(15,226)(16,227)(17,228)(18,229)(19,230)(20,231)(21,232)(22,233)(23,234)(24,235)(25,236)(26,237)(27,238)(28,239)(29,240)(30,241)(31,242)(32,243)(33,244)(34,245)(35,246)(36,247)(37,248)(38,249)(39,250)(40,251)(41,252)(42,253)(43,254)(44,255)(45,256)(46,257)(47,258)(48,259)(49,260)(50,261)(51,262)(52,263)(53,264)(54,265)(55,266)(56,267)(57,268)(58,269)(59,270)(60,271)(61,272)(62,273)(63,274)(64,275)(65,276)(66,277)(67,278)(68,279)(69,280)(70,281)(71,282)(72,283)(73,284)(74,285)(75,286)(76,287)(77,288)(78,289)(79,290)(80,291)(81,292)(82,293)(83,294)(84,295)(85,296)(86,297)(87,298)(88,299)(89,300)(90,301)(91,302)(92,303)(93,304)(94,305)(95,306)(96,307)(97,308)(98,309)(99,310)(100,311)(101,312)(102,313)(103,314)(104,315)(105,316)(106,317)(107,318)(108,319)(109,320)(110,321)(111,322)(112,323)(113,438)(114,439)(115,440)(116,441)(117,442)(118,443)(119,444)(120,445)(121,446)(122,447)(123,448)(124,337)(125,338)(126,339)(127,340)(128,341)(129,342)(130,343)(131,344)(132,345)(133,346)(134,347)(135,348)(136,349)(137,350)(138,351)(139,352)(140,353)(141,354)(142,355)(143,356)(144,357)(145,358)(146,359)(147,360)(148,361)(149,362)(150,363)(151,364)(152,365)(153,366)(154,367)(155,368)(156,369)(157,370)(158,371)(159,372)(160,373)(161,374)(162,375)(163,376)(164,377)(165,378)(166,379)(167,380)(168,381)(169,382)(170,383)(171,384)(172,385)(173,386)(174,387)(175,388)(176,389)(177,390)(178,391)(179,392)(180,393)(181,394)(182,395)(183,396)(184,397)(185,398)(186,399)(187,400)(188,401)(189,402)(190,403)(191,404)(192,405)(193,406)(194,407)(195,408)(196,409)(197,410)(198,411)(199,412)(200,413)(201,414)(202,415)(203,416)(204,417)(205,418)(206,419)(207,420)(208,421)(209,422)(210,423)(211,424)(212,425)(213,426)(214,427)(215,428)(216,429)(217,430)(218,431)(219,432)(220,433)(221,434)(222,435)(223,436)(224,437), (1,448)(2,337)(3,338)(4,339)(5,340)(6,341)(7,342)(8,343)(9,344)(10,345)(11,346)(12,347)(13,348)(14,349)(15,350)(16,351)(17,352)(18,353)(19,354)(20,355)(21,356)(22,357)(23,358)(24,359)(25,360)(26,361)(27,362)(28,363)(29,364)(30,365)(31,366)(32,367)(33,368)(34,369)(35,370)(36,371)(37,372)(38,373)(39,374)(40,375)(41,376)(42,377)(43,378)(44,379)(45,380)(46,381)(47,382)(48,383)(49,384)(50,385)(51,386)(52,387)(53,388)(54,389)(55,390)(56,391)(57,392)(58,393)(59,394)(60,395)(61,396)(62,397)(63,398)(64,399)(65,400)(66,401)(67,402)(68,403)(69,404)(70,405)(71,406)(72,407)(73,408)(74,409)(75,410)(76,411)(77,412)(78,413)(79,414)(80,415)(81,416)(82,417)(83,418)(84,419)(85,420)(86,421)(87,422)(88,423)(89,424)(90,425)(91,426)(92,427)(93,428)(94,429)(95,430)(96,431)(97,432)(98,433)(99,434)(100,435)(101,436)(102,437)(103,438)(104,439)(105,440)(106,441)(107,442)(108,443)(109,444)(110,445)(111,446)(112,447)(113,314)(114,315)(115,316)(116,317)(117,318)(118,319)(119,320)(120,321)(121,322)(122,323)(123,324)(124,325)(125,326)(126,327)(127,328)(128,329)(129,330)(130,331)(131,332)(132,333)(133,334)(134,335)(135,336)(136,225)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,238)(150,239)(151,240)(152,241)(153,242)(154,243)(155,244)(156,245)(157,246)(158,247)(159,248)(160,249)(161,250)(162,251)(163,252)(164,253)(165,254)(166,255)(167,256)(168,257)(169,258)(170,259)(171,260)(172,261)(173,262)(174,263)(175,264)(176,265)(177,266)(178,267)(179,268)(180,269)(181,270)(182,271)(183,272)(184,273)(185,274)(186,275)(187,276)(188,277)(189,278)(190,279)(191,280)(192,281)(193,282)(194,283)(195,284)(196,285)(197,286)(198,287)(199,288)(200,289)(201,290)(202,291)(203,292)(204,293)(205,294)(206,295)(207,296)(208,297)(209,298)(210,299)(211,300)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309)(221,310)(222,311)(223,312)(224,313), (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([(1,324),(2,325),(3,326),(4,327),(5,328),(6,329),(7,330),(8,331),(9,332),(10,333),(11,334),(12,335),(13,336),(14,225),(15,226),(16,227),(17,228),(18,229),(19,230),(20,231),(21,232),(22,233),(23,234),(24,235),(25,236),(26,237),(27,238),(28,239),(29,240),(30,241),(31,242),(32,243),(33,244),(34,245),(35,246),(36,247),(37,248),(38,249),(39,250),(40,251),(41,252),(42,253),(43,254),(44,255),(45,256),(46,257),(47,258),(48,259),(49,260),(50,261),(51,262),(52,263),(53,264),(54,265),(55,266),(56,267),(57,268),(58,269),(59,270),(60,271),(61,272),(62,273),(63,274),(64,275),(65,276),(66,277),(67,278),(68,279),(69,280),(70,281),(71,282),(72,283),(73,284),(74,285),(75,286),(76,287),(77,288),(78,289),(79,290),(80,291),(81,292),(82,293),(83,294),(84,295),(85,296),(86,297),(87,298),(88,299),(89,300),(90,301),(91,302),(92,303),(93,304),(94,305),(95,306),(96,307),(97,308),(98,309),(99,310),(100,311),(101,312),(102,313),(103,314),(104,315),(105,316),(106,317),(107,318),(108,319),(109,320),(110,321),(111,322),(112,323),(113,438),(114,439),(115,440),(116,441),(117,442),(118,443),(119,444),(120,445),(121,446),(122,447),(123,448),(124,337),(125,338),(126,339),(127,340),(128,341),(129,342),(130,343),(131,344),(132,345),(133,346),(134,347),(135,348),(136,349),(137,350),(138,351),(139,352),(140,353),(141,354),(142,355),(143,356),(144,357),(145,358),(146,359),(147,360),(148,361),(149,362),(150,363),(151,364),(152,365),(153,366),(154,367),(155,368),(156,369),(157,370),(158,371),(159,372),(160,373),(161,374),(162,375),(163,376),(164,377),(165,378),(166,379),(167,380),(168,381),(169,382),(170,383),(171,384),(172,385),(173,386),(174,387),(175,388),(176,389),(177,390),(178,391),(179,392),(180,393),(181,394),(182,395),(183,396),(184,397),(185,398),(186,399),(187,400),(188,401),(189,402),(190,403),(191,404),(192,405),(193,406),(194,407),(195,408),(196,409),(197,410),(198,411),(199,412),(200,413),(201,414),(202,415),(203,416),(204,417),(205,418),(206,419),(207,420),(208,421),(209,422),(210,423),(211,424),(212,425),(213,426),(214,427),(215,428),(216,429),(217,430),(218,431),(219,432),(220,433),(221,434),(222,435),(223,436),(224,437)], [(1,448),(2,337),(3,338),(4,339),(5,340),(6,341),(7,342),(8,343),(9,344),(10,345),(11,346),(12,347),(13,348),(14,349),(15,350),(16,351),(17,352),(18,353),(19,354),(20,355),(21,356),(22,357),(23,358),(24,359),(25,360),(26,361),(27,362),(28,363),(29,364),(30,365),(31,366),(32,367),(33,368),(34,369),(35,370),(36,371),(37,372),(38,373),(39,374),(40,375),(41,376),(42,377),(43,378),(44,379),(45,380),(46,381),(47,382),(48,383),(49,384),(50,385),(51,386),(52,387),(53,388),(54,389),(55,390),(56,391),(57,392),(58,393),(59,394),(60,395),(61,396),(62,397),(63,398),(64,399),(65,400),(66,401),(67,402),(68,403),(69,404),(70,405),(71,406),(72,407),(73,408),(74,409),(75,410),(76,411),(77,412),(78,413),(79,414),(80,415),(81,416),(82,417),(83,418),(84,419),(85,420),(86,421),(87,422),(88,423),(89,424),(90,425),(91,426),(92,427),(93,428),(94,429),(95,430),(96,431),(97,432),(98,433),(99,434),(100,435),(101,436),(102,437),(103,438),(104,439),(105,440),(106,441),(107,442),(108,443),(109,444),(110,445),(111,446),(112,447),(113,314),(114,315),(115,316),(116,317),(117,318),(118,319),(119,320),(120,321),(121,322),(122,323),(123,324),(124,325),(125,326),(126,327),(127,328),(128,329),(129,330),(130,331),(131,332),(132,333),(133,334),(134,335),(135,336),(136,225),(137,226),(138,227),(139,228),(140,229),(141,230),(142,231),(143,232),(144,233),(145,234),(146,235),(147,236),(148,237),(149,238),(150,239),(151,240),(152,241),(153,242),(154,243),(155,244),(156,245),(157,246),(158,247),(159,248),(160,249),(161,250),(162,251),(163,252),(164,253),(165,254),(166,255),(167,256),(168,257),(169,258),(170,259),(171,260),(172,261),(173,262),(174,263),(175,264),(176,265),(177,266),(178,267),(179,268),(180,269),(181,270),(182,271),(183,272),(184,273),(185,274),(186,275),(187,276),(188,277),(189,278),(190,279),(191,280),(192,281),(193,282),(194,283),(195,284),(196,285),(197,286),(198,287),(199,288),(200,289),(201,290),(202,291),(203,292),(204,293),(205,294),(206,295),(207,296),(208,297),(209,298),(210,299),(211,300),(212,301),(213,302),(214,303),(215,304),(216,305),(217,306),(218,307),(219,308),(220,309),(221,310),(222,311),(223,312),(224,313)], [(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···2G4A···4H7A···7F8A···8P14A···14AP16A···16AF28A···28AV56A···56CR112A···112GJ
order12···24···47···78···814···1416···1628···2856···56112···112
size11···11···11···11···11···11···11···11···11···1

448 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C4C4C7C8C8C14C14C16C28C28C56C56C112
kernelC22×C112C2×C112C22×C56C2×C56C22×C28C22×C16C2×C28C22×C14C2×C16C22×C8C2×C14C2×C8C22×C4C2×C4C23C22
# reps1616261243663236127224192

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

11200
01120
00112
,
100
01120
00112
,
10200
0910
0020
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

׿
×
𝔽