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G = C22×C4×C28order 448 = 26·7

Abelian group of type [2,2,4,28]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C4×C28, SmallGroup(448,1294)

Series: Derived Chief Lower central Upper central

C1 — C22×C4×C28
C1C2C22C2×C14C2×C28C4×C28C2×C4×C28 — C22×C4×C28
C1 — C22×C4×C28
C1 — C22×C4×C28

Generators and relations for C22×C4×C28
 G = < a,b,c,d | a2=b2=c4=d28=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 498, all normal (8 characteristic)
C1, C2 [×15], C4 [×24], C22, C22 [×34], C7, C2×C4 [×84], C23 [×15], C14 [×15], C42 [×16], C22×C4 [×42], C24, C28 [×24], C2×C14, C2×C14 [×34], C2×C42 [×12], C23×C4 [×3], C2×C28 [×84], C22×C14 [×15], C22×C42, C4×C28 [×16], C22×C28 [×42], C23×C14, C2×C4×C28 [×12], C23×C28 [×3], C22×C4×C28
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C7, C2×C4 [×84], C23 [×15], C14 [×15], C42 [×16], C22×C4 [×42], C24, C28 [×24], C2×C14 [×35], C2×C42 [×12], C23×C4 [×3], C2×C28 [×84], C22×C14 [×15], C22×C42, C4×C28 [×16], C22×C28 [×42], C23×C14, C2×C4×C28 [×12], C23×C28 [×3], C22×C4×C28

Smallest permutation representation of C22×C4×C28
Regular action on 448 points
Generators in S448
(1 445)(2 446)(3 447)(4 448)(5 421)(6 422)(7 423)(8 424)(9 425)(10 426)(11 427)(12 428)(13 429)(14 430)(15 431)(16 432)(17 433)(18 434)(19 435)(20 436)(21 437)(22 438)(23 439)(24 440)(25 441)(26 442)(27 443)(28 444)(29 156)(30 157)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 164)(38 165)(39 166)(40 167)(41 168)(42 141)(43 142)(44 143)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 150)(52 151)(53 152)(54 153)(55 154)(56 155)(57 260)(58 261)(59 262)(60 263)(61 264)(62 265)(63 266)(64 267)(65 268)(66 269)(67 270)(68 271)(69 272)(70 273)(71 274)(72 275)(73 276)(74 277)(75 278)(76 279)(77 280)(78 253)(79 254)(80 255)(81 256)(82 257)(83 258)(84 259)(85 392)(86 365)(87 366)(88 367)(89 368)(90 369)(91 370)(92 371)(93 372)(94 373)(95 374)(96 375)(97 376)(98 377)(99 378)(100 379)(101 380)(102 381)(103 382)(104 383)(105 384)(106 385)(107 386)(108 387)(109 388)(110 389)(111 390)(112 391)(113 303)(114 304)(115 305)(116 306)(117 307)(118 308)(119 281)(120 282)(121 283)(122 284)(123 285)(124 286)(125 287)(126 288)(127 289)(128 290)(129 291)(130 292)(131 293)(132 294)(133 295)(134 296)(135 297)(136 298)(137 299)(138 300)(139 301)(140 302)(169 213)(170 214)(171 215)(172 216)(173 217)(174 218)(175 219)(176 220)(177 221)(178 222)(179 223)(180 224)(181 197)(182 198)(183 199)(184 200)(185 201)(186 202)(187 203)(188 204)(189 205)(190 206)(191 207)(192 208)(193 209)(194 210)(195 211)(196 212)(225 318)(226 319)(227 320)(228 321)(229 322)(230 323)(231 324)(232 325)(233 326)(234 327)(235 328)(236 329)(237 330)(238 331)(239 332)(240 333)(241 334)(242 335)(243 336)(244 309)(245 310)(246 311)(247 312)(248 313)(249 314)(250 315)(251 316)(252 317)(337 404)(338 405)(339 406)(340 407)(341 408)(342 409)(343 410)(344 411)(345 412)(346 413)(347 414)(348 415)(349 416)(350 417)(351 418)(352 419)(353 420)(354 393)(355 394)(356 395)(357 396)(358 397)(359 398)(360 399)(361 400)(362 401)(363 402)(364 403)
(1 222)(2 223)(3 224)(4 197)(5 198)(6 199)(7 200)(8 201)(9 202)(10 203)(11 204)(12 205)(13 206)(14 207)(15 208)(16 209)(17 210)(18 211)(19 212)(20 213)(21 214)(22 215)(23 216)(24 217)(25 218)(26 219)(27 220)(28 221)(29 329)(30 330)(31 331)(32 332)(33 333)(34 334)(35 335)(36 336)(37 309)(38 310)(39 311)(40 312)(41 313)(42 314)(43 315)(44 316)(45 317)(46 318)(47 319)(48 320)(49 321)(50 322)(51 323)(52 324)(53 325)(54 326)(55 327)(56 328)(57 394)(58 395)(59 396)(60 397)(61 398)(62 399)(63 400)(64 401)(65 402)(66 403)(67 404)(68 405)(69 406)(70 407)(71 408)(72 409)(73 410)(74 411)(75 412)(76 413)(77 414)(78 415)(79 416)(80 417)(81 418)(82 419)(83 420)(84 393)(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)(98 281)(99 282)(100 283)(101 284)(102 285)(103 286)(104 287)(105 288)(106 289)(107 290)(108 291)(109 292)(110 293)(111 294)(112 295)(113 371)(114 372)(115 373)(116 374)(117 375)(118 376)(119 377)(120 378)(121 379)(122 380)(123 381)(124 382)(125 383)(126 384)(127 385)(128 386)(129 387)(130 388)(131 389)(132 390)(133 391)(134 392)(135 365)(136 366)(137 367)(138 368)(139 369)(140 370)(141 249)(142 250)(143 251)(144 252)(145 225)(146 226)(147 227)(148 228)(149 229)(150 230)(151 231)(152 232)(153 233)(154 234)(155 235)(156 236)(157 237)(158 238)(159 239)(160 240)(161 241)(162 242)(163 243)(164 244)(165 245)(166 246)(167 247)(168 248)(169 436)(170 437)(171 438)(172 439)(173 440)(174 441)(175 442)(176 443)(177 444)(178 445)(179 446)(180 447)(181 448)(182 421)(183 422)(184 423)(185 424)(186 425)(187 426)(188 427)(189 428)(190 429)(191 430)(192 431)(193 432)(194 433)(195 434)(196 435)(253 348)(254 349)(255 350)(256 351)(257 352)(258 353)(259 354)(260 355)(261 356)(262 357)(263 358)(264 359)(265 360)(266 361)(267 362)(268 363)(269 364)(270 337)(271 338)(272 339)(273 340)(274 341)(275 342)(276 343)(277 344)(278 345)(279 346)(280 347)
(1 242 374 359)(2 243 375 360)(3 244 376 361)(4 245 377 362)(5 246 378 363)(6 247 379 364)(7 248 380 337)(8 249 381 338)(9 250 382 339)(10 251 383 340)(11 252 384 341)(12 225 385 342)(13 226 386 343)(14 227 387 344)(15 228 388 345)(16 229 389 346)(17 230 390 347)(18 231 391 348)(19 232 392 349)(20 233 365 350)(21 234 366 351)(22 235 367 352)(23 236 368 353)(24 237 369 354)(25 238 370 355)(26 239 371 356)(27 240 372 357)(28 241 373 358)(29 300 83 172)(30 301 84 173)(31 302 57 174)(32 303 58 175)(33 304 59 176)(34 305 60 177)(35 306 61 178)(36 307 62 179)(37 308 63 180)(38 281 64 181)(39 282 65 182)(40 283 66 183)(41 284 67 184)(42 285 68 185)(43 286 69 186)(44 287 70 187)(45 288 71 188)(46 289 72 189)(47 290 73 190)(48 291 74 191)(49 292 75 192)(50 293 76 193)(51 294 77 194)(52 295 78 195)(53 296 79 196)(54 297 80 169)(55 298 81 170)(56 299 82 171)(85 416 435 325)(86 417 436 326)(87 418 437 327)(88 419 438 328)(89 420 439 329)(90 393 440 330)(91 394 441 331)(92 395 442 332)(93 396 443 333)(94 397 444 334)(95 398 445 335)(96 399 446 336)(97 400 447 309)(98 401 448 310)(99 402 421 311)(100 403 422 312)(101 404 423 313)(102 405 424 314)(103 406 425 315)(104 407 426 316)(105 408 427 317)(106 409 428 318)(107 410 429 319)(108 411 430 320)(109 412 431 321)(110 413 432 322)(111 414 433 323)(112 415 434 324)(113 261 219 159)(114 262 220 160)(115 263 221 161)(116 264 222 162)(117 265 223 163)(118 266 224 164)(119 267 197 165)(120 268 198 166)(121 269 199 167)(122 270 200 168)(123 271 201 141)(124 272 202 142)(125 273 203 143)(126 274 204 144)(127 275 205 145)(128 276 206 146)(129 277 207 147)(130 278 208 148)(131 279 209 149)(132 280 210 150)(133 253 211 151)(134 254 212 152)(135 255 213 153)(136 256 214 154)(137 257 215 155)(138 258 216 156)(139 259 217 157)(140 260 218 158)
(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)| (1,445)(2,446)(3,447)(4,448)(5,421)(6,422)(7,423)(8,424)(9,425)(10,426)(11,427)(12,428)(13,429)(14,430)(15,431)(16,432)(17,433)(18,434)(19,435)(20,436)(21,437)(22,438)(23,439)(24,440)(25,441)(26,442)(27,443)(28,444)(29,156)(30,157)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,166)(40,167)(41,168)(42,141)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,155)(57,260)(58,261)(59,262)(60,263)(61,264)(62,265)(63,266)(64,267)(65,268)(66,269)(67,270)(68,271)(69,272)(70,273)(71,274)(72,275)(73,276)(74,277)(75,278)(76,279)(77,280)(78,253)(79,254)(80,255)(81,256)(82,257)(83,258)(84,259)(85,392)(86,365)(87,366)(88,367)(89,368)(90,369)(91,370)(92,371)(93,372)(94,373)(95,374)(96,375)(97,376)(98,377)(99,378)(100,379)(101,380)(102,381)(103,382)(104,383)(105,384)(106,385)(107,386)(108,387)(109,388)(110,389)(111,390)(112,391)(113,303)(114,304)(115,305)(116,306)(117,307)(118,308)(119,281)(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)(137,299)(138,300)(139,301)(140,302)(169,213)(170,214)(171,215)(172,216)(173,217)(174,218)(175,219)(176,220)(177,221)(178,222)(179,223)(180,224)(181,197)(182,198)(183,199)(184,200)(185,201)(186,202)(187,203)(188,204)(189,205)(190,206)(191,207)(192,208)(193,209)(194,210)(195,211)(196,212)(225,318)(226,319)(227,320)(228,321)(229,322)(230,323)(231,324)(232,325)(233,326)(234,327)(235,328)(236,329)(237,330)(238,331)(239,332)(240,333)(241,334)(242,335)(243,336)(244,309)(245,310)(246,311)(247,312)(248,313)(249,314)(250,315)(251,316)(252,317)(337,404)(338,405)(339,406)(340,407)(341,408)(342,409)(343,410)(344,411)(345,412)(346,413)(347,414)(348,415)(349,416)(350,417)(351,418)(352,419)(353,420)(354,393)(355,394)(356,395)(357,396)(358,397)(359,398)(360,399)(361,400)(362,401)(363,402)(364,403), (1,222)(2,223)(3,224)(4,197)(5,198)(6,199)(7,200)(8,201)(9,202)(10,203)(11,204)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,329)(30,330)(31,331)(32,332)(33,333)(34,334)(35,335)(36,336)(37,309)(38,310)(39,311)(40,312)(41,313)(42,314)(43,315)(44,316)(45,317)(46,318)(47,319)(48,320)(49,321)(50,322)(51,323)(52,324)(53,325)(54,326)(55,327)(56,328)(57,394)(58,395)(59,396)(60,397)(61,398)(62,399)(63,400)(64,401)(65,402)(66,403)(67,404)(68,405)(69,406)(70,407)(71,408)(72,409)(73,410)(74,411)(75,412)(76,413)(77,414)(78,415)(79,416)(80,417)(81,418)(82,419)(83,420)(84,393)(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)(98,281)(99,282)(100,283)(101,284)(102,285)(103,286)(104,287)(105,288)(106,289)(107,290)(108,291)(109,292)(110,293)(111,294)(112,295)(113,371)(114,372)(115,373)(116,374)(117,375)(118,376)(119,377)(120,378)(121,379)(122,380)(123,381)(124,382)(125,383)(126,384)(127,385)(128,386)(129,387)(130,388)(131,389)(132,390)(133,391)(134,392)(135,365)(136,366)(137,367)(138,368)(139,369)(140,370)(141,249)(142,250)(143,251)(144,252)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,235)(156,236)(157,237)(158,238)(159,239)(160,240)(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)(169,436)(170,437)(171,438)(172,439)(173,440)(174,441)(175,442)(176,443)(177,444)(178,445)(179,446)(180,447)(181,448)(182,421)(183,422)(184,423)(185,424)(186,425)(187,426)(188,427)(189,428)(190,429)(191,430)(192,431)(193,432)(194,433)(195,434)(196,435)(253,348)(254,349)(255,350)(256,351)(257,352)(258,353)(259,354)(260,355)(261,356)(262,357)(263,358)(264,359)(265,360)(266,361)(267,362)(268,363)(269,364)(270,337)(271,338)(272,339)(273,340)(274,341)(275,342)(276,343)(277,344)(278,345)(279,346)(280,347), (1,242,374,359)(2,243,375,360)(3,244,376,361)(4,245,377,362)(5,246,378,363)(6,247,379,364)(7,248,380,337)(8,249,381,338)(9,250,382,339)(10,251,383,340)(11,252,384,341)(12,225,385,342)(13,226,386,343)(14,227,387,344)(15,228,388,345)(16,229,389,346)(17,230,390,347)(18,231,391,348)(19,232,392,349)(20,233,365,350)(21,234,366,351)(22,235,367,352)(23,236,368,353)(24,237,369,354)(25,238,370,355)(26,239,371,356)(27,240,372,357)(28,241,373,358)(29,300,83,172)(30,301,84,173)(31,302,57,174)(32,303,58,175)(33,304,59,176)(34,305,60,177)(35,306,61,178)(36,307,62,179)(37,308,63,180)(38,281,64,181)(39,282,65,182)(40,283,66,183)(41,284,67,184)(42,285,68,185)(43,286,69,186)(44,287,70,187)(45,288,71,188)(46,289,72,189)(47,290,73,190)(48,291,74,191)(49,292,75,192)(50,293,76,193)(51,294,77,194)(52,295,78,195)(53,296,79,196)(54,297,80,169)(55,298,81,170)(56,299,82,171)(85,416,435,325)(86,417,436,326)(87,418,437,327)(88,419,438,328)(89,420,439,329)(90,393,440,330)(91,394,441,331)(92,395,442,332)(93,396,443,333)(94,397,444,334)(95,398,445,335)(96,399,446,336)(97,400,447,309)(98,401,448,310)(99,402,421,311)(100,403,422,312)(101,404,423,313)(102,405,424,314)(103,406,425,315)(104,407,426,316)(105,408,427,317)(106,409,428,318)(107,410,429,319)(108,411,430,320)(109,412,431,321)(110,413,432,322)(111,414,433,323)(112,415,434,324)(113,261,219,159)(114,262,220,160)(115,263,221,161)(116,264,222,162)(117,265,223,163)(118,266,224,164)(119,267,197,165)(120,268,198,166)(121,269,199,167)(122,270,200,168)(123,271,201,141)(124,272,202,142)(125,273,203,143)(126,274,204,144)(127,275,205,145)(128,276,206,146)(129,277,207,147)(130,278,208,148)(131,279,209,149)(132,280,210,150)(133,253,211,151)(134,254,212,152)(135,255,213,153)(136,256,214,154)(137,257,215,155)(138,258,216,156)(139,259,217,157)(140,260,218,158), (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( (1,445)(2,446)(3,447)(4,448)(5,421)(6,422)(7,423)(8,424)(9,425)(10,426)(11,427)(12,428)(13,429)(14,430)(15,431)(16,432)(17,433)(18,434)(19,435)(20,436)(21,437)(22,438)(23,439)(24,440)(25,441)(26,442)(27,443)(28,444)(29,156)(30,157)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,166)(40,167)(41,168)(42,141)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,155)(57,260)(58,261)(59,262)(60,263)(61,264)(62,265)(63,266)(64,267)(65,268)(66,269)(67,270)(68,271)(69,272)(70,273)(71,274)(72,275)(73,276)(74,277)(75,278)(76,279)(77,280)(78,253)(79,254)(80,255)(81,256)(82,257)(83,258)(84,259)(85,392)(86,365)(87,366)(88,367)(89,368)(90,369)(91,370)(92,371)(93,372)(94,373)(95,374)(96,375)(97,376)(98,377)(99,378)(100,379)(101,380)(102,381)(103,382)(104,383)(105,384)(106,385)(107,386)(108,387)(109,388)(110,389)(111,390)(112,391)(113,303)(114,304)(115,305)(116,306)(117,307)(118,308)(119,281)(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)(137,299)(138,300)(139,301)(140,302)(169,213)(170,214)(171,215)(172,216)(173,217)(174,218)(175,219)(176,220)(177,221)(178,222)(179,223)(180,224)(181,197)(182,198)(183,199)(184,200)(185,201)(186,202)(187,203)(188,204)(189,205)(190,206)(191,207)(192,208)(193,209)(194,210)(195,211)(196,212)(225,318)(226,319)(227,320)(228,321)(229,322)(230,323)(231,324)(232,325)(233,326)(234,327)(235,328)(236,329)(237,330)(238,331)(239,332)(240,333)(241,334)(242,335)(243,336)(244,309)(245,310)(246,311)(247,312)(248,313)(249,314)(250,315)(251,316)(252,317)(337,404)(338,405)(339,406)(340,407)(341,408)(342,409)(343,410)(344,411)(345,412)(346,413)(347,414)(348,415)(349,416)(350,417)(351,418)(352,419)(353,420)(354,393)(355,394)(356,395)(357,396)(358,397)(359,398)(360,399)(361,400)(362,401)(363,402)(364,403), (1,222)(2,223)(3,224)(4,197)(5,198)(6,199)(7,200)(8,201)(9,202)(10,203)(11,204)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,329)(30,330)(31,331)(32,332)(33,333)(34,334)(35,335)(36,336)(37,309)(38,310)(39,311)(40,312)(41,313)(42,314)(43,315)(44,316)(45,317)(46,318)(47,319)(48,320)(49,321)(50,322)(51,323)(52,324)(53,325)(54,326)(55,327)(56,328)(57,394)(58,395)(59,396)(60,397)(61,398)(62,399)(63,400)(64,401)(65,402)(66,403)(67,404)(68,405)(69,406)(70,407)(71,408)(72,409)(73,410)(74,411)(75,412)(76,413)(77,414)(78,415)(79,416)(80,417)(81,418)(82,419)(83,420)(84,393)(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)(98,281)(99,282)(100,283)(101,284)(102,285)(103,286)(104,287)(105,288)(106,289)(107,290)(108,291)(109,292)(110,293)(111,294)(112,295)(113,371)(114,372)(115,373)(116,374)(117,375)(118,376)(119,377)(120,378)(121,379)(122,380)(123,381)(124,382)(125,383)(126,384)(127,385)(128,386)(129,387)(130,388)(131,389)(132,390)(133,391)(134,392)(135,365)(136,366)(137,367)(138,368)(139,369)(140,370)(141,249)(142,250)(143,251)(144,252)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,235)(156,236)(157,237)(158,238)(159,239)(160,240)(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)(169,436)(170,437)(171,438)(172,439)(173,440)(174,441)(175,442)(176,443)(177,444)(178,445)(179,446)(180,447)(181,448)(182,421)(183,422)(184,423)(185,424)(186,425)(187,426)(188,427)(189,428)(190,429)(191,430)(192,431)(193,432)(194,433)(195,434)(196,435)(253,348)(254,349)(255,350)(256,351)(257,352)(258,353)(259,354)(260,355)(261,356)(262,357)(263,358)(264,359)(265,360)(266,361)(267,362)(268,363)(269,364)(270,337)(271,338)(272,339)(273,340)(274,341)(275,342)(276,343)(277,344)(278,345)(279,346)(280,347), (1,242,374,359)(2,243,375,360)(3,244,376,361)(4,245,377,362)(5,246,378,363)(6,247,379,364)(7,248,380,337)(8,249,381,338)(9,250,382,339)(10,251,383,340)(11,252,384,341)(12,225,385,342)(13,226,386,343)(14,227,387,344)(15,228,388,345)(16,229,389,346)(17,230,390,347)(18,231,391,348)(19,232,392,349)(20,233,365,350)(21,234,366,351)(22,235,367,352)(23,236,368,353)(24,237,369,354)(25,238,370,355)(26,239,371,356)(27,240,372,357)(28,241,373,358)(29,300,83,172)(30,301,84,173)(31,302,57,174)(32,303,58,175)(33,304,59,176)(34,305,60,177)(35,306,61,178)(36,307,62,179)(37,308,63,180)(38,281,64,181)(39,282,65,182)(40,283,66,183)(41,284,67,184)(42,285,68,185)(43,286,69,186)(44,287,70,187)(45,288,71,188)(46,289,72,189)(47,290,73,190)(48,291,74,191)(49,292,75,192)(50,293,76,193)(51,294,77,194)(52,295,78,195)(53,296,79,196)(54,297,80,169)(55,298,81,170)(56,299,82,171)(85,416,435,325)(86,417,436,326)(87,418,437,327)(88,419,438,328)(89,420,439,329)(90,393,440,330)(91,394,441,331)(92,395,442,332)(93,396,443,333)(94,397,444,334)(95,398,445,335)(96,399,446,336)(97,400,447,309)(98,401,448,310)(99,402,421,311)(100,403,422,312)(101,404,423,313)(102,405,424,314)(103,406,425,315)(104,407,426,316)(105,408,427,317)(106,409,428,318)(107,410,429,319)(108,411,430,320)(109,412,431,321)(110,413,432,322)(111,414,433,323)(112,415,434,324)(113,261,219,159)(114,262,220,160)(115,263,221,161)(116,264,222,162)(117,265,223,163)(118,266,224,164)(119,267,197,165)(120,268,198,166)(121,269,199,167)(122,270,200,168)(123,271,201,141)(124,272,202,142)(125,273,203,143)(126,274,204,144)(127,275,205,145)(128,276,206,146)(129,277,207,147)(130,278,208,148)(131,279,209,149)(132,280,210,150)(133,253,211,151)(134,254,212,152)(135,255,213,153)(136,256,214,154)(137,257,215,155)(138,258,216,156)(139,259,217,157)(140,260,218,158), (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([(1,445),(2,446),(3,447),(4,448),(5,421),(6,422),(7,423),(8,424),(9,425),(10,426),(11,427),(12,428),(13,429),(14,430),(15,431),(16,432),(17,433),(18,434),(19,435),(20,436),(21,437),(22,438),(23,439),(24,440),(25,441),(26,442),(27,443),(28,444),(29,156),(30,157),(31,158),(32,159),(33,160),(34,161),(35,162),(36,163),(37,164),(38,165),(39,166),(40,167),(41,168),(42,141),(43,142),(44,143),(45,144),(46,145),(47,146),(48,147),(49,148),(50,149),(51,150),(52,151),(53,152),(54,153),(55,154),(56,155),(57,260),(58,261),(59,262),(60,263),(61,264),(62,265),(63,266),(64,267),(65,268),(66,269),(67,270),(68,271),(69,272),(70,273),(71,274),(72,275),(73,276),(74,277),(75,278),(76,279),(77,280),(78,253),(79,254),(80,255),(81,256),(82,257),(83,258),(84,259),(85,392),(86,365),(87,366),(88,367),(89,368),(90,369),(91,370),(92,371),(93,372),(94,373),(95,374),(96,375),(97,376),(98,377),(99,378),(100,379),(101,380),(102,381),(103,382),(104,383),(105,384),(106,385),(107,386),(108,387),(109,388),(110,389),(111,390),(112,391),(113,303),(114,304),(115,305),(116,306),(117,307),(118,308),(119,281),(120,282),(121,283),(122,284),(123,285),(124,286),(125,287),(126,288),(127,289),(128,290),(129,291),(130,292),(131,293),(132,294),(133,295),(134,296),(135,297),(136,298),(137,299),(138,300),(139,301),(140,302),(169,213),(170,214),(171,215),(172,216),(173,217),(174,218),(175,219),(176,220),(177,221),(178,222),(179,223),(180,224),(181,197),(182,198),(183,199),(184,200),(185,201),(186,202),(187,203),(188,204),(189,205),(190,206),(191,207),(192,208),(193,209),(194,210),(195,211),(196,212),(225,318),(226,319),(227,320),(228,321),(229,322),(230,323),(231,324),(232,325),(233,326),(234,327),(235,328),(236,329),(237,330),(238,331),(239,332),(240,333),(241,334),(242,335),(243,336),(244,309),(245,310),(246,311),(247,312),(248,313),(249,314),(250,315),(251,316),(252,317),(337,404),(338,405),(339,406),(340,407),(341,408),(342,409),(343,410),(344,411),(345,412),(346,413),(347,414),(348,415),(349,416),(350,417),(351,418),(352,419),(353,420),(354,393),(355,394),(356,395),(357,396),(358,397),(359,398),(360,399),(361,400),(362,401),(363,402),(364,403)], [(1,222),(2,223),(3,224),(4,197),(5,198),(6,199),(7,200),(8,201),(9,202),(10,203),(11,204),(12,205),(13,206),(14,207),(15,208),(16,209),(17,210),(18,211),(19,212),(20,213),(21,214),(22,215),(23,216),(24,217),(25,218),(26,219),(27,220),(28,221),(29,329),(30,330),(31,331),(32,332),(33,333),(34,334),(35,335),(36,336),(37,309),(38,310),(39,311),(40,312),(41,313),(42,314),(43,315),(44,316),(45,317),(46,318),(47,319),(48,320),(49,321),(50,322),(51,323),(52,324),(53,325),(54,326),(55,327),(56,328),(57,394),(58,395),(59,396),(60,397),(61,398),(62,399),(63,400),(64,401),(65,402),(66,403),(67,404),(68,405),(69,406),(70,407),(71,408),(72,409),(73,410),(74,411),(75,412),(76,413),(77,414),(78,415),(79,416),(80,417),(81,418),(82,419),(83,420),(84,393),(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),(98,281),(99,282),(100,283),(101,284),(102,285),(103,286),(104,287),(105,288),(106,289),(107,290),(108,291),(109,292),(110,293),(111,294),(112,295),(113,371),(114,372),(115,373),(116,374),(117,375),(118,376),(119,377),(120,378),(121,379),(122,380),(123,381),(124,382),(125,383),(126,384),(127,385),(128,386),(129,387),(130,388),(131,389),(132,390),(133,391),(134,392),(135,365),(136,366),(137,367),(138,368),(139,369),(140,370),(141,249),(142,250),(143,251),(144,252),(145,225),(146,226),(147,227),(148,228),(149,229),(150,230),(151,231),(152,232),(153,233),(154,234),(155,235),(156,236),(157,237),(158,238),(159,239),(160,240),(161,241),(162,242),(163,243),(164,244),(165,245),(166,246),(167,247),(168,248),(169,436),(170,437),(171,438),(172,439),(173,440),(174,441),(175,442),(176,443),(177,444),(178,445),(179,446),(180,447),(181,448),(182,421),(183,422),(184,423),(185,424),(186,425),(187,426),(188,427),(189,428),(190,429),(191,430),(192,431),(193,432),(194,433),(195,434),(196,435),(253,348),(254,349),(255,350),(256,351),(257,352),(258,353),(259,354),(260,355),(261,356),(262,357),(263,358),(264,359),(265,360),(266,361),(267,362),(268,363),(269,364),(270,337),(271,338),(272,339),(273,340),(274,341),(275,342),(276,343),(277,344),(278,345),(279,346),(280,347)], [(1,242,374,359),(2,243,375,360),(3,244,376,361),(4,245,377,362),(5,246,378,363),(6,247,379,364),(7,248,380,337),(8,249,381,338),(9,250,382,339),(10,251,383,340),(11,252,384,341),(12,225,385,342),(13,226,386,343),(14,227,387,344),(15,228,388,345),(16,229,389,346),(17,230,390,347),(18,231,391,348),(19,232,392,349),(20,233,365,350),(21,234,366,351),(22,235,367,352),(23,236,368,353),(24,237,369,354),(25,238,370,355),(26,239,371,356),(27,240,372,357),(28,241,373,358),(29,300,83,172),(30,301,84,173),(31,302,57,174),(32,303,58,175),(33,304,59,176),(34,305,60,177),(35,306,61,178),(36,307,62,179),(37,308,63,180),(38,281,64,181),(39,282,65,182),(40,283,66,183),(41,284,67,184),(42,285,68,185),(43,286,69,186),(44,287,70,187),(45,288,71,188),(46,289,72,189),(47,290,73,190),(48,291,74,191),(49,292,75,192),(50,293,76,193),(51,294,77,194),(52,295,78,195),(53,296,79,196),(54,297,80,169),(55,298,81,170),(56,299,82,171),(85,416,435,325),(86,417,436,326),(87,418,437,327),(88,419,438,328),(89,420,439,329),(90,393,440,330),(91,394,441,331),(92,395,442,332),(93,396,443,333),(94,397,444,334),(95,398,445,335),(96,399,446,336),(97,400,447,309),(98,401,448,310),(99,402,421,311),(100,403,422,312),(101,404,423,313),(102,405,424,314),(103,406,425,315),(104,407,426,316),(105,408,427,317),(106,409,428,318),(107,410,429,319),(108,411,430,320),(109,412,431,321),(110,413,432,322),(111,414,433,323),(112,415,434,324),(113,261,219,159),(114,262,220,160),(115,263,221,161),(116,264,222,162),(117,265,223,163),(118,266,224,164),(119,267,197,165),(120,268,198,166),(121,269,199,167),(122,270,200,168),(123,271,201,141),(124,272,202,142),(125,273,203,143),(126,274,204,144),(127,275,205,145),(128,276,206,146),(129,277,207,147),(130,278,208,148),(131,279,209,149),(132,280,210,150),(133,253,211,151),(134,254,212,152),(135,255,213,153),(136,256,214,154),(137,257,215,155),(138,258,216,156),(139,259,217,157),(140,260,218,158)], [(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···2O4A···4AV7A···7F14A···14CL28A···28KB
order12···24···47···714···1428···28
size11···11···11···11···11···1

448 irreducible representations

dim11111111
type+++
imageC1C2C2C4C7C14C14C28
kernelC22×C4×C28C2×C4×C28C23×C28C22×C28C22×C42C2×C42C23×C4C22×C4
# reps11234867218288

Matrix representation of C22×C4×C28 in GL4(𝔽29) generated by

28000
0100
0010
00028
,
1000
0100
00280
0001
,
17000
01700
00120
00012
,
14000
01500
00240
00024
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,1],[17,0,0,0,0,17,0,0,0,0,12,0,0,0,0,12],[14,0,0,0,0,15,0,0,0,0,24,0,0,0,0,24] >;

C22×C4×C28 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times C_{28}
% in TeX

G:=Group("C2^2xC4xC28");
// GroupNames label

G:=SmallGroup(448,1294);
// by ID

G=gap.SmallGroup(448,1294);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1576]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^4=d^28=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

׿
×
𝔽