direct product, abelian, monomial, 2-elementary
Aliases: C22×C4×C28, SmallGroup(448,1294)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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, C4, C22, C22, C7, C2×C4, C23, C14, C42, C22×C4, C24, C28, C2×C14, C2×C14, C2×C42, C23×C4, C2×C28, C22×C14, C22×C42, C4×C28, C22×C28, C23×C14, C2×C4×C28, C23×C28, C22×C4×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C42, C22×C4, C24, C28, C2×C14, C2×C42, C23×C4, C2×C28, C22×C14, C22×C42, C4×C28, C22×C28, C23×C14, C2×C4×C28, C23×C28, C22×C4×C28
►Regular action on 448 points
Generators in S448
(1 142)(2 143)(3 144)(4 145)(5 146)(6 147)(7 148)(8 149)(9 150)(10 151)(11 152)(12 153)(13 154)(14 155)(15 156)(16 157)(17 158)(18 159)(19 160)(20 161)(21 162)(22 163)(23 164)(24 165)(25 166)(26 167)(27 168)(28 141)(29 205)(30 206)(31 207)(32 208)(33 209)(34 210)(35 211)(36 212)(37 213)(38 214)(39 215)(40 216)(41 217)(42 218)(43 219)(44 220)(45 221)(46 222)(47 223)(48 224)(49 197)(50 198)(51 199)(52 200)(53 201)(54 202)(55 203)(56 204)(57 371)(58 372)(59 373)(60 374)(61 375)(62 376)(63 377)(64 378)(65 379)(66 380)(67 381)(68 382)(69 383)(70 384)(71 385)(72 386)(73 387)(74 388)(75 389)(76 390)(77 391)(78 392)(79 365)(80 366)(81 367)(82 368)(83 369)(84 370)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(169 342)(170 343)(171 344)(172 345)(173 346)(174 347)(175 348)(176 349)(177 350)(178 351)(179 352)(180 353)(181 354)(182 355)(183 356)(184 357)(185 358)(186 359)(187 360)(188 361)(189 362)(190 363)(191 364)(192 337)(193 338)(194 339)(195 340)(196 341)(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)(253 444)(254 445)(255 446)(256 447)(257 448)(258 421)(259 422)(260 423)(261 424)(262 425)(263 426)(264 427)(265 428)(266 429)(267 430)(268 431)(269 432)(270 433)(271 434)(272 435)(273 436)(274 437)(275 438)(276 439)(277 440)(278 441)(279 442)(280 443)(281 414)(282 415)(283 416)(284 417)(285 418)(286 419)(287 420)(288 393)(289 394)(290 395)(291 396)(292 397)(293 398)(294 399)(295 400)(296 401)(297 402)(298 403)(299 404)(300 405)(301 406)(302 407)(303 408)(304 409)(305 410)(306 411)(307 412)(308 413)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 242)(86 243)(87 244)(88 245)(89 246)(90 247)(91 248)(92 249)(93 250)(94 251)(95 252)(96 225)(97 226)(98 227)(99 228)(100 229)(101 230)(102 231)(103 232)(104 233)(105 234)(106 235)(107 236)(108 237)(109 238)(110 239)(111 240)(112 241)(113 313)(114 314)(115 315)(116 316)(117 317)(118 318)(119 319)(120 320)(121 321)(122 322)(123 323)(124 324)(125 325)(126 326)(127 327)(128 328)(129 329)(130 330)(131 331)(132 332)(133 333)(134 334)(135 335)(136 336)(137 309)(138 310)(139 311)(140 312)(141 213)(142 214)(143 215)(144 216)(145 217)(146 218)(147 219)(148 220)(149 221)(150 222)(151 223)(152 224)(153 197)(154 198)(155 199)(156 200)(157 201)(158 202)(159 203)(160 204)(161 205)(162 206)(163 207)(164 208)(165 209)(166 210)(167 211)(168 212)(253 295)(254 296)(255 297)(256 298)(257 299)(258 300)(259 301)(260 302)(261 303)(262 304)(263 305)(264 306)(265 307)(266 308)(267 281)(268 282)(269 283)(270 284)(271 285)(272 286)(273 287)(274 288)(275 289)(276 290)(277 291)(278 292)(279 293)(280 294)(337 366)(338 367)(339 368)(340 369)(341 370)(342 371)(343 372)(344 373)(345 374)(346 375)(347 376)(348 377)(349 378)(350 379)(351 380)(352 381)(353 382)(354 383)(355 384)(356 385)(357 386)(358 387)(359 388)(360 389)(361 390)(362 391)(363 392)(364 365)(393 437)(394 438)(395 439)(396 440)(397 441)(398 442)(399 443)(400 444)(401 445)(402 446)(403 447)(404 448)(405 421)(406 422)(407 423)(408 424)(409 425)(410 426)(411 427)(412 428)(413 429)(414 430)(415 431)(416 432)(417 433)(418 434)(419 435)(420 436)
(1 96 176 406)(2 97 177 407)(3 98 178 408)(4 99 179 409)(5 100 180 410)(6 101 181 411)(7 102 182 412)(8 103 183 413)(9 104 184 414)(10 105 185 415)(11 106 186 416)(12 107 187 417)(13 108 188 418)(14 109 189 419)(15 110 190 420)(16 111 191 393)(17 112 192 394)(18 85 193 395)(19 86 194 396)(20 87 195 397)(21 88 196 398)(22 89 169 399)(23 90 170 400)(24 91 171 401)(25 92 172 402)(26 93 173 403)(27 94 174 404)(28 95 175 405)(29 244 83 441)(30 245 84 442)(31 246 57 443)(32 247 58 444)(33 248 59 445)(34 249 60 446)(35 250 61 447)(36 251 62 448)(37 252 63 421)(38 225 64 422)(39 226 65 423)(40 227 66 424)(41 228 67 425)(42 229 68 426)(43 230 69 427)(44 231 70 428)(45 232 71 429)(46 233 72 430)(47 234 73 431)(48 235 74 432)(49 236 75 433)(50 237 76 434)(51 238 77 435)(52 239 78 436)(53 240 79 437)(54 241 80 438)(55 242 81 439)(56 243 82 440)(113 344 296 165)(114 345 297 166)(115 346 298 167)(116 347 299 168)(117 348 300 141)(118 349 301 142)(119 350 302 143)(120 351 303 144)(121 352 304 145)(122 353 305 146)(123 354 306 147)(124 355 307 148)(125 356 308 149)(126 357 281 150)(127 358 282 151)(128 359 283 152)(129 360 284 153)(130 361 285 154)(131 362 286 155)(132 363 287 156)(133 364 288 157)(134 337 289 158)(135 338 290 159)(136 339 291 160)(137 340 292 161)(138 341 293 162)(139 342 294 163)(140 343 295 164)(197 329 389 270)(198 330 390 271)(199 331 391 272)(200 332 392 273)(201 333 365 274)(202 334 366 275)(203 335 367 276)(204 336 368 277)(205 309 369 278)(206 310 370 279)(207 311 371 280)(208 312 372 253)(209 313 373 254)(210 314 374 255)(211 315 375 256)(212 316 376 257)(213 317 377 258)(214 318 378 259)(215 319 379 260)(216 320 380 261)(217 321 381 262)(218 322 382 263)(219 323 383 264)(220 324 384 265)(221 325 385 266)(222 326 386 267)(223 327 387 268)(224 328 388 269)
(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)| (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,166)(26,167)(27,168)(28,141)(29,205)(30,206)(31,207)(32,208)(33,209)(34,210)(35,211)(36,212)(37,213)(38,214)(39,215)(40,216)(41,217)(42,218)(43,219)(44,220)(45,221)(46,222)(47,223)(48,224)(49,197)(50,198)(51,199)(52,200)(53,201)(54,202)(55,203)(56,204)(57,371)(58,372)(59,373)(60,374)(61,375)(62,376)(63,377)(64,378)(65,379)(66,380)(67,381)(68,382)(69,383)(70,384)(71,385)(72,386)(73,387)(74,388)(75,389)(76,390)(77,391)(78,392)(79,365)(80,366)(81,367)(82,368)(83,369)(84,370)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(169,342)(170,343)(171,344)(172,345)(173,346)(174,347)(175,348)(176,349)(177,350)(178,351)(179,352)(180,353)(181,354)(182,355)(183,356)(184,357)(185,358)(186,359)(187,360)(188,361)(189,362)(190,363)(191,364)(192,337)(193,338)(194,339)(195,340)(196,341)(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)(253,444)(254,445)(255,446)(256,447)(257,448)(258,421)(259,422)(260,423)(261,424)(262,425)(263,426)(264,427)(265,428)(266,429)(267,430)(268,431)(269,432)(270,433)(271,434)(272,435)(273,436)(274,437)(275,438)(276,439)(277,440)(278,441)(279,442)(280,443)(281,414)(282,415)(283,416)(284,417)(285,418)(286,419)(287,420)(288,393)(289,394)(290,395)(291,396)(292,397)(293,398)(294,399)(295,400)(296,401)(297,402)(298,403)(299,404)(300,405)(301,406)(302,407)(303,408)(304,409)(305,410)(306,411)(307,412)(308,413), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,242)(86,243)(87,244)(88,245)(89,246)(90,247)(91,248)(92,249)(93,250)(94,251)(95,252)(96,225)(97,226)(98,227)(99,228)(100,229)(101,230)(102,231)(103,232)(104,233)(105,234)(106,235)(107,236)(108,237)(109,238)(110,239)(111,240)(112,241)(113,313)(114,314)(115,315)(116,316)(117,317)(118,318)(119,319)(120,320)(121,321)(122,322)(123,323)(124,324)(125,325)(126,326)(127,327)(128,328)(129,329)(130,330)(131,331)(132,332)(133,333)(134,334)(135,335)(136,336)(137,309)(138,310)(139,311)(140,312)(141,213)(142,214)(143,215)(144,216)(145,217)(146,218)(147,219)(148,220)(149,221)(150,222)(151,223)(152,224)(153,197)(154,198)(155,199)(156,200)(157,201)(158,202)(159,203)(160,204)(161,205)(162,206)(163,207)(164,208)(165,209)(166,210)(167,211)(168,212)(253,295)(254,296)(255,297)(256,298)(257,299)(258,300)(259,301)(260,302)(261,303)(262,304)(263,305)(264,306)(265,307)(266,308)(267,281)(268,282)(269,283)(270,284)(271,285)(272,286)(273,287)(274,288)(275,289)(276,290)(277,291)(278,292)(279,293)(280,294)(337,366)(338,367)(339,368)(340,369)(341,370)(342,371)(343,372)(344,373)(345,374)(346,375)(347,376)(348,377)(349,378)(350,379)(351,380)(352,381)(353,382)(354,383)(355,384)(356,385)(357,386)(358,387)(359,388)(360,389)(361,390)(362,391)(363,392)(364,365)(393,437)(394,438)(395,439)(396,440)(397,441)(398,442)(399,443)(400,444)(401,445)(402,446)(403,447)(404,448)(405,421)(406,422)(407,423)(408,424)(409,425)(410,426)(411,427)(412,428)(413,429)(414,430)(415,431)(416,432)(417,433)(418,434)(419,435)(420,436), (1,96,176,406)(2,97,177,407)(3,98,178,408)(4,99,179,409)(5,100,180,410)(6,101,181,411)(7,102,182,412)(8,103,183,413)(9,104,184,414)(10,105,185,415)(11,106,186,416)(12,107,187,417)(13,108,188,418)(14,109,189,419)(15,110,190,420)(16,111,191,393)(17,112,192,394)(18,85,193,395)(19,86,194,396)(20,87,195,397)(21,88,196,398)(22,89,169,399)(23,90,170,400)(24,91,171,401)(25,92,172,402)(26,93,173,403)(27,94,174,404)(28,95,175,405)(29,244,83,441)(30,245,84,442)(31,246,57,443)(32,247,58,444)(33,248,59,445)(34,249,60,446)(35,250,61,447)(36,251,62,448)(37,252,63,421)(38,225,64,422)(39,226,65,423)(40,227,66,424)(41,228,67,425)(42,229,68,426)(43,230,69,427)(44,231,70,428)(45,232,71,429)(46,233,72,430)(47,234,73,431)(48,235,74,432)(49,236,75,433)(50,237,76,434)(51,238,77,435)(52,239,78,436)(53,240,79,437)(54,241,80,438)(55,242,81,439)(56,243,82,440)(113,344,296,165)(114,345,297,166)(115,346,298,167)(116,347,299,168)(117,348,300,141)(118,349,301,142)(119,350,302,143)(120,351,303,144)(121,352,304,145)(122,353,305,146)(123,354,306,147)(124,355,307,148)(125,356,308,149)(126,357,281,150)(127,358,282,151)(128,359,283,152)(129,360,284,153)(130,361,285,154)(131,362,286,155)(132,363,287,156)(133,364,288,157)(134,337,289,158)(135,338,290,159)(136,339,291,160)(137,340,292,161)(138,341,293,162)(139,342,294,163)(140,343,295,164)(197,329,389,270)(198,330,390,271)(199,331,391,272)(200,332,392,273)(201,333,365,274)(202,334,366,275)(203,335,367,276)(204,336,368,277)(205,309,369,278)(206,310,370,279)(207,311,371,280)(208,312,372,253)(209,313,373,254)(210,314,374,255)(211,315,375,256)(212,316,376,257)(213,317,377,258)(214,318,378,259)(215,319,379,260)(216,320,380,261)(217,321,381,262)(218,322,382,263)(219,323,383,264)(220,324,384,265)(221,325,385,266)(222,326,386,267)(223,327,387,268)(224,328,388,269), (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( (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,166)(26,167)(27,168)(28,141)(29,205)(30,206)(31,207)(32,208)(33,209)(34,210)(35,211)(36,212)(37,213)(38,214)(39,215)(40,216)(41,217)(42,218)(43,219)(44,220)(45,221)(46,222)(47,223)(48,224)(49,197)(50,198)(51,199)(52,200)(53,201)(54,202)(55,203)(56,204)(57,371)(58,372)(59,373)(60,374)(61,375)(62,376)(63,377)(64,378)(65,379)(66,380)(67,381)(68,382)(69,383)(70,384)(71,385)(72,386)(73,387)(74,388)(75,389)(76,390)(77,391)(78,392)(79,365)(80,366)(81,367)(82,368)(83,369)(84,370)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(169,342)(170,343)(171,344)(172,345)(173,346)(174,347)(175,348)(176,349)(177,350)(178,351)(179,352)(180,353)(181,354)(182,355)(183,356)(184,357)(185,358)(186,359)(187,360)(188,361)(189,362)(190,363)(191,364)(192,337)(193,338)(194,339)(195,340)(196,341)(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)(253,444)(254,445)(255,446)(256,447)(257,448)(258,421)(259,422)(260,423)(261,424)(262,425)(263,426)(264,427)(265,428)(266,429)(267,430)(268,431)(269,432)(270,433)(271,434)(272,435)(273,436)(274,437)(275,438)(276,439)(277,440)(278,441)(279,442)(280,443)(281,414)(282,415)(283,416)(284,417)(285,418)(286,419)(287,420)(288,393)(289,394)(290,395)(291,396)(292,397)(293,398)(294,399)(295,400)(296,401)(297,402)(298,403)(299,404)(300,405)(301,406)(302,407)(303,408)(304,409)(305,410)(306,411)(307,412)(308,413), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,242)(86,243)(87,244)(88,245)(89,246)(90,247)(91,248)(92,249)(93,250)(94,251)(95,252)(96,225)(97,226)(98,227)(99,228)(100,229)(101,230)(102,231)(103,232)(104,233)(105,234)(106,235)(107,236)(108,237)(109,238)(110,239)(111,240)(112,241)(113,313)(114,314)(115,315)(116,316)(117,317)(118,318)(119,319)(120,320)(121,321)(122,322)(123,323)(124,324)(125,325)(126,326)(127,327)(128,328)(129,329)(130,330)(131,331)(132,332)(133,333)(134,334)(135,335)(136,336)(137,309)(138,310)(139,311)(140,312)(141,213)(142,214)(143,215)(144,216)(145,217)(146,218)(147,219)(148,220)(149,221)(150,222)(151,223)(152,224)(153,197)(154,198)(155,199)(156,200)(157,201)(158,202)(159,203)(160,204)(161,205)(162,206)(163,207)(164,208)(165,209)(166,210)(167,211)(168,212)(253,295)(254,296)(255,297)(256,298)(257,299)(258,300)(259,301)(260,302)(261,303)(262,304)(263,305)(264,306)(265,307)(266,308)(267,281)(268,282)(269,283)(270,284)(271,285)(272,286)(273,287)(274,288)(275,289)(276,290)(277,291)(278,292)(279,293)(280,294)(337,366)(338,367)(339,368)(340,369)(341,370)(342,371)(343,372)(344,373)(345,374)(346,375)(347,376)(348,377)(349,378)(350,379)(351,380)(352,381)(353,382)(354,383)(355,384)(356,385)(357,386)(358,387)(359,388)(360,389)(361,390)(362,391)(363,392)(364,365)(393,437)(394,438)(395,439)(396,440)(397,441)(398,442)(399,443)(400,444)(401,445)(402,446)(403,447)(404,448)(405,421)(406,422)(407,423)(408,424)(409,425)(410,426)(411,427)(412,428)(413,429)(414,430)(415,431)(416,432)(417,433)(418,434)(419,435)(420,436), (1,96,176,406)(2,97,177,407)(3,98,178,408)(4,99,179,409)(5,100,180,410)(6,101,181,411)(7,102,182,412)(8,103,183,413)(9,104,184,414)(10,105,185,415)(11,106,186,416)(12,107,187,417)(13,108,188,418)(14,109,189,419)(15,110,190,420)(16,111,191,393)(17,112,192,394)(18,85,193,395)(19,86,194,396)(20,87,195,397)(21,88,196,398)(22,89,169,399)(23,90,170,400)(24,91,171,401)(25,92,172,402)(26,93,173,403)(27,94,174,404)(28,95,175,405)(29,244,83,441)(30,245,84,442)(31,246,57,443)(32,247,58,444)(33,248,59,445)(34,249,60,446)(35,250,61,447)(36,251,62,448)(37,252,63,421)(38,225,64,422)(39,226,65,423)(40,227,66,424)(41,228,67,425)(42,229,68,426)(43,230,69,427)(44,231,70,428)(45,232,71,429)(46,233,72,430)(47,234,73,431)(48,235,74,432)(49,236,75,433)(50,237,76,434)(51,238,77,435)(52,239,78,436)(53,240,79,437)(54,241,80,438)(55,242,81,439)(56,243,82,440)(113,344,296,165)(114,345,297,166)(115,346,298,167)(116,347,299,168)(117,348,300,141)(118,349,301,142)(119,350,302,143)(120,351,303,144)(121,352,304,145)(122,353,305,146)(123,354,306,147)(124,355,307,148)(125,356,308,149)(126,357,281,150)(127,358,282,151)(128,359,283,152)(129,360,284,153)(130,361,285,154)(131,362,286,155)(132,363,287,156)(133,364,288,157)(134,337,289,158)(135,338,290,159)(136,339,291,160)(137,340,292,161)(138,341,293,162)(139,342,294,163)(140,343,295,164)(197,329,389,270)(198,330,390,271)(199,331,391,272)(200,332,392,273)(201,333,365,274)(202,334,366,275)(203,335,367,276)(204,336,368,277)(205,309,369,278)(206,310,370,279)(207,311,371,280)(208,312,372,253)(209,313,373,254)(210,314,374,255)(211,315,375,256)(212,316,376,257)(213,317,377,258)(214,318,378,259)(215,319,379,260)(216,320,380,261)(217,321,381,262)(218,322,382,263)(219,323,383,264)(220,324,384,265)(221,325,385,266)(222,326,386,267)(223,327,387,268)(224,328,388,269), (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([[(1,142),(2,143),(3,144),(4,145),(5,146),(6,147),(7,148),(8,149),(9,150),(10,151),(11,152),(12,153),(13,154),(14,155),(15,156),(16,157),(17,158),(18,159),(19,160),(20,161),(21,162),(22,163),(23,164),(24,165),(25,166),(26,167),(27,168),(28,141),(29,205),(30,206),(31,207),(32,208),(33,209),(34,210),(35,211),(36,212),(37,213),(38,214),(39,215),(40,216),(41,217),(42,218),(43,219),(44,220),(45,221),(46,222),(47,223),(48,224),(49,197),(50,198),(51,199),(52,200),(53,201),(54,202),(55,203),(56,204),(57,371),(58,372),(59,373),(60,374),(61,375),(62,376),(63,377),(64,378),(65,379),(66,380),(67,381),(68,382),(69,383),(70,384),(71,385),(72,386),(73,387),(74,388),(75,389),(76,390),(77,391),(78,392),(79,365),(80,366),(81,367),(82,368),(83,369),(84,370),(85,135),(86,136),(87,137),(88,138),(89,139),(90,140),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134),(169,342),(170,343),(171,344),(172,345),(173,346),(174,347),(175,348),(176,349),(177,350),(178,351),(179,352),(180,353),(181,354),(182,355),(183,356),(184,357),(185,358),(186,359),(187,360),(188,361),(189,362),(190,363),(191,364),(192,337),(193,338),(194,339),(195,340),(196,341),(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),(253,444),(254,445),(255,446),(256,447),(257,448),(258,421),(259,422),(260,423),(261,424),(262,425),(263,426),(264,427),(265,428),(266,429),(267,430),(268,431),(269,432),(270,433),(271,434),(272,435),(273,436),(274,437),(275,438),(276,439),(277,440),(278,441),(279,442),(280,443),(281,414),(282,415),(283,416),(284,417),(285,418),(286,419),(287,420),(288,393),(289,394),(290,395),(291,396),(292,397),(293,398),(294,399),(295,400),(296,401),(297,402),(298,403),(299,404),(300,405),(301,406),(302,407),(303,408),(304,409),(305,410),(306,411),(307,412),(308,413)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,242),(86,243),(87,244),(88,245),(89,246),(90,247),(91,248),(92,249),(93,250),(94,251),(95,252),(96,225),(97,226),(98,227),(99,228),(100,229),(101,230),(102,231),(103,232),(104,233),(105,234),(106,235),(107,236),(108,237),(109,238),(110,239),(111,240),(112,241),(113,313),(114,314),(115,315),(116,316),(117,317),(118,318),(119,319),(120,320),(121,321),(122,322),(123,323),(124,324),(125,325),(126,326),(127,327),(128,328),(129,329),(130,330),(131,331),(132,332),(133,333),(134,334),(135,335),(136,336),(137,309),(138,310),(139,311),(140,312),(141,213),(142,214),(143,215),(144,216),(145,217),(146,218),(147,219),(148,220),(149,221),(150,222),(151,223),(152,224),(153,197),(154,198),(155,199),(156,200),(157,201),(158,202),(159,203),(160,204),(161,205),(162,206),(163,207),(164,208),(165,209),(166,210),(167,211),(168,212),(253,295),(254,296),(255,297),(256,298),(257,299),(258,300),(259,301),(260,302),(261,303),(262,304),(263,305),(264,306),(265,307),(266,308),(267,281),(268,282),(269,283),(270,284),(271,285),(272,286),(273,287),(274,288),(275,289),(276,290),(277,291),(278,292),(279,293),(280,294),(337,366),(338,367),(339,368),(340,369),(341,370),(342,371),(343,372),(344,373),(345,374),(346,375),(347,376),(348,377),(349,378),(350,379),(351,380),(352,381),(353,382),(354,383),(355,384),(356,385),(357,386),(358,387),(359,388),(360,389),(361,390),(362,391),(363,392),(364,365),(393,437),(394,438),(395,439),(396,440),(397,441),(398,442),(399,443),(400,444),(401,445),(402,446),(403,447),(404,448),(405,421),(406,422),(407,423),(408,424),(409,425),(410,426),(411,427),(412,428),(413,429),(414,430),(415,431),(416,432),(417,433),(418,434),(419,435),(420,436)], [(1,96,176,406),(2,97,177,407),(3,98,178,408),(4,99,179,409),(5,100,180,410),(6,101,181,411),(7,102,182,412),(8,103,183,413),(9,104,184,414),(10,105,185,415),(11,106,186,416),(12,107,187,417),(13,108,188,418),(14,109,189,419),(15,110,190,420),(16,111,191,393),(17,112,192,394),(18,85,193,395),(19,86,194,396),(20,87,195,397),(21,88,196,398),(22,89,169,399),(23,90,170,400),(24,91,171,401),(25,92,172,402),(26,93,173,403),(27,94,174,404),(28,95,175,405),(29,244,83,441),(30,245,84,442),(31,246,57,443),(32,247,58,444),(33,248,59,445),(34,249,60,446),(35,250,61,447),(36,251,62,448),(37,252,63,421),(38,225,64,422),(39,226,65,423),(40,227,66,424),(41,228,67,425),(42,229,68,426),(43,230,69,427),(44,231,70,428),(45,232,71,429),(46,233,72,430),(47,234,73,431),(48,235,74,432),(49,236,75,433),(50,237,76,434),(51,238,77,435),(52,239,78,436),(53,240,79,437),(54,241,80,438),(55,242,81,439),(56,243,82,440),(113,344,296,165),(114,345,297,166),(115,346,298,167),(116,347,299,168),(117,348,300,141),(118,349,301,142),(119,350,302,143),(120,351,303,144),(121,352,304,145),(122,353,305,146),(123,354,306,147),(124,355,307,148),(125,356,308,149),(126,357,281,150),(127,358,282,151),(128,359,283,152),(129,360,284,153),(130,361,285,154),(131,362,286,155),(132,363,287,156),(133,364,288,157),(134,337,289,158),(135,338,290,159),(136,339,291,160),(137,340,292,161),(138,341,293,162),(139,342,294,163),(140,343,295,164),(197,329,389,270),(198,330,390,271),(199,331,391,272),(200,332,392,273),(201,333,365,274),(202,334,366,275),(203,335,367,276),(204,336,368,277),(205,309,369,278),(206,310,370,279),(207,311,371,280),(208,312,372,253),(209,313,373,254),(210,314,374,255),(211,315,375,256),(212,316,376,257),(213,317,377,258),(214,318,378,259),(215,319,379,260),(216,320,380,261),(217,321,381,262),(218,322,382,263),(219,323,383,264),(220,324,384,265),(221,325,385,266),(222,326,386,267),(223,327,387,268),(224,328,388,269)], [(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 | ··· | 2O | 4A | ··· | 4AV | 7A | ··· | 7F | 14A | ··· | 14CL | 28A | ··· | 28KB |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
448 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 |
| kernel | C22×C4×C28 | C2×C4×C28 | C23×C28 | C22×C28 | C22×C42 | C2×C42 | C23×C4 | C22×C4 |
| # reps | 1 | 12 | 3 | 48 | 6 | 72 | 18 | 288 |
Matrix representation of C22×C4×C28 ►in GL4(𝔽29) generated by
| 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 |
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