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G = C22×C4×C28  order 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

Derived series
C1 — C22×C4×C28
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — C4×C28 — C2×C4×C28 — C22×C4×C28
Lower central
C1 — C22×C4×C28
Upper central
C1 — C22×C4×C28

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

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

׿
×
⋊
ℤ
𝔽
○
≀