metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.47D8, C28.2Q16, C4.6Dic28, C42.4D14, C28.46SD16, C4⋊C8.4D7, C4⋊Dic7.2C4, C28⋊C8.9C2, C4.19(D4⋊D7), (C2×C28).464D4, (C2×C4).122D28, C7⋊1(C4.10D8), C4.11(Q8⋊D7), C28⋊2Q8.8C2, C4.10(C56⋊C2), (C4×C28).40C22, C2.4(C14.D8), C14.2(D4⋊C4), C14.8(Q8⋊C4), C22.61(D14⋊C4), C2.5(C28.44D4), C2.4(C4.12D28), C14.3(C4.10D4), (C7×C4⋊C8).4C2, (C2×C4).15(C4×D7), (C2×C28).27(C2×C4), (C2×C4).228(C7⋊D4), (C2×C14).44(C22⋊C4), SmallGroup(448,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.47D8
G = < a,b,c | a28=b8=1, c2=a14, bab-1=cac-1=a-1, cbc-1=a21b-1 >
Subgroups: 324 in 64 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C4⋊C8, C4⋊C8, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C4.10D8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28⋊C8, C7×C4⋊C8, C28⋊2Q8, C28.47D8
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Q16, D14, C4.10D4, D4⋊C4, Q8⋊C4, C4×D7, D28, C7⋊D4, C4.10D8, C56⋊C2, Dic28, D14⋊C4, D4⋊D7, Q8⋊D7, C14.D8, C28.44D4, C4.12D28, C28.47D8
►Regular action on 448 points
Generators in S448
(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)
(1 293 401 138 318 435 258 352)(2 292 402 137 319 434 259 351)(3 291 403 136 320 433 260 350)(4 290 404 135 321 432 261 349)(5 289 405 134 322 431 262 348)(6 288 406 133 323 430 263 347)(7 287 407 132 324 429 264 346)(8 286 408 131 325 428 265 345)(9 285 409 130 326 427 266 344)(10 284 410 129 327 426 267 343)(11 283 411 128 328 425 268 342)(12 282 412 127 329 424 269 341)(13 281 413 126 330 423 270 340)(14 308 414 125 331 422 271 339)(15 307 415 124 332 421 272 338)(16 306 416 123 333 448 273 337)(17 305 417 122 334 447 274 364)(18 304 418 121 335 446 275 363)(19 303 419 120 336 445 276 362)(20 302 420 119 309 444 277 361)(21 301 393 118 310 443 278 360)(22 300 394 117 311 442 279 359)(23 299 395 116 312 441 280 358)(24 298 396 115 313 440 253 357)(25 297 397 114 314 439 254 356)(26 296 398 113 315 438 255 355)(27 295 399 140 316 437 256 354)(28 294 400 139 317 436 257 353)(29 171 102 230 164 204 68 371)(30 170 103 229 165 203 69 370)(31 169 104 228 166 202 70 369)(32 196 105 227 167 201 71 368)(33 195 106 226 168 200 72 367)(34 194 107 225 141 199 73 366)(35 193 108 252 142 198 74 365)(36 192 109 251 143 197 75 392)(37 191 110 250 144 224 76 391)(38 190 111 249 145 223 77 390)(39 189 112 248 146 222 78 389)(40 188 85 247 147 221 79 388)(41 187 86 246 148 220 80 387)(42 186 87 245 149 219 81 386)(43 185 88 244 150 218 82 385)(44 184 89 243 151 217 83 384)(45 183 90 242 152 216 84 383)(46 182 91 241 153 215 57 382)(47 181 92 240 154 214 58 381)(48 180 93 239 155 213 59 380)(49 179 94 238 156 212 60 379)(50 178 95 237 157 211 61 378)(51 177 96 236 158 210 62 377)(52 176 97 235 159 209 63 376)(53 175 98 234 160 208 64 375)(54 174 99 233 161 207 65 374)(55 173 100 232 162 206 66 373)(56 172 101 231 163 205 67 372)
(1 66 15 80)(2 65 16 79)(3 64 17 78)(4 63 18 77)(5 62 19 76)(6 61 20 75)(7 60 21 74)(8 59 22 73)(9 58 23 72)(10 57 24 71)(11 84 25 70)(12 83 26 69)(13 82 27 68)(14 81 28 67)(29 270 43 256)(30 269 44 255)(31 268 45 254)(32 267 46 253)(33 266 47 280)(34 265 48 279)(35 264 49 278)(36 263 50 277)(37 262 51 276)(38 261 52 275)(39 260 53 274)(40 259 54 273)(41 258 55 272)(42 257 56 271)(85 319 99 333)(86 318 100 332)(87 317 101 331)(88 316 102 330)(89 315 103 329)(90 314 104 328)(91 313 105 327)(92 312 106 326)(93 311 107 325)(94 310 108 324)(95 309 109 323)(96 336 110 322)(97 335 111 321)(98 334 112 320)(113 236 127 250)(114 235 128 249)(115 234 129 248)(116 233 130 247)(117 232 131 246)(118 231 132 245)(119 230 133 244)(120 229 134 243)(121 228 135 242)(122 227 136 241)(123 226 137 240)(124 225 138 239)(125 252 139 238)(126 251 140 237)(141 408 155 394)(142 407 156 393)(143 406 157 420)(144 405 158 419)(145 404 159 418)(146 403 160 417)(147 402 161 416)(148 401 162 415)(149 400 163 414)(150 399 164 413)(151 398 165 412)(152 397 166 411)(153 396 167 410)(154 395 168 409)(169 432 183 446)(170 431 184 445)(171 430 185 444)(172 429 186 443)(173 428 187 442)(174 427 188 441)(175 426 189 440)(176 425 190 439)(177 424 191 438)(178 423 192 437)(179 422 193 436)(180 421 194 435)(181 448 195 434)(182 447 196 433)(197 295 211 281)(198 294 212 308)(199 293 213 307)(200 292 214 306)(201 291 215 305)(202 290 216 304)(203 289 217 303)(204 288 218 302)(205 287 219 301)(206 286 220 300)(207 285 221 299)(208 284 222 298)(209 283 223 297)(210 282 224 296)(337 367 351 381)(338 366 352 380)(339 365 353 379)(340 392 354 378)(341 391 355 377)(342 390 356 376)(343 389 357 375)(344 388 358 374)(345 387 359 373)(346 386 360 372)(347 385 361 371)(348 384 362 370)(349 383 363 369)(350 382 364 368)
G:=sub<Sym(448)| (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), (1,293,401,138,318,435,258,352)(2,292,402,137,319,434,259,351)(3,291,403,136,320,433,260,350)(4,290,404,135,321,432,261,349)(5,289,405,134,322,431,262,348)(6,288,406,133,323,430,263,347)(7,287,407,132,324,429,264,346)(8,286,408,131,325,428,265,345)(9,285,409,130,326,427,266,344)(10,284,410,129,327,426,267,343)(11,283,411,128,328,425,268,342)(12,282,412,127,329,424,269,341)(13,281,413,126,330,423,270,340)(14,308,414,125,331,422,271,339)(15,307,415,124,332,421,272,338)(16,306,416,123,333,448,273,337)(17,305,417,122,334,447,274,364)(18,304,418,121,335,446,275,363)(19,303,419,120,336,445,276,362)(20,302,420,119,309,444,277,361)(21,301,393,118,310,443,278,360)(22,300,394,117,311,442,279,359)(23,299,395,116,312,441,280,358)(24,298,396,115,313,440,253,357)(25,297,397,114,314,439,254,356)(26,296,398,113,315,438,255,355)(27,295,399,140,316,437,256,354)(28,294,400,139,317,436,257,353)(29,171,102,230,164,204,68,371)(30,170,103,229,165,203,69,370)(31,169,104,228,166,202,70,369)(32,196,105,227,167,201,71,368)(33,195,106,226,168,200,72,367)(34,194,107,225,141,199,73,366)(35,193,108,252,142,198,74,365)(36,192,109,251,143,197,75,392)(37,191,110,250,144,224,76,391)(38,190,111,249,145,223,77,390)(39,189,112,248,146,222,78,389)(40,188,85,247,147,221,79,388)(41,187,86,246,148,220,80,387)(42,186,87,245,149,219,81,386)(43,185,88,244,150,218,82,385)(44,184,89,243,151,217,83,384)(45,183,90,242,152,216,84,383)(46,182,91,241,153,215,57,382)(47,181,92,240,154,214,58,381)(48,180,93,239,155,213,59,380)(49,179,94,238,156,212,60,379)(50,178,95,237,157,211,61,378)(51,177,96,236,158,210,62,377)(52,176,97,235,159,209,63,376)(53,175,98,234,160,208,64,375)(54,174,99,233,161,207,65,374)(55,173,100,232,162,206,66,373)(56,172,101,231,163,205,67,372), (1,66,15,80)(2,65,16,79)(3,64,17,78)(4,63,18,77)(5,62,19,76)(6,61,20,75)(7,60,21,74)(8,59,22,73)(9,58,23,72)(10,57,24,71)(11,84,25,70)(12,83,26,69)(13,82,27,68)(14,81,28,67)(29,270,43,256)(30,269,44,255)(31,268,45,254)(32,267,46,253)(33,266,47,280)(34,265,48,279)(35,264,49,278)(36,263,50,277)(37,262,51,276)(38,261,52,275)(39,260,53,274)(40,259,54,273)(41,258,55,272)(42,257,56,271)(85,319,99,333)(86,318,100,332)(87,317,101,331)(88,316,102,330)(89,315,103,329)(90,314,104,328)(91,313,105,327)(92,312,106,326)(93,311,107,325)(94,310,108,324)(95,309,109,323)(96,336,110,322)(97,335,111,321)(98,334,112,320)(113,236,127,250)(114,235,128,249)(115,234,129,248)(116,233,130,247)(117,232,131,246)(118,231,132,245)(119,230,133,244)(120,229,134,243)(121,228,135,242)(122,227,136,241)(123,226,137,240)(124,225,138,239)(125,252,139,238)(126,251,140,237)(141,408,155,394)(142,407,156,393)(143,406,157,420)(144,405,158,419)(145,404,159,418)(146,403,160,417)(147,402,161,416)(148,401,162,415)(149,400,163,414)(150,399,164,413)(151,398,165,412)(152,397,166,411)(153,396,167,410)(154,395,168,409)(169,432,183,446)(170,431,184,445)(171,430,185,444)(172,429,186,443)(173,428,187,442)(174,427,188,441)(175,426,189,440)(176,425,190,439)(177,424,191,438)(178,423,192,437)(179,422,193,436)(180,421,194,435)(181,448,195,434)(182,447,196,433)(197,295,211,281)(198,294,212,308)(199,293,213,307)(200,292,214,306)(201,291,215,305)(202,290,216,304)(203,289,217,303)(204,288,218,302)(205,287,219,301)(206,286,220,300)(207,285,221,299)(208,284,222,298)(209,283,223,297)(210,282,224,296)(337,367,351,381)(338,366,352,380)(339,365,353,379)(340,392,354,378)(341,391,355,377)(342,390,356,376)(343,389,357,375)(344,388,358,374)(345,387,359,373)(346,386,360,372)(347,385,361,371)(348,384,362,370)(349,383,363,369)(350,382,364,368)>;
G:=Group( (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), (1,293,401,138,318,435,258,352)(2,292,402,137,319,434,259,351)(3,291,403,136,320,433,260,350)(4,290,404,135,321,432,261,349)(5,289,405,134,322,431,262,348)(6,288,406,133,323,430,263,347)(7,287,407,132,324,429,264,346)(8,286,408,131,325,428,265,345)(9,285,409,130,326,427,266,344)(10,284,410,129,327,426,267,343)(11,283,411,128,328,425,268,342)(12,282,412,127,329,424,269,341)(13,281,413,126,330,423,270,340)(14,308,414,125,331,422,271,339)(15,307,415,124,332,421,272,338)(16,306,416,123,333,448,273,337)(17,305,417,122,334,447,274,364)(18,304,418,121,335,446,275,363)(19,303,419,120,336,445,276,362)(20,302,420,119,309,444,277,361)(21,301,393,118,310,443,278,360)(22,300,394,117,311,442,279,359)(23,299,395,116,312,441,280,358)(24,298,396,115,313,440,253,357)(25,297,397,114,314,439,254,356)(26,296,398,113,315,438,255,355)(27,295,399,140,316,437,256,354)(28,294,400,139,317,436,257,353)(29,171,102,230,164,204,68,371)(30,170,103,229,165,203,69,370)(31,169,104,228,166,202,70,369)(32,196,105,227,167,201,71,368)(33,195,106,226,168,200,72,367)(34,194,107,225,141,199,73,366)(35,193,108,252,142,198,74,365)(36,192,109,251,143,197,75,392)(37,191,110,250,144,224,76,391)(38,190,111,249,145,223,77,390)(39,189,112,248,146,222,78,389)(40,188,85,247,147,221,79,388)(41,187,86,246,148,220,80,387)(42,186,87,245,149,219,81,386)(43,185,88,244,150,218,82,385)(44,184,89,243,151,217,83,384)(45,183,90,242,152,216,84,383)(46,182,91,241,153,215,57,382)(47,181,92,240,154,214,58,381)(48,180,93,239,155,213,59,380)(49,179,94,238,156,212,60,379)(50,178,95,237,157,211,61,378)(51,177,96,236,158,210,62,377)(52,176,97,235,159,209,63,376)(53,175,98,234,160,208,64,375)(54,174,99,233,161,207,65,374)(55,173,100,232,162,206,66,373)(56,172,101,231,163,205,67,372), (1,66,15,80)(2,65,16,79)(3,64,17,78)(4,63,18,77)(5,62,19,76)(6,61,20,75)(7,60,21,74)(8,59,22,73)(9,58,23,72)(10,57,24,71)(11,84,25,70)(12,83,26,69)(13,82,27,68)(14,81,28,67)(29,270,43,256)(30,269,44,255)(31,268,45,254)(32,267,46,253)(33,266,47,280)(34,265,48,279)(35,264,49,278)(36,263,50,277)(37,262,51,276)(38,261,52,275)(39,260,53,274)(40,259,54,273)(41,258,55,272)(42,257,56,271)(85,319,99,333)(86,318,100,332)(87,317,101,331)(88,316,102,330)(89,315,103,329)(90,314,104,328)(91,313,105,327)(92,312,106,326)(93,311,107,325)(94,310,108,324)(95,309,109,323)(96,336,110,322)(97,335,111,321)(98,334,112,320)(113,236,127,250)(114,235,128,249)(115,234,129,248)(116,233,130,247)(117,232,131,246)(118,231,132,245)(119,230,133,244)(120,229,134,243)(121,228,135,242)(122,227,136,241)(123,226,137,240)(124,225,138,239)(125,252,139,238)(126,251,140,237)(141,408,155,394)(142,407,156,393)(143,406,157,420)(144,405,158,419)(145,404,159,418)(146,403,160,417)(147,402,161,416)(148,401,162,415)(149,400,163,414)(150,399,164,413)(151,398,165,412)(152,397,166,411)(153,396,167,410)(154,395,168,409)(169,432,183,446)(170,431,184,445)(171,430,185,444)(172,429,186,443)(173,428,187,442)(174,427,188,441)(175,426,189,440)(176,425,190,439)(177,424,191,438)(178,423,192,437)(179,422,193,436)(180,421,194,435)(181,448,195,434)(182,447,196,433)(197,295,211,281)(198,294,212,308)(199,293,213,307)(200,292,214,306)(201,291,215,305)(202,290,216,304)(203,289,217,303)(204,288,218,302)(205,287,219,301)(206,286,220,300)(207,285,221,299)(208,284,222,298)(209,283,223,297)(210,282,224,296)(337,367,351,381)(338,366,352,380)(339,365,353,379)(340,392,354,378)(341,391,355,377)(342,390,356,376)(343,389,357,375)(344,388,358,374)(345,387,359,373)(346,386,360,372)(347,385,361,371)(348,384,362,370)(349,383,363,369)(350,382,364,368) );
G=PermutationGroup([[(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)], [(1,293,401,138,318,435,258,352),(2,292,402,137,319,434,259,351),(3,291,403,136,320,433,260,350),(4,290,404,135,321,432,261,349),(5,289,405,134,322,431,262,348),(6,288,406,133,323,430,263,347),(7,287,407,132,324,429,264,346),(8,286,408,131,325,428,265,345),(9,285,409,130,326,427,266,344),(10,284,410,129,327,426,267,343),(11,283,411,128,328,425,268,342),(12,282,412,127,329,424,269,341),(13,281,413,126,330,423,270,340),(14,308,414,125,331,422,271,339),(15,307,415,124,332,421,272,338),(16,306,416,123,333,448,273,337),(17,305,417,122,334,447,274,364),(18,304,418,121,335,446,275,363),(19,303,419,120,336,445,276,362),(20,302,420,119,309,444,277,361),(21,301,393,118,310,443,278,360),(22,300,394,117,311,442,279,359),(23,299,395,116,312,441,280,358),(24,298,396,115,313,440,253,357),(25,297,397,114,314,439,254,356),(26,296,398,113,315,438,255,355),(27,295,399,140,316,437,256,354),(28,294,400,139,317,436,257,353),(29,171,102,230,164,204,68,371),(30,170,103,229,165,203,69,370),(31,169,104,228,166,202,70,369),(32,196,105,227,167,201,71,368),(33,195,106,226,168,200,72,367),(34,194,107,225,141,199,73,366),(35,193,108,252,142,198,74,365),(36,192,109,251,143,197,75,392),(37,191,110,250,144,224,76,391),(38,190,111,249,145,223,77,390),(39,189,112,248,146,222,78,389),(40,188,85,247,147,221,79,388),(41,187,86,246,148,220,80,387),(42,186,87,245,149,219,81,386),(43,185,88,244,150,218,82,385),(44,184,89,243,151,217,83,384),(45,183,90,242,152,216,84,383),(46,182,91,241,153,215,57,382),(47,181,92,240,154,214,58,381),(48,180,93,239,155,213,59,380),(49,179,94,238,156,212,60,379),(50,178,95,237,157,211,61,378),(51,177,96,236,158,210,62,377),(52,176,97,235,159,209,63,376),(53,175,98,234,160,208,64,375),(54,174,99,233,161,207,65,374),(55,173,100,232,162,206,66,373),(56,172,101,231,163,205,67,372)], [(1,66,15,80),(2,65,16,79),(3,64,17,78),(4,63,18,77),(5,62,19,76),(6,61,20,75),(7,60,21,74),(8,59,22,73),(9,58,23,72),(10,57,24,71),(11,84,25,70),(12,83,26,69),(13,82,27,68),(14,81,28,67),(29,270,43,256),(30,269,44,255),(31,268,45,254),(32,267,46,253),(33,266,47,280),(34,265,48,279),(35,264,49,278),(36,263,50,277),(37,262,51,276),(38,261,52,275),(39,260,53,274),(40,259,54,273),(41,258,55,272),(42,257,56,271),(85,319,99,333),(86,318,100,332),(87,317,101,331),(88,316,102,330),(89,315,103,329),(90,314,104,328),(91,313,105,327),(92,312,106,326),(93,311,107,325),(94,310,108,324),(95,309,109,323),(96,336,110,322),(97,335,111,321),(98,334,112,320),(113,236,127,250),(114,235,128,249),(115,234,129,248),(116,233,130,247),(117,232,131,246),(118,231,132,245),(119,230,133,244),(120,229,134,243),(121,228,135,242),(122,227,136,241),(123,226,137,240),(124,225,138,239),(125,252,139,238),(126,251,140,237),(141,408,155,394),(142,407,156,393),(143,406,157,420),(144,405,158,419),(145,404,159,418),(146,403,160,417),(147,402,161,416),(148,401,162,415),(149,400,163,414),(150,399,164,413),(151,398,165,412),(152,397,166,411),(153,396,167,410),(154,395,168,409),(169,432,183,446),(170,431,184,445),(171,430,185,444),(172,429,186,443),(173,428,187,442),(174,427,188,441),(175,426,189,440),(176,425,190,439),(177,424,191,438),(178,423,192,437),(179,422,193,436),(180,421,194,435),(181,448,195,434),(182,447,196,433),(197,295,211,281),(198,294,212,308),(199,293,213,307),(200,292,214,306),(201,291,215,305),(202,290,216,304),(203,289,217,303),(204,288,218,302),(205,287,219,301),(206,286,220,300),(207,285,221,299),(208,284,222,298),(209,283,223,297),(210,282,224,296),(337,367,351,381),(338,366,352,380),(339,365,353,379),(340,392,354,378),(341,391,355,377),(342,390,356,376),(343,389,357,375),(344,388,358,374),(345,387,359,373),(346,386,360,372),(347,385,361,371),(348,384,362,370),(349,383,363,369),(350,382,364,368)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D7 | D8 | SD16 | Q16 | D14 | C4×D7 | D28 | C7⋊D4 | C56⋊C2 | Dic28 | C4.10D4 | D4⋊D7 | Q8⋊D7 | C4.12D28 |
| kernel | C28.47D8 | C28⋊C8 | C7×C4⋊C8 | C28⋊2Q8 | C4⋊Dic7 | C2×C28 | C4⋊C8 | C28 | C28 | C28 | C42 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 2 | 4 | 2 | 3 | 6 | 6 | 6 | 12 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of C28.47D8 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 105 | 13 |
| 0 | 0 | 56 | 36 |
| 100 | 13 | 0 | 0 |
| 100 | 100 | 0 | 0 |
| 0 | 0 | 49 | 22 |
| 0 | 0 | 45 | 64 |
| 103 | 50 | 0 | 0 |
| 50 | 10 | 0 | 0 |
| 0 | 0 | 6 | 91 |
| 0 | 0 | 89 | 107 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,105,56,0,0,13,36],[100,100,0,0,13,100,0,0,0,0,49,45,0,0,22,64],[103,50,0,0,50,10,0,0,0,0,6,89,0,0,91,107] >;
C28.47D8 in GAP, Magma, Sage, TeX
C_{28}._{47}D_8 % in TeX
G:=Group("C28.47D8"); // GroupNames label
G:=SmallGroup(448,39);
// by ID
G=gap.SmallGroup(448,39);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,85,316,422,387,268,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^21*b^-1>;
// generators/relations