metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.53D8, C28.24Q16, C28.1M4(2), C42.190D14, C7⋊C8⋊1C8, C4⋊C8.1D7, C7⋊2(C8⋊1C8), C28.1(C2×C8), C4.11(C8×D7), C14.6(C4⋊C8), (C2×C28).31Q8, C28⋊C8.6C2, C4.26(D4⋊D7), (C2×C28).485D4, C4.7(C8⋊D7), C14.1(C2.D8), C2.3(Dic7⋊C8), (C4×C28).37C22, (C2×C4).18Dic14, C4.12(C7⋊Q16), C14.4(C8.C4), C2.1(C28.Q8), C2.1(C28.53D4), C22.18(Dic7⋊C4), (C2×C7⋊C8).4C4, (C4×C7⋊C8).1C2, (C7×C4⋊C8).1C2, (C2×C28).46(C2×C4), (C2×C4).133(C4×D7), (C2×C14).31(C4⋊C4), (C2×C4).263(C7⋊D4), SmallGroup(448,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.53D8
G = < a,b,c | a28=b8=1, c2=a21, bab-1=cac-1=a13, cbc-1=b-1 >
Smallest permutation representation of C28.53D8
►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)
(1 415 358 186 96 303 373 262)(2 400 359 171 97 288 374 275)(3 413 360 184 98 301 375 260)(4 398 361 169 99 286 376 273)(5 411 362 182 100 299 377 258)(6 396 363 195 101 284 378 271)(7 409 364 180 102 297 379 256)(8 394 337 193 103 282 380 269)(9 407 338 178 104 295 381 254)(10 420 339 191 105 308 382 267)(11 405 340 176 106 293 383 280)(12 418 341 189 107 306 384 265)(13 403 342 174 108 291 385 278)(14 416 343 187 109 304 386 263)(15 401 344 172 110 289 387 276)(16 414 345 185 111 302 388 261)(17 399 346 170 112 287 389 274)(18 412 347 183 85 300 390 259)(19 397 348 196 86 285 391 272)(20 410 349 181 87 298 392 257)(21 395 350 194 88 283 365 270)(22 408 351 179 89 296 366 255)(23 393 352 192 90 281 367 268)(24 406 353 177 91 294 368 253)(25 419 354 190 92 307 369 266)(26 404 355 175 93 292 370 279)(27 417 356 188 94 305 371 264)(28 402 357 173 95 290 372 277)(29 239 119 448 69 336 205 144)(30 252 120 433 70 321 206 157)(31 237 121 446 71 334 207 142)(32 250 122 431 72 319 208 155)(33 235 123 444 73 332 209 168)(34 248 124 429 74 317 210 153)(35 233 125 442 75 330 211 166)(36 246 126 427 76 315 212 151)(37 231 127 440 77 328 213 164)(38 244 128 425 78 313 214 149)(39 229 129 438 79 326 215 162)(40 242 130 423 80 311 216 147)(41 227 131 436 81 324 217 160)(42 240 132 421 82 309 218 145)(43 225 133 434 83 322 219 158)(44 238 134 447 84 335 220 143)(45 251 135 432 57 320 221 156)(46 236 136 445 58 333 222 141)(47 249 137 430 59 318 223 154)(48 234 138 443 60 331 224 167)(49 247 139 428 61 316 197 152)(50 232 140 441 62 329 198 165)(51 245 113 426 63 314 199 150)(52 230 114 439 64 327 200 163)(53 243 115 424 65 312 201 148)(54 228 116 437 66 325 202 161)(55 241 117 422 67 310 203 146)(56 226 118 435 68 323 204 159)
(1 51 22 44 15 37 8 30)(2 36 23 29 16 50 9 43)(3 49 24 42 17 35 10 56)(4 34 25 55 18 48 11 41)(5 47 26 40 19 33 12 54)(6 32 27 53 20 46 13 39)(7 45 28 38 21 31 14 52)(57 95 78 88 71 109 64 102)(58 108 79 101 72 94 65 87)(59 93 80 86 73 107 66 100)(60 106 81 99 74 92 67 85)(61 91 82 112 75 105 68 98)(62 104 83 97 76 90 69 111)(63 89 84 110 77 103 70 96)(113 366 134 387 127 380 120 373)(114 379 135 372 128 365 121 386)(115 392 136 385 129 378 122 371)(116 377 137 370 130 391 123 384)(117 390 138 383 131 376 124 369)(118 375 139 368 132 389 125 382)(119 388 140 381 133 374 126 367)(141 403 162 396 155 417 148 410)(142 416 163 409 156 402 149 395)(143 401 164 394 157 415 150 408)(144 414 165 407 158 400 151 393)(145 399 166 420 159 413 152 406)(146 412 167 405 160 398 153 419)(147 397 168 418 161 411 154 404)(169 317 190 310 183 331 176 324)(170 330 191 323 184 316 177 309)(171 315 192 336 185 329 178 322)(172 328 193 321 186 314 179 335)(173 313 194 334 187 327 180 320)(174 326 195 319 188 312 181 333)(175 311 196 332 189 325 182 318)(197 353 218 346 211 339 204 360)(198 338 219 359 212 352 205 345)(199 351 220 344 213 337 206 358)(200 364 221 357 214 350 207 343)(201 349 222 342 215 363 208 356)(202 362 223 355 216 348 209 341)(203 347 224 340 217 361 210 354)(225 275 246 268 239 261 232 254)(226 260 247 253 240 274 233 267)(227 273 248 266 241 259 234 280)(228 258 249 279 242 272 235 265)(229 271 250 264 243 257 236 278)(230 256 251 277 244 270 237 263)(231 269 252 262 245 255 238 276)(281 448 302 441 295 434 288 427)(282 433 303 426 296 447 289 440)(283 446 304 439 297 432 290 425)(284 431 305 424 298 445 291 438)(285 444 306 437 299 430 292 423)(286 429 307 422 300 443 293 436)(287 442 308 435 301 428 294 421)
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), (1,415,358,186,96,303,373,262)(2,400,359,171,97,288,374,275)(3,413,360,184,98,301,375,260)(4,398,361,169,99,286,376,273)(5,411,362,182,100,299,377,258)(6,396,363,195,101,284,378,271)(7,409,364,180,102,297,379,256)(8,394,337,193,103,282,380,269)(9,407,338,178,104,295,381,254)(10,420,339,191,105,308,382,267)(11,405,340,176,106,293,383,280)(12,418,341,189,107,306,384,265)(13,403,342,174,108,291,385,278)(14,416,343,187,109,304,386,263)(15,401,344,172,110,289,387,276)(16,414,345,185,111,302,388,261)(17,399,346,170,112,287,389,274)(18,412,347,183,85,300,390,259)(19,397,348,196,86,285,391,272)(20,410,349,181,87,298,392,257)(21,395,350,194,88,283,365,270)(22,408,351,179,89,296,366,255)(23,393,352,192,90,281,367,268)(24,406,353,177,91,294,368,253)(25,419,354,190,92,307,369,266)(26,404,355,175,93,292,370,279)(27,417,356,188,94,305,371,264)(28,402,357,173,95,290,372,277)(29,239,119,448,69,336,205,144)(30,252,120,433,70,321,206,157)(31,237,121,446,71,334,207,142)(32,250,122,431,72,319,208,155)(33,235,123,444,73,332,209,168)(34,248,124,429,74,317,210,153)(35,233,125,442,75,330,211,166)(36,246,126,427,76,315,212,151)(37,231,127,440,77,328,213,164)(38,244,128,425,78,313,214,149)(39,229,129,438,79,326,215,162)(40,242,130,423,80,311,216,147)(41,227,131,436,81,324,217,160)(42,240,132,421,82,309,218,145)(43,225,133,434,83,322,219,158)(44,238,134,447,84,335,220,143)(45,251,135,432,57,320,221,156)(46,236,136,445,58,333,222,141)(47,249,137,430,59,318,223,154)(48,234,138,443,60,331,224,167)(49,247,139,428,61,316,197,152)(50,232,140,441,62,329,198,165)(51,245,113,426,63,314,199,150)(52,230,114,439,64,327,200,163)(53,243,115,424,65,312,201,148)(54,228,116,437,66,325,202,161)(55,241,117,422,67,310,203,146)(56,226,118,435,68,323,204,159), (1,51,22,44,15,37,8,30)(2,36,23,29,16,50,9,43)(3,49,24,42,17,35,10,56)(4,34,25,55,18,48,11,41)(5,47,26,40,19,33,12,54)(6,32,27,53,20,46,13,39)(7,45,28,38,21,31,14,52)(57,95,78,88,71,109,64,102)(58,108,79,101,72,94,65,87)(59,93,80,86,73,107,66,100)(60,106,81,99,74,92,67,85)(61,91,82,112,75,105,68,98)(62,104,83,97,76,90,69,111)(63,89,84,110,77,103,70,96)(113,366,134,387,127,380,120,373)(114,379,135,372,128,365,121,386)(115,392,136,385,129,378,122,371)(116,377,137,370,130,391,123,384)(117,390,138,383,131,376,124,369)(118,375,139,368,132,389,125,382)(119,388,140,381,133,374,126,367)(141,403,162,396,155,417,148,410)(142,416,163,409,156,402,149,395)(143,401,164,394,157,415,150,408)(144,414,165,407,158,400,151,393)(145,399,166,420,159,413,152,406)(146,412,167,405,160,398,153,419)(147,397,168,418,161,411,154,404)(169,317,190,310,183,331,176,324)(170,330,191,323,184,316,177,309)(171,315,192,336,185,329,178,322)(172,328,193,321,186,314,179,335)(173,313,194,334,187,327,180,320)(174,326,195,319,188,312,181,333)(175,311,196,332,189,325,182,318)(197,353,218,346,211,339,204,360)(198,338,219,359,212,352,205,345)(199,351,220,344,213,337,206,358)(200,364,221,357,214,350,207,343)(201,349,222,342,215,363,208,356)(202,362,223,355,216,348,209,341)(203,347,224,340,217,361,210,354)(225,275,246,268,239,261,232,254)(226,260,247,253,240,274,233,267)(227,273,248,266,241,259,234,280)(228,258,249,279,242,272,235,265)(229,271,250,264,243,257,236,278)(230,256,251,277,244,270,237,263)(231,269,252,262,245,255,238,276)(281,448,302,441,295,434,288,427)(282,433,303,426,296,447,289,440)(283,446,304,439,297,432,290,425)(284,431,305,424,298,445,291,438)(285,444,306,437,299,430,292,423)(286,429,307,422,300,443,293,436)(287,442,308,435,301,428,294,421)>;
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), (1,415,358,186,96,303,373,262)(2,400,359,171,97,288,374,275)(3,413,360,184,98,301,375,260)(4,398,361,169,99,286,376,273)(5,411,362,182,100,299,377,258)(6,396,363,195,101,284,378,271)(7,409,364,180,102,297,379,256)(8,394,337,193,103,282,380,269)(9,407,338,178,104,295,381,254)(10,420,339,191,105,308,382,267)(11,405,340,176,106,293,383,280)(12,418,341,189,107,306,384,265)(13,403,342,174,108,291,385,278)(14,416,343,187,109,304,386,263)(15,401,344,172,110,289,387,276)(16,414,345,185,111,302,388,261)(17,399,346,170,112,287,389,274)(18,412,347,183,85,300,390,259)(19,397,348,196,86,285,391,272)(20,410,349,181,87,298,392,257)(21,395,350,194,88,283,365,270)(22,408,351,179,89,296,366,255)(23,393,352,192,90,281,367,268)(24,406,353,177,91,294,368,253)(25,419,354,190,92,307,369,266)(26,404,355,175,93,292,370,279)(27,417,356,188,94,305,371,264)(28,402,357,173,95,290,372,277)(29,239,119,448,69,336,205,144)(30,252,120,433,70,321,206,157)(31,237,121,446,71,334,207,142)(32,250,122,431,72,319,208,155)(33,235,123,444,73,332,209,168)(34,248,124,429,74,317,210,153)(35,233,125,442,75,330,211,166)(36,246,126,427,76,315,212,151)(37,231,127,440,77,328,213,164)(38,244,128,425,78,313,214,149)(39,229,129,438,79,326,215,162)(40,242,130,423,80,311,216,147)(41,227,131,436,81,324,217,160)(42,240,132,421,82,309,218,145)(43,225,133,434,83,322,219,158)(44,238,134,447,84,335,220,143)(45,251,135,432,57,320,221,156)(46,236,136,445,58,333,222,141)(47,249,137,430,59,318,223,154)(48,234,138,443,60,331,224,167)(49,247,139,428,61,316,197,152)(50,232,140,441,62,329,198,165)(51,245,113,426,63,314,199,150)(52,230,114,439,64,327,200,163)(53,243,115,424,65,312,201,148)(54,228,116,437,66,325,202,161)(55,241,117,422,67,310,203,146)(56,226,118,435,68,323,204,159), (1,51,22,44,15,37,8,30)(2,36,23,29,16,50,9,43)(3,49,24,42,17,35,10,56)(4,34,25,55,18,48,11,41)(5,47,26,40,19,33,12,54)(6,32,27,53,20,46,13,39)(7,45,28,38,21,31,14,52)(57,95,78,88,71,109,64,102)(58,108,79,101,72,94,65,87)(59,93,80,86,73,107,66,100)(60,106,81,99,74,92,67,85)(61,91,82,112,75,105,68,98)(62,104,83,97,76,90,69,111)(63,89,84,110,77,103,70,96)(113,366,134,387,127,380,120,373)(114,379,135,372,128,365,121,386)(115,392,136,385,129,378,122,371)(116,377,137,370,130,391,123,384)(117,390,138,383,131,376,124,369)(118,375,139,368,132,389,125,382)(119,388,140,381,133,374,126,367)(141,403,162,396,155,417,148,410)(142,416,163,409,156,402,149,395)(143,401,164,394,157,415,150,408)(144,414,165,407,158,400,151,393)(145,399,166,420,159,413,152,406)(146,412,167,405,160,398,153,419)(147,397,168,418,161,411,154,404)(169,317,190,310,183,331,176,324)(170,330,191,323,184,316,177,309)(171,315,192,336,185,329,178,322)(172,328,193,321,186,314,179,335)(173,313,194,334,187,327,180,320)(174,326,195,319,188,312,181,333)(175,311,196,332,189,325,182,318)(197,353,218,346,211,339,204,360)(198,338,219,359,212,352,205,345)(199,351,220,344,213,337,206,358)(200,364,221,357,214,350,207,343)(201,349,222,342,215,363,208,356)(202,362,223,355,216,348,209,341)(203,347,224,340,217,361,210,354)(225,275,246,268,239,261,232,254)(226,260,247,253,240,274,233,267)(227,273,248,266,241,259,234,280)(228,258,249,279,242,272,235,265)(229,271,250,264,243,257,236,278)(230,256,251,277,244,270,237,263)(231,269,252,262,245,255,238,276)(281,448,302,441,295,434,288,427)(282,433,303,426,296,447,289,440)(283,446,304,439,297,432,290,425)(284,431,305,424,298,445,291,438)(285,444,306,437,299,430,292,423)(286,429,307,422,300,443,293,436)(287,442,308,435,301,428,294,421) );
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)], [(1,415,358,186,96,303,373,262),(2,400,359,171,97,288,374,275),(3,413,360,184,98,301,375,260),(4,398,361,169,99,286,376,273),(5,411,362,182,100,299,377,258),(6,396,363,195,101,284,378,271),(7,409,364,180,102,297,379,256),(8,394,337,193,103,282,380,269),(9,407,338,178,104,295,381,254),(10,420,339,191,105,308,382,267),(11,405,340,176,106,293,383,280),(12,418,341,189,107,306,384,265),(13,403,342,174,108,291,385,278),(14,416,343,187,109,304,386,263),(15,401,344,172,110,289,387,276),(16,414,345,185,111,302,388,261),(17,399,346,170,112,287,389,274),(18,412,347,183,85,300,390,259),(19,397,348,196,86,285,391,272),(20,410,349,181,87,298,392,257),(21,395,350,194,88,283,365,270),(22,408,351,179,89,296,366,255),(23,393,352,192,90,281,367,268),(24,406,353,177,91,294,368,253),(25,419,354,190,92,307,369,266),(26,404,355,175,93,292,370,279),(27,417,356,188,94,305,371,264),(28,402,357,173,95,290,372,277),(29,239,119,448,69,336,205,144),(30,252,120,433,70,321,206,157),(31,237,121,446,71,334,207,142),(32,250,122,431,72,319,208,155),(33,235,123,444,73,332,209,168),(34,248,124,429,74,317,210,153),(35,233,125,442,75,330,211,166),(36,246,126,427,76,315,212,151),(37,231,127,440,77,328,213,164),(38,244,128,425,78,313,214,149),(39,229,129,438,79,326,215,162),(40,242,130,423,80,311,216,147),(41,227,131,436,81,324,217,160),(42,240,132,421,82,309,218,145),(43,225,133,434,83,322,219,158),(44,238,134,447,84,335,220,143),(45,251,135,432,57,320,221,156),(46,236,136,445,58,333,222,141),(47,249,137,430,59,318,223,154),(48,234,138,443,60,331,224,167),(49,247,139,428,61,316,197,152),(50,232,140,441,62,329,198,165),(51,245,113,426,63,314,199,150),(52,230,114,439,64,327,200,163),(53,243,115,424,65,312,201,148),(54,228,116,437,66,325,202,161),(55,241,117,422,67,310,203,146),(56,226,118,435,68,323,204,159)], [(1,51,22,44,15,37,8,30),(2,36,23,29,16,50,9,43),(3,49,24,42,17,35,10,56),(4,34,25,55,18,48,11,41),(5,47,26,40,19,33,12,54),(6,32,27,53,20,46,13,39),(7,45,28,38,21,31,14,52),(57,95,78,88,71,109,64,102),(58,108,79,101,72,94,65,87),(59,93,80,86,73,107,66,100),(60,106,81,99,74,92,67,85),(61,91,82,112,75,105,68,98),(62,104,83,97,76,90,69,111),(63,89,84,110,77,103,70,96),(113,366,134,387,127,380,120,373),(114,379,135,372,128,365,121,386),(115,392,136,385,129,378,122,371),(116,377,137,370,130,391,123,384),(117,390,138,383,131,376,124,369),(118,375,139,368,132,389,125,382),(119,388,140,381,133,374,126,367),(141,403,162,396,155,417,148,410),(142,416,163,409,156,402,149,395),(143,401,164,394,157,415,150,408),(144,414,165,407,158,400,151,393),(145,399,166,420,159,413,152,406),(146,412,167,405,160,398,153,419),(147,397,168,418,161,411,154,404),(169,317,190,310,183,331,176,324),(170,330,191,323,184,316,177,309),(171,315,192,336,185,329,178,322),(172,328,193,321,186,314,179,335),(173,313,194,334,187,327,180,320),(174,326,195,319,188,312,181,333),(175,311,196,332,189,325,182,318),(197,353,218,346,211,339,204,360),(198,338,219,359,212,352,205,345),(199,351,220,344,213,337,206,358),(200,364,221,357,214,350,207,343),(201,349,222,342,215,363,208,356),(202,362,223,355,216,348,209,341),(203,347,224,340,217,361,210,354),(225,275,246,268,239,261,232,254),(226,260,247,253,240,274,233,267),(227,273,248,266,241,259,234,280),(228,258,249,279,242,272,235,265),(229,271,250,264,243,257,236,278),(230,256,251,277,244,270,237,263),(231,269,252,262,245,255,238,276),(281,448,302,441,295,434,288,427),(282,433,303,426,296,447,289,440),(283,446,304,439,297,432,290,425),(284,431,305,424,298,445,291,438),(285,444,306,437,299,430,292,423),(286,429,307,422,300,443,293,436),(287,442,308,435,301,428,294,421)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 8M | 8N | 8O | 8P | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | D4 | Q8 | D7 | M4(2) | D8 | Q16 | D14 | C8.C4 | Dic14 | C4×D7 | C7⋊D4 | C8×D7 | C8⋊D7 | D4⋊D7 | C7⋊Q16 | C28.53D4 |
| kernel | C28.53D8 | C4×C7⋊C8 | C28⋊C8 | C7×C4⋊C8 | C2×C7⋊C8 | C7⋊C8 | C2×C28 | C2×C28 | C4⋊C8 | C28 | C28 | C28 | C42 | C14 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 3 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 3 | 3 | 6 |
Matrix representation of C28.53D8 ►in GL4(𝔽113) generated by
| 98 | 21 | 0 | 0 |
| 92 | 37 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 51 | 46 | 0 | 0 |
| 27 | 62 | 0 | 0 |
| 0 | 0 | 51 | 15 |
| 0 | 0 | 15 | 0 |
| 97 | 103 | 0 | 0 |
| 58 | 16 | 0 | 0 |
| 0 | 0 | 54 | 91 |
| 0 | 0 | 87 | 59 |
G:=sub<GL(4,GF(113))| [98,92,0,0,21,37,0,0,0,0,98,0,0,0,0,98],[51,27,0,0,46,62,0,0,0,0,51,15,0,0,15,0],[97,58,0,0,103,16,0,0,0,0,54,87,0,0,91,59] >;
C28.53D8 in GAP, Magma, Sage, TeX
C_{28}._{53}D_8 % in TeX
G:=Group("C28.53D8"); // GroupNames label
G:=SmallGroup(448,36);
// by ID
G=gap.SmallGroup(448,36);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,141,36,100,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^21,b*a*b^-1=c*a*c^-1=a^13,c*b*c^-1=b^-1>;
// generators/relations
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