metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.39SD16, C28.2M4(2), C42.191D14, C7⋊C8⋊2C8, C4⋊C8.2D7, C7⋊2(C8⋊2C8), C28.2(C2×C8), C4.12(C8×D7), C14.7(C4⋊C8), (C2×C28).32Q8, C28⋊C8.7C2, (C2×C28).486D4, C4.8(C8⋊D7), C4.14(Q8⋊D7), C14.1(C4.Q8), C2.4(Dic7⋊C8), C4.14(D4.D7), (C4×C28).38C22, (C2×C4).19Dic14, C14.5(C8.C4), C2.1(C4.Dic14), C2.2(C28.53D4), C22.19(Dic7⋊C4), (C4×C7⋊C8).2C2, (C2×C7⋊C8).5C4, (C7×C4⋊C8).2C2, (C2×C28).47(C2×C4), (C2×C4).134(C4×D7), (C2×C14).32(C4⋊C4), (C2×C4).264(C7⋊D4), SmallGroup(448,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.39SD16
G = < a,b,c | a28=b8=1, c2=a21, bab-1=a13, ac=ca, cbc-1=b3 >
Smallest permutation representation of C28.39SD16
►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)
(1 299 379 165 47 63 239 343)(2 284 380 150 48 76 240 356)(3 297 381 163 49 61 241 341)(4 282 382 148 50 74 242 354)(5 295 383 161 51 59 243 339)(6 308 384 146 52 72 244 352)(7 293 385 159 53 57 245 337)(8 306 386 144 54 70 246 350)(9 291 387 157 55 83 247 363)(10 304 388 142 56 68 248 348)(11 289 389 155 29 81 249 361)(12 302 390 168 30 66 250 346)(13 287 391 153 31 79 251 359)(14 300 392 166 32 64 252 344)(15 285 365 151 33 77 225 357)(16 298 366 164 34 62 226 342)(17 283 367 149 35 75 227 355)(18 296 368 162 36 60 228 340)(19 281 369 147 37 73 229 353)(20 294 370 160 38 58 230 338)(21 307 371 145 39 71 231 351)(22 292 372 158 40 84 232 364)(23 305 373 143 41 69 233 349)(24 290 374 156 42 82 234 362)(25 303 375 141 43 67 235 347)(26 288 376 154 44 80 236 360)(27 301 377 167 45 65 237 345)(28 286 378 152 46 78 238 358)(85 319 216 412 428 126 193 273)(86 332 217 397 429 139 194 258)(87 317 218 410 430 124 195 271)(88 330 219 395 431 137 196 256)(89 315 220 408 432 122 169 269)(90 328 221 393 433 135 170 254)(91 313 222 406 434 120 171 267)(92 326 223 419 435 133 172 280)(93 311 224 404 436 118 173 265)(94 324 197 417 437 131 174 278)(95 309 198 402 438 116 175 263)(96 322 199 415 439 129 176 276)(97 335 200 400 440 114 177 261)(98 320 201 413 441 127 178 274)(99 333 202 398 442 140 179 259)(100 318 203 411 443 125 180 272)(101 331 204 396 444 138 181 257)(102 316 205 409 445 123 182 270)(103 329 206 394 446 136 183 255)(104 314 207 407 447 121 184 268)(105 327 208 420 448 134 185 253)(106 312 209 405 421 119 186 266)(107 325 210 418 422 132 187 279)(108 310 211 403 423 117 188 264)(109 323 212 416 424 130 189 277)(110 336 213 401 425 115 190 262)(111 321 214 414 426 128 191 275)(112 334 215 399 427 113 192 260)
(1 132 22 125 15 118 8 139)(2 133 23 126 16 119 9 140)(3 134 24 127 17 120 10 113)(4 135 25 128 18 121 11 114)(5 136 26 129 19 122 12 115)(6 137 27 130 20 123 13 116)(7 138 28 131 21 124 14 117)(29 335 50 328 43 321 36 314)(30 336 51 329 44 322 37 315)(31 309 52 330 45 323 38 316)(32 310 53 331 46 324 39 317)(33 311 54 332 47 325 40 318)(34 312 55 333 48 326 41 319)(35 313 56 334 49 327 42 320)(57 444 78 437 71 430 64 423)(58 445 79 438 72 431 65 424)(59 446 80 439 73 432 66 425)(60 447 81 440 74 433 67 426)(61 448 82 441 75 434 68 427)(62 421 83 442 76 435 69 428)(63 422 84 443 77 436 70 429)(85 298 106 291 99 284 92 305)(86 299 107 292 100 285 93 306)(87 300 108 293 101 286 94 307)(88 301 109 294 102 287 95 308)(89 302 110 295 103 288 96 281)(90 303 111 296 104 289 97 282)(91 304 112 297 105 290 98 283)(141 191 162 184 155 177 148 170)(142 192 163 185 156 178 149 171)(143 193 164 186 157 179 150 172)(144 194 165 187 158 180 151 173)(145 195 166 188 159 181 152 174)(146 196 167 189 160 182 153 175)(147 169 168 190 161 183 154 176)(197 351 218 344 211 337 204 358)(198 352 219 345 212 338 205 359)(199 353 220 346 213 339 206 360)(200 354 221 347 214 340 207 361)(201 355 222 348 215 341 208 362)(202 356 223 349 216 342 209 363)(203 357 224 350 217 343 210 364)(225 265 246 258 239 279 232 272)(226 266 247 259 240 280 233 273)(227 267 248 260 241 253 234 274)(228 268 249 261 242 254 235 275)(229 269 250 262 243 255 236 276)(230 270 251 263 244 256 237 277)(231 271 252 264 245 257 238 278)(365 404 386 397 379 418 372 411)(366 405 387 398 380 419 373 412)(367 406 388 399 381 420 374 413)(368 407 389 400 382 393 375 414)(369 408 390 401 383 394 376 415)(370 409 391 402 384 395 377 416)(371 410 392 403 385 396 378 417)
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), (1,299,379,165,47,63,239,343)(2,284,380,150,48,76,240,356)(3,297,381,163,49,61,241,341)(4,282,382,148,50,74,242,354)(5,295,383,161,51,59,243,339)(6,308,384,146,52,72,244,352)(7,293,385,159,53,57,245,337)(8,306,386,144,54,70,246,350)(9,291,387,157,55,83,247,363)(10,304,388,142,56,68,248,348)(11,289,389,155,29,81,249,361)(12,302,390,168,30,66,250,346)(13,287,391,153,31,79,251,359)(14,300,392,166,32,64,252,344)(15,285,365,151,33,77,225,357)(16,298,366,164,34,62,226,342)(17,283,367,149,35,75,227,355)(18,296,368,162,36,60,228,340)(19,281,369,147,37,73,229,353)(20,294,370,160,38,58,230,338)(21,307,371,145,39,71,231,351)(22,292,372,158,40,84,232,364)(23,305,373,143,41,69,233,349)(24,290,374,156,42,82,234,362)(25,303,375,141,43,67,235,347)(26,288,376,154,44,80,236,360)(27,301,377,167,45,65,237,345)(28,286,378,152,46,78,238,358)(85,319,216,412,428,126,193,273)(86,332,217,397,429,139,194,258)(87,317,218,410,430,124,195,271)(88,330,219,395,431,137,196,256)(89,315,220,408,432,122,169,269)(90,328,221,393,433,135,170,254)(91,313,222,406,434,120,171,267)(92,326,223,419,435,133,172,280)(93,311,224,404,436,118,173,265)(94,324,197,417,437,131,174,278)(95,309,198,402,438,116,175,263)(96,322,199,415,439,129,176,276)(97,335,200,400,440,114,177,261)(98,320,201,413,441,127,178,274)(99,333,202,398,442,140,179,259)(100,318,203,411,443,125,180,272)(101,331,204,396,444,138,181,257)(102,316,205,409,445,123,182,270)(103,329,206,394,446,136,183,255)(104,314,207,407,447,121,184,268)(105,327,208,420,448,134,185,253)(106,312,209,405,421,119,186,266)(107,325,210,418,422,132,187,279)(108,310,211,403,423,117,188,264)(109,323,212,416,424,130,189,277)(110,336,213,401,425,115,190,262)(111,321,214,414,426,128,191,275)(112,334,215,399,427,113,192,260), (1,132,22,125,15,118,8,139)(2,133,23,126,16,119,9,140)(3,134,24,127,17,120,10,113)(4,135,25,128,18,121,11,114)(5,136,26,129,19,122,12,115)(6,137,27,130,20,123,13,116)(7,138,28,131,21,124,14,117)(29,335,50,328,43,321,36,314)(30,336,51,329,44,322,37,315)(31,309,52,330,45,323,38,316)(32,310,53,331,46,324,39,317)(33,311,54,332,47,325,40,318)(34,312,55,333,48,326,41,319)(35,313,56,334,49,327,42,320)(57,444,78,437,71,430,64,423)(58,445,79,438,72,431,65,424)(59,446,80,439,73,432,66,425)(60,447,81,440,74,433,67,426)(61,448,82,441,75,434,68,427)(62,421,83,442,76,435,69,428)(63,422,84,443,77,436,70,429)(85,298,106,291,99,284,92,305)(86,299,107,292,100,285,93,306)(87,300,108,293,101,286,94,307)(88,301,109,294,102,287,95,308)(89,302,110,295,103,288,96,281)(90,303,111,296,104,289,97,282)(91,304,112,297,105,290,98,283)(141,191,162,184,155,177,148,170)(142,192,163,185,156,178,149,171)(143,193,164,186,157,179,150,172)(144,194,165,187,158,180,151,173)(145,195,166,188,159,181,152,174)(146,196,167,189,160,182,153,175)(147,169,168,190,161,183,154,176)(197,351,218,344,211,337,204,358)(198,352,219,345,212,338,205,359)(199,353,220,346,213,339,206,360)(200,354,221,347,214,340,207,361)(201,355,222,348,215,341,208,362)(202,356,223,349,216,342,209,363)(203,357,224,350,217,343,210,364)(225,265,246,258,239,279,232,272)(226,266,247,259,240,280,233,273)(227,267,248,260,241,253,234,274)(228,268,249,261,242,254,235,275)(229,269,250,262,243,255,236,276)(230,270,251,263,244,256,237,277)(231,271,252,264,245,257,238,278)(365,404,386,397,379,418,372,411)(366,405,387,398,380,419,373,412)(367,406,388,399,381,420,374,413)(368,407,389,400,382,393,375,414)(369,408,390,401,383,394,376,415)(370,409,391,402,384,395,377,416)(371,410,392,403,385,396,378,417)>;
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), (1,299,379,165,47,63,239,343)(2,284,380,150,48,76,240,356)(3,297,381,163,49,61,241,341)(4,282,382,148,50,74,242,354)(5,295,383,161,51,59,243,339)(6,308,384,146,52,72,244,352)(7,293,385,159,53,57,245,337)(8,306,386,144,54,70,246,350)(9,291,387,157,55,83,247,363)(10,304,388,142,56,68,248,348)(11,289,389,155,29,81,249,361)(12,302,390,168,30,66,250,346)(13,287,391,153,31,79,251,359)(14,300,392,166,32,64,252,344)(15,285,365,151,33,77,225,357)(16,298,366,164,34,62,226,342)(17,283,367,149,35,75,227,355)(18,296,368,162,36,60,228,340)(19,281,369,147,37,73,229,353)(20,294,370,160,38,58,230,338)(21,307,371,145,39,71,231,351)(22,292,372,158,40,84,232,364)(23,305,373,143,41,69,233,349)(24,290,374,156,42,82,234,362)(25,303,375,141,43,67,235,347)(26,288,376,154,44,80,236,360)(27,301,377,167,45,65,237,345)(28,286,378,152,46,78,238,358)(85,319,216,412,428,126,193,273)(86,332,217,397,429,139,194,258)(87,317,218,410,430,124,195,271)(88,330,219,395,431,137,196,256)(89,315,220,408,432,122,169,269)(90,328,221,393,433,135,170,254)(91,313,222,406,434,120,171,267)(92,326,223,419,435,133,172,280)(93,311,224,404,436,118,173,265)(94,324,197,417,437,131,174,278)(95,309,198,402,438,116,175,263)(96,322,199,415,439,129,176,276)(97,335,200,400,440,114,177,261)(98,320,201,413,441,127,178,274)(99,333,202,398,442,140,179,259)(100,318,203,411,443,125,180,272)(101,331,204,396,444,138,181,257)(102,316,205,409,445,123,182,270)(103,329,206,394,446,136,183,255)(104,314,207,407,447,121,184,268)(105,327,208,420,448,134,185,253)(106,312,209,405,421,119,186,266)(107,325,210,418,422,132,187,279)(108,310,211,403,423,117,188,264)(109,323,212,416,424,130,189,277)(110,336,213,401,425,115,190,262)(111,321,214,414,426,128,191,275)(112,334,215,399,427,113,192,260), (1,132,22,125,15,118,8,139)(2,133,23,126,16,119,9,140)(3,134,24,127,17,120,10,113)(4,135,25,128,18,121,11,114)(5,136,26,129,19,122,12,115)(6,137,27,130,20,123,13,116)(7,138,28,131,21,124,14,117)(29,335,50,328,43,321,36,314)(30,336,51,329,44,322,37,315)(31,309,52,330,45,323,38,316)(32,310,53,331,46,324,39,317)(33,311,54,332,47,325,40,318)(34,312,55,333,48,326,41,319)(35,313,56,334,49,327,42,320)(57,444,78,437,71,430,64,423)(58,445,79,438,72,431,65,424)(59,446,80,439,73,432,66,425)(60,447,81,440,74,433,67,426)(61,448,82,441,75,434,68,427)(62,421,83,442,76,435,69,428)(63,422,84,443,77,436,70,429)(85,298,106,291,99,284,92,305)(86,299,107,292,100,285,93,306)(87,300,108,293,101,286,94,307)(88,301,109,294,102,287,95,308)(89,302,110,295,103,288,96,281)(90,303,111,296,104,289,97,282)(91,304,112,297,105,290,98,283)(141,191,162,184,155,177,148,170)(142,192,163,185,156,178,149,171)(143,193,164,186,157,179,150,172)(144,194,165,187,158,180,151,173)(145,195,166,188,159,181,152,174)(146,196,167,189,160,182,153,175)(147,169,168,190,161,183,154,176)(197,351,218,344,211,337,204,358)(198,352,219,345,212,338,205,359)(199,353,220,346,213,339,206,360)(200,354,221,347,214,340,207,361)(201,355,222,348,215,341,208,362)(202,356,223,349,216,342,209,363)(203,357,224,350,217,343,210,364)(225,265,246,258,239,279,232,272)(226,266,247,259,240,280,233,273)(227,267,248,260,241,253,234,274)(228,268,249,261,242,254,235,275)(229,269,250,262,243,255,236,276)(230,270,251,263,244,256,237,277)(231,271,252,264,245,257,238,278)(365,404,386,397,379,418,372,411)(366,405,387,398,380,419,373,412)(367,406,388,399,381,420,374,413)(368,407,389,400,382,393,375,414)(369,408,390,401,383,394,376,415)(370,409,391,402,384,395,377,416)(371,410,392,403,385,396,378,417) );
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)], [(1,299,379,165,47,63,239,343),(2,284,380,150,48,76,240,356),(3,297,381,163,49,61,241,341),(4,282,382,148,50,74,242,354),(5,295,383,161,51,59,243,339),(6,308,384,146,52,72,244,352),(7,293,385,159,53,57,245,337),(8,306,386,144,54,70,246,350),(9,291,387,157,55,83,247,363),(10,304,388,142,56,68,248,348),(11,289,389,155,29,81,249,361),(12,302,390,168,30,66,250,346),(13,287,391,153,31,79,251,359),(14,300,392,166,32,64,252,344),(15,285,365,151,33,77,225,357),(16,298,366,164,34,62,226,342),(17,283,367,149,35,75,227,355),(18,296,368,162,36,60,228,340),(19,281,369,147,37,73,229,353),(20,294,370,160,38,58,230,338),(21,307,371,145,39,71,231,351),(22,292,372,158,40,84,232,364),(23,305,373,143,41,69,233,349),(24,290,374,156,42,82,234,362),(25,303,375,141,43,67,235,347),(26,288,376,154,44,80,236,360),(27,301,377,167,45,65,237,345),(28,286,378,152,46,78,238,358),(85,319,216,412,428,126,193,273),(86,332,217,397,429,139,194,258),(87,317,218,410,430,124,195,271),(88,330,219,395,431,137,196,256),(89,315,220,408,432,122,169,269),(90,328,221,393,433,135,170,254),(91,313,222,406,434,120,171,267),(92,326,223,419,435,133,172,280),(93,311,224,404,436,118,173,265),(94,324,197,417,437,131,174,278),(95,309,198,402,438,116,175,263),(96,322,199,415,439,129,176,276),(97,335,200,400,440,114,177,261),(98,320,201,413,441,127,178,274),(99,333,202,398,442,140,179,259),(100,318,203,411,443,125,180,272),(101,331,204,396,444,138,181,257),(102,316,205,409,445,123,182,270),(103,329,206,394,446,136,183,255),(104,314,207,407,447,121,184,268),(105,327,208,420,448,134,185,253),(106,312,209,405,421,119,186,266),(107,325,210,418,422,132,187,279),(108,310,211,403,423,117,188,264),(109,323,212,416,424,130,189,277),(110,336,213,401,425,115,190,262),(111,321,214,414,426,128,191,275),(112,334,215,399,427,113,192,260)], [(1,132,22,125,15,118,8,139),(2,133,23,126,16,119,9,140),(3,134,24,127,17,120,10,113),(4,135,25,128,18,121,11,114),(5,136,26,129,19,122,12,115),(6,137,27,130,20,123,13,116),(7,138,28,131,21,124,14,117),(29,335,50,328,43,321,36,314),(30,336,51,329,44,322,37,315),(31,309,52,330,45,323,38,316),(32,310,53,331,46,324,39,317),(33,311,54,332,47,325,40,318),(34,312,55,333,48,326,41,319),(35,313,56,334,49,327,42,320),(57,444,78,437,71,430,64,423),(58,445,79,438,72,431,65,424),(59,446,80,439,73,432,66,425),(60,447,81,440,74,433,67,426),(61,448,82,441,75,434,68,427),(62,421,83,442,76,435,69,428),(63,422,84,443,77,436,70,429),(85,298,106,291,99,284,92,305),(86,299,107,292,100,285,93,306),(87,300,108,293,101,286,94,307),(88,301,109,294,102,287,95,308),(89,302,110,295,103,288,96,281),(90,303,111,296,104,289,97,282),(91,304,112,297,105,290,98,283),(141,191,162,184,155,177,148,170),(142,192,163,185,156,178,149,171),(143,193,164,186,157,179,150,172),(144,194,165,187,158,180,151,173),(145,195,166,188,159,181,152,174),(146,196,167,189,160,182,153,175),(147,169,168,190,161,183,154,176),(197,351,218,344,211,337,204,358),(198,352,219,345,212,338,205,359),(199,353,220,346,213,339,206,360),(200,354,221,347,214,340,207,361),(201,355,222,348,215,341,208,362),(202,356,223,349,216,342,209,363),(203,357,224,350,217,343,210,364),(225,265,246,258,239,279,232,272),(226,266,247,259,240,280,233,273),(227,267,248,260,241,253,234,274),(228,268,249,261,242,254,235,275),(229,269,250,262,243,255,236,276),(230,270,251,263,244,256,237,277),(231,271,252,264,245,257,238,278),(365,404,386,397,379,418,372,411),(366,405,387,398,380,419,373,412),(367,406,388,399,381,420,374,413),(368,407,389,400,382,393,375,414),(369,408,390,401,383,394,376,415),(370,409,391,402,384,395,377,416),(371,410,392,403,385,396,378,417)]])
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 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | D4 | Q8 | D7 | M4(2) | SD16 | D14 | C8.C4 | Dic14 | C4×D7 | C7⋊D4 | C8×D7 | C8⋊D7 | D4.D7 | Q8⋊D7 | C28.53D4 |
| kernel | C28.39SD16 | C4×C7⋊C8 | C28⋊C8 | C7×C4⋊C8 | C2×C7⋊C8 | C7⋊C8 | C2×C28 | C2×C28 | C4⋊C8 | 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 | 4 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 3 | 3 | 6 |
Matrix representation of C28.39SD16 ►in GL4(𝔽113) generated by
| 55 | 55 | 0 | 0 |
| 3 | 79 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 24 | 14 | 0 | 0 |
| 96 | 89 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 81 | 26 |
| 70 | 37 | 0 | 0 |
| 39 | 43 | 0 | 0 |
| 0 | 0 | 14 | 72 |
| 0 | 0 | 52 | 99 |
G:=sub<GL(4,GF(113))| [55,3,0,0,55,79,0,0,0,0,98,0,0,0,0,98],[24,96,0,0,14,89,0,0,0,0,0,81,0,0,60,26],[70,39,0,0,37,43,0,0,0,0,14,52,0,0,72,99] >;
C28.39SD16 in GAP, Magma, Sage, TeX
C_{28}._{39}{\rm SD}_{16} % in TeX
G:=Group("C28.39SD16"); // GroupNames label
G:=SmallGroup(448,37);
// by ID
G=gap.SmallGroup(448,37);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,589,36,100,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^21,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export