metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊1C16, C8.28D28, C56.64D4, C56.15Q8, C8.15Dic14, C14.5M5(2), C42.8Dic7, C28.28M4(2), C4⋊(C7⋊C16), C7⋊1(C4⋊C16), (C4×C8).3D7, (C2×C28).4C8, C14.3(C4⋊C8), (C4×C28).13C4, (C2×C56).10C4, C14.7(C2×C16), (C4×C56).20C2, C28.38(C4⋊C4), (C2×C8).5Dic7, (C2×C8).331D14, C2.1(C28⋊C8), C4.18(C4⋊Dic7), C2.2(C28.C8), (C2×C56).396C22, C4.10(C4.Dic7), C2.3(C2×C7⋊C16), (C2×C7⋊C16).8C2, (C2×C4).3(C7⋊C8), C22.9(C2×C7⋊C8), (C2×C14).27(C2×C8), (C2×C28).311(C2×C4), (C2×C4).92(C2×Dic7), SmallGroup(448,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊C16
G = < a,b | a28=b16=1, bab-1=a-1 >
Smallest permutation representation of C28⋊C16
►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)
(1 249 147 295 334 222 88 194 74 42 425 397 257 380 136 364)(2 248 148 294 335 221 89 193 75 41 426 396 258 379 137 363)(3 247 149 293 336 220 90 192 76 40 427 395 259 378 138 362)(4 246 150 292 309 219 91 191 77 39 428 394 260 377 139 361)(5 245 151 291 310 218 92 190 78 38 429 393 261 376 140 360)(6 244 152 290 311 217 93 189 79 37 430 420 262 375 113 359)(7 243 153 289 312 216 94 188 80 36 431 419 263 374 114 358)(8 242 154 288 313 215 95 187 81 35 432 418 264 373 115 357)(9 241 155 287 314 214 96 186 82 34 433 417 265 372 116 356)(10 240 156 286 315 213 97 185 83 33 434 416 266 371 117 355)(11 239 157 285 316 212 98 184 84 32 435 415 267 370 118 354)(12 238 158 284 317 211 99 183 57 31 436 414 268 369 119 353)(13 237 159 283 318 210 100 182 58 30 437 413 269 368 120 352)(14 236 160 282 319 209 101 181 59 29 438 412 270 367 121 351)(15 235 161 281 320 208 102 180 60 56 439 411 271 366 122 350)(16 234 162 308 321 207 103 179 61 55 440 410 272 365 123 349)(17 233 163 307 322 206 104 178 62 54 441 409 273 392 124 348)(18 232 164 306 323 205 105 177 63 53 442 408 274 391 125 347)(19 231 165 305 324 204 106 176 64 52 443 407 275 390 126 346)(20 230 166 304 325 203 107 175 65 51 444 406 276 389 127 345)(21 229 167 303 326 202 108 174 66 50 445 405 277 388 128 344)(22 228 168 302 327 201 109 173 67 49 446 404 278 387 129 343)(23 227 141 301 328 200 110 172 68 48 447 403 279 386 130 342)(24 226 142 300 329 199 111 171 69 47 448 402 280 385 131 341)(25 225 143 299 330 198 112 170 70 46 421 401 253 384 132 340)(26 252 144 298 331 197 85 169 71 45 422 400 254 383 133 339)(27 251 145 297 332 224 86 196 72 44 423 399 255 382 134 338)(28 250 146 296 333 223 87 195 73 43 424 398 256 381 135 337)
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), (1,249,147,295,334,222,88,194,74,42,425,397,257,380,136,364)(2,248,148,294,335,221,89,193,75,41,426,396,258,379,137,363)(3,247,149,293,336,220,90,192,76,40,427,395,259,378,138,362)(4,246,150,292,309,219,91,191,77,39,428,394,260,377,139,361)(5,245,151,291,310,218,92,190,78,38,429,393,261,376,140,360)(6,244,152,290,311,217,93,189,79,37,430,420,262,375,113,359)(7,243,153,289,312,216,94,188,80,36,431,419,263,374,114,358)(8,242,154,288,313,215,95,187,81,35,432,418,264,373,115,357)(9,241,155,287,314,214,96,186,82,34,433,417,265,372,116,356)(10,240,156,286,315,213,97,185,83,33,434,416,266,371,117,355)(11,239,157,285,316,212,98,184,84,32,435,415,267,370,118,354)(12,238,158,284,317,211,99,183,57,31,436,414,268,369,119,353)(13,237,159,283,318,210,100,182,58,30,437,413,269,368,120,352)(14,236,160,282,319,209,101,181,59,29,438,412,270,367,121,351)(15,235,161,281,320,208,102,180,60,56,439,411,271,366,122,350)(16,234,162,308,321,207,103,179,61,55,440,410,272,365,123,349)(17,233,163,307,322,206,104,178,62,54,441,409,273,392,124,348)(18,232,164,306,323,205,105,177,63,53,442,408,274,391,125,347)(19,231,165,305,324,204,106,176,64,52,443,407,275,390,126,346)(20,230,166,304,325,203,107,175,65,51,444,406,276,389,127,345)(21,229,167,303,326,202,108,174,66,50,445,405,277,388,128,344)(22,228,168,302,327,201,109,173,67,49,446,404,278,387,129,343)(23,227,141,301,328,200,110,172,68,48,447,403,279,386,130,342)(24,226,142,300,329,199,111,171,69,47,448,402,280,385,131,341)(25,225,143,299,330,198,112,170,70,46,421,401,253,384,132,340)(26,252,144,298,331,197,85,169,71,45,422,400,254,383,133,339)(27,251,145,297,332,224,86,196,72,44,423,399,255,382,134,338)(28,250,146,296,333,223,87,195,73,43,424,398,256,381,135,337)>;
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), (1,249,147,295,334,222,88,194,74,42,425,397,257,380,136,364)(2,248,148,294,335,221,89,193,75,41,426,396,258,379,137,363)(3,247,149,293,336,220,90,192,76,40,427,395,259,378,138,362)(4,246,150,292,309,219,91,191,77,39,428,394,260,377,139,361)(5,245,151,291,310,218,92,190,78,38,429,393,261,376,140,360)(6,244,152,290,311,217,93,189,79,37,430,420,262,375,113,359)(7,243,153,289,312,216,94,188,80,36,431,419,263,374,114,358)(8,242,154,288,313,215,95,187,81,35,432,418,264,373,115,357)(9,241,155,287,314,214,96,186,82,34,433,417,265,372,116,356)(10,240,156,286,315,213,97,185,83,33,434,416,266,371,117,355)(11,239,157,285,316,212,98,184,84,32,435,415,267,370,118,354)(12,238,158,284,317,211,99,183,57,31,436,414,268,369,119,353)(13,237,159,283,318,210,100,182,58,30,437,413,269,368,120,352)(14,236,160,282,319,209,101,181,59,29,438,412,270,367,121,351)(15,235,161,281,320,208,102,180,60,56,439,411,271,366,122,350)(16,234,162,308,321,207,103,179,61,55,440,410,272,365,123,349)(17,233,163,307,322,206,104,178,62,54,441,409,273,392,124,348)(18,232,164,306,323,205,105,177,63,53,442,408,274,391,125,347)(19,231,165,305,324,204,106,176,64,52,443,407,275,390,126,346)(20,230,166,304,325,203,107,175,65,51,444,406,276,389,127,345)(21,229,167,303,326,202,108,174,66,50,445,405,277,388,128,344)(22,228,168,302,327,201,109,173,67,49,446,404,278,387,129,343)(23,227,141,301,328,200,110,172,68,48,447,403,279,386,130,342)(24,226,142,300,329,199,111,171,69,47,448,402,280,385,131,341)(25,225,143,299,330,198,112,170,70,46,421,401,253,384,132,340)(26,252,144,298,331,197,85,169,71,45,422,400,254,383,133,339)(27,251,145,297,332,224,86,196,72,44,423,399,255,382,134,338)(28,250,146,296,333,223,87,195,73,43,424,398,256,381,135,337) );
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)], [(1,249,147,295,334,222,88,194,74,42,425,397,257,380,136,364),(2,248,148,294,335,221,89,193,75,41,426,396,258,379,137,363),(3,247,149,293,336,220,90,192,76,40,427,395,259,378,138,362),(4,246,150,292,309,219,91,191,77,39,428,394,260,377,139,361),(5,245,151,291,310,218,92,190,78,38,429,393,261,376,140,360),(6,244,152,290,311,217,93,189,79,37,430,420,262,375,113,359),(7,243,153,289,312,216,94,188,80,36,431,419,263,374,114,358),(8,242,154,288,313,215,95,187,81,35,432,418,264,373,115,357),(9,241,155,287,314,214,96,186,82,34,433,417,265,372,116,356),(10,240,156,286,315,213,97,185,83,33,434,416,266,371,117,355),(11,239,157,285,316,212,98,184,84,32,435,415,267,370,118,354),(12,238,158,284,317,211,99,183,57,31,436,414,268,369,119,353),(13,237,159,283,318,210,100,182,58,30,437,413,269,368,120,352),(14,236,160,282,319,209,101,181,59,29,438,412,270,367,121,351),(15,235,161,281,320,208,102,180,60,56,439,411,271,366,122,350),(16,234,162,308,321,207,103,179,61,55,440,410,272,365,123,349),(17,233,163,307,322,206,104,178,62,54,441,409,273,392,124,348),(18,232,164,306,323,205,105,177,63,53,442,408,274,391,125,347),(19,231,165,305,324,204,106,176,64,52,443,407,275,390,126,346),(20,230,166,304,325,203,107,175,65,51,444,406,276,389,127,345),(21,229,167,303,326,202,108,174,66,50,445,405,277,388,128,344),(22,228,168,302,327,201,109,173,67,49,446,404,278,387,129,343),(23,227,141,301,328,200,110,172,68,48,447,403,279,386,130,342),(24,226,142,300,329,199,111,171,69,47,448,402,280,385,131,341),(25,225,143,299,330,198,112,170,70,46,421,401,253,384,132,340),(26,252,144,298,331,197,85,169,71,45,422,400,254,383,133,339),(27,251,145,297,332,224,86,196,72,44,423,399,255,382,134,338),(28,250,146,296,333,223,87,195,73,43,424,398,256,381,135,337)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 16A | ··· | 16P | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C16 | D4 | Q8 | D7 | M4(2) | Dic7 | Dic7 | D14 | M5(2) | Dic14 | D28 | C7⋊C8 | C7⋊C16 | C4.Dic7 | C28.C8 |
| kernel | C28⋊C16 | C2×C7⋊C16 | C4×C56 | C4×C28 | C2×C56 | C2×C28 | C28 | C56 | C56 | C4×C8 | C28 | C42 | C2×C8 | C2×C8 | C14 | C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 16 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 4 | 6 | 6 | 12 | 24 | 12 | 24 |
Matrix representation of C28⋊C16 ►in GL4(𝔽113) generated by
| 1 | 89 | 0 | 0 |
| 24 | 103 | 0 | 0 |
| 0 | 0 | 36 | 13 |
| 0 | 0 | 56 | 105 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 83 | 64 |
| 0 | 0 | 104 | 30 |
G:=sub<GL(4,GF(113))| [1,24,0,0,89,103,0,0,0,0,36,56,0,0,13,105],[0,40,0,0,40,0,0,0,0,0,83,104,0,0,64,30] >;
C28⋊C16 in GAP, Magma, Sage, TeX
C_{28}\rtimes C_{16} % in TeX
G:=Group("C28:C16"); // GroupNames label
G:=SmallGroup(448,19);
// by ID
G=gap.SmallGroup(448,19);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,64,100,102,18822]);
// Polycyclic
G:=Group<a,b|a^28=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
Export