metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊2Q16, C42.83D14, Dic14.24D4, C4⋊Q8.9D7, C4.57(D4×D7), C7⋊4(C4⋊2Q16), C28.39(C2×D4), C4⋊2(C7⋊Q16), (C2×C28).159D4, (C2×Q8).47D14, C14.42(C2×Q16), C28⋊C8.21C2, C28.85(C4○D4), C4.6(D4⋊2D7), C2.15(C28⋊2D4), (C2×C28).408C23, (C4×C28).137C22, (C4×Dic14).17C2, Q8⋊Dic7.13C2, (Q8×C14).65C22, C14.106(C4⋊D4), C14.99(C8.C22), C4⋊Dic7.349C22, C2.20(C28.C23), (C2×Dic14).275C22, (C7×C4⋊Q8).9C2, (C2×C7⋊Q16).6C2, C2.13(C2×C7⋊Q16), (C2×C14).539(C2×D4), (C2×C7⋊C8).140C22, (C2×C4).190(C7⋊D4), (C2×C4).505(C22×D7), C22.211(C2×C7⋊D4), SmallGroup(448,624)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊Q16
G = < a,b,c | a28=b8=1, c2=b4, bab-1=a-1, cac-1=a15, cbc-1=b-1 >
Subgroups: 428 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, Dic7, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C4⋊2Q16, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7⋊Q16, C4×C28, C7×C4⋊C4, C2×Dic14, Q8×C14, C28⋊C8, Q8⋊Dic7, C4×Dic14, C2×C7⋊Q16, C7×C4⋊Q8, C28⋊Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C4○D4, D14, C4⋊D4, C2×Q16, C8.C22, C7⋊D4, C22×D7, C4⋊2Q16, C7⋊Q16, D4×D7, D4⋊2D7, C2×C7⋊D4, C28⋊2D4, C28.C23, C2×C7⋊Q16, C28⋊Q16
►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)
(1 126 342 262 400 336 53 238)(2 125 343 261 401 335 54 237)(3 124 344 260 402 334 55 236)(4 123 345 259 403 333 56 235)(5 122 346 258 404 332 29 234)(6 121 347 257 405 331 30 233)(7 120 348 256 406 330 31 232)(8 119 349 255 407 329 32 231)(9 118 350 254 408 328 33 230)(10 117 351 253 409 327 34 229)(11 116 352 280 410 326 35 228)(12 115 353 279 411 325 36 227)(13 114 354 278 412 324 37 226)(14 113 355 277 413 323 38 225)(15 140 356 276 414 322 39 252)(16 139 357 275 415 321 40 251)(17 138 358 274 416 320 41 250)(18 137 359 273 417 319 42 249)(19 136 360 272 418 318 43 248)(20 135 361 271 419 317 44 247)(21 134 362 270 420 316 45 246)(22 133 363 269 393 315 46 245)(23 132 364 268 394 314 47 244)(24 131 337 267 395 313 48 243)(25 130 338 266 396 312 49 242)(26 129 339 265 397 311 50 241)(27 128 340 264 398 310 51 240)(28 127 341 263 399 309 52 239)(57 218 100 158 173 386 306 430)(58 217 101 157 174 385 307 429)(59 216 102 156 175 384 308 428)(60 215 103 155 176 383 281 427)(61 214 104 154 177 382 282 426)(62 213 105 153 178 381 283 425)(63 212 106 152 179 380 284 424)(64 211 107 151 180 379 285 423)(65 210 108 150 181 378 286 422)(66 209 109 149 182 377 287 421)(67 208 110 148 183 376 288 448)(68 207 111 147 184 375 289 447)(69 206 112 146 185 374 290 446)(70 205 85 145 186 373 291 445)(71 204 86 144 187 372 292 444)(72 203 87 143 188 371 293 443)(73 202 88 142 189 370 294 442)(74 201 89 141 190 369 295 441)(75 200 90 168 191 368 296 440)(76 199 91 167 192 367 297 439)(77 198 92 166 193 366 298 438)(78 197 93 165 194 365 299 437)(79 224 94 164 195 392 300 436)(80 223 95 163 196 391 301 435)(81 222 96 162 169 390 302 434)(82 221 97 161 170 389 303 433)(83 220 98 160 171 388 304 432)(84 219 99 159 172 387 305 431)
(1 95 400 301)(2 110 401 288)(3 97 402 303)(4 112 403 290)(5 99 404 305)(6 86 405 292)(7 101 406 307)(8 88 407 294)(9 103 408 281)(10 90 409 296)(11 105 410 283)(12 92 411 298)(13 107 412 285)(14 94 413 300)(15 109 414 287)(16 96 415 302)(17 111 416 289)(18 98 417 304)(19 85 418 291)(20 100 419 306)(21 87 420 293)(22 102 393 308)(23 89 394 295)(24 104 395 282)(25 91 396 297)(26 106 397 284)(27 93 398 299)(28 108 399 286)(29 172 346 84)(30 187 347 71)(31 174 348 58)(32 189 349 73)(33 176 350 60)(34 191 351 75)(35 178 352 62)(36 193 353 77)(37 180 354 64)(38 195 355 79)(39 182 356 66)(40 169 357 81)(41 184 358 68)(42 171 359 83)(43 186 360 70)(44 173 361 57)(45 188 362 72)(46 175 363 59)(47 190 364 74)(48 177 337 61)(49 192 338 76)(50 179 339 63)(51 194 340 78)(52 181 341 65)(53 196 342 80)(54 183 343 67)(55 170 344 82)(56 185 345 69)(113 224 323 392)(114 211 324 379)(115 198 325 366)(116 213 326 381)(117 200 327 368)(118 215 328 383)(119 202 329 370)(120 217 330 385)(121 204 331 372)(122 219 332 387)(123 206 333 374)(124 221 334 389)(125 208 335 376)(126 223 336 391)(127 210 309 378)(128 197 310 365)(129 212 311 380)(130 199 312 367)(131 214 313 382)(132 201 314 369)(133 216 315 384)(134 203 316 371)(135 218 317 386)(136 205 318 373)(137 220 319 388)(138 207 320 375)(139 222 321 390)(140 209 322 377)(141 268 441 244)(142 255 442 231)(143 270 443 246)(144 257 444 233)(145 272 445 248)(146 259 446 235)(147 274 447 250)(148 261 448 237)(149 276 421 252)(150 263 422 239)(151 278 423 226)(152 265 424 241)(153 280 425 228)(154 267 426 243)(155 254 427 230)(156 269 428 245)(157 256 429 232)(158 271 430 247)(159 258 431 234)(160 273 432 249)(161 260 433 236)(162 275 434 251)(163 262 435 238)(164 277 436 225)(165 264 437 240)(166 279 438 227)(167 266 439 242)(168 253 440 229)
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), (1,126,342,262,400,336,53,238)(2,125,343,261,401,335,54,237)(3,124,344,260,402,334,55,236)(4,123,345,259,403,333,56,235)(5,122,346,258,404,332,29,234)(6,121,347,257,405,331,30,233)(7,120,348,256,406,330,31,232)(8,119,349,255,407,329,32,231)(9,118,350,254,408,328,33,230)(10,117,351,253,409,327,34,229)(11,116,352,280,410,326,35,228)(12,115,353,279,411,325,36,227)(13,114,354,278,412,324,37,226)(14,113,355,277,413,323,38,225)(15,140,356,276,414,322,39,252)(16,139,357,275,415,321,40,251)(17,138,358,274,416,320,41,250)(18,137,359,273,417,319,42,249)(19,136,360,272,418,318,43,248)(20,135,361,271,419,317,44,247)(21,134,362,270,420,316,45,246)(22,133,363,269,393,315,46,245)(23,132,364,268,394,314,47,244)(24,131,337,267,395,313,48,243)(25,130,338,266,396,312,49,242)(26,129,339,265,397,311,50,241)(27,128,340,264,398,310,51,240)(28,127,341,263,399,309,52,239)(57,218,100,158,173,386,306,430)(58,217,101,157,174,385,307,429)(59,216,102,156,175,384,308,428)(60,215,103,155,176,383,281,427)(61,214,104,154,177,382,282,426)(62,213,105,153,178,381,283,425)(63,212,106,152,179,380,284,424)(64,211,107,151,180,379,285,423)(65,210,108,150,181,378,286,422)(66,209,109,149,182,377,287,421)(67,208,110,148,183,376,288,448)(68,207,111,147,184,375,289,447)(69,206,112,146,185,374,290,446)(70,205,85,145,186,373,291,445)(71,204,86,144,187,372,292,444)(72,203,87,143,188,371,293,443)(73,202,88,142,189,370,294,442)(74,201,89,141,190,369,295,441)(75,200,90,168,191,368,296,440)(76,199,91,167,192,367,297,439)(77,198,92,166,193,366,298,438)(78,197,93,165,194,365,299,437)(79,224,94,164,195,392,300,436)(80,223,95,163,196,391,301,435)(81,222,96,162,169,390,302,434)(82,221,97,161,170,389,303,433)(83,220,98,160,171,388,304,432)(84,219,99,159,172,387,305,431), (1,95,400,301)(2,110,401,288)(3,97,402,303)(4,112,403,290)(5,99,404,305)(6,86,405,292)(7,101,406,307)(8,88,407,294)(9,103,408,281)(10,90,409,296)(11,105,410,283)(12,92,411,298)(13,107,412,285)(14,94,413,300)(15,109,414,287)(16,96,415,302)(17,111,416,289)(18,98,417,304)(19,85,418,291)(20,100,419,306)(21,87,420,293)(22,102,393,308)(23,89,394,295)(24,104,395,282)(25,91,396,297)(26,106,397,284)(27,93,398,299)(28,108,399,286)(29,172,346,84)(30,187,347,71)(31,174,348,58)(32,189,349,73)(33,176,350,60)(34,191,351,75)(35,178,352,62)(36,193,353,77)(37,180,354,64)(38,195,355,79)(39,182,356,66)(40,169,357,81)(41,184,358,68)(42,171,359,83)(43,186,360,70)(44,173,361,57)(45,188,362,72)(46,175,363,59)(47,190,364,74)(48,177,337,61)(49,192,338,76)(50,179,339,63)(51,194,340,78)(52,181,341,65)(53,196,342,80)(54,183,343,67)(55,170,344,82)(56,185,345,69)(113,224,323,392)(114,211,324,379)(115,198,325,366)(116,213,326,381)(117,200,327,368)(118,215,328,383)(119,202,329,370)(120,217,330,385)(121,204,331,372)(122,219,332,387)(123,206,333,374)(124,221,334,389)(125,208,335,376)(126,223,336,391)(127,210,309,378)(128,197,310,365)(129,212,311,380)(130,199,312,367)(131,214,313,382)(132,201,314,369)(133,216,315,384)(134,203,316,371)(135,218,317,386)(136,205,318,373)(137,220,319,388)(138,207,320,375)(139,222,321,390)(140,209,322,377)(141,268,441,244)(142,255,442,231)(143,270,443,246)(144,257,444,233)(145,272,445,248)(146,259,446,235)(147,274,447,250)(148,261,448,237)(149,276,421,252)(150,263,422,239)(151,278,423,226)(152,265,424,241)(153,280,425,228)(154,267,426,243)(155,254,427,230)(156,269,428,245)(157,256,429,232)(158,271,430,247)(159,258,431,234)(160,273,432,249)(161,260,433,236)(162,275,434,251)(163,262,435,238)(164,277,436,225)(165,264,437,240)(166,279,438,227)(167,266,439,242)(168,253,440,229)>;
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), (1,126,342,262,400,336,53,238)(2,125,343,261,401,335,54,237)(3,124,344,260,402,334,55,236)(4,123,345,259,403,333,56,235)(5,122,346,258,404,332,29,234)(6,121,347,257,405,331,30,233)(7,120,348,256,406,330,31,232)(8,119,349,255,407,329,32,231)(9,118,350,254,408,328,33,230)(10,117,351,253,409,327,34,229)(11,116,352,280,410,326,35,228)(12,115,353,279,411,325,36,227)(13,114,354,278,412,324,37,226)(14,113,355,277,413,323,38,225)(15,140,356,276,414,322,39,252)(16,139,357,275,415,321,40,251)(17,138,358,274,416,320,41,250)(18,137,359,273,417,319,42,249)(19,136,360,272,418,318,43,248)(20,135,361,271,419,317,44,247)(21,134,362,270,420,316,45,246)(22,133,363,269,393,315,46,245)(23,132,364,268,394,314,47,244)(24,131,337,267,395,313,48,243)(25,130,338,266,396,312,49,242)(26,129,339,265,397,311,50,241)(27,128,340,264,398,310,51,240)(28,127,341,263,399,309,52,239)(57,218,100,158,173,386,306,430)(58,217,101,157,174,385,307,429)(59,216,102,156,175,384,308,428)(60,215,103,155,176,383,281,427)(61,214,104,154,177,382,282,426)(62,213,105,153,178,381,283,425)(63,212,106,152,179,380,284,424)(64,211,107,151,180,379,285,423)(65,210,108,150,181,378,286,422)(66,209,109,149,182,377,287,421)(67,208,110,148,183,376,288,448)(68,207,111,147,184,375,289,447)(69,206,112,146,185,374,290,446)(70,205,85,145,186,373,291,445)(71,204,86,144,187,372,292,444)(72,203,87,143,188,371,293,443)(73,202,88,142,189,370,294,442)(74,201,89,141,190,369,295,441)(75,200,90,168,191,368,296,440)(76,199,91,167,192,367,297,439)(77,198,92,166,193,366,298,438)(78,197,93,165,194,365,299,437)(79,224,94,164,195,392,300,436)(80,223,95,163,196,391,301,435)(81,222,96,162,169,390,302,434)(82,221,97,161,170,389,303,433)(83,220,98,160,171,388,304,432)(84,219,99,159,172,387,305,431), (1,95,400,301)(2,110,401,288)(3,97,402,303)(4,112,403,290)(5,99,404,305)(6,86,405,292)(7,101,406,307)(8,88,407,294)(9,103,408,281)(10,90,409,296)(11,105,410,283)(12,92,411,298)(13,107,412,285)(14,94,413,300)(15,109,414,287)(16,96,415,302)(17,111,416,289)(18,98,417,304)(19,85,418,291)(20,100,419,306)(21,87,420,293)(22,102,393,308)(23,89,394,295)(24,104,395,282)(25,91,396,297)(26,106,397,284)(27,93,398,299)(28,108,399,286)(29,172,346,84)(30,187,347,71)(31,174,348,58)(32,189,349,73)(33,176,350,60)(34,191,351,75)(35,178,352,62)(36,193,353,77)(37,180,354,64)(38,195,355,79)(39,182,356,66)(40,169,357,81)(41,184,358,68)(42,171,359,83)(43,186,360,70)(44,173,361,57)(45,188,362,72)(46,175,363,59)(47,190,364,74)(48,177,337,61)(49,192,338,76)(50,179,339,63)(51,194,340,78)(52,181,341,65)(53,196,342,80)(54,183,343,67)(55,170,344,82)(56,185,345,69)(113,224,323,392)(114,211,324,379)(115,198,325,366)(116,213,326,381)(117,200,327,368)(118,215,328,383)(119,202,329,370)(120,217,330,385)(121,204,331,372)(122,219,332,387)(123,206,333,374)(124,221,334,389)(125,208,335,376)(126,223,336,391)(127,210,309,378)(128,197,310,365)(129,212,311,380)(130,199,312,367)(131,214,313,382)(132,201,314,369)(133,216,315,384)(134,203,316,371)(135,218,317,386)(136,205,318,373)(137,220,319,388)(138,207,320,375)(139,222,321,390)(140,209,322,377)(141,268,441,244)(142,255,442,231)(143,270,443,246)(144,257,444,233)(145,272,445,248)(146,259,446,235)(147,274,447,250)(148,261,448,237)(149,276,421,252)(150,263,422,239)(151,278,423,226)(152,265,424,241)(153,280,425,228)(154,267,426,243)(155,254,427,230)(156,269,428,245)(157,256,429,232)(158,271,430,247)(159,258,431,234)(160,273,432,249)(161,260,433,236)(162,275,434,251)(163,262,435,238)(164,277,436,225)(165,264,437,240)(166,279,438,227)(167,266,439,242)(168,253,440,229) );
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)], [(1,126,342,262,400,336,53,238),(2,125,343,261,401,335,54,237),(3,124,344,260,402,334,55,236),(4,123,345,259,403,333,56,235),(5,122,346,258,404,332,29,234),(6,121,347,257,405,331,30,233),(7,120,348,256,406,330,31,232),(8,119,349,255,407,329,32,231),(9,118,350,254,408,328,33,230),(10,117,351,253,409,327,34,229),(11,116,352,280,410,326,35,228),(12,115,353,279,411,325,36,227),(13,114,354,278,412,324,37,226),(14,113,355,277,413,323,38,225),(15,140,356,276,414,322,39,252),(16,139,357,275,415,321,40,251),(17,138,358,274,416,320,41,250),(18,137,359,273,417,319,42,249),(19,136,360,272,418,318,43,248),(20,135,361,271,419,317,44,247),(21,134,362,270,420,316,45,246),(22,133,363,269,393,315,46,245),(23,132,364,268,394,314,47,244),(24,131,337,267,395,313,48,243),(25,130,338,266,396,312,49,242),(26,129,339,265,397,311,50,241),(27,128,340,264,398,310,51,240),(28,127,341,263,399,309,52,239),(57,218,100,158,173,386,306,430),(58,217,101,157,174,385,307,429),(59,216,102,156,175,384,308,428),(60,215,103,155,176,383,281,427),(61,214,104,154,177,382,282,426),(62,213,105,153,178,381,283,425),(63,212,106,152,179,380,284,424),(64,211,107,151,180,379,285,423),(65,210,108,150,181,378,286,422),(66,209,109,149,182,377,287,421),(67,208,110,148,183,376,288,448),(68,207,111,147,184,375,289,447),(69,206,112,146,185,374,290,446),(70,205,85,145,186,373,291,445),(71,204,86,144,187,372,292,444),(72,203,87,143,188,371,293,443),(73,202,88,142,189,370,294,442),(74,201,89,141,190,369,295,441),(75,200,90,168,191,368,296,440),(76,199,91,167,192,367,297,439),(77,198,92,166,193,366,298,438),(78,197,93,165,194,365,299,437),(79,224,94,164,195,392,300,436),(80,223,95,163,196,391,301,435),(81,222,96,162,169,390,302,434),(82,221,97,161,170,389,303,433),(83,220,98,160,171,388,304,432),(84,219,99,159,172,387,305,431)], [(1,95,400,301),(2,110,401,288),(3,97,402,303),(4,112,403,290),(5,99,404,305),(6,86,405,292),(7,101,406,307),(8,88,407,294),(9,103,408,281),(10,90,409,296),(11,105,410,283),(12,92,411,298),(13,107,412,285),(14,94,413,300),(15,109,414,287),(16,96,415,302),(17,111,416,289),(18,98,417,304),(19,85,418,291),(20,100,419,306),(21,87,420,293),(22,102,393,308),(23,89,394,295),(24,104,395,282),(25,91,396,297),(26,106,397,284),(27,93,398,299),(28,108,399,286),(29,172,346,84),(30,187,347,71),(31,174,348,58),(32,189,349,73),(33,176,350,60),(34,191,351,75),(35,178,352,62),(36,193,353,77),(37,180,354,64),(38,195,355,79),(39,182,356,66),(40,169,357,81),(41,184,358,68),(42,171,359,83),(43,186,360,70),(44,173,361,57),(45,188,362,72),(46,175,363,59),(47,190,364,74),(48,177,337,61),(49,192,338,76),(50,179,339,63),(51,194,340,78),(52,181,341,65),(53,196,342,80),(54,183,343,67),(55,170,344,82),(56,185,345,69),(113,224,323,392),(114,211,324,379),(115,198,325,366),(116,213,326,381),(117,200,327,368),(118,215,328,383),(119,202,329,370),(120,217,330,385),(121,204,331,372),(122,219,332,387),(123,206,333,374),(124,221,334,389),(125,208,335,376),(126,223,336,391),(127,210,309,378),(128,197,310,365),(129,212,311,380),(130,199,312,367),(131,214,313,382),(132,201,314,369),(133,216,315,384),(134,203,316,371),(135,218,317,386),(136,205,318,373),(137,220,319,388),(138,207,320,375),(139,222,321,390),(140,209,322,377),(141,268,441,244),(142,255,442,231),(143,270,443,246),(144,257,444,233),(145,272,445,248),(146,259,446,235),(147,274,447,250),(148,261,448,237),(149,276,421,252),(150,263,422,239),(151,278,423,226),(152,265,424,241),(153,280,425,228),(154,267,426,243),(155,254,427,230),(156,269,428,245),(157,256,429,232),(158,271,430,247),(159,258,431,234),(160,273,432,249),(161,260,433,236),(162,275,434,251),(163,262,435,238),(164,277,436,225),(165,264,437,240),(166,279,438,227),(167,266,439,242),(168,253,440,229)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | C7⋊D4 | C8.C22 | C7⋊Q16 | D4×D7 | D4⋊2D7 | C28.C23 |
| kernel | C28⋊Q16 | C28⋊C8 | Q8⋊Dic7 | C4×Dic14 | C2×C7⋊Q16 | C7×C4⋊Q8 | Dic14 | C2×C28 | C4⋊Q8 | C28 | C28 | C42 | C2×Q8 | C2×C4 | C14 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 1 | 6 | 3 | 3 | 6 |
Matrix representation of C28⋊Q16 ►in GL6(𝔽113)
| 25 | 79 | 0 | 0 | 0 | 0 |
| 34 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 17 | 0 | 0 |
| 0 | 0 | 103 | 100 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 37 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 112 | 0 | 0 |
| 0 | 0 | 40 | 70 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 62 |
| 0 | 0 | 0 | 0 | 82 | 0 |
| 91 | 101 | 0 | 0 | 0 | 0 |
| 12 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 70 | 1 | 0 | 0 |
| 0 | 0 | 73 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 34 |
| 0 | 0 | 0 | 0 | 106 | 89 |
G:=sub<GL(6,GF(113))| [25,34,0,0,0,0,79,112,0,0,0,0,0,0,13,103,0,0,0,0,17,100,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,37,0,0,0,0,3,112,0,0,0,0,0,0,43,40,0,0,0,0,112,70,0,0,0,0,0,0,62,82,0,0,0,0,62,0],[91,12,0,0,0,0,101,22,0,0,0,0,0,0,70,73,0,0,0,0,1,43,0,0,0,0,0,0,24,106,0,0,0,0,34,89] >;
C28⋊Q16 in GAP, Magma, Sage, TeX
C_{28}\rtimes Q_{16} % in TeX
G:=Group("C28:Q16"); // GroupNames label
G:=SmallGroup(448,624);
// by ID
G=gap.SmallGroup(448,624);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,254,219,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=b^4,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=b^-1>;
// generators/relations