metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.9Q16, C28.17SD16, C42.222D14, C4⋊Q8.7D7, C4.4(Q8⋊D7), (C2×C28).153D4, (C2×Q8).42D14, C14.41(C2×Q16), C4.4(C7⋊Q16), C7⋊3(C4.SD16), C28.78(C4○D4), C28⋊2Q8.20C2, C14.74(C2×SD16), C4.24(D4⋊2D7), (C4×C28).128C22, (C2×C28).399C23, Q8⋊Dic7.11C2, (Q8×C14).60C22, C14.45(C4.4D4), C4⋊Dic7.159C22, C2.12(C28.17D4), (C4×C7⋊C8).13C2, (C7×C4⋊Q8).7C2, C2.12(C2×Q8⋊D7), C2.12(C2×C7⋊Q16), (C2×C14).530(C2×D4), (C2×C7⋊C8).261C22, (C2×C4).135(C7⋊D4), (C2×C4).496(C22×D7), C22.202(C2×C7⋊D4), SmallGroup(448,615)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.Q16
G = < a,b,c | a28=b8=1, c2=b4, bab-1=a13, cac-1=a15, cbc-1=a14b-1 >
Subgroups: 396 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×C8, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C4×C8, Q8⋊C4, C4⋊Q8, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C4.SD16, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C7×C4⋊C4, C2×Dic14, Q8×C14, C4×C7⋊C8, Q8⋊Dic7, C28⋊2Q8, C7×C4⋊Q8, C28.Q16
Quotients: C1, C2, C22, D4, C23, D7, SD16, Q16, C2×D4, C4○D4, D14, C4.4D4, C2×SD16, C2×Q16, C7⋊D4, C22×D7, C4.SD16, Q8⋊D7, C7⋊Q16, D4⋊2D7, C2×C7⋊D4, C28.17D4, C2×Q8⋊D7, 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)
(1 207 144 289 78 269 358 381)(2 220 145 302 79 254 359 366)(3 205 146 287 80 267 360 379)(4 218 147 300 81 280 361 392)(5 203 148 285 82 265 362 377)(6 216 149 298 83 278 363 390)(7 201 150 283 84 263 364 375)(8 214 151 296 57 276 337 388)(9 199 152 281 58 261 338 373)(10 212 153 294 59 274 339 386)(11 197 154 307 60 259 340 371)(12 210 155 292 61 272 341 384)(13 223 156 305 62 257 342 369)(14 208 157 290 63 270 343 382)(15 221 158 303 64 255 344 367)(16 206 159 288 65 268 345 380)(17 219 160 301 66 253 346 365)(18 204 161 286 67 266 347 378)(19 217 162 299 68 279 348 391)(20 202 163 284 69 264 349 376)(21 215 164 297 70 277 350 389)(22 200 165 282 71 262 351 374)(23 213 166 295 72 275 352 387)(24 198 167 308 73 260 353 372)(25 211 168 293 74 273 354 385)(26 224 141 306 75 258 355 370)(27 209 142 291 76 271 356 383)(28 222 143 304 77 256 357 368)(29 94 178 440 230 320 114 404)(30 107 179 425 231 333 115 417)(31 92 180 438 232 318 116 402)(32 105 181 423 233 331 117 415)(33 90 182 436 234 316 118 400)(34 103 183 421 235 329 119 413)(35 88 184 434 236 314 120 398)(36 101 185 447 237 327 121 411)(37 86 186 432 238 312 122 396)(38 99 187 445 239 325 123 409)(39 112 188 430 240 310 124 394)(40 97 189 443 241 323 125 407)(41 110 190 428 242 336 126 420)(42 95 191 441 243 321 127 405)(43 108 192 426 244 334 128 418)(44 93 193 439 245 319 129 403)(45 106 194 424 246 332 130 416)(46 91 195 437 247 317 131 401)(47 104 196 422 248 330 132 414)(48 89 169 435 249 315 133 399)(49 102 170 448 250 328 134 412)(50 87 171 433 251 313 135 397)(51 100 172 446 252 326 136 410)(52 85 173 431 225 311 137 395)(53 98 174 444 226 324 138 408)(54 111 175 429 227 309 139 393)(55 96 176 442 228 322 140 406)(56 109 177 427 229 335 113 419)
(1 31 78 232)(2 46 79 247)(3 33 80 234)(4 48 81 249)(5 35 82 236)(6 50 83 251)(7 37 84 238)(8 52 57 225)(9 39 58 240)(10 54 59 227)(11 41 60 242)(12 56 61 229)(13 43 62 244)(14 30 63 231)(15 45 64 246)(16 32 65 233)(17 47 66 248)(18 34 67 235)(19 49 68 250)(20 36 69 237)(21 51 70 252)(22 38 71 239)(23 53 72 226)(24 40 73 241)(25 55 74 228)(26 42 75 243)(27 29 76 230)(28 44 77 245)(85 282 311 374)(86 297 312 389)(87 284 313 376)(88 299 314 391)(89 286 315 378)(90 301 316 365)(91 288 317 380)(92 303 318 367)(93 290 319 382)(94 305 320 369)(95 292 321 384)(96 307 322 371)(97 294 323 386)(98 281 324 373)(99 296 325 388)(100 283 326 375)(101 298 327 390)(102 285 328 377)(103 300 329 392)(104 287 330 379)(105 302 331 366)(106 289 332 381)(107 304 333 368)(108 291 334 383)(109 306 335 370)(110 293 336 385)(111 308 309 372)(112 295 310 387)(113 341 177 155)(114 356 178 142)(115 343 179 157)(116 358 180 144)(117 345 181 159)(118 360 182 146)(119 347 183 161)(120 362 184 148)(121 349 185 163)(122 364 186 150)(123 351 187 165)(124 338 188 152)(125 353 189 167)(126 340 190 154)(127 355 191 141)(128 342 192 156)(129 357 193 143)(130 344 194 158)(131 359 195 145)(132 346 196 160)(133 361 169 147)(134 348 170 162)(135 363 171 149)(136 350 172 164)(137 337 173 151)(138 352 174 166)(139 339 175 153)(140 354 176 168)(197 406 259 442)(198 393 260 429)(199 408 261 444)(200 395 262 431)(201 410 263 446)(202 397 264 433)(203 412 265 448)(204 399 266 435)(205 414 267 422)(206 401 268 437)(207 416 269 424)(208 403 270 439)(209 418 271 426)(210 405 272 441)(211 420 273 428)(212 407 274 443)(213 394 275 430)(214 409 276 445)(215 396 277 432)(216 411 278 447)(217 398 279 434)(218 413 280 421)(219 400 253 436)(220 415 254 423)(221 402 255 438)(222 417 256 425)(223 404 257 440)(224 419 258 427)
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), (1,207,144,289,78,269,358,381)(2,220,145,302,79,254,359,366)(3,205,146,287,80,267,360,379)(4,218,147,300,81,280,361,392)(5,203,148,285,82,265,362,377)(6,216,149,298,83,278,363,390)(7,201,150,283,84,263,364,375)(8,214,151,296,57,276,337,388)(9,199,152,281,58,261,338,373)(10,212,153,294,59,274,339,386)(11,197,154,307,60,259,340,371)(12,210,155,292,61,272,341,384)(13,223,156,305,62,257,342,369)(14,208,157,290,63,270,343,382)(15,221,158,303,64,255,344,367)(16,206,159,288,65,268,345,380)(17,219,160,301,66,253,346,365)(18,204,161,286,67,266,347,378)(19,217,162,299,68,279,348,391)(20,202,163,284,69,264,349,376)(21,215,164,297,70,277,350,389)(22,200,165,282,71,262,351,374)(23,213,166,295,72,275,352,387)(24,198,167,308,73,260,353,372)(25,211,168,293,74,273,354,385)(26,224,141,306,75,258,355,370)(27,209,142,291,76,271,356,383)(28,222,143,304,77,256,357,368)(29,94,178,440,230,320,114,404)(30,107,179,425,231,333,115,417)(31,92,180,438,232,318,116,402)(32,105,181,423,233,331,117,415)(33,90,182,436,234,316,118,400)(34,103,183,421,235,329,119,413)(35,88,184,434,236,314,120,398)(36,101,185,447,237,327,121,411)(37,86,186,432,238,312,122,396)(38,99,187,445,239,325,123,409)(39,112,188,430,240,310,124,394)(40,97,189,443,241,323,125,407)(41,110,190,428,242,336,126,420)(42,95,191,441,243,321,127,405)(43,108,192,426,244,334,128,418)(44,93,193,439,245,319,129,403)(45,106,194,424,246,332,130,416)(46,91,195,437,247,317,131,401)(47,104,196,422,248,330,132,414)(48,89,169,435,249,315,133,399)(49,102,170,448,250,328,134,412)(50,87,171,433,251,313,135,397)(51,100,172,446,252,326,136,410)(52,85,173,431,225,311,137,395)(53,98,174,444,226,324,138,408)(54,111,175,429,227,309,139,393)(55,96,176,442,228,322,140,406)(56,109,177,427,229,335,113,419), (1,31,78,232)(2,46,79,247)(3,33,80,234)(4,48,81,249)(5,35,82,236)(6,50,83,251)(7,37,84,238)(8,52,57,225)(9,39,58,240)(10,54,59,227)(11,41,60,242)(12,56,61,229)(13,43,62,244)(14,30,63,231)(15,45,64,246)(16,32,65,233)(17,47,66,248)(18,34,67,235)(19,49,68,250)(20,36,69,237)(21,51,70,252)(22,38,71,239)(23,53,72,226)(24,40,73,241)(25,55,74,228)(26,42,75,243)(27,29,76,230)(28,44,77,245)(85,282,311,374)(86,297,312,389)(87,284,313,376)(88,299,314,391)(89,286,315,378)(90,301,316,365)(91,288,317,380)(92,303,318,367)(93,290,319,382)(94,305,320,369)(95,292,321,384)(96,307,322,371)(97,294,323,386)(98,281,324,373)(99,296,325,388)(100,283,326,375)(101,298,327,390)(102,285,328,377)(103,300,329,392)(104,287,330,379)(105,302,331,366)(106,289,332,381)(107,304,333,368)(108,291,334,383)(109,306,335,370)(110,293,336,385)(111,308,309,372)(112,295,310,387)(113,341,177,155)(114,356,178,142)(115,343,179,157)(116,358,180,144)(117,345,181,159)(118,360,182,146)(119,347,183,161)(120,362,184,148)(121,349,185,163)(122,364,186,150)(123,351,187,165)(124,338,188,152)(125,353,189,167)(126,340,190,154)(127,355,191,141)(128,342,192,156)(129,357,193,143)(130,344,194,158)(131,359,195,145)(132,346,196,160)(133,361,169,147)(134,348,170,162)(135,363,171,149)(136,350,172,164)(137,337,173,151)(138,352,174,166)(139,339,175,153)(140,354,176,168)(197,406,259,442)(198,393,260,429)(199,408,261,444)(200,395,262,431)(201,410,263,446)(202,397,264,433)(203,412,265,448)(204,399,266,435)(205,414,267,422)(206,401,268,437)(207,416,269,424)(208,403,270,439)(209,418,271,426)(210,405,272,441)(211,420,273,428)(212,407,274,443)(213,394,275,430)(214,409,276,445)(215,396,277,432)(216,411,278,447)(217,398,279,434)(218,413,280,421)(219,400,253,436)(220,415,254,423)(221,402,255,438)(222,417,256,425)(223,404,257,440)(224,419,258,427)>;
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), (1,207,144,289,78,269,358,381)(2,220,145,302,79,254,359,366)(3,205,146,287,80,267,360,379)(4,218,147,300,81,280,361,392)(5,203,148,285,82,265,362,377)(6,216,149,298,83,278,363,390)(7,201,150,283,84,263,364,375)(8,214,151,296,57,276,337,388)(9,199,152,281,58,261,338,373)(10,212,153,294,59,274,339,386)(11,197,154,307,60,259,340,371)(12,210,155,292,61,272,341,384)(13,223,156,305,62,257,342,369)(14,208,157,290,63,270,343,382)(15,221,158,303,64,255,344,367)(16,206,159,288,65,268,345,380)(17,219,160,301,66,253,346,365)(18,204,161,286,67,266,347,378)(19,217,162,299,68,279,348,391)(20,202,163,284,69,264,349,376)(21,215,164,297,70,277,350,389)(22,200,165,282,71,262,351,374)(23,213,166,295,72,275,352,387)(24,198,167,308,73,260,353,372)(25,211,168,293,74,273,354,385)(26,224,141,306,75,258,355,370)(27,209,142,291,76,271,356,383)(28,222,143,304,77,256,357,368)(29,94,178,440,230,320,114,404)(30,107,179,425,231,333,115,417)(31,92,180,438,232,318,116,402)(32,105,181,423,233,331,117,415)(33,90,182,436,234,316,118,400)(34,103,183,421,235,329,119,413)(35,88,184,434,236,314,120,398)(36,101,185,447,237,327,121,411)(37,86,186,432,238,312,122,396)(38,99,187,445,239,325,123,409)(39,112,188,430,240,310,124,394)(40,97,189,443,241,323,125,407)(41,110,190,428,242,336,126,420)(42,95,191,441,243,321,127,405)(43,108,192,426,244,334,128,418)(44,93,193,439,245,319,129,403)(45,106,194,424,246,332,130,416)(46,91,195,437,247,317,131,401)(47,104,196,422,248,330,132,414)(48,89,169,435,249,315,133,399)(49,102,170,448,250,328,134,412)(50,87,171,433,251,313,135,397)(51,100,172,446,252,326,136,410)(52,85,173,431,225,311,137,395)(53,98,174,444,226,324,138,408)(54,111,175,429,227,309,139,393)(55,96,176,442,228,322,140,406)(56,109,177,427,229,335,113,419), (1,31,78,232)(2,46,79,247)(3,33,80,234)(4,48,81,249)(5,35,82,236)(6,50,83,251)(7,37,84,238)(8,52,57,225)(9,39,58,240)(10,54,59,227)(11,41,60,242)(12,56,61,229)(13,43,62,244)(14,30,63,231)(15,45,64,246)(16,32,65,233)(17,47,66,248)(18,34,67,235)(19,49,68,250)(20,36,69,237)(21,51,70,252)(22,38,71,239)(23,53,72,226)(24,40,73,241)(25,55,74,228)(26,42,75,243)(27,29,76,230)(28,44,77,245)(85,282,311,374)(86,297,312,389)(87,284,313,376)(88,299,314,391)(89,286,315,378)(90,301,316,365)(91,288,317,380)(92,303,318,367)(93,290,319,382)(94,305,320,369)(95,292,321,384)(96,307,322,371)(97,294,323,386)(98,281,324,373)(99,296,325,388)(100,283,326,375)(101,298,327,390)(102,285,328,377)(103,300,329,392)(104,287,330,379)(105,302,331,366)(106,289,332,381)(107,304,333,368)(108,291,334,383)(109,306,335,370)(110,293,336,385)(111,308,309,372)(112,295,310,387)(113,341,177,155)(114,356,178,142)(115,343,179,157)(116,358,180,144)(117,345,181,159)(118,360,182,146)(119,347,183,161)(120,362,184,148)(121,349,185,163)(122,364,186,150)(123,351,187,165)(124,338,188,152)(125,353,189,167)(126,340,190,154)(127,355,191,141)(128,342,192,156)(129,357,193,143)(130,344,194,158)(131,359,195,145)(132,346,196,160)(133,361,169,147)(134,348,170,162)(135,363,171,149)(136,350,172,164)(137,337,173,151)(138,352,174,166)(139,339,175,153)(140,354,176,168)(197,406,259,442)(198,393,260,429)(199,408,261,444)(200,395,262,431)(201,410,263,446)(202,397,264,433)(203,412,265,448)(204,399,266,435)(205,414,267,422)(206,401,268,437)(207,416,269,424)(208,403,270,439)(209,418,271,426)(210,405,272,441)(211,420,273,428)(212,407,274,443)(213,394,275,430)(214,409,276,445)(215,396,277,432)(216,411,278,447)(217,398,279,434)(218,413,280,421)(219,400,253,436)(220,415,254,423)(221,402,255,438)(222,417,256,425)(223,404,257,440)(224,419,258,427) );
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)], [(1,207,144,289,78,269,358,381),(2,220,145,302,79,254,359,366),(3,205,146,287,80,267,360,379),(4,218,147,300,81,280,361,392),(5,203,148,285,82,265,362,377),(6,216,149,298,83,278,363,390),(7,201,150,283,84,263,364,375),(8,214,151,296,57,276,337,388),(9,199,152,281,58,261,338,373),(10,212,153,294,59,274,339,386),(11,197,154,307,60,259,340,371),(12,210,155,292,61,272,341,384),(13,223,156,305,62,257,342,369),(14,208,157,290,63,270,343,382),(15,221,158,303,64,255,344,367),(16,206,159,288,65,268,345,380),(17,219,160,301,66,253,346,365),(18,204,161,286,67,266,347,378),(19,217,162,299,68,279,348,391),(20,202,163,284,69,264,349,376),(21,215,164,297,70,277,350,389),(22,200,165,282,71,262,351,374),(23,213,166,295,72,275,352,387),(24,198,167,308,73,260,353,372),(25,211,168,293,74,273,354,385),(26,224,141,306,75,258,355,370),(27,209,142,291,76,271,356,383),(28,222,143,304,77,256,357,368),(29,94,178,440,230,320,114,404),(30,107,179,425,231,333,115,417),(31,92,180,438,232,318,116,402),(32,105,181,423,233,331,117,415),(33,90,182,436,234,316,118,400),(34,103,183,421,235,329,119,413),(35,88,184,434,236,314,120,398),(36,101,185,447,237,327,121,411),(37,86,186,432,238,312,122,396),(38,99,187,445,239,325,123,409),(39,112,188,430,240,310,124,394),(40,97,189,443,241,323,125,407),(41,110,190,428,242,336,126,420),(42,95,191,441,243,321,127,405),(43,108,192,426,244,334,128,418),(44,93,193,439,245,319,129,403),(45,106,194,424,246,332,130,416),(46,91,195,437,247,317,131,401),(47,104,196,422,248,330,132,414),(48,89,169,435,249,315,133,399),(49,102,170,448,250,328,134,412),(50,87,171,433,251,313,135,397),(51,100,172,446,252,326,136,410),(52,85,173,431,225,311,137,395),(53,98,174,444,226,324,138,408),(54,111,175,429,227,309,139,393),(55,96,176,442,228,322,140,406),(56,109,177,427,229,335,113,419)], [(1,31,78,232),(2,46,79,247),(3,33,80,234),(4,48,81,249),(5,35,82,236),(6,50,83,251),(7,37,84,238),(8,52,57,225),(9,39,58,240),(10,54,59,227),(11,41,60,242),(12,56,61,229),(13,43,62,244),(14,30,63,231),(15,45,64,246),(16,32,65,233),(17,47,66,248),(18,34,67,235),(19,49,68,250),(20,36,69,237),(21,51,70,252),(22,38,71,239),(23,53,72,226),(24,40,73,241),(25,55,74,228),(26,42,75,243),(27,29,76,230),(28,44,77,245),(85,282,311,374),(86,297,312,389),(87,284,313,376),(88,299,314,391),(89,286,315,378),(90,301,316,365),(91,288,317,380),(92,303,318,367),(93,290,319,382),(94,305,320,369),(95,292,321,384),(96,307,322,371),(97,294,323,386),(98,281,324,373),(99,296,325,388),(100,283,326,375),(101,298,327,390),(102,285,328,377),(103,300,329,392),(104,287,330,379),(105,302,331,366),(106,289,332,381),(107,304,333,368),(108,291,334,383),(109,306,335,370),(110,293,336,385),(111,308,309,372),(112,295,310,387),(113,341,177,155),(114,356,178,142),(115,343,179,157),(116,358,180,144),(117,345,181,159),(118,360,182,146),(119,347,183,161),(120,362,184,148),(121,349,185,163),(122,364,186,150),(123,351,187,165),(124,338,188,152),(125,353,189,167),(126,340,190,154),(127,355,191,141),(128,342,192,156),(129,357,193,143),(130,344,194,158),(131,359,195,145),(132,346,196,160),(133,361,169,147),(134,348,170,162),(135,363,171,149),(136,350,172,164),(137,337,173,151),(138,352,174,166),(139,339,175,153),(140,354,176,168),(197,406,259,442),(198,393,260,429),(199,408,261,444),(200,395,262,431),(201,410,263,446),(202,397,264,433),(203,412,265,448),(204,399,266,435),(205,414,267,422),(206,401,268,437),(207,416,269,424),(208,403,270,439),(209,418,271,426),(210,405,272,441),(211,420,273,428),(212,407,274,443),(213,394,275,430),(214,409,276,445),(215,396,277,432),(216,411,278,447),(217,398,279,434),(218,413,280,421),(219,400,253,436),(220,415,254,423),(221,402,255,438),(222,417,256,425),(223,404,257,440),(224,419,258,427)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | SD16 | Q16 | C4○D4 | D14 | D14 | C7⋊D4 | Q8⋊D7 | C7⋊Q16 | D4⋊2D7 |
| kernel | C28.Q16 | C4×C7⋊C8 | Q8⋊Dic7 | C28⋊2Q8 | C7×C4⋊Q8 | C2×C28 | C4⋊Q8 | C28 | C28 | C28 | C42 | C2×Q8 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 4 | 4 | 4 | 3 | 6 | 12 | 6 | 6 | 6 |
Matrix representation of C28.Q16 ►in GL6(𝔽113)
| 32 | 102 | 0 | 0 | 0 | 0 |
| 11 | 81 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 103 | 9 |
| 0 | 0 | 0 | 0 | 14 | 89 |
| 28 | 61 | 0 | 0 | 0 | 0 |
| 52 | 85 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 51 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 23 |
| 0 | 0 | 0 | 0 | 49 | 111 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 64 | 0 | 0 |
| 0 | 0 | 52 | 93 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [32,11,0,0,0,0,102,81,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,14,0,0,0,0,9,89],[28,52,0,0,0,0,61,85,0,0,0,0,0,0,62,31,0,0,0,0,51,0,0,0,0,0,0,0,2,49,0,0,0,0,23,111],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,20,52,0,0,0,0,64,93,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;
C28.Q16 in GAP, Magma, Sage, TeX
C_{28}.Q_{16} % in TeX
G:=Group("C28.Q16"); // GroupNames label
G:=SmallGroup(448,615);
// by ID
G=gap.SmallGroup(448,615);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,64,590,135,184,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=b^4,b*a*b^-1=a^13,c*a*c^-1=a^15,c*b*c^-1=a^14*b^-1>;
// generators/relations