metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.11Q16, C28.20SD16, C42.226D14, C4⋊Q8.12D7, C4⋊C4.87D14, (C2×C28).162D4, C4.5(D4.D7), C14.45(C2×Q16), C4.5(C7⋊Q16), C7⋊4(C4.SD16), C28.87(C4○D4), C28⋊2Q8.22C2, C14.61(C2×SD16), (C2×C28).411C23, (C4×C28).140C22, C4.18(Q8⋊2D7), C14.Q16.15C2, C14.60(C4.4D4), C2.13(C28.23D4), (C2×Dic14).116C22, (C4×C7⋊C8).15C2, (C7×C4⋊Q8).12C2, C2.15(C2×D4.D7), C2.16(C2×C7⋊Q16), (C2×C14).542(C2×D4), (C2×C7⋊C8).265C22, (C2×C4).139(C7⋊D4), (C7×C4⋊C4).134C22, (C2×C4).508(C22×D7), C22.214(C2×C7⋊D4), SmallGroup(448,627)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.11Q16
G = < a,b,c | a28=b8=1, c2=a14b4, 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, C4⋊C4, C2×C8, 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×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C4×C7⋊C8, C14.Q16, C28⋊2Q8, C7×C4⋊Q8, C28.11Q16
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, D4.D7, C7⋊Q16, Q8⋊2D7, C2×C7⋊D4, C2×D4.D7, C2×C7⋊Q16, C28.23D4, C28.11Q16
►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)
(1 380 168 265 47 171 244 354)(2 365 141 278 48 184 245 339)(3 378 142 263 49 169 246 352)(4 391 143 276 50 182 247 337)(5 376 144 261 51 195 248 350)(6 389 145 274 52 180 249 363)(7 374 146 259 53 193 250 348)(8 387 147 272 54 178 251 361)(9 372 148 257 55 191 252 346)(10 385 149 270 56 176 225 359)(11 370 150 255 29 189 226 344)(12 383 151 268 30 174 227 357)(13 368 152 253 31 187 228 342)(14 381 153 266 32 172 229 355)(15 366 154 279 33 185 230 340)(16 379 155 264 34 170 231 353)(17 392 156 277 35 183 232 338)(18 377 157 262 36 196 233 351)(19 390 158 275 37 181 234 364)(20 375 159 260 38 194 235 349)(21 388 160 273 39 179 236 362)(22 373 161 258 40 192 237 347)(23 386 162 271 41 177 238 360)(24 371 163 256 42 190 239 345)(25 384 164 269 43 175 240 358)(26 369 165 254 44 188 241 343)(27 382 166 267 45 173 242 356)(28 367 167 280 46 186 243 341)(57 398 428 129 319 296 224 104)(58 411 429 114 320 281 197 89)(59 396 430 127 321 294 198 102)(60 409 431 140 322 307 199 87)(61 394 432 125 323 292 200 100)(62 407 433 138 324 305 201 85)(63 420 434 123 325 290 202 98)(64 405 435 136 326 303 203 111)(65 418 436 121 327 288 204 96)(66 403 437 134 328 301 205 109)(67 416 438 119 329 286 206 94)(68 401 439 132 330 299 207 107)(69 414 440 117 331 284 208 92)(70 399 441 130 332 297 209 105)(71 412 442 115 333 282 210 90)(72 397 443 128 334 295 211 103)(73 410 444 113 335 308 212 88)(74 395 445 126 336 293 213 101)(75 408 446 139 309 306 214 86)(76 393 447 124 310 291 215 99)(77 406 448 137 311 304 216 112)(78 419 421 122 312 289 217 97)(79 404 422 135 313 302 218 110)(80 417 423 120 314 287 219 95)(81 402 424 133 315 300 220 108)(82 415 425 118 316 285 221 93)(83 400 426 131 317 298 222 106)(84 413 427 116 318 283 223 91)
(1 90 33 129)(2 105 34 116)(3 92 35 131)(4 107 36 118)(5 94 37 133)(6 109 38 120)(7 96 39 135)(8 111 40 122)(9 98 41 137)(10 85 42 124)(11 100 43 139)(12 87 44 126)(13 102 45 113)(14 89 46 128)(15 104 47 115)(16 91 48 130)(17 106 49 117)(18 93 50 132)(19 108 51 119)(20 95 52 134)(21 110 53 121)(22 97 54 136)(23 112 55 123)(24 99 56 138)(25 86 29 125)(26 101 30 140)(27 88 31 127)(28 103 32 114)(57 279 333 354)(58 266 334 341)(59 253 335 356)(60 268 336 343)(61 255 309 358)(62 270 310 345)(63 257 311 360)(64 272 312 347)(65 259 313 362)(66 274 314 349)(67 261 315 364)(68 276 316 351)(69 263 317 338)(70 278 318 353)(71 265 319 340)(72 280 320 355)(73 267 321 342)(74 254 322 357)(75 269 323 344)(76 256 324 359)(77 271 325 346)(78 258 326 361)(79 273 327 348)(80 260 328 363)(81 275 329 350)(82 262 330 337)(83 277 331 352)(84 264 332 339)(141 297 231 413)(142 284 232 400)(143 299 233 415)(144 286 234 402)(145 301 235 417)(146 288 236 404)(147 303 237 419)(148 290 238 406)(149 305 239 393)(150 292 240 408)(151 307 241 395)(152 294 242 410)(153 281 243 397)(154 296 244 412)(155 283 245 399)(156 298 246 414)(157 285 247 401)(158 300 248 416)(159 287 249 403)(160 302 250 418)(161 289 251 405)(162 304 252 420)(163 291 225 407)(164 306 226 394)(165 293 227 409)(166 308 228 396)(167 295 229 411)(168 282 230 398)(169 426 392 208)(170 441 365 223)(171 428 366 210)(172 443 367 197)(173 430 368 212)(174 445 369 199)(175 432 370 214)(176 447 371 201)(177 434 372 216)(178 421 373 203)(179 436 374 218)(180 423 375 205)(181 438 376 220)(182 425 377 207)(183 440 378 222)(184 427 379 209)(185 442 380 224)(186 429 381 211)(187 444 382 198)(188 431 383 213)(189 446 384 200)(190 433 385 215)(191 448 386 202)(192 435 387 217)(193 422 388 204)(194 437 389 219)(195 424 390 206)(196 439 391 221)
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), (1,380,168,265,47,171,244,354)(2,365,141,278,48,184,245,339)(3,378,142,263,49,169,246,352)(4,391,143,276,50,182,247,337)(5,376,144,261,51,195,248,350)(6,389,145,274,52,180,249,363)(7,374,146,259,53,193,250,348)(8,387,147,272,54,178,251,361)(9,372,148,257,55,191,252,346)(10,385,149,270,56,176,225,359)(11,370,150,255,29,189,226,344)(12,383,151,268,30,174,227,357)(13,368,152,253,31,187,228,342)(14,381,153,266,32,172,229,355)(15,366,154,279,33,185,230,340)(16,379,155,264,34,170,231,353)(17,392,156,277,35,183,232,338)(18,377,157,262,36,196,233,351)(19,390,158,275,37,181,234,364)(20,375,159,260,38,194,235,349)(21,388,160,273,39,179,236,362)(22,373,161,258,40,192,237,347)(23,386,162,271,41,177,238,360)(24,371,163,256,42,190,239,345)(25,384,164,269,43,175,240,358)(26,369,165,254,44,188,241,343)(27,382,166,267,45,173,242,356)(28,367,167,280,46,186,243,341)(57,398,428,129,319,296,224,104)(58,411,429,114,320,281,197,89)(59,396,430,127,321,294,198,102)(60,409,431,140,322,307,199,87)(61,394,432,125,323,292,200,100)(62,407,433,138,324,305,201,85)(63,420,434,123,325,290,202,98)(64,405,435,136,326,303,203,111)(65,418,436,121,327,288,204,96)(66,403,437,134,328,301,205,109)(67,416,438,119,329,286,206,94)(68,401,439,132,330,299,207,107)(69,414,440,117,331,284,208,92)(70,399,441,130,332,297,209,105)(71,412,442,115,333,282,210,90)(72,397,443,128,334,295,211,103)(73,410,444,113,335,308,212,88)(74,395,445,126,336,293,213,101)(75,408,446,139,309,306,214,86)(76,393,447,124,310,291,215,99)(77,406,448,137,311,304,216,112)(78,419,421,122,312,289,217,97)(79,404,422,135,313,302,218,110)(80,417,423,120,314,287,219,95)(81,402,424,133,315,300,220,108)(82,415,425,118,316,285,221,93)(83,400,426,131,317,298,222,106)(84,413,427,116,318,283,223,91), (1,90,33,129)(2,105,34,116)(3,92,35,131)(4,107,36,118)(5,94,37,133)(6,109,38,120)(7,96,39,135)(8,111,40,122)(9,98,41,137)(10,85,42,124)(11,100,43,139)(12,87,44,126)(13,102,45,113)(14,89,46,128)(15,104,47,115)(16,91,48,130)(17,106,49,117)(18,93,50,132)(19,108,51,119)(20,95,52,134)(21,110,53,121)(22,97,54,136)(23,112,55,123)(24,99,56,138)(25,86,29,125)(26,101,30,140)(27,88,31,127)(28,103,32,114)(57,279,333,354)(58,266,334,341)(59,253,335,356)(60,268,336,343)(61,255,309,358)(62,270,310,345)(63,257,311,360)(64,272,312,347)(65,259,313,362)(66,274,314,349)(67,261,315,364)(68,276,316,351)(69,263,317,338)(70,278,318,353)(71,265,319,340)(72,280,320,355)(73,267,321,342)(74,254,322,357)(75,269,323,344)(76,256,324,359)(77,271,325,346)(78,258,326,361)(79,273,327,348)(80,260,328,363)(81,275,329,350)(82,262,330,337)(83,277,331,352)(84,264,332,339)(141,297,231,413)(142,284,232,400)(143,299,233,415)(144,286,234,402)(145,301,235,417)(146,288,236,404)(147,303,237,419)(148,290,238,406)(149,305,239,393)(150,292,240,408)(151,307,241,395)(152,294,242,410)(153,281,243,397)(154,296,244,412)(155,283,245,399)(156,298,246,414)(157,285,247,401)(158,300,248,416)(159,287,249,403)(160,302,250,418)(161,289,251,405)(162,304,252,420)(163,291,225,407)(164,306,226,394)(165,293,227,409)(166,308,228,396)(167,295,229,411)(168,282,230,398)(169,426,392,208)(170,441,365,223)(171,428,366,210)(172,443,367,197)(173,430,368,212)(174,445,369,199)(175,432,370,214)(176,447,371,201)(177,434,372,216)(178,421,373,203)(179,436,374,218)(180,423,375,205)(181,438,376,220)(182,425,377,207)(183,440,378,222)(184,427,379,209)(185,442,380,224)(186,429,381,211)(187,444,382,198)(188,431,383,213)(189,446,384,200)(190,433,385,215)(191,448,386,202)(192,435,387,217)(193,422,388,204)(194,437,389,219)(195,424,390,206)(196,439,391,221)>;
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), (1,380,168,265,47,171,244,354)(2,365,141,278,48,184,245,339)(3,378,142,263,49,169,246,352)(4,391,143,276,50,182,247,337)(5,376,144,261,51,195,248,350)(6,389,145,274,52,180,249,363)(7,374,146,259,53,193,250,348)(8,387,147,272,54,178,251,361)(9,372,148,257,55,191,252,346)(10,385,149,270,56,176,225,359)(11,370,150,255,29,189,226,344)(12,383,151,268,30,174,227,357)(13,368,152,253,31,187,228,342)(14,381,153,266,32,172,229,355)(15,366,154,279,33,185,230,340)(16,379,155,264,34,170,231,353)(17,392,156,277,35,183,232,338)(18,377,157,262,36,196,233,351)(19,390,158,275,37,181,234,364)(20,375,159,260,38,194,235,349)(21,388,160,273,39,179,236,362)(22,373,161,258,40,192,237,347)(23,386,162,271,41,177,238,360)(24,371,163,256,42,190,239,345)(25,384,164,269,43,175,240,358)(26,369,165,254,44,188,241,343)(27,382,166,267,45,173,242,356)(28,367,167,280,46,186,243,341)(57,398,428,129,319,296,224,104)(58,411,429,114,320,281,197,89)(59,396,430,127,321,294,198,102)(60,409,431,140,322,307,199,87)(61,394,432,125,323,292,200,100)(62,407,433,138,324,305,201,85)(63,420,434,123,325,290,202,98)(64,405,435,136,326,303,203,111)(65,418,436,121,327,288,204,96)(66,403,437,134,328,301,205,109)(67,416,438,119,329,286,206,94)(68,401,439,132,330,299,207,107)(69,414,440,117,331,284,208,92)(70,399,441,130,332,297,209,105)(71,412,442,115,333,282,210,90)(72,397,443,128,334,295,211,103)(73,410,444,113,335,308,212,88)(74,395,445,126,336,293,213,101)(75,408,446,139,309,306,214,86)(76,393,447,124,310,291,215,99)(77,406,448,137,311,304,216,112)(78,419,421,122,312,289,217,97)(79,404,422,135,313,302,218,110)(80,417,423,120,314,287,219,95)(81,402,424,133,315,300,220,108)(82,415,425,118,316,285,221,93)(83,400,426,131,317,298,222,106)(84,413,427,116,318,283,223,91), (1,90,33,129)(2,105,34,116)(3,92,35,131)(4,107,36,118)(5,94,37,133)(6,109,38,120)(7,96,39,135)(8,111,40,122)(9,98,41,137)(10,85,42,124)(11,100,43,139)(12,87,44,126)(13,102,45,113)(14,89,46,128)(15,104,47,115)(16,91,48,130)(17,106,49,117)(18,93,50,132)(19,108,51,119)(20,95,52,134)(21,110,53,121)(22,97,54,136)(23,112,55,123)(24,99,56,138)(25,86,29,125)(26,101,30,140)(27,88,31,127)(28,103,32,114)(57,279,333,354)(58,266,334,341)(59,253,335,356)(60,268,336,343)(61,255,309,358)(62,270,310,345)(63,257,311,360)(64,272,312,347)(65,259,313,362)(66,274,314,349)(67,261,315,364)(68,276,316,351)(69,263,317,338)(70,278,318,353)(71,265,319,340)(72,280,320,355)(73,267,321,342)(74,254,322,357)(75,269,323,344)(76,256,324,359)(77,271,325,346)(78,258,326,361)(79,273,327,348)(80,260,328,363)(81,275,329,350)(82,262,330,337)(83,277,331,352)(84,264,332,339)(141,297,231,413)(142,284,232,400)(143,299,233,415)(144,286,234,402)(145,301,235,417)(146,288,236,404)(147,303,237,419)(148,290,238,406)(149,305,239,393)(150,292,240,408)(151,307,241,395)(152,294,242,410)(153,281,243,397)(154,296,244,412)(155,283,245,399)(156,298,246,414)(157,285,247,401)(158,300,248,416)(159,287,249,403)(160,302,250,418)(161,289,251,405)(162,304,252,420)(163,291,225,407)(164,306,226,394)(165,293,227,409)(166,308,228,396)(167,295,229,411)(168,282,230,398)(169,426,392,208)(170,441,365,223)(171,428,366,210)(172,443,367,197)(173,430,368,212)(174,445,369,199)(175,432,370,214)(176,447,371,201)(177,434,372,216)(178,421,373,203)(179,436,374,218)(180,423,375,205)(181,438,376,220)(182,425,377,207)(183,440,378,222)(184,427,379,209)(185,442,380,224)(186,429,381,211)(187,444,382,198)(188,431,383,213)(189,446,384,200)(190,433,385,215)(191,448,386,202)(192,435,387,217)(193,422,388,204)(194,437,389,219)(195,424,390,206)(196,439,391,221) );
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)], [(1,380,168,265,47,171,244,354),(2,365,141,278,48,184,245,339),(3,378,142,263,49,169,246,352),(4,391,143,276,50,182,247,337),(5,376,144,261,51,195,248,350),(6,389,145,274,52,180,249,363),(7,374,146,259,53,193,250,348),(8,387,147,272,54,178,251,361),(9,372,148,257,55,191,252,346),(10,385,149,270,56,176,225,359),(11,370,150,255,29,189,226,344),(12,383,151,268,30,174,227,357),(13,368,152,253,31,187,228,342),(14,381,153,266,32,172,229,355),(15,366,154,279,33,185,230,340),(16,379,155,264,34,170,231,353),(17,392,156,277,35,183,232,338),(18,377,157,262,36,196,233,351),(19,390,158,275,37,181,234,364),(20,375,159,260,38,194,235,349),(21,388,160,273,39,179,236,362),(22,373,161,258,40,192,237,347),(23,386,162,271,41,177,238,360),(24,371,163,256,42,190,239,345),(25,384,164,269,43,175,240,358),(26,369,165,254,44,188,241,343),(27,382,166,267,45,173,242,356),(28,367,167,280,46,186,243,341),(57,398,428,129,319,296,224,104),(58,411,429,114,320,281,197,89),(59,396,430,127,321,294,198,102),(60,409,431,140,322,307,199,87),(61,394,432,125,323,292,200,100),(62,407,433,138,324,305,201,85),(63,420,434,123,325,290,202,98),(64,405,435,136,326,303,203,111),(65,418,436,121,327,288,204,96),(66,403,437,134,328,301,205,109),(67,416,438,119,329,286,206,94),(68,401,439,132,330,299,207,107),(69,414,440,117,331,284,208,92),(70,399,441,130,332,297,209,105),(71,412,442,115,333,282,210,90),(72,397,443,128,334,295,211,103),(73,410,444,113,335,308,212,88),(74,395,445,126,336,293,213,101),(75,408,446,139,309,306,214,86),(76,393,447,124,310,291,215,99),(77,406,448,137,311,304,216,112),(78,419,421,122,312,289,217,97),(79,404,422,135,313,302,218,110),(80,417,423,120,314,287,219,95),(81,402,424,133,315,300,220,108),(82,415,425,118,316,285,221,93),(83,400,426,131,317,298,222,106),(84,413,427,116,318,283,223,91)], [(1,90,33,129),(2,105,34,116),(3,92,35,131),(4,107,36,118),(5,94,37,133),(6,109,38,120),(7,96,39,135),(8,111,40,122),(9,98,41,137),(10,85,42,124),(11,100,43,139),(12,87,44,126),(13,102,45,113),(14,89,46,128),(15,104,47,115),(16,91,48,130),(17,106,49,117),(18,93,50,132),(19,108,51,119),(20,95,52,134),(21,110,53,121),(22,97,54,136),(23,112,55,123),(24,99,56,138),(25,86,29,125),(26,101,30,140),(27,88,31,127),(28,103,32,114),(57,279,333,354),(58,266,334,341),(59,253,335,356),(60,268,336,343),(61,255,309,358),(62,270,310,345),(63,257,311,360),(64,272,312,347),(65,259,313,362),(66,274,314,349),(67,261,315,364),(68,276,316,351),(69,263,317,338),(70,278,318,353),(71,265,319,340),(72,280,320,355),(73,267,321,342),(74,254,322,357),(75,269,323,344),(76,256,324,359),(77,271,325,346),(78,258,326,361),(79,273,327,348),(80,260,328,363),(81,275,329,350),(82,262,330,337),(83,277,331,352),(84,264,332,339),(141,297,231,413),(142,284,232,400),(143,299,233,415),(144,286,234,402),(145,301,235,417),(146,288,236,404),(147,303,237,419),(148,290,238,406),(149,305,239,393),(150,292,240,408),(151,307,241,395),(152,294,242,410),(153,281,243,397),(154,296,244,412),(155,283,245,399),(156,298,246,414),(157,285,247,401),(158,300,248,416),(159,287,249,403),(160,302,250,418),(161,289,251,405),(162,304,252,420),(163,291,225,407),(164,306,226,394),(165,293,227,409),(166,308,228,396),(167,295,229,411),(168,282,230,398),(169,426,392,208),(170,441,365,223),(171,428,366,210),(172,443,367,197),(173,430,368,212),(174,445,369,199),(175,432,370,214),(176,447,371,201),(177,434,372,216),(178,421,373,203),(179,436,374,218),(180,423,375,205),(181,438,376,220),(182,425,377,207),(183,440,378,222),(184,427,379,209),(185,442,380,224),(186,429,381,211),(187,444,382,198),(188,431,383,213),(189,446,384,200),(190,433,385,215),(191,448,386,202),(192,435,387,217),(193,422,388,204),(194,437,389,219),(195,424,390,206),(196,439,391,221)]])
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 | D4.D7 | C7⋊Q16 | Q8⋊2D7 |
| kernel | C28.11Q16 | C4×C7⋊C8 | C14.Q16 | C28⋊2Q8 | C7×C4⋊Q8 | C2×C28 | C4⋊Q8 | C28 | C28 | C28 | C42 | C4⋊C4 | 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 C28.11Q16 ►in GL6(𝔽113)
| 56 | 88 | 0 | 0 | 0 | 0 |
| 17 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 79 | 79 | 0 | 0 |
| 0 | 0 | 68 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 49 | 77 | 0 | 0 | 0 | 0 |
| 29 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 100 | 0 | 0 |
| 0 | 0 | 58 | 90 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 15 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 72 | 111 | 0 | 0 | 0 | 0 |
| 50 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 22 |
| 0 | 0 | 0 | 0 | 99 | 104 |
G:=sub<GL(6,GF(113))| [56,17,0,0,0,0,88,57,0,0,0,0,0,0,79,68,0,0,0,0,79,58,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[49,29,0,0,0,0,77,64,0,0,0,0,0,0,23,58,0,0,0,0,100,90,0,0,0,0,0,0,62,15,0,0,0,0,15,0],[72,50,0,0,0,0,111,41,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,9,99,0,0,0,0,22,104] >;
C28.11Q16 in GAP, Magma, Sage, TeX
C_{28}._{11}Q_{16} % in TeX
G:=Group("C28.11Q16"); // GroupNames label
G:=SmallGroup(448,627);
// by ID
G=gap.SmallGroup(448,627);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,219,100,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14*b^4,b*a*b^-1=a^13,c*a*c^-1=a^15,c*b*c^-1=a^14*b^-1>;
// generators/relations