metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊3Q16, C42.225D14, C7⋊C8.18D4, C4.19(D4×D7), C4⋊Q8.11D7, C4⋊1(C7⋊Q16), C7⋊2(C4⋊Q16), C28.40(C2×D4), (C2×C28).161D4, (C2×Q8).48D14, C14.44(C2×Q16), C28⋊2Q8.21C2, C2.16(C28⋊D4), C14.25(C4⋊1D4), (C2×C28).410C23, (C4×C28).139C22, (Q8×C14).66C22, (C2×Dic14).115C22, (C4×C7⋊C8).14C2, (C7×C4⋊Q8).11C2, (C2×C7⋊Q16).7C2, C2.15(C2×C7⋊Q16), (C2×C14).541(C2×D4), (C2×C7⋊C8).264C22, (C2×C4).138(C7⋊D4), (C2×C4).507(C22×D7), C22.213(C2×C7⋊D4), SmallGroup(448,626)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊3Q16
G = < a,b,c | a28=b8=1, c2=b4, bab-1=a13, cac-1=a15, cbc-1=b-1 >
Subgroups: 524 in 122 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C4×C8, C4⋊Q8, C4⋊Q8, C2×Q16, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×Q8, C4⋊Q16, C2×C7⋊C8, C4⋊Dic7, C7⋊Q16, C4×C28, C7×C4⋊C4, C2×Dic14, Q8×C14, C4×C7⋊C8, C28⋊2Q8, C2×C7⋊Q16, C7×C4⋊Q8, C28⋊3Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C4⋊1D4, C2×Q16, C7⋊D4, C22×D7, C4⋊Q16, C7⋊Q16, D4×D7, C2×C7⋊D4, C28⋊D4, C2×C7⋊Q16, C28⋊3Q16
►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)
(1 178 267 285 105 440 408 238)(2 191 268 298 106 425 409 251)(3 176 269 283 107 438 410 236)(4 189 270 296 108 423 411 249)(5 174 271 281 109 436 412 234)(6 187 272 294 110 421 413 247)(7 172 273 307 111 434 414 232)(8 185 274 292 112 447 415 245)(9 170 275 305 85 432 416 230)(10 183 276 290 86 445 417 243)(11 196 277 303 87 430 418 228)(12 181 278 288 88 443 419 241)(13 194 279 301 89 428 420 226)(14 179 280 286 90 441 393 239)(15 192 253 299 91 426 394 252)(16 177 254 284 92 439 395 237)(17 190 255 297 93 424 396 250)(18 175 256 282 94 437 397 235)(19 188 257 295 95 422 398 248)(20 173 258 308 96 435 399 233)(21 186 259 293 97 448 400 246)(22 171 260 306 98 433 401 231)(23 184 261 291 99 446 402 244)(24 169 262 304 100 431 403 229)(25 182 263 289 101 444 404 242)(26 195 264 302 102 429 405 227)(27 180 265 287 103 442 406 240)(28 193 266 300 104 427 407 225)(29 375 353 312 138 164 199 58)(30 388 354 325 139 149 200 71)(31 373 355 310 140 162 201 84)(32 386 356 323 113 147 202 69)(33 371 357 336 114 160 203 82)(34 384 358 321 115 145 204 67)(35 369 359 334 116 158 205 80)(36 382 360 319 117 143 206 65)(37 367 361 332 118 156 207 78)(38 380 362 317 119 141 208 63)(39 365 363 330 120 154 209 76)(40 378 364 315 121 167 210 61)(41 391 337 328 122 152 211 74)(42 376 338 313 123 165 212 59)(43 389 339 326 124 150 213 72)(44 374 340 311 125 163 214 57)(45 387 341 324 126 148 215 70)(46 372 342 309 127 161 216 83)(47 385 343 322 128 146 217 68)(48 370 344 335 129 159 218 81)(49 383 345 320 130 144 219 66)(50 368 346 333 131 157 220 79)(51 381 347 318 132 142 221 64)(52 366 348 331 133 155 222 77)(53 379 349 316 134 168 223 62)(54 392 350 329 135 153 224 75)(55 377 351 314 136 166 197 60)(56 390 352 327 137 151 198 73)
(1 144 105 383)(2 159 106 370)(3 146 107 385)(4 161 108 372)(5 148 109 387)(6 163 110 374)(7 150 111 389)(8 165 112 376)(9 152 85 391)(10 167 86 378)(11 154 87 365)(12 141 88 380)(13 156 89 367)(14 143 90 382)(15 158 91 369)(16 145 92 384)(17 160 93 371)(18 147 94 386)(19 162 95 373)(20 149 96 388)(21 164 97 375)(22 151 98 390)(23 166 99 377)(24 153 100 392)(25 168 101 379)(26 155 102 366)(27 142 103 381)(28 157 104 368)(29 186 138 448)(30 173 139 435)(31 188 140 422)(32 175 113 437)(33 190 114 424)(34 177 115 439)(35 192 116 426)(36 179 117 441)(37 194 118 428)(38 181 119 443)(39 196 120 430)(40 183 121 445)(41 170 122 432)(42 185 123 447)(43 172 124 434)(44 187 125 421)(45 174 126 436)(46 189 127 423)(47 176 128 438)(48 191 129 425)(49 178 130 440)(50 193 131 427)(51 180 132 442)(52 195 133 429)(53 182 134 444)(54 169 135 431)(55 184 136 446)(56 171 137 433)(57 272 311 413)(58 259 312 400)(59 274 313 415)(60 261 314 402)(61 276 315 417)(62 263 316 404)(63 278 317 419)(64 265 318 406)(65 280 319 393)(66 267 320 408)(67 254 321 395)(68 269 322 410)(69 256 323 397)(70 271 324 412)(71 258 325 399)(72 273 326 414)(73 260 327 401)(74 275 328 416)(75 262 329 403)(76 277 330 418)(77 264 331 405)(78 279 332 420)(79 266 333 407)(80 253 334 394)(81 268 335 409)(82 255 336 396)(83 270 309 411)(84 257 310 398)(197 291 351 244)(198 306 352 231)(199 293 353 246)(200 308 354 233)(201 295 355 248)(202 282 356 235)(203 297 357 250)(204 284 358 237)(205 299 359 252)(206 286 360 239)(207 301 361 226)(208 288 362 241)(209 303 363 228)(210 290 364 243)(211 305 337 230)(212 292 338 245)(213 307 339 232)(214 294 340 247)(215 281 341 234)(216 296 342 249)(217 283 343 236)(218 298 344 251)(219 285 345 238)(220 300 346 225)(221 287 347 240)(222 302 348 227)(223 289 349 242)(224 304 350 229)
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), (1,178,267,285,105,440,408,238)(2,191,268,298,106,425,409,251)(3,176,269,283,107,438,410,236)(4,189,270,296,108,423,411,249)(5,174,271,281,109,436,412,234)(6,187,272,294,110,421,413,247)(7,172,273,307,111,434,414,232)(8,185,274,292,112,447,415,245)(9,170,275,305,85,432,416,230)(10,183,276,290,86,445,417,243)(11,196,277,303,87,430,418,228)(12,181,278,288,88,443,419,241)(13,194,279,301,89,428,420,226)(14,179,280,286,90,441,393,239)(15,192,253,299,91,426,394,252)(16,177,254,284,92,439,395,237)(17,190,255,297,93,424,396,250)(18,175,256,282,94,437,397,235)(19,188,257,295,95,422,398,248)(20,173,258,308,96,435,399,233)(21,186,259,293,97,448,400,246)(22,171,260,306,98,433,401,231)(23,184,261,291,99,446,402,244)(24,169,262,304,100,431,403,229)(25,182,263,289,101,444,404,242)(26,195,264,302,102,429,405,227)(27,180,265,287,103,442,406,240)(28,193,266,300,104,427,407,225)(29,375,353,312,138,164,199,58)(30,388,354,325,139,149,200,71)(31,373,355,310,140,162,201,84)(32,386,356,323,113,147,202,69)(33,371,357,336,114,160,203,82)(34,384,358,321,115,145,204,67)(35,369,359,334,116,158,205,80)(36,382,360,319,117,143,206,65)(37,367,361,332,118,156,207,78)(38,380,362,317,119,141,208,63)(39,365,363,330,120,154,209,76)(40,378,364,315,121,167,210,61)(41,391,337,328,122,152,211,74)(42,376,338,313,123,165,212,59)(43,389,339,326,124,150,213,72)(44,374,340,311,125,163,214,57)(45,387,341,324,126,148,215,70)(46,372,342,309,127,161,216,83)(47,385,343,322,128,146,217,68)(48,370,344,335,129,159,218,81)(49,383,345,320,130,144,219,66)(50,368,346,333,131,157,220,79)(51,381,347,318,132,142,221,64)(52,366,348,331,133,155,222,77)(53,379,349,316,134,168,223,62)(54,392,350,329,135,153,224,75)(55,377,351,314,136,166,197,60)(56,390,352,327,137,151,198,73), (1,144,105,383)(2,159,106,370)(3,146,107,385)(4,161,108,372)(5,148,109,387)(6,163,110,374)(7,150,111,389)(8,165,112,376)(9,152,85,391)(10,167,86,378)(11,154,87,365)(12,141,88,380)(13,156,89,367)(14,143,90,382)(15,158,91,369)(16,145,92,384)(17,160,93,371)(18,147,94,386)(19,162,95,373)(20,149,96,388)(21,164,97,375)(22,151,98,390)(23,166,99,377)(24,153,100,392)(25,168,101,379)(26,155,102,366)(27,142,103,381)(28,157,104,368)(29,186,138,448)(30,173,139,435)(31,188,140,422)(32,175,113,437)(33,190,114,424)(34,177,115,439)(35,192,116,426)(36,179,117,441)(37,194,118,428)(38,181,119,443)(39,196,120,430)(40,183,121,445)(41,170,122,432)(42,185,123,447)(43,172,124,434)(44,187,125,421)(45,174,126,436)(46,189,127,423)(47,176,128,438)(48,191,129,425)(49,178,130,440)(50,193,131,427)(51,180,132,442)(52,195,133,429)(53,182,134,444)(54,169,135,431)(55,184,136,446)(56,171,137,433)(57,272,311,413)(58,259,312,400)(59,274,313,415)(60,261,314,402)(61,276,315,417)(62,263,316,404)(63,278,317,419)(64,265,318,406)(65,280,319,393)(66,267,320,408)(67,254,321,395)(68,269,322,410)(69,256,323,397)(70,271,324,412)(71,258,325,399)(72,273,326,414)(73,260,327,401)(74,275,328,416)(75,262,329,403)(76,277,330,418)(77,264,331,405)(78,279,332,420)(79,266,333,407)(80,253,334,394)(81,268,335,409)(82,255,336,396)(83,270,309,411)(84,257,310,398)(197,291,351,244)(198,306,352,231)(199,293,353,246)(200,308,354,233)(201,295,355,248)(202,282,356,235)(203,297,357,250)(204,284,358,237)(205,299,359,252)(206,286,360,239)(207,301,361,226)(208,288,362,241)(209,303,363,228)(210,290,364,243)(211,305,337,230)(212,292,338,245)(213,307,339,232)(214,294,340,247)(215,281,341,234)(216,296,342,249)(217,283,343,236)(218,298,344,251)(219,285,345,238)(220,300,346,225)(221,287,347,240)(222,302,348,227)(223,289,349,242)(224,304,350,229)>;
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), (1,178,267,285,105,440,408,238)(2,191,268,298,106,425,409,251)(3,176,269,283,107,438,410,236)(4,189,270,296,108,423,411,249)(5,174,271,281,109,436,412,234)(6,187,272,294,110,421,413,247)(7,172,273,307,111,434,414,232)(8,185,274,292,112,447,415,245)(9,170,275,305,85,432,416,230)(10,183,276,290,86,445,417,243)(11,196,277,303,87,430,418,228)(12,181,278,288,88,443,419,241)(13,194,279,301,89,428,420,226)(14,179,280,286,90,441,393,239)(15,192,253,299,91,426,394,252)(16,177,254,284,92,439,395,237)(17,190,255,297,93,424,396,250)(18,175,256,282,94,437,397,235)(19,188,257,295,95,422,398,248)(20,173,258,308,96,435,399,233)(21,186,259,293,97,448,400,246)(22,171,260,306,98,433,401,231)(23,184,261,291,99,446,402,244)(24,169,262,304,100,431,403,229)(25,182,263,289,101,444,404,242)(26,195,264,302,102,429,405,227)(27,180,265,287,103,442,406,240)(28,193,266,300,104,427,407,225)(29,375,353,312,138,164,199,58)(30,388,354,325,139,149,200,71)(31,373,355,310,140,162,201,84)(32,386,356,323,113,147,202,69)(33,371,357,336,114,160,203,82)(34,384,358,321,115,145,204,67)(35,369,359,334,116,158,205,80)(36,382,360,319,117,143,206,65)(37,367,361,332,118,156,207,78)(38,380,362,317,119,141,208,63)(39,365,363,330,120,154,209,76)(40,378,364,315,121,167,210,61)(41,391,337,328,122,152,211,74)(42,376,338,313,123,165,212,59)(43,389,339,326,124,150,213,72)(44,374,340,311,125,163,214,57)(45,387,341,324,126,148,215,70)(46,372,342,309,127,161,216,83)(47,385,343,322,128,146,217,68)(48,370,344,335,129,159,218,81)(49,383,345,320,130,144,219,66)(50,368,346,333,131,157,220,79)(51,381,347,318,132,142,221,64)(52,366,348,331,133,155,222,77)(53,379,349,316,134,168,223,62)(54,392,350,329,135,153,224,75)(55,377,351,314,136,166,197,60)(56,390,352,327,137,151,198,73), (1,144,105,383)(2,159,106,370)(3,146,107,385)(4,161,108,372)(5,148,109,387)(6,163,110,374)(7,150,111,389)(8,165,112,376)(9,152,85,391)(10,167,86,378)(11,154,87,365)(12,141,88,380)(13,156,89,367)(14,143,90,382)(15,158,91,369)(16,145,92,384)(17,160,93,371)(18,147,94,386)(19,162,95,373)(20,149,96,388)(21,164,97,375)(22,151,98,390)(23,166,99,377)(24,153,100,392)(25,168,101,379)(26,155,102,366)(27,142,103,381)(28,157,104,368)(29,186,138,448)(30,173,139,435)(31,188,140,422)(32,175,113,437)(33,190,114,424)(34,177,115,439)(35,192,116,426)(36,179,117,441)(37,194,118,428)(38,181,119,443)(39,196,120,430)(40,183,121,445)(41,170,122,432)(42,185,123,447)(43,172,124,434)(44,187,125,421)(45,174,126,436)(46,189,127,423)(47,176,128,438)(48,191,129,425)(49,178,130,440)(50,193,131,427)(51,180,132,442)(52,195,133,429)(53,182,134,444)(54,169,135,431)(55,184,136,446)(56,171,137,433)(57,272,311,413)(58,259,312,400)(59,274,313,415)(60,261,314,402)(61,276,315,417)(62,263,316,404)(63,278,317,419)(64,265,318,406)(65,280,319,393)(66,267,320,408)(67,254,321,395)(68,269,322,410)(69,256,323,397)(70,271,324,412)(71,258,325,399)(72,273,326,414)(73,260,327,401)(74,275,328,416)(75,262,329,403)(76,277,330,418)(77,264,331,405)(78,279,332,420)(79,266,333,407)(80,253,334,394)(81,268,335,409)(82,255,336,396)(83,270,309,411)(84,257,310,398)(197,291,351,244)(198,306,352,231)(199,293,353,246)(200,308,354,233)(201,295,355,248)(202,282,356,235)(203,297,357,250)(204,284,358,237)(205,299,359,252)(206,286,360,239)(207,301,361,226)(208,288,362,241)(209,303,363,228)(210,290,364,243)(211,305,337,230)(212,292,338,245)(213,307,339,232)(214,294,340,247)(215,281,341,234)(216,296,342,249)(217,283,343,236)(218,298,344,251)(219,285,345,238)(220,300,346,225)(221,287,347,240)(222,302,348,227)(223,289,349,242)(224,304,350,229) );
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)], [(1,178,267,285,105,440,408,238),(2,191,268,298,106,425,409,251),(3,176,269,283,107,438,410,236),(4,189,270,296,108,423,411,249),(5,174,271,281,109,436,412,234),(6,187,272,294,110,421,413,247),(7,172,273,307,111,434,414,232),(8,185,274,292,112,447,415,245),(9,170,275,305,85,432,416,230),(10,183,276,290,86,445,417,243),(11,196,277,303,87,430,418,228),(12,181,278,288,88,443,419,241),(13,194,279,301,89,428,420,226),(14,179,280,286,90,441,393,239),(15,192,253,299,91,426,394,252),(16,177,254,284,92,439,395,237),(17,190,255,297,93,424,396,250),(18,175,256,282,94,437,397,235),(19,188,257,295,95,422,398,248),(20,173,258,308,96,435,399,233),(21,186,259,293,97,448,400,246),(22,171,260,306,98,433,401,231),(23,184,261,291,99,446,402,244),(24,169,262,304,100,431,403,229),(25,182,263,289,101,444,404,242),(26,195,264,302,102,429,405,227),(27,180,265,287,103,442,406,240),(28,193,266,300,104,427,407,225),(29,375,353,312,138,164,199,58),(30,388,354,325,139,149,200,71),(31,373,355,310,140,162,201,84),(32,386,356,323,113,147,202,69),(33,371,357,336,114,160,203,82),(34,384,358,321,115,145,204,67),(35,369,359,334,116,158,205,80),(36,382,360,319,117,143,206,65),(37,367,361,332,118,156,207,78),(38,380,362,317,119,141,208,63),(39,365,363,330,120,154,209,76),(40,378,364,315,121,167,210,61),(41,391,337,328,122,152,211,74),(42,376,338,313,123,165,212,59),(43,389,339,326,124,150,213,72),(44,374,340,311,125,163,214,57),(45,387,341,324,126,148,215,70),(46,372,342,309,127,161,216,83),(47,385,343,322,128,146,217,68),(48,370,344,335,129,159,218,81),(49,383,345,320,130,144,219,66),(50,368,346,333,131,157,220,79),(51,381,347,318,132,142,221,64),(52,366,348,331,133,155,222,77),(53,379,349,316,134,168,223,62),(54,392,350,329,135,153,224,75),(55,377,351,314,136,166,197,60),(56,390,352,327,137,151,198,73)], [(1,144,105,383),(2,159,106,370),(3,146,107,385),(4,161,108,372),(5,148,109,387),(6,163,110,374),(7,150,111,389),(8,165,112,376),(9,152,85,391),(10,167,86,378),(11,154,87,365),(12,141,88,380),(13,156,89,367),(14,143,90,382),(15,158,91,369),(16,145,92,384),(17,160,93,371),(18,147,94,386),(19,162,95,373),(20,149,96,388),(21,164,97,375),(22,151,98,390),(23,166,99,377),(24,153,100,392),(25,168,101,379),(26,155,102,366),(27,142,103,381),(28,157,104,368),(29,186,138,448),(30,173,139,435),(31,188,140,422),(32,175,113,437),(33,190,114,424),(34,177,115,439),(35,192,116,426),(36,179,117,441),(37,194,118,428),(38,181,119,443),(39,196,120,430),(40,183,121,445),(41,170,122,432),(42,185,123,447),(43,172,124,434),(44,187,125,421),(45,174,126,436),(46,189,127,423),(47,176,128,438),(48,191,129,425),(49,178,130,440),(50,193,131,427),(51,180,132,442),(52,195,133,429),(53,182,134,444),(54,169,135,431),(55,184,136,446),(56,171,137,433),(57,272,311,413),(58,259,312,400),(59,274,313,415),(60,261,314,402),(61,276,315,417),(62,263,316,404),(63,278,317,419),(64,265,318,406),(65,280,319,393),(66,267,320,408),(67,254,321,395),(68,269,322,410),(69,256,323,397),(70,271,324,412),(71,258,325,399),(72,273,326,414),(73,260,327,401),(74,275,328,416),(75,262,329,403),(76,277,330,418),(77,264,331,405),(78,279,332,420),(79,266,333,407),(80,253,334,394),(81,268,335,409),(82,255,336,396),(83,270,309,411),(84,257,310,398),(197,291,351,244),(198,306,352,231),(199,293,353,246),(200,308,354,233),(201,295,355,248),(202,282,356,235),(203,297,357,250),(204,284,358,237),(205,299,359,252),(206,286,360,239),(207,301,361,226),(208,288,362,241),(209,303,363,228),(210,290,364,243),(211,305,337,230),(212,292,338,245),(213,307,339,232),(214,294,340,247),(215,281,341,234),(216,296,342,249),(217,283,343,236),(218,298,344,251),(219,285,345,238),(220,300,346,225),(221,287,347,240),(222,302,348,227),(223,289,349,242),(224,304,350,229)]])
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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | Q16 | D14 | D14 | C7⋊D4 | C7⋊Q16 | D4×D7 |
| kernel | C28⋊3Q16 | C4×C7⋊C8 | C28⋊2Q8 | C2×C7⋊Q16 | C7×C4⋊Q8 | C7⋊C8 | C2×C28 | C4⋊Q8 | C28 | C42 | C2×Q8 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 3 | 8 | 3 | 6 | 12 | 12 | 6 |
Matrix representation of C28⋊3Q16 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 89 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 29 |
| 0 | 0 | 0 | 0 | 3 | 108 |
| 82 | 31 | 0 | 0 | 0 | 0 |
| 82 | 82 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 90 | 0 | 0 |
| 0 | 0 | 17 | 77 | 0 | 0 |
| 0 | 0 | 0 | 0 | 108 | 84 |
| 0 | 0 | 0 | 0 | 110 | 5 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 13 | 100 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 41 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,24,0,0,0,0,89,1,0,0,0,0,0,0,5,3,0,0,0,0,29,108],[82,82,0,0,0,0,31,82,0,0,0,0,0,0,36,17,0,0,0,0,90,77,0,0,0,0,0,0,108,110,0,0,0,0,84,5],[13,13,0,0,0,0,13,100,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,41,1] >;
C28⋊3Q16 in GAP, Magma, Sage, TeX
C_{28}\rtimes_3Q_{16} % in TeX
G:=Group("C28:3Q16"); // GroupNames label
G:=SmallGroup(448,626);
// by ID
G=gap.SmallGroup(448,626);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,219,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=b^4,b*a*b^-1=a^13,c*a*c^-1=a^15,c*b*c^-1=b^-1>;
// generators/relations