direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C28, (C4×C8).6C14, C8.10(C2×C28), C56.66(C2×C4), (C4×C56).24C2, C2.14(D4×C28), (C4×Q8).2C14, Q8.1(C2×C28), C2.3(C14×Q16), C2.D8.9C14, C14.116(C4×D4), (C2×C28).362D4, (Q8×C28).15C2, (C2×Q16).7C14, C14.50(C2×Q16), C4.11(C22×C28), C42.72(C2×C14), Q8⋊C4.8C14, (C14×Q16).14C2, C22.53(D4×C14), C14.118(C4○D8), C28.258(C4○D4), (C2×C56).422C22, (C2×C28).906C23, C28.156(C22×C4), (C4×C28).357C22, (Q8×C14).255C22, C4.3(C7×C4○D4), C2.5(C7×C4○D8), (C2×C4).52(C7×D4), C4⋊C4.47(C2×C14), (C2×C8).67(C2×C14), (C7×Q8).19(C2×C4), (C7×C2.D8).18C2, (C2×C14).629(C2×D4), (C2×Q8).40(C2×C14), (C7×C4⋊C4).368C22, (C2×C4).81(C22×C14), (C7×Q8⋊C4).17C2, SmallGroup(448,847)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C28
G = < a,b,c | a28=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 154 in 110 conjugacy classes, 74 normal (34 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C28, C28, C28, C2×C14, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C56, C56, C2×C28, C2×C28, C7×Q8, C7×Q8, C4×Q16, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×Q16, Q8×C14, C4×C56, C7×Q8⋊C4, C7×C2.D8, Q8×C28, C14×Q16, Q16×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, Q16, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×Q16, C4○D8, C2×C28, C7×D4, C22×C14, C4×Q16, C7×Q16, C22×C28, D4×C14, C7×C4○D4, D4×C28, C14×Q16, C7×C4○D8, Q16×C28
►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)
(1 436 63 33 356 374 119 179)(2 437 64 34 357 375 120 180)(3 438 65 35 358 376 121 181)(4 439 66 36 359 377 122 182)(5 440 67 37 360 378 123 183)(6 441 68 38 361 379 124 184)(7 442 69 39 362 380 125 185)(8 443 70 40 363 381 126 186)(9 444 71 41 364 382 127 187)(10 445 72 42 337 383 128 188)(11 446 73 43 338 384 129 189)(12 447 74 44 339 385 130 190)(13 448 75 45 340 386 131 191)(14 421 76 46 341 387 132 192)(15 422 77 47 342 388 133 193)(16 423 78 48 343 389 134 194)(17 424 79 49 344 390 135 195)(18 425 80 50 345 391 136 196)(19 426 81 51 346 392 137 169)(20 427 82 52 347 365 138 170)(21 428 83 53 348 366 139 171)(22 429 84 54 349 367 140 172)(23 430 57 55 350 368 113 173)(24 431 58 56 351 369 114 174)(25 432 59 29 352 370 115 175)(26 433 60 30 353 371 116 176)(27 434 61 31 354 372 117 177)(28 435 62 32 355 373 118 178)(85 238 335 213 289 412 255 160)(86 239 336 214 290 413 256 161)(87 240 309 215 291 414 257 162)(88 241 310 216 292 415 258 163)(89 242 311 217 293 416 259 164)(90 243 312 218 294 417 260 165)(91 244 313 219 295 418 261 166)(92 245 314 220 296 419 262 167)(93 246 315 221 297 420 263 168)(94 247 316 222 298 393 264 141)(95 248 317 223 299 394 265 142)(96 249 318 224 300 395 266 143)(97 250 319 197 301 396 267 144)(98 251 320 198 302 397 268 145)(99 252 321 199 303 398 269 146)(100 225 322 200 304 399 270 147)(101 226 323 201 305 400 271 148)(102 227 324 202 306 401 272 149)(103 228 325 203 307 402 273 150)(104 229 326 204 308 403 274 151)(105 230 327 205 281 404 275 152)(106 231 328 206 282 405 276 153)(107 232 329 207 283 406 277 154)(108 233 330 208 284 407 278 155)(109 234 331 209 285 408 279 156)(110 235 332 210 286 409 280 157)(111 236 333 211 287 410 253 158)(112 237 334 212 288 411 254 159)
(1 284 356 108)(2 285 357 109)(3 286 358 110)(4 287 359 111)(5 288 360 112)(6 289 361 85)(7 290 362 86)(8 291 363 87)(9 292 364 88)(10 293 337 89)(11 294 338 90)(12 295 339 91)(13 296 340 92)(14 297 341 93)(15 298 342 94)(16 299 343 95)(17 300 344 96)(18 301 345 97)(19 302 346 98)(20 303 347 99)(21 304 348 100)(22 305 349 101)(23 306 350 102)(24 307 351 103)(25 308 352 104)(26 281 353 105)(27 282 354 106)(28 283 355 107)(29 229 175 403)(30 230 176 404)(31 231 177 405)(32 232 178 406)(33 233 179 407)(34 234 180 408)(35 235 181 409)(36 236 182 410)(37 237 183 411)(38 238 184 412)(39 239 185 413)(40 240 186 414)(41 241 187 415)(42 242 188 416)(43 243 189 417)(44 244 190 418)(45 245 191 419)(46 246 192 420)(47 247 193 393)(48 248 194 394)(49 249 195 395)(50 250 196 396)(51 251 169 397)(52 252 170 398)(53 225 171 399)(54 226 172 400)(55 227 173 401)(56 228 174 402)(57 324 113 272)(58 325 114 273)(59 326 115 274)(60 327 116 275)(61 328 117 276)(62 329 118 277)(63 330 119 278)(64 331 120 279)(65 332 121 280)(66 333 122 253)(67 334 123 254)(68 335 124 255)(69 336 125 256)(70 309 126 257)(71 310 127 258)(72 311 128 259)(73 312 129 260)(74 313 130 261)(75 314 131 262)(76 315 132 263)(77 316 133 264)(78 317 134 265)(79 318 135 266)(80 319 136 267)(81 320 137 268)(82 321 138 269)(83 322 139 270)(84 323 140 271)(141 422 222 388)(142 423 223 389)(143 424 224 390)(144 425 197 391)(145 426 198 392)(146 427 199 365)(147 428 200 366)(148 429 201 367)(149 430 202 368)(150 431 203 369)(151 432 204 370)(152 433 205 371)(153 434 206 372)(154 435 207 373)(155 436 208 374)(156 437 209 375)(157 438 210 376)(158 439 211 377)(159 440 212 378)(160 441 213 379)(161 442 214 380)(162 443 215 381)(163 444 216 382)(164 445 217 383)(165 446 218 384)(166 447 219 385)(167 448 220 386)(168 421 221 387)
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), (1,436,63,33,356,374,119,179)(2,437,64,34,357,375,120,180)(3,438,65,35,358,376,121,181)(4,439,66,36,359,377,122,182)(5,440,67,37,360,378,123,183)(6,441,68,38,361,379,124,184)(7,442,69,39,362,380,125,185)(8,443,70,40,363,381,126,186)(9,444,71,41,364,382,127,187)(10,445,72,42,337,383,128,188)(11,446,73,43,338,384,129,189)(12,447,74,44,339,385,130,190)(13,448,75,45,340,386,131,191)(14,421,76,46,341,387,132,192)(15,422,77,47,342,388,133,193)(16,423,78,48,343,389,134,194)(17,424,79,49,344,390,135,195)(18,425,80,50,345,391,136,196)(19,426,81,51,346,392,137,169)(20,427,82,52,347,365,138,170)(21,428,83,53,348,366,139,171)(22,429,84,54,349,367,140,172)(23,430,57,55,350,368,113,173)(24,431,58,56,351,369,114,174)(25,432,59,29,352,370,115,175)(26,433,60,30,353,371,116,176)(27,434,61,31,354,372,117,177)(28,435,62,32,355,373,118,178)(85,238,335,213,289,412,255,160)(86,239,336,214,290,413,256,161)(87,240,309,215,291,414,257,162)(88,241,310,216,292,415,258,163)(89,242,311,217,293,416,259,164)(90,243,312,218,294,417,260,165)(91,244,313,219,295,418,261,166)(92,245,314,220,296,419,262,167)(93,246,315,221,297,420,263,168)(94,247,316,222,298,393,264,141)(95,248,317,223,299,394,265,142)(96,249,318,224,300,395,266,143)(97,250,319,197,301,396,267,144)(98,251,320,198,302,397,268,145)(99,252,321,199,303,398,269,146)(100,225,322,200,304,399,270,147)(101,226,323,201,305,400,271,148)(102,227,324,202,306,401,272,149)(103,228,325,203,307,402,273,150)(104,229,326,204,308,403,274,151)(105,230,327,205,281,404,275,152)(106,231,328,206,282,405,276,153)(107,232,329,207,283,406,277,154)(108,233,330,208,284,407,278,155)(109,234,331,209,285,408,279,156)(110,235,332,210,286,409,280,157)(111,236,333,211,287,410,253,158)(112,237,334,212,288,411,254,159), (1,284,356,108)(2,285,357,109)(3,286,358,110)(4,287,359,111)(5,288,360,112)(6,289,361,85)(7,290,362,86)(8,291,363,87)(9,292,364,88)(10,293,337,89)(11,294,338,90)(12,295,339,91)(13,296,340,92)(14,297,341,93)(15,298,342,94)(16,299,343,95)(17,300,344,96)(18,301,345,97)(19,302,346,98)(20,303,347,99)(21,304,348,100)(22,305,349,101)(23,306,350,102)(24,307,351,103)(25,308,352,104)(26,281,353,105)(27,282,354,106)(28,283,355,107)(29,229,175,403)(30,230,176,404)(31,231,177,405)(32,232,178,406)(33,233,179,407)(34,234,180,408)(35,235,181,409)(36,236,182,410)(37,237,183,411)(38,238,184,412)(39,239,185,413)(40,240,186,414)(41,241,187,415)(42,242,188,416)(43,243,189,417)(44,244,190,418)(45,245,191,419)(46,246,192,420)(47,247,193,393)(48,248,194,394)(49,249,195,395)(50,250,196,396)(51,251,169,397)(52,252,170,398)(53,225,171,399)(54,226,172,400)(55,227,173,401)(56,228,174,402)(57,324,113,272)(58,325,114,273)(59,326,115,274)(60,327,116,275)(61,328,117,276)(62,329,118,277)(63,330,119,278)(64,331,120,279)(65,332,121,280)(66,333,122,253)(67,334,123,254)(68,335,124,255)(69,336,125,256)(70,309,126,257)(71,310,127,258)(72,311,128,259)(73,312,129,260)(74,313,130,261)(75,314,131,262)(76,315,132,263)(77,316,133,264)(78,317,134,265)(79,318,135,266)(80,319,136,267)(81,320,137,268)(82,321,138,269)(83,322,139,270)(84,323,140,271)(141,422,222,388)(142,423,223,389)(143,424,224,390)(144,425,197,391)(145,426,198,392)(146,427,199,365)(147,428,200,366)(148,429,201,367)(149,430,202,368)(150,431,203,369)(151,432,204,370)(152,433,205,371)(153,434,206,372)(154,435,207,373)(155,436,208,374)(156,437,209,375)(157,438,210,376)(158,439,211,377)(159,440,212,378)(160,441,213,379)(161,442,214,380)(162,443,215,381)(163,444,216,382)(164,445,217,383)(165,446,218,384)(166,447,219,385)(167,448,220,386)(168,421,221,387)>;
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), (1,436,63,33,356,374,119,179)(2,437,64,34,357,375,120,180)(3,438,65,35,358,376,121,181)(4,439,66,36,359,377,122,182)(5,440,67,37,360,378,123,183)(6,441,68,38,361,379,124,184)(7,442,69,39,362,380,125,185)(8,443,70,40,363,381,126,186)(9,444,71,41,364,382,127,187)(10,445,72,42,337,383,128,188)(11,446,73,43,338,384,129,189)(12,447,74,44,339,385,130,190)(13,448,75,45,340,386,131,191)(14,421,76,46,341,387,132,192)(15,422,77,47,342,388,133,193)(16,423,78,48,343,389,134,194)(17,424,79,49,344,390,135,195)(18,425,80,50,345,391,136,196)(19,426,81,51,346,392,137,169)(20,427,82,52,347,365,138,170)(21,428,83,53,348,366,139,171)(22,429,84,54,349,367,140,172)(23,430,57,55,350,368,113,173)(24,431,58,56,351,369,114,174)(25,432,59,29,352,370,115,175)(26,433,60,30,353,371,116,176)(27,434,61,31,354,372,117,177)(28,435,62,32,355,373,118,178)(85,238,335,213,289,412,255,160)(86,239,336,214,290,413,256,161)(87,240,309,215,291,414,257,162)(88,241,310,216,292,415,258,163)(89,242,311,217,293,416,259,164)(90,243,312,218,294,417,260,165)(91,244,313,219,295,418,261,166)(92,245,314,220,296,419,262,167)(93,246,315,221,297,420,263,168)(94,247,316,222,298,393,264,141)(95,248,317,223,299,394,265,142)(96,249,318,224,300,395,266,143)(97,250,319,197,301,396,267,144)(98,251,320,198,302,397,268,145)(99,252,321,199,303,398,269,146)(100,225,322,200,304,399,270,147)(101,226,323,201,305,400,271,148)(102,227,324,202,306,401,272,149)(103,228,325,203,307,402,273,150)(104,229,326,204,308,403,274,151)(105,230,327,205,281,404,275,152)(106,231,328,206,282,405,276,153)(107,232,329,207,283,406,277,154)(108,233,330,208,284,407,278,155)(109,234,331,209,285,408,279,156)(110,235,332,210,286,409,280,157)(111,236,333,211,287,410,253,158)(112,237,334,212,288,411,254,159), (1,284,356,108)(2,285,357,109)(3,286,358,110)(4,287,359,111)(5,288,360,112)(6,289,361,85)(7,290,362,86)(8,291,363,87)(9,292,364,88)(10,293,337,89)(11,294,338,90)(12,295,339,91)(13,296,340,92)(14,297,341,93)(15,298,342,94)(16,299,343,95)(17,300,344,96)(18,301,345,97)(19,302,346,98)(20,303,347,99)(21,304,348,100)(22,305,349,101)(23,306,350,102)(24,307,351,103)(25,308,352,104)(26,281,353,105)(27,282,354,106)(28,283,355,107)(29,229,175,403)(30,230,176,404)(31,231,177,405)(32,232,178,406)(33,233,179,407)(34,234,180,408)(35,235,181,409)(36,236,182,410)(37,237,183,411)(38,238,184,412)(39,239,185,413)(40,240,186,414)(41,241,187,415)(42,242,188,416)(43,243,189,417)(44,244,190,418)(45,245,191,419)(46,246,192,420)(47,247,193,393)(48,248,194,394)(49,249,195,395)(50,250,196,396)(51,251,169,397)(52,252,170,398)(53,225,171,399)(54,226,172,400)(55,227,173,401)(56,228,174,402)(57,324,113,272)(58,325,114,273)(59,326,115,274)(60,327,116,275)(61,328,117,276)(62,329,118,277)(63,330,119,278)(64,331,120,279)(65,332,121,280)(66,333,122,253)(67,334,123,254)(68,335,124,255)(69,336,125,256)(70,309,126,257)(71,310,127,258)(72,311,128,259)(73,312,129,260)(74,313,130,261)(75,314,131,262)(76,315,132,263)(77,316,133,264)(78,317,134,265)(79,318,135,266)(80,319,136,267)(81,320,137,268)(82,321,138,269)(83,322,139,270)(84,323,140,271)(141,422,222,388)(142,423,223,389)(143,424,224,390)(144,425,197,391)(145,426,198,392)(146,427,199,365)(147,428,200,366)(148,429,201,367)(149,430,202,368)(150,431,203,369)(151,432,204,370)(152,433,205,371)(153,434,206,372)(154,435,207,373)(155,436,208,374)(156,437,209,375)(157,438,210,376)(158,439,211,377)(159,440,212,378)(160,441,213,379)(161,442,214,380)(162,443,215,381)(163,444,216,382)(164,445,217,383)(165,446,218,384)(166,447,219,385)(167,448,220,386)(168,421,221,387) );
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)], [(1,436,63,33,356,374,119,179),(2,437,64,34,357,375,120,180),(3,438,65,35,358,376,121,181),(4,439,66,36,359,377,122,182),(5,440,67,37,360,378,123,183),(6,441,68,38,361,379,124,184),(7,442,69,39,362,380,125,185),(8,443,70,40,363,381,126,186),(9,444,71,41,364,382,127,187),(10,445,72,42,337,383,128,188),(11,446,73,43,338,384,129,189),(12,447,74,44,339,385,130,190),(13,448,75,45,340,386,131,191),(14,421,76,46,341,387,132,192),(15,422,77,47,342,388,133,193),(16,423,78,48,343,389,134,194),(17,424,79,49,344,390,135,195),(18,425,80,50,345,391,136,196),(19,426,81,51,346,392,137,169),(20,427,82,52,347,365,138,170),(21,428,83,53,348,366,139,171),(22,429,84,54,349,367,140,172),(23,430,57,55,350,368,113,173),(24,431,58,56,351,369,114,174),(25,432,59,29,352,370,115,175),(26,433,60,30,353,371,116,176),(27,434,61,31,354,372,117,177),(28,435,62,32,355,373,118,178),(85,238,335,213,289,412,255,160),(86,239,336,214,290,413,256,161),(87,240,309,215,291,414,257,162),(88,241,310,216,292,415,258,163),(89,242,311,217,293,416,259,164),(90,243,312,218,294,417,260,165),(91,244,313,219,295,418,261,166),(92,245,314,220,296,419,262,167),(93,246,315,221,297,420,263,168),(94,247,316,222,298,393,264,141),(95,248,317,223,299,394,265,142),(96,249,318,224,300,395,266,143),(97,250,319,197,301,396,267,144),(98,251,320,198,302,397,268,145),(99,252,321,199,303,398,269,146),(100,225,322,200,304,399,270,147),(101,226,323,201,305,400,271,148),(102,227,324,202,306,401,272,149),(103,228,325,203,307,402,273,150),(104,229,326,204,308,403,274,151),(105,230,327,205,281,404,275,152),(106,231,328,206,282,405,276,153),(107,232,329,207,283,406,277,154),(108,233,330,208,284,407,278,155),(109,234,331,209,285,408,279,156),(110,235,332,210,286,409,280,157),(111,236,333,211,287,410,253,158),(112,237,334,212,288,411,254,159)], [(1,284,356,108),(2,285,357,109),(3,286,358,110),(4,287,359,111),(5,288,360,112),(6,289,361,85),(7,290,362,86),(8,291,363,87),(9,292,364,88),(10,293,337,89),(11,294,338,90),(12,295,339,91),(13,296,340,92),(14,297,341,93),(15,298,342,94),(16,299,343,95),(17,300,344,96),(18,301,345,97),(19,302,346,98),(20,303,347,99),(21,304,348,100),(22,305,349,101),(23,306,350,102),(24,307,351,103),(25,308,352,104),(26,281,353,105),(27,282,354,106),(28,283,355,107),(29,229,175,403),(30,230,176,404),(31,231,177,405),(32,232,178,406),(33,233,179,407),(34,234,180,408),(35,235,181,409),(36,236,182,410),(37,237,183,411),(38,238,184,412),(39,239,185,413),(40,240,186,414),(41,241,187,415),(42,242,188,416),(43,243,189,417),(44,244,190,418),(45,245,191,419),(46,246,192,420),(47,247,193,393),(48,248,194,394),(49,249,195,395),(50,250,196,396),(51,251,169,397),(52,252,170,398),(53,225,171,399),(54,226,172,400),(55,227,173,401),(56,228,174,402),(57,324,113,272),(58,325,114,273),(59,326,115,274),(60,327,116,275),(61,328,117,276),(62,329,118,277),(63,330,119,278),(64,331,120,279),(65,332,121,280),(66,333,122,253),(67,334,123,254),(68,335,124,255),(69,336,125,256),(70,309,126,257),(71,310,127,258),(72,311,128,259),(73,312,129,260),(74,313,130,261),(75,314,131,262),(76,315,132,263),(77,316,133,264),(78,317,134,265),(79,318,135,266),(80,319,136,267),(81,320,137,268),(82,321,138,269),(83,322,139,270),(84,323,140,271),(141,422,222,388),(142,423,223,389),(143,424,224,390),(144,425,197,391),(145,426,198,392),(146,427,199,365),(147,428,200,366),(148,429,201,367),(149,430,202,368),(150,431,203,369),(151,432,204,370),(152,433,205,371),(153,434,206,372),(154,435,207,373),(155,436,208,374),(156,437,209,375),(157,438,210,376),(158,439,211,377),(159,440,212,378),(160,441,213,379),(161,442,214,380),(162,443,215,381),(163,444,216,382),(164,445,217,383),(165,446,218,384),(166,447,219,385),(167,448,220,386),(168,421,221,387)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28CR | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | Q16 | C4○D4 | C4○D8 | C7×D4 | C7×Q16 | C7×C4○D4 | C7×C4○D8 |
| kernel | Q16×C28 | C4×C56 | C7×Q8⋊C4 | C7×C2.D8 | Q8×C28 | C14×Q16 | C7×Q16 | C4×Q16 | C4×C8 | Q8⋊C4 | C2.D8 | C4×Q8 | C2×Q16 | Q16 | C2×C28 | C28 | C28 | C14 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 6 | 6 | 12 | 6 | 12 | 6 | 48 | 2 | 4 | 2 | 4 | 12 | 24 | 12 | 24 |
Matrix representation of Q16×C28 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 14 |
| 112 | 0 | 0 |
| 0 | 31 | 82 |
| 0 | 31 | 31 |
| 1 | 0 | 0 |
| 0 | 27 | 20 |
| 0 | 20 | 86 |
G:=sub<GL(3,GF(113))| [98,0,0,0,14,0,0,0,14],[112,0,0,0,31,31,0,82,31],[1,0,0,0,27,20,0,20,86] >;
Q16×C28 in GAP, Magma, Sage, TeX
Q_{16}\times C_{28} % in TeX
G:=Group("Q16xC28"); // GroupNames label
G:=SmallGroup(448,847);
// by ID
G=gap.SmallGroup(448,847);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576,1780,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations