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## G = Q16×C2×C14order 448 = 26·7

### Direct product of C2×C14 and Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — Q16×C2×C14
 Chief series C1 — C2 — C4 — C28 — C7×Q8 — C7×Q16 — C14×Q16 — Q16×C2×C14
 Lower central C1 — C2 — C4 — Q16×C2×C14
 Upper central C1 — C22×C14 — C22×C28 — Q16×C2×C14

Generators and relations for Q16×C2×C14
G = < a,b,c,d | a2=b14=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 338 in 258 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C23, C14, C14, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C28, C28, C28, C2×C14, C22×C8, C2×Q16, C22×Q8, C56, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×C14, C22×Q16, C2×C56, C7×Q16, C22×C28, C22×C28, Q8×C14, Q8×C14, C22×C56, C14×Q16, Q8×C2×C14, Q16×C2×C14
Quotients: C1, C2, C22, C7, D4, C23, C14, Q16, C2×D4, C24, C2×C14, C2×Q16, C22×D4, C7×D4, C22×C14, C22×Q16, C7×Q16, D4×C14, C23×C14, C14×Q16, D4×C2×C14, Q16×C2×C14

Smallest permutation representation of Q16×C2×C14
Regular action on 448 points
Generators in S448
(1 208)(2 209)(3 210)(4 197)(5 198)(6 199)(7 200)(8 201)(9 202)(10 203)(11 204)(12 205)(13 206)(14 207)(15 346)(16 347)(17 348)(18 349)(19 350)(20 337)(21 338)(22 339)(23 340)(24 341)(25 342)(26 343)(27 344)(28 345)(29 410)(30 411)(31 412)(32 413)(33 414)(34 415)(35 416)(36 417)(37 418)(38 419)(39 420)(40 407)(41 408)(42 409)(43 427)(44 428)(45 429)(46 430)(47 431)(48 432)(49 433)(50 434)(51 421)(52 422)(53 423)(54 424)(55 425)(56 426)(57 396)(58 397)(59 398)(60 399)(61 400)(62 401)(63 402)(64 403)(65 404)(66 405)(67 406)(68 393)(69 394)(70 395)(71 332)(72 333)(73 334)(74 335)(75 336)(76 323)(77 324)(78 325)(79 326)(80 327)(81 328)(82 329)(83 330)(84 331)(85 322)(86 309)(87 310)(88 311)(89 312)(90 313)(91 314)(92 315)(93 316)(94 317)(95 318)(96 319)(97 320)(98 321)(99 161)(100 162)(101 163)(102 164)(103 165)(104 166)(105 167)(106 168)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 304)(114 305)(115 306)(116 307)(117 308)(118 295)(119 296)(120 297)(121 298)(122 299)(123 300)(124 301)(125 302)(126 303)(127 232)(128 233)(129 234)(130 235)(131 236)(132 237)(133 238)(134 225)(135 226)(136 227)(137 228)(138 229)(139 230)(140 231)(141 445)(142 446)(143 447)(144 448)(145 435)(146 436)(147 437)(148 438)(149 439)(150 440)(151 441)(152 442)(153 443)(154 444)(169 213)(170 214)(171 215)(172 216)(173 217)(174 218)(175 219)(176 220)(177 221)(178 222)(179 223)(180 224)(181 211)(182 212)(183 370)(184 371)(185 372)(186 373)(187 374)(188 375)(189 376)(190 377)(191 378)(192 365)(193 366)(194 367)(195 368)(196 369)(239 274)(240 275)(241 276)(242 277)(243 278)(244 279)(245 280)(246 267)(247 268)(248 269)(249 270)(250 271)(251 272)(252 273)(253 353)(254 354)(255 355)(256 356)(257 357)(258 358)(259 359)(260 360)(261 361)(262 362)(263 363)(264 364)(265 351)(266 352)(281 391)(282 392)(283 379)(284 380)(285 381)(286 382)(287 383)(288 384)(289 385)(290 386)(291 387)(292 388)(293 389)(294 390)
(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)
(1 381 222 237 39 184 310 399)(2 382 223 238 40 185 311 400)(3 383 224 225 41 186 312 401)(4 384 211 226 42 187 313 402)(5 385 212 227 29 188 314 403)(6 386 213 228 30 189 315 404)(7 387 214 229 31 190 316 405)(8 388 215 230 32 191 317 406)(9 389 216 231 33 192 318 393)(10 390 217 232 34 193 319 394)(11 391 218 233 35 194 320 395)(12 392 219 234 36 195 321 396)(13 379 220 235 37 196 322 397)(14 380 221 236 38 183 309 398)(15 351 110 302 273 324 43 448)(16 352 111 303 274 325 44 435)(17 353 112 304 275 326 45 436)(18 354 99 305 276 327 46 437)(19 355 100 306 277 328 47 438)(20 356 101 307 278 329 48 439)(21 357 102 308 279 330 49 440)(22 358 103 295 280 331 50 441)(23 359 104 296 267 332 51 442)(24 360 105 297 268 333 52 443)(25 361 106 298 269 334 53 444)(26 362 107 299 270 335 54 445)(27 363 108 300 271 336 55 446)(28 364 109 301 272 323 56 447)(57 205 282 175 129 417 368 98)(58 206 283 176 130 418 369 85)(59 207 284 177 131 419 370 86)(60 208 285 178 132 420 371 87)(61 209 286 179 133 407 372 88)(62 210 287 180 134 408 373 89)(63 197 288 181 135 409 374 90)(64 198 289 182 136 410 375 91)(65 199 290 169 137 411 376 92)(66 200 291 170 138 412 377 93)(67 201 292 171 139 413 378 94)(68 202 293 172 140 414 365 95)(69 203 294 173 127 415 366 96)(70 204 281 174 128 416 367 97)(71 421 152 340 259 166 119 246)(72 422 153 341 260 167 120 247)(73 423 154 342 261 168 121 248)(74 424 141 343 262 155 122 249)(75 425 142 344 263 156 123 250)(76 426 143 345 264 157 124 251)(77 427 144 346 265 158 125 252)(78 428 145 347 266 159 126 239)(79 429 146 348 253 160 113 240)(80 430 147 349 254 161 114 241)(81 431 148 350 255 162 115 242)(82 432 149 337 256 163 116 243)(83 433 150 338 257 164 117 244)(84 434 151 339 258 165 118 245)
(1 300 39 446)(2 301 40 447)(3 302 41 448)(4 303 42 435)(5 304 29 436)(6 305 30 437)(7 306 31 438)(8 307 32 439)(9 308 33 440)(10 295 34 441)(11 296 35 442)(12 297 36 443)(13 298 37 444)(14 299 38 445)(15 401 273 225)(16 402 274 226)(17 403 275 227)(18 404 276 228)(19 405 277 229)(20 406 278 230)(21 393 279 231)(22 394 280 232)(23 395 267 233)(24 396 268 234)(25 397 269 235)(26 398 270 236)(27 399 271 237)(28 400 272 238)(43 383 110 186)(44 384 111 187)(45 385 112 188)(46 386 99 189)(47 387 100 190)(48 388 101 191)(49 389 102 192)(50 390 103 193)(51 391 104 194)(52 392 105 195)(53 379 106 196)(54 380 107 183)(55 381 108 184)(56 382 109 185)(57 247 129 341)(58 248 130 342)(59 249 131 343)(60 250 132 344)(61 251 133 345)(62 252 134 346)(63 239 135 347)(64 240 136 348)(65 241 137 349)(66 242 138 350)(67 243 139 337)(68 244 140 338)(69 245 127 339)(70 246 128 340)(71 174 259 97)(72 175 260 98)(73 176 261 85)(74 177 262 86)(75 178 263 87)(76 179 264 88)(77 180 265 89)(78 181 266 90)(79 182 253 91)(80 169 254 92)(81 170 255 93)(82 171 256 94)(83 172 257 95)(84 173 258 96)(113 410 146 198)(114 411 147 199)(115 412 148 200)(116 413 149 201)(117 414 150 202)(118 415 151 203)(119 416 152 204)(120 417 153 205)(121 418 154 206)(122 419 141 207)(123 420 142 208)(124 407 143 209)(125 408 144 210)(126 409 145 197)(155 370 424 284)(156 371 425 285)(157 372 426 286)(158 373 427 287)(159 374 428 288)(160 375 429 289)(161 376 430 290)(162 377 431 291)(163 378 432 292)(164 365 433 293)(165 366 434 294)(166 367 421 281)(167 368 422 282)(168 369 423 283)(211 352 313 325)(212 353 314 326)(213 354 315 327)(214 355 316 328)(215 356 317 329)(216 357 318 330)(217 358 319 331)(218 359 320 332)(219 360 321 333)(220 361 322 334)(221 362 309 335)(222 363 310 336)(223 364 311 323)(224 351 312 324)

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

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

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

196 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 7A ··· 7F 8A ··· 8H 14A ··· 14AP 28A ··· 28X 28Y ··· 28BT 56A ··· 56AV order 1 2 ··· 2 4 4 4 4 4 ··· 4 7 ··· 7 8 ··· 8 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C7 C14 C14 C14 D4 D4 Q16 C7×D4 C7×D4 C7×Q16 kernel Q16×C2×C14 C22×C56 C14×Q16 Q8×C2×C14 C22×Q16 C22×C8 C2×Q16 C22×Q8 C2×C28 C22×C14 C2×C14 C2×C4 C23 C22 # reps 1 1 12 2 6 6 72 12 3 1 8 18 6 48

Matrix representation of Q16×C2×C14 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 112 0 0 0 0 109 0 0 0 0 109
,
 112 0 0 0 0 112 0 0 0 0 31 82 0 0 31 31
,
 112 0 0 0 0 1 0 0 0 0 108 58 0 0 58 5
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,112,0,0,0,0,109,0,0,0,0,109],[112,0,0,0,0,112,0,0,0,0,31,31,0,0,82,31],[112,0,0,0,0,1,0,0,0,0,108,58,0,0,58,5] >;

Q16×C2×C14 in GAP, Magma, Sage, TeX

Q_{16}\times C_2\times C_{14}
% in TeX

G:=Group("Q16xC2xC14");
// GroupNames label

G:=SmallGroup(448,1354);
// by ID

G=gap.SmallGroup(448,1354);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,1576,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽