direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C2×C14, C28.80C24, C56.78C23, C4.20(D4×C14), (C2×C28).434D4, C28.327(C2×D4), C8.9(C22×C14), C4.3(C23×C14), C23.62(C7×D4), (C22×C8).10C14, (C22×C56).28C2, C22.67(D4×C14), Q8.1(C22×C14), (C7×Q8).35C23, (C22×Q8).9C14, (C2×C28).973C23, (C2×C56).429C22, (C22×C14).223D4, C14.201(C22×D4), (Q8×C14).280C22, (C22×C28).603C22, C2.25(D4×C2×C14), (C2×C4).90(C7×D4), (Q8×C2×C14).19C2, (C2×C8).87(C2×C14), (C2×C14).688(C2×D4), (C2×Q8).68(C2×C14), (C2×C4).143(C22×C14), (C22×C4).130(C2×C14), SmallGroup(448,1354)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 338 in 258 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C7, C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×8], Q8 [×12], C23, C14, C14 [×6], C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×12], C2×Q8 [×6], C28, C28 [×3], C28 [×8], C2×C14 [×7], C22×C8, C2×Q16 [×12], C22×Q8 [×2], C56 [×4], C2×C28 [×6], C2×C28 [×12], C7×Q8 [×8], C7×Q8 [×12], C22×C14, C22×Q16, C2×C56 [×6], C7×Q16 [×16], C22×C28, C22×C28 [×2], Q8×C14 [×12], Q8×C14 [×6], C22×C56, C14×Q16 [×12], Q8×C2×C14 [×2], Q16×C2×C14
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], Q16 [×4], C2×D4 [×6], C24, C2×C14 [×35], C2×Q16 [×6], C22×D4, C7×D4 [×4], C22×C14 [×15], C22×Q16, C7×Q16 [×4], D4×C14 [×6], C23×C14, C14×Q16 [×6], D4×C2×C14, Q16×C2×C14
Generators and relations
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 >
►Regular action on 448 points
Generators in S448
(1 198)(2 199)(3 200)(4 201)(5 202)(6 203)(7 204)(8 205)(9 206)(10 207)(11 208)(12 209)(13 210)(14 197)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 349)(30 350)(31 337)(32 338)(33 339)(34 340)(35 341)(36 342)(37 343)(38 344)(39 345)(40 346)(41 347)(42 348)(43 352)(44 353)(45 354)(46 355)(47 356)(48 357)(49 358)(50 359)(51 360)(52 361)(53 362)(54 363)(55 364)(56 351)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 155)(78 156)(79 157)(80 158)(81 159)(82 160)(83 161)(84 162)(85 291)(86 292)(87 293)(88 294)(89 281)(90 282)(91 283)(92 284)(93 285)(94 286)(95 287)(96 288)(97 289)(98 290)(99 272)(100 273)(101 274)(102 275)(103 276)(104 277)(105 278)(106 279)(107 280)(108 267)(109 268)(110 269)(111 270)(112 271)(113 232)(114 233)(115 234)(116 235)(117 236)(118 237)(119 238)(120 225)(121 226)(122 227)(123 228)(124 229)(125 230)(126 231)(127 153)(128 154)(129 141)(130 142)(131 143)(132 144)(133 145)(134 146)(135 147)(136 148)(137 149)(138 150)(139 151)(140 152)(169 330)(170 331)(171 332)(172 333)(173 334)(174 335)(175 336)(176 323)(177 324)(178 325)(179 326)(180 327)(181 328)(182 329)(183 424)(184 425)(185 426)(186 427)(187 428)(188 429)(189 430)(190 431)(191 432)(192 433)(193 434)(194 421)(195 422)(196 423)(211 372)(212 373)(213 374)(214 375)(215 376)(216 377)(217 378)(218 365)(219 366)(220 367)(221 368)(222 369)(223 370)(224 371)(239 381)(240 382)(241 383)(242 384)(243 385)(244 386)(245 387)(246 388)(247 389)(248 390)(249 391)(250 392)(251 379)(252 380)(253 396)(254 397)(255 398)(256 399)(257 400)(258 401)(259 402)(260 403)(261 404)(262 405)(263 406)(264 393)(265 394)(266 395)(295 411)(296 412)(297 413)(298 414)(299 415)(300 416)(301 417)(302 418)(303 419)(304 420)(305 407)(306 408)(307 409)(308 410)(309 440)(310 441)(311 442)(312 443)(313 444)(314 445)(315 446)(316 447)(317 448)(318 435)(319 436)(320 437)(321 438)(322 439)
(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 76 172 48 65 314 412 266)(2 77 173 49 66 315 413 253)(3 78 174 50 67 316 414 254)(4 79 175 51 68 317 415 255)(5 80 176 52 69 318 416 256)(6 81 177 53 70 319 417 257)(7 82 178 54 57 320 418 258)(8 83 179 55 58 321 419 259)(9 84 180 56 59 322 420 260)(10 71 181 43 60 309 407 261)(11 72 182 44 61 310 408 262)(12 73 169 45 62 311 409 263)(13 74 170 46 63 312 410 264)(14 75 171 47 64 313 411 265)(15 444 295 394 197 167 332 356)(16 445 296 395 198 168 333 357)(17 446 297 396 199 155 334 358)(18 447 298 397 200 156 335 359)(19 448 299 398 201 157 336 360)(20 435 300 399 202 158 323 361)(21 436 301 400 203 159 324 362)(22 437 302 401 204 160 325 363)(23 438 303 402 205 161 326 364)(24 439 304 403 206 162 327 351)(25 440 305 404 207 163 328 352)(26 441 306 405 208 164 329 353)(27 442 307 406 209 165 330 354)(28 443 308 393 210 166 331 355)(29 390 108 124 367 145 97 183)(30 391 109 125 368 146 98 184)(31 392 110 126 369 147 85 185)(32 379 111 113 370 148 86 186)(33 380 112 114 371 149 87 187)(34 381 99 115 372 150 88 188)(35 382 100 116 373 151 89 189)(36 383 101 117 374 152 90 190)(37 384 102 118 375 153 91 191)(38 385 103 119 376 154 92 192)(39 386 104 120 377 141 93 193)(40 387 105 121 378 142 94 194)(41 388 106 122 365 143 95 195)(42 389 107 123 366 144 96 196)(127 283 432 343 242 275 237 214)(128 284 433 344 243 276 238 215)(129 285 434 345 244 277 225 216)(130 286 421 346 245 278 226 217)(131 287 422 347 246 279 227 218)(132 288 423 348 247 280 228 219)(133 289 424 349 248 267 229 220)(134 290 425 350 249 268 230 221)(135 291 426 337 250 269 231 222)(136 292 427 338 251 270 232 223)(137 293 428 339 252 271 233 224)(138 294 429 340 239 272 234 211)(139 281 430 341 240 273 235 212)(140 282 431 342 241 274 236 213)
(1 372 65 34)(2 373 66 35)(3 374 67 36)(4 375 68 37)(5 376 69 38)(6 377 70 39)(7 378 57 40)(8 365 58 41)(9 366 59 42)(10 367 60 29)(11 368 61 30)(12 369 62 31)(13 370 63 32)(14 371 64 33)(15 339 197 224)(16 340 198 211)(17 341 199 212)(18 342 200 213)(19 343 201 214)(20 344 202 215)(21 345 203 216)(22 346 204 217)(23 347 205 218)(24 348 206 219)(25 349 207 220)(26 350 208 221)(27 337 209 222)(28 338 210 223)(43 390 261 145)(44 391 262 146)(45 392 263 147)(46 379 264 148)(47 380 265 149)(48 381 266 150)(49 382 253 151)(50 383 254 152)(51 384 255 153)(52 385 256 154)(53 386 257 141)(54 387 258 142)(55 388 259 143)(56 389 260 144)(71 124 309 183)(72 125 310 184)(73 126 311 185)(74 113 312 186)(75 114 313 187)(76 115 314 188)(77 116 315 189)(78 117 316 190)(79 118 317 191)(80 119 318 192)(81 120 319 193)(82 121 320 194)(83 122 321 195)(84 123 322 196)(85 169 110 409)(86 170 111 410)(87 171 112 411)(88 172 99 412)(89 173 100 413)(90 174 101 414)(91 175 102 415)(92 176 103 416)(93 177 104 417)(94 178 105 418)(95 179 106 419)(96 180 107 420)(97 181 108 407)(98 182 109 408)(127 360 242 398)(128 361 243 399)(129 362 244 400)(130 363 245 401)(131 364 246 402)(132 351 247 403)(133 352 248 404)(134 353 249 405)(135 354 250 406)(136 355 251 393)(137 356 252 394)(138 357 239 395)(139 358 240 396)(140 359 241 397)(155 235 446 430)(156 236 447 431)(157 237 448 432)(158 238 435 433)(159 225 436 434)(160 226 437 421)(161 227 438 422)(162 228 439 423)(163 229 440 424)(164 230 441 425)(165 231 442 426)(166 232 443 427)(167 233 444 428)(168 234 445 429)(267 305 289 328)(268 306 290 329)(269 307 291 330)(270 308 292 331)(271 295 293 332)(272 296 294 333)(273 297 281 334)(274 298 282 335)(275 299 283 336)(276 300 284 323)(277 301 285 324)(278 302 286 325)(279 303 287 326)(280 304 288 327)
G:=sub<Sym(448)| (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,197)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,349)(30,350)(31,337)(32,338)(33,339)(34,340)(35,341)(36,342)(37,343)(38,344)(39,345)(40,346)(41,347)(42,348)(43,352)(44,353)(45,354)(46,355)(47,356)(48,357)(49,358)(50,359)(51,360)(52,361)(53,362)(54,363)(55,364)(56,351)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,291)(86,292)(87,293)(88,294)(89,281)(90,282)(91,283)(92,284)(93,285)(94,286)(95,287)(96,288)(97,289)(98,290)(99,272)(100,273)(101,274)(102,275)(103,276)(104,277)(105,278)(106,279)(107,280)(108,267)(109,268)(110,269)(111,270)(112,271)(113,232)(114,233)(115,234)(116,235)(117,236)(118,237)(119,238)(120,225)(121,226)(122,227)(123,228)(124,229)(125,230)(126,231)(127,153)(128,154)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,151)(140,152)(169,330)(170,331)(171,332)(172,333)(173,334)(174,335)(175,336)(176,323)(177,324)(178,325)(179,326)(180,327)(181,328)(182,329)(183,424)(184,425)(185,426)(186,427)(187,428)(188,429)(189,430)(190,431)(191,432)(192,433)(193,434)(194,421)(195,422)(196,423)(211,372)(212,373)(213,374)(214,375)(215,376)(216,377)(217,378)(218,365)(219,366)(220,367)(221,368)(222,369)(223,370)(224,371)(239,381)(240,382)(241,383)(242,384)(243,385)(244,386)(245,387)(246,388)(247,389)(248,390)(249,391)(250,392)(251,379)(252,380)(253,396)(254,397)(255,398)(256,399)(257,400)(258,401)(259,402)(260,403)(261,404)(262,405)(263,406)(264,393)(265,394)(266,395)(295,411)(296,412)(297,413)(298,414)(299,415)(300,416)(301,417)(302,418)(303,419)(304,420)(305,407)(306,408)(307,409)(308,410)(309,440)(310,441)(311,442)(312,443)(313,444)(314,445)(315,446)(316,447)(317,448)(318,435)(319,436)(320,437)(321,438)(322,439), (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), (1,76,172,48,65,314,412,266)(2,77,173,49,66,315,413,253)(3,78,174,50,67,316,414,254)(4,79,175,51,68,317,415,255)(5,80,176,52,69,318,416,256)(6,81,177,53,70,319,417,257)(7,82,178,54,57,320,418,258)(8,83,179,55,58,321,419,259)(9,84,180,56,59,322,420,260)(10,71,181,43,60,309,407,261)(11,72,182,44,61,310,408,262)(12,73,169,45,62,311,409,263)(13,74,170,46,63,312,410,264)(14,75,171,47,64,313,411,265)(15,444,295,394,197,167,332,356)(16,445,296,395,198,168,333,357)(17,446,297,396,199,155,334,358)(18,447,298,397,200,156,335,359)(19,448,299,398,201,157,336,360)(20,435,300,399,202,158,323,361)(21,436,301,400,203,159,324,362)(22,437,302,401,204,160,325,363)(23,438,303,402,205,161,326,364)(24,439,304,403,206,162,327,351)(25,440,305,404,207,163,328,352)(26,441,306,405,208,164,329,353)(27,442,307,406,209,165,330,354)(28,443,308,393,210,166,331,355)(29,390,108,124,367,145,97,183)(30,391,109,125,368,146,98,184)(31,392,110,126,369,147,85,185)(32,379,111,113,370,148,86,186)(33,380,112,114,371,149,87,187)(34,381,99,115,372,150,88,188)(35,382,100,116,373,151,89,189)(36,383,101,117,374,152,90,190)(37,384,102,118,375,153,91,191)(38,385,103,119,376,154,92,192)(39,386,104,120,377,141,93,193)(40,387,105,121,378,142,94,194)(41,388,106,122,365,143,95,195)(42,389,107,123,366,144,96,196)(127,283,432,343,242,275,237,214)(128,284,433,344,243,276,238,215)(129,285,434,345,244,277,225,216)(130,286,421,346,245,278,226,217)(131,287,422,347,246,279,227,218)(132,288,423,348,247,280,228,219)(133,289,424,349,248,267,229,220)(134,290,425,350,249,268,230,221)(135,291,426,337,250,269,231,222)(136,292,427,338,251,270,232,223)(137,293,428,339,252,271,233,224)(138,294,429,340,239,272,234,211)(139,281,430,341,240,273,235,212)(140,282,431,342,241,274,236,213), (1,372,65,34)(2,373,66,35)(3,374,67,36)(4,375,68,37)(5,376,69,38)(6,377,70,39)(7,378,57,40)(8,365,58,41)(9,366,59,42)(10,367,60,29)(11,368,61,30)(12,369,62,31)(13,370,63,32)(14,371,64,33)(15,339,197,224)(16,340,198,211)(17,341,199,212)(18,342,200,213)(19,343,201,214)(20,344,202,215)(21,345,203,216)(22,346,204,217)(23,347,205,218)(24,348,206,219)(25,349,207,220)(26,350,208,221)(27,337,209,222)(28,338,210,223)(43,390,261,145)(44,391,262,146)(45,392,263,147)(46,379,264,148)(47,380,265,149)(48,381,266,150)(49,382,253,151)(50,383,254,152)(51,384,255,153)(52,385,256,154)(53,386,257,141)(54,387,258,142)(55,388,259,143)(56,389,260,144)(71,124,309,183)(72,125,310,184)(73,126,311,185)(74,113,312,186)(75,114,313,187)(76,115,314,188)(77,116,315,189)(78,117,316,190)(79,118,317,191)(80,119,318,192)(81,120,319,193)(82,121,320,194)(83,122,321,195)(84,123,322,196)(85,169,110,409)(86,170,111,410)(87,171,112,411)(88,172,99,412)(89,173,100,413)(90,174,101,414)(91,175,102,415)(92,176,103,416)(93,177,104,417)(94,178,105,418)(95,179,106,419)(96,180,107,420)(97,181,108,407)(98,182,109,408)(127,360,242,398)(128,361,243,399)(129,362,244,400)(130,363,245,401)(131,364,246,402)(132,351,247,403)(133,352,248,404)(134,353,249,405)(135,354,250,406)(136,355,251,393)(137,356,252,394)(138,357,239,395)(139,358,240,396)(140,359,241,397)(155,235,446,430)(156,236,447,431)(157,237,448,432)(158,238,435,433)(159,225,436,434)(160,226,437,421)(161,227,438,422)(162,228,439,423)(163,229,440,424)(164,230,441,425)(165,231,442,426)(166,232,443,427)(167,233,444,428)(168,234,445,429)(267,305,289,328)(268,306,290,329)(269,307,291,330)(270,308,292,331)(271,295,293,332)(272,296,294,333)(273,297,281,334)(274,298,282,335)(275,299,283,336)(276,300,284,323)(277,301,285,324)(278,302,286,325)(279,303,287,326)(280,304,288,327)>;
G:=Group( (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,197)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,349)(30,350)(31,337)(32,338)(33,339)(34,340)(35,341)(36,342)(37,343)(38,344)(39,345)(40,346)(41,347)(42,348)(43,352)(44,353)(45,354)(46,355)(47,356)(48,357)(49,358)(50,359)(51,360)(52,361)(53,362)(54,363)(55,364)(56,351)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,291)(86,292)(87,293)(88,294)(89,281)(90,282)(91,283)(92,284)(93,285)(94,286)(95,287)(96,288)(97,289)(98,290)(99,272)(100,273)(101,274)(102,275)(103,276)(104,277)(105,278)(106,279)(107,280)(108,267)(109,268)(110,269)(111,270)(112,271)(113,232)(114,233)(115,234)(116,235)(117,236)(118,237)(119,238)(120,225)(121,226)(122,227)(123,228)(124,229)(125,230)(126,231)(127,153)(128,154)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,151)(140,152)(169,330)(170,331)(171,332)(172,333)(173,334)(174,335)(175,336)(176,323)(177,324)(178,325)(179,326)(180,327)(181,328)(182,329)(183,424)(184,425)(185,426)(186,427)(187,428)(188,429)(189,430)(190,431)(191,432)(192,433)(193,434)(194,421)(195,422)(196,423)(211,372)(212,373)(213,374)(214,375)(215,376)(216,377)(217,378)(218,365)(219,366)(220,367)(221,368)(222,369)(223,370)(224,371)(239,381)(240,382)(241,383)(242,384)(243,385)(244,386)(245,387)(246,388)(247,389)(248,390)(249,391)(250,392)(251,379)(252,380)(253,396)(254,397)(255,398)(256,399)(257,400)(258,401)(259,402)(260,403)(261,404)(262,405)(263,406)(264,393)(265,394)(266,395)(295,411)(296,412)(297,413)(298,414)(299,415)(300,416)(301,417)(302,418)(303,419)(304,420)(305,407)(306,408)(307,409)(308,410)(309,440)(310,441)(311,442)(312,443)(313,444)(314,445)(315,446)(316,447)(317,448)(318,435)(319,436)(320,437)(321,438)(322,439), (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), (1,76,172,48,65,314,412,266)(2,77,173,49,66,315,413,253)(3,78,174,50,67,316,414,254)(4,79,175,51,68,317,415,255)(5,80,176,52,69,318,416,256)(6,81,177,53,70,319,417,257)(7,82,178,54,57,320,418,258)(8,83,179,55,58,321,419,259)(9,84,180,56,59,322,420,260)(10,71,181,43,60,309,407,261)(11,72,182,44,61,310,408,262)(12,73,169,45,62,311,409,263)(13,74,170,46,63,312,410,264)(14,75,171,47,64,313,411,265)(15,444,295,394,197,167,332,356)(16,445,296,395,198,168,333,357)(17,446,297,396,199,155,334,358)(18,447,298,397,200,156,335,359)(19,448,299,398,201,157,336,360)(20,435,300,399,202,158,323,361)(21,436,301,400,203,159,324,362)(22,437,302,401,204,160,325,363)(23,438,303,402,205,161,326,364)(24,439,304,403,206,162,327,351)(25,440,305,404,207,163,328,352)(26,441,306,405,208,164,329,353)(27,442,307,406,209,165,330,354)(28,443,308,393,210,166,331,355)(29,390,108,124,367,145,97,183)(30,391,109,125,368,146,98,184)(31,392,110,126,369,147,85,185)(32,379,111,113,370,148,86,186)(33,380,112,114,371,149,87,187)(34,381,99,115,372,150,88,188)(35,382,100,116,373,151,89,189)(36,383,101,117,374,152,90,190)(37,384,102,118,375,153,91,191)(38,385,103,119,376,154,92,192)(39,386,104,120,377,141,93,193)(40,387,105,121,378,142,94,194)(41,388,106,122,365,143,95,195)(42,389,107,123,366,144,96,196)(127,283,432,343,242,275,237,214)(128,284,433,344,243,276,238,215)(129,285,434,345,244,277,225,216)(130,286,421,346,245,278,226,217)(131,287,422,347,246,279,227,218)(132,288,423,348,247,280,228,219)(133,289,424,349,248,267,229,220)(134,290,425,350,249,268,230,221)(135,291,426,337,250,269,231,222)(136,292,427,338,251,270,232,223)(137,293,428,339,252,271,233,224)(138,294,429,340,239,272,234,211)(139,281,430,341,240,273,235,212)(140,282,431,342,241,274,236,213), (1,372,65,34)(2,373,66,35)(3,374,67,36)(4,375,68,37)(5,376,69,38)(6,377,70,39)(7,378,57,40)(8,365,58,41)(9,366,59,42)(10,367,60,29)(11,368,61,30)(12,369,62,31)(13,370,63,32)(14,371,64,33)(15,339,197,224)(16,340,198,211)(17,341,199,212)(18,342,200,213)(19,343,201,214)(20,344,202,215)(21,345,203,216)(22,346,204,217)(23,347,205,218)(24,348,206,219)(25,349,207,220)(26,350,208,221)(27,337,209,222)(28,338,210,223)(43,390,261,145)(44,391,262,146)(45,392,263,147)(46,379,264,148)(47,380,265,149)(48,381,266,150)(49,382,253,151)(50,383,254,152)(51,384,255,153)(52,385,256,154)(53,386,257,141)(54,387,258,142)(55,388,259,143)(56,389,260,144)(71,124,309,183)(72,125,310,184)(73,126,311,185)(74,113,312,186)(75,114,313,187)(76,115,314,188)(77,116,315,189)(78,117,316,190)(79,118,317,191)(80,119,318,192)(81,120,319,193)(82,121,320,194)(83,122,321,195)(84,123,322,196)(85,169,110,409)(86,170,111,410)(87,171,112,411)(88,172,99,412)(89,173,100,413)(90,174,101,414)(91,175,102,415)(92,176,103,416)(93,177,104,417)(94,178,105,418)(95,179,106,419)(96,180,107,420)(97,181,108,407)(98,182,109,408)(127,360,242,398)(128,361,243,399)(129,362,244,400)(130,363,245,401)(131,364,246,402)(132,351,247,403)(133,352,248,404)(134,353,249,405)(135,354,250,406)(136,355,251,393)(137,356,252,394)(138,357,239,395)(139,358,240,396)(140,359,241,397)(155,235,446,430)(156,236,447,431)(157,237,448,432)(158,238,435,433)(159,225,436,434)(160,226,437,421)(161,227,438,422)(162,228,439,423)(163,229,440,424)(164,230,441,425)(165,231,442,426)(166,232,443,427)(167,233,444,428)(168,234,445,429)(267,305,289,328)(268,306,290,329)(269,307,291,330)(270,308,292,331)(271,295,293,332)(272,296,294,333)(273,297,281,334)(274,298,282,335)(275,299,283,336)(276,300,284,323)(277,301,285,324)(278,302,286,325)(279,303,287,326)(280,304,288,327) );
G=PermutationGroup([(1,198),(2,199),(3,200),(4,201),(5,202),(6,203),(7,204),(8,205),(9,206),(10,207),(11,208),(12,209),(13,210),(14,197),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,349),(30,350),(31,337),(32,338),(33,339),(34,340),(35,341),(36,342),(37,343),(38,344),(39,345),(40,346),(41,347),(42,348),(43,352),(44,353),(45,354),(46,355),(47,356),(48,357),(49,358),(50,359),(51,360),(52,361),(53,362),(54,363),(55,364),(56,351),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,155),(78,156),(79,157),(80,158),(81,159),(82,160),(83,161),(84,162),(85,291),(86,292),(87,293),(88,294),(89,281),(90,282),(91,283),(92,284),(93,285),(94,286),(95,287),(96,288),(97,289),(98,290),(99,272),(100,273),(101,274),(102,275),(103,276),(104,277),(105,278),(106,279),(107,280),(108,267),(109,268),(110,269),(111,270),(112,271),(113,232),(114,233),(115,234),(116,235),(117,236),(118,237),(119,238),(120,225),(121,226),(122,227),(123,228),(124,229),(125,230),(126,231),(127,153),(128,154),(129,141),(130,142),(131,143),(132,144),(133,145),(134,146),(135,147),(136,148),(137,149),(138,150),(139,151),(140,152),(169,330),(170,331),(171,332),(172,333),(173,334),(174,335),(175,336),(176,323),(177,324),(178,325),(179,326),(180,327),(181,328),(182,329),(183,424),(184,425),(185,426),(186,427),(187,428),(188,429),(189,430),(190,431),(191,432),(192,433),(193,434),(194,421),(195,422),(196,423),(211,372),(212,373),(213,374),(214,375),(215,376),(216,377),(217,378),(218,365),(219,366),(220,367),(221,368),(222,369),(223,370),(224,371),(239,381),(240,382),(241,383),(242,384),(243,385),(244,386),(245,387),(246,388),(247,389),(248,390),(249,391),(250,392),(251,379),(252,380),(253,396),(254,397),(255,398),(256,399),(257,400),(258,401),(259,402),(260,403),(261,404),(262,405),(263,406),(264,393),(265,394),(266,395),(295,411),(296,412),(297,413),(298,414),(299,415),(300,416),(301,417),(302,418),(303,419),(304,420),(305,407),(306,408),(307,409),(308,410),(309,440),(310,441),(311,442),(312,443),(313,444),(314,445),(315,446),(316,447),(317,448),(318,435),(319,436),(320,437),(321,438),(322,439)], [(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)], [(1,76,172,48,65,314,412,266),(2,77,173,49,66,315,413,253),(3,78,174,50,67,316,414,254),(4,79,175,51,68,317,415,255),(5,80,176,52,69,318,416,256),(6,81,177,53,70,319,417,257),(7,82,178,54,57,320,418,258),(8,83,179,55,58,321,419,259),(9,84,180,56,59,322,420,260),(10,71,181,43,60,309,407,261),(11,72,182,44,61,310,408,262),(12,73,169,45,62,311,409,263),(13,74,170,46,63,312,410,264),(14,75,171,47,64,313,411,265),(15,444,295,394,197,167,332,356),(16,445,296,395,198,168,333,357),(17,446,297,396,199,155,334,358),(18,447,298,397,200,156,335,359),(19,448,299,398,201,157,336,360),(20,435,300,399,202,158,323,361),(21,436,301,400,203,159,324,362),(22,437,302,401,204,160,325,363),(23,438,303,402,205,161,326,364),(24,439,304,403,206,162,327,351),(25,440,305,404,207,163,328,352),(26,441,306,405,208,164,329,353),(27,442,307,406,209,165,330,354),(28,443,308,393,210,166,331,355),(29,390,108,124,367,145,97,183),(30,391,109,125,368,146,98,184),(31,392,110,126,369,147,85,185),(32,379,111,113,370,148,86,186),(33,380,112,114,371,149,87,187),(34,381,99,115,372,150,88,188),(35,382,100,116,373,151,89,189),(36,383,101,117,374,152,90,190),(37,384,102,118,375,153,91,191),(38,385,103,119,376,154,92,192),(39,386,104,120,377,141,93,193),(40,387,105,121,378,142,94,194),(41,388,106,122,365,143,95,195),(42,389,107,123,366,144,96,196),(127,283,432,343,242,275,237,214),(128,284,433,344,243,276,238,215),(129,285,434,345,244,277,225,216),(130,286,421,346,245,278,226,217),(131,287,422,347,246,279,227,218),(132,288,423,348,247,280,228,219),(133,289,424,349,248,267,229,220),(134,290,425,350,249,268,230,221),(135,291,426,337,250,269,231,222),(136,292,427,338,251,270,232,223),(137,293,428,339,252,271,233,224),(138,294,429,340,239,272,234,211),(139,281,430,341,240,273,235,212),(140,282,431,342,241,274,236,213)], [(1,372,65,34),(2,373,66,35),(3,374,67,36),(4,375,68,37),(5,376,69,38),(6,377,70,39),(7,378,57,40),(8,365,58,41),(9,366,59,42),(10,367,60,29),(11,368,61,30),(12,369,62,31),(13,370,63,32),(14,371,64,33),(15,339,197,224),(16,340,198,211),(17,341,199,212),(18,342,200,213),(19,343,201,214),(20,344,202,215),(21,345,203,216),(22,346,204,217),(23,347,205,218),(24,348,206,219),(25,349,207,220),(26,350,208,221),(27,337,209,222),(28,338,210,223),(43,390,261,145),(44,391,262,146),(45,392,263,147),(46,379,264,148),(47,380,265,149),(48,381,266,150),(49,382,253,151),(50,383,254,152),(51,384,255,153),(52,385,256,154),(53,386,257,141),(54,387,258,142),(55,388,259,143),(56,389,260,144),(71,124,309,183),(72,125,310,184),(73,126,311,185),(74,113,312,186),(75,114,313,187),(76,115,314,188),(77,116,315,189),(78,117,316,190),(79,118,317,191),(80,119,318,192),(81,120,319,193),(82,121,320,194),(83,122,321,195),(84,123,322,196),(85,169,110,409),(86,170,111,410),(87,171,112,411),(88,172,99,412),(89,173,100,413),(90,174,101,414),(91,175,102,415),(92,176,103,416),(93,177,104,417),(94,178,105,418),(95,179,106,419),(96,180,107,420),(97,181,108,407),(98,182,109,408),(127,360,242,398),(128,361,243,399),(129,362,244,400),(130,363,245,401),(131,364,246,402),(132,351,247,403),(133,352,248,404),(134,353,249,405),(135,354,250,406),(136,355,251,393),(137,356,252,394),(138,357,239,395),(139,358,240,396),(140,359,241,397),(155,235,446,430),(156,236,447,431),(157,237,448,432),(158,238,435,433),(159,225,436,434),(160,226,437,421),(161,227,438,422),(162,228,439,423),(163,229,440,424),(164,230,441,425),(165,231,442,426),(166,232,443,427),(167,233,444,428),(168,234,445,429),(267,305,289,328),(268,306,290,329),(269,307,291,330),(270,308,292,331),(271,295,293,332),(272,296,294,333),(273,297,281,334),(274,298,282,335),(275,299,283,336),(276,300,284,323),(277,301,285,324),(278,302,286,325),(279,303,287,326),(280,304,288,327)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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