metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊9Q8, C42.170D14, C14.812+ (1+4), C7⋊2(Q82), C4⋊Q8.15D7, C4.18(Q8×D7), C28⋊Q8.13C2, C28.53(C2×Q8), C4⋊C4.215D14, (C2×Q8).85D14, (C2×C28).99C23, Dic7.15(C2×Q8), C14.45(C22×Q8), (C2×C14).266C24, (C4×C28).207C22, Dic7⋊Q8.9C2, (C4×Dic14).26C2, C2.85(D4⋊6D14), Dic7⋊3Q8.13C2, Dic7⋊C4.58C22, C4⋊Dic7.383C22, (Q8×C14).133C22, C22.287(C23×D7), (C4×Dic7).158C22, (C2×Dic7).271C23, (C2×Dic14).186C22, C2.28(C2×Q8×D7), (C7×C4⋊Q8).15C2, (C2×C4).91(C22×D7), (C7×C4⋊C4).209C22, SmallGroup(448,1175)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 812 in 212 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2 [×2], C4 [×4], C4 [×17], C22, C7, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×14], C14, C14 [×2], C42, C42 [×8], C4⋊C4 [×4], C4⋊C4 [×14], C2×Q8 [×2], C2×Q8 [×6], Dic7 [×8], Dic7 [×4], C28 [×4], C28 [×5], C2×C14, C4×Q8 [×6], C4⋊Q8, C4⋊Q8 [×8], Dic14 [×8], Dic14 [×4], C2×Dic7 [×8], C2×C28, C2×C28 [×6], C7×Q8 [×2], Q82, C4×Dic7 [×8], Dic7⋊C4 [×12], C4⋊Dic7 [×2], C4×C28, C7×C4⋊C4 [×4], C2×Dic14 [×6], Q8×C14 [×2], C4×Dic14 [×2], Dic7⋊3Q8 [×4], C28⋊Q8 [×4], Dic7⋊Q8 [×4], C7×C4⋊Q8, Dic14⋊9Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×8], C23 [×15], D7, C2×Q8 [×12], C24, D14 [×7], C22×Q8 [×2], 2+ (1+4), C22×D7 [×7], Q82, Q8×D7 [×4], C23×D7, D4⋊6D14, C2×Q8×D7 [×2], Dic14⋊9Q8
Generators and relations
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, cac-1=a15, ad=da, bc=cb, dbd-1=a14b, dcd-1=c-1 >
►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)
(1 137 15 123)(2 136 16 122)(3 135 17 121)(4 134 18 120)(5 133 19 119)(6 132 20 118)(7 131 21 117)(8 130 22 116)(9 129 23 115)(10 128 24 114)(11 127 25 113)(12 126 26 140)(13 125 27 139)(14 124 28 138)(29 163 43 149)(30 162 44 148)(31 161 45 147)(32 160 46 146)(33 159 47 145)(34 158 48 144)(35 157 49 143)(36 156 50 142)(37 155 51 141)(38 154 52 168)(39 153 53 167)(40 152 54 166)(41 151 55 165)(42 150 56 164)(57 86 71 100)(58 85 72 99)(59 112 73 98)(60 111 74 97)(61 110 75 96)(62 109 76 95)(63 108 77 94)(64 107 78 93)(65 106 79 92)(66 105 80 91)(67 104 81 90)(68 103 82 89)(69 102 83 88)(70 101 84 87)(169 323 183 309)(170 322 184 336)(171 321 185 335)(172 320 186 334)(173 319 187 333)(174 318 188 332)(175 317 189 331)(176 316 190 330)(177 315 191 329)(178 314 192 328)(179 313 193 327)(180 312 194 326)(181 311 195 325)(182 310 196 324)(197 448 211 434)(198 447 212 433)(199 446 213 432)(200 445 214 431)(201 444 215 430)(202 443 216 429)(203 442 217 428)(204 441 218 427)(205 440 219 426)(206 439 220 425)(207 438 221 424)(208 437 222 423)(209 436 223 422)(210 435 224 421)(225 403 239 417)(226 402 240 416)(227 401 241 415)(228 400 242 414)(229 399 243 413)(230 398 244 412)(231 397 245 411)(232 396 246 410)(233 395 247 409)(234 394 248 408)(235 393 249 407)(236 420 250 406)(237 419 251 405)(238 418 252 404)(253 385 267 371)(254 384 268 370)(255 383 269 369)(256 382 270 368)(257 381 271 367)(258 380 272 366)(259 379 273 365)(260 378 274 392)(261 377 275 391)(262 376 276 390)(263 375 277 389)(264 374 278 388)(265 373 279 387)(266 372 280 386)(281 355 295 341)(282 354 296 340)(283 353 297 339)(284 352 298 338)(285 351 299 337)(286 350 300 364)(287 349 301 363)(288 348 302 362)(289 347 303 361)(290 346 304 360)(291 345 305 359)(292 344 306 358)(293 343 307 357)(294 342 308 356)
(1 203 395 86)(2 218 396 101)(3 205 397 88)(4 220 398 103)(5 207 399 90)(6 222 400 105)(7 209 401 92)(8 224 402 107)(9 211 403 94)(10 198 404 109)(11 213 405 96)(12 200 406 111)(13 215 407 98)(14 202 408 85)(15 217 409 100)(16 204 410 87)(17 219 411 102)(18 206 412 89)(19 221 413 104)(20 208 414 91)(21 223 415 106)(22 210 416 93)(23 197 417 108)(24 212 418 95)(25 199 419 110)(26 214 420 97)(27 201 393 112)(28 216 394 99)(29 318 300 275)(30 333 301 262)(31 320 302 277)(32 335 303 264)(33 322 304 279)(34 309 305 266)(35 324 306 253)(36 311 307 268)(37 326 308 255)(38 313 281 270)(39 328 282 257)(40 315 283 272)(41 330 284 259)(42 317 285 274)(43 332 286 261)(44 319 287 276)(45 334 288 263)(46 321 289 278)(47 336 290 265)(48 323 291 280)(49 310 292 267)(50 325 293 254)(51 312 294 269)(52 327 295 256)(53 314 296 271)(54 329 297 258)(55 316 298 273)(56 331 299 260)(57 123 428 233)(58 138 429 248)(59 125 430 235)(60 140 431 250)(61 127 432 237)(62 114 433 252)(63 129 434 239)(64 116 435 226)(65 131 436 241)(66 118 437 228)(67 133 438 243)(68 120 439 230)(69 135 440 245)(70 122 441 232)(71 137 442 247)(72 124 443 234)(73 139 444 249)(74 126 445 236)(75 113 446 251)(76 128 447 238)(77 115 448 225)(78 130 421 240)(79 117 422 227)(80 132 423 242)(81 119 424 229)(82 134 425 244)(83 121 426 231)(84 136 427 246)(141 194 342 369)(142 181 343 384)(143 196 344 371)(144 183 345 386)(145 170 346 373)(146 185 347 388)(147 172 348 375)(148 187 349 390)(149 174 350 377)(150 189 351 392)(151 176 352 379)(152 191 353 366)(153 178 354 381)(154 193 355 368)(155 180 356 383)(156 195 357 370)(157 182 358 385)(158 169 359 372)(159 184 360 387)(160 171 361 374)(161 186 362 389)(162 173 363 376)(163 188 364 391)(164 175 337 378)(165 190 338 365)(166 177 339 380)(167 192 340 367)(168 179 341 382)
(1 278 395 321)(2 279 396 322)(3 280 397 323)(4 253 398 324)(5 254 399 325)(6 255 400 326)(7 256 401 327)(8 257 402 328)(9 258 403 329)(10 259 404 330)(11 260 405 331)(12 261 406 332)(13 262 407 333)(14 263 408 334)(15 264 409 335)(16 265 410 336)(17 266 411 309)(18 267 412 310)(19 268 413 311)(20 269 414 312)(21 270 415 313)(22 271 416 314)(23 272 417 315)(24 273 418 316)(25 274 419 317)(26 275 420 318)(27 276 393 319)(28 277 394 320)(29 214 300 97)(30 215 301 98)(31 216 302 99)(32 217 303 100)(33 218 304 101)(34 219 305 102)(35 220 306 103)(36 221 307 104)(37 222 308 105)(38 223 281 106)(39 224 282 107)(40 197 283 108)(41 198 284 109)(42 199 285 110)(43 200 286 111)(44 201 287 112)(45 202 288 85)(46 203 289 86)(47 204 290 87)(48 205 291 88)(49 206 292 89)(50 207 293 90)(51 208 294 91)(52 209 295 92)(53 210 296 93)(54 211 297 94)(55 212 298 95)(56 213 299 96)(57 146 428 347)(58 147 429 348)(59 148 430 349)(60 149 431 350)(61 150 432 351)(62 151 433 352)(63 152 434 353)(64 153 435 354)(65 154 436 355)(66 155 437 356)(67 156 438 357)(68 157 439 358)(69 158 440 359)(70 159 441 360)(71 160 442 361)(72 161 443 362)(73 162 444 363)(74 163 445 364)(75 164 446 337)(76 165 447 338)(77 166 448 339)(78 167 421 340)(79 168 422 341)(80 141 423 342)(81 142 424 343)(82 143 425 344)(83 144 426 345)(84 145 427 346)(113 378 251 175)(114 379 252 176)(115 380 225 177)(116 381 226 178)(117 382 227 179)(118 383 228 180)(119 384 229 181)(120 385 230 182)(121 386 231 183)(122 387 232 184)(123 388 233 185)(124 389 234 186)(125 390 235 187)(126 391 236 188)(127 392 237 189)(128 365 238 190)(129 366 239 191)(130 367 240 192)(131 368 241 193)(132 369 242 194)(133 370 243 195)(134 371 244 196)(135 372 245 169)(136 373 246 170)(137 374 247 171)(138 375 248 172)(139 376 249 173)(140 377 250 174)
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), (1,137,15,123)(2,136,16,122)(3,135,17,121)(4,134,18,120)(5,133,19,119)(6,132,20,118)(7,131,21,117)(8,130,22,116)(9,129,23,115)(10,128,24,114)(11,127,25,113)(12,126,26,140)(13,125,27,139)(14,124,28,138)(29,163,43,149)(30,162,44,148)(31,161,45,147)(32,160,46,146)(33,159,47,145)(34,158,48,144)(35,157,49,143)(36,156,50,142)(37,155,51,141)(38,154,52,168)(39,153,53,167)(40,152,54,166)(41,151,55,165)(42,150,56,164)(57,86,71,100)(58,85,72,99)(59,112,73,98)(60,111,74,97)(61,110,75,96)(62,109,76,95)(63,108,77,94)(64,107,78,93)(65,106,79,92)(66,105,80,91)(67,104,81,90)(68,103,82,89)(69,102,83,88)(70,101,84,87)(169,323,183,309)(170,322,184,336)(171,321,185,335)(172,320,186,334)(173,319,187,333)(174,318,188,332)(175,317,189,331)(176,316,190,330)(177,315,191,329)(178,314,192,328)(179,313,193,327)(180,312,194,326)(181,311,195,325)(182,310,196,324)(197,448,211,434)(198,447,212,433)(199,446,213,432)(200,445,214,431)(201,444,215,430)(202,443,216,429)(203,442,217,428)(204,441,218,427)(205,440,219,426)(206,439,220,425)(207,438,221,424)(208,437,222,423)(209,436,223,422)(210,435,224,421)(225,403,239,417)(226,402,240,416)(227,401,241,415)(228,400,242,414)(229,399,243,413)(230,398,244,412)(231,397,245,411)(232,396,246,410)(233,395,247,409)(234,394,248,408)(235,393,249,407)(236,420,250,406)(237,419,251,405)(238,418,252,404)(253,385,267,371)(254,384,268,370)(255,383,269,369)(256,382,270,368)(257,381,271,367)(258,380,272,366)(259,379,273,365)(260,378,274,392)(261,377,275,391)(262,376,276,390)(263,375,277,389)(264,374,278,388)(265,373,279,387)(266,372,280,386)(281,355,295,341)(282,354,296,340)(283,353,297,339)(284,352,298,338)(285,351,299,337)(286,350,300,364)(287,349,301,363)(288,348,302,362)(289,347,303,361)(290,346,304,360)(291,345,305,359)(292,344,306,358)(293,343,307,357)(294,342,308,356), (1,203,395,86)(2,218,396,101)(3,205,397,88)(4,220,398,103)(5,207,399,90)(6,222,400,105)(7,209,401,92)(8,224,402,107)(9,211,403,94)(10,198,404,109)(11,213,405,96)(12,200,406,111)(13,215,407,98)(14,202,408,85)(15,217,409,100)(16,204,410,87)(17,219,411,102)(18,206,412,89)(19,221,413,104)(20,208,414,91)(21,223,415,106)(22,210,416,93)(23,197,417,108)(24,212,418,95)(25,199,419,110)(26,214,420,97)(27,201,393,112)(28,216,394,99)(29,318,300,275)(30,333,301,262)(31,320,302,277)(32,335,303,264)(33,322,304,279)(34,309,305,266)(35,324,306,253)(36,311,307,268)(37,326,308,255)(38,313,281,270)(39,328,282,257)(40,315,283,272)(41,330,284,259)(42,317,285,274)(43,332,286,261)(44,319,287,276)(45,334,288,263)(46,321,289,278)(47,336,290,265)(48,323,291,280)(49,310,292,267)(50,325,293,254)(51,312,294,269)(52,327,295,256)(53,314,296,271)(54,329,297,258)(55,316,298,273)(56,331,299,260)(57,123,428,233)(58,138,429,248)(59,125,430,235)(60,140,431,250)(61,127,432,237)(62,114,433,252)(63,129,434,239)(64,116,435,226)(65,131,436,241)(66,118,437,228)(67,133,438,243)(68,120,439,230)(69,135,440,245)(70,122,441,232)(71,137,442,247)(72,124,443,234)(73,139,444,249)(74,126,445,236)(75,113,446,251)(76,128,447,238)(77,115,448,225)(78,130,421,240)(79,117,422,227)(80,132,423,242)(81,119,424,229)(82,134,425,244)(83,121,426,231)(84,136,427,246)(141,194,342,369)(142,181,343,384)(143,196,344,371)(144,183,345,386)(145,170,346,373)(146,185,347,388)(147,172,348,375)(148,187,349,390)(149,174,350,377)(150,189,351,392)(151,176,352,379)(152,191,353,366)(153,178,354,381)(154,193,355,368)(155,180,356,383)(156,195,357,370)(157,182,358,385)(158,169,359,372)(159,184,360,387)(160,171,361,374)(161,186,362,389)(162,173,363,376)(163,188,364,391)(164,175,337,378)(165,190,338,365)(166,177,339,380)(167,192,340,367)(168,179,341,382), (1,278,395,321)(2,279,396,322)(3,280,397,323)(4,253,398,324)(5,254,399,325)(6,255,400,326)(7,256,401,327)(8,257,402,328)(9,258,403,329)(10,259,404,330)(11,260,405,331)(12,261,406,332)(13,262,407,333)(14,263,408,334)(15,264,409,335)(16,265,410,336)(17,266,411,309)(18,267,412,310)(19,268,413,311)(20,269,414,312)(21,270,415,313)(22,271,416,314)(23,272,417,315)(24,273,418,316)(25,274,419,317)(26,275,420,318)(27,276,393,319)(28,277,394,320)(29,214,300,97)(30,215,301,98)(31,216,302,99)(32,217,303,100)(33,218,304,101)(34,219,305,102)(35,220,306,103)(36,221,307,104)(37,222,308,105)(38,223,281,106)(39,224,282,107)(40,197,283,108)(41,198,284,109)(42,199,285,110)(43,200,286,111)(44,201,287,112)(45,202,288,85)(46,203,289,86)(47,204,290,87)(48,205,291,88)(49,206,292,89)(50,207,293,90)(51,208,294,91)(52,209,295,92)(53,210,296,93)(54,211,297,94)(55,212,298,95)(56,213,299,96)(57,146,428,347)(58,147,429,348)(59,148,430,349)(60,149,431,350)(61,150,432,351)(62,151,433,352)(63,152,434,353)(64,153,435,354)(65,154,436,355)(66,155,437,356)(67,156,438,357)(68,157,439,358)(69,158,440,359)(70,159,441,360)(71,160,442,361)(72,161,443,362)(73,162,444,363)(74,163,445,364)(75,164,446,337)(76,165,447,338)(77,166,448,339)(78,167,421,340)(79,168,422,341)(80,141,423,342)(81,142,424,343)(82,143,425,344)(83,144,426,345)(84,145,427,346)(113,378,251,175)(114,379,252,176)(115,380,225,177)(116,381,226,178)(117,382,227,179)(118,383,228,180)(119,384,229,181)(120,385,230,182)(121,386,231,183)(122,387,232,184)(123,388,233,185)(124,389,234,186)(125,390,235,187)(126,391,236,188)(127,392,237,189)(128,365,238,190)(129,366,239,191)(130,367,240,192)(131,368,241,193)(132,369,242,194)(133,370,243,195)(134,371,244,196)(135,372,245,169)(136,373,246,170)(137,374,247,171)(138,375,248,172)(139,376,249,173)(140,377,250,174)>;
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), (1,137,15,123)(2,136,16,122)(3,135,17,121)(4,134,18,120)(5,133,19,119)(6,132,20,118)(7,131,21,117)(8,130,22,116)(9,129,23,115)(10,128,24,114)(11,127,25,113)(12,126,26,140)(13,125,27,139)(14,124,28,138)(29,163,43,149)(30,162,44,148)(31,161,45,147)(32,160,46,146)(33,159,47,145)(34,158,48,144)(35,157,49,143)(36,156,50,142)(37,155,51,141)(38,154,52,168)(39,153,53,167)(40,152,54,166)(41,151,55,165)(42,150,56,164)(57,86,71,100)(58,85,72,99)(59,112,73,98)(60,111,74,97)(61,110,75,96)(62,109,76,95)(63,108,77,94)(64,107,78,93)(65,106,79,92)(66,105,80,91)(67,104,81,90)(68,103,82,89)(69,102,83,88)(70,101,84,87)(169,323,183,309)(170,322,184,336)(171,321,185,335)(172,320,186,334)(173,319,187,333)(174,318,188,332)(175,317,189,331)(176,316,190,330)(177,315,191,329)(178,314,192,328)(179,313,193,327)(180,312,194,326)(181,311,195,325)(182,310,196,324)(197,448,211,434)(198,447,212,433)(199,446,213,432)(200,445,214,431)(201,444,215,430)(202,443,216,429)(203,442,217,428)(204,441,218,427)(205,440,219,426)(206,439,220,425)(207,438,221,424)(208,437,222,423)(209,436,223,422)(210,435,224,421)(225,403,239,417)(226,402,240,416)(227,401,241,415)(228,400,242,414)(229,399,243,413)(230,398,244,412)(231,397,245,411)(232,396,246,410)(233,395,247,409)(234,394,248,408)(235,393,249,407)(236,420,250,406)(237,419,251,405)(238,418,252,404)(253,385,267,371)(254,384,268,370)(255,383,269,369)(256,382,270,368)(257,381,271,367)(258,380,272,366)(259,379,273,365)(260,378,274,392)(261,377,275,391)(262,376,276,390)(263,375,277,389)(264,374,278,388)(265,373,279,387)(266,372,280,386)(281,355,295,341)(282,354,296,340)(283,353,297,339)(284,352,298,338)(285,351,299,337)(286,350,300,364)(287,349,301,363)(288,348,302,362)(289,347,303,361)(290,346,304,360)(291,345,305,359)(292,344,306,358)(293,343,307,357)(294,342,308,356), (1,203,395,86)(2,218,396,101)(3,205,397,88)(4,220,398,103)(5,207,399,90)(6,222,400,105)(7,209,401,92)(8,224,402,107)(9,211,403,94)(10,198,404,109)(11,213,405,96)(12,200,406,111)(13,215,407,98)(14,202,408,85)(15,217,409,100)(16,204,410,87)(17,219,411,102)(18,206,412,89)(19,221,413,104)(20,208,414,91)(21,223,415,106)(22,210,416,93)(23,197,417,108)(24,212,418,95)(25,199,419,110)(26,214,420,97)(27,201,393,112)(28,216,394,99)(29,318,300,275)(30,333,301,262)(31,320,302,277)(32,335,303,264)(33,322,304,279)(34,309,305,266)(35,324,306,253)(36,311,307,268)(37,326,308,255)(38,313,281,270)(39,328,282,257)(40,315,283,272)(41,330,284,259)(42,317,285,274)(43,332,286,261)(44,319,287,276)(45,334,288,263)(46,321,289,278)(47,336,290,265)(48,323,291,280)(49,310,292,267)(50,325,293,254)(51,312,294,269)(52,327,295,256)(53,314,296,271)(54,329,297,258)(55,316,298,273)(56,331,299,260)(57,123,428,233)(58,138,429,248)(59,125,430,235)(60,140,431,250)(61,127,432,237)(62,114,433,252)(63,129,434,239)(64,116,435,226)(65,131,436,241)(66,118,437,228)(67,133,438,243)(68,120,439,230)(69,135,440,245)(70,122,441,232)(71,137,442,247)(72,124,443,234)(73,139,444,249)(74,126,445,236)(75,113,446,251)(76,128,447,238)(77,115,448,225)(78,130,421,240)(79,117,422,227)(80,132,423,242)(81,119,424,229)(82,134,425,244)(83,121,426,231)(84,136,427,246)(141,194,342,369)(142,181,343,384)(143,196,344,371)(144,183,345,386)(145,170,346,373)(146,185,347,388)(147,172,348,375)(148,187,349,390)(149,174,350,377)(150,189,351,392)(151,176,352,379)(152,191,353,366)(153,178,354,381)(154,193,355,368)(155,180,356,383)(156,195,357,370)(157,182,358,385)(158,169,359,372)(159,184,360,387)(160,171,361,374)(161,186,362,389)(162,173,363,376)(163,188,364,391)(164,175,337,378)(165,190,338,365)(166,177,339,380)(167,192,340,367)(168,179,341,382), (1,278,395,321)(2,279,396,322)(3,280,397,323)(4,253,398,324)(5,254,399,325)(6,255,400,326)(7,256,401,327)(8,257,402,328)(9,258,403,329)(10,259,404,330)(11,260,405,331)(12,261,406,332)(13,262,407,333)(14,263,408,334)(15,264,409,335)(16,265,410,336)(17,266,411,309)(18,267,412,310)(19,268,413,311)(20,269,414,312)(21,270,415,313)(22,271,416,314)(23,272,417,315)(24,273,418,316)(25,274,419,317)(26,275,420,318)(27,276,393,319)(28,277,394,320)(29,214,300,97)(30,215,301,98)(31,216,302,99)(32,217,303,100)(33,218,304,101)(34,219,305,102)(35,220,306,103)(36,221,307,104)(37,222,308,105)(38,223,281,106)(39,224,282,107)(40,197,283,108)(41,198,284,109)(42,199,285,110)(43,200,286,111)(44,201,287,112)(45,202,288,85)(46,203,289,86)(47,204,290,87)(48,205,291,88)(49,206,292,89)(50,207,293,90)(51,208,294,91)(52,209,295,92)(53,210,296,93)(54,211,297,94)(55,212,298,95)(56,213,299,96)(57,146,428,347)(58,147,429,348)(59,148,430,349)(60,149,431,350)(61,150,432,351)(62,151,433,352)(63,152,434,353)(64,153,435,354)(65,154,436,355)(66,155,437,356)(67,156,438,357)(68,157,439,358)(69,158,440,359)(70,159,441,360)(71,160,442,361)(72,161,443,362)(73,162,444,363)(74,163,445,364)(75,164,446,337)(76,165,447,338)(77,166,448,339)(78,167,421,340)(79,168,422,341)(80,141,423,342)(81,142,424,343)(82,143,425,344)(83,144,426,345)(84,145,427,346)(113,378,251,175)(114,379,252,176)(115,380,225,177)(116,381,226,178)(117,382,227,179)(118,383,228,180)(119,384,229,181)(120,385,230,182)(121,386,231,183)(122,387,232,184)(123,388,233,185)(124,389,234,186)(125,390,235,187)(126,391,236,188)(127,392,237,189)(128,365,238,190)(129,366,239,191)(130,367,240,192)(131,368,241,193)(132,369,242,194)(133,370,243,195)(134,371,244,196)(135,372,245,169)(136,373,246,170)(137,374,247,171)(138,375,248,172)(139,376,249,173)(140,377,250,174) );
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)], [(1,137,15,123),(2,136,16,122),(3,135,17,121),(4,134,18,120),(5,133,19,119),(6,132,20,118),(7,131,21,117),(8,130,22,116),(9,129,23,115),(10,128,24,114),(11,127,25,113),(12,126,26,140),(13,125,27,139),(14,124,28,138),(29,163,43,149),(30,162,44,148),(31,161,45,147),(32,160,46,146),(33,159,47,145),(34,158,48,144),(35,157,49,143),(36,156,50,142),(37,155,51,141),(38,154,52,168),(39,153,53,167),(40,152,54,166),(41,151,55,165),(42,150,56,164),(57,86,71,100),(58,85,72,99),(59,112,73,98),(60,111,74,97),(61,110,75,96),(62,109,76,95),(63,108,77,94),(64,107,78,93),(65,106,79,92),(66,105,80,91),(67,104,81,90),(68,103,82,89),(69,102,83,88),(70,101,84,87),(169,323,183,309),(170,322,184,336),(171,321,185,335),(172,320,186,334),(173,319,187,333),(174,318,188,332),(175,317,189,331),(176,316,190,330),(177,315,191,329),(178,314,192,328),(179,313,193,327),(180,312,194,326),(181,311,195,325),(182,310,196,324),(197,448,211,434),(198,447,212,433),(199,446,213,432),(200,445,214,431),(201,444,215,430),(202,443,216,429),(203,442,217,428),(204,441,218,427),(205,440,219,426),(206,439,220,425),(207,438,221,424),(208,437,222,423),(209,436,223,422),(210,435,224,421),(225,403,239,417),(226,402,240,416),(227,401,241,415),(228,400,242,414),(229,399,243,413),(230,398,244,412),(231,397,245,411),(232,396,246,410),(233,395,247,409),(234,394,248,408),(235,393,249,407),(236,420,250,406),(237,419,251,405),(238,418,252,404),(253,385,267,371),(254,384,268,370),(255,383,269,369),(256,382,270,368),(257,381,271,367),(258,380,272,366),(259,379,273,365),(260,378,274,392),(261,377,275,391),(262,376,276,390),(263,375,277,389),(264,374,278,388),(265,373,279,387),(266,372,280,386),(281,355,295,341),(282,354,296,340),(283,353,297,339),(284,352,298,338),(285,351,299,337),(286,350,300,364),(287,349,301,363),(288,348,302,362),(289,347,303,361),(290,346,304,360),(291,345,305,359),(292,344,306,358),(293,343,307,357),(294,342,308,356)], [(1,203,395,86),(2,218,396,101),(3,205,397,88),(4,220,398,103),(5,207,399,90),(6,222,400,105),(7,209,401,92),(8,224,402,107),(9,211,403,94),(10,198,404,109),(11,213,405,96),(12,200,406,111),(13,215,407,98),(14,202,408,85),(15,217,409,100),(16,204,410,87),(17,219,411,102),(18,206,412,89),(19,221,413,104),(20,208,414,91),(21,223,415,106),(22,210,416,93),(23,197,417,108),(24,212,418,95),(25,199,419,110),(26,214,420,97),(27,201,393,112),(28,216,394,99),(29,318,300,275),(30,333,301,262),(31,320,302,277),(32,335,303,264),(33,322,304,279),(34,309,305,266),(35,324,306,253),(36,311,307,268),(37,326,308,255),(38,313,281,270),(39,328,282,257),(40,315,283,272),(41,330,284,259),(42,317,285,274),(43,332,286,261),(44,319,287,276),(45,334,288,263),(46,321,289,278),(47,336,290,265),(48,323,291,280),(49,310,292,267),(50,325,293,254),(51,312,294,269),(52,327,295,256),(53,314,296,271),(54,329,297,258),(55,316,298,273),(56,331,299,260),(57,123,428,233),(58,138,429,248),(59,125,430,235),(60,140,431,250),(61,127,432,237),(62,114,433,252),(63,129,434,239),(64,116,435,226),(65,131,436,241),(66,118,437,228),(67,133,438,243),(68,120,439,230),(69,135,440,245),(70,122,441,232),(71,137,442,247),(72,124,443,234),(73,139,444,249),(74,126,445,236),(75,113,446,251),(76,128,447,238),(77,115,448,225),(78,130,421,240),(79,117,422,227),(80,132,423,242),(81,119,424,229),(82,134,425,244),(83,121,426,231),(84,136,427,246),(141,194,342,369),(142,181,343,384),(143,196,344,371),(144,183,345,386),(145,170,346,373),(146,185,347,388),(147,172,348,375),(148,187,349,390),(149,174,350,377),(150,189,351,392),(151,176,352,379),(152,191,353,366),(153,178,354,381),(154,193,355,368),(155,180,356,383),(156,195,357,370),(157,182,358,385),(158,169,359,372),(159,184,360,387),(160,171,361,374),(161,186,362,389),(162,173,363,376),(163,188,364,391),(164,175,337,378),(165,190,338,365),(166,177,339,380),(167,192,340,367),(168,179,341,382)], [(1,278,395,321),(2,279,396,322),(3,280,397,323),(4,253,398,324),(5,254,399,325),(6,255,400,326),(7,256,401,327),(8,257,402,328),(9,258,403,329),(10,259,404,330),(11,260,405,331),(12,261,406,332),(13,262,407,333),(14,263,408,334),(15,264,409,335),(16,265,410,336),(17,266,411,309),(18,267,412,310),(19,268,413,311),(20,269,414,312),(21,270,415,313),(22,271,416,314),(23,272,417,315),(24,273,418,316),(25,274,419,317),(26,275,420,318),(27,276,393,319),(28,277,394,320),(29,214,300,97),(30,215,301,98),(31,216,302,99),(32,217,303,100),(33,218,304,101),(34,219,305,102),(35,220,306,103),(36,221,307,104),(37,222,308,105),(38,223,281,106),(39,224,282,107),(40,197,283,108),(41,198,284,109),(42,199,285,110),(43,200,286,111),(44,201,287,112),(45,202,288,85),(46,203,289,86),(47,204,290,87),(48,205,291,88),(49,206,292,89),(50,207,293,90),(51,208,294,91),(52,209,295,92),(53,210,296,93),(54,211,297,94),(55,212,298,95),(56,213,299,96),(57,146,428,347),(58,147,429,348),(59,148,430,349),(60,149,431,350),(61,150,432,351),(62,151,433,352),(63,152,434,353),(64,153,435,354),(65,154,436,355),(66,155,437,356),(67,156,438,357),(68,157,439,358),(69,158,440,359),(70,159,441,360),(71,160,442,361),(72,161,443,362),(73,162,444,363),(74,163,445,364),(75,164,446,337),(76,165,447,338),(77,166,448,339),(78,167,421,340),(79,168,422,341),(80,141,423,342),(81,142,424,343),(82,143,425,344),(83,144,426,345),(84,145,427,346),(113,378,251,175),(114,379,252,176),(115,380,225,177),(116,381,226,178),(117,382,227,179),(118,383,228,180),(119,384,229,181),(120,385,230,182),(121,386,231,183),(122,387,232,184),(123,388,233,185),(124,389,234,186),(125,390,235,187),(126,391,236,188),(127,392,237,189),(128,365,238,190),(129,366,239,191),(130,367,240,192),(131,368,241,193),(132,369,242,194),(133,370,243,195),(134,371,244,196),(135,372,245,169),(136,373,246,170),(137,374,247,171),(138,375,248,172),(139,376,249,173),(140,377,250,174)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 14 |
| 0 | 0 | 0 | 0 | 14 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 5 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 27 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 24 | 2 | 0 | 0 | 0 | 0 |
| 16 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 15 |
| 0 | 0 | 0 | 0 | 15 | 8 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,28,11,0,0,0,0,0,0,8,14,0,0,0,0,14,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,5,0,0,0,0,7,13,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[17,27,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[24,16,0,0,0,0,2,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,15,0,0,0,0,15,8] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | D14 | D14 | D14 | 2+ (1+4) | Q8×D7 | D4⋊6D14 |
| kernel | Dic14⋊9Q8 | C4×Dic14 | Dic7⋊3Q8 | C28⋊Q8 | Dic7⋊Q8 | C7×C4⋊Q8 | Dic14 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C14 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 8 | 3 | 3 | 12 | 6 | 1 | 12 | 6 |
In GAP, Magma, Sage, TeX
Dic_{14}\rtimes_9Q_8 % in TeX
G:=Group("Dic14:9Q8"); // GroupNames label
G:=SmallGroup(448,1175);
// by ID
G=gap.SmallGroup(448,1175);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,477,232,100,570,185,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=c^4=1,b^2=a^14,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^15,a*d=d*a,b*c=c*b,d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations
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