metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14.4Q8, C42.67D14, C4.9(Q8×D7), C7⋊7(Q8.Q8), C4⋊C4.72D14, C28.31(C2×Q8), (C2×C28).273D4, C28⋊C8.20C2, C42.C2.2D7, C28.69(C4○D4), C14.107(C4○D8), (C2×C28).381C23, (C4×C28).111C22, C4.32(Q8⋊2D7), (C4×Dic14).16C2, C14.Q16.12C2, C28.Q8.12C2, C14.73(C22⋊Q8), C4.Dic14.13C2, C2.10(D14⋊3Q8), C4⋊Dic7.342C22, C2.26(D4.8D14), C2.20(D4.9D14), C14.121(C8.C22), (C2×Dic14).273C22, (C2×C14).512(C2×D4), (C2×C4).64(C7⋊D4), (C2×C7⋊C8).124C22, (C7×C42.C2).1C2, (C7×C4⋊C4).119C22, (C2×C4).479(C22×D7), C22.185(C2×C7⋊D4), SmallGroup(448,597)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14.4Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=dad-1=a-1, cac-1=a15, cbc-1=a21b, bd=db, dcd-1=a7c-1 >
Subgroups: 348 in 90 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, Q8.Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C28⋊C8, C28.Q8, C4.Dic14, C14.Q16, C4×Dic14, C7×C42.C2, Dic14.4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C4○D8, C8.C22, C7⋊D4, C22×D7, Q8.Q8, Q8×D7, Q8⋊2D7, C2×C7⋊D4, D14⋊3Q8, D4.8D14, D4.9D14, Dic14.4Q8
►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)
(1 368 15 382)(2 367 16 381)(3 366 17 380)(4 365 18 379)(5 392 19 378)(6 391 20 377)(7 390 21 376)(8 389 22 375)(9 388 23 374)(10 387 24 373)(11 386 25 372)(12 385 26 371)(13 384 27 370)(14 383 28 369)(29 81 43 67)(30 80 44 66)(31 79 45 65)(32 78 46 64)(33 77 47 63)(34 76 48 62)(35 75 49 61)(36 74 50 60)(37 73 51 59)(38 72 52 58)(39 71 53 57)(40 70 54 84)(41 69 55 83)(42 68 56 82)(85 439 99 425)(86 438 100 424)(87 437 101 423)(88 436 102 422)(89 435 103 421)(90 434 104 448)(91 433 105 447)(92 432 106 446)(93 431 107 445)(94 430 108 444)(95 429 109 443)(96 428 110 442)(97 427 111 441)(98 426 112 440)(113 190 127 176)(114 189 128 175)(115 188 129 174)(116 187 130 173)(117 186 131 172)(118 185 132 171)(119 184 133 170)(120 183 134 169)(121 182 135 196)(122 181 136 195)(123 180 137 194)(124 179 138 193)(125 178 139 192)(126 177 140 191)(141 362 155 348)(142 361 156 347)(143 360 157 346)(144 359 158 345)(145 358 159 344)(146 357 160 343)(147 356 161 342)(148 355 162 341)(149 354 163 340)(150 353 164 339)(151 352 165 338)(152 351 166 337)(153 350 167 364)(154 349 168 363)(197 261 211 275)(198 260 212 274)(199 259 213 273)(200 258 214 272)(201 257 215 271)(202 256 216 270)(203 255 217 269)(204 254 218 268)(205 253 219 267)(206 280 220 266)(207 279 221 265)(208 278 222 264)(209 277 223 263)(210 276 224 262)(225 333 239 319)(226 332 240 318)(227 331 241 317)(228 330 242 316)(229 329 243 315)(230 328 244 314)(231 327 245 313)(232 326 246 312)(233 325 247 311)(234 324 248 310)(235 323 249 309)(236 322 250 336)(237 321 251 335)(238 320 252 334)(281 397 295 411)(282 396 296 410)(283 395 297 409)(284 394 298 408)(285 393 299 407)(286 420 300 406)(287 419 301 405)(288 418 302 404)(289 417 303 403)(290 416 304 402)(291 415 305 401)(292 414 306 400)(293 413 307 399)(294 412 308 398)
(1 90 119 292)(2 105 120 307)(3 92 121 294)(4 107 122 281)(5 94 123 296)(6 109 124 283)(7 96 125 298)(8 111 126 285)(9 98 127 300)(10 85 128 287)(11 100 129 302)(12 87 130 289)(13 102 131 304)(14 89 132 291)(15 104 133 306)(16 91 134 293)(17 106 135 308)(18 93 136 295)(19 108 137 282)(20 95 138 297)(21 110 139 284)(22 97 140 299)(23 112 113 286)(24 99 114 301)(25 86 115 288)(26 101 116 303)(27 88 117 290)(28 103 118 305)(29 280 315 352)(30 267 316 339)(31 254 317 354)(32 269 318 341)(33 256 319 356)(34 271 320 343)(35 258 321 358)(36 273 322 345)(37 260 323 360)(38 275 324 347)(39 262 325 362)(40 277 326 349)(41 264 327 364)(42 279 328 351)(43 266 329 338)(44 253 330 353)(45 268 331 340)(46 255 332 355)(47 270 333 342)(48 257 334 357)(49 272 335 344)(50 259 336 359)(51 274 309 346)(52 261 310 361)(53 276 311 348)(54 263 312 363)(55 278 313 350)(56 265 314 337)(57 203 233 148)(58 218 234 163)(59 205 235 150)(60 220 236 165)(61 207 237 152)(62 222 238 167)(63 209 239 154)(64 224 240 141)(65 211 241 156)(66 198 242 143)(67 213 243 158)(68 200 244 145)(69 215 245 160)(70 202 246 147)(71 217 247 162)(72 204 248 149)(73 219 249 164)(74 206 250 151)(75 221 251 166)(76 208 252 153)(77 223 225 168)(78 210 226 155)(79 197 227 142)(80 212 228 157)(81 199 229 144)(82 214 230 159)(83 201 231 146)(84 216 232 161)(169 420 381 440)(170 407 382 427)(171 394 383 442)(172 409 384 429)(173 396 385 444)(174 411 386 431)(175 398 387 446)(176 413 388 433)(177 400 389 448)(178 415 390 435)(179 402 391 422)(180 417 392 437)(181 404 365 424)(182 419 366 439)(183 406 367 426)(184 393 368 441)(185 408 369 428)(186 395 370 443)(187 410 371 430)(188 397 372 445)(189 412 373 432)(190 399 374 447)(191 414 375 434)(192 401 376 421)(193 416 377 436)(194 403 378 423)(195 418 379 438)(196 405 380 425)
(1 310 119 52)(2 309 120 51)(3 336 121 50)(4 335 122 49)(5 334 123 48)(6 333 124 47)(7 332 125 46)(8 331 126 45)(9 330 127 44)(10 329 128 43)(11 328 129 42)(12 327 130 41)(13 326 131 40)(14 325 132 39)(15 324 133 38)(16 323 134 37)(17 322 135 36)(18 321 136 35)(19 320 137 34)(20 319 138 33)(21 318 139 32)(22 317 140 31)(23 316 113 30)(24 315 114 29)(25 314 115 56)(26 313 116 55)(27 312 117 54)(28 311 118 53)(57 369 233 185)(58 368 234 184)(59 367 235 183)(60 366 236 182)(61 365 237 181)(62 392 238 180)(63 391 239 179)(64 390 240 178)(65 389 241 177)(66 388 242 176)(67 387 243 175)(68 386 244 174)(69 385 245 173)(70 384 246 172)(71 383 247 171)(72 382 248 170)(73 381 249 169)(74 380 250 196)(75 379 251 195)(76 378 252 194)(77 377 225 193)(78 376 226 192)(79 375 227 191)(80 374 228 190)(81 373 229 189)(82 372 230 188)(83 371 231 187)(84 370 232 186)(85 259 287 359)(86 258 288 358)(87 257 289 357)(88 256 290 356)(89 255 291 355)(90 254 292 354)(91 253 293 353)(92 280 294 352)(93 279 295 351)(94 278 296 350)(95 277 297 349)(96 276 298 348)(97 275 299 347)(98 274 300 346)(99 273 301 345)(100 272 302 344)(101 271 303 343)(102 270 304 342)(103 269 305 341)(104 268 306 340)(105 267 307 339)(106 266 308 338)(107 265 281 337)(108 264 282 364)(109 263 283 363)(110 262 284 362)(111 261 285 361)(112 260 286 360)(141 428 224 408)(142 427 197 407)(143 426 198 406)(144 425 199 405)(145 424 200 404)(146 423 201 403)(147 422 202 402)(148 421 203 401)(149 448 204 400)(150 447 205 399)(151 446 206 398)(152 445 207 397)(153 444 208 396)(154 443 209 395)(155 442 210 394)(156 441 211 393)(157 440 212 420)(158 439 213 419)(159 438 214 418)(160 437 215 417)(161 436 216 416)(162 435 217 415)(163 434 218 414)(164 433 219 413)(165 432 220 412)(166 431 221 411)(167 430 222 410)(168 429 223 409)
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), (1,368,15,382)(2,367,16,381)(3,366,17,380)(4,365,18,379)(5,392,19,378)(6,391,20,377)(7,390,21,376)(8,389,22,375)(9,388,23,374)(10,387,24,373)(11,386,25,372)(12,385,26,371)(13,384,27,370)(14,383,28,369)(29,81,43,67)(30,80,44,66)(31,79,45,65)(32,78,46,64)(33,77,47,63)(34,76,48,62)(35,75,49,61)(36,74,50,60)(37,73,51,59)(38,72,52,58)(39,71,53,57)(40,70,54,84)(41,69,55,83)(42,68,56,82)(85,439,99,425)(86,438,100,424)(87,437,101,423)(88,436,102,422)(89,435,103,421)(90,434,104,448)(91,433,105,447)(92,432,106,446)(93,431,107,445)(94,430,108,444)(95,429,109,443)(96,428,110,442)(97,427,111,441)(98,426,112,440)(113,190,127,176)(114,189,128,175)(115,188,129,174)(116,187,130,173)(117,186,131,172)(118,185,132,171)(119,184,133,170)(120,183,134,169)(121,182,135,196)(122,181,136,195)(123,180,137,194)(124,179,138,193)(125,178,139,192)(126,177,140,191)(141,362,155,348)(142,361,156,347)(143,360,157,346)(144,359,158,345)(145,358,159,344)(146,357,160,343)(147,356,161,342)(148,355,162,341)(149,354,163,340)(150,353,164,339)(151,352,165,338)(152,351,166,337)(153,350,167,364)(154,349,168,363)(197,261,211,275)(198,260,212,274)(199,259,213,273)(200,258,214,272)(201,257,215,271)(202,256,216,270)(203,255,217,269)(204,254,218,268)(205,253,219,267)(206,280,220,266)(207,279,221,265)(208,278,222,264)(209,277,223,263)(210,276,224,262)(225,333,239,319)(226,332,240,318)(227,331,241,317)(228,330,242,316)(229,329,243,315)(230,328,244,314)(231,327,245,313)(232,326,246,312)(233,325,247,311)(234,324,248,310)(235,323,249,309)(236,322,250,336)(237,321,251,335)(238,320,252,334)(281,397,295,411)(282,396,296,410)(283,395,297,409)(284,394,298,408)(285,393,299,407)(286,420,300,406)(287,419,301,405)(288,418,302,404)(289,417,303,403)(290,416,304,402)(291,415,305,401)(292,414,306,400)(293,413,307,399)(294,412,308,398), (1,90,119,292)(2,105,120,307)(3,92,121,294)(4,107,122,281)(5,94,123,296)(6,109,124,283)(7,96,125,298)(8,111,126,285)(9,98,127,300)(10,85,128,287)(11,100,129,302)(12,87,130,289)(13,102,131,304)(14,89,132,291)(15,104,133,306)(16,91,134,293)(17,106,135,308)(18,93,136,295)(19,108,137,282)(20,95,138,297)(21,110,139,284)(22,97,140,299)(23,112,113,286)(24,99,114,301)(25,86,115,288)(26,101,116,303)(27,88,117,290)(28,103,118,305)(29,280,315,352)(30,267,316,339)(31,254,317,354)(32,269,318,341)(33,256,319,356)(34,271,320,343)(35,258,321,358)(36,273,322,345)(37,260,323,360)(38,275,324,347)(39,262,325,362)(40,277,326,349)(41,264,327,364)(42,279,328,351)(43,266,329,338)(44,253,330,353)(45,268,331,340)(46,255,332,355)(47,270,333,342)(48,257,334,357)(49,272,335,344)(50,259,336,359)(51,274,309,346)(52,261,310,361)(53,276,311,348)(54,263,312,363)(55,278,313,350)(56,265,314,337)(57,203,233,148)(58,218,234,163)(59,205,235,150)(60,220,236,165)(61,207,237,152)(62,222,238,167)(63,209,239,154)(64,224,240,141)(65,211,241,156)(66,198,242,143)(67,213,243,158)(68,200,244,145)(69,215,245,160)(70,202,246,147)(71,217,247,162)(72,204,248,149)(73,219,249,164)(74,206,250,151)(75,221,251,166)(76,208,252,153)(77,223,225,168)(78,210,226,155)(79,197,227,142)(80,212,228,157)(81,199,229,144)(82,214,230,159)(83,201,231,146)(84,216,232,161)(169,420,381,440)(170,407,382,427)(171,394,383,442)(172,409,384,429)(173,396,385,444)(174,411,386,431)(175,398,387,446)(176,413,388,433)(177,400,389,448)(178,415,390,435)(179,402,391,422)(180,417,392,437)(181,404,365,424)(182,419,366,439)(183,406,367,426)(184,393,368,441)(185,408,369,428)(186,395,370,443)(187,410,371,430)(188,397,372,445)(189,412,373,432)(190,399,374,447)(191,414,375,434)(192,401,376,421)(193,416,377,436)(194,403,378,423)(195,418,379,438)(196,405,380,425), (1,310,119,52)(2,309,120,51)(3,336,121,50)(4,335,122,49)(5,334,123,48)(6,333,124,47)(7,332,125,46)(8,331,126,45)(9,330,127,44)(10,329,128,43)(11,328,129,42)(12,327,130,41)(13,326,131,40)(14,325,132,39)(15,324,133,38)(16,323,134,37)(17,322,135,36)(18,321,136,35)(19,320,137,34)(20,319,138,33)(21,318,139,32)(22,317,140,31)(23,316,113,30)(24,315,114,29)(25,314,115,56)(26,313,116,55)(27,312,117,54)(28,311,118,53)(57,369,233,185)(58,368,234,184)(59,367,235,183)(60,366,236,182)(61,365,237,181)(62,392,238,180)(63,391,239,179)(64,390,240,178)(65,389,241,177)(66,388,242,176)(67,387,243,175)(68,386,244,174)(69,385,245,173)(70,384,246,172)(71,383,247,171)(72,382,248,170)(73,381,249,169)(74,380,250,196)(75,379,251,195)(76,378,252,194)(77,377,225,193)(78,376,226,192)(79,375,227,191)(80,374,228,190)(81,373,229,189)(82,372,230,188)(83,371,231,187)(84,370,232,186)(85,259,287,359)(86,258,288,358)(87,257,289,357)(88,256,290,356)(89,255,291,355)(90,254,292,354)(91,253,293,353)(92,280,294,352)(93,279,295,351)(94,278,296,350)(95,277,297,349)(96,276,298,348)(97,275,299,347)(98,274,300,346)(99,273,301,345)(100,272,302,344)(101,271,303,343)(102,270,304,342)(103,269,305,341)(104,268,306,340)(105,267,307,339)(106,266,308,338)(107,265,281,337)(108,264,282,364)(109,263,283,363)(110,262,284,362)(111,261,285,361)(112,260,286,360)(141,428,224,408)(142,427,197,407)(143,426,198,406)(144,425,199,405)(145,424,200,404)(146,423,201,403)(147,422,202,402)(148,421,203,401)(149,448,204,400)(150,447,205,399)(151,446,206,398)(152,445,207,397)(153,444,208,396)(154,443,209,395)(155,442,210,394)(156,441,211,393)(157,440,212,420)(158,439,213,419)(159,438,214,418)(160,437,215,417)(161,436,216,416)(162,435,217,415)(163,434,218,414)(164,433,219,413)(165,432,220,412)(166,431,221,411)(167,430,222,410)(168,429,223,409)>;
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), (1,368,15,382)(2,367,16,381)(3,366,17,380)(4,365,18,379)(5,392,19,378)(6,391,20,377)(7,390,21,376)(8,389,22,375)(9,388,23,374)(10,387,24,373)(11,386,25,372)(12,385,26,371)(13,384,27,370)(14,383,28,369)(29,81,43,67)(30,80,44,66)(31,79,45,65)(32,78,46,64)(33,77,47,63)(34,76,48,62)(35,75,49,61)(36,74,50,60)(37,73,51,59)(38,72,52,58)(39,71,53,57)(40,70,54,84)(41,69,55,83)(42,68,56,82)(85,439,99,425)(86,438,100,424)(87,437,101,423)(88,436,102,422)(89,435,103,421)(90,434,104,448)(91,433,105,447)(92,432,106,446)(93,431,107,445)(94,430,108,444)(95,429,109,443)(96,428,110,442)(97,427,111,441)(98,426,112,440)(113,190,127,176)(114,189,128,175)(115,188,129,174)(116,187,130,173)(117,186,131,172)(118,185,132,171)(119,184,133,170)(120,183,134,169)(121,182,135,196)(122,181,136,195)(123,180,137,194)(124,179,138,193)(125,178,139,192)(126,177,140,191)(141,362,155,348)(142,361,156,347)(143,360,157,346)(144,359,158,345)(145,358,159,344)(146,357,160,343)(147,356,161,342)(148,355,162,341)(149,354,163,340)(150,353,164,339)(151,352,165,338)(152,351,166,337)(153,350,167,364)(154,349,168,363)(197,261,211,275)(198,260,212,274)(199,259,213,273)(200,258,214,272)(201,257,215,271)(202,256,216,270)(203,255,217,269)(204,254,218,268)(205,253,219,267)(206,280,220,266)(207,279,221,265)(208,278,222,264)(209,277,223,263)(210,276,224,262)(225,333,239,319)(226,332,240,318)(227,331,241,317)(228,330,242,316)(229,329,243,315)(230,328,244,314)(231,327,245,313)(232,326,246,312)(233,325,247,311)(234,324,248,310)(235,323,249,309)(236,322,250,336)(237,321,251,335)(238,320,252,334)(281,397,295,411)(282,396,296,410)(283,395,297,409)(284,394,298,408)(285,393,299,407)(286,420,300,406)(287,419,301,405)(288,418,302,404)(289,417,303,403)(290,416,304,402)(291,415,305,401)(292,414,306,400)(293,413,307,399)(294,412,308,398), (1,90,119,292)(2,105,120,307)(3,92,121,294)(4,107,122,281)(5,94,123,296)(6,109,124,283)(7,96,125,298)(8,111,126,285)(9,98,127,300)(10,85,128,287)(11,100,129,302)(12,87,130,289)(13,102,131,304)(14,89,132,291)(15,104,133,306)(16,91,134,293)(17,106,135,308)(18,93,136,295)(19,108,137,282)(20,95,138,297)(21,110,139,284)(22,97,140,299)(23,112,113,286)(24,99,114,301)(25,86,115,288)(26,101,116,303)(27,88,117,290)(28,103,118,305)(29,280,315,352)(30,267,316,339)(31,254,317,354)(32,269,318,341)(33,256,319,356)(34,271,320,343)(35,258,321,358)(36,273,322,345)(37,260,323,360)(38,275,324,347)(39,262,325,362)(40,277,326,349)(41,264,327,364)(42,279,328,351)(43,266,329,338)(44,253,330,353)(45,268,331,340)(46,255,332,355)(47,270,333,342)(48,257,334,357)(49,272,335,344)(50,259,336,359)(51,274,309,346)(52,261,310,361)(53,276,311,348)(54,263,312,363)(55,278,313,350)(56,265,314,337)(57,203,233,148)(58,218,234,163)(59,205,235,150)(60,220,236,165)(61,207,237,152)(62,222,238,167)(63,209,239,154)(64,224,240,141)(65,211,241,156)(66,198,242,143)(67,213,243,158)(68,200,244,145)(69,215,245,160)(70,202,246,147)(71,217,247,162)(72,204,248,149)(73,219,249,164)(74,206,250,151)(75,221,251,166)(76,208,252,153)(77,223,225,168)(78,210,226,155)(79,197,227,142)(80,212,228,157)(81,199,229,144)(82,214,230,159)(83,201,231,146)(84,216,232,161)(169,420,381,440)(170,407,382,427)(171,394,383,442)(172,409,384,429)(173,396,385,444)(174,411,386,431)(175,398,387,446)(176,413,388,433)(177,400,389,448)(178,415,390,435)(179,402,391,422)(180,417,392,437)(181,404,365,424)(182,419,366,439)(183,406,367,426)(184,393,368,441)(185,408,369,428)(186,395,370,443)(187,410,371,430)(188,397,372,445)(189,412,373,432)(190,399,374,447)(191,414,375,434)(192,401,376,421)(193,416,377,436)(194,403,378,423)(195,418,379,438)(196,405,380,425), (1,310,119,52)(2,309,120,51)(3,336,121,50)(4,335,122,49)(5,334,123,48)(6,333,124,47)(7,332,125,46)(8,331,126,45)(9,330,127,44)(10,329,128,43)(11,328,129,42)(12,327,130,41)(13,326,131,40)(14,325,132,39)(15,324,133,38)(16,323,134,37)(17,322,135,36)(18,321,136,35)(19,320,137,34)(20,319,138,33)(21,318,139,32)(22,317,140,31)(23,316,113,30)(24,315,114,29)(25,314,115,56)(26,313,116,55)(27,312,117,54)(28,311,118,53)(57,369,233,185)(58,368,234,184)(59,367,235,183)(60,366,236,182)(61,365,237,181)(62,392,238,180)(63,391,239,179)(64,390,240,178)(65,389,241,177)(66,388,242,176)(67,387,243,175)(68,386,244,174)(69,385,245,173)(70,384,246,172)(71,383,247,171)(72,382,248,170)(73,381,249,169)(74,380,250,196)(75,379,251,195)(76,378,252,194)(77,377,225,193)(78,376,226,192)(79,375,227,191)(80,374,228,190)(81,373,229,189)(82,372,230,188)(83,371,231,187)(84,370,232,186)(85,259,287,359)(86,258,288,358)(87,257,289,357)(88,256,290,356)(89,255,291,355)(90,254,292,354)(91,253,293,353)(92,280,294,352)(93,279,295,351)(94,278,296,350)(95,277,297,349)(96,276,298,348)(97,275,299,347)(98,274,300,346)(99,273,301,345)(100,272,302,344)(101,271,303,343)(102,270,304,342)(103,269,305,341)(104,268,306,340)(105,267,307,339)(106,266,308,338)(107,265,281,337)(108,264,282,364)(109,263,283,363)(110,262,284,362)(111,261,285,361)(112,260,286,360)(141,428,224,408)(142,427,197,407)(143,426,198,406)(144,425,199,405)(145,424,200,404)(146,423,201,403)(147,422,202,402)(148,421,203,401)(149,448,204,400)(150,447,205,399)(151,446,206,398)(152,445,207,397)(153,444,208,396)(154,443,209,395)(155,442,210,394)(156,441,211,393)(157,440,212,420)(158,439,213,419)(159,438,214,418)(160,437,215,417)(161,436,216,416)(162,435,217,415)(163,434,218,414)(164,433,219,413)(165,432,220,412)(166,431,221,411)(167,430,222,410)(168,429,223,409) );
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)], [(1,368,15,382),(2,367,16,381),(3,366,17,380),(4,365,18,379),(5,392,19,378),(6,391,20,377),(7,390,21,376),(8,389,22,375),(9,388,23,374),(10,387,24,373),(11,386,25,372),(12,385,26,371),(13,384,27,370),(14,383,28,369),(29,81,43,67),(30,80,44,66),(31,79,45,65),(32,78,46,64),(33,77,47,63),(34,76,48,62),(35,75,49,61),(36,74,50,60),(37,73,51,59),(38,72,52,58),(39,71,53,57),(40,70,54,84),(41,69,55,83),(42,68,56,82),(85,439,99,425),(86,438,100,424),(87,437,101,423),(88,436,102,422),(89,435,103,421),(90,434,104,448),(91,433,105,447),(92,432,106,446),(93,431,107,445),(94,430,108,444),(95,429,109,443),(96,428,110,442),(97,427,111,441),(98,426,112,440),(113,190,127,176),(114,189,128,175),(115,188,129,174),(116,187,130,173),(117,186,131,172),(118,185,132,171),(119,184,133,170),(120,183,134,169),(121,182,135,196),(122,181,136,195),(123,180,137,194),(124,179,138,193),(125,178,139,192),(126,177,140,191),(141,362,155,348),(142,361,156,347),(143,360,157,346),(144,359,158,345),(145,358,159,344),(146,357,160,343),(147,356,161,342),(148,355,162,341),(149,354,163,340),(150,353,164,339),(151,352,165,338),(152,351,166,337),(153,350,167,364),(154,349,168,363),(197,261,211,275),(198,260,212,274),(199,259,213,273),(200,258,214,272),(201,257,215,271),(202,256,216,270),(203,255,217,269),(204,254,218,268),(205,253,219,267),(206,280,220,266),(207,279,221,265),(208,278,222,264),(209,277,223,263),(210,276,224,262),(225,333,239,319),(226,332,240,318),(227,331,241,317),(228,330,242,316),(229,329,243,315),(230,328,244,314),(231,327,245,313),(232,326,246,312),(233,325,247,311),(234,324,248,310),(235,323,249,309),(236,322,250,336),(237,321,251,335),(238,320,252,334),(281,397,295,411),(282,396,296,410),(283,395,297,409),(284,394,298,408),(285,393,299,407),(286,420,300,406),(287,419,301,405),(288,418,302,404),(289,417,303,403),(290,416,304,402),(291,415,305,401),(292,414,306,400),(293,413,307,399),(294,412,308,398)], [(1,90,119,292),(2,105,120,307),(3,92,121,294),(4,107,122,281),(5,94,123,296),(6,109,124,283),(7,96,125,298),(8,111,126,285),(9,98,127,300),(10,85,128,287),(11,100,129,302),(12,87,130,289),(13,102,131,304),(14,89,132,291),(15,104,133,306),(16,91,134,293),(17,106,135,308),(18,93,136,295),(19,108,137,282),(20,95,138,297),(21,110,139,284),(22,97,140,299),(23,112,113,286),(24,99,114,301),(25,86,115,288),(26,101,116,303),(27,88,117,290),(28,103,118,305),(29,280,315,352),(30,267,316,339),(31,254,317,354),(32,269,318,341),(33,256,319,356),(34,271,320,343),(35,258,321,358),(36,273,322,345),(37,260,323,360),(38,275,324,347),(39,262,325,362),(40,277,326,349),(41,264,327,364),(42,279,328,351),(43,266,329,338),(44,253,330,353),(45,268,331,340),(46,255,332,355),(47,270,333,342),(48,257,334,357),(49,272,335,344),(50,259,336,359),(51,274,309,346),(52,261,310,361),(53,276,311,348),(54,263,312,363),(55,278,313,350),(56,265,314,337),(57,203,233,148),(58,218,234,163),(59,205,235,150),(60,220,236,165),(61,207,237,152),(62,222,238,167),(63,209,239,154),(64,224,240,141),(65,211,241,156),(66,198,242,143),(67,213,243,158),(68,200,244,145),(69,215,245,160),(70,202,246,147),(71,217,247,162),(72,204,248,149),(73,219,249,164),(74,206,250,151),(75,221,251,166),(76,208,252,153),(77,223,225,168),(78,210,226,155),(79,197,227,142),(80,212,228,157),(81,199,229,144),(82,214,230,159),(83,201,231,146),(84,216,232,161),(169,420,381,440),(170,407,382,427),(171,394,383,442),(172,409,384,429),(173,396,385,444),(174,411,386,431),(175,398,387,446),(176,413,388,433),(177,400,389,448),(178,415,390,435),(179,402,391,422),(180,417,392,437),(181,404,365,424),(182,419,366,439),(183,406,367,426),(184,393,368,441),(185,408,369,428),(186,395,370,443),(187,410,371,430),(188,397,372,445),(189,412,373,432),(190,399,374,447),(191,414,375,434),(192,401,376,421),(193,416,377,436),(194,403,378,423),(195,418,379,438),(196,405,380,425)], [(1,310,119,52),(2,309,120,51),(3,336,121,50),(4,335,122,49),(5,334,123,48),(6,333,124,47),(7,332,125,46),(8,331,126,45),(9,330,127,44),(10,329,128,43),(11,328,129,42),(12,327,130,41),(13,326,131,40),(14,325,132,39),(15,324,133,38),(16,323,134,37),(17,322,135,36),(18,321,136,35),(19,320,137,34),(20,319,138,33),(21,318,139,32),(22,317,140,31),(23,316,113,30),(24,315,114,29),(25,314,115,56),(26,313,116,55),(27,312,117,54),(28,311,118,53),(57,369,233,185),(58,368,234,184),(59,367,235,183),(60,366,236,182),(61,365,237,181),(62,392,238,180),(63,391,239,179),(64,390,240,178),(65,389,241,177),(66,388,242,176),(67,387,243,175),(68,386,244,174),(69,385,245,173),(70,384,246,172),(71,383,247,171),(72,382,248,170),(73,381,249,169),(74,380,250,196),(75,379,251,195),(76,378,252,194),(77,377,225,193),(78,376,226,192),(79,375,227,191),(80,374,228,190),(81,373,229,189),(82,372,230,188),(83,371,231,187),(84,370,232,186),(85,259,287,359),(86,258,288,358),(87,257,289,357),(88,256,290,356),(89,255,291,355),(90,254,292,354),(91,253,293,353),(92,280,294,352),(93,279,295,351),(94,278,296,350),(95,277,297,349),(96,276,298,348),(97,275,299,347),(98,274,300,346),(99,273,301,345),(100,272,302,344),(101,271,303,343),(102,270,304,342),(103,269,305,341),(104,268,306,340),(105,267,307,339),(106,266,308,338),(107,265,281,337),(108,264,282,364),(109,263,283,363),(110,262,284,362),(111,261,285,361),(112,260,286,360),(141,428,224,408),(142,427,197,407),(143,426,198,406),(144,425,199,405),(145,424,200,404),(146,423,201,403),(147,422,202,402),(148,421,203,401),(149,448,204,400),(150,447,205,399),(151,446,206,398),(152,445,207,397),(153,444,208,396),(154,443,209,395),(155,442,210,394),(156,441,211,393),(157,440,212,420),(158,439,213,419),(159,438,214,418),(160,437,215,417),(161,436,216,416),(162,435,217,415),(163,434,218,414),(164,433,219,413),(165,432,220,412),(166,431,221,411),(167,430,222,410),(168,429,223,409)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | C4○D4 | D14 | D14 | C4○D8 | C7⋊D4 | C8.C22 | Q8×D7 | Q8⋊2D7 | D4.8D14 | D4.9D14 |
| kernel | Dic14.4Q8 | C28⋊C8 | C28.Q8 | C4.Dic14 | C14.Q16 | C4×Dic14 | C7×C42.C2 | Dic14 | C2×C28 | C42.C2 | C28 | C42 | C4⋊C4 | C14 | C2×C4 | C14 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 6 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Dic14.4Q8 ►in GL6(𝔽113)
| 112 | 1 | 0 | 0 | 0 | 0 |
| 87 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 40 | 106 | 0 | 0 | 0 | 0 |
| 67 | 73 | 0 | 0 | 0 | 0 |
| 0 | 0 | 82 | 30 | 0 | 0 |
| 0 | 0 | 81 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 106 |
| 0 | 0 | 0 | 0 | 106 | 96 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 82 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 86 | 17 |
| 0 | 0 | 0 | 0 | 17 | 27 |
| 40 | 106 | 0 | 0 | 0 | 0 |
| 67 | 73 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 84 | 105 |
| 0 | 0 | 0 | 0 | 105 | 29 |
G:=sub<GL(6,GF(113))| [112,87,0,0,0,0,1,25,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[40,67,0,0,0,0,106,73,0,0,0,0,0,0,82,81,0,0,0,0,30,31,0,0,0,0,0,0,17,106,0,0,0,0,106,96],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,98,82,0,0,0,0,0,15,0,0,0,0,0,0,86,17,0,0,0,0,17,27],[40,67,0,0,0,0,106,73,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,84,105,0,0,0,0,105,29] >;
Dic14.4Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}._4Q_8 % in TeX
G:=Group("Dic14.4Q8"); // GroupNames label
G:=SmallGroup(448,597);
// by ID
G=gap.SmallGroup(448,597);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,254,219,100,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=c^4=1,b^2=a^14,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^21*b,b*d=d*b,d*c*d^-1=a^7*c^-1>;
// generators/relations