metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊7Q8, C42.146D14, C14.942- 1+4, C4.15(Q8×D7), C7⋊4(Q8⋊3Q8), C28⋊Q8.12C2, C28.47(C2×Q8), C4⋊C4.202D14, C42.C2.6D7, (C2×C28).84C23, Dic7.13(C2×Q8), Dic7.Q8.2C2, C14.39(C22×Q8), (C4×C28).190C22, (C2×C14).229C24, (C4×Dic14).24C2, C28.3Q8.12C2, Dic7.40(C4○D4), Dic7⋊3Q8.11C2, C4⋊Dic7.378C22, C22.250(C23×D7), Dic7⋊C4.143C22, (C2×Dic7).119C23, (C4×Dic7).137C22, C2.55(D4.10D14), (C2×Dic14).298C22, C2.22(C2×Q8×D7), C2.82(D7×C4○D4), C14.193(C2×C4○D4), (C2×C4).75(C22×D7), (C7×C42.C2).5C2, (C7×C4⋊C4).184C22, SmallGroup(448,1138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊7Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, ac=ca, dad-1=a13, cbc-1=dbd-1=a14b, dcd-1=c-1 >
Subgroups: 732 in 200 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×Q8, C42.C2, C42.C2, C4⋊Q8, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, Q8⋊3Q8, C4×Dic7, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C4×Dic14, Dic7⋊3Q8, Dic7⋊3Q8, C28⋊Q8, C28⋊Q8, Dic7.Q8, C28.3Q8, C7×C42.C2, Dic14⋊7Q8
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, C24, D14, C22×Q8, C2×C4○D4, 2- 1+4, C22×D7, Q8⋊3Q8, Q8×D7, C23×D7, C2×Q8×D7, D7×C4○D4, D4.10D14, Dic14⋊7Q8
►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 31 15 45)(2 30 16 44)(3 29 17 43)(4 56 18 42)(5 55 19 41)(6 54 20 40)(7 53 21 39)(8 52 22 38)(9 51 23 37)(10 50 24 36)(11 49 25 35)(12 48 26 34)(13 47 27 33)(14 46 28 32)(57 146 71 160)(58 145 72 159)(59 144 73 158)(60 143 74 157)(61 142 75 156)(62 141 76 155)(63 168 77 154)(64 167 78 153)(65 166 79 152)(66 165 80 151)(67 164 81 150)(68 163 82 149)(69 162 83 148)(70 161 84 147)(85 341 99 355)(86 340 100 354)(87 339 101 353)(88 338 102 352)(89 337 103 351)(90 364 104 350)(91 363 105 349)(92 362 106 348)(93 361 107 347)(94 360 108 346)(95 359 109 345)(96 358 110 344)(97 357 111 343)(98 356 112 342)(113 280 127 266)(114 279 128 265)(115 278 129 264)(116 277 130 263)(117 276 131 262)(118 275 132 261)(119 274 133 260)(120 273 134 259)(121 272 135 258)(122 271 136 257)(123 270 137 256)(124 269 138 255)(125 268 139 254)(126 267 140 253)(169 203 183 217)(170 202 184 216)(171 201 185 215)(172 200 186 214)(173 199 187 213)(174 198 188 212)(175 197 189 211)(176 224 190 210)(177 223 191 209)(178 222 192 208)(179 221 193 207)(180 220 194 206)(181 219 195 205)(182 218 196 204)(225 396 239 410)(226 395 240 409)(227 394 241 408)(228 393 242 407)(229 420 243 406)(230 419 244 405)(231 418 245 404)(232 417 246 403)(233 416 247 402)(234 415 248 401)(235 414 249 400)(236 413 250 399)(237 412 251 398)(238 411 252 397)(281 446 295 432)(282 445 296 431)(283 444 297 430)(284 443 298 429)(285 442 299 428)(286 441 300 427)(287 440 301 426)(288 439 302 425)(289 438 303 424)(290 437 304 423)(291 436 305 422)(292 435 306 421)(293 434 307 448)(294 433 308 447)(309 391 323 377)(310 390 324 376)(311 389 325 375)(312 388 326 374)(313 387 327 373)(314 386 328 372)(315 385 329 371)(316 384 330 370)(317 383 331 369)(318 382 332 368)(319 381 333 367)(320 380 334 366)(321 379 335 365)(322 378 336 392)
(1 404 197 71)(2 405 198 72)(3 406 199 73)(4 407 200 74)(5 408 201 75)(6 409 202 76)(7 410 203 77)(8 411 204 78)(9 412 205 79)(10 413 206 80)(11 414 207 81)(12 415 208 82)(13 416 209 83)(14 417 210 84)(15 418 211 57)(16 419 212 58)(17 420 213 59)(18 393 214 60)(19 394 215 61)(20 395 216 62)(21 396 217 63)(22 397 218 64)(23 398 219 65)(24 399 220 66)(25 400 221 67)(26 401 222 68)(27 402 223 69)(28 403 224 70)(29 243 187 144)(30 244 188 145)(31 245 189 146)(32 246 190 147)(33 247 191 148)(34 248 192 149)(35 249 193 150)(36 250 194 151)(37 251 195 152)(38 252 196 153)(39 225 169 154)(40 226 170 155)(41 227 171 156)(42 228 172 157)(43 229 173 158)(44 230 174 159)(45 231 175 160)(46 232 176 161)(47 233 177 162)(48 234 178 163)(49 235 179 164)(50 236 180 165)(51 237 181 166)(52 238 182 167)(53 239 183 168)(54 240 184 141)(55 241 185 142)(56 242 186 143)(85 281 318 264)(86 282 319 265)(87 283 320 266)(88 284 321 267)(89 285 322 268)(90 286 323 269)(91 287 324 270)(92 288 325 271)(93 289 326 272)(94 290 327 273)(95 291 328 274)(96 292 329 275)(97 293 330 276)(98 294 331 277)(99 295 332 278)(100 296 333 279)(101 297 334 280)(102 298 335 253)(103 299 336 254)(104 300 309 255)(105 301 310 256)(106 302 311 257)(107 303 312 258)(108 304 313 259)(109 305 314 260)(110 306 315 261)(111 307 316 262)(112 308 317 263)(113 353 444 366)(114 354 445 367)(115 355 446 368)(116 356 447 369)(117 357 448 370)(118 358 421 371)(119 359 422 372)(120 360 423 373)(121 361 424 374)(122 362 425 375)(123 363 426 376)(124 364 427 377)(125 337 428 378)(126 338 429 379)(127 339 430 380)(128 340 431 381)(129 341 432 382)(130 342 433 383)(131 343 434 384)(132 344 435 385)(133 345 436 386)(134 346 437 387)(135 347 438 388)(136 348 439 389)(137 349 440 390)(138 350 441 391)(139 351 442 392)(140 352 443 365)
(1 268 197 285)(2 253 198 298)(3 266 199 283)(4 279 200 296)(5 264 201 281)(6 277 202 294)(7 262 203 307)(8 275 204 292)(9 260 205 305)(10 273 206 290)(11 258 207 303)(12 271 208 288)(13 256 209 301)(14 269 210 286)(15 254 211 299)(16 267 212 284)(17 280 213 297)(18 265 214 282)(19 278 215 295)(20 263 216 308)(21 276 217 293)(22 261 218 306)(23 274 219 291)(24 259 220 304)(25 272 221 289)(26 257 222 302)(27 270 223 287)(28 255 224 300)(29 127 187 430)(30 140 188 443)(31 125 189 428)(32 138 190 441)(33 123 191 426)(34 136 192 439)(35 121 193 424)(36 134 194 437)(37 119 195 422)(38 132 196 435)(39 117 169 448)(40 130 170 433)(41 115 171 446)(42 128 172 431)(43 113 173 444)(44 126 174 429)(45 139 175 442)(46 124 176 427)(47 137 177 440)(48 122 178 425)(49 135 179 438)(50 120 180 423)(51 133 181 436)(52 118 182 421)(53 131 183 434)(54 116 184 447)(55 129 185 432)(56 114 186 445)(57 103 418 336)(58 88 419 321)(59 101 420 334)(60 86 393 319)(61 99 394 332)(62 112 395 317)(63 97 396 330)(64 110 397 315)(65 95 398 328)(66 108 399 313)(67 93 400 326)(68 106 401 311)(69 91 402 324)(70 104 403 309)(71 89 404 322)(72 102 405 335)(73 87 406 320)(74 100 407 333)(75 85 408 318)(76 98 409 331)(77 111 410 316)(78 96 411 329)(79 109 412 314)(80 94 413 327)(81 107 414 312)(82 92 415 325)(83 105 416 310)(84 90 417 323)(141 356 240 369)(142 341 241 382)(143 354 242 367)(144 339 243 380)(145 352 244 365)(146 337 245 378)(147 350 246 391)(148 363 247 376)(149 348 248 389)(150 361 249 374)(151 346 250 387)(152 359 251 372)(153 344 252 385)(154 357 225 370)(155 342 226 383)(156 355 227 368)(157 340 228 381)(158 353 229 366)(159 338 230 379)(160 351 231 392)(161 364 232 377)(162 349 233 390)(163 362 234 375)(164 347 235 388)(165 360 236 373)(166 345 237 386)(167 358 238 371)(168 343 239 384)
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), (1,31,15,45)(2,30,16,44)(3,29,17,43)(4,56,18,42)(5,55,19,41)(6,54,20,40)(7,53,21,39)(8,52,22,38)(9,51,23,37)(10,50,24,36)(11,49,25,35)(12,48,26,34)(13,47,27,33)(14,46,28,32)(57,146,71,160)(58,145,72,159)(59,144,73,158)(60,143,74,157)(61,142,75,156)(62,141,76,155)(63,168,77,154)(64,167,78,153)(65,166,79,152)(66,165,80,151)(67,164,81,150)(68,163,82,149)(69,162,83,148)(70,161,84,147)(85,341,99,355)(86,340,100,354)(87,339,101,353)(88,338,102,352)(89,337,103,351)(90,364,104,350)(91,363,105,349)(92,362,106,348)(93,361,107,347)(94,360,108,346)(95,359,109,345)(96,358,110,344)(97,357,111,343)(98,356,112,342)(113,280,127,266)(114,279,128,265)(115,278,129,264)(116,277,130,263)(117,276,131,262)(118,275,132,261)(119,274,133,260)(120,273,134,259)(121,272,135,258)(122,271,136,257)(123,270,137,256)(124,269,138,255)(125,268,139,254)(126,267,140,253)(169,203,183,217)(170,202,184,216)(171,201,185,215)(172,200,186,214)(173,199,187,213)(174,198,188,212)(175,197,189,211)(176,224,190,210)(177,223,191,209)(178,222,192,208)(179,221,193,207)(180,220,194,206)(181,219,195,205)(182,218,196,204)(225,396,239,410)(226,395,240,409)(227,394,241,408)(228,393,242,407)(229,420,243,406)(230,419,244,405)(231,418,245,404)(232,417,246,403)(233,416,247,402)(234,415,248,401)(235,414,249,400)(236,413,250,399)(237,412,251,398)(238,411,252,397)(281,446,295,432)(282,445,296,431)(283,444,297,430)(284,443,298,429)(285,442,299,428)(286,441,300,427)(287,440,301,426)(288,439,302,425)(289,438,303,424)(290,437,304,423)(291,436,305,422)(292,435,306,421)(293,434,307,448)(294,433,308,447)(309,391,323,377)(310,390,324,376)(311,389,325,375)(312,388,326,374)(313,387,327,373)(314,386,328,372)(315,385,329,371)(316,384,330,370)(317,383,331,369)(318,382,332,368)(319,381,333,367)(320,380,334,366)(321,379,335,365)(322,378,336,392), (1,404,197,71)(2,405,198,72)(3,406,199,73)(4,407,200,74)(5,408,201,75)(6,409,202,76)(7,410,203,77)(8,411,204,78)(9,412,205,79)(10,413,206,80)(11,414,207,81)(12,415,208,82)(13,416,209,83)(14,417,210,84)(15,418,211,57)(16,419,212,58)(17,420,213,59)(18,393,214,60)(19,394,215,61)(20,395,216,62)(21,396,217,63)(22,397,218,64)(23,398,219,65)(24,399,220,66)(25,400,221,67)(26,401,222,68)(27,402,223,69)(28,403,224,70)(29,243,187,144)(30,244,188,145)(31,245,189,146)(32,246,190,147)(33,247,191,148)(34,248,192,149)(35,249,193,150)(36,250,194,151)(37,251,195,152)(38,252,196,153)(39,225,169,154)(40,226,170,155)(41,227,171,156)(42,228,172,157)(43,229,173,158)(44,230,174,159)(45,231,175,160)(46,232,176,161)(47,233,177,162)(48,234,178,163)(49,235,179,164)(50,236,180,165)(51,237,181,166)(52,238,182,167)(53,239,183,168)(54,240,184,141)(55,241,185,142)(56,242,186,143)(85,281,318,264)(86,282,319,265)(87,283,320,266)(88,284,321,267)(89,285,322,268)(90,286,323,269)(91,287,324,270)(92,288,325,271)(93,289,326,272)(94,290,327,273)(95,291,328,274)(96,292,329,275)(97,293,330,276)(98,294,331,277)(99,295,332,278)(100,296,333,279)(101,297,334,280)(102,298,335,253)(103,299,336,254)(104,300,309,255)(105,301,310,256)(106,302,311,257)(107,303,312,258)(108,304,313,259)(109,305,314,260)(110,306,315,261)(111,307,316,262)(112,308,317,263)(113,353,444,366)(114,354,445,367)(115,355,446,368)(116,356,447,369)(117,357,448,370)(118,358,421,371)(119,359,422,372)(120,360,423,373)(121,361,424,374)(122,362,425,375)(123,363,426,376)(124,364,427,377)(125,337,428,378)(126,338,429,379)(127,339,430,380)(128,340,431,381)(129,341,432,382)(130,342,433,383)(131,343,434,384)(132,344,435,385)(133,345,436,386)(134,346,437,387)(135,347,438,388)(136,348,439,389)(137,349,440,390)(138,350,441,391)(139,351,442,392)(140,352,443,365), (1,268,197,285)(2,253,198,298)(3,266,199,283)(4,279,200,296)(5,264,201,281)(6,277,202,294)(7,262,203,307)(8,275,204,292)(9,260,205,305)(10,273,206,290)(11,258,207,303)(12,271,208,288)(13,256,209,301)(14,269,210,286)(15,254,211,299)(16,267,212,284)(17,280,213,297)(18,265,214,282)(19,278,215,295)(20,263,216,308)(21,276,217,293)(22,261,218,306)(23,274,219,291)(24,259,220,304)(25,272,221,289)(26,257,222,302)(27,270,223,287)(28,255,224,300)(29,127,187,430)(30,140,188,443)(31,125,189,428)(32,138,190,441)(33,123,191,426)(34,136,192,439)(35,121,193,424)(36,134,194,437)(37,119,195,422)(38,132,196,435)(39,117,169,448)(40,130,170,433)(41,115,171,446)(42,128,172,431)(43,113,173,444)(44,126,174,429)(45,139,175,442)(46,124,176,427)(47,137,177,440)(48,122,178,425)(49,135,179,438)(50,120,180,423)(51,133,181,436)(52,118,182,421)(53,131,183,434)(54,116,184,447)(55,129,185,432)(56,114,186,445)(57,103,418,336)(58,88,419,321)(59,101,420,334)(60,86,393,319)(61,99,394,332)(62,112,395,317)(63,97,396,330)(64,110,397,315)(65,95,398,328)(66,108,399,313)(67,93,400,326)(68,106,401,311)(69,91,402,324)(70,104,403,309)(71,89,404,322)(72,102,405,335)(73,87,406,320)(74,100,407,333)(75,85,408,318)(76,98,409,331)(77,111,410,316)(78,96,411,329)(79,109,412,314)(80,94,413,327)(81,107,414,312)(82,92,415,325)(83,105,416,310)(84,90,417,323)(141,356,240,369)(142,341,241,382)(143,354,242,367)(144,339,243,380)(145,352,244,365)(146,337,245,378)(147,350,246,391)(148,363,247,376)(149,348,248,389)(150,361,249,374)(151,346,250,387)(152,359,251,372)(153,344,252,385)(154,357,225,370)(155,342,226,383)(156,355,227,368)(157,340,228,381)(158,353,229,366)(159,338,230,379)(160,351,231,392)(161,364,232,377)(162,349,233,390)(163,362,234,375)(164,347,235,388)(165,360,236,373)(166,345,237,386)(167,358,238,371)(168,343,239,384)>;
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), (1,31,15,45)(2,30,16,44)(3,29,17,43)(4,56,18,42)(5,55,19,41)(6,54,20,40)(7,53,21,39)(8,52,22,38)(9,51,23,37)(10,50,24,36)(11,49,25,35)(12,48,26,34)(13,47,27,33)(14,46,28,32)(57,146,71,160)(58,145,72,159)(59,144,73,158)(60,143,74,157)(61,142,75,156)(62,141,76,155)(63,168,77,154)(64,167,78,153)(65,166,79,152)(66,165,80,151)(67,164,81,150)(68,163,82,149)(69,162,83,148)(70,161,84,147)(85,341,99,355)(86,340,100,354)(87,339,101,353)(88,338,102,352)(89,337,103,351)(90,364,104,350)(91,363,105,349)(92,362,106,348)(93,361,107,347)(94,360,108,346)(95,359,109,345)(96,358,110,344)(97,357,111,343)(98,356,112,342)(113,280,127,266)(114,279,128,265)(115,278,129,264)(116,277,130,263)(117,276,131,262)(118,275,132,261)(119,274,133,260)(120,273,134,259)(121,272,135,258)(122,271,136,257)(123,270,137,256)(124,269,138,255)(125,268,139,254)(126,267,140,253)(169,203,183,217)(170,202,184,216)(171,201,185,215)(172,200,186,214)(173,199,187,213)(174,198,188,212)(175,197,189,211)(176,224,190,210)(177,223,191,209)(178,222,192,208)(179,221,193,207)(180,220,194,206)(181,219,195,205)(182,218,196,204)(225,396,239,410)(226,395,240,409)(227,394,241,408)(228,393,242,407)(229,420,243,406)(230,419,244,405)(231,418,245,404)(232,417,246,403)(233,416,247,402)(234,415,248,401)(235,414,249,400)(236,413,250,399)(237,412,251,398)(238,411,252,397)(281,446,295,432)(282,445,296,431)(283,444,297,430)(284,443,298,429)(285,442,299,428)(286,441,300,427)(287,440,301,426)(288,439,302,425)(289,438,303,424)(290,437,304,423)(291,436,305,422)(292,435,306,421)(293,434,307,448)(294,433,308,447)(309,391,323,377)(310,390,324,376)(311,389,325,375)(312,388,326,374)(313,387,327,373)(314,386,328,372)(315,385,329,371)(316,384,330,370)(317,383,331,369)(318,382,332,368)(319,381,333,367)(320,380,334,366)(321,379,335,365)(322,378,336,392), (1,404,197,71)(2,405,198,72)(3,406,199,73)(4,407,200,74)(5,408,201,75)(6,409,202,76)(7,410,203,77)(8,411,204,78)(9,412,205,79)(10,413,206,80)(11,414,207,81)(12,415,208,82)(13,416,209,83)(14,417,210,84)(15,418,211,57)(16,419,212,58)(17,420,213,59)(18,393,214,60)(19,394,215,61)(20,395,216,62)(21,396,217,63)(22,397,218,64)(23,398,219,65)(24,399,220,66)(25,400,221,67)(26,401,222,68)(27,402,223,69)(28,403,224,70)(29,243,187,144)(30,244,188,145)(31,245,189,146)(32,246,190,147)(33,247,191,148)(34,248,192,149)(35,249,193,150)(36,250,194,151)(37,251,195,152)(38,252,196,153)(39,225,169,154)(40,226,170,155)(41,227,171,156)(42,228,172,157)(43,229,173,158)(44,230,174,159)(45,231,175,160)(46,232,176,161)(47,233,177,162)(48,234,178,163)(49,235,179,164)(50,236,180,165)(51,237,181,166)(52,238,182,167)(53,239,183,168)(54,240,184,141)(55,241,185,142)(56,242,186,143)(85,281,318,264)(86,282,319,265)(87,283,320,266)(88,284,321,267)(89,285,322,268)(90,286,323,269)(91,287,324,270)(92,288,325,271)(93,289,326,272)(94,290,327,273)(95,291,328,274)(96,292,329,275)(97,293,330,276)(98,294,331,277)(99,295,332,278)(100,296,333,279)(101,297,334,280)(102,298,335,253)(103,299,336,254)(104,300,309,255)(105,301,310,256)(106,302,311,257)(107,303,312,258)(108,304,313,259)(109,305,314,260)(110,306,315,261)(111,307,316,262)(112,308,317,263)(113,353,444,366)(114,354,445,367)(115,355,446,368)(116,356,447,369)(117,357,448,370)(118,358,421,371)(119,359,422,372)(120,360,423,373)(121,361,424,374)(122,362,425,375)(123,363,426,376)(124,364,427,377)(125,337,428,378)(126,338,429,379)(127,339,430,380)(128,340,431,381)(129,341,432,382)(130,342,433,383)(131,343,434,384)(132,344,435,385)(133,345,436,386)(134,346,437,387)(135,347,438,388)(136,348,439,389)(137,349,440,390)(138,350,441,391)(139,351,442,392)(140,352,443,365), (1,268,197,285)(2,253,198,298)(3,266,199,283)(4,279,200,296)(5,264,201,281)(6,277,202,294)(7,262,203,307)(8,275,204,292)(9,260,205,305)(10,273,206,290)(11,258,207,303)(12,271,208,288)(13,256,209,301)(14,269,210,286)(15,254,211,299)(16,267,212,284)(17,280,213,297)(18,265,214,282)(19,278,215,295)(20,263,216,308)(21,276,217,293)(22,261,218,306)(23,274,219,291)(24,259,220,304)(25,272,221,289)(26,257,222,302)(27,270,223,287)(28,255,224,300)(29,127,187,430)(30,140,188,443)(31,125,189,428)(32,138,190,441)(33,123,191,426)(34,136,192,439)(35,121,193,424)(36,134,194,437)(37,119,195,422)(38,132,196,435)(39,117,169,448)(40,130,170,433)(41,115,171,446)(42,128,172,431)(43,113,173,444)(44,126,174,429)(45,139,175,442)(46,124,176,427)(47,137,177,440)(48,122,178,425)(49,135,179,438)(50,120,180,423)(51,133,181,436)(52,118,182,421)(53,131,183,434)(54,116,184,447)(55,129,185,432)(56,114,186,445)(57,103,418,336)(58,88,419,321)(59,101,420,334)(60,86,393,319)(61,99,394,332)(62,112,395,317)(63,97,396,330)(64,110,397,315)(65,95,398,328)(66,108,399,313)(67,93,400,326)(68,106,401,311)(69,91,402,324)(70,104,403,309)(71,89,404,322)(72,102,405,335)(73,87,406,320)(74,100,407,333)(75,85,408,318)(76,98,409,331)(77,111,410,316)(78,96,411,329)(79,109,412,314)(80,94,413,327)(81,107,414,312)(82,92,415,325)(83,105,416,310)(84,90,417,323)(141,356,240,369)(142,341,241,382)(143,354,242,367)(144,339,243,380)(145,352,244,365)(146,337,245,378)(147,350,246,391)(148,363,247,376)(149,348,248,389)(150,361,249,374)(151,346,250,387)(152,359,251,372)(153,344,252,385)(154,357,225,370)(155,342,226,383)(156,355,227,368)(157,340,228,381)(158,353,229,366)(159,338,230,379)(160,351,231,392)(161,364,232,377)(162,349,233,390)(163,362,234,375)(164,347,235,388)(165,360,236,373)(166,345,237,386)(167,358,238,371)(168,343,239,384) );
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)], [(1,31,15,45),(2,30,16,44),(3,29,17,43),(4,56,18,42),(5,55,19,41),(6,54,20,40),(7,53,21,39),(8,52,22,38),(9,51,23,37),(10,50,24,36),(11,49,25,35),(12,48,26,34),(13,47,27,33),(14,46,28,32),(57,146,71,160),(58,145,72,159),(59,144,73,158),(60,143,74,157),(61,142,75,156),(62,141,76,155),(63,168,77,154),(64,167,78,153),(65,166,79,152),(66,165,80,151),(67,164,81,150),(68,163,82,149),(69,162,83,148),(70,161,84,147),(85,341,99,355),(86,340,100,354),(87,339,101,353),(88,338,102,352),(89,337,103,351),(90,364,104,350),(91,363,105,349),(92,362,106,348),(93,361,107,347),(94,360,108,346),(95,359,109,345),(96,358,110,344),(97,357,111,343),(98,356,112,342),(113,280,127,266),(114,279,128,265),(115,278,129,264),(116,277,130,263),(117,276,131,262),(118,275,132,261),(119,274,133,260),(120,273,134,259),(121,272,135,258),(122,271,136,257),(123,270,137,256),(124,269,138,255),(125,268,139,254),(126,267,140,253),(169,203,183,217),(170,202,184,216),(171,201,185,215),(172,200,186,214),(173,199,187,213),(174,198,188,212),(175,197,189,211),(176,224,190,210),(177,223,191,209),(178,222,192,208),(179,221,193,207),(180,220,194,206),(181,219,195,205),(182,218,196,204),(225,396,239,410),(226,395,240,409),(227,394,241,408),(228,393,242,407),(229,420,243,406),(230,419,244,405),(231,418,245,404),(232,417,246,403),(233,416,247,402),(234,415,248,401),(235,414,249,400),(236,413,250,399),(237,412,251,398),(238,411,252,397),(281,446,295,432),(282,445,296,431),(283,444,297,430),(284,443,298,429),(285,442,299,428),(286,441,300,427),(287,440,301,426),(288,439,302,425),(289,438,303,424),(290,437,304,423),(291,436,305,422),(292,435,306,421),(293,434,307,448),(294,433,308,447),(309,391,323,377),(310,390,324,376),(311,389,325,375),(312,388,326,374),(313,387,327,373),(314,386,328,372),(315,385,329,371),(316,384,330,370),(317,383,331,369),(318,382,332,368),(319,381,333,367),(320,380,334,366),(321,379,335,365),(322,378,336,392)], [(1,404,197,71),(2,405,198,72),(3,406,199,73),(4,407,200,74),(5,408,201,75),(6,409,202,76),(7,410,203,77),(8,411,204,78),(9,412,205,79),(10,413,206,80),(11,414,207,81),(12,415,208,82),(13,416,209,83),(14,417,210,84),(15,418,211,57),(16,419,212,58),(17,420,213,59),(18,393,214,60),(19,394,215,61),(20,395,216,62),(21,396,217,63),(22,397,218,64),(23,398,219,65),(24,399,220,66),(25,400,221,67),(26,401,222,68),(27,402,223,69),(28,403,224,70),(29,243,187,144),(30,244,188,145),(31,245,189,146),(32,246,190,147),(33,247,191,148),(34,248,192,149),(35,249,193,150),(36,250,194,151),(37,251,195,152),(38,252,196,153),(39,225,169,154),(40,226,170,155),(41,227,171,156),(42,228,172,157),(43,229,173,158),(44,230,174,159),(45,231,175,160),(46,232,176,161),(47,233,177,162),(48,234,178,163),(49,235,179,164),(50,236,180,165),(51,237,181,166),(52,238,182,167),(53,239,183,168),(54,240,184,141),(55,241,185,142),(56,242,186,143),(85,281,318,264),(86,282,319,265),(87,283,320,266),(88,284,321,267),(89,285,322,268),(90,286,323,269),(91,287,324,270),(92,288,325,271),(93,289,326,272),(94,290,327,273),(95,291,328,274),(96,292,329,275),(97,293,330,276),(98,294,331,277),(99,295,332,278),(100,296,333,279),(101,297,334,280),(102,298,335,253),(103,299,336,254),(104,300,309,255),(105,301,310,256),(106,302,311,257),(107,303,312,258),(108,304,313,259),(109,305,314,260),(110,306,315,261),(111,307,316,262),(112,308,317,263),(113,353,444,366),(114,354,445,367),(115,355,446,368),(116,356,447,369),(117,357,448,370),(118,358,421,371),(119,359,422,372),(120,360,423,373),(121,361,424,374),(122,362,425,375),(123,363,426,376),(124,364,427,377),(125,337,428,378),(126,338,429,379),(127,339,430,380),(128,340,431,381),(129,341,432,382),(130,342,433,383),(131,343,434,384),(132,344,435,385),(133,345,436,386),(134,346,437,387),(135,347,438,388),(136,348,439,389),(137,349,440,390),(138,350,441,391),(139,351,442,392),(140,352,443,365)], [(1,268,197,285),(2,253,198,298),(3,266,199,283),(4,279,200,296),(5,264,201,281),(6,277,202,294),(7,262,203,307),(8,275,204,292),(9,260,205,305),(10,273,206,290),(11,258,207,303),(12,271,208,288),(13,256,209,301),(14,269,210,286),(15,254,211,299),(16,267,212,284),(17,280,213,297),(18,265,214,282),(19,278,215,295),(20,263,216,308),(21,276,217,293),(22,261,218,306),(23,274,219,291),(24,259,220,304),(25,272,221,289),(26,257,222,302),(27,270,223,287),(28,255,224,300),(29,127,187,430),(30,140,188,443),(31,125,189,428),(32,138,190,441),(33,123,191,426),(34,136,192,439),(35,121,193,424),(36,134,194,437),(37,119,195,422),(38,132,196,435),(39,117,169,448),(40,130,170,433),(41,115,171,446),(42,128,172,431),(43,113,173,444),(44,126,174,429),(45,139,175,442),(46,124,176,427),(47,137,177,440),(48,122,178,425),(49,135,179,438),(50,120,180,423),(51,133,181,436),(52,118,182,421),(53,131,183,434),(54,116,184,447),(55,129,185,432),(56,114,186,445),(57,103,418,336),(58,88,419,321),(59,101,420,334),(60,86,393,319),(61,99,394,332),(62,112,395,317),(63,97,396,330),(64,110,397,315),(65,95,398,328),(66,108,399,313),(67,93,400,326),(68,106,401,311),(69,91,402,324),(70,104,403,309),(71,89,404,322),(72,102,405,335),(73,87,406,320),(74,100,407,333),(75,85,408,318),(76,98,409,331),(77,111,410,316),(78,96,411,329),(79,109,412,314),(80,94,413,327),(81,107,414,312),(82,92,415,325),(83,105,416,310),(84,90,417,323),(141,356,240,369),(142,341,241,382),(143,354,242,367),(144,339,243,380),(145,352,244,365),(146,337,245,378),(147,350,246,391),(148,363,247,376),(149,348,248,389),(150,361,249,374),(151,346,250,387),(152,359,251,372),(153,344,252,385),(154,357,225,370),(155,342,226,383),(156,355,227,368),(157,340,228,381),(158,353,229,366),(159,338,230,379),(160,351,231,392),(161,364,232,377),(162,349,233,390),(163,362,234,375),(164,347,235,388),(165,360,236,373),(166,345,237,386),(167,358,238,371),(168,343,239,384)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | 2- 1+4 | Q8×D7 | D7×C4○D4 | D4.10D14 |
| kernel | Dic14⋊7Q8 | C4×Dic14 | Dic7⋊3Q8 | C28⋊Q8 | Dic7.Q8 | C28.3Q8 | C7×C42.C2 | Dic14 | C42.C2 | Dic7 | C42 | C4⋊C4 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 4 | 3 | 4 | 1 | 1 | 4 | 3 | 4 | 3 | 18 | 1 | 6 | 6 | 6 |
Matrix representation of Dic14⋊7Q8 ►in GL6(𝔽29)
| 10 | 23 | 0 | 0 | 0 | 0 |
| 12 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 13 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 8 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 21 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 25 | 14 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 12 |
| 0 | 0 | 0 | 0 | 10 | 16 |
| 25 | 14 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 21 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 18 |
| 0 | 0 | 0 | 0 | 23 | 22 |
G:=sub<GL(6,GF(29))| [10,12,0,0,0,0,23,19,0,0,0,0,0,0,11,13,0,0,0,0,1,25,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[8,8,0,0,0,0,10,21,0,0,0,0,0,0,22,6,0,0,0,0,21,7,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[25,1,0,0,0,0,14,4,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,13,10,0,0,0,0,12,16],[25,1,0,0,0,0,14,4,0,0,0,0,0,0,22,6,0,0,0,0,21,7,0,0,0,0,0,0,7,23,0,0,0,0,18,22] >;
Dic14⋊7Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes_7Q_8 % in TeX
G:=Group("Dic14:7Q8"); // GroupNames label
G:=SmallGroup(448,1138);
// by ID
G=gap.SmallGroup(448,1138);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,219,268,1571,297,192,18822]);
// 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,a*c=c*a,d*a*d^-1=a^13,c*b*c^-1=d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations