metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊2C8, C28.25Q16, C28.40SD16, C28.4M4(2), C42.193D14, C7⋊2(Q8⋊C8), C4⋊C8.5D7, C4.2(C8×D7), C28.4(C2×C8), C14.11C4≀C2, C2.8(D14⋊C8), (C2×C28).226D4, (C2×C4).110D28, C4.2(C8⋊D7), C4⋊Dic7.15C4, C14.6(C22⋊C8), C4.15(D4.D7), (C4×C28).42C22, (C4×Dic14).6C2, C4.13(C7⋊Q16), C2.2(D28⋊4C4), C14.2(Q8⋊C4), (C2×Dic14).11C4, C2.1(C14.Q16), C22.35(D14⋊C4), (C4×C7⋊C8).3C2, (C7×C4⋊C8).5C2, (C2×C4).66(C4×D7), (C2×C28).49(C2×C4), (C2×C4).266(C7⋊D4), (C2×C14).46(C22⋊C4), SmallGroup(448,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊2C8
G = < a,b,c | a28=c8=1, b2=a14, bab-1=a-1, cac-1=a15, cbc-1=a21b >
Subgroups: 292 in 70 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C4×C8, C4⋊C8, C4×Q8, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×C28, Q8⋊C8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C4×C7⋊C8, C7×C4⋊C8, C4×Dic14, Dic14⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), SD16, Q16, D14, C22⋊C8, Q8⋊C4, C4≀C2, C4×D7, D28, C7⋊D4, Q8⋊C8, C8×D7, C8⋊D7, D14⋊C4, D4.D7, C7⋊Q16, C14.Q16, D14⋊C8, D28⋊4C4, Dic14⋊2C8
►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)
(1 168 15 154)(2 167 16 153)(3 166 17 152)(4 165 18 151)(5 164 19 150)(6 163 20 149)(7 162 21 148)(8 161 22 147)(9 160 23 146)(10 159 24 145)(11 158 25 144)(12 157 26 143)(13 156 27 142)(14 155 28 141)(29 260 43 274)(30 259 44 273)(31 258 45 272)(32 257 46 271)(33 256 47 270)(34 255 48 269)(35 254 49 268)(36 253 50 267)(37 280 51 266)(38 279 52 265)(39 278 53 264)(40 277 54 263)(41 276 55 262)(42 275 56 261)(57 402 71 416)(58 401 72 415)(59 400 73 414)(60 399 74 413)(61 398 75 412)(62 397 76 411)(63 396 77 410)(64 395 78 409)(65 394 79 408)(66 393 80 407)(67 420 81 406)(68 419 82 405)(69 418 83 404)(70 417 84 403)(85 388 99 374)(86 387 100 373)(87 386 101 372)(88 385 102 371)(89 384 103 370)(90 383 104 369)(91 382 105 368)(92 381 106 367)(93 380 107 366)(94 379 108 365)(95 378 109 392)(96 377 110 391)(97 376 111 390)(98 375 112 389)(113 178 127 192)(114 177 128 191)(115 176 129 190)(116 175 130 189)(117 174 131 188)(118 173 132 187)(119 172 133 186)(120 171 134 185)(121 170 135 184)(122 169 136 183)(123 196 137 182)(124 195 138 181)(125 194 139 180)(126 193 140 179)(197 433 211 447)(198 432 212 446)(199 431 213 445)(200 430 214 444)(201 429 215 443)(202 428 216 442)(203 427 217 441)(204 426 218 440)(205 425 219 439)(206 424 220 438)(207 423 221 437)(208 422 222 436)(209 421 223 435)(210 448 224 434)(225 284 239 298)(226 283 240 297)(227 282 241 296)(228 281 242 295)(229 308 243 294)(230 307 244 293)(231 306 245 292)(232 305 246 291)(233 304 247 290)(234 303 248 289)(235 302 249 288)(236 301 250 287)(237 300 251 286)(238 299 252 285)(309 364 323 350)(310 363 324 349)(311 362 325 348)(312 361 326 347)(313 360 327 346)(314 359 328 345)(315 358 329 344)(316 357 330 343)(317 356 331 342)(318 355 332 341)(319 354 333 340)(320 353 334 339)(321 352 335 338)(322 351 336 337)
(1 358 420 381 210 173 303 30)(2 345 393 368 211 188 304 45)(3 360 394 383 212 175 305 32)(4 347 395 370 213 190 306 47)(5 362 396 385 214 177 307 34)(6 349 397 372 215 192 308 49)(7 364 398 387 216 179 281 36)(8 351 399 374 217 194 282 51)(9 338 400 389 218 181 283 38)(10 353 401 376 219 196 284 53)(11 340 402 391 220 183 285 40)(12 355 403 378 221 170 286 55)(13 342 404 365 222 185 287 42)(14 357 405 380 223 172 288 29)(15 344 406 367 224 187 289 44)(16 359 407 382 197 174 290 31)(17 346 408 369 198 189 291 46)(18 361 409 384 199 176 292 33)(19 348 410 371 200 191 293 48)(20 363 411 386 201 178 294 35)(21 350 412 373 202 193 295 50)(22 337 413 388 203 180 296 37)(23 352 414 375 204 195 297 52)(24 339 415 390 205 182 298 39)(25 354 416 377 206 169 299 54)(26 341 417 392 207 184 300 41)(27 356 418 379 208 171 301 56)(28 343 419 366 209 186 302 43)(57 89 424 115 252 270 144 312)(58 104 425 130 225 257 145 327)(59 91 426 117 226 272 146 314)(60 106 427 132 227 259 147 329)(61 93 428 119 228 274 148 316)(62 108 429 134 229 261 149 331)(63 95 430 121 230 276 150 318)(64 110 431 136 231 263 151 333)(65 97 432 123 232 278 152 320)(66 112 433 138 233 265 153 335)(67 99 434 125 234 280 154 322)(68 86 435 140 235 267 155 309)(69 101 436 127 236 254 156 324)(70 88 437 114 237 269 157 311)(71 103 438 129 238 256 158 326)(72 90 439 116 239 271 159 313)(73 105 440 131 240 258 160 328)(74 92 441 118 241 273 161 315)(75 107 442 133 242 260 162 330)(76 94 443 120 243 275 163 317)(77 109 444 135 244 262 164 332)(78 96 445 122 245 277 165 319)(79 111 446 137 246 264 166 334)(80 98 447 124 247 279 167 321)(81 85 448 139 248 266 168 336)(82 100 421 126 249 253 141 323)(83 87 422 113 250 268 142 310)(84 102 423 128 251 255 143 325)
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), (1,168,15,154)(2,167,16,153)(3,166,17,152)(4,165,18,151)(5,164,19,150)(6,163,20,149)(7,162,21,148)(8,161,22,147)(9,160,23,146)(10,159,24,145)(11,158,25,144)(12,157,26,143)(13,156,27,142)(14,155,28,141)(29,260,43,274)(30,259,44,273)(31,258,45,272)(32,257,46,271)(33,256,47,270)(34,255,48,269)(35,254,49,268)(36,253,50,267)(37,280,51,266)(38,279,52,265)(39,278,53,264)(40,277,54,263)(41,276,55,262)(42,275,56,261)(57,402,71,416)(58,401,72,415)(59,400,73,414)(60,399,74,413)(61,398,75,412)(62,397,76,411)(63,396,77,410)(64,395,78,409)(65,394,79,408)(66,393,80,407)(67,420,81,406)(68,419,82,405)(69,418,83,404)(70,417,84,403)(85,388,99,374)(86,387,100,373)(87,386,101,372)(88,385,102,371)(89,384,103,370)(90,383,104,369)(91,382,105,368)(92,381,106,367)(93,380,107,366)(94,379,108,365)(95,378,109,392)(96,377,110,391)(97,376,111,390)(98,375,112,389)(113,178,127,192)(114,177,128,191)(115,176,129,190)(116,175,130,189)(117,174,131,188)(118,173,132,187)(119,172,133,186)(120,171,134,185)(121,170,135,184)(122,169,136,183)(123,196,137,182)(124,195,138,181)(125,194,139,180)(126,193,140,179)(197,433,211,447)(198,432,212,446)(199,431,213,445)(200,430,214,444)(201,429,215,443)(202,428,216,442)(203,427,217,441)(204,426,218,440)(205,425,219,439)(206,424,220,438)(207,423,221,437)(208,422,222,436)(209,421,223,435)(210,448,224,434)(225,284,239,298)(226,283,240,297)(227,282,241,296)(228,281,242,295)(229,308,243,294)(230,307,244,293)(231,306,245,292)(232,305,246,291)(233,304,247,290)(234,303,248,289)(235,302,249,288)(236,301,250,287)(237,300,251,286)(238,299,252,285)(309,364,323,350)(310,363,324,349)(311,362,325,348)(312,361,326,347)(313,360,327,346)(314,359,328,345)(315,358,329,344)(316,357,330,343)(317,356,331,342)(318,355,332,341)(319,354,333,340)(320,353,334,339)(321,352,335,338)(322,351,336,337), (1,358,420,381,210,173,303,30)(2,345,393,368,211,188,304,45)(3,360,394,383,212,175,305,32)(4,347,395,370,213,190,306,47)(5,362,396,385,214,177,307,34)(6,349,397,372,215,192,308,49)(7,364,398,387,216,179,281,36)(8,351,399,374,217,194,282,51)(9,338,400,389,218,181,283,38)(10,353,401,376,219,196,284,53)(11,340,402,391,220,183,285,40)(12,355,403,378,221,170,286,55)(13,342,404,365,222,185,287,42)(14,357,405,380,223,172,288,29)(15,344,406,367,224,187,289,44)(16,359,407,382,197,174,290,31)(17,346,408,369,198,189,291,46)(18,361,409,384,199,176,292,33)(19,348,410,371,200,191,293,48)(20,363,411,386,201,178,294,35)(21,350,412,373,202,193,295,50)(22,337,413,388,203,180,296,37)(23,352,414,375,204,195,297,52)(24,339,415,390,205,182,298,39)(25,354,416,377,206,169,299,54)(26,341,417,392,207,184,300,41)(27,356,418,379,208,171,301,56)(28,343,419,366,209,186,302,43)(57,89,424,115,252,270,144,312)(58,104,425,130,225,257,145,327)(59,91,426,117,226,272,146,314)(60,106,427,132,227,259,147,329)(61,93,428,119,228,274,148,316)(62,108,429,134,229,261,149,331)(63,95,430,121,230,276,150,318)(64,110,431,136,231,263,151,333)(65,97,432,123,232,278,152,320)(66,112,433,138,233,265,153,335)(67,99,434,125,234,280,154,322)(68,86,435,140,235,267,155,309)(69,101,436,127,236,254,156,324)(70,88,437,114,237,269,157,311)(71,103,438,129,238,256,158,326)(72,90,439,116,239,271,159,313)(73,105,440,131,240,258,160,328)(74,92,441,118,241,273,161,315)(75,107,442,133,242,260,162,330)(76,94,443,120,243,275,163,317)(77,109,444,135,244,262,164,332)(78,96,445,122,245,277,165,319)(79,111,446,137,246,264,166,334)(80,98,447,124,247,279,167,321)(81,85,448,139,248,266,168,336)(82,100,421,126,249,253,141,323)(83,87,422,113,250,268,142,310)(84,102,423,128,251,255,143,325)>;
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), (1,168,15,154)(2,167,16,153)(3,166,17,152)(4,165,18,151)(5,164,19,150)(6,163,20,149)(7,162,21,148)(8,161,22,147)(9,160,23,146)(10,159,24,145)(11,158,25,144)(12,157,26,143)(13,156,27,142)(14,155,28,141)(29,260,43,274)(30,259,44,273)(31,258,45,272)(32,257,46,271)(33,256,47,270)(34,255,48,269)(35,254,49,268)(36,253,50,267)(37,280,51,266)(38,279,52,265)(39,278,53,264)(40,277,54,263)(41,276,55,262)(42,275,56,261)(57,402,71,416)(58,401,72,415)(59,400,73,414)(60,399,74,413)(61,398,75,412)(62,397,76,411)(63,396,77,410)(64,395,78,409)(65,394,79,408)(66,393,80,407)(67,420,81,406)(68,419,82,405)(69,418,83,404)(70,417,84,403)(85,388,99,374)(86,387,100,373)(87,386,101,372)(88,385,102,371)(89,384,103,370)(90,383,104,369)(91,382,105,368)(92,381,106,367)(93,380,107,366)(94,379,108,365)(95,378,109,392)(96,377,110,391)(97,376,111,390)(98,375,112,389)(113,178,127,192)(114,177,128,191)(115,176,129,190)(116,175,130,189)(117,174,131,188)(118,173,132,187)(119,172,133,186)(120,171,134,185)(121,170,135,184)(122,169,136,183)(123,196,137,182)(124,195,138,181)(125,194,139,180)(126,193,140,179)(197,433,211,447)(198,432,212,446)(199,431,213,445)(200,430,214,444)(201,429,215,443)(202,428,216,442)(203,427,217,441)(204,426,218,440)(205,425,219,439)(206,424,220,438)(207,423,221,437)(208,422,222,436)(209,421,223,435)(210,448,224,434)(225,284,239,298)(226,283,240,297)(227,282,241,296)(228,281,242,295)(229,308,243,294)(230,307,244,293)(231,306,245,292)(232,305,246,291)(233,304,247,290)(234,303,248,289)(235,302,249,288)(236,301,250,287)(237,300,251,286)(238,299,252,285)(309,364,323,350)(310,363,324,349)(311,362,325,348)(312,361,326,347)(313,360,327,346)(314,359,328,345)(315,358,329,344)(316,357,330,343)(317,356,331,342)(318,355,332,341)(319,354,333,340)(320,353,334,339)(321,352,335,338)(322,351,336,337), (1,358,420,381,210,173,303,30)(2,345,393,368,211,188,304,45)(3,360,394,383,212,175,305,32)(4,347,395,370,213,190,306,47)(5,362,396,385,214,177,307,34)(6,349,397,372,215,192,308,49)(7,364,398,387,216,179,281,36)(8,351,399,374,217,194,282,51)(9,338,400,389,218,181,283,38)(10,353,401,376,219,196,284,53)(11,340,402,391,220,183,285,40)(12,355,403,378,221,170,286,55)(13,342,404,365,222,185,287,42)(14,357,405,380,223,172,288,29)(15,344,406,367,224,187,289,44)(16,359,407,382,197,174,290,31)(17,346,408,369,198,189,291,46)(18,361,409,384,199,176,292,33)(19,348,410,371,200,191,293,48)(20,363,411,386,201,178,294,35)(21,350,412,373,202,193,295,50)(22,337,413,388,203,180,296,37)(23,352,414,375,204,195,297,52)(24,339,415,390,205,182,298,39)(25,354,416,377,206,169,299,54)(26,341,417,392,207,184,300,41)(27,356,418,379,208,171,301,56)(28,343,419,366,209,186,302,43)(57,89,424,115,252,270,144,312)(58,104,425,130,225,257,145,327)(59,91,426,117,226,272,146,314)(60,106,427,132,227,259,147,329)(61,93,428,119,228,274,148,316)(62,108,429,134,229,261,149,331)(63,95,430,121,230,276,150,318)(64,110,431,136,231,263,151,333)(65,97,432,123,232,278,152,320)(66,112,433,138,233,265,153,335)(67,99,434,125,234,280,154,322)(68,86,435,140,235,267,155,309)(69,101,436,127,236,254,156,324)(70,88,437,114,237,269,157,311)(71,103,438,129,238,256,158,326)(72,90,439,116,239,271,159,313)(73,105,440,131,240,258,160,328)(74,92,441,118,241,273,161,315)(75,107,442,133,242,260,162,330)(76,94,443,120,243,275,163,317)(77,109,444,135,244,262,164,332)(78,96,445,122,245,277,165,319)(79,111,446,137,246,264,166,334)(80,98,447,124,247,279,167,321)(81,85,448,139,248,266,168,336)(82,100,421,126,249,253,141,323)(83,87,422,113,250,268,142,310)(84,102,423,128,251,255,143,325) );
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)], [(1,168,15,154),(2,167,16,153),(3,166,17,152),(4,165,18,151),(5,164,19,150),(6,163,20,149),(7,162,21,148),(8,161,22,147),(9,160,23,146),(10,159,24,145),(11,158,25,144),(12,157,26,143),(13,156,27,142),(14,155,28,141),(29,260,43,274),(30,259,44,273),(31,258,45,272),(32,257,46,271),(33,256,47,270),(34,255,48,269),(35,254,49,268),(36,253,50,267),(37,280,51,266),(38,279,52,265),(39,278,53,264),(40,277,54,263),(41,276,55,262),(42,275,56,261),(57,402,71,416),(58,401,72,415),(59,400,73,414),(60,399,74,413),(61,398,75,412),(62,397,76,411),(63,396,77,410),(64,395,78,409),(65,394,79,408),(66,393,80,407),(67,420,81,406),(68,419,82,405),(69,418,83,404),(70,417,84,403),(85,388,99,374),(86,387,100,373),(87,386,101,372),(88,385,102,371),(89,384,103,370),(90,383,104,369),(91,382,105,368),(92,381,106,367),(93,380,107,366),(94,379,108,365),(95,378,109,392),(96,377,110,391),(97,376,111,390),(98,375,112,389),(113,178,127,192),(114,177,128,191),(115,176,129,190),(116,175,130,189),(117,174,131,188),(118,173,132,187),(119,172,133,186),(120,171,134,185),(121,170,135,184),(122,169,136,183),(123,196,137,182),(124,195,138,181),(125,194,139,180),(126,193,140,179),(197,433,211,447),(198,432,212,446),(199,431,213,445),(200,430,214,444),(201,429,215,443),(202,428,216,442),(203,427,217,441),(204,426,218,440),(205,425,219,439),(206,424,220,438),(207,423,221,437),(208,422,222,436),(209,421,223,435),(210,448,224,434),(225,284,239,298),(226,283,240,297),(227,282,241,296),(228,281,242,295),(229,308,243,294),(230,307,244,293),(231,306,245,292),(232,305,246,291),(233,304,247,290),(234,303,248,289),(235,302,249,288),(236,301,250,287),(237,300,251,286),(238,299,252,285),(309,364,323,350),(310,363,324,349),(311,362,325,348),(312,361,326,347),(313,360,327,346),(314,359,328,345),(315,358,329,344),(316,357,330,343),(317,356,331,342),(318,355,332,341),(319,354,333,340),(320,353,334,339),(321,352,335,338),(322,351,336,337)], [(1,358,420,381,210,173,303,30),(2,345,393,368,211,188,304,45),(3,360,394,383,212,175,305,32),(4,347,395,370,213,190,306,47),(5,362,396,385,214,177,307,34),(6,349,397,372,215,192,308,49),(7,364,398,387,216,179,281,36),(8,351,399,374,217,194,282,51),(9,338,400,389,218,181,283,38),(10,353,401,376,219,196,284,53),(11,340,402,391,220,183,285,40),(12,355,403,378,221,170,286,55),(13,342,404,365,222,185,287,42),(14,357,405,380,223,172,288,29),(15,344,406,367,224,187,289,44),(16,359,407,382,197,174,290,31),(17,346,408,369,198,189,291,46),(18,361,409,384,199,176,292,33),(19,348,410,371,200,191,293,48),(20,363,411,386,201,178,294,35),(21,350,412,373,202,193,295,50),(22,337,413,388,203,180,296,37),(23,352,414,375,204,195,297,52),(24,339,415,390,205,182,298,39),(25,354,416,377,206,169,299,54),(26,341,417,392,207,184,300,41),(27,356,418,379,208,171,301,56),(28,343,419,366,209,186,302,43),(57,89,424,115,252,270,144,312),(58,104,425,130,225,257,145,327),(59,91,426,117,226,272,146,314),(60,106,427,132,227,259,147,329),(61,93,428,119,228,274,148,316),(62,108,429,134,229,261,149,331),(63,95,430,121,230,276,150,318),(64,110,431,136,231,263,151,333),(65,97,432,123,232,278,152,320),(66,112,433,138,233,265,153,335),(67,99,434,125,234,280,154,322),(68,86,435,140,235,267,155,309),(69,101,436,127,236,254,156,324),(70,88,437,114,237,269,157,311),(71,103,438,129,238,256,158,326),(72,90,439,116,239,271,159,313),(73,105,440,131,240,258,160,328),(74,92,441,118,241,273,161,315),(75,107,442,133,242,260,162,330),(76,94,443,120,243,275,163,317),(77,109,444,135,244,262,164,332),(78,96,445,122,245,277,165,319),(79,111,446,137,246,264,166,334),(80,98,447,124,247,279,167,321),(81,85,448,139,248,266,168,336),(82,100,421,126,249,253,141,323),(83,87,422,113,250,268,142,310),(84,102,423,128,251,255,143,325)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D7 | M4(2) | SD16 | Q16 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | C8×D7 | C8⋊D7 | D4.D7 | C7⋊Q16 | D28⋊4C4 |
| kernel | Dic14⋊2C8 | C4×C7⋊C8 | C7×C4⋊C8 | C4×Dic14 | C4⋊Dic7 | C2×Dic14 | Dic14 | C2×C28 | C4⋊C8 | C28 | C28 | C28 | C42 | C14 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 3 | 3 | 6 |
Matrix representation of Dic14⋊2C8 ►in GL4(𝔽113) generated by
| 34 | 1 | 0 | 0 |
| 111 | 103 | 0 | 0 |
| 0 | 0 | 1 | 22 |
| 0 | 0 | 41 | 112 |
| 33 | 46 | 0 | 0 |
| 109 | 80 | 0 | 0 |
| 0 | 0 | 110 | 85 |
| 0 | 0 | 73 | 3 |
| 44 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 79 | 76 |
| 0 | 0 | 108 | 34 |
G:=sub<GL(4,GF(113))| [34,111,0,0,1,103,0,0,0,0,1,41,0,0,22,112],[33,109,0,0,46,80,0,0,0,0,110,73,0,0,85,3],[44,0,0,0,0,44,0,0,0,0,79,108,0,0,76,34] >;
Dic14⋊2C8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes_2C_8 % in TeX
G:=Group("Dic14:2C8"); // GroupNames label
G:=SmallGroup(448,41);
// by ID
G=gap.SmallGroup(448,41);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,100,1123,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=c^8=1,b^2=a^14,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^21*b>;
// generators/relations