Dic14.2Q8 - GroupNames

G = Dic₁₄.₂Q₈ order 448 = 2⁶·7

2^nd non-split extension by Dic₁₄ of Q₈ acting via Q₈/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₁₄.₂Q₈, C₄.₆(Q₈×D₇), C₇⋊₅(Q₈.Q₈), C₄⋊C₄.₄₅D₁₄, (C₂×C₈).₂₆D₁₄, C₂.D₈.₆D₇, C₂₈.₁₉(C₂×Q₈), C₁₄.₂₇(C₄○D₈), C₄.₇₉(C₄○D₂₈), Dic₇⋊C₈.₁₁C₂, (C₂×Dic₇).₄₇D₄, C₁₄.Q₁₆.₉C₂, C₂².₂₂₆(D₄×D₇), C₄.Dic₁₄.₈C₂, C₂₈.₃Q₈.₈C₂, C₂₈.₁₇₁(C₄○D₄), C₂.₁₂(D₈⋊₃D₇), (C₂×C₅₆).₂₄₀C₂², (C₂×C₂₈).₂₉₃C₂³, Dic₇⋊₃Q₈.₉C₂, C₁₄.₄₀(C₂²⋊Q₈), C₂.₂₁(Q₁₆⋊D₇), C₂.₁₇(D₁₄⋊Q₈), C₂₈.₄₄D₄.₁₁C₂, C₁₄.₆₈(C₈.C₂²), C₄⋊Dic₇.₁₁₉C₂², (C₄×Dic₇).₃₆C₂², (C₂×Dic₁₄).₈₈C₂², (C₂×C₇⋊C₈).₆₇C₂², (C₇×C₂.D₈).₁₂C₂, (C₂×C₁₄).₂₉₈(C₂×D₄), (C₇×C₄⋊C₄).₈₆C₂², (C₂×C₄).₃₉₆(C₂²×D₇), SmallGroup(448,411)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — Dic₁₄.₂Q₈

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄×Dic₇ — Dic₇⋊₃Q₈ — Dic₁₄.₂Q₈

Lower central

C₇ — C₁₄ — C₂×C₂₈ — Dic₁₄.₂Q₈

Upper central

C₁ — C₂² — C₂×C₄ — C₂.D₈

Generators and relations for Dic₁₄.₂Q₈
G = < a,b,c,d | a²⁸=c⁴=1, b²=a¹⁴, d²=a¹⁴c², bab^-1=a^-1, cac^-1=a¹⁵, dad^-1=a¹³, cbc^-1=a⁷b, bd=db, dcd^-1=c^-1 >

Subgroups: 396 in 90 conjugacy classes, 39 normal (37 characteristic)
C₁, C₂, C₄, C₄, C₂², C₇, C₈, C₂×C₄, C₂×C₄, Q₈, C₁₄, C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×Q₈, Dic₇, C₂₈, C₂₈, C₂×C₁₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₄×Q₈, C₄².C₂, C₇⋊C₈, C₅₆, Dic₁₄, Dic₁₄, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, Q₈.Q₈, C₂×C₇⋊C₈, C₄×Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₄⋊Dic₇, C₇×C₄⋊C₄, C₂×C₅₆, C₂×Dic₁₄, C₄.Dic₁₄, C₁₄.Q₁₆, Dic₇⋊C₈, C₂₈.₄₄D₄, C₇×C₂.D₈, Dic₇⋊₃Q₈, C₂₈.₃Q₈, Dic₁₄.₂Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₇, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₄, C₂²⋊Q₈, C₄○D₈, C₈.C₂², C₂²×D₇, Q₈.Q₈, C₄○D₂₈, D₄×D₇, Q₈×D₇, D₁₄⋊Q₈, D₈⋊₃D₇, Q₁₆⋊D₇, Dic₁₄.₂Q₈

Smallest permutation representation of Dic₁₄.₂Q₈
►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)

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

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

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

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

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

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

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	···	28F	28G	···	28R	56A	···	56L
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	56	···	56
size	1	1	1	1	2	2	4	4	8	14	14	28	28	28	56	2	2	2	4	4	28	28	2	···	2	4	···	4	8	···	8	4	···	4

61 irreducible representations

dim	1	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	C₂	Q₈	D₄	D₇	C₄○D₄	D₁₄	D₁₄	C₄○D₈	C₄○D₂₈	C₈.C₂²	Q₈×D₇	D₄×D₇	D₈⋊₃D₇	Q₁₆⋊D₇
kernel	Dic₁₄.₂Q₈	C₄.Dic₁₄	C₁₄.Q₁₆	Dic₇⋊C₈	C₂₈.₄₄D₄	C₇×C₂.D₈	Dic₇⋊₃Q₈	C₂₈.₃Q₈	Dic₁₄	C₂×Dic₇	C₂.D₈	C₂₈	C₄⋊C₄	C₂×C₈	C₁₄	C₄	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	2	2	3	2	6	3	4	12	1	3	3	6	6

Matrix representation of Dic₁₄.₂Q₈ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	88	1	0	0
0	0	53	34	0	0
0	0	0	0	1	111
0	0	0	0	1	112

1	0	0	0	0	0
0	1	0	0	0	0
0	0	29	65	0	0
0	0	74	84	0	0
0	0	0	0	88	111
0	0	0	0	87	25

12	36	0	0	0	0
87	101	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	87	24
0	0	0	0	99	26

91	64	0	0	0	0
86	22	0	0	0	0
0	0	29	65	0	0
0	0	74	84	0	0
0	0	0	0	98	0
0	0	0	0	0	98

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,88,53,0,0,0,0,1,34,0,0,0,0,0,0,1,1,0,0,0,0,111,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,74,0,0,0,0,65,84,0,0,0,0,0,0,88,87,0,0,0,0,111,25],[12,87,0,0,0,0,36,101,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,87,99,0,0,0,0,24,26],[91,86,0,0,0,0,64,22,0,0,0,0,0,0,29,74,0,0,0,0,65,84,0,0,0,0,0,0,98,0,0,0,0,0,0,98] >;

Dic₁₄.₂Q₈ in GAP, Magma, Sage, TeX

{\rm Dic}_{14}._2Q_8

% in TeX

G:=Group("Dic14.2Q8");

// GroupNames label

G:=SmallGroup(448,411);

// by ID

G=gap.SmallGroup(448,411);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,1094,135,268,570,297,136,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^28=c^4=1,b^2=a^14,d^2=a^14*c^2,b*a*b^-1=a^-1,c*a*c^-1=a^15,d*a*d^-1=a^13,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

G = Dic14.2Q8 order 448 = 26·7

2nd non-split extension by Dic14 of Q8 acting via Q8/C2=C22

G = Dic₁₄.₂Q₈ order 448 = 2⁶·7

2^nd non-split extension by Dic₁₄ of Q₈ acting via Q₈/C₂=C₂²