Dic14:8Q8 - GroupNames

G = Dic₁₄⋊₈Q₈ order 448 = 2⁶·7

6^th semidirect product of Dic₁₄ and Q₈ acting via Q₈/C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₁₄⋊₈Q₈, C₄².₁₆₉D₁₄, C₁₄.₃₃2_-¹⁺⁴, C₄⋊Q₈.₁₄D₇, C₄.₁₇(Q₈×D₇), C₇⋊₅(Q₈⋊₃Q₈), C₂₈.₅₂(C₂×Q₈), C₄⋊C₄.₁₂₁D₁₄, (C₂×C₂₈).₉₈C₂³, (C₂×Q₈).₁₄₁D₁₄, (Q₈×Dic₇).₁₃C₂, Dic₇.₁₄(C₂×Q₈), Dic₇.Q₈.₄C₂, C₂₈.₁₃₅(C₄○D₄), C₄.₁₈(D₄⋊₂D₇), C₁₄.₄₄(C₂²×Q₈), (C₄×C₂₈).₂₀₆C₂², (C₂×C₁₄).₂₆₅C₂⁴, (C₄×Dic₁₄).₂₅C₂, Dic₇⋊Q₈.₈C₂, C₂₈.₃Q₈.₁₄C₂, Dic₇⋊₃Q₈.₁₂C₂, Dic₇⋊C₄.₅₇C₂², C₄⋊Dic₇.₃₈₂C₂², (Q₈×C₁₄).₁₃₂C₂², C₂².₂₈₆(C₂³×D₇), (C₂×Dic₇).₂₇₀C₂³, (C₄×Dic₇).₁₅₇C₂², C₂.₃₄(Q₈.₁₀D₁₄), (C₂×Dic₁₄).₃₀₂C₂², C₂.₂₇(C₂×Q₈×D₇), (C₇×C₄⋊Q₈).₁₄C₂, C₁₄.₉₉(C₂×C₄○D₄), C₂.₆₃(C₂×D₄⋊₂D₇), (C₂×C₄).₉₀(C₂²×D₇), (C₇×C₄⋊C₄).₂₀₈C₂², SmallGroup(448,1174)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — Dic₁₄⋊₈Q₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×Dic₇ — C₄×Dic₇ — Dic₇⋊₃Q₈ — Dic₁₄⋊₈Q₈

Lower central

C₇ — C₂×C₁₄ — Dic₁₄⋊₈Q₈

Upper central

C₁ — C₂² — C₄⋊Q₈

Generators and relations for Dic₁₄⋊₈Q₈
G = < a,b,c,d | a²⁸=c⁴=1, b²=a¹⁴, d²=c², bab^-1=a^-1, ac=ca, dad^-1=a¹⁵, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 684 in 200 conjugacy classes, 107 normal (27 characteristic)
C₁, C₂, C₄, C₄, C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₁₄, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×Q₈, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, C₂×C₁₄, C₄×Q₈, C₄².C₂, C₄⋊Q₈, C₄⋊Q₈, Dic₁₄, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, Q₈⋊₃Q₈, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₄⋊Dic₇, C₄×C₂₈, C₇×C₄⋊C₄, C₂×Dic₁₄, Q₈×C₁₄, C₄×Dic₁₄, Dic₇⋊₃Q₈, Dic₇.Q₈, C₂₈.₃Q₈, Dic₇⋊Q₈, Q₈×Dic₇, C₇×C₄⋊Q₈, Dic₁₄⋊₈Q₈
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, D₄⋊₂D₇, Q₈×D₇, C₂³×D₇, C₂×D₄⋊₂D₇, C₂×Q₈×D₇, Q₈.₁₀D₁₄, Dic₁₄⋊₈Q₈

Smallest permutation representation of Dic₁₄⋊₈Q₈
►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)

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

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

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

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

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

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

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	1	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	-	+		+	+	+	-	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	2_-¹⁺⁴	D₄⋊₂D₇	Q₈×D₇	Q₈.₁₀D₁₄
kernel	Dic₁₄⋊₈Q₈	C₄×Dic₁₄	Dic₇⋊₃Q₈	Dic₇.Q₈	C₂₈.₃Q₈	Dic₇⋊Q₈	Q₈×Dic₇	C₇×C₄⋊Q₈	Dic₁₄	C₄⋊Q₈	C₂₈	C₄²	C₄⋊C₄	C₂×Q₈	C₁₄	C₄	C₄	C₂
# reps	1	2	2	4	2	2	2	1	4	3	4	3	12	6	1	6	6	6

Matrix representation of Dic₁₄⋊₈Q₈ ►in GL₆(𝔽₂₉)

0	28	0	0	0	0
1	0	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	0	1
0	0	0	0	28	3

20	11	0	0	0	0
11	9	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	17	12
0	0	0	0	5	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	28	0	0	0
0	0	0	0	28	0
0	0	0	0	0	28

8	16	0	0	0	0
16	21	0	0	0	0
0	0	16	2	0	0
0	0	2	13	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(29))| [0,1,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,3],[20,11,0,0,0,0,11,9,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,5,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[8,16,0,0,0,0,16,21,0,0,0,0,0,0,16,2,0,0,0,0,2,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic₁₄⋊₈Q₈ in GAP, Magma, Sage, TeX

{\rm Dic}_{14}\rtimes_8Q_8

% in TeX

G:=Group("Dic14:8Q8");

// GroupNames label

G:=SmallGroup(448,1174);

// by ID

G=gap.SmallGroup(448,1174);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,219,268,1571,297,136,18822]);

// Polycyclic

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

// generators/relations

G = Dic14⋊8Q8 order 448 = 26·7

6th semidirect product of Dic14 and Q8 acting via Q8/C4=C2

G = Dic₁₄⋊₈Q₈ order 448 = 2⁶·7

6^th semidirect product of Dic₁₄ and Q₈ acting via Q₈/C₄=C₂