Dic14.Q8 - GroupNames

G = Dic₁₄.Q₈ order 448 = 2⁶·7

1^st non-split extension by Dic₁₄ of Q₈ acting via Q₈/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₁₄.₁Q₈, C₄.₂(Q₈×D₇), C₇⋊₄(Q₈.Q₈), C₄⋊C₄.₃₄D₁₄, C₄.Q₈.₆D₇, C₂₈.₁₁(C₂×Q₈), (C₂×C₈).₁₃₆D₁₄, C₁₄.₅₂(C₄○D₈), C₄.₇₁(C₄○D₂₈), Dic₇⋊C₈.₁₄C₂, (C₂×Dic₇).₃₉D₄, C₂₈.Q₈.₅C₂, C₁₄.Q₁₆.₆C₂, C₂².₂₁₂(D₄×D₇), C₂₈.₃Q₈.₅C₂, C₂₈.₁₆₇(C₄○D₄), (C₂×C₅₆).₂₈₃C₂², (C₂×C₂₈).₂₇₃C₂³, Dic₇⋊₃Q₈.₆C₂, C₁₄.₃₆(C₂²⋊Q₈), C₂.₁₃(D₁₄⋊Q₈), C₂₈.₄₄D₄.₁₄C₂, C₂.₂₂(SD₁₆⋊D₇), C₁₄.₄₀(C₈.C₂²), C₄⋊Dic₇.₁₀₅C₂², (C₄×Dic₇).₃₀C₂², C₂.₂₀(SD₁₆⋊₃D₇), (C₂×Dic₁₄).₈₁C₂², (C₂×C₇⋊C₈).₅₄C₂², (C₇×C₄.Q₈).₁₀C₂, (C₂×C₁₄).₂₇₈(C₂×D₄), (C₇×C₄⋊C₄).₆₆C₂², (C₂×C₄).₃₇₆(C₂²×D₇), SmallGroup(448,391)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — Dic₁₄.Q₈

Chief series

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

Lower central

C₇ — C₁₄ — C₂×C₂₈ — Dic₁₄.Q₈

Upper central

C₁ — C₂² — C₂×C₄ — C₄.Q₈

Generators and relations for Dic₁₄.Q₈
G = < a,b,c,d | a²⁸=c⁴=1, b²=a¹⁴, d²=c², bab^-1=a^-1, cac^-1=a¹⁵, dad^-1=a¹³, cbc^-1=a⁷b, bd=db, dcd^-1=a¹⁴c^-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, C₂₈.Q₈, C₁₄.Q₁₆, Dic₇⋊C₈, C₂₈.₄₄D₄, C₇×C₄.Q₈, Dic₇⋊₃Q₈, C₂₈.₃Q₈, Dic₁₄.Q₈
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, SD₁₆⋊D₇, SD₁₆⋊₃D₇, Dic₁₄.Q₈

Smallest permutation representation of Dic₁₄.Q₈
►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)

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

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

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

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

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

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

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	SD₁₆⋊D₇	SD₁₆⋊₃D₇
kernel	Dic₁₄.Q₈	C₂₈.Q₈	C₁₄.Q₁₆	Dic₇⋊C₈	C₂₈.₄₄D₄	C₇×C₄.Q₈	Dic₇⋊₃Q₈	C₂₈.₃Q₈	Dic₁₄	C₂×Dic₇	C₄.Q₈	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.Q₈ ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	112	1	0	0
0	0	102	10	0	0
0	0	0	0	112	2
0	0	0	0	112	1

12	14	0	0	0	0
14	101	0	0	0	0
0	0	0	80	0	0
0	0	89	0	0	0
0	0	0	0	87	26
0	0	0	0	100	26

0	1	0	0	0	0
112	0	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	15	83
0	0	0	0	0	98

67	97	0	0	0	0
97	46	0	0	0	0
0	0	0	80	0	0
0	0	89	0	0	0
0	0	0	0	15	0
0	0	0	0	0	15

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,102,0,0,0,0,1,10,0,0,0,0,0,0,112,112,0,0,0,0,2,1],[12,14,0,0,0,0,14,101,0,0,0,0,0,0,0,89,0,0,0,0,80,0,0,0,0,0,0,0,87,100,0,0,0,0,26,26],[0,112,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,0,0,0,0,83,98],[67,97,0,0,0,0,97,46,0,0,0,0,0,0,0,89,0,0,0,0,80,0,0,0,0,0,0,0,15,0,0,0,0,0,0,15] >;

Dic₁₄.Q₈ in GAP, Magma, Sage, TeX

{\rm Dic}_{14}.Q_8

% in TeX

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

// GroupNames label

G:=SmallGroup(448,391);

// by ID

G=gap.SmallGroup(448,391);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,1094,135,100,570,297,136,18822]);

// 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,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=a^14*c^-1>;

// generators/relations

G = Dic14.Q8 order 448 = 26·7

1st non-split extension by Dic14 of Q8 acting via Q8/C2=C22

G = Dic₁₄.Q₈ order 448 = 2⁶·7

1^st non-split extension by Dic₁₄ of Q₈ acting via Q₈/C₂=C₂²