Copied to
clipboard

## G = Dic14.3Q8order 448 = 26·7

### 1st non-split extension by Dic14 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — Dic14.3Q8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4⋊Dic7 — C4×Dic14 — Dic14.3Q8
 Lower central C7 — C14 — C2×C28 — Dic14.3Q8
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for Dic14.3Q8
G = < a,b,c,d | a28=1, b2=c4=a14, d2=a7c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a7b, bd=db, dcd-1=c3 >

Subgroups: 420 in 90 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C56, Dic14, Dic14, C2×Dic7, C2×C28, Q8.Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28.44D4, C8⋊Dic7, C561C4, C7×C4⋊C8, C4×Dic14, C28.6Q8, Dic14.3Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C4○D8, C8.C22, D28, C22×D7, Q8.Q8, C2×D28, D42D7, Q8×D7, D142Q8, D567C2, C8.D14, Dic14.3Q8

Smallest permutation representation of Dic14.3Q8
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 75 15 61)(2 74 16 60)(3 73 17 59)(4 72 18 58)(5 71 19 57)(6 70 20 84)(7 69 21 83)(8 68 22 82)(9 67 23 81)(10 66 24 80)(11 65 25 79)(12 64 26 78)(13 63 27 77)(14 62 28 76)(29 202 43 216)(30 201 44 215)(31 200 45 214)(32 199 46 213)(33 198 47 212)(34 197 48 211)(35 224 49 210)(36 223 50 209)(37 222 51 208)(38 221 52 207)(39 220 53 206)(40 219 54 205)(41 218 55 204)(42 217 56 203)(85 268 99 254)(86 267 100 253)(87 266 101 280)(88 265 102 279)(89 264 103 278)(90 263 104 277)(91 262 105 276)(92 261 106 275)(93 260 107 274)(94 259 108 273)(95 258 109 272)(96 257 110 271)(97 256 111 270)(98 255 112 269)(113 323 127 309)(114 322 128 336)(115 321 129 335)(116 320 130 334)(117 319 131 333)(118 318 132 332)(119 317 133 331)(120 316 134 330)(121 315 135 329)(122 314 136 328)(123 313 137 327)(124 312 138 326)(125 311 139 325)(126 310 140 324)(141 340 155 354)(142 339 156 353)(143 338 157 352)(144 337 158 351)(145 364 159 350)(146 363 160 349)(147 362 161 348)(148 361 162 347)(149 360 163 346)(150 359 164 345)(151 358 165 344)(152 357 166 343)(153 356 167 342)(154 355 168 341)(169 410 183 396)(170 409 184 395)(171 408 185 394)(172 407 186 393)(173 406 187 420)(174 405 188 419)(175 404 189 418)(176 403 190 417)(177 402 191 416)(178 401 192 415)(179 400 193 414)(180 399 194 413)(181 398 195 412)(182 397 196 411)(225 378 239 392)(226 377 240 391)(227 376 241 390)(228 375 242 389)(229 374 243 388)(230 373 244 387)(231 372 245 386)(232 371 246 385)(233 370 247 384)(234 369 248 383)(235 368 249 382)(236 367 250 381)(237 366 251 380)(238 365 252 379)(281 447 295 433)(282 446 296 432)(283 445 297 431)(284 444 298 430)(285 443 299 429)(286 442 300 428)(287 441 301 427)(288 440 302 426)(289 439 303 425)(290 438 304 424)(291 437 305 423)(292 436 306 422)(293 435 307 421)(294 434 308 448)
(1 434 114 409 15 448 128 395)(2 435 115 410 16 421 129 396)(3 436 116 411 17 422 130 397)(4 437 117 412 18 423 131 398)(5 438 118 413 19 424 132 399)(6 439 119 414 20 425 133 400)(7 440 120 415 21 426 134 401)(8 441 121 416 22 427 135 402)(9 442 122 417 23 428 136 403)(10 443 123 418 24 429 137 404)(11 444 124 419 25 430 138 405)(12 445 125 420 26 431 139 406)(13 446 126 393 27 432 140 407)(14 447 127 394 28 433 113 408)(29 390 348 254 43 376 362 268)(30 391 349 255 44 377 363 269)(31 392 350 256 45 378 364 270)(32 365 351 257 46 379 337 271)(33 366 352 258 47 380 338 272)(34 367 353 259 48 381 339 273)(35 368 354 260 49 382 340 274)(36 369 355 261 50 383 341 275)(37 370 356 262 51 384 342 276)(38 371 357 263 52 385 343 277)(39 372 358 264 53 386 344 278)(40 373 359 265 54 387 345 279)(41 374 360 266 55 388 346 280)(42 375 361 267 56 389 347 253)(57 297 318 187 71 283 332 173)(58 298 319 188 72 284 333 174)(59 299 320 189 73 285 334 175)(60 300 321 190 74 286 335 176)(61 301 322 191 75 287 336 177)(62 302 323 192 76 288 309 178)(63 303 324 193 77 289 310 179)(64 304 325 194 78 290 311 180)(65 305 326 195 79 291 312 181)(66 306 327 196 80 292 313 182)(67 307 328 169 81 293 314 183)(68 308 329 170 82 294 315 184)(69 281 330 171 83 295 316 185)(70 282 331 172 84 296 317 186)(85 223 227 154 99 209 241 168)(86 224 228 155 100 210 242 141)(87 197 229 156 101 211 243 142)(88 198 230 157 102 212 244 143)(89 199 231 158 103 213 245 144)(90 200 232 159 104 214 246 145)(91 201 233 160 105 215 247 146)(92 202 234 161 106 216 248 147)(93 203 235 162 107 217 249 148)(94 204 236 163 108 218 250 149)(95 205 237 164 109 219 251 150)(96 206 238 165 110 220 252 151)(97 207 239 166 111 221 225 152)(98 208 240 167 112 222 226 153)
(1 91 121 240)(2 90 122 239)(3 89 123 238)(4 88 124 237)(5 87 125 236)(6 86 126 235)(7 85 127 234)(8 112 128 233)(9 111 129 232)(10 110 130 231)(11 109 131 230)(12 108 132 229)(13 107 133 228)(14 106 134 227)(15 105 135 226)(16 104 136 225)(17 103 137 252)(18 102 138 251)(19 101 139 250)(20 100 140 249)(21 99 113 248)(22 98 114 247)(23 97 115 246)(24 96 116 245)(25 95 117 244)(26 94 118 243)(27 93 119 242)(28 92 120 241)(29 302 355 171)(30 301 356 170)(31 300 357 169)(32 299 358 196)(33 298 359 195)(34 297 360 194)(35 296 361 193)(36 295 362 192)(37 294 363 191)(38 293 364 190)(39 292 337 189)(40 291 338 188)(41 290 339 187)(42 289 340 186)(43 288 341 185)(44 287 342 184)(45 286 343 183)(46 285 344 182)(47 284 345 181)(48 283 346 180)(49 282 347 179)(50 281 348 178)(51 308 349 177)(52 307 350 176)(53 306 351 175)(54 305 352 174)(55 304 353 173)(56 303 354 172)(57 280 325 381)(58 279 326 380)(59 278 327 379)(60 277 328 378)(61 276 329 377)(62 275 330 376)(63 274 331 375)(64 273 332 374)(65 272 333 373)(66 271 334 372)(67 270 335 371)(68 269 336 370)(69 268 309 369)(70 267 310 368)(71 266 311 367)(72 265 312 366)(73 264 313 365)(74 263 314 392)(75 262 315 391)(76 261 316 390)(77 260 317 389)(78 259 318 388)(79 258 319 387)(80 257 320 386)(81 256 321 385)(82 255 322 384)(83 254 323 383)(84 253 324 382)(141 407 203 425)(142 406 204 424)(143 405 205 423)(144 404 206 422)(145 403 207 421)(146 402 208 448)(147 401 209 447)(148 400 210 446)(149 399 211 445)(150 398 212 444)(151 397 213 443)(152 396 214 442)(153 395 215 441)(154 394 216 440)(155 393 217 439)(156 420 218 438)(157 419 219 437)(158 418 220 436)(159 417 221 435)(160 416 222 434)(161 415 223 433)(162 414 224 432)(163 413 197 431)(164 412 198 430)(165 411 199 429)(166 410 200 428)(167 409 201 427)(168 408 202 426)

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

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

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

79 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 ··· 28L 28M ··· 28X 56A ··· 56X 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 2 2 4 28 28 28 28 56 56 2 2 2 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

79 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + + + - - - - image C1 C2 C2 C2 C2 C2 C2 Q8 D4 D7 C4○D4 D14 D14 C4○D8 D28 D56⋊7C2 C8.C22 D4⋊2D7 Q8×D7 C8.D14 kernel Dic14.3Q8 C28.44D4 C8⋊Dic7 C56⋊1C4 C7×C4⋊C8 C4×Dic14 C28.6Q8 Dic14 C2×C28 C4⋊C8 C28 C42 C2×C8 C14 C2×C4 C2 C14 C4 C4 C2 # reps 1 2 1 1 1 1 1 2 2 3 2 3 6 4 12 24 1 3 3 6

Matrix representation of Dic14.3Q8 in GL6(𝔽113)

 1 36 0 0 0 0 69 112 0 0 0 0 0 0 80 112 0 0 0 0 72 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 104 13 0 0 0 0 98 9 0 0 0 0 0 0 66 29 0 0 0 0 107 47 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 0 16 0 0 0 0 106 87 0 0 0 0 0 0 55 8 0 0 0 0 102 58 0 0 0 0 0 0 0 1 0 0 0 0 112 0
,
 91 82 0 0 0 0 1 22 0 0 0 0 0 0 66 29 0 0 0 0 107 47 0 0 0 0 0 0 46 97 0 0 0 0 97 67

G:=sub<GL(6,GF(113))| [1,69,0,0,0,0,36,112,0,0,0,0,0,0,80,72,0,0,0,0,112,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[104,98,0,0,0,0,13,9,0,0,0,0,0,0,66,107,0,0,0,0,29,47,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[0,106,0,0,0,0,16,87,0,0,0,0,0,0,55,102,0,0,0,0,8,58,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[91,1,0,0,0,0,82,22,0,0,0,0,0,0,66,107,0,0,0,0,29,47,0,0,0,0,0,0,46,97,0,0,0,0,97,67] >;

Dic14.3Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{14}._3Q_8
% in TeX

G:=Group("Dic14.3Q8");
// GroupNames label

G:=SmallGroup(448,363);
// by ID

G=gap.SmallGroup(448,363);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,254,219,58,1123,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=1,b^2=c^4=a^14,d^2=a^7*c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

׿
×
𝔽