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G = Dic14⋊8Q8  order 448 = 26·7

6th 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, C4.Dic14.14C2, Dic7⋊3Q8.12C2, Dic7⋊C4.57C22, C4⋊Dic7.382C22, (Q8×C14).132C22, C22.286(C23×D7), (C4×Dic7).157C22, (C2×Dic7).270C23, 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

Subgroups: 684 in 200 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C4 [×4], C4 [×15], C22, C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C14 [×3], C42, C42 [×8], C4⋊C4 [×4], C4⋊C4 [×18], C2×Q8 [×2], C2×Q8 [×2], Dic7 [×4], Dic7 [×6], C28 [×4], C28 [×5], C2×C14, C4×Q8 [×6], C42.C2 [×6], C4⋊Q8, C4⋊Q8 [×2], Dic14 [×4], Dic14 [×2], C2×Dic7 [×8], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×4], Q8⋊3Q8, C4×Dic7 [×8], Dic7⋊C4 [×12], C4⋊Dic7 [×2], C4⋊Dic7 [×4], C4×C28, C7×C4⋊C4 [×4], C2×Dic14 [×2], Q8×C14 [×2], C4×Dic14 [×2], Dic7⋊3Q8 [×2], Dic7.Q8 [×4], C4.Dic14 [×2], Dic7⋊Q8 [×2], Q8×Dic7 [×2], C7×C4⋊Q8, Dic14⋊8Q8

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×2], C24, D14 [×7], C22×Q8, C2×C4○D4, 2- (1+4), C22×D7 [×7], Q8⋊3Q8, D4⋊2D7 [×2], Q8×D7 [×2], C23×D7, C2×D4⋊2D7, C2×Q8×D7, Q8.10D14, Dic14⋊8Q8

Generators and relations
 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 >

Smallest permutation representation
►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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

0280000
100000
0028000
0002800
000001
0000283
,
20110000
1190000
0028000
0002800
00001712
0000512
,
100000
010000
000100
0028000
0000280
0000028
,
8160000
16210000
0016200
0021300
000010
000001

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] >;
 

67 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4I4J···4Q4R4S4T4U7A7B7C14A···14I28A···28R28S···28AD
order122244444···44···4444477714···1428···2828···28
size111122224···414···14282828282222···24···48···8

67 irreducible representations

dim111111112222224444
type++++++++-++++---
imageC1C2C2C2C2C2C2C2Q8D7C4○D4D14D14D142- (1+4)D4⋊2D7Q8×D7Q8.10D14
kernelDic14⋊8Q8C4×Dic14Dic7⋊3Q8Dic7.Q8C4.Dic14Dic7⋊Q8Q8×Dic7C7×C4⋊Q8Dic14C4⋊Q8C28C42C4⋊C4C2×Q8C14C4C4C2
# reps1224222143431261666

In GAP, Magma, Sage, TeX 

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
 

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