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G = Q8⋊5Dic14  order 448 = 26·7

1st semidirect product of Q8 and Dic14 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊5Dic14, C42.123D14, C14.82- (1+4), (C7×Q8)⋊6Q8, C7⋊2(Q8⋊3Q8), C28⋊Q8.11C2, (C4×Q8).11D7, C28.45(C2×Q8), C4⋊C4.292D14, (Q8×C28).12C2, (C2×Q8).197D14, (Q8×Dic7).11C2, C4.18(C2×Dic14), C14.16(C22×Q8), (C2×C28).493C23, (C2×C14).113C24, (C4×C28).165C22, (C4×Dic14).22C2, Dic7⋊C4.9C22, C28.6Q8.11C2, C4.Dic14.10C2, Dic7.36(C4○D4), C4⋊Dic7.304C22, (Q8×C14).213C22, (C2×Dic7).51C23, (C4×Dic7).82C22, C2.18(C22×Dic14), C22.138(C23×D7), C2.11(Q8.10D14), (C2×Dic14).241C22, C2.28(D7×C4○D4), C14.143(C2×C4○D4), (C7×C4⋊C4).341C22, (C2×C4).167(C22×D7), SmallGroup(448,1022)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — Q8⋊5Dic14
Chief series
C1 — C7 — C14 — C2×C14 — C2×Dic7 — C4×Dic7 — Q8×Dic7 — Q8⋊5Dic14
Lower central
C7 — C2×C14 — Q8⋊5Dic14

Subgroups: 708 in 200 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C4 [×6], C4 [×13], C22, C7, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×6], C14 [×3], C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic7 [×2], Dic7 [×7], C28 [×6], C28 [×4], C2×C14, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic14 [×6], C2×Dic7 [×2], C2×Dic7 [×6], C2×C28, C2×C28 [×6], C7×Q8 [×4], Q8⋊3Q8, C4×Dic7 [×6], Dic7⋊C4, Dic7⋊C4 [×9], C4⋊Dic7 [×9], C4×C28 [×3], C7×C4⋊C4 [×3], C2×Dic14 [×3], Q8×C14, C4×Dic14 [×3], C28.6Q8 [×3], C28⋊Q8 [×3], C4.Dic14 [×3], Q8×Dic7 [×2], Q8×C28, Q8⋊5Dic14

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), Dic14 [×4], C22×D7 [×7], Q8⋊3Q8, C2×Dic14 [×6], C23×D7, C22×Dic14, Q8.10D14, D7×C4○D4, Q8⋊5Dic14

Generators and relations
 G = < a,b,c,d | a4=c28=1, b2=a2, d2=c14, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >

Smallest permutation representation
►Regular action on 448 points
Generators in S448
(1 338 414 309)(2 339 415 310)(3 340 416 311)(4 341 417 312)(5 342 418 313)(6 343 419 314)(7 344 420 315)(8 345 393 316)(9 346 394 317)(10 347 395 318)(11 348 396 319)(12 349 397 320)(13 350 398 321)(14 351 399 322)(15 352 400 323)(16 353 401 324)(17 354 402 325)(18 355 403 326)(19 356 404 327)(20 357 405 328)(21 358 406 329)(22 359 407 330)(23 360 408 331)(24 361 409 332)(25 362 410 333)(26 363 411 334)(27 364 412 335)(28 337 413 336)(29 305 147 131)(30 306 148 132)(31 307 149 133)(32 308 150 134)(33 281 151 135)(34 282 152 136)(35 283 153 137)(36 284 154 138)(37 285 155 139)(38 286 156 140)(39 287 157 113)(40 288 158 114)(41 289 159 115)(42 290 160 116)(43 291 161 117)(44 292 162 118)(45 293 163 119)(46 294 164 120)(47 295 165 121)(48 296 166 122)(49 297 167 123)(50 298 168 124)(51 299 141 125)(52 300 142 126)(53 301 143 127)(54 302 144 128)(55 303 145 129)(56 304 146 130)(57 232 94 376)(58 233 95 377)(59 234 96 378)(60 235 97 379)(61 236 98 380)(62 237 99 381)(63 238 100 382)(64 239 101 383)(65 240 102 384)(66 241 103 385)(67 242 104 386)(68 243 105 387)(69 244 106 388)(70 245 107 389)(71 246 108 390)(72 247 109 391)(73 248 110 392)(74 249 111 365)(75 250 112 366)(76 251 85 367)(77 252 86 368)(78 225 87 369)(79 226 88 370)(80 227 89 371)(81 228 90 372)(82 229 91 373)(83 230 92 374)(84 231 93 375)(169 438 270 219)(170 439 271 220)(171 440 272 221)(172 441 273 222)(173 442 274 223)(174 443 275 224)(175 444 276 197)(176 445 277 198)(177 446 278 199)(178 447 279 200)(179 448 280 201)(180 421 253 202)(181 422 254 203)(182 423 255 204)(183 424 256 205)(184 425 257 206)(185 426 258 207)(186 427 259 208)(187 428 260 209)(188 429 261 210)(189 430 262 211)(190 431 263 212)(191 432 264 213)(192 433 265 214)(193 434 266 215)(194 435 267 216)(195 436 268 217)(196 437 269 218)

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

(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)

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

Matrix representation ►G ⊆ GL4(𝔽29) generated by

1000
0100
0001
00280
,
1000
0100
00017
00170
,
21000
131800
00012
00170
,
102000
81900
00280
00028
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,17,0,0,17,0],[21,13,0,0,0,18,0,0,0,0,0,17,0,0,12,0],[10,8,0,0,20,19,0,0,0,0,28,0,0,0,0,28] >;
 

85 conjugacy classes

class 1 2A2B2C4A···4H4I4J4K4L4M4N4O4P···4U7A7B7C14A···14I28A···28L28M···28AV
order12224···444444444···477714···1428···2828···28
size11112···24441414141428···282222···22···24···4

85 irreducible representations

dim11111112222222444
type+++++++-++++--
imageC1C2C2C2C2C2C2Q8D7C4○D4D14D14D14Dic142- (1+4)Q8.10D14D7×C4○D4
kernelQ8⋊5Dic14C4×Dic14C28.6Q8C28⋊Q8C4.Dic14Q8×Dic7Q8×C28C7×Q8C4×Q8Dic7C42C4⋊C4C2×Q8Q8C14C2C2
# reps133332143499324166

In GAP, Magma, Sage, TeX 

Q_8\rtimes_5Dic_{14}
 
% in TeX
 
G:=Group("Q8:5Dic14");
 
// GroupNames label
 
G:=SmallGroup(448,1022);
 
// by ID
 
G=gap.SmallGroup(448,1022);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,387,184,675,192,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^28=1,b^2=a^2,d^2=c^14,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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×
⋊
ℤ
𝔽
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