direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×Dic14, C42.120D14, C14.1082+ (1+4), C7⋊1(Q82), (C7×Q8)⋊5Q8, (C4×Q8).9D7, C4.48(Q8×D7), C28⋊Q8.10C2, C28.44(C2×Q8), C4⋊C4.290D14, (Q8×C28).10C2, Dic7.9(C2×Q8), (C2×Q8).196D14, C28⋊2Q8.23C2, (Q8×Dic7).10C2, C4.17(C2×Dic14), C14.15(C22×Q8), (C4×C28).163C22, (C2×C14).110C24, (C2×C28).167C23, (C4×Dic14).20C2, C2.21(D4⋊8D14), C4⋊Dic7.201C22, (Q8×C14).210C22, (C4×Dic7).79C22, C2.17(C22×Dic14), C22.135(C23×D7), Dic7⋊C4.113C22, (C2×Dic14).30C22, (C2×Dic7).210C23, C2.10(C2×Q8×D7), (C7×C4⋊C4).338C22, (C2×C4).582(C22×D7), SmallGroup(448,1019)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 836 in 212 conjugacy classes, 123 normal (18 characteristic)
C1, C2 [×3], C4 [×8], C4 [×13], C22, C7, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×10], C14 [×3], C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×15], C2×Q8, C2×Q8 [×7], Dic7 [×4], Dic7 [×6], C28 [×8], C28 [×3], C2×C14, C4×Q8, C4×Q8 [×5], C4⋊Q8 [×9], Dic14 [×4], Dic14 [×6], C2×Dic7 [×8], C2×C28, C2×C28 [×6], C7×Q8 [×4], Q82, C4×Dic7 [×6], Dic7⋊C4 [×6], C4⋊Dic7 [×9], C4×C28 [×3], C7×C4⋊C4 [×3], C2×Dic14, C2×Dic14 [×6], Q8×C14, C4×Dic14 [×3], C28⋊2Q8 [×3], C28⋊Q8 [×6], Q8×Dic7 [×2], Q8×C28, Q8×Dic14
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×8], C23 [×15], D7, C2×Q8 [×12], C24, D14 [×7], C22×Q8 [×2], 2+ (1+4), Dic14 [×4], C22×D7 [×7], Q82, C2×Dic14 [×6], Q8×D7 [×2], C23×D7, C22×Dic14, C2×Q8×D7, D4⋊8D14, Q8×Dic14
Generators and relations
G = < a,b,c,d | a4=c28=1, b2=a2, d2=c14, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 448 points
Generators in S448
(1 33 392 444)(2 34 365 445)(3 35 366 446)(4 36 367 447)(5 37 368 448)(6 38 369 421)(7 39 370 422)(8 40 371 423)(9 41 372 424)(10 42 373 425)(11 43 374 426)(12 44 375 427)(13 45 376 428)(14 46 377 429)(15 47 378 430)(16 48 379 431)(17 49 380 432)(18 50 381 433)(19 51 382 434)(20 52 383 435)(21 53 384 436)(22 54 385 437)(23 55 386 438)(24 56 387 439)(25 29 388 440)(26 30 389 441)(27 31 390 442)(28 32 391 443)(57 397 195 148)(58 398 196 149)(59 399 169 150)(60 400 170 151)(61 401 171 152)(62 402 172 153)(63 403 173 154)(64 404 174 155)(65 405 175 156)(66 406 176 157)(67 407 177 158)(68 408 178 159)(69 409 179 160)(70 410 180 161)(71 411 181 162)(72 412 182 163)(73 413 183 164)(74 414 184 165)(75 415 185 166)(76 416 186 167)(77 417 187 168)(78 418 188 141)(79 419 189 142)(80 420 190 143)(81 393 191 144)(82 394 192 145)(83 395 193 146)(84 396 194 147)(85 282 238 264)(86 283 239 265)(87 284 240 266)(88 285 241 267)(89 286 242 268)(90 287 243 269)(91 288 244 270)(92 289 245 271)(93 290 246 272)(94 291 247 273)(95 292 248 274)(96 293 249 275)(97 294 250 276)(98 295 251 277)(99 296 252 278)(100 297 225 279)(101 298 226 280)(102 299 227 253)(103 300 228 254)(104 301 229 255)(105 302 230 256)(106 303 231 257)(107 304 232 258)(108 305 233 259)(109 306 234 260)(110 307 235 261)(111 308 236 262)(112 281 237 263)(113 313 341 219)(114 314 342 220)(115 315 343 221)(116 316 344 222)(117 317 345 223)(118 318 346 224)(119 319 347 197)(120 320 348 198)(121 321 349 199)(122 322 350 200)(123 323 351 201)(124 324 352 202)(125 325 353 203)(126 326 354 204)(127 327 355 205)(128 328 356 206)(129 329 357 207)(130 330 358 208)(131 331 359 209)(132 332 360 210)(133 333 361 211)(134 334 362 212)(135 335 363 213)(136 336 364 214)(137 309 337 215)(138 310 338 216)(139 311 339 217)(140 312 340 218)
(1 73 392 183)(2 74 365 184)(3 75 366 185)(4 76 367 186)(5 77 368 187)(6 78 369 188)(7 79 370 189)(8 80 371 190)(9 81 372 191)(10 82 373 192)(11 83 374 193)(12 84 375 194)(13 57 376 195)(14 58 377 196)(15 59 378 169)(16 60 379 170)(17 61 380 171)(18 62 381 172)(19 63 382 173)(20 64 383 174)(21 65 384 175)(22 66 385 176)(23 67 386 177)(24 68 387 178)(25 69 388 179)(26 70 389 180)(27 71 390 181)(28 72 391 182)(29 160 440 409)(30 161 441 410)(31 162 442 411)(32 163 443 412)(33 164 444 413)(34 165 445 414)(35 166 446 415)(36 167 447 416)(37 168 448 417)(38 141 421 418)(39 142 422 419)(40 143 423 420)(41 144 424 393)(42 145 425 394)(43 146 426 395)(44 147 427 396)(45 148 428 397)(46 149 429 398)(47 150 430 399)(48 151 431 400)(49 152 432 401)(50 153 433 402)(51 154 434 403)(52 155 435 404)(53 156 436 405)(54 157 437 406)(55 158 438 407)(56 159 439 408)(85 114 238 342)(86 115 239 343)(87 116 240 344)(88 117 241 345)(89 118 242 346)(90 119 243 347)(91 120 244 348)(92 121 245 349)(93 122 246 350)(94 123 247 351)(95 124 248 352)(96 125 249 353)(97 126 250 354)(98 127 251 355)(99 128 252 356)(100 129 225 357)(101 130 226 358)(102 131 227 359)(103 132 228 360)(104 133 229 361)(105 134 230 362)(106 135 231 363)(107 136 232 364)(108 137 233 337)(109 138 234 338)(110 139 235 339)(111 140 236 340)(112 113 237 341)(197 269 319 287)(198 270 320 288)(199 271 321 289)(200 272 322 290)(201 273 323 291)(202 274 324 292)(203 275 325 293)(204 276 326 294)(205 277 327 295)(206 278 328 296)(207 279 329 297)(208 280 330 298)(209 253 331 299)(210 254 332 300)(211 255 333 301)(212 256 334 302)(213 257 335 303)(214 258 336 304)(215 259 309 305)(216 260 310 306)(217 261 311 307)(218 262 312 308)(219 263 313 281)(220 264 314 282)(221 265 315 283)(222 266 316 284)(223 267 317 285)(224 268 318 286)
(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)
(1 207 15 221)(2 206 16 220)(3 205 17 219)(4 204 18 218)(5 203 19 217)(6 202 20 216)(7 201 21 215)(8 200 22 214)(9 199 23 213)(10 198 24 212)(11 197 25 211)(12 224 26 210)(13 223 27 209)(14 222 28 208)(29 133 43 119)(30 132 44 118)(31 131 45 117)(32 130 46 116)(33 129 47 115)(34 128 48 114)(35 127 49 113)(36 126 50 140)(37 125 51 139)(38 124 52 138)(39 123 53 137)(40 122 54 136)(41 121 55 135)(42 120 56 134)(57 267 71 253)(58 266 72 280)(59 265 73 279)(60 264 74 278)(61 263 75 277)(62 262 76 276)(63 261 77 275)(64 260 78 274)(65 259 79 273)(66 258 80 272)(67 257 81 271)(68 256 82 270)(69 255 83 269)(70 254 84 268)(85 414 99 400)(86 413 100 399)(87 412 101 398)(88 411 102 397)(89 410 103 396)(90 409 104 395)(91 408 105 394)(92 407 106 393)(93 406 107 420)(94 405 108 419)(95 404 109 418)(96 403 110 417)(97 402 111 416)(98 401 112 415)(141 248 155 234)(142 247 156 233)(143 246 157 232)(144 245 158 231)(145 244 159 230)(146 243 160 229)(147 242 161 228)(148 241 162 227)(149 240 163 226)(150 239 164 225)(151 238 165 252)(152 237 166 251)(153 236 167 250)(154 235 168 249)(169 283 183 297)(170 282 184 296)(171 281 185 295)(172 308 186 294)(173 307 187 293)(174 306 188 292)(175 305 189 291)(176 304 190 290)(177 303 191 289)(178 302 192 288)(179 301 193 287)(180 300 194 286)(181 299 195 285)(182 298 196 284)(309 370 323 384)(310 369 324 383)(311 368 325 382)(312 367 326 381)(313 366 327 380)(314 365 328 379)(315 392 329 378)(316 391 330 377)(317 390 331 376)(318 389 332 375)(319 388 333 374)(320 387 334 373)(321 386 335 372)(322 385 336 371)(337 422 351 436)(338 421 352 435)(339 448 353 434)(340 447 354 433)(341 446 355 432)(342 445 356 431)(343 444 357 430)(344 443 358 429)(345 442 359 428)(346 441 360 427)(347 440 361 426)(348 439 362 425)(349 438 363 424)(350 437 364 423)
G:=sub<Sym(448)| (1,33,392,444)(2,34,365,445)(3,35,366,446)(4,36,367,447)(5,37,368,448)(6,38,369,421)(7,39,370,422)(8,40,371,423)(9,41,372,424)(10,42,373,425)(11,43,374,426)(12,44,375,427)(13,45,376,428)(14,46,377,429)(15,47,378,430)(16,48,379,431)(17,49,380,432)(18,50,381,433)(19,51,382,434)(20,52,383,435)(21,53,384,436)(22,54,385,437)(23,55,386,438)(24,56,387,439)(25,29,388,440)(26,30,389,441)(27,31,390,442)(28,32,391,443)(57,397,195,148)(58,398,196,149)(59,399,169,150)(60,400,170,151)(61,401,171,152)(62,402,172,153)(63,403,173,154)(64,404,174,155)(65,405,175,156)(66,406,176,157)(67,407,177,158)(68,408,178,159)(69,409,179,160)(70,410,180,161)(71,411,181,162)(72,412,182,163)(73,413,183,164)(74,414,184,165)(75,415,185,166)(76,416,186,167)(77,417,187,168)(78,418,188,141)(79,419,189,142)(80,420,190,143)(81,393,191,144)(82,394,192,145)(83,395,193,146)(84,396,194,147)(85,282,238,264)(86,283,239,265)(87,284,240,266)(88,285,241,267)(89,286,242,268)(90,287,243,269)(91,288,244,270)(92,289,245,271)(93,290,246,272)(94,291,247,273)(95,292,248,274)(96,293,249,275)(97,294,250,276)(98,295,251,277)(99,296,252,278)(100,297,225,279)(101,298,226,280)(102,299,227,253)(103,300,228,254)(104,301,229,255)(105,302,230,256)(106,303,231,257)(107,304,232,258)(108,305,233,259)(109,306,234,260)(110,307,235,261)(111,308,236,262)(112,281,237,263)(113,313,341,219)(114,314,342,220)(115,315,343,221)(116,316,344,222)(117,317,345,223)(118,318,346,224)(119,319,347,197)(120,320,348,198)(121,321,349,199)(122,322,350,200)(123,323,351,201)(124,324,352,202)(125,325,353,203)(126,326,354,204)(127,327,355,205)(128,328,356,206)(129,329,357,207)(130,330,358,208)(131,331,359,209)(132,332,360,210)(133,333,361,211)(134,334,362,212)(135,335,363,213)(136,336,364,214)(137,309,337,215)(138,310,338,216)(139,311,339,217)(140,312,340,218), (1,73,392,183)(2,74,365,184)(3,75,366,185)(4,76,367,186)(5,77,368,187)(6,78,369,188)(7,79,370,189)(8,80,371,190)(9,81,372,191)(10,82,373,192)(11,83,374,193)(12,84,375,194)(13,57,376,195)(14,58,377,196)(15,59,378,169)(16,60,379,170)(17,61,380,171)(18,62,381,172)(19,63,382,173)(20,64,383,174)(21,65,384,175)(22,66,385,176)(23,67,386,177)(24,68,387,178)(25,69,388,179)(26,70,389,180)(27,71,390,181)(28,72,391,182)(29,160,440,409)(30,161,441,410)(31,162,442,411)(32,163,443,412)(33,164,444,413)(34,165,445,414)(35,166,446,415)(36,167,447,416)(37,168,448,417)(38,141,421,418)(39,142,422,419)(40,143,423,420)(41,144,424,393)(42,145,425,394)(43,146,426,395)(44,147,427,396)(45,148,428,397)(46,149,429,398)(47,150,430,399)(48,151,431,400)(49,152,432,401)(50,153,433,402)(51,154,434,403)(52,155,435,404)(53,156,436,405)(54,157,437,406)(55,158,438,407)(56,159,439,408)(85,114,238,342)(86,115,239,343)(87,116,240,344)(88,117,241,345)(89,118,242,346)(90,119,243,347)(91,120,244,348)(92,121,245,349)(93,122,246,350)(94,123,247,351)(95,124,248,352)(96,125,249,353)(97,126,250,354)(98,127,251,355)(99,128,252,356)(100,129,225,357)(101,130,226,358)(102,131,227,359)(103,132,228,360)(104,133,229,361)(105,134,230,362)(106,135,231,363)(107,136,232,364)(108,137,233,337)(109,138,234,338)(110,139,235,339)(111,140,236,340)(112,113,237,341)(197,269,319,287)(198,270,320,288)(199,271,321,289)(200,272,322,290)(201,273,323,291)(202,274,324,292)(203,275,325,293)(204,276,326,294)(205,277,327,295)(206,278,328,296)(207,279,329,297)(208,280,330,298)(209,253,331,299)(210,254,332,300)(211,255,333,301)(212,256,334,302)(213,257,335,303)(214,258,336,304)(215,259,309,305)(216,260,310,306)(217,261,311,307)(218,262,312,308)(219,263,313,281)(220,264,314,282)(221,265,315,283)(222,266,316,284)(223,267,317,285)(224,268,318,286), (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), (1,207,15,221)(2,206,16,220)(3,205,17,219)(4,204,18,218)(5,203,19,217)(6,202,20,216)(7,201,21,215)(8,200,22,214)(9,199,23,213)(10,198,24,212)(11,197,25,211)(12,224,26,210)(13,223,27,209)(14,222,28,208)(29,133,43,119)(30,132,44,118)(31,131,45,117)(32,130,46,116)(33,129,47,115)(34,128,48,114)(35,127,49,113)(36,126,50,140)(37,125,51,139)(38,124,52,138)(39,123,53,137)(40,122,54,136)(41,121,55,135)(42,120,56,134)(57,267,71,253)(58,266,72,280)(59,265,73,279)(60,264,74,278)(61,263,75,277)(62,262,76,276)(63,261,77,275)(64,260,78,274)(65,259,79,273)(66,258,80,272)(67,257,81,271)(68,256,82,270)(69,255,83,269)(70,254,84,268)(85,414,99,400)(86,413,100,399)(87,412,101,398)(88,411,102,397)(89,410,103,396)(90,409,104,395)(91,408,105,394)(92,407,106,393)(93,406,107,420)(94,405,108,419)(95,404,109,418)(96,403,110,417)(97,402,111,416)(98,401,112,415)(141,248,155,234)(142,247,156,233)(143,246,157,232)(144,245,158,231)(145,244,159,230)(146,243,160,229)(147,242,161,228)(148,241,162,227)(149,240,163,226)(150,239,164,225)(151,238,165,252)(152,237,166,251)(153,236,167,250)(154,235,168,249)(169,283,183,297)(170,282,184,296)(171,281,185,295)(172,308,186,294)(173,307,187,293)(174,306,188,292)(175,305,189,291)(176,304,190,290)(177,303,191,289)(178,302,192,288)(179,301,193,287)(180,300,194,286)(181,299,195,285)(182,298,196,284)(309,370,323,384)(310,369,324,383)(311,368,325,382)(312,367,326,381)(313,366,327,380)(314,365,328,379)(315,392,329,378)(316,391,330,377)(317,390,331,376)(318,389,332,375)(319,388,333,374)(320,387,334,373)(321,386,335,372)(322,385,336,371)(337,422,351,436)(338,421,352,435)(339,448,353,434)(340,447,354,433)(341,446,355,432)(342,445,356,431)(343,444,357,430)(344,443,358,429)(345,442,359,428)(346,441,360,427)(347,440,361,426)(348,439,362,425)(349,438,363,424)(350,437,364,423)>;
G:=Group( (1,33,392,444)(2,34,365,445)(3,35,366,446)(4,36,367,447)(5,37,368,448)(6,38,369,421)(7,39,370,422)(8,40,371,423)(9,41,372,424)(10,42,373,425)(11,43,374,426)(12,44,375,427)(13,45,376,428)(14,46,377,429)(15,47,378,430)(16,48,379,431)(17,49,380,432)(18,50,381,433)(19,51,382,434)(20,52,383,435)(21,53,384,436)(22,54,385,437)(23,55,386,438)(24,56,387,439)(25,29,388,440)(26,30,389,441)(27,31,390,442)(28,32,391,443)(57,397,195,148)(58,398,196,149)(59,399,169,150)(60,400,170,151)(61,401,171,152)(62,402,172,153)(63,403,173,154)(64,404,174,155)(65,405,175,156)(66,406,176,157)(67,407,177,158)(68,408,178,159)(69,409,179,160)(70,410,180,161)(71,411,181,162)(72,412,182,163)(73,413,183,164)(74,414,184,165)(75,415,185,166)(76,416,186,167)(77,417,187,168)(78,418,188,141)(79,419,189,142)(80,420,190,143)(81,393,191,144)(82,394,192,145)(83,395,193,146)(84,396,194,147)(85,282,238,264)(86,283,239,265)(87,284,240,266)(88,285,241,267)(89,286,242,268)(90,287,243,269)(91,288,244,270)(92,289,245,271)(93,290,246,272)(94,291,247,273)(95,292,248,274)(96,293,249,275)(97,294,250,276)(98,295,251,277)(99,296,252,278)(100,297,225,279)(101,298,226,280)(102,299,227,253)(103,300,228,254)(104,301,229,255)(105,302,230,256)(106,303,231,257)(107,304,232,258)(108,305,233,259)(109,306,234,260)(110,307,235,261)(111,308,236,262)(112,281,237,263)(113,313,341,219)(114,314,342,220)(115,315,343,221)(116,316,344,222)(117,317,345,223)(118,318,346,224)(119,319,347,197)(120,320,348,198)(121,321,349,199)(122,322,350,200)(123,323,351,201)(124,324,352,202)(125,325,353,203)(126,326,354,204)(127,327,355,205)(128,328,356,206)(129,329,357,207)(130,330,358,208)(131,331,359,209)(132,332,360,210)(133,333,361,211)(134,334,362,212)(135,335,363,213)(136,336,364,214)(137,309,337,215)(138,310,338,216)(139,311,339,217)(140,312,340,218), (1,73,392,183)(2,74,365,184)(3,75,366,185)(4,76,367,186)(5,77,368,187)(6,78,369,188)(7,79,370,189)(8,80,371,190)(9,81,372,191)(10,82,373,192)(11,83,374,193)(12,84,375,194)(13,57,376,195)(14,58,377,196)(15,59,378,169)(16,60,379,170)(17,61,380,171)(18,62,381,172)(19,63,382,173)(20,64,383,174)(21,65,384,175)(22,66,385,176)(23,67,386,177)(24,68,387,178)(25,69,388,179)(26,70,389,180)(27,71,390,181)(28,72,391,182)(29,160,440,409)(30,161,441,410)(31,162,442,411)(32,163,443,412)(33,164,444,413)(34,165,445,414)(35,166,446,415)(36,167,447,416)(37,168,448,417)(38,141,421,418)(39,142,422,419)(40,143,423,420)(41,144,424,393)(42,145,425,394)(43,146,426,395)(44,147,427,396)(45,148,428,397)(46,149,429,398)(47,150,430,399)(48,151,431,400)(49,152,432,401)(50,153,433,402)(51,154,434,403)(52,155,435,404)(53,156,436,405)(54,157,437,406)(55,158,438,407)(56,159,439,408)(85,114,238,342)(86,115,239,343)(87,116,240,344)(88,117,241,345)(89,118,242,346)(90,119,243,347)(91,120,244,348)(92,121,245,349)(93,122,246,350)(94,123,247,351)(95,124,248,352)(96,125,249,353)(97,126,250,354)(98,127,251,355)(99,128,252,356)(100,129,225,357)(101,130,226,358)(102,131,227,359)(103,132,228,360)(104,133,229,361)(105,134,230,362)(106,135,231,363)(107,136,232,364)(108,137,233,337)(109,138,234,338)(110,139,235,339)(111,140,236,340)(112,113,237,341)(197,269,319,287)(198,270,320,288)(199,271,321,289)(200,272,322,290)(201,273,323,291)(202,274,324,292)(203,275,325,293)(204,276,326,294)(205,277,327,295)(206,278,328,296)(207,279,329,297)(208,280,330,298)(209,253,331,299)(210,254,332,300)(211,255,333,301)(212,256,334,302)(213,257,335,303)(214,258,336,304)(215,259,309,305)(216,260,310,306)(217,261,311,307)(218,262,312,308)(219,263,313,281)(220,264,314,282)(221,265,315,283)(222,266,316,284)(223,267,317,285)(224,268,318,286), (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), (1,207,15,221)(2,206,16,220)(3,205,17,219)(4,204,18,218)(5,203,19,217)(6,202,20,216)(7,201,21,215)(8,200,22,214)(9,199,23,213)(10,198,24,212)(11,197,25,211)(12,224,26,210)(13,223,27,209)(14,222,28,208)(29,133,43,119)(30,132,44,118)(31,131,45,117)(32,130,46,116)(33,129,47,115)(34,128,48,114)(35,127,49,113)(36,126,50,140)(37,125,51,139)(38,124,52,138)(39,123,53,137)(40,122,54,136)(41,121,55,135)(42,120,56,134)(57,267,71,253)(58,266,72,280)(59,265,73,279)(60,264,74,278)(61,263,75,277)(62,262,76,276)(63,261,77,275)(64,260,78,274)(65,259,79,273)(66,258,80,272)(67,257,81,271)(68,256,82,270)(69,255,83,269)(70,254,84,268)(85,414,99,400)(86,413,100,399)(87,412,101,398)(88,411,102,397)(89,410,103,396)(90,409,104,395)(91,408,105,394)(92,407,106,393)(93,406,107,420)(94,405,108,419)(95,404,109,418)(96,403,110,417)(97,402,111,416)(98,401,112,415)(141,248,155,234)(142,247,156,233)(143,246,157,232)(144,245,158,231)(145,244,159,230)(146,243,160,229)(147,242,161,228)(148,241,162,227)(149,240,163,226)(150,239,164,225)(151,238,165,252)(152,237,166,251)(153,236,167,250)(154,235,168,249)(169,283,183,297)(170,282,184,296)(171,281,185,295)(172,308,186,294)(173,307,187,293)(174,306,188,292)(175,305,189,291)(176,304,190,290)(177,303,191,289)(178,302,192,288)(179,301,193,287)(180,300,194,286)(181,299,195,285)(182,298,196,284)(309,370,323,384)(310,369,324,383)(311,368,325,382)(312,367,326,381)(313,366,327,380)(314,365,328,379)(315,392,329,378)(316,391,330,377)(317,390,331,376)(318,389,332,375)(319,388,333,374)(320,387,334,373)(321,386,335,372)(322,385,336,371)(337,422,351,436)(338,421,352,435)(339,448,353,434)(340,447,354,433)(341,446,355,432)(342,445,356,431)(343,444,357,430)(344,443,358,429)(345,442,359,428)(346,441,360,427)(347,440,361,426)(348,439,362,425)(349,438,363,424)(350,437,364,423) );
G=PermutationGroup([(1,33,392,444),(2,34,365,445),(3,35,366,446),(4,36,367,447),(5,37,368,448),(6,38,369,421),(7,39,370,422),(8,40,371,423),(9,41,372,424),(10,42,373,425),(11,43,374,426),(12,44,375,427),(13,45,376,428),(14,46,377,429),(15,47,378,430),(16,48,379,431),(17,49,380,432),(18,50,381,433),(19,51,382,434),(20,52,383,435),(21,53,384,436),(22,54,385,437),(23,55,386,438),(24,56,387,439),(25,29,388,440),(26,30,389,441),(27,31,390,442),(28,32,391,443),(57,397,195,148),(58,398,196,149),(59,399,169,150),(60,400,170,151),(61,401,171,152),(62,402,172,153),(63,403,173,154),(64,404,174,155),(65,405,175,156),(66,406,176,157),(67,407,177,158),(68,408,178,159),(69,409,179,160),(70,410,180,161),(71,411,181,162),(72,412,182,163),(73,413,183,164),(74,414,184,165),(75,415,185,166),(76,416,186,167),(77,417,187,168),(78,418,188,141),(79,419,189,142),(80,420,190,143),(81,393,191,144),(82,394,192,145),(83,395,193,146),(84,396,194,147),(85,282,238,264),(86,283,239,265),(87,284,240,266),(88,285,241,267),(89,286,242,268),(90,287,243,269),(91,288,244,270),(92,289,245,271),(93,290,246,272),(94,291,247,273),(95,292,248,274),(96,293,249,275),(97,294,250,276),(98,295,251,277),(99,296,252,278),(100,297,225,279),(101,298,226,280),(102,299,227,253),(103,300,228,254),(104,301,229,255),(105,302,230,256),(106,303,231,257),(107,304,232,258),(108,305,233,259),(109,306,234,260),(110,307,235,261),(111,308,236,262),(112,281,237,263),(113,313,341,219),(114,314,342,220),(115,315,343,221),(116,316,344,222),(117,317,345,223),(118,318,346,224),(119,319,347,197),(120,320,348,198),(121,321,349,199),(122,322,350,200),(123,323,351,201),(124,324,352,202),(125,325,353,203),(126,326,354,204),(127,327,355,205),(128,328,356,206),(129,329,357,207),(130,330,358,208),(131,331,359,209),(132,332,360,210),(133,333,361,211),(134,334,362,212),(135,335,363,213),(136,336,364,214),(137,309,337,215),(138,310,338,216),(139,311,339,217),(140,312,340,218)], [(1,73,392,183),(2,74,365,184),(3,75,366,185),(4,76,367,186),(5,77,368,187),(6,78,369,188),(7,79,370,189),(8,80,371,190),(9,81,372,191),(10,82,373,192),(11,83,374,193),(12,84,375,194),(13,57,376,195),(14,58,377,196),(15,59,378,169),(16,60,379,170),(17,61,380,171),(18,62,381,172),(19,63,382,173),(20,64,383,174),(21,65,384,175),(22,66,385,176),(23,67,386,177),(24,68,387,178),(25,69,388,179),(26,70,389,180),(27,71,390,181),(28,72,391,182),(29,160,440,409),(30,161,441,410),(31,162,442,411),(32,163,443,412),(33,164,444,413),(34,165,445,414),(35,166,446,415),(36,167,447,416),(37,168,448,417),(38,141,421,418),(39,142,422,419),(40,143,423,420),(41,144,424,393),(42,145,425,394),(43,146,426,395),(44,147,427,396),(45,148,428,397),(46,149,429,398),(47,150,430,399),(48,151,431,400),(49,152,432,401),(50,153,433,402),(51,154,434,403),(52,155,435,404),(53,156,436,405),(54,157,437,406),(55,158,438,407),(56,159,439,408),(85,114,238,342),(86,115,239,343),(87,116,240,344),(88,117,241,345),(89,118,242,346),(90,119,243,347),(91,120,244,348),(92,121,245,349),(93,122,246,350),(94,123,247,351),(95,124,248,352),(96,125,249,353),(97,126,250,354),(98,127,251,355),(99,128,252,356),(100,129,225,357),(101,130,226,358),(102,131,227,359),(103,132,228,360),(104,133,229,361),(105,134,230,362),(106,135,231,363),(107,136,232,364),(108,137,233,337),(109,138,234,338),(110,139,235,339),(111,140,236,340),(112,113,237,341),(197,269,319,287),(198,270,320,288),(199,271,321,289),(200,272,322,290),(201,273,323,291),(202,274,324,292),(203,275,325,293),(204,276,326,294),(205,277,327,295),(206,278,328,296),(207,279,329,297),(208,280,330,298),(209,253,331,299),(210,254,332,300),(211,255,333,301),(212,256,334,302),(213,257,335,303),(214,258,336,304),(215,259,309,305),(216,260,310,306),(217,261,311,307),(218,262,312,308),(219,263,313,281),(220,264,314,282),(221,265,315,283),(222,266,316,284),(223,267,317,285),(224,268,318,286)], [(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)], [(1,207,15,221),(2,206,16,220),(3,205,17,219),(4,204,18,218),(5,203,19,217),(6,202,20,216),(7,201,21,215),(8,200,22,214),(9,199,23,213),(10,198,24,212),(11,197,25,211),(12,224,26,210),(13,223,27,209),(14,222,28,208),(29,133,43,119),(30,132,44,118),(31,131,45,117),(32,130,46,116),(33,129,47,115),(34,128,48,114),(35,127,49,113),(36,126,50,140),(37,125,51,139),(38,124,52,138),(39,123,53,137),(40,122,54,136),(41,121,55,135),(42,120,56,134),(57,267,71,253),(58,266,72,280),(59,265,73,279),(60,264,74,278),(61,263,75,277),(62,262,76,276),(63,261,77,275),(64,260,78,274),(65,259,79,273),(66,258,80,272),(67,257,81,271),(68,256,82,270),(69,255,83,269),(70,254,84,268),(85,414,99,400),(86,413,100,399),(87,412,101,398),(88,411,102,397),(89,410,103,396),(90,409,104,395),(91,408,105,394),(92,407,106,393),(93,406,107,420),(94,405,108,419),(95,404,109,418),(96,403,110,417),(97,402,111,416),(98,401,112,415),(141,248,155,234),(142,247,156,233),(143,246,157,232),(144,245,158,231),(145,244,159,230),(146,243,160,229),(147,242,161,228),(148,241,162,227),(149,240,163,226),(150,239,164,225),(151,238,165,252),(152,237,166,251),(153,236,167,250),(154,235,168,249),(169,283,183,297),(170,282,184,296),(171,281,185,295),(172,308,186,294),(173,307,187,293),(174,306,188,292),(175,305,189,291),(176,304,190,290),(177,303,191,289),(178,302,192,288),(179,301,193,287),(180,300,194,286),(181,299,195,285),(182,298,196,284),(309,370,323,384),(310,369,324,383),(311,368,325,382),(312,367,326,381),(313,366,327,380),(314,365,328,379),(315,392,329,378),(316,391,330,377),(317,390,331,376),(318,389,332,375),(319,388,333,374),(320,387,334,373),(321,386,335,372),(322,385,336,371),(337,422,351,436),(338,421,352,435),(339,448,353,434),(340,447,354,433),(341,446,355,432),(342,445,356,431),(343,444,357,430),(344,443,358,429),(345,442,359,428),(346,441,360,427),(347,440,361,426),(348,439,362,425),(349,438,363,424),(350,437,364,423)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 27 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 27 | 14 |
| 2 | 21 | 0 | 0 |
| 14 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 26 | 0 | 0 |
| 8 | 9 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,1,0,0,27,1],[1,0,0,0,0,1,0,0,0,0,15,27,0,0,26,14],[2,14,0,0,21,17,0,0,0,0,1,0,0,0,0,1],[20,8,0,0,26,9,0,0,0,0,28,0,0,0,0,28] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | ··· | 4U | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | + | + | - | + | - | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | Q8 | D7 | D14 | D14 | D14 | Dic14 | 2+ (1+4) | Q8×D7 | D4⋊8D14 |
| kernel | Q8×Dic14 | C4×Dic14 | C28⋊2Q8 | C28⋊Q8 | Q8×Dic7 | Q8×C28 | Dic14 | C7×Q8 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C14 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 4 | 3 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
Q_8\times Dic_{14} % in TeX
G:=Group("Q8xDic14"); // GroupNames label
G:=SmallGroup(448,1019);
// by ID
G=gap.SmallGroup(448,1019);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,387,184,675,80,18822]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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