metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊7Q8, C42.146D14, C14.942- (1+4), C4.15(Q8×D7), C7⋊4(Q8⋊3Q8), C28⋊Q8.12C2, C28.47(C2×Q8), C4⋊C4.202D14, C42.C2.6D7, (C2×C28).84C23, Dic7.13(C2×Q8), Dic7.Q8.2C2, C14.39(C22×Q8), (C4×C28).190C22, (C2×C14).229C24, (C4×Dic14).24C2, C4.Dic14.12C2, Dic7.40(C4○D4), Dic7⋊3Q8.11C2, C4⋊Dic7.378C22, C22.250(C23×D7), Dic7⋊C4.143C22, (C2×Dic7).119C23, (C4×Dic7).137C22, C2.55(D4.10D14), (C2×Dic14).298C22, C2.22(C2×Q8×D7), C2.82(D7×C4○D4), C14.193(C2×C4○D4), (C2×C4).75(C22×D7), (C7×C42.C2).5C2, (C7×C4⋊C4).184C22, SmallGroup(448,1138)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 732 in 200 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C4 [×2], C4 [×17], C22, C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C14 [×3], C42, C42 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×16], C2×Q8 [×4], Dic7 [×6], Dic7 [×5], C28 [×2], C28 [×6], C2×C14, C4×Q8 [×6], C42.C2, C42.C2 [×5], C4⋊Q8 [×3], Dic14 [×4], Dic14 [×6], C2×Dic7 [×4], C2×Dic7 [×4], C2×C28 [×3], C2×C28 [×4], Q8⋊3Q8, C4×Dic7 [×2], C4×Dic7 [×6], Dic7⋊C4 [×2], Dic7⋊C4 [×10], C4⋊Dic7 [×2], C4⋊Dic7 [×2], C4×C28, C7×C4⋊C4 [×2], C7×C4⋊C4 [×4], C2×Dic14 [×2], C2×Dic14 [×2], C4×Dic14 [×2], Dic7⋊3Q8 [×2], Dic7⋊3Q8 [×2], C28⋊Q8, C28⋊Q8 [×2], Dic7.Q8 [×4], C4.Dic14, C7×C42.C2, Dic14⋊7Q8
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, Q8×D7 [×2], C23×D7, C2×Q8×D7, D7×C4○D4, D4.10D14, Dic14⋊7Q8
Generators and relations
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, ac=ca, dad-1=a13, cbc-1=dbd-1=a14b, dcd-1=c-1 >
►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)
(1 149 15 163)(2 148 16 162)(3 147 17 161)(4 146 18 160)(5 145 19 159)(6 144 20 158)(7 143 21 157)(8 142 22 156)(9 141 23 155)(10 168 24 154)(11 167 25 153)(12 166 26 152)(13 165 27 151)(14 164 28 150)(29 266 43 280)(30 265 44 279)(31 264 45 278)(32 263 46 277)(33 262 47 276)(34 261 48 275)(35 260 49 274)(36 259 50 273)(37 258 51 272)(38 257 52 271)(39 256 53 270)(40 255 54 269)(41 254 55 268)(42 253 56 267)(57 237 71 251)(58 236 72 250)(59 235 73 249)(60 234 74 248)(61 233 75 247)(62 232 76 246)(63 231 77 245)(64 230 78 244)(65 229 79 243)(66 228 80 242)(67 227 81 241)(68 226 82 240)(69 225 83 239)(70 252 84 238)(85 197 99 211)(86 224 100 210)(87 223 101 209)(88 222 102 208)(89 221 103 207)(90 220 104 206)(91 219 105 205)(92 218 106 204)(93 217 107 203)(94 216 108 202)(95 215 109 201)(96 214 110 200)(97 213 111 199)(98 212 112 198)(113 176 127 190)(114 175 128 189)(115 174 129 188)(116 173 130 187)(117 172 131 186)(118 171 132 185)(119 170 133 184)(120 169 134 183)(121 196 135 182)(122 195 136 181)(123 194 137 180)(124 193 138 179)(125 192 139 178)(126 191 140 177)(281 427 295 441)(282 426 296 440)(283 425 297 439)(284 424 298 438)(285 423 299 437)(286 422 300 436)(287 421 301 435)(288 448 302 434)(289 447 303 433)(290 446 304 432)(291 445 305 431)(292 444 306 430)(293 443 307 429)(294 442 308 428)(309 383 323 369)(310 382 324 368)(311 381 325 367)(312 380 326 366)(313 379 327 365)(314 378 328 392)(315 377 329 391)(316 376 330 390)(317 375 331 389)(318 374 332 388)(319 373 333 387)(320 372 334 386)(321 371 335 385)(322 370 336 384)(337 417 351 403)(338 416 352 402)(339 415 353 401)(340 414 354 400)(341 413 355 399)(342 412 356 398)(343 411 357 397)(344 410 358 396)(345 409 359 395)(346 408 360 394)(347 407 361 393)(348 406 362 420)(349 405 363 419)(350 404 364 418)
(1 180 277 215)(2 181 278 216)(3 182 279 217)(4 183 280 218)(5 184 253 219)(6 185 254 220)(7 186 255 221)(8 187 256 222)(9 188 257 223)(10 189 258 224)(11 190 259 197)(12 191 260 198)(13 192 261 199)(14 193 262 200)(15 194 263 201)(16 195 264 202)(17 196 265 203)(18 169 266 204)(19 170 267 205)(20 171 268 206)(21 172 269 207)(22 173 270 208)(23 174 271 209)(24 175 272 210)(25 176 273 211)(26 177 274 212)(27 178 275 213)(28 179 276 214)(29 92 146 134)(30 93 147 135)(31 94 148 136)(32 95 149 137)(33 96 150 138)(34 97 151 139)(35 98 152 140)(36 99 153 113)(37 100 154 114)(38 101 155 115)(39 102 156 116)(40 103 157 117)(41 104 158 118)(42 105 159 119)(43 106 160 120)(44 107 161 121)(45 108 162 122)(46 109 163 123)(47 110 164 124)(48 111 165 125)(49 112 166 126)(50 85 167 127)(51 86 168 128)(52 87 141 129)(53 88 142 130)(54 89 143 131)(55 90 144 132)(56 91 145 133)(57 407 368 446)(58 408 369 447)(59 409 370 448)(60 410 371 421)(61 411 372 422)(62 412 373 423)(63 413 374 424)(64 414 375 425)(65 415 376 426)(66 416 377 427)(67 417 378 428)(68 418 379 429)(69 419 380 430)(70 420 381 431)(71 393 382 432)(72 394 383 433)(73 395 384 434)(74 396 385 435)(75 397 386 436)(76 398 387 437)(77 399 388 438)(78 400 389 439)(79 401 390 440)(80 402 391 441)(81 403 392 442)(82 404 365 443)(83 405 366 444)(84 406 367 445)(225 363 326 306)(226 364 327 307)(227 337 328 308)(228 338 329 281)(229 339 330 282)(230 340 331 283)(231 341 332 284)(232 342 333 285)(233 343 334 286)(234 344 335 287)(235 345 336 288)(236 346 309 289)(237 347 310 290)(238 348 311 291)(239 349 312 292)(240 350 313 293)(241 351 314 294)(242 352 315 295)(243 353 316 296)(244 354 317 297)(245 355 318 298)(246 356 319 299)(247 357 320 300)(248 358 321 301)(249 359 322 302)(250 360 323 303)(251 361 324 304)(252 362 325 305)
(1 364 277 307)(2 349 278 292)(3 362 279 305)(4 347 280 290)(5 360 253 303)(6 345 254 288)(7 358 255 301)(8 343 256 286)(9 356 257 299)(10 341 258 284)(11 354 259 297)(12 339 260 282)(13 352 261 295)(14 337 262 308)(15 350 263 293)(16 363 264 306)(17 348 265 291)(18 361 266 304)(19 346 267 289)(20 359 268 302)(21 344 269 287)(22 357 270 300)(23 342 271 285)(24 355 272 298)(25 340 273 283)(26 353 274 296)(27 338 275 281)(28 351 276 294)(29 432 146 393)(30 445 147 406)(31 430 148 419)(32 443 149 404)(33 428 150 417)(34 441 151 402)(35 426 152 415)(36 439 153 400)(37 424 154 413)(38 437 155 398)(39 422 156 411)(40 435 157 396)(41 448 158 409)(42 433 159 394)(43 446 160 407)(44 431 161 420)(45 444 162 405)(46 429 163 418)(47 442 164 403)(48 427 165 416)(49 440 166 401)(50 425 167 414)(51 438 168 399)(52 423 141 412)(53 436 142 397)(54 421 143 410)(55 434 144 395)(56 447 145 408)(57 106 368 120)(58 91 369 133)(59 104 370 118)(60 89 371 131)(61 102 372 116)(62 87 373 129)(63 100 374 114)(64 85 375 127)(65 98 376 140)(66 111 377 125)(67 96 378 138)(68 109 379 123)(69 94 380 136)(70 107 381 121)(71 92 382 134)(72 105 383 119)(73 90 384 132)(74 103 385 117)(75 88 386 130)(76 101 387 115)(77 86 388 128)(78 99 389 113)(79 112 390 126)(80 97 391 139)(81 110 392 124)(82 95 365 137)(83 108 366 122)(84 93 367 135)(169 251 204 324)(170 236 205 309)(171 249 206 322)(172 234 207 335)(173 247 208 320)(174 232 209 333)(175 245 210 318)(176 230 211 331)(177 243 212 316)(178 228 213 329)(179 241 214 314)(180 226 215 327)(181 239 216 312)(182 252 217 325)(183 237 218 310)(184 250 219 323)(185 235 220 336)(186 248 221 321)(187 233 222 334)(188 246 223 319)(189 231 224 332)(190 244 197 317)(191 229 198 330)(192 242 199 315)(193 227 200 328)(194 240 201 313)(195 225 202 326)(196 238 203 311)
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), (1,149,15,163)(2,148,16,162)(3,147,17,161)(4,146,18,160)(5,145,19,159)(6,144,20,158)(7,143,21,157)(8,142,22,156)(9,141,23,155)(10,168,24,154)(11,167,25,153)(12,166,26,152)(13,165,27,151)(14,164,28,150)(29,266,43,280)(30,265,44,279)(31,264,45,278)(32,263,46,277)(33,262,47,276)(34,261,48,275)(35,260,49,274)(36,259,50,273)(37,258,51,272)(38,257,52,271)(39,256,53,270)(40,255,54,269)(41,254,55,268)(42,253,56,267)(57,237,71,251)(58,236,72,250)(59,235,73,249)(60,234,74,248)(61,233,75,247)(62,232,76,246)(63,231,77,245)(64,230,78,244)(65,229,79,243)(66,228,80,242)(67,227,81,241)(68,226,82,240)(69,225,83,239)(70,252,84,238)(85,197,99,211)(86,224,100,210)(87,223,101,209)(88,222,102,208)(89,221,103,207)(90,220,104,206)(91,219,105,205)(92,218,106,204)(93,217,107,203)(94,216,108,202)(95,215,109,201)(96,214,110,200)(97,213,111,199)(98,212,112,198)(113,176,127,190)(114,175,128,189)(115,174,129,188)(116,173,130,187)(117,172,131,186)(118,171,132,185)(119,170,133,184)(120,169,134,183)(121,196,135,182)(122,195,136,181)(123,194,137,180)(124,193,138,179)(125,192,139,178)(126,191,140,177)(281,427,295,441)(282,426,296,440)(283,425,297,439)(284,424,298,438)(285,423,299,437)(286,422,300,436)(287,421,301,435)(288,448,302,434)(289,447,303,433)(290,446,304,432)(291,445,305,431)(292,444,306,430)(293,443,307,429)(294,442,308,428)(309,383,323,369)(310,382,324,368)(311,381,325,367)(312,380,326,366)(313,379,327,365)(314,378,328,392)(315,377,329,391)(316,376,330,390)(317,375,331,389)(318,374,332,388)(319,373,333,387)(320,372,334,386)(321,371,335,385)(322,370,336,384)(337,417,351,403)(338,416,352,402)(339,415,353,401)(340,414,354,400)(341,413,355,399)(342,412,356,398)(343,411,357,397)(344,410,358,396)(345,409,359,395)(346,408,360,394)(347,407,361,393)(348,406,362,420)(349,405,363,419)(350,404,364,418), (1,180,277,215)(2,181,278,216)(3,182,279,217)(4,183,280,218)(5,184,253,219)(6,185,254,220)(7,186,255,221)(8,187,256,222)(9,188,257,223)(10,189,258,224)(11,190,259,197)(12,191,260,198)(13,192,261,199)(14,193,262,200)(15,194,263,201)(16,195,264,202)(17,196,265,203)(18,169,266,204)(19,170,267,205)(20,171,268,206)(21,172,269,207)(22,173,270,208)(23,174,271,209)(24,175,272,210)(25,176,273,211)(26,177,274,212)(27,178,275,213)(28,179,276,214)(29,92,146,134)(30,93,147,135)(31,94,148,136)(32,95,149,137)(33,96,150,138)(34,97,151,139)(35,98,152,140)(36,99,153,113)(37,100,154,114)(38,101,155,115)(39,102,156,116)(40,103,157,117)(41,104,158,118)(42,105,159,119)(43,106,160,120)(44,107,161,121)(45,108,162,122)(46,109,163,123)(47,110,164,124)(48,111,165,125)(49,112,166,126)(50,85,167,127)(51,86,168,128)(52,87,141,129)(53,88,142,130)(54,89,143,131)(55,90,144,132)(56,91,145,133)(57,407,368,446)(58,408,369,447)(59,409,370,448)(60,410,371,421)(61,411,372,422)(62,412,373,423)(63,413,374,424)(64,414,375,425)(65,415,376,426)(66,416,377,427)(67,417,378,428)(68,418,379,429)(69,419,380,430)(70,420,381,431)(71,393,382,432)(72,394,383,433)(73,395,384,434)(74,396,385,435)(75,397,386,436)(76,398,387,437)(77,399,388,438)(78,400,389,439)(79,401,390,440)(80,402,391,441)(81,403,392,442)(82,404,365,443)(83,405,366,444)(84,406,367,445)(225,363,326,306)(226,364,327,307)(227,337,328,308)(228,338,329,281)(229,339,330,282)(230,340,331,283)(231,341,332,284)(232,342,333,285)(233,343,334,286)(234,344,335,287)(235,345,336,288)(236,346,309,289)(237,347,310,290)(238,348,311,291)(239,349,312,292)(240,350,313,293)(241,351,314,294)(242,352,315,295)(243,353,316,296)(244,354,317,297)(245,355,318,298)(246,356,319,299)(247,357,320,300)(248,358,321,301)(249,359,322,302)(250,360,323,303)(251,361,324,304)(252,362,325,305), (1,364,277,307)(2,349,278,292)(3,362,279,305)(4,347,280,290)(5,360,253,303)(6,345,254,288)(7,358,255,301)(8,343,256,286)(9,356,257,299)(10,341,258,284)(11,354,259,297)(12,339,260,282)(13,352,261,295)(14,337,262,308)(15,350,263,293)(16,363,264,306)(17,348,265,291)(18,361,266,304)(19,346,267,289)(20,359,268,302)(21,344,269,287)(22,357,270,300)(23,342,271,285)(24,355,272,298)(25,340,273,283)(26,353,274,296)(27,338,275,281)(28,351,276,294)(29,432,146,393)(30,445,147,406)(31,430,148,419)(32,443,149,404)(33,428,150,417)(34,441,151,402)(35,426,152,415)(36,439,153,400)(37,424,154,413)(38,437,155,398)(39,422,156,411)(40,435,157,396)(41,448,158,409)(42,433,159,394)(43,446,160,407)(44,431,161,420)(45,444,162,405)(46,429,163,418)(47,442,164,403)(48,427,165,416)(49,440,166,401)(50,425,167,414)(51,438,168,399)(52,423,141,412)(53,436,142,397)(54,421,143,410)(55,434,144,395)(56,447,145,408)(57,106,368,120)(58,91,369,133)(59,104,370,118)(60,89,371,131)(61,102,372,116)(62,87,373,129)(63,100,374,114)(64,85,375,127)(65,98,376,140)(66,111,377,125)(67,96,378,138)(68,109,379,123)(69,94,380,136)(70,107,381,121)(71,92,382,134)(72,105,383,119)(73,90,384,132)(74,103,385,117)(75,88,386,130)(76,101,387,115)(77,86,388,128)(78,99,389,113)(79,112,390,126)(80,97,391,139)(81,110,392,124)(82,95,365,137)(83,108,366,122)(84,93,367,135)(169,251,204,324)(170,236,205,309)(171,249,206,322)(172,234,207,335)(173,247,208,320)(174,232,209,333)(175,245,210,318)(176,230,211,331)(177,243,212,316)(178,228,213,329)(179,241,214,314)(180,226,215,327)(181,239,216,312)(182,252,217,325)(183,237,218,310)(184,250,219,323)(185,235,220,336)(186,248,221,321)(187,233,222,334)(188,246,223,319)(189,231,224,332)(190,244,197,317)(191,229,198,330)(192,242,199,315)(193,227,200,328)(194,240,201,313)(195,225,202,326)(196,238,203,311)>;
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), (1,149,15,163)(2,148,16,162)(3,147,17,161)(4,146,18,160)(5,145,19,159)(6,144,20,158)(7,143,21,157)(8,142,22,156)(9,141,23,155)(10,168,24,154)(11,167,25,153)(12,166,26,152)(13,165,27,151)(14,164,28,150)(29,266,43,280)(30,265,44,279)(31,264,45,278)(32,263,46,277)(33,262,47,276)(34,261,48,275)(35,260,49,274)(36,259,50,273)(37,258,51,272)(38,257,52,271)(39,256,53,270)(40,255,54,269)(41,254,55,268)(42,253,56,267)(57,237,71,251)(58,236,72,250)(59,235,73,249)(60,234,74,248)(61,233,75,247)(62,232,76,246)(63,231,77,245)(64,230,78,244)(65,229,79,243)(66,228,80,242)(67,227,81,241)(68,226,82,240)(69,225,83,239)(70,252,84,238)(85,197,99,211)(86,224,100,210)(87,223,101,209)(88,222,102,208)(89,221,103,207)(90,220,104,206)(91,219,105,205)(92,218,106,204)(93,217,107,203)(94,216,108,202)(95,215,109,201)(96,214,110,200)(97,213,111,199)(98,212,112,198)(113,176,127,190)(114,175,128,189)(115,174,129,188)(116,173,130,187)(117,172,131,186)(118,171,132,185)(119,170,133,184)(120,169,134,183)(121,196,135,182)(122,195,136,181)(123,194,137,180)(124,193,138,179)(125,192,139,178)(126,191,140,177)(281,427,295,441)(282,426,296,440)(283,425,297,439)(284,424,298,438)(285,423,299,437)(286,422,300,436)(287,421,301,435)(288,448,302,434)(289,447,303,433)(290,446,304,432)(291,445,305,431)(292,444,306,430)(293,443,307,429)(294,442,308,428)(309,383,323,369)(310,382,324,368)(311,381,325,367)(312,380,326,366)(313,379,327,365)(314,378,328,392)(315,377,329,391)(316,376,330,390)(317,375,331,389)(318,374,332,388)(319,373,333,387)(320,372,334,386)(321,371,335,385)(322,370,336,384)(337,417,351,403)(338,416,352,402)(339,415,353,401)(340,414,354,400)(341,413,355,399)(342,412,356,398)(343,411,357,397)(344,410,358,396)(345,409,359,395)(346,408,360,394)(347,407,361,393)(348,406,362,420)(349,405,363,419)(350,404,364,418), (1,180,277,215)(2,181,278,216)(3,182,279,217)(4,183,280,218)(5,184,253,219)(6,185,254,220)(7,186,255,221)(8,187,256,222)(9,188,257,223)(10,189,258,224)(11,190,259,197)(12,191,260,198)(13,192,261,199)(14,193,262,200)(15,194,263,201)(16,195,264,202)(17,196,265,203)(18,169,266,204)(19,170,267,205)(20,171,268,206)(21,172,269,207)(22,173,270,208)(23,174,271,209)(24,175,272,210)(25,176,273,211)(26,177,274,212)(27,178,275,213)(28,179,276,214)(29,92,146,134)(30,93,147,135)(31,94,148,136)(32,95,149,137)(33,96,150,138)(34,97,151,139)(35,98,152,140)(36,99,153,113)(37,100,154,114)(38,101,155,115)(39,102,156,116)(40,103,157,117)(41,104,158,118)(42,105,159,119)(43,106,160,120)(44,107,161,121)(45,108,162,122)(46,109,163,123)(47,110,164,124)(48,111,165,125)(49,112,166,126)(50,85,167,127)(51,86,168,128)(52,87,141,129)(53,88,142,130)(54,89,143,131)(55,90,144,132)(56,91,145,133)(57,407,368,446)(58,408,369,447)(59,409,370,448)(60,410,371,421)(61,411,372,422)(62,412,373,423)(63,413,374,424)(64,414,375,425)(65,415,376,426)(66,416,377,427)(67,417,378,428)(68,418,379,429)(69,419,380,430)(70,420,381,431)(71,393,382,432)(72,394,383,433)(73,395,384,434)(74,396,385,435)(75,397,386,436)(76,398,387,437)(77,399,388,438)(78,400,389,439)(79,401,390,440)(80,402,391,441)(81,403,392,442)(82,404,365,443)(83,405,366,444)(84,406,367,445)(225,363,326,306)(226,364,327,307)(227,337,328,308)(228,338,329,281)(229,339,330,282)(230,340,331,283)(231,341,332,284)(232,342,333,285)(233,343,334,286)(234,344,335,287)(235,345,336,288)(236,346,309,289)(237,347,310,290)(238,348,311,291)(239,349,312,292)(240,350,313,293)(241,351,314,294)(242,352,315,295)(243,353,316,296)(244,354,317,297)(245,355,318,298)(246,356,319,299)(247,357,320,300)(248,358,321,301)(249,359,322,302)(250,360,323,303)(251,361,324,304)(252,362,325,305), (1,364,277,307)(2,349,278,292)(3,362,279,305)(4,347,280,290)(5,360,253,303)(6,345,254,288)(7,358,255,301)(8,343,256,286)(9,356,257,299)(10,341,258,284)(11,354,259,297)(12,339,260,282)(13,352,261,295)(14,337,262,308)(15,350,263,293)(16,363,264,306)(17,348,265,291)(18,361,266,304)(19,346,267,289)(20,359,268,302)(21,344,269,287)(22,357,270,300)(23,342,271,285)(24,355,272,298)(25,340,273,283)(26,353,274,296)(27,338,275,281)(28,351,276,294)(29,432,146,393)(30,445,147,406)(31,430,148,419)(32,443,149,404)(33,428,150,417)(34,441,151,402)(35,426,152,415)(36,439,153,400)(37,424,154,413)(38,437,155,398)(39,422,156,411)(40,435,157,396)(41,448,158,409)(42,433,159,394)(43,446,160,407)(44,431,161,420)(45,444,162,405)(46,429,163,418)(47,442,164,403)(48,427,165,416)(49,440,166,401)(50,425,167,414)(51,438,168,399)(52,423,141,412)(53,436,142,397)(54,421,143,410)(55,434,144,395)(56,447,145,408)(57,106,368,120)(58,91,369,133)(59,104,370,118)(60,89,371,131)(61,102,372,116)(62,87,373,129)(63,100,374,114)(64,85,375,127)(65,98,376,140)(66,111,377,125)(67,96,378,138)(68,109,379,123)(69,94,380,136)(70,107,381,121)(71,92,382,134)(72,105,383,119)(73,90,384,132)(74,103,385,117)(75,88,386,130)(76,101,387,115)(77,86,388,128)(78,99,389,113)(79,112,390,126)(80,97,391,139)(81,110,392,124)(82,95,365,137)(83,108,366,122)(84,93,367,135)(169,251,204,324)(170,236,205,309)(171,249,206,322)(172,234,207,335)(173,247,208,320)(174,232,209,333)(175,245,210,318)(176,230,211,331)(177,243,212,316)(178,228,213,329)(179,241,214,314)(180,226,215,327)(181,239,216,312)(182,252,217,325)(183,237,218,310)(184,250,219,323)(185,235,220,336)(186,248,221,321)(187,233,222,334)(188,246,223,319)(189,231,224,332)(190,244,197,317)(191,229,198,330)(192,242,199,315)(193,227,200,328)(194,240,201,313)(195,225,202,326)(196,238,203,311) );
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)], [(1,149,15,163),(2,148,16,162),(3,147,17,161),(4,146,18,160),(5,145,19,159),(6,144,20,158),(7,143,21,157),(8,142,22,156),(9,141,23,155),(10,168,24,154),(11,167,25,153),(12,166,26,152),(13,165,27,151),(14,164,28,150),(29,266,43,280),(30,265,44,279),(31,264,45,278),(32,263,46,277),(33,262,47,276),(34,261,48,275),(35,260,49,274),(36,259,50,273),(37,258,51,272),(38,257,52,271),(39,256,53,270),(40,255,54,269),(41,254,55,268),(42,253,56,267),(57,237,71,251),(58,236,72,250),(59,235,73,249),(60,234,74,248),(61,233,75,247),(62,232,76,246),(63,231,77,245),(64,230,78,244),(65,229,79,243),(66,228,80,242),(67,227,81,241),(68,226,82,240),(69,225,83,239),(70,252,84,238),(85,197,99,211),(86,224,100,210),(87,223,101,209),(88,222,102,208),(89,221,103,207),(90,220,104,206),(91,219,105,205),(92,218,106,204),(93,217,107,203),(94,216,108,202),(95,215,109,201),(96,214,110,200),(97,213,111,199),(98,212,112,198),(113,176,127,190),(114,175,128,189),(115,174,129,188),(116,173,130,187),(117,172,131,186),(118,171,132,185),(119,170,133,184),(120,169,134,183),(121,196,135,182),(122,195,136,181),(123,194,137,180),(124,193,138,179),(125,192,139,178),(126,191,140,177),(281,427,295,441),(282,426,296,440),(283,425,297,439),(284,424,298,438),(285,423,299,437),(286,422,300,436),(287,421,301,435),(288,448,302,434),(289,447,303,433),(290,446,304,432),(291,445,305,431),(292,444,306,430),(293,443,307,429),(294,442,308,428),(309,383,323,369),(310,382,324,368),(311,381,325,367),(312,380,326,366),(313,379,327,365),(314,378,328,392),(315,377,329,391),(316,376,330,390),(317,375,331,389),(318,374,332,388),(319,373,333,387),(320,372,334,386),(321,371,335,385),(322,370,336,384),(337,417,351,403),(338,416,352,402),(339,415,353,401),(340,414,354,400),(341,413,355,399),(342,412,356,398),(343,411,357,397),(344,410,358,396),(345,409,359,395),(346,408,360,394),(347,407,361,393),(348,406,362,420),(349,405,363,419),(350,404,364,418)], [(1,180,277,215),(2,181,278,216),(3,182,279,217),(4,183,280,218),(5,184,253,219),(6,185,254,220),(7,186,255,221),(8,187,256,222),(9,188,257,223),(10,189,258,224),(11,190,259,197),(12,191,260,198),(13,192,261,199),(14,193,262,200),(15,194,263,201),(16,195,264,202),(17,196,265,203),(18,169,266,204),(19,170,267,205),(20,171,268,206),(21,172,269,207),(22,173,270,208),(23,174,271,209),(24,175,272,210),(25,176,273,211),(26,177,274,212),(27,178,275,213),(28,179,276,214),(29,92,146,134),(30,93,147,135),(31,94,148,136),(32,95,149,137),(33,96,150,138),(34,97,151,139),(35,98,152,140),(36,99,153,113),(37,100,154,114),(38,101,155,115),(39,102,156,116),(40,103,157,117),(41,104,158,118),(42,105,159,119),(43,106,160,120),(44,107,161,121),(45,108,162,122),(46,109,163,123),(47,110,164,124),(48,111,165,125),(49,112,166,126),(50,85,167,127),(51,86,168,128),(52,87,141,129),(53,88,142,130),(54,89,143,131),(55,90,144,132),(56,91,145,133),(57,407,368,446),(58,408,369,447),(59,409,370,448),(60,410,371,421),(61,411,372,422),(62,412,373,423),(63,413,374,424),(64,414,375,425),(65,415,376,426),(66,416,377,427),(67,417,378,428),(68,418,379,429),(69,419,380,430),(70,420,381,431),(71,393,382,432),(72,394,383,433),(73,395,384,434),(74,396,385,435),(75,397,386,436),(76,398,387,437),(77,399,388,438),(78,400,389,439),(79,401,390,440),(80,402,391,441),(81,403,392,442),(82,404,365,443),(83,405,366,444),(84,406,367,445),(225,363,326,306),(226,364,327,307),(227,337,328,308),(228,338,329,281),(229,339,330,282),(230,340,331,283),(231,341,332,284),(232,342,333,285),(233,343,334,286),(234,344,335,287),(235,345,336,288),(236,346,309,289),(237,347,310,290),(238,348,311,291),(239,349,312,292),(240,350,313,293),(241,351,314,294),(242,352,315,295),(243,353,316,296),(244,354,317,297),(245,355,318,298),(246,356,319,299),(247,357,320,300),(248,358,321,301),(249,359,322,302),(250,360,323,303),(251,361,324,304),(252,362,325,305)], [(1,364,277,307),(2,349,278,292),(3,362,279,305),(4,347,280,290),(5,360,253,303),(6,345,254,288),(7,358,255,301),(8,343,256,286),(9,356,257,299),(10,341,258,284),(11,354,259,297),(12,339,260,282),(13,352,261,295),(14,337,262,308),(15,350,263,293),(16,363,264,306),(17,348,265,291),(18,361,266,304),(19,346,267,289),(20,359,268,302),(21,344,269,287),(22,357,270,300),(23,342,271,285),(24,355,272,298),(25,340,273,283),(26,353,274,296),(27,338,275,281),(28,351,276,294),(29,432,146,393),(30,445,147,406),(31,430,148,419),(32,443,149,404),(33,428,150,417),(34,441,151,402),(35,426,152,415),(36,439,153,400),(37,424,154,413),(38,437,155,398),(39,422,156,411),(40,435,157,396),(41,448,158,409),(42,433,159,394),(43,446,160,407),(44,431,161,420),(45,444,162,405),(46,429,163,418),(47,442,164,403),(48,427,165,416),(49,440,166,401),(50,425,167,414),(51,438,168,399),(52,423,141,412),(53,436,142,397),(54,421,143,410),(55,434,144,395),(56,447,145,408),(57,106,368,120),(58,91,369,133),(59,104,370,118),(60,89,371,131),(61,102,372,116),(62,87,373,129),(63,100,374,114),(64,85,375,127),(65,98,376,140),(66,111,377,125),(67,96,378,138),(68,109,379,123),(69,94,380,136),(70,107,381,121),(71,92,382,134),(72,105,383,119),(73,90,384,132),(74,103,385,117),(75,88,386,130),(76,101,387,115),(77,86,388,128),(78,99,389,113),(79,112,390,126),(80,97,391,139),(81,110,392,124),(82,95,365,137),(83,108,366,122),(84,93,367,135),(169,251,204,324),(170,236,205,309),(171,249,206,322),(172,234,207,335),(173,247,208,320),(174,232,209,333),(175,245,210,318),(176,230,211,331),(177,243,212,316),(178,228,213,329),(179,241,214,314),(180,226,215,327),(181,239,216,312),(182,252,217,325),(183,237,218,310),(184,250,219,323),(185,235,220,336),(186,248,221,321),(187,233,222,334),(188,246,223,319),(189,231,224,332),(190,244,197,317),(191,229,198,330),(192,242,199,315),(193,227,200,328),(194,240,201,313),(195,225,202,326),(196,238,203,311)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 10 | 23 | 0 | 0 | 0 | 0 |
| 12 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 13 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 8 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 21 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 25 | 14 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 12 |
| 0 | 0 | 0 | 0 | 10 | 16 |
| 25 | 14 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 21 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 18 |
| 0 | 0 | 0 | 0 | 23 | 22 |
G:=sub<GL(6,GF(29))| [10,12,0,0,0,0,23,19,0,0,0,0,0,0,11,13,0,0,0,0,1,25,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[8,8,0,0,0,0,10,21,0,0,0,0,0,0,22,6,0,0,0,0,21,7,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[25,1,0,0,0,0,14,4,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,13,10,0,0,0,0,12,16],[25,1,0,0,0,0,14,4,0,0,0,0,0,0,22,6,0,0,0,0,21,7,0,0,0,0,0,0,7,23,0,0,0,0,18,22] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | 2- (1+4) | Q8×D7 | D7×C4○D4 | D4.10D14 |
| kernel | Dic14⋊7Q8 | C4×Dic14 | Dic7⋊3Q8 | C28⋊Q8 | Dic7.Q8 | C4.Dic14 | C7×C42.C2 | Dic14 | C42.C2 | Dic7 | C42 | C4⋊C4 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 4 | 3 | 4 | 1 | 1 | 4 | 3 | 4 | 3 | 18 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
Dic_{14}\rtimes_7Q_8 % in TeX
G:=Group("Dic14:7Q8"); // GroupNames label
G:=SmallGroup(448,1138);
// by ID
G=gap.SmallGroup(448,1138);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,219,268,1571,297,192,18822]);
// 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^13,c*b*c^-1=d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations
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