metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊2Q8, Dic7.4Q16, C4.5(Q8×D7), C28⋊Q8.8C2, C7⋊3(C4.Q16), C2.D8.5D7, (C2×C8).25D14, C2.13(D7×Q16), C28.18(C2×Q8), C4⋊C4.168D14, C14.22(C2×Q16), C4.78(C4○D28), Dic7⋊C8.10C2, C14.Q16.8C2, C28.Q8.8C2, C22.224(D4×D7), C28.170(C4○D4), C2.20(D8⋊D7), C14.38(C8⋊C22), (C2×C56).239C22, (C2×C28).291C23, Dic7⋊3Q8.8C2, (C2×Dic7).166D4, C14.39(C22⋊Q8), C2.16(D14⋊Q8), C28.44D4.10C2, C4⋊Dic7.117C22, (C4×Dic7).34C22, (C2×Dic14).87C22, (C2×C7⋊C8).65C22, (C7×C2.D8).11C2, (C2×C14).296(C2×D4), (C7×C4⋊C4).84C22, (C2×C4).394(C22×D7), SmallGroup(448,409)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊2Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, cac-1=a15, dad-1=a13, cbc-1=a21b, dbd-1=a14b, dcd-1=c-1 >
Subgroups: 460 in 96 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C2.D8, C2.D8, C4×Q8, C4⋊Q8, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C4.Q16, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×Dic14, C28.Q8, C14.Q16, Dic7⋊C8, C28.44D4, C7×C2.D8, Dic7⋊3Q8, C28⋊Q8, Dic14⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, Q16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×Q16, C8⋊C22, C22×D7, C4.Q16, C4○D28, D4×D7, Q8×D7, D14⋊Q8, D8⋊D7, D7×Q16, Dic14⋊2Q8
►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)
(1 387 15 373)(2 386 16 372)(3 385 17 371)(4 384 18 370)(5 383 19 369)(6 382 20 368)(7 381 21 367)(8 380 22 366)(9 379 23 365)(10 378 24 392)(11 377 25 391)(12 376 26 390)(13 375 27 389)(14 374 28 388)(29 165 43 151)(30 164 44 150)(31 163 45 149)(32 162 46 148)(33 161 47 147)(34 160 48 146)(35 159 49 145)(36 158 50 144)(37 157 51 143)(38 156 52 142)(39 155 53 141)(40 154 54 168)(41 153 55 167)(42 152 56 166)(57 331 71 317)(58 330 72 316)(59 329 73 315)(60 328 74 314)(61 327 75 313)(62 326 76 312)(63 325 77 311)(64 324 78 310)(65 323 79 309)(66 322 80 336)(67 321 81 335)(68 320 82 334)(69 319 83 333)(70 318 84 332)(85 421 99 435)(86 448 100 434)(87 447 101 433)(88 446 102 432)(89 445 103 431)(90 444 104 430)(91 443 105 429)(92 442 106 428)(93 441 107 427)(94 440 108 426)(95 439 109 425)(96 438 110 424)(97 437 111 423)(98 436 112 422)(113 227 127 241)(114 226 128 240)(115 225 129 239)(116 252 130 238)(117 251 131 237)(118 250 132 236)(119 249 133 235)(120 248 134 234)(121 247 135 233)(122 246 136 232)(123 245 137 231)(124 244 138 230)(125 243 139 229)(126 242 140 228)(169 350 183 364)(170 349 184 363)(171 348 185 362)(172 347 186 361)(173 346 187 360)(174 345 188 359)(175 344 189 358)(176 343 190 357)(177 342 191 356)(178 341 192 355)(179 340 193 354)(180 339 194 353)(181 338 195 352)(182 337 196 351)(197 258 211 272)(198 257 212 271)(199 256 213 270)(200 255 214 269)(201 254 215 268)(202 253 216 267)(203 280 217 266)(204 279 218 265)(205 278 219 264)(206 277 220 263)(207 276 221 262)(208 275 222 261)(209 274 223 260)(210 273 224 259)(281 415 295 401)(282 414 296 400)(283 413 297 399)(284 412 298 398)(285 411 299 397)(286 410 300 396)(287 409 301 395)(288 408 302 394)(289 407 303 393)(290 406 304 420)(291 405 305 419)(292 404 306 418)(293 403 307 417)(294 402 308 416)
(1 218 125 283)(2 205 126 298)(3 220 127 285)(4 207 128 300)(5 222 129 287)(6 209 130 302)(7 224 131 289)(8 211 132 304)(9 198 133 291)(10 213 134 306)(11 200 135 293)(12 215 136 308)(13 202 137 295)(14 217 138 282)(15 204 139 297)(16 219 140 284)(17 206 113 299)(18 221 114 286)(19 208 115 301)(20 223 116 288)(21 210 117 303)(22 197 118 290)(23 212 119 305)(24 199 120 292)(25 214 121 307)(26 201 122 294)(27 216 123 281)(28 203 124 296)(29 447 68 181)(30 434 69 196)(31 421 70 183)(32 436 71 170)(33 423 72 185)(34 438 73 172)(35 425 74 187)(36 440 75 174)(37 427 76 189)(38 442 77 176)(39 429 78 191)(40 444 79 178)(41 431 80 193)(42 446 81 180)(43 433 82 195)(44 448 83 182)(45 435 84 169)(46 422 57 184)(47 437 58 171)(48 424 59 186)(49 439 60 173)(50 426 61 188)(51 441 62 175)(52 428 63 190)(53 443 64 177)(54 430 65 192)(55 445 66 179)(56 432 67 194)(85 311 350 156)(86 326 351 143)(87 313 352 158)(88 328 353 145)(89 315 354 160)(90 330 355 147)(91 317 356 162)(92 332 357 149)(93 319 358 164)(94 334 359 151)(95 321 360 166)(96 336 361 153)(97 323 362 168)(98 310 363 155)(99 325 364 142)(100 312 337 157)(101 327 338 144)(102 314 339 159)(103 329 340 146)(104 316 341 161)(105 331 342 148)(106 318 343 163)(107 333 344 150)(108 320 345 165)(109 335 346 152)(110 322 347 167)(111 309 348 154)(112 324 349 141)(225 402 369 254)(226 417 370 269)(227 404 371 256)(228 419 372 271)(229 406 373 258)(230 393 374 273)(231 408 375 260)(232 395 376 275)(233 410 377 262)(234 397 378 277)(235 412 379 264)(236 399 380 279)(237 414 381 266)(238 401 382 253)(239 416 383 268)(240 403 384 255)(241 418 385 270)(242 405 386 257)(243 420 387 272)(244 407 388 259)(245 394 389 274)(246 409 390 261)(247 396 391 276)(248 411 392 263)(249 398 365 278)(250 413 366 265)(251 400 367 280)(252 415 368 267)
(1 323 125 168)(2 336 126 153)(3 321 127 166)(4 334 128 151)(5 319 129 164)(6 332 130 149)(7 317 131 162)(8 330 132 147)(9 315 133 160)(10 328 134 145)(11 313 135 158)(12 326 136 143)(13 311 137 156)(14 324 138 141)(15 309 139 154)(16 322 140 167)(17 335 113 152)(18 320 114 165)(19 333 115 150)(20 318 116 163)(21 331 117 148)(22 316 118 161)(23 329 119 146)(24 314 120 159)(25 327 121 144)(26 312 122 157)(27 325 123 142)(28 310 124 155)(29 370 68 226)(30 383 69 239)(31 368 70 252)(32 381 71 237)(33 366 72 250)(34 379 73 235)(35 392 74 248)(36 377 75 233)(37 390 76 246)(38 375 77 231)(39 388 78 244)(40 373 79 229)(41 386 80 242)(42 371 81 227)(43 384 82 240)(44 369 83 225)(45 382 84 238)(46 367 57 251)(47 380 58 236)(48 365 59 249)(49 378 60 234)(50 391 61 247)(51 376 62 232)(52 389 63 245)(53 374 64 230)(54 387 65 243)(55 372 66 228)(56 385 67 241)(85 295 350 202)(86 308 351 215)(87 293 352 200)(88 306 353 213)(89 291 354 198)(90 304 355 211)(91 289 356 224)(92 302 357 209)(93 287 358 222)(94 300 359 207)(95 285 360 220)(96 298 361 205)(97 283 362 218)(98 296 363 203)(99 281 364 216)(100 294 337 201)(101 307 338 214)(102 292 339 199)(103 305 340 212)(104 290 341 197)(105 303 342 210)(106 288 343 223)(107 301 344 208)(108 286 345 221)(109 299 346 206)(110 284 347 219)(111 297 348 204)(112 282 349 217)(169 253 435 401)(170 266 436 414)(171 279 437 399)(172 264 438 412)(173 277 439 397)(174 262 440 410)(175 275 441 395)(176 260 442 408)(177 273 443 393)(178 258 444 406)(179 271 445 419)(180 256 446 404)(181 269 447 417)(182 254 448 402)(183 267 421 415)(184 280 422 400)(185 265 423 413)(186 278 424 398)(187 263 425 411)(188 276 426 396)(189 261 427 409)(190 274 428 394)(191 259 429 407)(192 272 430 420)(193 257 431 405)(194 270 432 418)(195 255 433 403)(196 268 434 416)
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), (1,387,15,373)(2,386,16,372)(3,385,17,371)(4,384,18,370)(5,383,19,369)(6,382,20,368)(7,381,21,367)(8,380,22,366)(9,379,23,365)(10,378,24,392)(11,377,25,391)(12,376,26,390)(13,375,27,389)(14,374,28,388)(29,165,43,151)(30,164,44,150)(31,163,45,149)(32,162,46,148)(33,161,47,147)(34,160,48,146)(35,159,49,145)(36,158,50,144)(37,157,51,143)(38,156,52,142)(39,155,53,141)(40,154,54,168)(41,153,55,167)(42,152,56,166)(57,331,71,317)(58,330,72,316)(59,329,73,315)(60,328,74,314)(61,327,75,313)(62,326,76,312)(63,325,77,311)(64,324,78,310)(65,323,79,309)(66,322,80,336)(67,321,81,335)(68,320,82,334)(69,319,83,333)(70,318,84,332)(85,421,99,435)(86,448,100,434)(87,447,101,433)(88,446,102,432)(89,445,103,431)(90,444,104,430)(91,443,105,429)(92,442,106,428)(93,441,107,427)(94,440,108,426)(95,439,109,425)(96,438,110,424)(97,437,111,423)(98,436,112,422)(113,227,127,241)(114,226,128,240)(115,225,129,239)(116,252,130,238)(117,251,131,237)(118,250,132,236)(119,249,133,235)(120,248,134,234)(121,247,135,233)(122,246,136,232)(123,245,137,231)(124,244,138,230)(125,243,139,229)(126,242,140,228)(169,350,183,364)(170,349,184,363)(171,348,185,362)(172,347,186,361)(173,346,187,360)(174,345,188,359)(175,344,189,358)(176,343,190,357)(177,342,191,356)(178,341,192,355)(179,340,193,354)(180,339,194,353)(181,338,195,352)(182,337,196,351)(197,258,211,272)(198,257,212,271)(199,256,213,270)(200,255,214,269)(201,254,215,268)(202,253,216,267)(203,280,217,266)(204,279,218,265)(205,278,219,264)(206,277,220,263)(207,276,221,262)(208,275,222,261)(209,274,223,260)(210,273,224,259)(281,415,295,401)(282,414,296,400)(283,413,297,399)(284,412,298,398)(285,411,299,397)(286,410,300,396)(287,409,301,395)(288,408,302,394)(289,407,303,393)(290,406,304,420)(291,405,305,419)(292,404,306,418)(293,403,307,417)(294,402,308,416), (1,218,125,283)(2,205,126,298)(3,220,127,285)(4,207,128,300)(5,222,129,287)(6,209,130,302)(7,224,131,289)(8,211,132,304)(9,198,133,291)(10,213,134,306)(11,200,135,293)(12,215,136,308)(13,202,137,295)(14,217,138,282)(15,204,139,297)(16,219,140,284)(17,206,113,299)(18,221,114,286)(19,208,115,301)(20,223,116,288)(21,210,117,303)(22,197,118,290)(23,212,119,305)(24,199,120,292)(25,214,121,307)(26,201,122,294)(27,216,123,281)(28,203,124,296)(29,447,68,181)(30,434,69,196)(31,421,70,183)(32,436,71,170)(33,423,72,185)(34,438,73,172)(35,425,74,187)(36,440,75,174)(37,427,76,189)(38,442,77,176)(39,429,78,191)(40,444,79,178)(41,431,80,193)(42,446,81,180)(43,433,82,195)(44,448,83,182)(45,435,84,169)(46,422,57,184)(47,437,58,171)(48,424,59,186)(49,439,60,173)(50,426,61,188)(51,441,62,175)(52,428,63,190)(53,443,64,177)(54,430,65,192)(55,445,66,179)(56,432,67,194)(85,311,350,156)(86,326,351,143)(87,313,352,158)(88,328,353,145)(89,315,354,160)(90,330,355,147)(91,317,356,162)(92,332,357,149)(93,319,358,164)(94,334,359,151)(95,321,360,166)(96,336,361,153)(97,323,362,168)(98,310,363,155)(99,325,364,142)(100,312,337,157)(101,327,338,144)(102,314,339,159)(103,329,340,146)(104,316,341,161)(105,331,342,148)(106,318,343,163)(107,333,344,150)(108,320,345,165)(109,335,346,152)(110,322,347,167)(111,309,348,154)(112,324,349,141)(225,402,369,254)(226,417,370,269)(227,404,371,256)(228,419,372,271)(229,406,373,258)(230,393,374,273)(231,408,375,260)(232,395,376,275)(233,410,377,262)(234,397,378,277)(235,412,379,264)(236,399,380,279)(237,414,381,266)(238,401,382,253)(239,416,383,268)(240,403,384,255)(241,418,385,270)(242,405,386,257)(243,420,387,272)(244,407,388,259)(245,394,389,274)(246,409,390,261)(247,396,391,276)(248,411,392,263)(249,398,365,278)(250,413,366,265)(251,400,367,280)(252,415,368,267), (1,323,125,168)(2,336,126,153)(3,321,127,166)(4,334,128,151)(5,319,129,164)(6,332,130,149)(7,317,131,162)(8,330,132,147)(9,315,133,160)(10,328,134,145)(11,313,135,158)(12,326,136,143)(13,311,137,156)(14,324,138,141)(15,309,139,154)(16,322,140,167)(17,335,113,152)(18,320,114,165)(19,333,115,150)(20,318,116,163)(21,331,117,148)(22,316,118,161)(23,329,119,146)(24,314,120,159)(25,327,121,144)(26,312,122,157)(27,325,123,142)(28,310,124,155)(29,370,68,226)(30,383,69,239)(31,368,70,252)(32,381,71,237)(33,366,72,250)(34,379,73,235)(35,392,74,248)(36,377,75,233)(37,390,76,246)(38,375,77,231)(39,388,78,244)(40,373,79,229)(41,386,80,242)(42,371,81,227)(43,384,82,240)(44,369,83,225)(45,382,84,238)(46,367,57,251)(47,380,58,236)(48,365,59,249)(49,378,60,234)(50,391,61,247)(51,376,62,232)(52,389,63,245)(53,374,64,230)(54,387,65,243)(55,372,66,228)(56,385,67,241)(85,295,350,202)(86,308,351,215)(87,293,352,200)(88,306,353,213)(89,291,354,198)(90,304,355,211)(91,289,356,224)(92,302,357,209)(93,287,358,222)(94,300,359,207)(95,285,360,220)(96,298,361,205)(97,283,362,218)(98,296,363,203)(99,281,364,216)(100,294,337,201)(101,307,338,214)(102,292,339,199)(103,305,340,212)(104,290,341,197)(105,303,342,210)(106,288,343,223)(107,301,344,208)(108,286,345,221)(109,299,346,206)(110,284,347,219)(111,297,348,204)(112,282,349,217)(169,253,435,401)(170,266,436,414)(171,279,437,399)(172,264,438,412)(173,277,439,397)(174,262,440,410)(175,275,441,395)(176,260,442,408)(177,273,443,393)(178,258,444,406)(179,271,445,419)(180,256,446,404)(181,269,447,417)(182,254,448,402)(183,267,421,415)(184,280,422,400)(185,265,423,413)(186,278,424,398)(187,263,425,411)(188,276,426,396)(189,261,427,409)(190,274,428,394)(191,259,429,407)(192,272,430,420)(193,257,431,405)(194,270,432,418)(195,255,433,403)(196,268,434,416)>;
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), (1,387,15,373)(2,386,16,372)(3,385,17,371)(4,384,18,370)(5,383,19,369)(6,382,20,368)(7,381,21,367)(8,380,22,366)(9,379,23,365)(10,378,24,392)(11,377,25,391)(12,376,26,390)(13,375,27,389)(14,374,28,388)(29,165,43,151)(30,164,44,150)(31,163,45,149)(32,162,46,148)(33,161,47,147)(34,160,48,146)(35,159,49,145)(36,158,50,144)(37,157,51,143)(38,156,52,142)(39,155,53,141)(40,154,54,168)(41,153,55,167)(42,152,56,166)(57,331,71,317)(58,330,72,316)(59,329,73,315)(60,328,74,314)(61,327,75,313)(62,326,76,312)(63,325,77,311)(64,324,78,310)(65,323,79,309)(66,322,80,336)(67,321,81,335)(68,320,82,334)(69,319,83,333)(70,318,84,332)(85,421,99,435)(86,448,100,434)(87,447,101,433)(88,446,102,432)(89,445,103,431)(90,444,104,430)(91,443,105,429)(92,442,106,428)(93,441,107,427)(94,440,108,426)(95,439,109,425)(96,438,110,424)(97,437,111,423)(98,436,112,422)(113,227,127,241)(114,226,128,240)(115,225,129,239)(116,252,130,238)(117,251,131,237)(118,250,132,236)(119,249,133,235)(120,248,134,234)(121,247,135,233)(122,246,136,232)(123,245,137,231)(124,244,138,230)(125,243,139,229)(126,242,140,228)(169,350,183,364)(170,349,184,363)(171,348,185,362)(172,347,186,361)(173,346,187,360)(174,345,188,359)(175,344,189,358)(176,343,190,357)(177,342,191,356)(178,341,192,355)(179,340,193,354)(180,339,194,353)(181,338,195,352)(182,337,196,351)(197,258,211,272)(198,257,212,271)(199,256,213,270)(200,255,214,269)(201,254,215,268)(202,253,216,267)(203,280,217,266)(204,279,218,265)(205,278,219,264)(206,277,220,263)(207,276,221,262)(208,275,222,261)(209,274,223,260)(210,273,224,259)(281,415,295,401)(282,414,296,400)(283,413,297,399)(284,412,298,398)(285,411,299,397)(286,410,300,396)(287,409,301,395)(288,408,302,394)(289,407,303,393)(290,406,304,420)(291,405,305,419)(292,404,306,418)(293,403,307,417)(294,402,308,416), (1,218,125,283)(2,205,126,298)(3,220,127,285)(4,207,128,300)(5,222,129,287)(6,209,130,302)(7,224,131,289)(8,211,132,304)(9,198,133,291)(10,213,134,306)(11,200,135,293)(12,215,136,308)(13,202,137,295)(14,217,138,282)(15,204,139,297)(16,219,140,284)(17,206,113,299)(18,221,114,286)(19,208,115,301)(20,223,116,288)(21,210,117,303)(22,197,118,290)(23,212,119,305)(24,199,120,292)(25,214,121,307)(26,201,122,294)(27,216,123,281)(28,203,124,296)(29,447,68,181)(30,434,69,196)(31,421,70,183)(32,436,71,170)(33,423,72,185)(34,438,73,172)(35,425,74,187)(36,440,75,174)(37,427,76,189)(38,442,77,176)(39,429,78,191)(40,444,79,178)(41,431,80,193)(42,446,81,180)(43,433,82,195)(44,448,83,182)(45,435,84,169)(46,422,57,184)(47,437,58,171)(48,424,59,186)(49,439,60,173)(50,426,61,188)(51,441,62,175)(52,428,63,190)(53,443,64,177)(54,430,65,192)(55,445,66,179)(56,432,67,194)(85,311,350,156)(86,326,351,143)(87,313,352,158)(88,328,353,145)(89,315,354,160)(90,330,355,147)(91,317,356,162)(92,332,357,149)(93,319,358,164)(94,334,359,151)(95,321,360,166)(96,336,361,153)(97,323,362,168)(98,310,363,155)(99,325,364,142)(100,312,337,157)(101,327,338,144)(102,314,339,159)(103,329,340,146)(104,316,341,161)(105,331,342,148)(106,318,343,163)(107,333,344,150)(108,320,345,165)(109,335,346,152)(110,322,347,167)(111,309,348,154)(112,324,349,141)(225,402,369,254)(226,417,370,269)(227,404,371,256)(228,419,372,271)(229,406,373,258)(230,393,374,273)(231,408,375,260)(232,395,376,275)(233,410,377,262)(234,397,378,277)(235,412,379,264)(236,399,380,279)(237,414,381,266)(238,401,382,253)(239,416,383,268)(240,403,384,255)(241,418,385,270)(242,405,386,257)(243,420,387,272)(244,407,388,259)(245,394,389,274)(246,409,390,261)(247,396,391,276)(248,411,392,263)(249,398,365,278)(250,413,366,265)(251,400,367,280)(252,415,368,267), (1,323,125,168)(2,336,126,153)(3,321,127,166)(4,334,128,151)(5,319,129,164)(6,332,130,149)(7,317,131,162)(8,330,132,147)(9,315,133,160)(10,328,134,145)(11,313,135,158)(12,326,136,143)(13,311,137,156)(14,324,138,141)(15,309,139,154)(16,322,140,167)(17,335,113,152)(18,320,114,165)(19,333,115,150)(20,318,116,163)(21,331,117,148)(22,316,118,161)(23,329,119,146)(24,314,120,159)(25,327,121,144)(26,312,122,157)(27,325,123,142)(28,310,124,155)(29,370,68,226)(30,383,69,239)(31,368,70,252)(32,381,71,237)(33,366,72,250)(34,379,73,235)(35,392,74,248)(36,377,75,233)(37,390,76,246)(38,375,77,231)(39,388,78,244)(40,373,79,229)(41,386,80,242)(42,371,81,227)(43,384,82,240)(44,369,83,225)(45,382,84,238)(46,367,57,251)(47,380,58,236)(48,365,59,249)(49,378,60,234)(50,391,61,247)(51,376,62,232)(52,389,63,245)(53,374,64,230)(54,387,65,243)(55,372,66,228)(56,385,67,241)(85,295,350,202)(86,308,351,215)(87,293,352,200)(88,306,353,213)(89,291,354,198)(90,304,355,211)(91,289,356,224)(92,302,357,209)(93,287,358,222)(94,300,359,207)(95,285,360,220)(96,298,361,205)(97,283,362,218)(98,296,363,203)(99,281,364,216)(100,294,337,201)(101,307,338,214)(102,292,339,199)(103,305,340,212)(104,290,341,197)(105,303,342,210)(106,288,343,223)(107,301,344,208)(108,286,345,221)(109,299,346,206)(110,284,347,219)(111,297,348,204)(112,282,349,217)(169,253,435,401)(170,266,436,414)(171,279,437,399)(172,264,438,412)(173,277,439,397)(174,262,440,410)(175,275,441,395)(176,260,442,408)(177,273,443,393)(178,258,444,406)(179,271,445,419)(180,256,446,404)(181,269,447,417)(182,254,448,402)(183,267,421,415)(184,280,422,400)(185,265,423,413)(186,278,424,398)(187,263,425,411)(188,276,426,396)(189,261,427,409)(190,274,428,394)(191,259,429,407)(192,272,430,420)(193,257,431,405)(194,270,432,418)(195,255,433,403)(196,268,434,416) );
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)], [(1,387,15,373),(2,386,16,372),(3,385,17,371),(4,384,18,370),(5,383,19,369),(6,382,20,368),(7,381,21,367),(8,380,22,366),(9,379,23,365),(10,378,24,392),(11,377,25,391),(12,376,26,390),(13,375,27,389),(14,374,28,388),(29,165,43,151),(30,164,44,150),(31,163,45,149),(32,162,46,148),(33,161,47,147),(34,160,48,146),(35,159,49,145),(36,158,50,144),(37,157,51,143),(38,156,52,142),(39,155,53,141),(40,154,54,168),(41,153,55,167),(42,152,56,166),(57,331,71,317),(58,330,72,316),(59,329,73,315),(60,328,74,314),(61,327,75,313),(62,326,76,312),(63,325,77,311),(64,324,78,310),(65,323,79,309),(66,322,80,336),(67,321,81,335),(68,320,82,334),(69,319,83,333),(70,318,84,332),(85,421,99,435),(86,448,100,434),(87,447,101,433),(88,446,102,432),(89,445,103,431),(90,444,104,430),(91,443,105,429),(92,442,106,428),(93,441,107,427),(94,440,108,426),(95,439,109,425),(96,438,110,424),(97,437,111,423),(98,436,112,422),(113,227,127,241),(114,226,128,240),(115,225,129,239),(116,252,130,238),(117,251,131,237),(118,250,132,236),(119,249,133,235),(120,248,134,234),(121,247,135,233),(122,246,136,232),(123,245,137,231),(124,244,138,230),(125,243,139,229),(126,242,140,228),(169,350,183,364),(170,349,184,363),(171,348,185,362),(172,347,186,361),(173,346,187,360),(174,345,188,359),(175,344,189,358),(176,343,190,357),(177,342,191,356),(178,341,192,355),(179,340,193,354),(180,339,194,353),(181,338,195,352),(182,337,196,351),(197,258,211,272),(198,257,212,271),(199,256,213,270),(200,255,214,269),(201,254,215,268),(202,253,216,267),(203,280,217,266),(204,279,218,265),(205,278,219,264),(206,277,220,263),(207,276,221,262),(208,275,222,261),(209,274,223,260),(210,273,224,259),(281,415,295,401),(282,414,296,400),(283,413,297,399),(284,412,298,398),(285,411,299,397),(286,410,300,396),(287,409,301,395),(288,408,302,394),(289,407,303,393),(290,406,304,420),(291,405,305,419),(292,404,306,418),(293,403,307,417),(294,402,308,416)], [(1,218,125,283),(2,205,126,298),(3,220,127,285),(4,207,128,300),(5,222,129,287),(6,209,130,302),(7,224,131,289),(8,211,132,304),(9,198,133,291),(10,213,134,306),(11,200,135,293),(12,215,136,308),(13,202,137,295),(14,217,138,282),(15,204,139,297),(16,219,140,284),(17,206,113,299),(18,221,114,286),(19,208,115,301),(20,223,116,288),(21,210,117,303),(22,197,118,290),(23,212,119,305),(24,199,120,292),(25,214,121,307),(26,201,122,294),(27,216,123,281),(28,203,124,296),(29,447,68,181),(30,434,69,196),(31,421,70,183),(32,436,71,170),(33,423,72,185),(34,438,73,172),(35,425,74,187),(36,440,75,174),(37,427,76,189),(38,442,77,176),(39,429,78,191),(40,444,79,178),(41,431,80,193),(42,446,81,180),(43,433,82,195),(44,448,83,182),(45,435,84,169),(46,422,57,184),(47,437,58,171),(48,424,59,186),(49,439,60,173),(50,426,61,188),(51,441,62,175),(52,428,63,190),(53,443,64,177),(54,430,65,192),(55,445,66,179),(56,432,67,194),(85,311,350,156),(86,326,351,143),(87,313,352,158),(88,328,353,145),(89,315,354,160),(90,330,355,147),(91,317,356,162),(92,332,357,149),(93,319,358,164),(94,334,359,151),(95,321,360,166),(96,336,361,153),(97,323,362,168),(98,310,363,155),(99,325,364,142),(100,312,337,157),(101,327,338,144),(102,314,339,159),(103,329,340,146),(104,316,341,161),(105,331,342,148),(106,318,343,163),(107,333,344,150),(108,320,345,165),(109,335,346,152),(110,322,347,167),(111,309,348,154),(112,324,349,141),(225,402,369,254),(226,417,370,269),(227,404,371,256),(228,419,372,271),(229,406,373,258),(230,393,374,273),(231,408,375,260),(232,395,376,275),(233,410,377,262),(234,397,378,277),(235,412,379,264),(236,399,380,279),(237,414,381,266),(238,401,382,253),(239,416,383,268),(240,403,384,255),(241,418,385,270),(242,405,386,257),(243,420,387,272),(244,407,388,259),(245,394,389,274),(246,409,390,261),(247,396,391,276),(248,411,392,263),(249,398,365,278),(250,413,366,265),(251,400,367,280),(252,415,368,267)], [(1,323,125,168),(2,336,126,153),(3,321,127,166),(4,334,128,151),(5,319,129,164),(6,332,130,149),(7,317,131,162),(8,330,132,147),(9,315,133,160),(10,328,134,145),(11,313,135,158),(12,326,136,143),(13,311,137,156),(14,324,138,141),(15,309,139,154),(16,322,140,167),(17,335,113,152),(18,320,114,165),(19,333,115,150),(20,318,116,163),(21,331,117,148),(22,316,118,161),(23,329,119,146),(24,314,120,159),(25,327,121,144),(26,312,122,157),(27,325,123,142),(28,310,124,155),(29,370,68,226),(30,383,69,239),(31,368,70,252),(32,381,71,237),(33,366,72,250),(34,379,73,235),(35,392,74,248),(36,377,75,233),(37,390,76,246),(38,375,77,231),(39,388,78,244),(40,373,79,229),(41,386,80,242),(42,371,81,227),(43,384,82,240),(44,369,83,225),(45,382,84,238),(46,367,57,251),(47,380,58,236),(48,365,59,249),(49,378,60,234),(50,391,61,247),(51,376,62,232),(52,389,63,245),(53,374,64,230),(54,387,65,243),(55,372,66,228),(56,385,67,241),(85,295,350,202),(86,308,351,215),(87,293,352,200),(88,306,353,213),(89,291,354,198),(90,304,355,211),(91,289,356,224),(92,302,357,209),(93,287,358,222),(94,300,359,207),(95,285,360,220),(96,298,361,205),(97,283,362,218),(98,296,363,203),(99,281,364,216),(100,294,337,201),(101,307,338,214),(102,292,339,199),(103,305,340,212),(104,290,341,197),(105,303,342,210),(106,288,343,223),(107,301,344,208),(108,286,345,221),(109,299,346,206),(110,284,347,219),(111,297,348,204),(112,282,349,217),(169,253,435,401),(170,266,436,414),(171,279,437,399),(172,264,438,412),(173,277,439,397),(174,262,440,410),(175,275,441,395),(176,260,442,408),(177,273,443,393),(178,258,444,406),(179,271,445,419),(180,256,446,404),(181,269,447,417),(182,254,448,402),(183,267,421,415),(184,280,422,400),(185,265,423,413),(186,278,424,398),(187,263,425,411),(188,276,426,396),(189,261,427,409),(190,274,428,394),(191,259,429,407),(192,272,430,420),(193,257,431,405),(194,270,432,418),(195,255,433,403),(196,268,434,416)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | C4○D28 | C8⋊C22 | Q8×D7 | D4×D7 | D8⋊D7 | D7×Q16 |
| kernel | Dic14⋊2Q8 | C28.Q8 | C14.Q16 | Dic7⋊C8 | C28.44D4 | C7×C2.D8 | Dic7⋊3Q8 | C28⋊Q8 | Dic14 | C2×Dic7 | C2.D8 | Dic7 | C28 | C4⋊C4 | C2×C8 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 6 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Dic14⋊2Q8 ►in GL4(𝔽113) generated by
| 1 | 22 | 0 | 0 |
| 41 | 112 | 0 | 0 |
| 0 | 0 | 24 | 112 |
| 0 | 0 | 100 | 10 |
| 64 | 98 | 0 | 0 |
| 17 | 49 | 0 | 0 |
| 0 | 0 | 10 | 88 |
| 0 | 0 | 13 | 103 |
| 68 | 83 | 0 | 0 |
| 60 | 45 | 0 | 0 |
| 0 | 0 | 96 | 67 |
| 0 | 0 | 80 | 17 |
| 112 | 91 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 76 | 36 |
| 0 | 0 | 31 | 37 |
G:=sub<GL(4,GF(113))| [1,41,0,0,22,112,0,0,0,0,24,100,0,0,112,10],[64,17,0,0,98,49,0,0,0,0,10,13,0,0,88,103],[68,60,0,0,83,45,0,0,0,0,96,80,0,0,67,17],[112,72,0,0,91,1,0,0,0,0,76,31,0,0,36,37] >;
Dic14⋊2Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes_2Q_8 % in TeX
G:=Group("Dic14:2Q8"); // GroupNames label
G:=SmallGroup(448,409);
// by ID
G=gap.SmallGroup(448,409);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,422,135,268,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=c^4=1,b^2=a^14,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^15,d*a*d^-1=a^13,c*b*c^-1=a^21*b,d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations