metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊6Q8, C28.21SD16, C42.85D14, C4⋊Q8.13D7, C4.14(Q8×D7), C4⋊C4.88D14, C7⋊5(Q8⋊Q8), C28.41(C2×Q8), (C2×C28).163D4, C4.8(D4.D7), C28⋊C8.23C2, C28.88(C4○D4), C14.62(C2×SD16), (C2×C28).412C23, (C4×C28).141C22, C4.37(Q8⋊2D7), (C4×Dic14).19C2, C14.Q16.16C2, C14.78(C22⋊Q8), C4.Dic14.18C2, C2.15(D14⋊3Q8), C4⋊Dic7.351C22, C2.21(C28.C23), C14.100(C8.C22), (C2×Dic14).277C22, (C7×C4⋊Q8).13C2, C2.16(C2×D4.D7), (C2×C14).543(C2×D4), (C2×C7⋊C8).142C22, (C2×C4).192(C7⋊D4), (C7×C4⋊C4).135C22, (C2×C4).509(C22×D7), C22.215(C2×C7⋊D4), SmallGroup(448,628)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊6Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, cac-1=a15, ad=da, cbc-1=a21b, bd=db, dcd-1=c-1 >
Subgroups: 364 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, Q8⋊Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C28⋊C8, C4.Dic14, C14.Q16, C4×Dic14, C7×C4⋊Q8, Dic14⋊6Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×SD16, C8.C22, C7⋊D4, C22×D7, Q8⋊Q8, D4.D7, Q8×D7, Q8⋊2D7, C2×C7⋊D4, C2×D4.D7, C28.C23, D14⋊3Q8, Dic14⋊6Q8
►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)
(1 192 15 178)(2 191 16 177)(3 190 17 176)(4 189 18 175)(5 188 19 174)(6 187 20 173)(7 186 21 172)(8 185 22 171)(9 184 23 170)(10 183 24 169)(11 182 25 196)(12 181 26 195)(13 180 27 194)(14 179 28 193)(29 210 43 224)(30 209 44 223)(31 208 45 222)(32 207 46 221)(33 206 47 220)(34 205 48 219)(35 204 49 218)(36 203 50 217)(37 202 51 216)(38 201 52 215)(39 200 53 214)(40 199 54 213)(41 198 55 212)(42 197 56 211)(57 388 71 374)(58 387 72 373)(59 386 73 372)(60 385 74 371)(61 384 75 370)(62 383 76 369)(63 382 77 368)(64 381 78 367)(65 380 79 366)(66 379 80 365)(67 378 81 392)(68 377 82 391)(69 376 83 390)(70 375 84 389)(85 319 99 333)(86 318 100 332)(87 317 101 331)(88 316 102 330)(89 315 103 329)(90 314 104 328)(91 313 105 327)(92 312 106 326)(93 311 107 325)(94 310 108 324)(95 309 109 323)(96 336 110 322)(97 335 111 321)(98 334 112 320)(113 236 127 250)(114 235 128 249)(115 234 129 248)(116 233 130 247)(117 232 131 246)(118 231 132 245)(119 230 133 244)(120 229 134 243)(121 228 135 242)(122 227 136 241)(123 226 137 240)(124 225 138 239)(125 252 139 238)(126 251 140 237)(141 297 155 283)(142 296 156 282)(143 295 157 281)(144 294 158 308)(145 293 159 307)(146 292 160 306)(147 291 161 305)(148 290 162 304)(149 289 163 303)(150 288 164 302)(151 287 165 301)(152 286 166 300)(153 285 167 299)(154 284 168 298)(253 356 267 342)(254 355 268 341)(255 354 269 340)(256 353 270 339)(257 352 271 338)(258 351 272 337)(259 350 273 364)(260 349 274 363)(261 348 275 362)(262 347 276 361)(263 346 277 360)(264 345 278 359)(265 344 279 358)(266 343 280 357)(393 422 407 436)(394 421 408 435)(395 448 409 434)(396 447 410 433)(397 446 411 432)(398 445 412 431)(399 444 413 430)(400 443 414 429)(401 442 415 428)(402 441 416 427)(403 440 417 426)(404 439 418 425)(405 438 419 424)(406 437 420 423)
(1 95 356 211)(2 110 357 198)(3 97 358 213)(4 112 359 200)(5 99 360 215)(6 86 361 202)(7 101 362 217)(8 88 363 204)(9 103 364 219)(10 90 337 206)(11 105 338 221)(12 92 339 208)(13 107 340 223)(14 94 341 210)(15 109 342 197)(16 96 343 212)(17 111 344 199)(18 98 345 214)(19 85 346 201)(20 100 347 216)(21 87 348 203)(22 102 349 218)(23 89 350 205)(24 104 351 220)(25 91 352 207)(26 106 353 222)(27 93 354 209)(28 108 355 224)(29 172 324 275)(30 187 325 262)(31 174 326 277)(32 189 327 264)(33 176 328 279)(34 191 329 266)(35 178 330 253)(36 193 331 268)(37 180 332 255)(38 195 333 270)(39 182 334 257)(40 169 335 272)(41 184 336 259)(42 171 309 274)(43 186 310 261)(44 173 311 276)(45 188 312 263)(46 175 313 278)(47 190 314 265)(48 177 315 280)(49 192 316 267)(50 179 317 254)(51 194 318 269)(52 181 319 256)(53 196 320 271)(54 183 321 258)(55 170 322 273)(56 185 323 260)(57 430 248 292)(58 445 249 307)(59 432 250 294)(60 447 251 281)(61 434 252 296)(62 421 225 283)(63 436 226 298)(64 423 227 285)(65 438 228 300)(66 425 229 287)(67 440 230 302)(68 427 231 289)(69 442 232 304)(70 429 233 291)(71 444 234 306)(72 431 235 293)(73 446 236 308)(74 433 237 295)(75 448 238 282)(76 435 239 297)(77 422 240 284)(78 437 241 299)(79 424 242 286)(80 439 243 301)(81 426 244 288)(82 441 245 303)(83 428 246 290)(84 443 247 305)(113 165 386 404)(114 152 387 419)(115 167 388 406)(116 154 389 393)(117 141 390 408)(118 156 391 395)(119 143 392 410)(120 158 365 397)(121 145 366 412)(122 160 367 399)(123 147 368 414)(124 162 369 401)(125 149 370 416)(126 164 371 403)(127 151 372 418)(128 166 373 405)(129 153 374 420)(130 168 375 407)(131 155 376 394)(132 142 377 409)(133 157 378 396)(134 144 379 411)(135 159 380 398)(136 146 381 413)(137 161 382 400)(138 148 383 415)(139 163 384 402)(140 150 385 417)
(1 289 356 427)(2 290 357 428)(3 291 358 429)(4 292 359 430)(5 293 360 431)(6 294 361 432)(7 295 362 433)(8 296 363 434)(9 297 364 435)(10 298 337 436)(11 299 338 437)(12 300 339 438)(13 301 340 439)(14 302 341 440)(15 303 342 441)(16 304 343 442)(17 305 344 443)(18 306 345 444)(19 307 346 445)(20 308 347 446)(21 281 348 447)(22 282 349 448)(23 283 350 421)(24 284 351 422)(25 285 352 423)(26 286 353 424)(27 287 354 425)(28 288 355 426)(29 392 324 119)(30 365 325 120)(31 366 326 121)(32 367 327 122)(33 368 328 123)(34 369 329 124)(35 370 330 125)(36 371 331 126)(37 372 332 127)(38 373 333 128)(39 374 334 129)(40 375 335 130)(41 376 336 131)(42 377 309 132)(43 378 310 133)(44 379 311 134)(45 380 312 135)(46 381 313 136)(47 382 314 137)(48 383 315 138)(49 384 316 139)(50 385 317 140)(51 386 318 113)(52 387 319 114)(53 388 320 115)(54 389 321 116)(55 390 322 117)(56 391 323 118)(57 112 248 200)(58 85 249 201)(59 86 250 202)(60 87 251 203)(61 88 252 204)(62 89 225 205)(63 90 226 206)(64 91 227 207)(65 92 228 208)(66 93 229 209)(67 94 230 210)(68 95 231 211)(69 96 232 212)(70 97 233 213)(71 98 234 214)(72 99 235 215)(73 100 236 216)(74 101 237 217)(75 102 238 218)(76 103 239 219)(77 104 240 220)(78 105 241 221)(79 106 242 222)(80 107 243 223)(81 108 244 224)(82 109 245 197)(83 110 246 198)(84 111 247 199)(141 273 408 170)(142 274 409 171)(143 275 410 172)(144 276 411 173)(145 277 412 174)(146 278 413 175)(147 279 414 176)(148 280 415 177)(149 253 416 178)(150 254 417 179)(151 255 418 180)(152 256 419 181)(153 257 420 182)(154 258 393 183)(155 259 394 184)(156 260 395 185)(157 261 396 186)(158 262 397 187)(159 263 398 188)(160 264 399 189)(161 265 400 190)(162 266 401 191)(163 267 402 192)(164 268 403 193)(165 269 404 194)(166 270 405 195)(167 271 406 196)(168 272 407 169)
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), (1,192,15,178)(2,191,16,177)(3,190,17,176)(4,189,18,175)(5,188,19,174)(6,187,20,173)(7,186,21,172)(8,185,22,171)(9,184,23,170)(10,183,24,169)(11,182,25,196)(12,181,26,195)(13,180,27,194)(14,179,28,193)(29,210,43,224)(30,209,44,223)(31,208,45,222)(32,207,46,221)(33,206,47,220)(34,205,48,219)(35,204,49,218)(36,203,50,217)(37,202,51,216)(38,201,52,215)(39,200,53,214)(40,199,54,213)(41,198,55,212)(42,197,56,211)(57,388,71,374)(58,387,72,373)(59,386,73,372)(60,385,74,371)(61,384,75,370)(62,383,76,369)(63,382,77,368)(64,381,78,367)(65,380,79,366)(66,379,80,365)(67,378,81,392)(68,377,82,391)(69,376,83,390)(70,375,84,389)(85,319,99,333)(86,318,100,332)(87,317,101,331)(88,316,102,330)(89,315,103,329)(90,314,104,328)(91,313,105,327)(92,312,106,326)(93,311,107,325)(94,310,108,324)(95,309,109,323)(96,336,110,322)(97,335,111,321)(98,334,112,320)(113,236,127,250)(114,235,128,249)(115,234,129,248)(116,233,130,247)(117,232,131,246)(118,231,132,245)(119,230,133,244)(120,229,134,243)(121,228,135,242)(122,227,136,241)(123,226,137,240)(124,225,138,239)(125,252,139,238)(126,251,140,237)(141,297,155,283)(142,296,156,282)(143,295,157,281)(144,294,158,308)(145,293,159,307)(146,292,160,306)(147,291,161,305)(148,290,162,304)(149,289,163,303)(150,288,164,302)(151,287,165,301)(152,286,166,300)(153,285,167,299)(154,284,168,298)(253,356,267,342)(254,355,268,341)(255,354,269,340)(256,353,270,339)(257,352,271,338)(258,351,272,337)(259,350,273,364)(260,349,274,363)(261,348,275,362)(262,347,276,361)(263,346,277,360)(264,345,278,359)(265,344,279,358)(266,343,280,357)(393,422,407,436)(394,421,408,435)(395,448,409,434)(396,447,410,433)(397,446,411,432)(398,445,412,431)(399,444,413,430)(400,443,414,429)(401,442,415,428)(402,441,416,427)(403,440,417,426)(404,439,418,425)(405,438,419,424)(406,437,420,423), (1,95,356,211)(2,110,357,198)(3,97,358,213)(4,112,359,200)(5,99,360,215)(6,86,361,202)(7,101,362,217)(8,88,363,204)(9,103,364,219)(10,90,337,206)(11,105,338,221)(12,92,339,208)(13,107,340,223)(14,94,341,210)(15,109,342,197)(16,96,343,212)(17,111,344,199)(18,98,345,214)(19,85,346,201)(20,100,347,216)(21,87,348,203)(22,102,349,218)(23,89,350,205)(24,104,351,220)(25,91,352,207)(26,106,353,222)(27,93,354,209)(28,108,355,224)(29,172,324,275)(30,187,325,262)(31,174,326,277)(32,189,327,264)(33,176,328,279)(34,191,329,266)(35,178,330,253)(36,193,331,268)(37,180,332,255)(38,195,333,270)(39,182,334,257)(40,169,335,272)(41,184,336,259)(42,171,309,274)(43,186,310,261)(44,173,311,276)(45,188,312,263)(46,175,313,278)(47,190,314,265)(48,177,315,280)(49,192,316,267)(50,179,317,254)(51,194,318,269)(52,181,319,256)(53,196,320,271)(54,183,321,258)(55,170,322,273)(56,185,323,260)(57,430,248,292)(58,445,249,307)(59,432,250,294)(60,447,251,281)(61,434,252,296)(62,421,225,283)(63,436,226,298)(64,423,227,285)(65,438,228,300)(66,425,229,287)(67,440,230,302)(68,427,231,289)(69,442,232,304)(70,429,233,291)(71,444,234,306)(72,431,235,293)(73,446,236,308)(74,433,237,295)(75,448,238,282)(76,435,239,297)(77,422,240,284)(78,437,241,299)(79,424,242,286)(80,439,243,301)(81,426,244,288)(82,441,245,303)(83,428,246,290)(84,443,247,305)(113,165,386,404)(114,152,387,419)(115,167,388,406)(116,154,389,393)(117,141,390,408)(118,156,391,395)(119,143,392,410)(120,158,365,397)(121,145,366,412)(122,160,367,399)(123,147,368,414)(124,162,369,401)(125,149,370,416)(126,164,371,403)(127,151,372,418)(128,166,373,405)(129,153,374,420)(130,168,375,407)(131,155,376,394)(132,142,377,409)(133,157,378,396)(134,144,379,411)(135,159,380,398)(136,146,381,413)(137,161,382,400)(138,148,383,415)(139,163,384,402)(140,150,385,417), (1,289,356,427)(2,290,357,428)(3,291,358,429)(4,292,359,430)(5,293,360,431)(6,294,361,432)(7,295,362,433)(8,296,363,434)(9,297,364,435)(10,298,337,436)(11,299,338,437)(12,300,339,438)(13,301,340,439)(14,302,341,440)(15,303,342,441)(16,304,343,442)(17,305,344,443)(18,306,345,444)(19,307,346,445)(20,308,347,446)(21,281,348,447)(22,282,349,448)(23,283,350,421)(24,284,351,422)(25,285,352,423)(26,286,353,424)(27,287,354,425)(28,288,355,426)(29,392,324,119)(30,365,325,120)(31,366,326,121)(32,367,327,122)(33,368,328,123)(34,369,329,124)(35,370,330,125)(36,371,331,126)(37,372,332,127)(38,373,333,128)(39,374,334,129)(40,375,335,130)(41,376,336,131)(42,377,309,132)(43,378,310,133)(44,379,311,134)(45,380,312,135)(46,381,313,136)(47,382,314,137)(48,383,315,138)(49,384,316,139)(50,385,317,140)(51,386,318,113)(52,387,319,114)(53,388,320,115)(54,389,321,116)(55,390,322,117)(56,391,323,118)(57,112,248,200)(58,85,249,201)(59,86,250,202)(60,87,251,203)(61,88,252,204)(62,89,225,205)(63,90,226,206)(64,91,227,207)(65,92,228,208)(66,93,229,209)(67,94,230,210)(68,95,231,211)(69,96,232,212)(70,97,233,213)(71,98,234,214)(72,99,235,215)(73,100,236,216)(74,101,237,217)(75,102,238,218)(76,103,239,219)(77,104,240,220)(78,105,241,221)(79,106,242,222)(80,107,243,223)(81,108,244,224)(82,109,245,197)(83,110,246,198)(84,111,247,199)(141,273,408,170)(142,274,409,171)(143,275,410,172)(144,276,411,173)(145,277,412,174)(146,278,413,175)(147,279,414,176)(148,280,415,177)(149,253,416,178)(150,254,417,179)(151,255,418,180)(152,256,419,181)(153,257,420,182)(154,258,393,183)(155,259,394,184)(156,260,395,185)(157,261,396,186)(158,262,397,187)(159,263,398,188)(160,264,399,189)(161,265,400,190)(162,266,401,191)(163,267,402,192)(164,268,403,193)(165,269,404,194)(166,270,405,195)(167,271,406,196)(168,272,407,169)>;
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), (1,192,15,178)(2,191,16,177)(3,190,17,176)(4,189,18,175)(5,188,19,174)(6,187,20,173)(7,186,21,172)(8,185,22,171)(9,184,23,170)(10,183,24,169)(11,182,25,196)(12,181,26,195)(13,180,27,194)(14,179,28,193)(29,210,43,224)(30,209,44,223)(31,208,45,222)(32,207,46,221)(33,206,47,220)(34,205,48,219)(35,204,49,218)(36,203,50,217)(37,202,51,216)(38,201,52,215)(39,200,53,214)(40,199,54,213)(41,198,55,212)(42,197,56,211)(57,388,71,374)(58,387,72,373)(59,386,73,372)(60,385,74,371)(61,384,75,370)(62,383,76,369)(63,382,77,368)(64,381,78,367)(65,380,79,366)(66,379,80,365)(67,378,81,392)(68,377,82,391)(69,376,83,390)(70,375,84,389)(85,319,99,333)(86,318,100,332)(87,317,101,331)(88,316,102,330)(89,315,103,329)(90,314,104,328)(91,313,105,327)(92,312,106,326)(93,311,107,325)(94,310,108,324)(95,309,109,323)(96,336,110,322)(97,335,111,321)(98,334,112,320)(113,236,127,250)(114,235,128,249)(115,234,129,248)(116,233,130,247)(117,232,131,246)(118,231,132,245)(119,230,133,244)(120,229,134,243)(121,228,135,242)(122,227,136,241)(123,226,137,240)(124,225,138,239)(125,252,139,238)(126,251,140,237)(141,297,155,283)(142,296,156,282)(143,295,157,281)(144,294,158,308)(145,293,159,307)(146,292,160,306)(147,291,161,305)(148,290,162,304)(149,289,163,303)(150,288,164,302)(151,287,165,301)(152,286,166,300)(153,285,167,299)(154,284,168,298)(253,356,267,342)(254,355,268,341)(255,354,269,340)(256,353,270,339)(257,352,271,338)(258,351,272,337)(259,350,273,364)(260,349,274,363)(261,348,275,362)(262,347,276,361)(263,346,277,360)(264,345,278,359)(265,344,279,358)(266,343,280,357)(393,422,407,436)(394,421,408,435)(395,448,409,434)(396,447,410,433)(397,446,411,432)(398,445,412,431)(399,444,413,430)(400,443,414,429)(401,442,415,428)(402,441,416,427)(403,440,417,426)(404,439,418,425)(405,438,419,424)(406,437,420,423), (1,95,356,211)(2,110,357,198)(3,97,358,213)(4,112,359,200)(5,99,360,215)(6,86,361,202)(7,101,362,217)(8,88,363,204)(9,103,364,219)(10,90,337,206)(11,105,338,221)(12,92,339,208)(13,107,340,223)(14,94,341,210)(15,109,342,197)(16,96,343,212)(17,111,344,199)(18,98,345,214)(19,85,346,201)(20,100,347,216)(21,87,348,203)(22,102,349,218)(23,89,350,205)(24,104,351,220)(25,91,352,207)(26,106,353,222)(27,93,354,209)(28,108,355,224)(29,172,324,275)(30,187,325,262)(31,174,326,277)(32,189,327,264)(33,176,328,279)(34,191,329,266)(35,178,330,253)(36,193,331,268)(37,180,332,255)(38,195,333,270)(39,182,334,257)(40,169,335,272)(41,184,336,259)(42,171,309,274)(43,186,310,261)(44,173,311,276)(45,188,312,263)(46,175,313,278)(47,190,314,265)(48,177,315,280)(49,192,316,267)(50,179,317,254)(51,194,318,269)(52,181,319,256)(53,196,320,271)(54,183,321,258)(55,170,322,273)(56,185,323,260)(57,430,248,292)(58,445,249,307)(59,432,250,294)(60,447,251,281)(61,434,252,296)(62,421,225,283)(63,436,226,298)(64,423,227,285)(65,438,228,300)(66,425,229,287)(67,440,230,302)(68,427,231,289)(69,442,232,304)(70,429,233,291)(71,444,234,306)(72,431,235,293)(73,446,236,308)(74,433,237,295)(75,448,238,282)(76,435,239,297)(77,422,240,284)(78,437,241,299)(79,424,242,286)(80,439,243,301)(81,426,244,288)(82,441,245,303)(83,428,246,290)(84,443,247,305)(113,165,386,404)(114,152,387,419)(115,167,388,406)(116,154,389,393)(117,141,390,408)(118,156,391,395)(119,143,392,410)(120,158,365,397)(121,145,366,412)(122,160,367,399)(123,147,368,414)(124,162,369,401)(125,149,370,416)(126,164,371,403)(127,151,372,418)(128,166,373,405)(129,153,374,420)(130,168,375,407)(131,155,376,394)(132,142,377,409)(133,157,378,396)(134,144,379,411)(135,159,380,398)(136,146,381,413)(137,161,382,400)(138,148,383,415)(139,163,384,402)(140,150,385,417), (1,289,356,427)(2,290,357,428)(3,291,358,429)(4,292,359,430)(5,293,360,431)(6,294,361,432)(7,295,362,433)(8,296,363,434)(9,297,364,435)(10,298,337,436)(11,299,338,437)(12,300,339,438)(13,301,340,439)(14,302,341,440)(15,303,342,441)(16,304,343,442)(17,305,344,443)(18,306,345,444)(19,307,346,445)(20,308,347,446)(21,281,348,447)(22,282,349,448)(23,283,350,421)(24,284,351,422)(25,285,352,423)(26,286,353,424)(27,287,354,425)(28,288,355,426)(29,392,324,119)(30,365,325,120)(31,366,326,121)(32,367,327,122)(33,368,328,123)(34,369,329,124)(35,370,330,125)(36,371,331,126)(37,372,332,127)(38,373,333,128)(39,374,334,129)(40,375,335,130)(41,376,336,131)(42,377,309,132)(43,378,310,133)(44,379,311,134)(45,380,312,135)(46,381,313,136)(47,382,314,137)(48,383,315,138)(49,384,316,139)(50,385,317,140)(51,386,318,113)(52,387,319,114)(53,388,320,115)(54,389,321,116)(55,390,322,117)(56,391,323,118)(57,112,248,200)(58,85,249,201)(59,86,250,202)(60,87,251,203)(61,88,252,204)(62,89,225,205)(63,90,226,206)(64,91,227,207)(65,92,228,208)(66,93,229,209)(67,94,230,210)(68,95,231,211)(69,96,232,212)(70,97,233,213)(71,98,234,214)(72,99,235,215)(73,100,236,216)(74,101,237,217)(75,102,238,218)(76,103,239,219)(77,104,240,220)(78,105,241,221)(79,106,242,222)(80,107,243,223)(81,108,244,224)(82,109,245,197)(83,110,246,198)(84,111,247,199)(141,273,408,170)(142,274,409,171)(143,275,410,172)(144,276,411,173)(145,277,412,174)(146,278,413,175)(147,279,414,176)(148,280,415,177)(149,253,416,178)(150,254,417,179)(151,255,418,180)(152,256,419,181)(153,257,420,182)(154,258,393,183)(155,259,394,184)(156,260,395,185)(157,261,396,186)(158,262,397,187)(159,263,398,188)(160,264,399,189)(161,265,400,190)(162,266,401,191)(163,267,402,192)(164,268,403,193)(165,269,404,194)(166,270,405,195)(167,271,406,196)(168,272,407,169) );
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)], [(1,192,15,178),(2,191,16,177),(3,190,17,176),(4,189,18,175),(5,188,19,174),(6,187,20,173),(7,186,21,172),(8,185,22,171),(9,184,23,170),(10,183,24,169),(11,182,25,196),(12,181,26,195),(13,180,27,194),(14,179,28,193),(29,210,43,224),(30,209,44,223),(31,208,45,222),(32,207,46,221),(33,206,47,220),(34,205,48,219),(35,204,49,218),(36,203,50,217),(37,202,51,216),(38,201,52,215),(39,200,53,214),(40,199,54,213),(41,198,55,212),(42,197,56,211),(57,388,71,374),(58,387,72,373),(59,386,73,372),(60,385,74,371),(61,384,75,370),(62,383,76,369),(63,382,77,368),(64,381,78,367),(65,380,79,366),(66,379,80,365),(67,378,81,392),(68,377,82,391),(69,376,83,390),(70,375,84,389),(85,319,99,333),(86,318,100,332),(87,317,101,331),(88,316,102,330),(89,315,103,329),(90,314,104,328),(91,313,105,327),(92,312,106,326),(93,311,107,325),(94,310,108,324),(95,309,109,323),(96,336,110,322),(97,335,111,321),(98,334,112,320),(113,236,127,250),(114,235,128,249),(115,234,129,248),(116,233,130,247),(117,232,131,246),(118,231,132,245),(119,230,133,244),(120,229,134,243),(121,228,135,242),(122,227,136,241),(123,226,137,240),(124,225,138,239),(125,252,139,238),(126,251,140,237),(141,297,155,283),(142,296,156,282),(143,295,157,281),(144,294,158,308),(145,293,159,307),(146,292,160,306),(147,291,161,305),(148,290,162,304),(149,289,163,303),(150,288,164,302),(151,287,165,301),(152,286,166,300),(153,285,167,299),(154,284,168,298),(253,356,267,342),(254,355,268,341),(255,354,269,340),(256,353,270,339),(257,352,271,338),(258,351,272,337),(259,350,273,364),(260,349,274,363),(261,348,275,362),(262,347,276,361),(263,346,277,360),(264,345,278,359),(265,344,279,358),(266,343,280,357),(393,422,407,436),(394,421,408,435),(395,448,409,434),(396,447,410,433),(397,446,411,432),(398,445,412,431),(399,444,413,430),(400,443,414,429),(401,442,415,428),(402,441,416,427),(403,440,417,426),(404,439,418,425),(405,438,419,424),(406,437,420,423)], [(1,95,356,211),(2,110,357,198),(3,97,358,213),(4,112,359,200),(5,99,360,215),(6,86,361,202),(7,101,362,217),(8,88,363,204),(9,103,364,219),(10,90,337,206),(11,105,338,221),(12,92,339,208),(13,107,340,223),(14,94,341,210),(15,109,342,197),(16,96,343,212),(17,111,344,199),(18,98,345,214),(19,85,346,201),(20,100,347,216),(21,87,348,203),(22,102,349,218),(23,89,350,205),(24,104,351,220),(25,91,352,207),(26,106,353,222),(27,93,354,209),(28,108,355,224),(29,172,324,275),(30,187,325,262),(31,174,326,277),(32,189,327,264),(33,176,328,279),(34,191,329,266),(35,178,330,253),(36,193,331,268),(37,180,332,255),(38,195,333,270),(39,182,334,257),(40,169,335,272),(41,184,336,259),(42,171,309,274),(43,186,310,261),(44,173,311,276),(45,188,312,263),(46,175,313,278),(47,190,314,265),(48,177,315,280),(49,192,316,267),(50,179,317,254),(51,194,318,269),(52,181,319,256),(53,196,320,271),(54,183,321,258),(55,170,322,273),(56,185,323,260),(57,430,248,292),(58,445,249,307),(59,432,250,294),(60,447,251,281),(61,434,252,296),(62,421,225,283),(63,436,226,298),(64,423,227,285),(65,438,228,300),(66,425,229,287),(67,440,230,302),(68,427,231,289),(69,442,232,304),(70,429,233,291),(71,444,234,306),(72,431,235,293),(73,446,236,308),(74,433,237,295),(75,448,238,282),(76,435,239,297),(77,422,240,284),(78,437,241,299),(79,424,242,286),(80,439,243,301),(81,426,244,288),(82,441,245,303),(83,428,246,290),(84,443,247,305),(113,165,386,404),(114,152,387,419),(115,167,388,406),(116,154,389,393),(117,141,390,408),(118,156,391,395),(119,143,392,410),(120,158,365,397),(121,145,366,412),(122,160,367,399),(123,147,368,414),(124,162,369,401),(125,149,370,416),(126,164,371,403),(127,151,372,418),(128,166,373,405),(129,153,374,420),(130,168,375,407),(131,155,376,394),(132,142,377,409),(133,157,378,396),(134,144,379,411),(135,159,380,398),(136,146,381,413),(137,161,382,400),(138,148,383,415),(139,163,384,402),(140,150,385,417)], [(1,289,356,427),(2,290,357,428),(3,291,358,429),(4,292,359,430),(5,293,360,431),(6,294,361,432),(7,295,362,433),(8,296,363,434),(9,297,364,435),(10,298,337,436),(11,299,338,437),(12,300,339,438),(13,301,340,439),(14,302,341,440),(15,303,342,441),(16,304,343,442),(17,305,344,443),(18,306,345,444),(19,307,346,445),(20,308,347,446),(21,281,348,447),(22,282,349,448),(23,283,350,421),(24,284,351,422),(25,285,352,423),(26,286,353,424),(27,287,354,425),(28,288,355,426),(29,392,324,119),(30,365,325,120),(31,366,326,121),(32,367,327,122),(33,368,328,123),(34,369,329,124),(35,370,330,125),(36,371,331,126),(37,372,332,127),(38,373,333,128),(39,374,334,129),(40,375,335,130),(41,376,336,131),(42,377,309,132),(43,378,310,133),(44,379,311,134),(45,380,312,135),(46,381,313,136),(47,382,314,137),(48,383,315,138),(49,384,316,139),(50,385,317,140),(51,386,318,113),(52,387,319,114),(53,388,320,115),(54,389,321,116),(55,390,322,117),(56,391,323,118),(57,112,248,200),(58,85,249,201),(59,86,250,202),(60,87,251,203),(61,88,252,204),(62,89,225,205),(63,90,226,206),(64,91,227,207),(65,92,228,208),(66,93,229,209),(67,94,230,210),(68,95,231,211),(69,96,232,212),(70,97,233,213),(71,98,234,214),(72,99,235,215),(73,100,236,216),(74,101,237,217),(75,102,238,218),(76,103,239,219),(77,104,240,220),(78,105,241,221),(79,106,242,222),(80,107,243,223),(81,108,244,224),(82,109,245,197),(83,110,246,198),(84,111,247,199),(141,273,408,170),(142,274,409,171),(143,275,410,172),(144,276,411,173),(145,277,412,174),(146,278,413,175),(147,279,414,176),(148,280,415,177),(149,253,416,178),(150,254,417,179),(151,255,418,180),(152,256,419,181),(153,257,420,182),(154,258,393,183),(155,259,394,184),(156,260,395,185),(157,261,396,186),(158,262,397,187),(159,263,398,188),(160,264,399,189),(161,265,400,190),(162,266,401,191),(163,267,402,192),(164,268,403,193),(165,269,404,194),(166,270,405,195),(167,271,406,196),(168,272,407,169)]])
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 | ··· | 28R | 28S | ··· | 28AD |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 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 | Q8 | D4 | D7 | SD16 | C4○D4 | D14 | D14 | C7⋊D4 | C8.C22 | D4.D7 | Q8×D7 | Q8⋊2D7 | C28.C23 |
| kernel | Dic14⋊6Q8 | C28⋊C8 | C4.Dic14 | C14.Q16 | C4×Dic14 | C7×C4⋊Q8 | Dic14 | C2×C28 | C4⋊Q8 | C28 | C28 | C42 | C4⋊C4 | C2×C4 | C14 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 1 | 6 | 3 | 3 | 6 |
Matrix representation of Dic14⋊6Q8 ►in GL6(𝔽113)
| 1 | 91 | 0 | 0 | 0 | 0 |
| 72 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 112 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 83 | 100 | 0 | 0 | 0 | 0 |
| 78 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 79 | 34 | 0 | 0 |
| 0 | 0 | 89 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 66 | 8 | 0 | 0 | 0 | 0 |
| 63 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 101 | 0 | 0 |
| 0 | 0 | 24 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 81 |
| 0 | 0 | 0 | 0 | 106 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 75 |
| 0 | 0 | 0 | 0 | 76 | 47 |
G:=sub<GL(6,GF(113))| [1,72,0,0,0,0,91,112,0,0,0,0,0,0,25,2,0,0,0,0,112,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[83,78,0,0,0,0,100,30,0,0,0,0,0,0,79,89,0,0,0,0,34,34,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[66,63,0,0,0,0,8,47,0,0,0,0,0,0,96,24,0,0,0,0,101,17,0,0,0,0,0,0,1,106,0,0,0,0,81,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,76,0,0,0,0,75,47] >;
Dic14⋊6Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes_6Q_8 % in TeX
G:=Group("Dic14:6Q8"); // GroupNames label
G:=SmallGroup(448,628);
// by ID
G=gap.SmallGroup(448,628);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,254,219,100,1123,297,136,18822]);
// 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,a*d=d*a,c*b*c^-1=a^21*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations