metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊10Q8, C42.121D14, C14.642- 1+4, C4.49(Q8×D7), C7⋊1(Q8⋊3Q8), (C4×Q8).10D7, C4⋊C4.321D14, (Q8×C28).11C2, C28.107(C2×Q8), C4.17(C4○D28), (C2×Q8).175D14, C28⋊2Q8.24C2, Dic7.10(C2×Q8), Dic7.Q8.1C2, C28.115(C4○D4), C14.27(C22×Q8), (C2×C14).111C24, (C4×C28).164C22, (C2×C28).589C23, (C4×Dic14).21C2, Dic7⋊Q8.7C2, C4⋊Dic7.42C22, C28.6Q8.10C2, Dic7⋊3Q8.10C2, (Q8×C14).211C22, (C2×Dic7).50C23, (C4×Dic7).80C22, C22.136(C23×D7), Dic7⋊C4.114C22, C2.21(D4.10D14), (C2×Dic14).146C22, C2.11(C2×Q8×D7), C2.59(C2×C4○D28), C14.52(C2×C4○D4), (C7×C4⋊C4).339C22, (C2×C4).166(C22×D7), SmallGroup(448,1020)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊10Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a14b, bd=db, dcd-1=c-1 >
Subgroups: 708 in 200 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C14, C42, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, Q8⋊3Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, Q8×C14, C4×Dic14, C4×Dic14, C28⋊2Q8, C28.6Q8, Dic7⋊3Q8, Dic7.Q8, Dic7⋊Q8, Q8×C28, Dic14⋊10Q8
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, C24, D14, C22×Q8, C2×C4○D4, 2- 1+4, C22×D7, Q8⋊3Q8, C4○D28, Q8×D7, C23×D7, C2×C4○D28, C2×Q8×D7, D4.10D14, Dic14⋊10Q8
►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)
(1 171 15 185)(2 170 16 184)(3 169 17 183)(4 196 18 182)(5 195 19 181)(6 194 20 180)(7 193 21 179)(8 192 22 178)(9 191 23 177)(10 190 24 176)(11 189 25 175)(12 188 26 174)(13 187 27 173)(14 186 28 172)(29 389 43 375)(30 388 44 374)(31 387 45 373)(32 386 46 372)(33 385 47 371)(34 384 48 370)(35 383 49 369)(36 382 50 368)(37 381 51 367)(38 380 52 366)(39 379 53 365)(40 378 54 392)(41 377 55 391)(42 376 56 390)(57 446 71 432)(58 445 72 431)(59 444 73 430)(60 443 74 429)(61 442 75 428)(62 441 76 427)(63 440 77 426)(64 439 78 425)(65 438 79 424)(66 437 80 423)(67 436 81 422)(68 435 82 421)(69 434 83 448)(70 433 84 447)(85 415 99 401)(86 414 100 400)(87 413 101 399)(88 412 102 398)(89 411 103 397)(90 410 104 396)(91 409 105 395)(92 408 106 394)(93 407 107 393)(94 406 108 420)(95 405 109 419)(96 404 110 418)(97 403 111 417)(98 402 112 416)(113 279 127 265)(114 278 128 264)(115 277 129 263)(116 276 130 262)(117 275 131 261)(118 274 132 260)(119 273 133 259)(120 272 134 258)(121 271 135 257)(122 270 136 256)(123 269 137 255)(124 268 138 254)(125 267 139 253)(126 266 140 280)(141 298 155 284)(142 297 156 283)(143 296 157 282)(144 295 158 281)(145 294 159 308)(146 293 160 307)(147 292 161 306)(148 291 162 305)(149 290 163 304)(150 289 164 303)(151 288 165 302)(152 287 166 301)(153 286 167 300)(154 285 168 299)(197 241 211 227)(198 240 212 226)(199 239 213 225)(200 238 214 252)(201 237 215 251)(202 236 216 250)(203 235 217 249)(204 234 218 248)(205 233 219 247)(206 232 220 246)(207 231 221 245)(208 230 222 244)(209 229 223 243)(210 228 224 242)(309 344 323 358)(310 343 324 357)(311 342 325 356)(312 341 326 355)(313 340 327 354)(314 339 328 353)(315 338 329 352)(316 337 330 351)(317 364 331 350)(318 363 332 349)(319 362 333 348)(320 361 334 347)(321 360 335 346)(322 359 336 345)
(1 342 293 257)(2 343 294 258)(3 344 295 259)(4 345 296 260)(5 346 297 261)(6 347 298 262)(7 348 299 263)(8 349 300 264)(9 350 301 265)(10 351 302 266)(11 352 303 267)(12 353 304 268)(13 354 305 269)(14 355 306 270)(15 356 307 271)(16 357 308 272)(17 358 281 273)(18 359 282 274)(19 360 283 275)(20 361 284 276)(21 362 285 277)(22 363 286 278)(23 364 287 279)(24 337 288 280)(25 338 289 253)(26 339 290 254)(27 340 291 255)(28 341 292 256)(29 410 250 439)(30 411 251 440)(31 412 252 441)(32 413 225 442)(33 414 226 443)(34 415 227 444)(35 416 228 445)(36 417 229 446)(37 418 230 447)(38 419 231 448)(39 420 232 421)(40 393 233 422)(41 394 234 423)(42 395 235 424)(43 396 236 425)(44 397 237 426)(45 398 238 427)(46 399 239 428)(47 400 240 429)(48 401 241 430)(49 402 242 431)(50 403 243 432)(51 404 244 433)(52 405 245 434)(53 406 246 435)(54 407 247 436)(55 408 248 437)(56 409 249 438)(57 382 111 223)(58 383 112 224)(59 384 85 197)(60 385 86 198)(61 386 87 199)(62 387 88 200)(63 388 89 201)(64 389 90 202)(65 390 91 203)(66 391 92 204)(67 392 93 205)(68 365 94 206)(69 366 95 207)(70 367 96 208)(71 368 97 209)(72 369 98 210)(73 370 99 211)(74 371 100 212)(75 372 101 213)(76 373 102 214)(77 374 103 215)(78 375 104 216)(79 376 105 217)(80 377 106 218)(81 378 107 219)(82 379 108 220)(83 380 109 221)(84 381 110 222)(113 177 317 166)(114 178 318 167)(115 179 319 168)(116 180 320 141)(117 181 321 142)(118 182 322 143)(119 183 323 144)(120 184 324 145)(121 185 325 146)(122 186 326 147)(123 187 327 148)(124 188 328 149)(125 189 329 150)(126 190 330 151)(127 191 331 152)(128 192 332 153)(129 193 333 154)(130 194 334 155)(131 195 335 156)(132 196 336 157)(133 169 309 158)(134 170 310 159)(135 171 311 160)(136 172 312 161)(137 173 313 162)(138 174 314 163)(139 175 315 164)(140 176 316 165)
(1 385 293 198)(2 386 294 199)(3 387 295 200)(4 388 296 201)(5 389 297 202)(6 390 298 203)(7 391 299 204)(8 392 300 205)(9 365 301 206)(10 366 302 207)(11 367 303 208)(12 368 304 209)(13 369 305 210)(14 370 306 211)(15 371 307 212)(16 372 308 213)(17 373 281 214)(18 374 282 215)(19 375 283 216)(20 376 284 217)(21 377 285 218)(22 378 286 219)(23 379 287 220)(24 380 288 221)(25 381 289 222)(26 382 290 223)(27 383 291 224)(28 384 292 197)(29 142 250 181)(30 143 251 182)(31 144 252 183)(32 145 225 184)(33 146 226 185)(34 147 227 186)(35 148 228 187)(36 149 229 188)(37 150 230 189)(38 151 231 190)(39 152 232 191)(40 153 233 192)(41 154 234 193)(42 155 235 194)(43 156 236 195)(44 157 237 196)(45 158 238 169)(46 159 239 170)(47 160 240 171)(48 161 241 172)(49 162 242 173)(50 163 243 174)(51 164 244 175)(52 165 245 176)(53 166 246 177)(54 167 247 178)(55 168 248 179)(56 141 249 180)(57 254 111 339)(58 255 112 340)(59 256 85 341)(60 257 86 342)(61 258 87 343)(62 259 88 344)(63 260 89 345)(64 261 90 346)(65 262 91 347)(66 263 92 348)(67 264 93 349)(68 265 94 350)(69 266 95 351)(70 267 96 352)(71 268 97 353)(72 269 98 354)(73 270 99 355)(74 271 100 356)(75 272 101 357)(76 273 102 358)(77 274 103 359)(78 275 104 360)(79 276 105 361)(80 277 106 362)(81 278 107 363)(82 279 108 364)(83 280 109 337)(84 253 110 338)(113 406 317 435)(114 407 318 436)(115 408 319 437)(116 409 320 438)(117 410 321 439)(118 411 322 440)(119 412 323 441)(120 413 324 442)(121 414 325 443)(122 415 326 444)(123 416 327 445)(124 417 328 446)(125 418 329 447)(126 419 330 448)(127 420 331 421)(128 393 332 422)(129 394 333 423)(130 395 334 424)(131 396 335 425)(132 397 336 426)(133 398 309 427)(134 399 310 428)(135 400 311 429)(136 401 312 430)(137 402 313 431)(138 403 314 432)(139 404 315 433)(140 405 316 434)
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), (1,171,15,185)(2,170,16,184)(3,169,17,183)(4,196,18,182)(5,195,19,181)(6,194,20,180)(7,193,21,179)(8,192,22,178)(9,191,23,177)(10,190,24,176)(11,189,25,175)(12,188,26,174)(13,187,27,173)(14,186,28,172)(29,389,43,375)(30,388,44,374)(31,387,45,373)(32,386,46,372)(33,385,47,371)(34,384,48,370)(35,383,49,369)(36,382,50,368)(37,381,51,367)(38,380,52,366)(39,379,53,365)(40,378,54,392)(41,377,55,391)(42,376,56,390)(57,446,71,432)(58,445,72,431)(59,444,73,430)(60,443,74,429)(61,442,75,428)(62,441,76,427)(63,440,77,426)(64,439,78,425)(65,438,79,424)(66,437,80,423)(67,436,81,422)(68,435,82,421)(69,434,83,448)(70,433,84,447)(85,415,99,401)(86,414,100,400)(87,413,101,399)(88,412,102,398)(89,411,103,397)(90,410,104,396)(91,409,105,395)(92,408,106,394)(93,407,107,393)(94,406,108,420)(95,405,109,419)(96,404,110,418)(97,403,111,417)(98,402,112,416)(113,279,127,265)(114,278,128,264)(115,277,129,263)(116,276,130,262)(117,275,131,261)(118,274,132,260)(119,273,133,259)(120,272,134,258)(121,271,135,257)(122,270,136,256)(123,269,137,255)(124,268,138,254)(125,267,139,253)(126,266,140,280)(141,298,155,284)(142,297,156,283)(143,296,157,282)(144,295,158,281)(145,294,159,308)(146,293,160,307)(147,292,161,306)(148,291,162,305)(149,290,163,304)(150,289,164,303)(151,288,165,302)(152,287,166,301)(153,286,167,300)(154,285,168,299)(197,241,211,227)(198,240,212,226)(199,239,213,225)(200,238,214,252)(201,237,215,251)(202,236,216,250)(203,235,217,249)(204,234,218,248)(205,233,219,247)(206,232,220,246)(207,231,221,245)(208,230,222,244)(209,229,223,243)(210,228,224,242)(309,344,323,358)(310,343,324,357)(311,342,325,356)(312,341,326,355)(313,340,327,354)(314,339,328,353)(315,338,329,352)(316,337,330,351)(317,364,331,350)(318,363,332,349)(319,362,333,348)(320,361,334,347)(321,360,335,346)(322,359,336,345), (1,342,293,257)(2,343,294,258)(3,344,295,259)(4,345,296,260)(5,346,297,261)(6,347,298,262)(7,348,299,263)(8,349,300,264)(9,350,301,265)(10,351,302,266)(11,352,303,267)(12,353,304,268)(13,354,305,269)(14,355,306,270)(15,356,307,271)(16,357,308,272)(17,358,281,273)(18,359,282,274)(19,360,283,275)(20,361,284,276)(21,362,285,277)(22,363,286,278)(23,364,287,279)(24,337,288,280)(25,338,289,253)(26,339,290,254)(27,340,291,255)(28,341,292,256)(29,410,250,439)(30,411,251,440)(31,412,252,441)(32,413,225,442)(33,414,226,443)(34,415,227,444)(35,416,228,445)(36,417,229,446)(37,418,230,447)(38,419,231,448)(39,420,232,421)(40,393,233,422)(41,394,234,423)(42,395,235,424)(43,396,236,425)(44,397,237,426)(45,398,238,427)(46,399,239,428)(47,400,240,429)(48,401,241,430)(49,402,242,431)(50,403,243,432)(51,404,244,433)(52,405,245,434)(53,406,246,435)(54,407,247,436)(55,408,248,437)(56,409,249,438)(57,382,111,223)(58,383,112,224)(59,384,85,197)(60,385,86,198)(61,386,87,199)(62,387,88,200)(63,388,89,201)(64,389,90,202)(65,390,91,203)(66,391,92,204)(67,392,93,205)(68,365,94,206)(69,366,95,207)(70,367,96,208)(71,368,97,209)(72,369,98,210)(73,370,99,211)(74,371,100,212)(75,372,101,213)(76,373,102,214)(77,374,103,215)(78,375,104,216)(79,376,105,217)(80,377,106,218)(81,378,107,219)(82,379,108,220)(83,380,109,221)(84,381,110,222)(113,177,317,166)(114,178,318,167)(115,179,319,168)(116,180,320,141)(117,181,321,142)(118,182,322,143)(119,183,323,144)(120,184,324,145)(121,185,325,146)(122,186,326,147)(123,187,327,148)(124,188,328,149)(125,189,329,150)(126,190,330,151)(127,191,331,152)(128,192,332,153)(129,193,333,154)(130,194,334,155)(131,195,335,156)(132,196,336,157)(133,169,309,158)(134,170,310,159)(135,171,311,160)(136,172,312,161)(137,173,313,162)(138,174,314,163)(139,175,315,164)(140,176,316,165), (1,385,293,198)(2,386,294,199)(3,387,295,200)(4,388,296,201)(5,389,297,202)(6,390,298,203)(7,391,299,204)(8,392,300,205)(9,365,301,206)(10,366,302,207)(11,367,303,208)(12,368,304,209)(13,369,305,210)(14,370,306,211)(15,371,307,212)(16,372,308,213)(17,373,281,214)(18,374,282,215)(19,375,283,216)(20,376,284,217)(21,377,285,218)(22,378,286,219)(23,379,287,220)(24,380,288,221)(25,381,289,222)(26,382,290,223)(27,383,291,224)(28,384,292,197)(29,142,250,181)(30,143,251,182)(31,144,252,183)(32,145,225,184)(33,146,226,185)(34,147,227,186)(35,148,228,187)(36,149,229,188)(37,150,230,189)(38,151,231,190)(39,152,232,191)(40,153,233,192)(41,154,234,193)(42,155,235,194)(43,156,236,195)(44,157,237,196)(45,158,238,169)(46,159,239,170)(47,160,240,171)(48,161,241,172)(49,162,242,173)(50,163,243,174)(51,164,244,175)(52,165,245,176)(53,166,246,177)(54,167,247,178)(55,168,248,179)(56,141,249,180)(57,254,111,339)(58,255,112,340)(59,256,85,341)(60,257,86,342)(61,258,87,343)(62,259,88,344)(63,260,89,345)(64,261,90,346)(65,262,91,347)(66,263,92,348)(67,264,93,349)(68,265,94,350)(69,266,95,351)(70,267,96,352)(71,268,97,353)(72,269,98,354)(73,270,99,355)(74,271,100,356)(75,272,101,357)(76,273,102,358)(77,274,103,359)(78,275,104,360)(79,276,105,361)(80,277,106,362)(81,278,107,363)(82,279,108,364)(83,280,109,337)(84,253,110,338)(113,406,317,435)(114,407,318,436)(115,408,319,437)(116,409,320,438)(117,410,321,439)(118,411,322,440)(119,412,323,441)(120,413,324,442)(121,414,325,443)(122,415,326,444)(123,416,327,445)(124,417,328,446)(125,418,329,447)(126,419,330,448)(127,420,331,421)(128,393,332,422)(129,394,333,423)(130,395,334,424)(131,396,335,425)(132,397,336,426)(133,398,309,427)(134,399,310,428)(135,400,311,429)(136,401,312,430)(137,402,313,431)(138,403,314,432)(139,404,315,433)(140,405,316,434)>;
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), (1,171,15,185)(2,170,16,184)(3,169,17,183)(4,196,18,182)(5,195,19,181)(6,194,20,180)(7,193,21,179)(8,192,22,178)(9,191,23,177)(10,190,24,176)(11,189,25,175)(12,188,26,174)(13,187,27,173)(14,186,28,172)(29,389,43,375)(30,388,44,374)(31,387,45,373)(32,386,46,372)(33,385,47,371)(34,384,48,370)(35,383,49,369)(36,382,50,368)(37,381,51,367)(38,380,52,366)(39,379,53,365)(40,378,54,392)(41,377,55,391)(42,376,56,390)(57,446,71,432)(58,445,72,431)(59,444,73,430)(60,443,74,429)(61,442,75,428)(62,441,76,427)(63,440,77,426)(64,439,78,425)(65,438,79,424)(66,437,80,423)(67,436,81,422)(68,435,82,421)(69,434,83,448)(70,433,84,447)(85,415,99,401)(86,414,100,400)(87,413,101,399)(88,412,102,398)(89,411,103,397)(90,410,104,396)(91,409,105,395)(92,408,106,394)(93,407,107,393)(94,406,108,420)(95,405,109,419)(96,404,110,418)(97,403,111,417)(98,402,112,416)(113,279,127,265)(114,278,128,264)(115,277,129,263)(116,276,130,262)(117,275,131,261)(118,274,132,260)(119,273,133,259)(120,272,134,258)(121,271,135,257)(122,270,136,256)(123,269,137,255)(124,268,138,254)(125,267,139,253)(126,266,140,280)(141,298,155,284)(142,297,156,283)(143,296,157,282)(144,295,158,281)(145,294,159,308)(146,293,160,307)(147,292,161,306)(148,291,162,305)(149,290,163,304)(150,289,164,303)(151,288,165,302)(152,287,166,301)(153,286,167,300)(154,285,168,299)(197,241,211,227)(198,240,212,226)(199,239,213,225)(200,238,214,252)(201,237,215,251)(202,236,216,250)(203,235,217,249)(204,234,218,248)(205,233,219,247)(206,232,220,246)(207,231,221,245)(208,230,222,244)(209,229,223,243)(210,228,224,242)(309,344,323,358)(310,343,324,357)(311,342,325,356)(312,341,326,355)(313,340,327,354)(314,339,328,353)(315,338,329,352)(316,337,330,351)(317,364,331,350)(318,363,332,349)(319,362,333,348)(320,361,334,347)(321,360,335,346)(322,359,336,345), (1,342,293,257)(2,343,294,258)(3,344,295,259)(4,345,296,260)(5,346,297,261)(6,347,298,262)(7,348,299,263)(8,349,300,264)(9,350,301,265)(10,351,302,266)(11,352,303,267)(12,353,304,268)(13,354,305,269)(14,355,306,270)(15,356,307,271)(16,357,308,272)(17,358,281,273)(18,359,282,274)(19,360,283,275)(20,361,284,276)(21,362,285,277)(22,363,286,278)(23,364,287,279)(24,337,288,280)(25,338,289,253)(26,339,290,254)(27,340,291,255)(28,341,292,256)(29,410,250,439)(30,411,251,440)(31,412,252,441)(32,413,225,442)(33,414,226,443)(34,415,227,444)(35,416,228,445)(36,417,229,446)(37,418,230,447)(38,419,231,448)(39,420,232,421)(40,393,233,422)(41,394,234,423)(42,395,235,424)(43,396,236,425)(44,397,237,426)(45,398,238,427)(46,399,239,428)(47,400,240,429)(48,401,241,430)(49,402,242,431)(50,403,243,432)(51,404,244,433)(52,405,245,434)(53,406,246,435)(54,407,247,436)(55,408,248,437)(56,409,249,438)(57,382,111,223)(58,383,112,224)(59,384,85,197)(60,385,86,198)(61,386,87,199)(62,387,88,200)(63,388,89,201)(64,389,90,202)(65,390,91,203)(66,391,92,204)(67,392,93,205)(68,365,94,206)(69,366,95,207)(70,367,96,208)(71,368,97,209)(72,369,98,210)(73,370,99,211)(74,371,100,212)(75,372,101,213)(76,373,102,214)(77,374,103,215)(78,375,104,216)(79,376,105,217)(80,377,106,218)(81,378,107,219)(82,379,108,220)(83,380,109,221)(84,381,110,222)(113,177,317,166)(114,178,318,167)(115,179,319,168)(116,180,320,141)(117,181,321,142)(118,182,322,143)(119,183,323,144)(120,184,324,145)(121,185,325,146)(122,186,326,147)(123,187,327,148)(124,188,328,149)(125,189,329,150)(126,190,330,151)(127,191,331,152)(128,192,332,153)(129,193,333,154)(130,194,334,155)(131,195,335,156)(132,196,336,157)(133,169,309,158)(134,170,310,159)(135,171,311,160)(136,172,312,161)(137,173,313,162)(138,174,314,163)(139,175,315,164)(140,176,316,165), (1,385,293,198)(2,386,294,199)(3,387,295,200)(4,388,296,201)(5,389,297,202)(6,390,298,203)(7,391,299,204)(8,392,300,205)(9,365,301,206)(10,366,302,207)(11,367,303,208)(12,368,304,209)(13,369,305,210)(14,370,306,211)(15,371,307,212)(16,372,308,213)(17,373,281,214)(18,374,282,215)(19,375,283,216)(20,376,284,217)(21,377,285,218)(22,378,286,219)(23,379,287,220)(24,380,288,221)(25,381,289,222)(26,382,290,223)(27,383,291,224)(28,384,292,197)(29,142,250,181)(30,143,251,182)(31,144,252,183)(32,145,225,184)(33,146,226,185)(34,147,227,186)(35,148,228,187)(36,149,229,188)(37,150,230,189)(38,151,231,190)(39,152,232,191)(40,153,233,192)(41,154,234,193)(42,155,235,194)(43,156,236,195)(44,157,237,196)(45,158,238,169)(46,159,239,170)(47,160,240,171)(48,161,241,172)(49,162,242,173)(50,163,243,174)(51,164,244,175)(52,165,245,176)(53,166,246,177)(54,167,247,178)(55,168,248,179)(56,141,249,180)(57,254,111,339)(58,255,112,340)(59,256,85,341)(60,257,86,342)(61,258,87,343)(62,259,88,344)(63,260,89,345)(64,261,90,346)(65,262,91,347)(66,263,92,348)(67,264,93,349)(68,265,94,350)(69,266,95,351)(70,267,96,352)(71,268,97,353)(72,269,98,354)(73,270,99,355)(74,271,100,356)(75,272,101,357)(76,273,102,358)(77,274,103,359)(78,275,104,360)(79,276,105,361)(80,277,106,362)(81,278,107,363)(82,279,108,364)(83,280,109,337)(84,253,110,338)(113,406,317,435)(114,407,318,436)(115,408,319,437)(116,409,320,438)(117,410,321,439)(118,411,322,440)(119,412,323,441)(120,413,324,442)(121,414,325,443)(122,415,326,444)(123,416,327,445)(124,417,328,446)(125,418,329,447)(126,419,330,448)(127,420,331,421)(128,393,332,422)(129,394,333,423)(130,395,334,424)(131,396,335,425)(132,397,336,426)(133,398,309,427)(134,399,310,428)(135,400,311,429)(136,401,312,430)(137,402,313,431)(138,403,314,432)(139,404,315,433)(140,405,316,434) );
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)], [(1,171,15,185),(2,170,16,184),(3,169,17,183),(4,196,18,182),(5,195,19,181),(6,194,20,180),(7,193,21,179),(8,192,22,178),(9,191,23,177),(10,190,24,176),(11,189,25,175),(12,188,26,174),(13,187,27,173),(14,186,28,172),(29,389,43,375),(30,388,44,374),(31,387,45,373),(32,386,46,372),(33,385,47,371),(34,384,48,370),(35,383,49,369),(36,382,50,368),(37,381,51,367),(38,380,52,366),(39,379,53,365),(40,378,54,392),(41,377,55,391),(42,376,56,390),(57,446,71,432),(58,445,72,431),(59,444,73,430),(60,443,74,429),(61,442,75,428),(62,441,76,427),(63,440,77,426),(64,439,78,425),(65,438,79,424),(66,437,80,423),(67,436,81,422),(68,435,82,421),(69,434,83,448),(70,433,84,447),(85,415,99,401),(86,414,100,400),(87,413,101,399),(88,412,102,398),(89,411,103,397),(90,410,104,396),(91,409,105,395),(92,408,106,394),(93,407,107,393),(94,406,108,420),(95,405,109,419),(96,404,110,418),(97,403,111,417),(98,402,112,416),(113,279,127,265),(114,278,128,264),(115,277,129,263),(116,276,130,262),(117,275,131,261),(118,274,132,260),(119,273,133,259),(120,272,134,258),(121,271,135,257),(122,270,136,256),(123,269,137,255),(124,268,138,254),(125,267,139,253),(126,266,140,280),(141,298,155,284),(142,297,156,283),(143,296,157,282),(144,295,158,281),(145,294,159,308),(146,293,160,307),(147,292,161,306),(148,291,162,305),(149,290,163,304),(150,289,164,303),(151,288,165,302),(152,287,166,301),(153,286,167,300),(154,285,168,299),(197,241,211,227),(198,240,212,226),(199,239,213,225),(200,238,214,252),(201,237,215,251),(202,236,216,250),(203,235,217,249),(204,234,218,248),(205,233,219,247),(206,232,220,246),(207,231,221,245),(208,230,222,244),(209,229,223,243),(210,228,224,242),(309,344,323,358),(310,343,324,357),(311,342,325,356),(312,341,326,355),(313,340,327,354),(314,339,328,353),(315,338,329,352),(316,337,330,351),(317,364,331,350),(318,363,332,349),(319,362,333,348),(320,361,334,347),(321,360,335,346),(322,359,336,345)], [(1,342,293,257),(2,343,294,258),(3,344,295,259),(4,345,296,260),(5,346,297,261),(6,347,298,262),(7,348,299,263),(8,349,300,264),(9,350,301,265),(10,351,302,266),(11,352,303,267),(12,353,304,268),(13,354,305,269),(14,355,306,270),(15,356,307,271),(16,357,308,272),(17,358,281,273),(18,359,282,274),(19,360,283,275),(20,361,284,276),(21,362,285,277),(22,363,286,278),(23,364,287,279),(24,337,288,280),(25,338,289,253),(26,339,290,254),(27,340,291,255),(28,341,292,256),(29,410,250,439),(30,411,251,440),(31,412,252,441),(32,413,225,442),(33,414,226,443),(34,415,227,444),(35,416,228,445),(36,417,229,446),(37,418,230,447),(38,419,231,448),(39,420,232,421),(40,393,233,422),(41,394,234,423),(42,395,235,424),(43,396,236,425),(44,397,237,426),(45,398,238,427),(46,399,239,428),(47,400,240,429),(48,401,241,430),(49,402,242,431),(50,403,243,432),(51,404,244,433),(52,405,245,434),(53,406,246,435),(54,407,247,436),(55,408,248,437),(56,409,249,438),(57,382,111,223),(58,383,112,224),(59,384,85,197),(60,385,86,198),(61,386,87,199),(62,387,88,200),(63,388,89,201),(64,389,90,202),(65,390,91,203),(66,391,92,204),(67,392,93,205),(68,365,94,206),(69,366,95,207),(70,367,96,208),(71,368,97,209),(72,369,98,210),(73,370,99,211),(74,371,100,212),(75,372,101,213),(76,373,102,214),(77,374,103,215),(78,375,104,216),(79,376,105,217),(80,377,106,218),(81,378,107,219),(82,379,108,220),(83,380,109,221),(84,381,110,222),(113,177,317,166),(114,178,318,167),(115,179,319,168),(116,180,320,141),(117,181,321,142),(118,182,322,143),(119,183,323,144),(120,184,324,145),(121,185,325,146),(122,186,326,147),(123,187,327,148),(124,188,328,149),(125,189,329,150),(126,190,330,151),(127,191,331,152),(128,192,332,153),(129,193,333,154),(130,194,334,155),(131,195,335,156),(132,196,336,157),(133,169,309,158),(134,170,310,159),(135,171,311,160),(136,172,312,161),(137,173,313,162),(138,174,314,163),(139,175,315,164),(140,176,316,165)], [(1,385,293,198),(2,386,294,199),(3,387,295,200),(4,388,296,201),(5,389,297,202),(6,390,298,203),(7,391,299,204),(8,392,300,205),(9,365,301,206),(10,366,302,207),(11,367,303,208),(12,368,304,209),(13,369,305,210),(14,370,306,211),(15,371,307,212),(16,372,308,213),(17,373,281,214),(18,374,282,215),(19,375,283,216),(20,376,284,217),(21,377,285,218),(22,378,286,219),(23,379,287,220),(24,380,288,221),(25,381,289,222),(26,382,290,223),(27,383,291,224),(28,384,292,197),(29,142,250,181),(30,143,251,182),(31,144,252,183),(32,145,225,184),(33,146,226,185),(34,147,227,186),(35,148,228,187),(36,149,229,188),(37,150,230,189),(38,151,231,190),(39,152,232,191),(40,153,233,192),(41,154,234,193),(42,155,235,194),(43,156,236,195),(44,157,237,196),(45,158,238,169),(46,159,239,170),(47,160,240,171),(48,161,241,172),(49,162,242,173),(50,163,243,174),(51,164,244,175),(52,165,245,176),(53,166,246,177),(54,167,247,178),(55,168,248,179),(56,141,249,180),(57,254,111,339),(58,255,112,340),(59,256,85,341),(60,257,86,342),(61,258,87,343),(62,259,88,344),(63,260,89,345),(64,261,90,346),(65,262,91,347),(66,263,92,348),(67,264,93,349),(68,265,94,350),(69,266,95,351),(70,267,96,352),(71,268,97,353),(72,269,98,354),(73,270,99,355),(74,271,100,356),(75,272,101,357),(76,273,102,358),(77,274,103,359),(78,275,104,360),(79,276,105,361),(80,277,106,362),(81,278,107,363),(82,279,108,364),(83,280,109,337),(84,253,110,338),(113,406,317,435),(114,407,318,436),(115,408,319,437),(116,409,320,438),(117,410,321,439),(118,411,322,440),(119,412,323,441),(120,413,324,442),(121,414,325,443),(122,415,326,444),(123,416,327,445),(124,417,328,446),(125,418,329,447),(126,419,330,448),(127,420,331,421),(128,393,332,422),(129,394,333,423),(130,395,334,424),(131,396,335,425),(132,397,336,426),(133,398,309,427),(134,399,310,428),(135,400,311,429),(136,401,312,430),(137,402,313,431),(138,403,314,432),(139,404,315,433),(140,405,316,434)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | ··· | 4U | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | 2- 1+4 | Q8×D7 | D4.10D14 |
| kernel | Dic14⋊10Q8 | C4×Dic14 | C28⋊2Q8 | C28.6Q8 | Dic7⋊3Q8 | Dic7.Q8 | Dic7⋊Q8 | Q8×C28 | Dic14 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C4 | C14 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 2 | 4 | 2 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
Matrix representation of Dic14⋊10Q8 ►in GL4(𝔽29) generated by
| 2 | 21 | 0 | 0 |
| 14 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 21 | 17 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 24 | 14 | 0 | 0 |
| 19 | 5 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 1 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 7 | 16 |
| 0 | 0 | 15 | 22 |
G:=sub<GL(4,GF(29))| [2,14,0,0,21,17,0,0,0,0,1,0,0,0,0,1],[12,21,0,0,0,17,0,0,0,0,28,0,0,0,0,28],[24,19,0,0,14,5,0,0,0,0,1,1,0,0,27,28],[28,0,0,0,0,28,0,0,0,0,7,15,0,0,16,22] >;
Dic14⋊10Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes_{10}Q_8 % in TeX
G:=Group("Dic14:10Q8"); // GroupNames label
G:=SmallGroup(448,1020);
// by ID
G=gap.SmallGroup(448,1020);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,477,232,100,185,192,18822]);
// 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,a*d=d*a,c*b*c^-1=a^14*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations