metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊5Q8, C28.10Q16, C42.84D14, C4⋊Q8.10D7, C4.13(Q8×D7), C4⋊C4.86D14, C7⋊5(C4.Q16), C28.40(C2×Q8), (C2×C28).160D4, C14.43(C2×Q16), C28⋊C8.22C2, C4.8(C7⋊Q16), C28.86(C4○D4), (C2×C28).409C23, (C4×C28).138C22, C4.36(Q8⋊2D7), (C4×Dic14).18C2, C14.Q16.14C2, C28.Q8.17C2, C14.77(C22⋊Q8), C14.100(C8⋊C22), C2.14(D14⋊3Q8), C4⋊Dic7.350C22, C2.21(D4.D14), (C2×Dic14).276C22, (C7×C4⋊Q8).10C2, C2.14(C2×C7⋊Q16), (C2×C14).540(C2×D4), (C2×C7⋊C8).141C22, (C2×C4).191(C7⋊D4), (C7×C4⋊C4).133C22, (C2×C4).506(C22×D7), C22.212(C2×C7⋊D4), SmallGroup(448,625)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊5Q8
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, cac-1=a15, ad=da, cbc-1=a21b, dbd-1=a14b, 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, C2.D8, C4×Q8, C4⋊Q8, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C4.Q16, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C28⋊C8, C28.Q8, C14.Q16, C4×Dic14, C7×C4⋊Q8, Dic14⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, Q16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×Q16, C8⋊C22, C7⋊D4, C22×D7, C4.Q16, C7⋊Q16, Q8×D7, Q8⋊2D7, C2×C7⋊D4, D4.D14, C2×C7⋊Q16, D14⋊3Q8, Dic14⋊5Q8
►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)
(1 374 15 388)(2 373 16 387)(3 372 17 386)(4 371 18 385)(5 370 19 384)(6 369 20 383)(7 368 21 382)(8 367 22 381)(9 366 23 380)(10 365 24 379)(11 392 25 378)(12 391 26 377)(13 390 27 376)(14 389 28 375)(29 418 43 404)(30 417 44 403)(31 416 45 402)(32 415 46 401)(33 414 47 400)(34 413 48 399)(35 412 49 398)(36 411 50 397)(37 410 51 396)(38 409 52 395)(39 408 53 394)(40 407 54 393)(41 406 55 420)(42 405 56 419)(57 111 71 97)(58 110 72 96)(59 109 73 95)(60 108 74 94)(61 107 75 93)(62 106 76 92)(63 105 77 91)(64 104 78 90)(65 103 79 89)(66 102 80 88)(67 101 81 87)(68 100 82 86)(69 99 83 85)(70 98 84 112)(113 320 127 334)(114 319 128 333)(115 318 129 332)(116 317 130 331)(117 316 131 330)(118 315 132 329)(119 314 133 328)(120 313 134 327)(121 312 135 326)(122 311 136 325)(123 310 137 324)(124 309 138 323)(125 336 139 322)(126 335 140 321)(141 269 155 255)(142 268 156 254)(143 267 157 253)(144 266 158 280)(145 265 159 279)(146 264 160 278)(147 263 161 277)(148 262 162 276)(149 261 163 275)(150 260 164 274)(151 259 165 273)(152 258 166 272)(153 257 167 271)(154 256 168 270)(169 240 183 226)(170 239 184 225)(171 238 185 252)(172 237 186 251)(173 236 187 250)(174 235 188 249)(175 234 189 248)(176 233 190 247)(177 232 191 246)(178 231 192 245)(179 230 193 244)(180 229 194 243)(181 228 195 242)(182 227 196 241)(197 299 211 285)(198 298 212 284)(199 297 213 283)(200 296 214 282)(201 295 215 281)(202 294 216 308)(203 293 217 307)(204 292 218 306)(205 291 219 305)(206 290 220 304)(207 289 221 303)(208 288 222 302)(209 287 223 301)(210 286 224 300)(337 446 351 432)(338 445 352 431)(339 444 353 430)(340 443 354 429)(341 442 355 428)(342 441 356 427)(343 440 357 426)(344 439 358 425)(345 438 359 424)(346 437 360 423)(347 436 361 422)(348 435 362 421)(349 434 363 448)(350 433 364 447)
(1 393 303 72)(2 408 304 59)(3 395 305 74)(4 410 306 61)(5 397 307 76)(6 412 308 63)(7 399 281 78)(8 414 282 65)(9 401 283 80)(10 416 284 67)(11 403 285 82)(12 418 286 69)(13 405 287 84)(14 420 288 71)(15 407 289 58)(16 394 290 73)(17 409 291 60)(18 396 292 75)(19 411 293 62)(20 398 294 77)(21 413 295 64)(22 400 296 79)(23 415 297 66)(24 402 298 81)(25 417 299 68)(26 404 300 83)(27 419 301 70)(28 406 302 57)(29 217 85 384)(30 204 86 371)(31 219 87 386)(32 206 88 373)(33 221 89 388)(34 208 90 375)(35 223 91 390)(36 210 92 377)(37 197 93 392)(38 212 94 379)(39 199 95 366)(40 214 96 381)(41 201 97 368)(42 216 98 383)(43 203 99 370)(44 218 100 385)(45 205 101 372)(46 220 102 387)(47 207 103 374)(48 222 104 389)(49 209 105 376)(50 224 106 391)(51 211 107 378)(52 198 108 365)(53 213 109 380)(54 200 110 367)(55 215 111 382)(56 202 112 369)(113 170 447 144)(114 185 448 159)(115 172 421 146)(116 187 422 161)(117 174 423 148)(118 189 424 163)(119 176 425 150)(120 191 426 165)(121 178 427 152)(122 193 428 167)(123 180 429 154)(124 195 430 141)(125 182 431 156)(126 169 432 143)(127 184 433 158)(128 171 434 145)(129 186 435 160)(130 173 436 147)(131 188 437 162)(132 175 438 149)(133 190 439 164)(134 177 440 151)(135 192 441 166)(136 179 442 153)(137 194 443 168)(138 181 444 155)(139 196 445 142)(140 183 446 157)(225 343 280 313)(226 358 253 328)(227 345 254 315)(228 360 255 330)(229 347 256 317)(230 362 257 332)(231 349 258 319)(232 364 259 334)(233 351 260 321)(234 338 261 336)(235 353 262 323)(236 340 263 310)(237 355 264 325)(238 342 265 312)(239 357 266 327)(240 344 267 314)(241 359 268 329)(242 346 269 316)(243 361 270 331)(244 348 271 318)(245 363 272 333)(246 350 273 320)(247 337 274 335)(248 352 275 322)(249 339 276 309)(250 354 277 324)(251 341 278 311)(252 356 279 326)
(1 331 303 361)(2 332 304 362)(3 333 305 363)(4 334 306 364)(5 335 307 337)(6 336 308 338)(7 309 281 339)(8 310 282 340)(9 311 283 341)(10 312 284 342)(11 313 285 343)(12 314 286 344)(13 315 287 345)(14 316 288 346)(15 317 289 347)(16 318 290 348)(17 319 291 349)(18 320 292 350)(19 321 293 351)(20 322 294 352)(21 323 295 353)(22 324 296 354)(23 325 297 355)(24 326 298 356)(25 327 299 357)(26 328 300 358)(27 329 301 359)(28 330 302 360)(29 157 85 183)(30 158 86 184)(31 159 87 185)(32 160 88 186)(33 161 89 187)(34 162 90 188)(35 163 91 189)(36 164 92 190)(37 165 93 191)(38 166 94 192)(39 167 95 193)(40 168 96 194)(41 141 97 195)(42 142 98 196)(43 143 99 169)(44 144 100 170)(45 145 101 171)(46 146 102 172)(47 147 103 173)(48 148 104 174)(49 149 105 175)(50 150 106 176)(51 151 107 177)(52 152 108 178)(53 153 109 179)(54 154 110 180)(55 155 111 181)(56 156 112 182)(57 228 406 255)(58 229 407 256)(59 230 408 257)(60 231 409 258)(61 232 410 259)(62 233 411 260)(63 234 412 261)(64 235 413 262)(65 236 414 263)(66 237 415 264)(67 238 416 265)(68 239 417 266)(69 240 418 267)(70 241 419 268)(71 242 420 269)(72 243 393 270)(73 244 394 271)(74 245 395 272)(75 246 396 273)(76 247 397 274)(77 248 398 275)(78 249 399 276)(79 250 400 277)(80 251 401 278)(81 252 402 279)(82 225 403 280)(83 226 404 253)(84 227 405 254)(113 218 447 385)(114 219 448 386)(115 220 421 387)(116 221 422 388)(117 222 423 389)(118 223 424 390)(119 224 425 391)(120 197 426 392)(121 198 427 365)(122 199 428 366)(123 200 429 367)(124 201 430 368)(125 202 431 369)(126 203 432 370)(127 204 433 371)(128 205 434 372)(129 206 435 373)(130 207 436 374)(131 208 437 375)(132 209 438 376)(133 210 439 377)(134 211 440 378)(135 212 441 379)(136 213 442 380)(137 214 443 381)(138 215 444 382)(139 216 445 383)(140 217 446 384)
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), (1,374,15,388)(2,373,16,387)(3,372,17,386)(4,371,18,385)(5,370,19,384)(6,369,20,383)(7,368,21,382)(8,367,22,381)(9,366,23,380)(10,365,24,379)(11,392,25,378)(12,391,26,377)(13,390,27,376)(14,389,28,375)(29,418,43,404)(30,417,44,403)(31,416,45,402)(32,415,46,401)(33,414,47,400)(34,413,48,399)(35,412,49,398)(36,411,50,397)(37,410,51,396)(38,409,52,395)(39,408,53,394)(40,407,54,393)(41,406,55,420)(42,405,56,419)(57,111,71,97)(58,110,72,96)(59,109,73,95)(60,108,74,94)(61,107,75,93)(62,106,76,92)(63,105,77,91)(64,104,78,90)(65,103,79,89)(66,102,80,88)(67,101,81,87)(68,100,82,86)(69,99,83,85)(70,98,84,112)(113,320,127,334)(114,319,128,333)(115,318,129,332)(116,317,130,331)(117,316,131,330)(118,315,132,329)(119,314,133,328)(120,313,134,327)(121,312,135,326)(122,311,136,325)(123,310,137,324)(124,309,138,323)(125,336,139,322)(126,335,140,321)(141,269,155,255)(142,268,156,254)(143,267,157,253)(144,266,158,280)(145,265,159,279)(146,264,160,278)(147,263,161,277)(148,262,162,276)(149,261,163,275)(150,260,164,274)(151,259,165,273)(152,258,166,272)(153,257,167,271)(154,256,168,270)(169,240,183,226)(170,239,184,225)(171,238,185,252)(172,237,186,251)(173,236,187,250)(174,235,188,249)(175,234,189,248)(176,233,190,247)(177,232,191,246)(178,231,192,245)(179,230,193,244)(180,229,194,243)(181,228,195,242)(182,227,196,241)(197,299,211,285)(198,298,212,284)(199,297,213,283)(200,296,214,282)(201,295,215,281)(202,294,216,308)(203,293,217,307)(204,292,218,306)(205,291,219,305)(206,290,220,304)(207,289,221,303)(208,288,222,302)(209,287,223,301)(210,286,224,300)(337,446,351,432)(338,445,352,431)(339,444,353,430)(340,443,354,429)(341,442,355,428)(342,441,356,427)(343,440,357,426)(344,439,358,425)(345,438,359,424)(346,437,360,423)(347,436,361,422)(348,435,362,421)(349,434,363,448)(350,433,364,447), (1,393,303,72)(2,408,304,59)(3,395,305,74)(4,410,306,61)(5,397,307,76)(6,412,308,63)(7,399,281,78)(8,414,282,65)(9,401,283,80)(10,416,284,67)(11,403,285,82)(12,418,286,69)(13,405,287,84)(14,420,288,71)(15,407,289,58)(16,394,290,73)(17,409,291,60)(18,396,292,75)(19,411,293,62)(20,398,294,77)(21,413,295,64)(22,400,296,79)(23,415,297,66)(24,402,298,81)(25,417,299,68)(26,404,300,83)(27,419,301,70)(28,406,302,57)(29,217,85,384)(30,204,86,371)(31,219,87,386)(32,206,88,373)(33,221,89,388)(34,208,90,375)(35,223,91,390)(36,210,92,377)(37,197,93,392)(38,212,94,379)(39,199,95,366)(40,214,96,381)(41,201,97,368)(42,216,98,383)(43,203,99,370)(44,218,100,385)(45,205,101,372)(46,220,102,387)(47,207,103,374)(48,222,104,389)(49,209,105,376)(50,224,106,391)(51,211,107,378)(52,198,108,365)(53,213,109,380)(54,200,110,367)(55,215,111,382)(56,202,112,369)(113,170,447,144)(114,185,448,159)(115,172,421,146)(116,187,422,161)(117,174,423,148)(118,189,424,163)(119,176,425,150)(120,191,426,165)(121,178,427,152)(122,193,428,167)(123,180,429,154)(124,195,430,141)(125,182,431,156)(126,169,432,143)(127,184,433,158)(128,171,434,145)(129,186,435,160)(130,173,436,147)(131,188,437,162)(132,175,438,149)(133,190,439,164)(134,177,440,151)(135,192,441,166)(136,179,442,153)(137,194,443,168)(138,181,444,155)(139,196,445,142)(140,183,446,157)(225,343,280,313)(226,358,253,328)(227,345,254,315)(228,360,255,330)(229,347,256,317)(230,362,257,332)(231,349,258,319)(232,364,259,334)(233,351,260,321)(234,338,261,336)(235,353,262,323)(236,340,263,310)(237,355,264,325)(238,342,265,312)(239,357,266,327)(240,344,267,314)(241,359,268,329)(242,346,269,316)(243,361,270,331)(244,348,271,318)(245,363,272,333)(246,350,273,320)(247,337,274,335)(248,352,275,322)(249,339,276,309)(250,354,277,324)(251,341,278,311)(252,356,279,326), (1,331,303,361)(2,332,304,362)(3,333,305,363)(4,334,306,364)(5,335,307,337)(6,336,308,338)(7,309,281,339)(8,310,282,340)(9,311,283,341)(10,312,284,342)(11,313,285,343)(12,314,286,344)(13,315,287,345)(14,316,288,346)(15,317,289,347)(16,318,290,348)(17,319,291,349)(18,320,292,350)(19,321,293,351)(20,322,294,352)(21,323,295,353)(22,324,296,354)(23,325,297,355)(24,326,298,356)(25,327,299,357)(26,328,300,358)(27,329,301,359)(28,330,302,360)(29,157,85,183)(30,158,86,184)(31,159,87,185)(32,160,88,186)(33,161,89,187)(34,162,90,188)(35,163,91,189)(36,164,92,190)(37,165,93,191)(38,166,94,192)(39,167,95,193)(40,168,96,194)(41,141,97,195)(42,142,98,196)(43,143,99,169)(44,144,100,170)(45,145,101,171)(46,146,102,172)(47,147,103,173)(48,148,104,174)(49,149,105,175)(50,150,106,176)(51,151,107,177)(52,152,108,178)(53,153,109,179)(54,154,110,180)(55,155,111,181)(56,156,112,182)(57,228,406,255)(58,229,407,256)(59,230,408,257)(60,231,409,258)(61,232,410,259)(62,233,411,260)(63,234,412,261)(64,235,413,262)(65,236,414,263)(66,237,415,264)(67,238,416,265)(68,239,417,266)(69,240,418,267)(70,241,419,268)(71,242,420,269)(72,243,393,270)(73,244,394,271)(74,245,395,272)(75,246,396,273)(76,247,397,274)(77,248,398,275)(78,249,399,276)(79,250,400,277)(80,251,401,278)(81,252,402,279)(82,225,403,280)(83,226,404,253)(84,227,405,254)(113,218,447,385)(114,219,448,386)(115,220,421,387)(116,221,422,388)(117,222,423,389)(118,223,424,390)(119,224,425,391)(120,197,426,392)(121,198,427,365)(122,199,428,366)(123,200,429,367)(124,201,430,368)(125,202,431,369)(126,203,432,370)(127,204,433,371)(128,205,434,372)(129,206,435,373)(130,207,436,374)(131,208,437,375)(132,209,438,376)(133,210,439,377)(134,211,440,378)(135,212,441,379)(136,213,442,380)(137,214,443,381)(138,215,444,382)(139,216,445,383)(140,217,446,384)>;
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), (1,374,15,388)(2,373,16,387)(3,372,17,386)(4,371,18,385)(5,370,19,384)(6,369,20,383)(7,368,21,382)(8,367,22,381)(9,366,23,380)(10,365,24,379)(11,392,25,378)(12,391,26,377)(13,390,27,376)(14,389,28,375)(29,418,43,404)(30,417,44,403)(31,416,45,402)(32,415,46,401)(33,414,47,400)(34,413,48,399)(35,412,49,398)(36,411,50,397)(37,410,51,396)(38,409,52,395)(39,408,53,394)(40,407,54,393)(41,406,55,420)(42,405,56,419)(57,111,71,97)(58,110,72,96)(59,109,73,95)(60,108,74,94)(61,107,75,93)(62,106,76,92)(63,105,77,91)(64,104,78,90)(65,103,79,89)(66,102,80,88)(67,101,81,87)(68,100,82,86)(69,99,83,85)(70,98,84,112)(113,320,127,334)(114,319,128,333)(115,318,129,332)(116,317,130,331)(117,316,131,330)(118,315,132,329)(119,314,133,328)(120,313,134,327)(121,312,135,326)(122,311,136,325)(123,310,137,324)(124,309,138,323)(125,336,139,322)(126,335,140,321)(141,269,155,255)(142,268,156,254)(143,267,157,253)(144,266,158,280)(145,265,159,279)(146,264,160,278)(147,263,161,277)(148,262,162,276)(149,261,163,275)(150,260,164,274)(151,259,165,273)(152,258,166,272)(153,257,167,271)(154,256,168,270)(169,240,183,226)(170,239,184,225)(171,238,185,252)(172,237,186,251)(173,236,187,250)(174,235,188,249)(175,234,189,248)(176,233,190,247)(177,232,191,246)(178,231,192,245)(179,230,193,244)(180,229,194,243)(181,228,195,242)(182,227,196,241)(197,299,211,285)(198,298,212,284)(199,297,213,283)(200,296,214,282)(201,295,215,281)(202,294,216,308)(203,293,217,307)(204,292,218,306)(205,291,219,305)(206,290,220,304)(207,289,221,303)(208,288,222,302)(209,287,223,301)(210,286,224,300)(337,446,351,432)(338,445,352,431)(339,444,353,430)(340,443,354,429)(341,442,355,428)(342,441,356,427)(343,440,357,426)(344,439,358,425)(345,438,359,424)(346,437,360,423)(347,436,361,422)(348,435,362,421)(349,434,363,448)(350,433,364,447), (1,393,303,72)(2,408,304,59)(3,395,305,74)(4,410,306,61)(5,397,307,76)(6,412,308,63)(7,399,281,78)(8,414,282,65)(9,401,283,80)(10,416,284,67)(11,403,285,82)(12,418,286,69)(13,405,287,84)(14,420,288,71)(15,407,289,58)(16,394,290,73)(17,409,291,60)(18,396,292,75)(19,411,293,62)(20,398,294,77)(21,413,295,64)(22,400,296,79)(23,415,297,66)(24,402,298,81)(25,417,299,68)(26,404,300,83)(27,419,301,70)(28,406,302,57)(29,217,85,384)(30,204,86,371)(31,219,87,386)(32,206,88,373)(33,221,89,388)(34,208,90,375)(35,223,91,390)(36,210,92,377)(37,197,93,392)(38,212,94,379)(39,199,95,366)(40,214,96,381)(41,201,97,368)(42,216,98,383)(43,203,99,370)(44,218,100,385)(45,205,101,372)(46,220,102,387)(47,207,103,374)(48,222,104,389)(49,209,105,376)(50,224,106,391)(51,211,107,378)(52,198,108,365)(53,213,109,380)(54,200,110,367)(55,215,111,382)(56,202,112,369)(113,170,447,144)(114,185,448,159)(115,172,421,146)(116,187,422,161)(117,174,423,148)(118,189,424,163)(119,176,425,150)(120,191,426,165)(121,178,427,152)(122,193,428,167)(123,180,429,154)(124,195,430,141)(125,182,431,156)(126,169,432,143)(127,184,433,158)(128,171,434,145)(129,186,435,160)(130,173,436,147)(131,188,437,162)(132,175,438,149)(133,190,439,164)(134,177,440,151)(135,192,441,166)(136,179,442,153)(137,194,443,168)(138,181,444,155)(139,196,445,142)(140,183,446,157)(225,343,280,313)(226,358,253,328)(227,345,254,315)(228,360,255,330)(229,347,256,317)(230,362,257,332)(231,349,258,319)(232,364,259,334)(233,351,260,321)(234,338,261,336)(235,353,262,323)(236,340,263,310)(237,355,264,325)(238,342,265,312)(239,357,266,327)(240,344,267,314)(241,359,268,329)(242,346,269,316)(243,361,270,331)(244,348,271,318)(245,363,272,333)(246,350,273,320)(247,337,274,335)(248,352,275,322)(249,339,276,309)(250,354,277,324)(251,341,278,311)(252,356,279,326), (1,331,303,361)(2,332,304,362)(3,333,305,363)(4,334,306,364)(5,335,307,337)(6,336,308,338)(7,309,281,339)(8,310,282,340)(9,311,283,341)(10,312,284,342)(11,313,285,343)(12,314,286,344)(13,315,287,345)(14,316,288,346)(15,317,289,347)(16,318,290,348)(17,319,291,349)(18,320,292,350)(19,321,293,351)(20,322,294,352)(21,323,295,353)(22,324,296,354)(23,325,297,355)(24,326,298,356)(25,327,299,357)(26,328,300,358)(27,329,301,359)(28,330,302,360)(29,157,85,183)(30,158,86,184)(31,159,87,185)(32,160,88,186)(33,161,89,187)(34,162,90,188)(35,163,91,189)(36,164,92,190)(37,165,93,191)(38,166,94,192)(39,167,95,193)(40,168,96,194)(41,141,97,195)(42,142,98,196)(43,143,99,169)(44,144,100,170)(45,145,101,171)(46,146,102,172)(47,147,103,173)(48,148,104,174)(49,149,105,175)(50,150,106,176)(51,151,107,177)(52,152,108,178)(53,153,109,179)(54,154,110,180)(55,155,111,181)(56,156,112,182)(57,228,406,255)(58,229,407,256)(59,230,408,257)(60,231,409,258)(61,232,410,259)(62,233,411,260)(63,234,412,261)(64,235,413,262)(65,236,414,263)(66,237,415,264)(67,238,416,265)(68,239,417,266)(69,240,418,267)(70,241,419,268)(71,242,420,269)(72,243,393,270)(73,244,394,271)(74,245,395,272)(75,246,396,273)(76,247,397,274)(77,248,398,275)(78,249,399,276)(79,250,400,277)(80,251,401,278)(81,252,402,279)(82,225,403,280)(83,226,404,253)(84,227,405,254)(113,218,447,385)(114,219,448,386)(115,220,421,387)(116,221,422,388)(117,222,423,389)(118,223,424,390)(119,224,425,391)(120,197,426,392)(121,198,427,365)(122,199,428,366)(123,200,429,367)(124,201,430,368)(125,202,431,369)(126,203,432,370)(127,204,433,371)(128,205,434,372)(129,206,435,373)(130,207,436,374)(131,208,437,375)(132,209,438,376)(133,210,439,377)(134,211,440,378)(135,212,441,379)(136,213,442,380)(137,214,443,381)(138,215,444,382)(139,216,445,383)(140,217,446,384) );
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)], [(1,374,15,388),(2,373,16,387),(3,372,17,386),(4,371,18,385),(5,370,19,384),(6,369,20,383),(7,368,21,382),(8,367,22,381),(9,366,23,380),(10,365,24,379),(11,392,25,378),(12,391,26,377),(13,390,27,376),(14,389,28,375),(29,418,43,404),(30,417,44,403),(31,416,45,402),(32,415,46,401),(33,414,47,400),(34,413,48,399),(35,412,49,398),(36,411,50,397),(37,410,51,396),(38,409,52,395),(39,408,53,394),(40,407,54,393),(41,406,55,420),(42,405,56,419),(57,111,71,97),(58,110,72,96),(59,109,73,95),(60,108,74,94),(61,107,75,93),(62,106,76,92),(63,105,77,91),(64,104,78,90),(65,103,79,89),(66,102,80,88),(67,101,81,87),(68,100,82,86),(69,99,83,85),(70,98,84,112),(113,320,127,334),(114,319,128,333),(115,318,129,332),(116,317,130,331),(117,316,131,330),(118,315,132,329),(119,314,133,328),(120,313,134,327),(121,312,135,326),(122,311,136,325),(123,310,137,324),(124,309,138,323),(125,336,139,322),(126,335,140,321),(141,269,155,255),(142,268,156,254),(143,267,157,253),(144,266,158,280),(145,265,159,279),(146,264,160,278),(147,263,161,277),(148,262,162,276),(149,261,163,275),(150,260,164,274),(151,259,165,273),(152,258,166,272),(153,257,167,271),(154,256,168,270),(169,240,183,226),(170,239,184,225),(171,238,185,252),(172,237,186,251),(173,236,187,250),(174,235,188,249),(175,234,189,248),(176,233,190,247),(177,232,191,246),(178,231,192,245),(179,230,193,244),(180,229,194,243),(181,228,195,242),(182,227,196,241),(197,299,211,285),(198,298,212,284),(199,297,213,283),(200,296,214,282),(201,295,215,281),(202,294,216,308),(203,293,217,307),(204,292,218,306),(205,291,219,305),(206,290,220,304),(207,289,221,303),(208,288,222,302),(209,287,223,301),(210,286,224,300),(337,446,351,432),(338,445,352,431),(339,444,353,430),(340,443,354,429),(341,442,355,428),(342,441,356,427),(343,440,357,426),(344,439,358,425),(345,438,359,424),(346,437,360,423),(347,436,361,422),(348,435,362,421),(349,434,363,448),(350,433,364,447)], [(1,393,303,72),(2,408,304,59),(3,395,305,74),(4,410,306,61),(5,397,307,76),(6,412,308,63),(7,399,281,78),(8,414,282,65),(9,401,283,80),(10,416,284,67),(11,403,285,82),(12,418,286,69),(13,405,287,84),(14,420,288,71),(15,407,289,58),(16,394,290,73),(17,409,291,60),(18,396,292,75),(19,411,293,62),(20,398,294,77),(21,413,295,64),(22,400,296,79),(23,415,297,66),(24,402,298,81),(25,417,299,68),(26,404,300,83),(27,419,301,70),(28,406,302,57),(29,217,85,384),(30,204,86,371),(31,219,87,386),(32,206,88,373),(33,221,89,388),(34,208,90,375),(35,223,91,390),(36,210,92,377),(37,197,93,392),(38,212,94,379),(39,199,95,366),(40,214,96,381),(41,201,97,368),(42,216,98,383),(43,203,99,370),(44,218,100,385),(45,205,101,372),(46,220,102,387),(47,207,103,374),(48,222,104,389),(49,209,105,376),(50,224,106,391),(51,211,107,378),(52,198,108,365),(53,213,109,380),(54,200,110,367),(55,215,111,382),(56,202,112,369),(113,170,447,144),(114,185,448,159),(115,172,421,146),(116,187,422,161),(117,174,423,148),(118,189,424,163),(119,176,425,150),(120,191,426,165),(121,178,427,152),(122,193,428,167),(123,180,429,154),(124,195,430,141),(125,182,431,156),(126,169,432,143),(127,184,433,158),(128,171,434,145),(129,186,435,160),(130,173,436,147),(131,188,437,162),(132,175,438,149),(133,190,439,164),(134,177,440,151),(135,192,441,166),(136,179,442,153),(137,194,443,168),(138,181,444,155),(139,196,445,142),(140,183,446,157),(225,343,280,313),(226,358,253,328),(227,345,254,315),(228,360,255,330),(229,347,256,317),(230,362,257,332),(231,349,258,319),(232,364,259,334),(233,351,260,321),(234,338,261,336),(235,353,262,323),(236,340,263,310),(237,355,264,325),(238,342,265,312),(239,357,266,327),(240,344,267,314),(241,359,268,329),(242,346,269,316),(243,361,270,331),(244,348,271,318),(245,363,272,333),(246,350,273,320),(247,337,274,335),(248,352,275,322),(249,339,276,309),(250,354,277,324),(251,341,278,311),(252,356,279,326)], [(1,331,303,361),(2,332,304,362),(3,333,305,363),(4,334,306,364),(5,335,307,337),(6,336,308,338),(7,309,281,339),(8,310,282,340),(9,311,283,341),(10,312,284,342),(11,313,285,343),(12,314,286,344),(13,315,287,345),(14,316,288,346),(15,317,289,347),(16,318,290,348),(17,319,291,349),(18,320,292,350),(19,321,293,351),(20,322,294,352),(21,323,295,353),(22,324,296,354),(23,325,297,355),(24,326,298,356),(25,327,299,357),(26,328,300,358),(27,329,301,359),(28,330,302,360),(29,157,85,183),(30,158,86,184),(31,159,87,185),(32,160,88,186),(33,161,89,187),(34,162,90,188),(35,163,91,189),(36,164,92,190),(37,165,93,191),(38,166,94,192),(39,167,95,193),(40,168,96,194),(41,141,97,195),(42,142,98,196),(43,143,99,169),(44,144,100,170),(45,145,101,171),(46,146,102,172),(47,147,103,173),(48,148,104,174),(49,149,105,175),(50,150,106,176),(51,151,107,177),(52,152,108,178),(53,153,109,179),(54,154,110,180),(55,155,111,181),(56,156,112,182),(57,228,406,255),(58,229,407,256),(59,230,408,257),(60,231,409,258),(61,232,410,259),(62,233,411,260),(63,234,412,261),(64,235,413,262),(65,236,414,263),(66,237,415,264),(67,238,416,265),(68,239,417,266),(69,240,418,267),(70,241,419,268),(71,242,420,269),(72,243,393,270),(73,244,394,271),(74,245,395,272),(75,246,396,273),(76,247,397,274),(77,248,398,275),(78,249,399,276),(79,250,400,277),(80,251,401,278),(81,252,402,279),(82,225,403,280),(83,226,404,253),(84,227,405,254),(113,218,447,385),(114,219,448,386),(115,220,421,387),(116,221,422,388),(117,222,423,389),(118,223,424,390),(119,224,425,391),(120,197,426,392),(121,198,427,365),(122,199,428,366),(123,200,429,367),(124,201,430,368),(125,202,431,369),(126,203,432,370),(127,204,433,371),(128,205,434,372),(129,206,435,373),(130,207,436,374),(131,208,437,375),(132,209,438,376),(133,210,439,377),(134,211,440,378),(135,212,441,379),(136,213,442,380),(137,214,443,381),(138,215,444,382),(139,216,445,383),(140,217,446,384)]])
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 | Q16 | C4○D4 | D14 | D14 | C7⋊D4 | C8⋊C22 | C7⋊Q16 | Q8×D7 | Q8⋊2D7 | D4.D14 |
| kernel | Dic14⋊5Q8 | C28⋊C8 | C28.Q8 | 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⋊5Q8 ►in GL6(𝔽113)
| 1 | 112 | 0 | 0 | 0 | 0 |
| 11 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 111 | 0 | 0 |
| 0 | 0 | 1 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 89 | 104 | 0 | 0 | 0 | 0 |
| 89 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 34 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 84 | 108 | 0 | 0 | 0 | 0 |
| 55 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 95 | 55 | 0 | 0 |
| 0 | 0 | 66 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 2 | 0 | 0 |
| 0 | 0 | 112 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 63 |
| 0 | 0 | 0 | 0 | 63 | 95 |
G:=sub<GL(6,GF(113))| [1,11,0,0,0,0,112,103,0,0,0,0,0,0,1,1,0,0,0,0,111,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[89,89,0,0,0,0,104,24,0,0,0,0,0,0,103,7,0,0,0,0,34,10,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[84,55,0,0,0,0,108,29,0,0,0,0,0,0,95,66,0,0,0,0,55,18,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,112,0,0,0,0,2,1,0,0,0,0,0,0,18,63,0,0,0,0,63,95] >;
Dic14⋊5Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes_5Q_8 % in TeX
G:=Group("Dic14:5Q8"); // GroupNames label
G:=SmallGroup(448,625);
// by ID
G=gap.SmallGroup(448,625);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,254,219,268,1123,297,136,18822]);
// 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,d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations