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G = Q8×Dic14order 448 = 26·7

Direct product of Q8 and Dic14

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — Q8×Dic14
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C4×Dic7 — Q8×Dic7 — Q8×Dic14
 Lower central C7 — C2×C14 — Q8×Dic14
 Upper central C1 — C22 — C4×Q8

Generators and relations for Q8×Dic14
G = < a,b,c,d | a4=c28=1, b2=a2, d2=c14, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 836 in 212 conjugacy classes, 123 normal (18 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×Q8, C4×Q8, C4⋊Q8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, Q82, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C2×Dic14, C2×Dic14, Q8×C14, C4×Dic14, C282Q8, C28⋊Q8, Q8×Dic7, Q8×C28, Q8×Dic14
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C24, D14, C22×Q8, 2+ 1+4, Dic14, C22×D7, Q82, C2×Dic14, Q8×D7, C23×D7, C22×Dic14, C2×Q8×D7, D48D14, Q8×Dic14

Smallest permutation representation of Q8×Dic14
Regular action on 448 points
Generators in S448
(1 238 412 67)(2 239 413 68)(3 240 414 69)(4 241 415 70)(5 242 416 71)(6 243 417 72)(7 244 418 73)(8 245 419 74)(9 246 420 75)(10 247 393 76)(11 248 394 77)(12 249 395 78)(13 250 396 79)(14 251 397 80)(15 252 398 81)(16 225 399 82)(17 226 400 83)(18 227 401 84)(19 228 402 57)(20 229 403 58)(21 230 404 59)(22 231 405 60)(23 232 406 61)(24 233 407 62)(25 234 408 63)(26 235 409 64)(27 236 410 65)(28 237 411 66)(29 208 183 279)(30 209 184 280)(31 210 185 253)(32 211 186 254)(33 212 187 255)(34 213 188 256)(35 214 189 257)(36 215 190 258)(37 216 191 259)(38 217 192 260)(39 218 193 261)(40 219 194 262)(41 220 195 263)(42 221 196 264)(43 222 169 265)(44 223 170 266)(45 224 171 267)(46 197 172 268)(47 198 173 269)(48 199 174 270)(49 200 175 271)(50 201 176 272)(51 202 177 273)(52 203 178 274)(53 204 179 275)(54 205 180 276)(55 206 181 277)(56 207 182 278)(85 298 428 168)(86 299 429 141)(87 300 430 142)(88 301 431 143)(89 302 432 144)(90 303 433 145)(91 304 434 146)(92 305 435 147)(93 306 436 148)(94 307 437 149)(95 308 438 150)(96 281 439 151)(97 282 440 152)(98 283 441 153)(99 284 442 154)(100 285 443 155)(101 286 444 156)(102 287 445 157)(103 288 446 158)(104 289 447 159)(105 290 448 160)(106 291 421 161)(107 292 422 162)(108 293 423 163)(109 294 424 164)(110 295 425 165)(111 296 426 166)(112 297 427 167)(113 380 329 351)(114 381 330 352)(115 382 331 353)(116 383 332 354)(117 384 333 355)(118 385 334 356)(119 386 335 357)(120 387 336 358)(121 388 309 359)(122 389 310 360)(123 390 311 361)(124 391 312 362)(125 392 313 363)(126 365 314 364)(127 366 315 337)(128 367 316 338)(129 368 317 339)(130 369 318 340)(131 370 319 341)(132 371 320 342)(133 372 321 343)(134 373 322 344)(135 374 323 345)(136 375 324 346)(137 376 325 347)(138 377 326 348)(139 378 327 349)(140 379 328 350)
(1 260 412 217)(2 261 413 218)(3 262 414 219)(4 263 415 220)(5 264 416 221)(6 265 417 222)(7 266 418 223)(8 267 419 224)(9 268 420 197)(10 269 393 198)(11 270 394 199)(12 271 395 200)(13 272 396 201)(14 273 397 202)(15 274 398 203)(16 275 399 204)(17 276 400 205)(18 277 401 206)(19 278 402 207)(20 279 403 208)(21 280 404 209)(22 253 405 210)(23 254 406 211)(24 255 407 212)(25 256 408 213)(26 257 409 214)(27 258 410 215)(28 259 411 216)(29 229 183 58)(30 230 184 59)(31 231 185 60)(32 232 186 61)(33 233 187 62)(34 234 188 63)(35 235 189 64)(36 236 190 65)(37 237 191 66)(38 238 192 67)(39 239 193 68)(40 240 194 69)(41 241 195 70)(42 242 196 71)(43 243 169 72)(44 244 170 73)(45 245 171 74)(46 246 172 75)(47 247 173 76)(48 248 174 77)(49 249 175 78)(50 250 176 79)(51 251 177 80)(52 252 178 81)(53 225 179 82)(54 226 180 83)(55 227 181 84)(56 228 182 57)(85 319 428 131)(86 320 429 132)(87 321 430 133)(88 322 431 134)(89 323 432 135)(90 324 433 136)(91 325 434 137)(92 326 435 138)(93 327 436 139)(94 328 437 140)(95 329 438 113)(96 330 439 114)(97 331 440 115)(98 332 441 116)(99 333 442 117)(100 334 443 118)(101 335 444 119)(102 336 445 120)(103 309 446 121)(104 310 447 122)(105 311 448 123)(106 312 421 124)(107 313 422 125)(108 314 423 126)(109 315 424 127)(110 316 425 128)(111 317 426 129)(112 318 427 130)(141 342 299 371)(142 343 300 372)(143 344 301 373)(144 345 302 374)(145 346 303 375)(146 347 304 376)(147 348 305 377)(148 349 306 378)(149 350 307 379)(150 351 308 380)(151 352 281 381)(152 353 282 382)(153 354 283 383)(154 355 284 384)(155 356 285 385)(156 357 286 386)(157 358 287 387)(158 359 288 388)(159 360 289 389)(160 361 290 390)(161 362 291 391)(162 363 292 392)(163 364 293 365)(164 337 294 366)(165 338 295 367)(166 339 296 368)(167 340 297 369)(168 341 298 370)
(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)
(1 324 15 310)(2 323 16 309)(3 322 17 336)(4 321 18 335)(5 320 19 334)(6 319 20 333)(7 318 21 332)(8 317 22 331)(9 316 23 330)(10 315 24 329)(11 314 25 328)(12 313 26 327)(13 312 27 326)(14 311 28 325)(29 154 43 168)(30 153 44 167)(31 152 45 166)(32 151 46 165)(33 150 47 164)(34 149 48 163)(35 148 49 162)(36 147 50 161)(37 146 51 160)(38 145 52 159)(39 144 53 158)(40 143 54 157)(41 142 55 156)(42 141 56 155)(57 385 71 371)(58 384 72 370)(59 383 73 369)(60 382 74 368)(61 381 75 367)(62 380 76 366)(63 379 77 365)(64 378 78 392)(65 377 79 391)(66 376 80 390)(67 375 81 389)(68 374 82 388)(69 373 83 387)(70 372 84 386)(85 208 99 222)(86 207 100 221)(87 206 101 220)(88 205 102 219)(89 204 103 218)(90 203 104 217)(91 202 105 216)(92 201 106 215)(93 200 107 214)(94 199 108 213)(95 198 109 212)(96 197 110 211)(97 224 111 210)(98 223 112 209)(113 393 127 407)(114 420 128 406)(115 419 129 405)(116 418 130 404)(117 417 131 403)(118 416 132 402)(119 415 133 401)(120 414 134 400)(121 413 135 399)(122 412 136 398)(123 411 137 397)(124 410 138 396)(125 409 139 395)(126 408 140 394)(169 298 183 284)(170 297 184 283)(171 296 185 282)(172 295 186 281)(173 294 187 308)(174 293 188 307)(175 292 189 306)(176 291 190 305)(177 290 191 304)(178 289 192 303)(179 288 193 302)(180 287 194 301)(181 286 195 300)(182 285 196 299)(225 359 239 345)(226 358 240 344)(227 357 241 343)(228 356 242 342)(229 355 243 341)(230 354 244 340)(231 353 245 339)(232 352 246 338)(233 351 247 337)(234 350 248 364)(235 349 249 363)(236 348 250 362)(237 347 251 361)(238 346 252 360)(253 440 267 426)(254 439 268 425)(255 438 269 424)(256 437 270 423)(257 436 271 422)(258 435 272 421)(259 434 273 448)(260 433 274 447)(261 432 275 446)(262 431 276 445)(263 430 277 444)(264 429 278 443)(265 428 279 442)(266 427 280 441)

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

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

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

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 2 2 2 2 2 2 2 4 4 4 type + + + + + + - - + + + + - + - + image C1 C2 C2 C2 C2 C2 Q8 Q8 D7 D14 D14 D14 Dic14 2+ 1+4 Q8×D7 D4⋊8D14 kernel Q8×Dic14 C4×Dic14 C28⋊2Q8 C28⋊Q8 Q8×Dic7 Q8×C28 Dic14 C7×Q8 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C14 C4 C2 # reps 1 3 3 6 2 1 4 4 3 9 9 3 24 1 6 6

Matrix representation of Q8×Dic14 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 28 27 0 0 1 1
,
 1 0 0 0 0 1 0 0 0 0 15 26 0 0 27 14
,
 2 21 0 0 14 17 0 0 0 0 1 0 0 0 0 1
,
 20 26 0 0 8 9 0 0 0 0 28 0 0 0 0 28
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,1,0,0,27,1],[1,0,0,0,0,1,0,0,0,0,15,27,0,0,26,14],[2,14,0,0,21,17,0,0,0,0,1,0,0,0,0,1],[20,8,0,0,26,9,0,0,0,0,28,0,0,0,0,28] >;

Q8×Dic14 in GAP, Magma, Sage, TeX

Q_8\times {\rm Dic}_{14}
% in TeX

G:=Group("Q8xDic14");
// GroupNames label

G:=SmallGroup(448,1019);
// by ID

G=gap.SmallGroup(448,1019);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,387,184,675,80,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^28=1,b^2=a^2,d^2=c^14,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽