metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.3Dic14, C42.55D14, C7⋊6(Q8.Q8), (C4×Q8).4D7, (C7×Q8).3Q8, (C2×C28).65D4, (Q8×C28).4C2, C28.30(C2×Q8), C4⋊C4.250D14, C28⋊C8.16C2, C4.64(C4○D28), C14.91(C4○D8), C28.57(C4○D4), (C4×C28).93C22, (C2×Q8).157D14, C4.14(C2×Dic14), Q8⋊Dic7.9C2, C28.6Q8.6C2, (C2×C28).344C23, C28.Q8.11C2, C14.66(C22⋊Q8), C4.Dic14.11C2, C2.8(C28.C23), C14.86(C8.C22), C4⋊Dic7.141C22, (Q8×C14).192C22, C2.12(D4.8D14), C2.17(C28.48D4), (C2×C7⋊C8).99C22, (C2×C14).475(C2×D4), (C2×C4).220(C7⋊D4), (C7×C4⋊C4).281C22, (C2×C4).444(C22×D7), C22.154(C2×C7⋊D4), SmallGroup(448,556)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.3Dic14
G = < a,b,c,d | a4=c28=1, b2=a2, d2=a2c14, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 324 in 90 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, Q8.Q8, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, Q8×C14, C28⋊C8, C28.Q8, C4.Dic14, Q8⋊Dic7, C28.6Q8, Q8×C28, Q8.3Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C4○D8, C8.C22, Dic14, C7⋊D4, C22×D7, Q8.Q8, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4, C28.C23, D4.8D14, Q8.3Dic14
►Regular action on 448 points
Generators in S448
(1 93 399 83)(2 94 400 84)(3 95 401 57)(4 96 402 58)(5 97 403 59)(6 98 404 60)(7 99 405 61)(8 100 406 62)(9 101 407 63)(10 102 408 64)(11 103 409 65)(12 104 410 66)(13 105 411 67)(14 106 412 68)(15 107 413 69)(16 108 414 70)(17 109 415 71)(18 110 416 72)(19 111 417 73)(20 112 418 74)(21 85 419 75)(22 86 420 76)(23 87 393 77)(24 88 394 78)(25 89 395 79)(26 90 396 80)(27 91 397 81)(28 92 398 82)(29 376 141 203)(30 377 142 204)(31 378 143 205)(32 379 144 206)(33 380 145 207)(34 381 146 208)(35 382 147 209)(36 383 148 210)(37 384 149 211)(38 385 150 212)(39 386 151 213)(40 387 152 214)(41 388 153 215)(42 389 154 216)(43 390 155 217)(44 391 156 218)(45 392 157 219)(46 365 158 220)(47 366 159 221)(48 367 160 222)(49 368 161 223)(50 369 162 224)(51 370 163 197)(52 371 164 198)(53 372 165 199)(54 373 166 200)(55 374 167 201)(56 375 168 202)(113 230 339 294)(114 231 340 295)(115 232 341 296)(116 233 342 297)(117 234 343 298)(118 235 344 299)(119 236 345 300)(120 237 346 301)(121 238 347 302)(122 239 348 303)(123 240 349 304)(124 241 350 305)(125 242 351 306)(126 243 352 307)(127 244 353 308)(128 245 354 281)(129 246 355 282)(130 247 356 283)(131 248 357 284)(132 249 358 285)(133 250 359 286)(134 251 360 287)(135 252 361 288)(136 225 362 289)(137 226 363 290)(138 227 364 291)(139 228 337 292)(140 229 338 293)(169 313 439 262)(170 314 440 263)(171 315 441 264)(172 316 442 265)(173 317 443 266)(174 318 444 267)(175 319 445 268)(176 320 446 269)(177 321 447 270)(178 322 448 271)(179 323 421 272)(180 324 422 273)(181 325 423 274)(182 326 424 275)(183 327 425 276)(184 328 426 277)(185 329 427 278)(186 330 428 279)(187 331 429 280)(188 332 430 253)(189 333 431 254)(190 334 432 255)(191 335 433 256)(192 336 434 257)(193 309 435 258)(194 310 436 259)(195 311 437 260)(196 312 438 261)
(1 245 399 281)(2 246 400 282)(3 247 401 283)(4 248 402 284)(5 249 403 285)(6 250 404 286)(7 251 405 287)(8 252 406 288)(9 225 407 289)(10 226 408 290)(11 227 409 291)(12 228 410 292)(13 229 411 293)(14 230 412 294)(15 231 413 295)(16 232 414 296)(17 233 415 297)(18 234 416 298)(19 235 417 299)(20 236 418 300)(21 237 419 301)(22 238 420 302)(23 239 393 303)(24 240 394 304)(25 241 395 305)(26 242 396 306)(27 243 397 307)(28 244 398 308)(29 261 141 312)(30 262 142 313)(31 263 143 314)(32 264 144 315)(33 265 145 316)(34 266 146 317)(35 267 147 318)(36 268 148 319)(37 269 149 320)(38 270 150 321)(39 271 151 322)(40 272 152 323)(41 273 153 324)(42 274 154 325)(43 275 155 326)(44 276 156 327)(45 277 157 328)(46 278 158 329)(47 279 159 330)(48 280 160 331)(49 253 161 332)(50 254 162 333)(51 255 163 334)(52 256 164 335)(53 257 165 336)(54 258 166 309)(55 259 167 310)(56 260 168 311)(57 356 95 130)(58 357 96 131)(59 358 97 132)(60 359 98 133)(61 360 99 134)(62 361 100 135)(63 362 101 136)(64 363 102 137)(65 364 103 138)(66 337 104 139)(67 338 105 140)(68 339 106 113)(69 340 107 114)(70 341 108 115)(71 342 109 116)(72 343 110 117)(73 344 111 118)(74 345 112 119)(75 346 85 120)(76 347 86 121)(77 348 87 122)(78 349 88 123)(79 350 89 124)(80 351 90 125)(81 352 91 126)(82 353 92 127)(83 354 93 128)(84 355 94 129)(169 377 439 204)(170 378 440 205)(171 379 441 206)(172 380 442 207)(173 381 443 208)(174 382 444 209)(175 383 445 210)(176 384 446 211)(177 385 447 212)(178 386 448 213)(179 387 421 214)(180 388 422 215)(181 389 423 216)(182 390 424 217)(183 391 425 218)(184 392 426 219)(185 365 427 220)(186 366 428 221)(187 367 429 222)(188 368 430 223)(189 369 431 224)(190 370 432 197)(191 371 433 198)(192 372 434 199)(193 373 435 200)(194 374 436 201)(195 375 437 202)(196 376 438 203)
(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)
(1 211 413 370)(2 383 414 224)(3 209 415 368)(4 381 416 222)(5 207 417 366)(6 379 418 220)(7 205 419 392)(8 377 420 218)(9 203 393 390)(10 375 394 216)(11 201 395 388)(12 373 396 214)(13 199 397 386)(14 371 398 212)(15 197 399 384)(16 369 400 210)(17 223 401 382)(18 367 402 208)(19 221 403 380)(20 365 404 206)(21 219 405 378)(22 391 406 204)(23 217 407 376)(24 389 408 202)(25 215 409 374)(26 387 410 200)(27 213 411 372)(28 385 412 198)(29 87 155 63)(30 76 156 100)(31 85 157 61)(32 74 158 98)(33 111 159 59)(34 72 160 96)(35 109 161 57)(36 70 162 94)(37 107 163 83)(38 68 164 92)(39 105 165 81)(40 66 166 90)(41 103 167 79)(42 64 168 88)(43 101 141 77)(44 62 142 86)(45 99 143 75)(46 60 144 112)(47 97 145 73)(48 58 146 110)(49 95 147 71)(50 84 148 108)(51 93 149 69)(52 82 150 106)(53 91 151 67)(54 80 152 104)(55 89 153 65)(56 78 154 102)(113 191 353 447)(114 432 354 176)(115 189 355 445)(116 430 356 174)(117 187 357 443)(118 428 358 172)(119 185 359 441)(120 426 360 170)(121 183 361 439)(122 424 362 196)(123 181 363 437)(124 422 364 194)(125 179 337 435)(126 448 338 192)(127 177 339 433)(128 446 340 190)(129 175 341 431)(130 444 342 188)(131 173 343 429)(132 442 344 186)(133 171 345 427)(134 440 346 184)(135 169 347 425)(136 438 348 182)(137 195 349 423)(138 436 350 180)(139 193 351 421)(140 434 352 178)(225 312 303 275)(226 260 304 325)(227 310 305 273)(228 258 306 323)(229 336 307 271)(230 256 308 321)(231 334 281 269)(232 254 282 319)(233 332 283 267)(234 280 284 317)(235 330 285 265)(236 278 286 315)(237 328 287 263)(238 276 288 313)(239 326 289 261)(240 274 290 311)(241 324 291 259)(242 272 292 309)(243 322 293 257)(244 270 294 335)(245 320 295 255)(246 268 296 333)(247 318 297 253)(248 266 298 331)(249 316 299 279)(250 264 300 329)(251 314 301 277)(252 262 302 327)
G:=sub<Sym(448)| (1,93,399,83)(2,94,400,84)(3,95,401,57)(4,96,402,58)(5,97,403,59)(6,98,404,60)(7,99,405,61)(8,100,406,62)(9,101,407,63)(10,102,408,64)(11,103,409,65)(12,104,410,66)(13,105,411,67)(14,106,412,68)(15,107,413,69)(16,108,414,70)(17,109,415,71)(18,110,416,72)(19,111,417,73)(20,112,418,74)(21,85,419,75)(22,86,420,76)(23,87,393,77)(24,88,394,78)(25,89,395,79)(26,90,396,80)(27,91,397,81)(28,92,398,82)(29,376,141,203)(30,377,142,204)(31,378,143,205)(32,379,144,206)(33,380,145,207)(34,381,146,208)(35,382,147,209)(36,383,148,210)(37,384,149,211)(38,385,150,212)(39,386,151,213)(40,387,152,214)(41,388,153,215)(42,389,154,216)(43,390,155,217)(44,391,156,218)(45,392,157,219)(46,365,158,220)(47,366,159,221)(48,367,160,222)(49,368,161,223)(50,369,162,224)(51,370,163,197)(52,371,164,198)(53,372,165,199)(54,373,166,200)(55,374,167,201)(56,375,168,202)(113,230,339,294)(114,231,340,295)(115,232,341,296)(116,233,342,297)(117,234,343,298)(118,235,344,299)(119,236,345,300)(120,237,346,301)(121,238,347,302)(122,239,348,303)(123,240,349,304)(124,241,350,305)(125,242,351,306)(126,243,352,307)(127,244,353,308)(128,245,354,281)(129,246,355,282)(130,247,356,283)(131,248,357,284)(132,249,358,285)(133,250,359,286)(134,251,360,287)(135,252,361,288)(136,225,362,289)(137,226,363,290)(138,227,364,291)(139,228,337,292)(140,229,338,293)(169,313,439,262)(170,314,440,263)(171,315,441,264)(172,316,442,265)(173,317,443,266)(174,318,444,267)(175,319,445,268)(176,320,446,269)(177,321,447,270)(178,322,448,271)(179,323,421,272)(180,324,422,273)(181,325,423,274)(182,326,424,275)(183,327,425,276)(184,328,426,277)(185,329,427,278)(186,330,428,279)(187,331,429,280)(188,332,430,253)(189,333,431,254)(190,334,432,255)(191,335,433,256)(192,336,434,257)(193,309,435,258)(194,310,436,259)(195,311,437,260)(196,312,438,261), (1,245,399,281)(2,246,400,282)(3,247,401,283)(4,248,402,284)(5,249,403,285)(6,250,404,286)(7,251,405,287)(8,252,406,288)(9,225,407,289)(10,226,408,290)(11,227,409,291)(12,228,410,292)(13,229,411,293)(14,230,412,294)(15,231,413,295)(16,232,414,296)(17,233,415,297)(18,234,416,298)(19,235,417,299)(20,236,418,300)(21,237,419,301)(22,238,420,302)(23,239,393,303)(24,240,394,304)(25,241,395,305)(26,242,396,306)(27,243,397,307)(28,244,398,308)(29,261,141,312)(30,262,142,313)(31,263,143,314)(32,264,144,315)(33,265,145,316)(34,266,146,317)(35,267,147,318)(36,268,148,319)(37,269,149,320)(38,270,150,321)(39,271,151,322)(40,272,152,323)(41,273,153,324)(42,274,154,325)(43,275,155,326)(44,276,156,327)(45,277,157,328)(46,278,158,329)(47,279,159,330)(48,280,160,331)(49,253,161,332)(50,254,162,333)(51,255,163,334)(52,256,164,335)(53,257,165,336)(54,258,166,309)(55,259,167,310)(56,260,168,311)(57,356,95,130)(58,357,96,131)(59,358,97,132)(60,359,98,133)(61,360,99,134)(62,361,100,135)(63,362,101,136)(64,363,102,137)(65,364,103,138)(66,337,104,139)(67,338,105,140)(68,339,106,113)(69,340,107,114)(70,341,108,115)(71,342,109,116)(72,343,110,117)(73,344,111,118)(74,345,112,119)(75,346,85,120)(76,347,86,121)(77,348,87,122)(78,349,88,123)(79,350,89,124)(80,351,90,125)(81,352,91,126)(82,353,92,127)(83,354,93,128)(84,355,94,129)(169,377,439,204)(170,378,440,205)(171,379,441,206)(172,380,442,207)(173,381,443,208)(174,382,444,209)(175,383,445,210)(176,384,446,211)(177,385,447,212)(178,386,448,213)(179,387,421,214)(180,388,422,215)(181,389,423,216)(182,390,424,217)(183,391,425,218)(184,392,426,219)(185,365,427,220)(186,366,428,221)(187,367,429,222)(188,368,430,223)(189,369,431,224)(190,370,432,197)(191,371,433,198)(192,372,434,199)(193,373,435,200)(194,374,436,201)(195,375,437,202)(196,376,438,203), (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), (1,211,413,370)(2,383,414,224)(3,209,415,368)(4,381,416,222)(5,207,417,366)(6,379,418,220)(7,205,419,392)(8,377,420,218)(9,203,393,390)(10,375,394,216)(11,201,395,388)(12,373,396,214)(13,199,397,386)(14,371,398,212)(15,197,399,384)(16,369,400,210)(17,223,401,382)(18,367,402,208)(19,221,403,380)(20,365,404,206)(21,219,405,378)(22,391,406,204)(23,217,407,376)(24,389,408,202)(25,215,409,374)(26,387,410,200)(27,213,411,372)(28,385,412,198)(29,87,155,63)(30,76,156,100)(31,85,157,61)(32,74,158,98)(33,111,159,59)(34,72,160,96)(35,109,161,57)(36,70,162,94)(37,107,163,83)(38,68,164,92)(39,105,165,81)(40,66,166,90)(41,103,167,79)(42,64,168,88)(43,101,141,77)(44,62,142,86)(45,99,143,75)(46,60,144,112)(47,97,145,73)(48,58,146,110)(49,95,147,71)(50,84,148,108)(51,93,149,69)(52,82,150,106)(53,91,151,67)(54,80,152,104)(55,89,153,65)(56,78,154,102)(113,191,353,447)(114,432,354,176)(115,189,355,445)(116,430,356,174)(117,187,357,443)(118,428,358,172)(119,185,359,441)(120,426,360,170)(121,183,361,439)(122,424,362,196)(123,181,363,437)(124,422,364,194)(125,179,337,435)(126,448,338,192)(127,177,339,433)(128,446,340,190)(129,175,341,431)(130,444,342,188)(131,173,343,429)(132,442,344,186)(133,171,345,427)(134,440,346,184)(135,169,347,425)(136,438,348,182)(137,195,349,423)(138,436,350,180)(139,193,351,421)(140,434,352,178)(225,312,303,275)(226,260,304,325)(227,310,305,273)(228,258,306,323)(229,336,307,271)(230,256,308,321)(231,334,281,269)(232,254,282,319)(233,332,283,267)(234,280,284,317)(235,330,285,265)(236,278,286,315)(237,328,287,263)(238,276,288,313)(239,326,289,261)(240,274,290,311)(241,324,291,259)(242,272,292,309)(243,322,293,257)(244,270,294,335)(245,320,295,255)(246,268,296,333)(247,318,297,253)(248,266,298,331)(249,316,299,279)(250,264,300,329)(251,314,301,277)(252,262,302,327)>;
G:=Group( (1,93,399,83)(2,94,400,84)(3,95,401,57)(4,96,402,58)(5,97,403,59)(6,98,404,60)(7,99,405,61)(8,100,406,62)(9,101,407,63)(10,102,408,64)(11,103,409,65)(12,104,410,66)(13,105,411,67)(14,106,412,68)(15,107,413,69)(16,108,414,70)(17,109,415,71)(18,110,416,72)(19,111,417,73)(20,112,418,74)(21,85,419,75)(22,86,420,76)(23,87,393,77)(24,88,394,78)(25,89,395,79)(26,90,396,80)(27,91,397,81)(28,92,398,82)(29,376,141,203)(30,377,142,204)(31,378,143,205)(32,379,144,206)(33,380,145,207)(34,381,146,208)(35,382,147,209)(36,383,148,210)(37,384,149,211)(38,385,150,212)(39,386,151,213)(40,387,152,214)(41,388,153,215)(42,389,154,216)(43,390,155,217)(44,391,156,218)(45,392,157,219)(46,365,158,220)(47,366,159,221)(48,367,160,222)(49,368,161,223)(50,369,162,224)(51,370,163,197)(52,371,164,198)(53,372,165,199)(54,373,166,200)(55,374,167,201)(56,375,168,202)(113,230,339,294)(114,231,340,295)(115,232,341,296)(116,233,342,297)(117,234,343,298)(118,235,344,299)(119,236,345,300)(120,237,346,301)(121,238,347,302)(122,239,348,303)(123,240,349,304)(124,241,350,305)(125,242,351,306)(126,243,352,307)(127,244,353,308)(128,245,354,281)(129,246,355,282)(130,247,356,283)(131,248,357,284)(132,249,358,285)(133,250,359,286)(134,251,360,287)(135,252,361,288)(136,225,362,289)(137,226,363,290)(138,227,364,291)(139,228,337,292)(140,229,338,293)(169,313,439,262)(170,314,440,263)(171,315,441,264)(172,316,442,265)(173,317,443,266)(174,318,444,267)(175,319,445,268)(176,320,446,269)(177,321,447,270)(178,322,448,271)(179,323,421,272)(180,324,422,273)(181,325,423,274)(182,326,424,275)(183,327,425,276)(184,328,426,277)(185,329,427,278)(186,330,428,279)(187,331,429,280)(188,332,430,253)(189,333,431,254)(190,334,432,255)(191,335,433,256)(192,336,434,257)(193,309,435,258)(194,310,436,259)(195,311,437,260)(196,312,438,261), (1,245,399,281)(2,246,400,282)(3,247,401,283)(4,248,402,284)(5,249,403,285)(6,250,404,286)(7,251,405,287)(8,252,406,288)(9,225,407,289)(10,226,408,290)(11,227,409,291)(12,228,410,292)(13,229,411,293)(14,230,412,294)(15,231,413,295)(16,232,414,296)(17,233,415,297)(18,234,416,298)(19,235,417,299)(20,236,418,300)(21,237,419,301)(22,238,420,302)(23,239,393,303)(24,240,394,304)(25,241,395,305)(26,242,396,306)(27,243,397,307)(28,244,398,308)(29,261,141,312)(30,262,142,313)(31,263,143,314)(32,264,144,315)(33,265,145,316)(34,266,146,317)(35,267,147,318)(36,268,148,319)(37,269,149,320)(38,270,150,321)(39,271,151,322)(40,272,152,323)(41,273,153,324)(42,274,154,325)(43,275,155,326)(44,276,156,327)(45,277,157,328)(46,278,158,329)(47,279,159,330)(48,280,160,331)(49,253,161,332)(50,254,162,333)(51,255,163,334)(52,256,164,335)(53,257,165,336)(54,258,166,309)(55,259,167,310)(56,260,168,311)(57,356,95,130)(58,357,96,131)(59,358,97,132)(60,359,98,133)(61,360,99,134)(62,361,100,135)(63,362,101,136)(64,363,102,137)(65,364,103,138)(66,337,104,139)(67,338,105,140)(68,339,106,113)(69,340,107,114)(70,341,108,115)(71,342,109,116)(72,343,110,117)(73,344,111,118)(74,345,112,119)(75,346,85,120)(76,347,86,121)(77,348,87,122)(78,349,88,123)(79,350,89,124)(80,351,90,125)(81,352,91,126)(82,353,92,127)(83,354,93,128)(84,355,94,129)(169,377,439,204)(170,378,440,205)(171,379,441,206)(172,380,442,207)(173,381,443,208)(174,382,444,209)(175,383,445,210)(176,384,446,211)(177,385,447,212)(178,386,448,213)(179,387,421,214)(180,388,422,215)(181,389,423,216)(182,390,424,217)(183,391,425,218)(184,392,426,219)(185,365,427,220)(186,366,428,221)(187,367,429,222)(188,368,430,223)(189,369,431,224)(190,370,432,197)(191,371,433,198)(192,372,434,199)(193,373,435,200)(194,374,436,201)(195,375,437,202)(196,376,438,203), (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), (1,211,413,370)(2,383,414,224)(3,209,415,368)(4,381,416,222)(5,207,417,366)(6,379,418,220)(7,205,419,392)(8,377,420,218)(9,203,393,390)(10,375,394,216)(11,201,395,388)(12,373,396,214)(13,199,397,386)(14,371,398,212)(15,197,399,384)(16,369,400,210)(17,223,401,382)(18,367,402,208)(19,221,403,380)(20,365,404,206)(21,219,405,378)(22,391,406,204)(23,217,407,376)(24,389,408,202)(25,215,409,374)(26,387,410,200)(27,213,411,372)(28,385,412,198)(29,87,155,63)(30,76,156,100)(31,85,157,61)(32,74,158,98)(33,111,159,59)(34,72,160,96)(35,109,161,57)(36,70,162,94)(37,107,163,83)(38,68,164,92)(39,105,165,81)(40,66,166,90)(41,103,167,79)(42,64,168,88)(43,101,141,77)(44,62,142,86)(45,99,143,75)(46,60,144,112)(47,97,145,73)(48,58,146,110)(49,95,147,71)(50,84,148,108)(51,93,149,69)(52,82,150,106)(53,91,151,67)(54,80,152,104)(55,89,153,65)(56,78,154,102)(113,191,353,447)(114,432,354,176)(115,189,355,445)(116,430,356,174)(117,187,357,443)(118,428,358,172)(119,185,359,441)(120,426,360,170)(121,183,361,439)(122,424,362,196)(123,181,363,437)(124,422,364,194)(125,179,337,435)(126,448,338,192)(127,177,339,433)(128,446,340,190)(129,175,341,431)(130,444,342,188)(131,173,343,429)(132,442,344,186)(133,171,345,427)(134,440,346,184)(135,169,347,425)(136,438,348,182)(137,195,349,423)(138,436,350,180)(139,193,351,421)(140,434,352,178)(225,312,303,275)(226,260,304,325)(227,310,305,273)(228,258,306,323)(229,336,307,271)(230,256,308,321)(231,334,281,269)(232,254,282,319)(233,332,283,267)(234,280,284,317)(235,330,285,265)(236,278,286,315)(237,328,287,263)(238,276,288,313)(239,326,289,261)(240,274,290,311)(241,324,291,259)(242,272,292,309)(243,322,293,257)(244,270,294,335)(245,320,295,255)(246,268,296,333)(247,318,297,253)(248,266,298,331)(249,316,299,279)(250,264,300,329)(251,314,301,277)(252,262,302,327) );
G=PermutationGroup([[(1,93,399,83),(2,94,400,84),(3,95,401,57),(4,96,402,58),(5,97,403,59),(6,98,404,60),(7,99,405,61),(8,100,406,62),(9,101,407,63),(10,102,408,64),(11,103,409,65),(12,104,410,66),(13,105,411,67),(14,106,412,68),(15,107,413,69),(16,108,414,70),(17,109,415,71),(18,110,416,72),(19,111,417,73),(20,112,418,74),(21,85,419,75),(22,86,420,76),(23,87,393,77),(24,88,394,78),(25,89,395,79),(26,90,396,80),(27,91,397,81),(28,92,398,82),(29,376,141,203),(30,377,142,204),(31,378,143,205),(32,379,144,206),(33,380,145,207),(34,381,146,208),(35,382,147,209),(36,383,148,210),(37,384,149,211),(38,385,150,212),(39,386,151,213),(40,387,152,214),(41,388,153,215),(42,389,154,216),(43,390,155,217),(44,391,156,218),(45,392,157,219),(46,365,158,220),(47,366,159,221),(48,367,160,222),(49,368,161,223),(50,369,162,224),(51,370,163,197),(52,371,164,198),(53,372,165,199),(54,373,166,200),(55,374,167,201),(56,375,168,202),(113,230,339,294),(114,231,340,295),(115,232,341,296),(116,233,342,297),(117,234,343,298),(118,235,344,299),(119,236,345,300),(120,237,346,301),(121,238,347,302),(122,239,348,303),(123,240,349,304),(124,241,350,305),(125,242,351,306),(126,243,352,307),(127,244,353,308),(128,245,354,281),(129,246,355,282),(130,247,356,283),(131,248,357,284),(132,249,358,285),(133,250,359,286),(134,251,360,287),(135,252,361,288),(136,225,362,289),(137,226,363,290),(138,227,364,291),(139,228,337,292),(140,229,338,293),(169,313,439,262),(170,314,440,263),(171,315,441,264),(172,316,442,265),(173,317,443,266),(174,318,444,267),(175,319,445,268),(176,320,446,269),(177,321,447,270),(178,322,448,271),(179,323,421,272),(180,324,422,273),(181,325,423,274),(182,326,424,275),(183,327,425,276),(184,328,426,277),(185,329,427,278),(186,330,428,279),(187,331,429,280),(188,332,430,253),(189,333,431,254),(190,334,432,255),(191,335,433,256),(192,336,434,257),(193,309,435,258),(194,310,436,259),(195,311,437,260),(196,312,438,261)], [(1,245,399,281),(2,246,400,282),(3,247,401,283),(4,248,402,284),(5,249,403,285),(6,250,404,286),(7,251,405,287),(8,252,406,288),(9,225,407,289),(10,226,408,290),(11,227,409,291),(12,228,410,292),(13,229,411,293),(14,230,412,294),(15,231,413,295),(16,232,414,296),(17,233,415,297),(18,234,416,298),(19,235,417,299),(20,236,418,300),(21,237,419,301),(22,238,420,302),(23,239,393,303),(24,240,394,304),(25,241,395,305),(26,242,396,306),(27,243,397,307),(28,244,398,308),(29,261,141,312),(30,262,142,313),(31,263,143,314),(32,264,144,315),(33,265,145,316),(34,266,146,317),(35,267,147,318),(36,268,148,319),(37,269,149,320),(38,270,150,321),(39,271,151,322),(40,272,152,323),(41,273,153,324),(42,274,154,325),(43,275,155,326),(44,276,156,327),(45,277,157,328),(46,278,158,329),(47,279,159,330),(48,280,160,331),(49,253,161,332),(50,254,162,333),(51,255,163,334),(52,256,164,335),(53,257,165,336),(54,258,166,309),(55,259,167,310),(56,260,168,311),(57,356,95,130),(58,357,96,131),(59,358,97,132),(60,359,98,133),(61,360,99,134),(62,361,100,135),(63,362,101,136),(64,363,102,137),(65,364,103,138),(66,337,104,139),(67,338,105,140),(68,339,106,113),(69,340,107,114),(70,341,108,115),(71,342,109,116),(72,343,110,117),(73,344,111,118),(74,345,112,119),(75,346,85,120),(76,347,86,121),(77,348,87,122),(78,349,88,123),(79,350,89,124),(80,351,90,125),(81,352,91,126),(82,353,92,127),(83,354,93,128),(84,355,94,129),(169,377,439,204),(170,378,440,205),(171,379,441,206),(172,380,442,207),(173,381,443,208),(174,382,444,209),(175,383,445,210),(176,384,446,211),(177,385,447,212),(178,386,448,213),(179,387,421,214),(180,388,422,215),(181,389,423,216),(182,390,424,217),(183,391,425,218),(184,392,426,219),(185,365,427,220),(186,366,428,221),(187,367,429,222),(188,368,430,223),(189,369,431,224),(190,370,432,197),(191,371,433,198),(192,372,434,199),(193,373,435,200),(194,374,436,201),(195,375,437,202),(196,376,438,203)], [(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)], [(1,211,413,370),(2,383,414,224),(3,209,415,368),(4,381,416,222),(5,207,417,366),(6,379,418,220),(7,205,419,392),(8,377,420,218),(9,203,393,390),(10,375,394,216),(11,201,395,388),(12,373,396,214),(13,199,397,386),(14,371,398,212),(15,197,399,384),(16,369,400,210),(17,223,401,382),(18,367,402,208),(19,221,403,380),(20,365,404,206),(21,219,405,378),(22,391,406,204),(23,217,407,376),(24,389,408,202),(25,215,409,374),(26,387,410,200),(27,213,411,372),(28,385,412,198),(29,87,155,63),(30,76,156,100),(31,85,157,61),(32,74,158,98),(33,111,159,59),(34,72,160,96),(35,109,161,57),(36,70,162,94),(37,107,163,83),(38,68,164,92),(39,105,165,81),(40,66,166,90),(41,103,167,79),(42,64,168,88),(43,101,141,77),(44,62,142,86),(45,99,143,75),(46,60,144,112),(47,97,145,73),(48,58,146,110),(49,95,147,71),(50,84,148,108),(51,93,149,69),(52,82,150,106),(53,91,151,67),(54,80,152,104),(55,89,153,65),(56,78,154,102),(113,191,353,447),(114,432,354,176),(115,189,355,445),(116,430,356,174),(117,187,357,443),(118,428,358,172),(119,185,359,441),(120,426,360,170),(121,183,361,439),(122,424,362,196),(123,181,363,437),(124,422,364,194),(125,179,337,435),(126,448,338,192),(127,177,339,433),(128,446,340,190),(129,175,341,431),(130,444,342,188),(131,173,343,429),(132,442,344,186),(133,171,345,427),(134,440,346,184),(135,169,347,425),(136,438,348,182),(137,195,349,423),(138,436,350,180),(139,193,351,421),(140,434,352,178),(225,312,303,275),(226,260,304,325),(227,310,305,273),(228,258,306,323),(229,336,307,271),(230,256,308,321),(231,334,281,269),(232,254,282,319),(233,332,283,267),(234,280,284,317),(235,330,285,265),(236,278,286,315),(237,328,287,263),(238,276,288,313),(239,326,289,261),(240,274,290,311),(241,324,291,259),(242,272,292,309),(243,322,293,257),(244,270,294,335),(245,320,295,255),(246,268,296,333),(247,318,297,253),(248,266,298,331),(249,316,299,279),(250,264,300,329),(251,314,301,277),(252,262,302,327)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 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 | ··· | 4 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | Dic14 | C4○D28 | C8.C22 | C28.C23 | D4.8D14 |
| kernel | Q8.3Dic14 | C28⋊C8 | C28.Q8 | C4.Dic14 | Q8⋊Dic7 | C28.6Q8 | Q8×C28 | C2×C28 | C7×Q8 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C14 | C2×C4 | Q8 | C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 4 | 12 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of Q8.3Dic14 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 22 |
| 0 | 0 | 41 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 103 | 78 |
| 0 | 0 | 61 | 10 |
| 109 | 96 | 0 | 0 |
| 94 | 32 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 104 | 40 | 0 | 0 |
| 94 | 9 | 0 | 0 |
| 0 | 0 | 10 | 29 |
| 0 | 0 | 94 | 103 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,1,41,0,0,22,112],[112,0,0,0,0,112,0,0,0,0,103,61,0,0,78,10],[109,94,0,0,96,32,0,0,0,0,15,0,0,0,0,15],[104,94,0,0,40,9,0,0,0,0,10,94,0,0,29,103] >;
Q8.3Dic14 in GAP, Magma, Sage, TeX
Q_8._3{\rm Dic}_{14} % in TeX
G:=Group("Q8.3Dic14"); // GroupNames label
G:=SmallGroup(448,556);
// by ID
G=gap.SmallGroup(448,556);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,344,254,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^28=1,b^2=a^2,d^2=a^2*c^14,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations