metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.2Dic14, C7⋊2(Q8.Q8), (C7×Q8).2Q8, C28.8(C2×Q8), C4⋊C4.21D14, C8⋊Dic7.7C2, (C2×C8).121D14, Dic7⋊C8.6C2, Q8⋊C4.6D7, (Q8×Dic7).6C2, C4.8(C2×Dic14), C14.46(C4○D8), (C2×Q8).102D14, C28.Q8.3C2, Q8⋊Dic7.6C2, C22.193(D4×D7), C28.3Q8.3C2, C28.160(C4○D4), C4.85(D4⋊2D7), (C2×C56).132C22, (C2×C28).239C23, (C2×Dic7).151D4, C14.14(C22⋊Q8), C4⋊Dic7.87C22, (Q8×C14).22C22, C2.12(Q16⋊D7), C14.57(C8.C22), (C4×Dic7).23C22, C2.15(SD16⋊3D7), C2.19(C22⋊Dic14), (C2×C7⋊C8).34C22, (C2×C14).252(C2×D4), (C7×C4⋊C4).40C22, (C7×Q8⋊C4).6C2, (C2×C4).346(C22×D7), SmallGroup(448,333)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.2Dic14
G = < a,b,c,d | a4=c28=1, b2=a2, d2=a2c14, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >
Subgroups: 372 in 90 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, Q8.Q8, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C2×C56, Q8×C14, C28.Q8, Dic7⋊C8, C8⋊Dic7, Q8⋊Dic7, C7×Q8⋊C4, C28.3Q8, Q8×Dic7, Q8.2Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C4○D8, C8.C22, Dic14, C22×D7, Q8.Q8, C2×Dic14, D4×D7, D4⋊2D7, C22⋊Dic14, SD16⋊3D7, Q16⋊D7, Q8.2Dic14
►Regular action on 448 points
Generators in S448
(1 251 72 352)(2 353 73 252)(3 225 74 354)(4 355 75 226)(5 227 76 356)(6 357 77 228)(7 229 78 358)(8 359 79 230)(9 231 80 360)(10 361 81 232)(11 233 82 362)(12 363 83 234)(13 235 84 364)(14 337 57 236)(15 237 58 338)(16 339 59 238)(17 239 60 340)(18 341 61 240)(19 241 62 342)(20 343 63 242)(21 243 64 344)(22 345 65 244)(23 245 66 346)(24 347 67 246)(25 247 68 348)(26 349 69 248)(27 249 70 350)(28 351 71 250)(29 432 302 376)(30 377 303 433)(31 434 304 378)(32 379 305 435)(33 436 306 380)(34 381 307 437)(35 438 308 382)(36 383 281 439)(37 440 282 384)(38 385 283 441)(39 442 284 386)(40 387 285 443)(41 444 286 388)(42 389 287 445)(43 446 288 390)(44 391 289 447)(45 448 290 392)(46 365 291 421)(47 422 292 366)(48 367 293 423)(49 424 294 368)(50 369 295 425)(51 426 296 370)(52 371 297 427)(53 428 298 372)(54 373 299 429)(55 430 300 374)(56 375 301 431)(85 395 180 318)(86 319 181 396)(87 397 182 320)(88 321 183 398)(89 399 184 322)(90 323 185 400)(91 401 186 324)(92 325 187 402)(93 403 188 326)(94 327 189 404)(95 405 190 328)(96 329 191 406)(97 407 192 330)(98 331 193 408)(99 409 194 332)(100 333 195 410)(101 411 196 334)(102 335 169 412)(103 413 170 336)(104 309 171 414)(105 415 172 310)(106 311 173 416)(107 417 174 312)(108 313 175 418)(109 419 176 314)(110 315 177 420)(111 393 178 316)(112 317 179 394)(113 279 158 219)(114 220 159 280)(115 253 160 221)(116 222 161 254)(117 255 162 223)(118 224 163 256)(119 257 164 197)(120 198 165 258)(121 259 166 199)(122 200 167 260)(123 261 168 201)(124 202 141 262)(125 263 142 203)(126 204 143 264)(127 265 144 205)(128 206 145 266)(129 267 146 207)(130 208 147 268)(131 269 148 209)(132 210 149 270)(133 271 150 211)(134 212 151 272)(135 273 152 213)(136 214 153 274)(137 275 154 215)(138 216 155 276)(139 277 156 217)(140 218 157 278)
(1 260 72 200)(2 168 73 123)(3 262 74 202)(4 142 75 125)(5 264 76 204)(6 144 77 127)(7 266 78 206)(8 146 79 129)(9 268 80 208)(10 148 81 131)(11 270 82 210)(12 150 83 133)(13 272 84 212)(14 152 57 135)(15 274 58 214)(16 154 59 137)(17 276 60 216)(18 156 61 139)(19 278 62 218)(20 158 63 113)(21 280 64 220)(22 160 65 115)(23 254 66 222)(24 162 67 117)(25 256 68 224)(26 164 69 119)(27 258 70 198)(28 166 71 121)(29 192 302 97)(30 408 303 331)(31 194 304 99)(32 410 305 333)(33 196 306 101)(34 412 307 335)(35 170 308 103)(36 414 281 309)(37 172 282 105)(38 416 283 311)(39 174 284 107)(40 418 285 313)(41 176 286 109)(42 420 287 315)(43 178 288 111)(44 394 289 317)(45 180 290 85)(46 396 291 319)(47 182 292 87)(48 398 293 321)(49 184 294 89)(50 400 295 323)(51 186 296 91)(52 402 297 325)(53 188 298 93)(54 404 299 327)(55 190 300 95)(56 406 301 329)(86 365 181 421)(88 367 183 423)(90 369 185 425)(92 371 187 427)(94 373 189 429)(96 375 191 431)(98 377 193 433)(100 379 195 435)(102 381 169 437)(104 383 171 439)(106 385 173 441)(108 387 175 443)(110 389 177 445)(112 391 179 447)(114 243 159 344)(116 245 161 346)(118 247 163 348)(120 249 165 350)(122 251 167 352)(124 225 141 354)(126 227 143 356)(128 229 145 358)(130 231 147 360)(132 233 149 362)(134 235 151 364)(136 237 153 338)(138 239 155 340)(140 241 157 342)(197 349 257 248)(199 351 259 250)(201 353 261 252)(203 355 263 226)(205 357 265 228)(207 359 267 230)(209 361 269 232)(211 363 271 234)(213 337 273 236)(215 339 275 238)(217 341 277 240)(219 343 279 242)(221 345 253 244)(223 347 255 246)(310 440 415 384)(312 442 417 386)(314 444 419 388)(316 446 393 390)(318 448 395 392)(320 422 397 366)(322 424 399 368)(324 426 401 370)(326 428 403 372)(328 430 405 374)(330 432 407 376)(332 434 409 378)(334 436 411 380)(336 438 413 382)
(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 58 444)(2 373 59 443)(3 372 60 442)(4 371 61 441)(5 370 62 440)(6 369 63 439)(7 368 64 438)(8 367 65 437)(9 366 66 436)(10 365 67 435)(11 392 68 434)(12 391 69 433)(13 390 70 432)(14 389 71 431)(15 388 72 430)(16 387 73 429)(17 386 74 428)(18 385 75 427)(19 384 76 426)(20 383 77 425)(21 382 78 424)(22 381 79 423)(23 380 80 422)(24 379 81 421)(25 378 82 448)(26 377 83 447)(27 376 84 446)(28 375 57 445)(29 364 288 249)(30 363 289 248)(31 362 290 247)(32 361 291 246)(33 360 292 245)(34 359 293 244)(35 358 294 243)(36 357 295 242)(37 356 296 241)(38 355 297 240)(39 354 298 239)(40 353 299 238)(41 352 300 237)(42 351 301 236)(43 350 302 235)(44 349 303 234)(45 348 304 233)(46 347 305 232)(47 346 306 231)(48 345 307 230)(49 344 308 229)(50 343 281 228)(51 342 282 227)(52 341 283 226)(53 340 284 225)(54 339 285 252)(55 338 286 251)(56 337 287 250)(85 163 194 132)(86 162 195 131)(87 161 196 130)(88 160 169 129)(89 159 170 128)(90 158 171 127)(91 157 172 126)(92 156 173 125)(93 155 174 124)(94 154 175 123)(95 153 176 122)(96 152 177 121)(97 151 178 120)(98 150 179 119)(99 149 180 118)(100 148 181 117)(101 147 182 116)(102 146 183 115)(103 145 184 114)(104 144 185 113)(105 143 186 140)(106 142 187 139)(107 141 188 138)(108 168 189 137)(109 167 190 136)(110 166 191 135)(111 165 192 134)(112 164 193 133)(197 408 271 317)(198 407 272 316)(199 406 273 315)(200 405 274 314)(201 404 275 313)(202 403 276 312)(203 402 277 311)(204 401 278 310)(205 400 279 309)(206 399 280 336)(207 398 253 335)(208 397 254 334)(209 396 255 333)(210 395 256 332)(211 394 257 331)(212 393 258 330)(213 420 259 329)(214 419 260 328)(215 418 261 327)(216 417 262 326)(217 416 263 325)(218 415 264 324)(219 414 265 323)(220 413 266 322)(221 412 267 321)(222 411 268 320)(223 410 269 319)(224 409 270 318)
G:=sub<Sym(448)| (1,251,72,352)(2,353,73,252)(3,225,74,354)(4,355,75,226)(5,227,76,356)(6,357,77,228)(7,229,78,358)(8,359,79,230)(9,231,80,360)(10,361,81,232)(11,233,82,362)(12,363,83,234)(13,235,84,364)(14,337,57,236)(15,237,58,338)(16,339,59,238)(17,239,60,340)(18,341,61,240)(19,241,62,342)(20,343,63,242)(21,243,64,344)(22,345,65,244)(23,245,66,346)(24,347,67,246)(25,247,68,348)(26,349,69,248)(27,249,70,350)(28,351,71,250)(29,432,302,376)(30,377,303,433)(31,434,304,378)(32,379,305,435)(33,436,306,380)(34,381,307,437)(35,438,308,382)(36,383,281,439)(37,440,282,384)(38,385,283,441)(39,442,284,386)(40,387,285,443)(41,444,286,388)(42,389,287,445)(43,446,288,390)(44,391,289,447)(45,448,290,392)(46,365,291,421)(47,422,292,366)(48,367,293,423)(49,424,294,368)(50,369,295,425)(51,426,296,370)(52,371,297,427)(53,428,298,372)(54,373,299,429)(55,430,300,374)(56,375,301,431)(85,395,180,318)(86,319,181,396)(87,397,182,320)(88,321,183,398)(89,399,184,322)(90,323,185,400)(91,401,186,324)(92,325,187,402)(93,403,188,326)(94,327,189,404)(95,405,190,328)(96,329,191,406)(97,407,192,330)(98,331,193,408)(99,409,194,332)(100,333,195,410)(101,411,196,334)(102,335,169,412)(103,413,170,336)(104,309,171,414)(105,415,172,310)(106,311,173,416)(107,417,174,312)(108,313,175,418)(109,419,176,314)(110,315,177,420)(111,393,178,316)(112,317,179,394)(113,279,158,219)(114,220,159,280)(115,253,160,221)(116,222,161,254)(117,255,162,223)(118,224,163,256)(119,257,164,197)(120,198,165,258)(121,259,166,199)(122,200,167,260)(123,261,168,201)(124,202,141,262)(125,263,142,203)(126,204,143,264)(127,265,144,205)(128,206,145,266)(129,267,146,207)(130,208,147,268)(131,269,148,209)(132,210,149,270)(133,271,150,211)(134,212,151,272)(135,273,152,213)(136,214,153,274)(137,275,154,215)(138,216,155,276)(139,277,156,217)(140,218,157,278), (1,260,72,200)(2,168,73,123)(3,262,74,202)(4,142,75,125)(5,264,76,204)(6,144,77,127)(7,266,78,206)(8,146,79,129)(9,268,80,208)(10,148,81,131)(11,270,82,210)(12,150,83,133)(13,272,84,212)(14,152,57,135)(15,274,58,214)(16,154,59,137)(17,276,60,216)(18,156,61,139)(19,278,62,218)(20,158,63,113)(21,280,64,220)(22,160,65,115)(23,254,66,222)(24,162,67,117)(25,256,68,224)(26,164,69,119)(27,258,70,198)(28,166,71,121)(29,192,302,97)(30,408,303,331)(31,194,304,99)(32,410,305,333)(33,196,306,101)(34,412,307,335)(35,170,308,103)(36,414,281,309)(37,172,282,105)(38,416,283,311)(39,174,284,107)(40,418,285,313)(41,176,286,109)(42,420,287,315)(43,178,288,111)(44,394,289,317)(45,180,290,85)(46,396,291,319)(47,182,292,87)(48,398,293,321)(49,184,294,89)(50,400,295,323)(51,186,296,91)(52,402,297,325)(53,188,298,93)(54,404,299,327)(55,190,300,95)(56,406,301,329)(86,365,181,421)(88,367,183,423)(90,369,185,425)(92,371,187,427)(94,373,189,429)(96,375,191,431)(98,377,193,433)(100,379,195,435)(102,381,169,437)(104,383,171,439)(106,385,173,441)(108,387,175,443)(110,389,177,445)(112,391,179,447)(114,243,159,344)(116,245,161,346)(118,247,163,348)(120,249,165,350)(122,251,167,352)(124,225,141,354)(126,227,143,356)(128,229,145,358)(130,231,147,360)(132,233,149,362)(134,235,151,364)(136,237,153,338)(138,239,155,340)(140,241,157,342)(197,349,257,248)(199,351,259,250)(201,353,261,252)(203,355,263,226)(205,357,265,228)(207,359,267,230)(209,361,269,232)(211,363,271,234)(213,337,273,236)(215,339,275,238)(217,341,277,240)(219,343,279,242)(221,345,253,244)(223,347,255,246)(310,440,415,384)(312,442,417,386)(314,444,419,388)(316,446,393,390)(318,448,395,392)(320,422,397,366)(322,424,399,368)(324,426,401,370)(326,428,403,372)(328,430,405,374)(330,432,407,376)(332,434,409,378)(334,436,411,380)(336,438,413,382), (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), (1,374,58,444)(2,373,59,443)(3,372,60,442)(4,371,61,441)(5,370,62,440)(6,369,63,439)(7,368,64,438)(8,367,65,437)(9,366,66,436)(10,365,67,435)(11,392,68,434)(12,391,69,433)(13,390,70,432)(14,389,71,431)(15,388,72,430)(16,387,73,429)(17,386,74,428)(18,385,75,427)(19,384,76,426)(20,383,77,425)(21,382,78,424)(22,381,79,423)(23,380,80,422)(24,379,81,421)(25,378,82,448)(26,377,83,447)(27,376,84,446)(28,375,57,445)(29,364,288,249)(30,363,289,248)(31,362,290,247)(32,361,291,246)(33,360,292,245)(34,359,293,244)(35,358,294,243)(36,357,295,242)(37,356,296,241)(38,355,297,240)(39,354,298,239)(40,353,299,238)(41,352,300,237)(42,351,301,236)(43,350,302,235)(44,349,303,234)(45,348,304,233)(46,347,305,232)(47,346,306,231)(48,345,307,230)(49,344,308,229)(50,343,281,228)(51,342,282,227)(52,341,283,226)(53,340,284,225)(54,339,285,252)(55,338,286,251)(56,337,287,250)(85,163,194,132)(86,162,195,131)(87,161,196,130)(88,160,169,129)(89,159,170,128)(90,158,171,127)(91,157,172,126)(92,156,173,125)(93,155,174,124)(94,154,175,123)(95,153,176,122)(96,152,177,121)(97,151,178,120)(98,150,179,119)(99,149,180,118)(100,148,181,117)(101,147,182,116)(102,146,183,115)(103,145,184,114)(104,144,185,113)(105,143,186,140)(106,142,187,139)(107,141,188,138)(108,168,189,137)(109,167,190,136)(110,166,191,135)(111,165,192,134)(112,164,193,133)(197,408,271,317)(198,407,272,316)(199,406,273,315)(200,405,274,314)(201,404,275,313)(202,403,276,312)(203,402,277,311)(204,401,278,310)(205,400,279,309)(206,399,280,336)(207,398,253,335)(208,397,254,334)(209,396,255,333)(210,395,256,332)(211,394,257,331)(212,393,258,330)(213,420,259,329)(214,419,260,328)(215,418,261,327)(216,417,262,326)(217,416,263,325)(218,415,264,324)(219,414,265,323)(220,413,266,322)(221,412,267,321)(222,411,268,320)(223,410,269,319)(224,409,270,318)>;
G:=Group( (1,251,72,352)(2,353,73,252)(3,225,74,354)(4,355,75,226)(5,227,76,356)(6,357,77,228)(7,229,78,358)(8,359,79,230)(9,231,80,360)(10,361,81,232)(11,233,82,362)(12,363,83,234)(13,235,84,364)(14,337,57,236)(15,237,58,338)(16,339,59,238)(17,239,60,340)(18,341,61,240)(19,241,62,342)(20,343,63,242)(21,243,64,344)(22,345,65,244)(23,245,66,346)(24,347,67,246)(25,247,68,348)(26,349,69,248)(27,249,70,350)(28,351,71,250)(29,432,302,376)(30,377,303,433)(31,434,304,378)(32,379,305,435)(33,436,306,380)(34,381,307,437)(35,438,308,382)(36,383,281,439)(37,440,282,384)(38,385,283,441)(39,442,284,386)(40,387,285,443)(41,444,286,388)(42,389,287,445)(43,446,288,390)(44,391,289,447)(45,448,290,392)(46,365,291,421)(47,422,292,366)(48,367,293,423)(49,424,294,368)(50,369,295,425)(51,426,296,370)(52,371,297,427)(53,428,298,372)(54,373,299,429)(55,430,300,374)(56,375,301,431)(85,395,180,318)(86,319,181,396)(87,397,182,320)(88,321,183,398)(89,399,184,322)(90,323,185,400)(91,401,186,324)(92,325,187,402)(93,403,188,326)(94,327,189,404)(95,405,190,328)(96,329,191,406)(97,407,192,330)(98,331,193,408)(99,409,194,332)(100,333,195,410)(101,411,196,334)(102,335,169,412)(103,413,170,336)(104,309,171,414)(105,415,172,310)(106,311,173,416)(107,417,174,312)(108,313,175,418)(109,419,176,314)(110,315,177,420)(111,393,178,316)(112,317,179,394)(113,279,158,219)(114,220,159,280)(115,253,160,221)(116,222,161,254)(117,255,162,223)(118,224,163,256)(119,257,164,197)(120,198,165,258)(121,259,166,199)(122,200,167,260)(123,261,168,201)(124,202,141,262)(125,263,142,203)(126,204,143,264)(127,265,144,205)(128,206,145,266)(129,267,146,207)(130,208,147,268)(131,269,148,209)(132,210,149,270)(133,271,150,211)(134,212,151,272)(135,273,152,213)(136,214,153,274)(137,275,154,215)(138,216,155,276)(139,277,156,217)(140,218,157,278), (1,260,72,200)(2,168,73,123)(3,262,74,202)(4,142,75,125)(5,264,76,204)(6,144,77,127)(7,266,78,206)(8,146,79,129)(9,268,80,208)(10,148,81,131)(11,270,82,210)(12,150,83,133)(13,272,84,212)(14,152,57,135)(15,274,58,214)(16,154,59,137)(17,276,60,216)(18,156,61,139)(19,278,62,218)(20,158,63,113)(21,280,64,220)(22,160,65,115)(23,254,66,222)(24,162,67,117)(25,256,68,224)(26,164,69,119)(27,258,70,198)(28,166,71,121)(29,192,302,97)(30,408,303,331)(31,194,304,99)(32,410,305,333)(33,196,306,101)(34,412,307,335)(35,170,308,103)(36,414,281,309)(37,172,282,105)(38,416,283,311)(39,174,284,107)(40,418,285,313)(41,176,286,109)(42,420,287,315)(43,178,288,111)(44,394,289,317)(45,180,290,85)(46,396,291,319)(47,182,292,87)(48,398,293,321)(49,184,294,89)(50,400,295,323)(51,186,296,91)(52,402,297,325)(53,188,298,93)(54,404,299,327)(55,190,300,95)(56,406,301,329)(86,365,181,421)(88,367,183,423)(90,369,185,425)(92,371,187,427)(94,373,189,429)(96,375,191,431)(98,377,193,433)(100,379,195,435)(102,381,169,437)(104,383,171,439)(106,385,173,441)(108,387,175,443)(110,389,177,445)(112,391,179,447)(114,243,159,344)(116,245,161,346)(118,247,163,348)(120,249,165,350)(122,251,167,352)(124,225,141,354)(126,227,143,356)(128,229,145,358)(130,231,147,360)(132,233,149,362)(134,235,151,364)(136,237,153,338)(138,239,155,340)(140,241,157,342)(197,349,257,248)(199,351,259,250)(201,353,261,252)(203,355,263,226)(205,357,265,228)(207,359,267,230)(209,361,269,232)(211,363,271,234)(213,337,273,236)(215,339,275,238)(217,341,277,240)(219,343,279,242)(221,345,253,244)(223,347,255,246)(310,440,415,384)(312,442,417,386)(314,444,419,388)(316,446,393,390)(318,448,395,392)(320,422,397,366)(322,424,399,368)(324,426,401,370)(326,428,403,372)(328,430,405,374)(330,432,407,376)(332,434,409,378)(334,436,411,380)(336,438,413,382), (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), (1,374,58,444)(2,373,59,443)(3,372,60,442)(4,371,61,441)(5,370,62,440)(6,369,63,439)(7,368,64,438)(8,367,65,437)(9,366,66,436)(10,365,67,435)(11,392,68,434)(12,391,69,433)(13,390,70,432)(14,389,71,431)(15,388,72,430)(16,387,73,429)(17,386,74,428)(18,385,75,427)(19,384,76,426)(20,383,77,425)(21,382,78,424)(22,381,79,423)(23,380,80,422)(24,379,81,421)(25,378,82,448)(26,377,83,447)(27,376,84,446)(28,375,57,445)(29,364,288,249)(30,363,289,248)(31,362,290,247)(32,361,291,246)(33,360,292,245)(34,359,293,244)(35,358,294,243)(36,357,295,242)(37,356,296,241)(38,355,297,240)(39,354,298,239)(40,353,299,238)(41,352,300,237)(42,351,301,236)(43,350,302,235)(44,349,303,234)(45,348,304,233)(46,347,305,232)(47,346,306,231)(48,345,307,230)(49,344,308,229)(50,343,281,228)(51,342,282,227)(52,341,283,226)(53,340,284,225)(54,339,285,252)(55,338,286,251)(56,337,287,250)(85,163,194,132)(86,162,195,131)(87,161,196,130)(88,160,169,129)(89,159,170,128)(90,158,171,127)(91,157,172,126)(92,156,173,125)(93,155,174,124)(94,154,175,123)(95,153,176,122)(96,152,177,121)(97,151,178,120)(98,150,179,119)(99,149,180,118)(100,148,181,117)(101,147,182,116)(102,146,183,115)(103,145,184,114)(104,144,185,113)(105,143,186,140)(106,142,187,139)(107,141,188,138)(108,168,189,137)(109,167,190,136)(110,166,191,135)(111,165,192,134)(112,164,193,133)(197,408,271,317)(198,407,272,316)(199,406,273,315)(200,405,274,314)(201,404,275,313)(202,403,276,312)(203,402,277,311)(204,401,278,310)(205,400,279,309)(206,399,280,336)(207,398,253,335)(208,397,254,334)(209,396,255,333)(210,395,256,332)(211,394,257,331)(212,393,258,330)(213,420,259,329)(214,419,260,328)(215,418,261,327)(216,417,262,326)(217,416,263,325)(218,415,264,324)(219,414,265,323)(220,413,266,322)(221,412,267,321)(222,411,268,320)(223,410,269,319)(224,409,270,318) );
G=PermutationGroup([[(1,251,72,352),(2,353,73,252),(3,225,74,354),(4,355,75,226),(5,227,76,356),(6,357,77,228),(7,229,78,358),(8,359,79,230),(9,231,80,360),(10,361,81,232),(11,233,82,362),(12,363,83,234),(13,235,84,364),(14,337,57,236),(15,237,58,338),(16,339,59,238),(17,239,60,340),(18,341,61,240),(19,241,62,342),(20,343,63,242),(21,243,64,344),(22,345,65,244),(23,245,66,346),(24,347,67,246),(25,247,68,348),(26,349,69,248),(27,249,70,350),(28,351,71,250),(29,432,302,376),(30,377,303,433),(31,434,304,378),(32,379,305,435),(33,436,306,380),(34,381,307,437),(35,438,308,382),(36,383,281,439),(37,440,282,384),(38,385,283,441),(39,442,284,386),(40,387,285,443),(41,444,286,388),(42,389,287,445),(43,446,288,390),(44,391,289,447),(45,448,290,392),(46,365,291,421),(47,422,292,366),(48,367,293,423),(49,424,294,368),(50,369,295,425),(51,426,296,370),(52,371,297,427),(53,428,298,372),(54,373,299,429),(55,430,300,374),(56,375,301,431),(85,395,180,318),(86,319,181,396),(87,397,182,320),(88,321,183,398),(89,399,184,322),(90,323,185,400),(91,401,186,324),(92,325,187,402),(93,403,188,326),(94,327,189,404),(95,405,190,328),(96,329,191,406),(97,407,192,330),(98,331,193,408),(99,409,194,332),(100,333,195,410),(101,411,196,334),(102,335,169,412),(103,413,170,336),(104,309,171,414),(105,415,172,310),(106,311,173,416),(107,417,174,312),(108,313,175,418),(109,419,176,314),(110,315,177,420),(111,393,178,316),(112,317,179,394),(113,279,158,219),(114,220,159,280),(115,253,160,221),(116,222,161,254),(117,255,162,223),(118,224,163,256),(119,257,164,197),(120,198,165,258),(121,259,166,199),(122,200,167,260),(123,261,168,201),(124,202,141,262),(125,263,142,203),(126,204,143,264),(127,265,144,205),(128,206,145,266),(129,267,146,207),(130,208,147,268),(131,269,148,209),(132,210,149,270),(133,271,150,211),(134,212,151,272),(135,273,152,213),(136,214,153,274),(137,275,154,215),(138,216,155,276),(139,277,156,217),(140,218,157,278)], [(1,260,72,200),(2,168,73,123),(3,262,74,202),(4,142,75,125),(5,264,76,204),(6,144,77,127),(7,266,78,206),(8,146,79,129),(9,268,80,208),(10,148,81,131),(11,270,82,210),(12,150,83,133),(13,272,84,212),(14,152,57,135),(15,274,58,214),(16,154,59,137),(17,276,60,216),(18,156,61,139),(19,278,62,218),(20,158,63,113),(21,280,64,220),(22,160,65,115),(23,254,66,222),(24,162,67,117),(25,256,68,224),(26,164,69,119),(27,258,70,198),(28,166,71,121),(29,192,302,97),(30,408,303,331),(31,194,304,99),(32,410,305,333),(33,196,306,101),(34,412,307,335),(35,170,308,103),(36,414,281,309),(37,172,282,105),(38,416,283,311),(39,174,284,107),(40,418,285,313),(41,176,286,109),(42,420,287,315),(43,178,288,111),(44,394,289,317),(45,180,290,85),(46,396,291,319),(47,182,292,87),(48,398,293,321),(49,184,294,89),(50,400,295,323),(51,186,296,91),(52,402,297,325),(53,188,298,93),(54,404,299,327),(55,190,300,95),(56,406,301,329),(86,365,181,421),(88,367,183,423),(90,369,185,425),(92,371,187,427),(94,373,189,429),(96,375,191,431),(98,377,193,433),(100,379,195,435),(102,381,169,437),(104,383,171,439),(106,385,173,441),(108,387,175,443),(110,389,177,445),(112,391,179,447),(114,243,159,344),(116,245,161,346),(118,247,163,348),(120,249,165,350),(122,251,167,352),(124,225,141,354),(126,227,143,356),(128,229,145,358),(130,231,147,360),(132,233,149,362),(134,235,151,364),(136,237,153,338),(138,239,155,340),(140,241,157,342),(197,349,257,248),(199,351,259,250),(201,353,261,252),(203,355,263,226),(205,357,265,228),(207,359,267,230),(209,361,269,232),(211,363,271,234),(213,337,273,236),(215,339,275,238),(217,341,277,240),(219,343,279,242),(221,345,253,244),(223,347,255,246),(310,440,415,384),(312,442,417,386),(314,444,419,388),(316,446,393,390),(318,448,395,392),(320,422,397,366),(322,424,399,368),(324,426,401,370),(326,428,403,372),(328,430,405,374),(330,432,407,376),(332,434,409,378),(334,436,411,380),(336,438,413,382)], [(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)], [(1,374,58,444),(2,373,59,443),(3,372,60,442),(4,371,61,441),(5,370,62,440),(6,369,63,439),(7,368,64,438),(8,367,65,437),(9,366,66,436),(10,365,67,435),(11,392,68,434),(12,391,69,433),(13,390,70,432),(14,389,71,431),(15,388,72,430),(16,387,73,429),(17,386,74,428),(18,385,75,427),(19,384,76,426),(20,383,77,425),(21,382,78,424),(22,381,79,423),(23,380,80,422),(24,379,81,421),(25,378,82,448),(26,377,83,447),(27,376,84,446),(28,375,57,445),(29,364,288,249),(30,363,289,248),(31,362,290,247),(32,361,291,246),(33,360,292,245),(34,359,293,244),(35,358,294,243),(36,357,295,242),(37,356,296,241),(38,355,297,240),(39,354,298,239),(40,353,299,238),(41,352,300,237),(42,351,301,236),(43,350,302,235),(44,349,303,234),(45,348,304,233),(46,347,305,232),(47,346,306,231),(48,345,307,230),(49,344,308,229),(50,343,281,228),(51,342,282,227),(52,341,283,226),(53,340,284,225),(54,339,285,252),(55,338,286,251),(56,337,287,250),(85,163,194,132),(86,162,195,131),(87,161,196,130),(88,160,169,129),(89,159,170,128),(90,158,171,127),(91,157,172,126),(92,156,173,125),(93,155,174,124),(94,154,175,123),(95,153,176,122),(96,152,177,121),(97,151,178,120),(98,150,179,119),(99,149,180,118),(100,148,181,117),(101,147,182,116),(102,146,183,115),(103,145,184,114),(104,144,185,113),(105,143,186,140),(106,142,187,139),(107,141,188,138),(108,168,189,137),(109,167,190,136),(110,166,191,135),(111,165,192,134),(112,164,193,133),(197,408,271,317),(198,407,272,316),(199,406,273,315),(200,405,274,314),(201,404,275,313),(202,403,276,312),(203,402,277,311),(204,401,278,310),(205,400,279,309),(206,399,280,336),(207,398,253,335),(208,397,254,334),(209,396,255,333),(210,395,256,332),(211,394,257,331),(212,393,258,330),(213,420,259,329),(214,419,260,328),(215,418,261,327),(216,417,262,326),(217,416,263,325),(218,415,264,324),(219,414,265,323),(220,413,266,322),(221,412,267,321),(222,411,268,320),(223,410,269,319),(224,409,270,318)]])
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 | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| 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 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | C4○D8 | Dic14 | C8.C22 | D4⋊2D7 | D4×D7 | SD16⋊3D7 | Q16⋊D7 |
| kernel | Q8.2Dic14 | C28.Q8 | Dic7⋊C8 | C8⋊Dic7 | Q8⋊Dic7 | C7×Q8⋊C4 | C28.3Q8 | Q8×Dic7 | C2×Dic7 | C7×Q8 | Q8⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | C14 | Q8 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of Q8.2Dic14 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 69 |
| 0 | 0 | 36 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 29 | 61 |
| 0 | 0 | 64 | 84 |
| 8 | 17 | 0 | 0 |
| 96 | 77 | 0 | 0 |
| 0 | 0 | 72 | 27 |
| 0 | 0 | 101 | 41 |
| 15 | 92 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,112,36,0,0,69,1],[1,0,0,0,0,1,0,0,0,0,29,64,0,0,61,84],[8,96,0,0,17,77,0,0,0,0,72,101,0,0,27,41],[15,0,0,0,92,98,0,0,0,0,98,0,0,0,0,98] >;
Q8.2Dic14 in GAP, Magma, Sage, TeX
Q_8._2{\rm Dic}_{14} % in TeX
G:=Group("Q8.2Dic14"); // GroupNames label
G:=SmallGroup(448,333);
// by ID
G=gap.SmallGroup(448,333);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,926,219,226,851,438,102,18822]);
// 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=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations