metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊6Dic14, C42.124D14, C14.652- (1+4), (C7×Q8)⋊7Q8, C7⋊3(Q8⋊3Q8), (C4×Q8).12D7, C28.46(C2×Q8), C4⋊C4.293D14, (Q8×C28).13C2, (C2×Q8).198D14, C28⋊2Q8.25C2, (Q8×Dic7).12C2, C4.19(C2×Dic14), C28.333(C4○D4), C14.17(C22×Q8), (C2×C14).114C24, (C4×C28).166C22, (C2×C28).168C23, C4.49(Q8⋊2D7), (C4×Dic14).23C2, C4⋊Dic7.43C22, C4.Dic14.11C2, (Q8×C14).214C22, (C2×Dic7).52C23, (C4×Dic7).83C22, C2.19(C22×Dic14), C22.139(C23×D7), Dic7⋊C4.115C22, C2.22(D4.10D14), (C2×Dic14).242C22, C14.110(C2×C4○D4), C2.10(C2×Q8⋊2D7), (C7×C4⋊C4).342C22, (C2×C4).733(C22×D7), SmallGroup(448,1023)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 708 in 200 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C4 [×8], C4 [×11], C22, C7, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×6], C14 [×3], C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic7 [×8], C28 [×8], C28 [×3], C2×C14, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic14 [×6], C2×Dic7 [×8], C2×C28, C2×C28 [×6], C7×Q8 [×4], Q8⋊3Q8, C4×Dic7 [×6], Dic7⋊C4 [×6], C4⋊Dic7, C4⋊Dic7 [×12], C4×C28 [×3], C7×C4⋊C4 [×3], C2×Dic14 [×3], Q8×C14, C4×Dic14 [×3], C28⋊2Q8 [×3], C4.Dic14 [×6], Q8×Dic7 [×2], Q8×C28, Q8⋊6Dic14
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×2], C24, D14 [×7], C22×Q8, C2×C4○D4, 2- (1+4), Dic14 [×4], C22×D7 [×7], Q8⋊3Q8, C2×Dic14 [×6], Q8⋊2D7 [×2], C23×D7, C22×Dic14, C2×Q8⋊2D7, D4.10D14, Q8⋊6Dic14
Generators and relations
G = < a,b,c,d | a4=c28=1, b2=a2, d2=c14, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
►Regular action on 448 points
Generators in S448
(1 113 83 96)(2 114 84 97)(3 115 57 98)(4 116 58 99)(5 117 59 100)(6 118 60 101)(7 119 61 102)(8 120 62 103)(9 121 63 104)(10 122 64 105)(11 123 65 106)(12 124 66 107)(13 125 67 108)(14 126 68 109)(15 127 69 110)(16 128 70 111)(17 129 71 112)(18 130 72 85)(19 131 73 86)(20 132 74 87)(21 133 75 88)(22 134 76 89)(23 135 77 90)(24 136 78 91)(25 137 79 92)(26 138 80 93)(27 139 81 94)(28 140 82 95)(29 149 202 357)(30 150 203 358)(31 151 204 359)(32 152 205 360)(33 153 206 361)(34 154 207 362)(35 155 208 363)(36 156 209 364)(37 157 210 337)(38 158 211 338)(39 159 212 339)(40 160 213 340)(41 161 214 341)(42 162 215 342)(43 163 216 343)(44 164 217 344)(45 165 218 345)(46 166 219 346)(47 167 220 347)(48 168 221 348)(49 141 222 349)(50 142 223 350)(51 143 224 351)(52 144 197 352)(53 145 198 353)(54 146 199 354)(55 147 200 355)(56 148 201 356)(169 309 247 375)(170 310 248 376)(171 311 249 377)(172 312 250 378)(173 313 251 379)(174 314 252 380)(175 315 225 381)(176 316 226 382)(177 317 227 383)(178 318 228 384)(179 319 229 385)(180 320 230 386)(181 321 231 387)(182 322 232 388)(183 323 233 389)(184 324 234 390)(185 325 235 391)(186 326 236 392)(187 327 237 365)(188 328 238 366)(189 329 239 367)(190 330 240 368)(191 331 241 369)(192 332 242 370)(193 333 243 371)(194 334 244 372)(195 335 245 373)(196 336 246 374)(253 431 415 286)(254 432 416 287)(255 433 417 288)(256 434 418 289)(257 435 419 290)(258 436 420 291)(259 437 393 292)(260 438 394 293)(261 439 395 294)(262 440 396 295)(263 441 397 296)(264 442 398 297)(265 443 399 298)(266 444 400 299)(267 445 401 300)(268 446 402 301)(269 447 403 302)(270 448 404 303)(271 421 405 304)(272 422 406 305)(273 423 407 306)(274 424 408 307)(275 425 409 308)(276 426 410 281)(277 427 411 282)(278 428 412 283)(279 429 413 284)(280 430 414 285)
(1 44 83 217)(2 45 84 218)(3 46 57 219)(4 47 58 220)(5 48 59 221)(6 49 60 222)(7 50 61 223)(8 51 62 224)(9 52 63 197)(10 53 64 198)(11 54 65 199)(12 55 66 200)(13 56 67 201)(14 29 68 202)(15 30 69 203)(16 31 70 204)(17 32 71 205)(18 33 72 206)(19 34 73 207)(20 35 74 208)(21 36 75 209)(22 37 76 210)(23 38 77 211)(24 39 78 212)(25 40 79 213)(26 41 80 214)(27 42 81 215)(28 43 82 216)(85 153 130 361)(86 154 131 362)(87 155 132 363)(88 156 133 364)(89 157 134 337)(90 158 135 338)(91 159 136 339)(92 160 137 340)(93 161 138 341)(94 162 139 342)(95 163 140 343)(96 164 113 344)(97 165 114 345)(98 166 115 346)(99 167 116 347)(100 168 117 348)(101 141 118 349)(102 142 119 350)(103 143 120 351)(104 144 121 352)(105 145 122 353)(106 146 123 354)(107 147 124 355)(108 148 125 356)(109 149 126 357)(110 150 127 358)(111 151 128 359)(112 152 129 360)(169 416 247 254)(170 417 248 255)(171 418 249 256)(172 419 250 257)(173 420 251 258)(174 393 252 259)(175 394 225 260)(176 395 226 261)(177 396 227 262)(178 397 228 263)(179 398 229 264)(180 399 230 265)(181 400 231 266)(182 401 232 267)(183 402 233 268)(184 403 234 269)(185 404 235 270)(186 405 236 271)(187 406 237 272)(188 407 238 273)(189 408 239 274)(190 409 240 275)(191 410 241 276)(192 411 242 277)(193 412 243 278)(194 413 244 279)(195 414 245 280)(196 415 246 253)(281 331 426 369)(282 332 427 370)(283 333 428 371)(284 334 429 372)(285 335 430 373)(286 336 431 374)(287 309 432 375)(288 310 433 376)(289 311 434 377)(290 312 435 378)(291 313 436 379)(292 314 437 380)(293 315 438 381)(294 316 439 382)(295 317 440 383)(296 318 441 384)(297 319 442 385)(298 320 443 386)(299 321 444 387)(300 322 445 388)(301 323 446 389)(302 324 447 390)(303 325 448 391)(304 326 421 392)(305 327 422 365)(306 328 423 366)(307 329 424 367)(308 330 425 368)
(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)
(1 432 15 446)(2 431 16 445)(3 430 17 444)(4 429 18 443)(5 428 19 442)(6 427 20 441)(7 426 21 440)(8 425 22 439)(9 424 23 438)(10 423 24 437)(11 422 25 436)(12 421 26 435)(13 448 27 434)(14 447 28 433)(29 324 43 310)(30 323 44 309)(31 322 45 336)(32 321 46 335)(33 320 47 334)(34 319 48 333)(35 318 49 332)(36 317 50 331)(37 316 51 330)(38 315 52 329)(39 314 53 328)(40 313 54 327)(41 312 55 326)(42 311 56 325)(57 285 71 299)(58 284 72 298)(59 283 73 297)(60 282 74 296)(61 281 75 295)(62 308 76 294)(63 307 77 293)(64 306 78 292)(65 305 79 291)(66 304 80 290)(67 303 81 289)(68 302 82 288)(69 301 83 287)(70 300 84 286)(85 265 99 279)(86 264 100 278)(87 263 101 277)(88 262 102 276)(89 261 103 275)(90 260 104 274)(91 259 105 273)(92 258 106 272)(93 257 107 271)(94 256 108 270)(95 255 109 269)(96 254 110 268)(97 253 111 267)(98 280 112 266)(113 416 127 402)(114 415 128 401)(115 414 129 400)(116 413 130 399)(117 412 131 398)(118 411 132 397)(119 410 133 396)(120 409 134 395)(121 408 135 394)(122 407 136 393)(123 406 137 420)(124 405 138 419)(125 404 139 418)(126 403 140 417)(141 242 155 228)(142 241 156 227)(143 240 157 226)(144 239 158 225)(145 238 159 252)(146 237 160 251)(147 236 161 250)(148 235 162 249)(149 234 163 248)(150 233 164 247)(151 232 165 246)(152 231 166 245)(153 230 167 244)(154 229 168 243)(169 358 183 344)(170 357 184 343)(171 356 185 342)(172 355 186 341)(173 354 187 340)(174 353 188 339)(175 352 189 338)(176 351 190 337)(177 350 191 364)(178 349 192 363)(179 348 193 362)(180 347 194 361)(181 346 195 360)(182 345 196 359)(197 367 211 381)(198 366 212 380)(199 365 213 379)(200 392 214 378)(201 391 215 377)(202 390 216 376)(203 389 217 375)(204 388 218 374)(205 387 219 373)(206 386 220 372)(207 385 221 371)(208 384 222 370)(209 383 223 369)(210 382 224 368)
G:=sub<Sym(448)| (1,113,83,96)(2,114,84,97)(3,115,57,98)(4,116,58,99)(5,117,59,100)(6,118,60,101)(7,119,61,102)(8,120,62,103)(9,121,63,104)(10,122,64,105)(11,123,65,106)(12,124,66,107)(13,125,67,108)(14,126,68,109)(15,127,69,110)(16,128,70,111)(17,129,71,112)(18,130,72,85)(19,131,73,86)(20,132,74,87)(21,133,75,88)(22,134,76,89)(23,135,77,90)(24,136,78,91)(25,137,79,92)(26,138,80,93)(27,139,81,94)(28,140,82,95)(29,149,202,357)(30,150,203,358)(31,151,204,359)(32,152,205,360)(33,153,206,361)(34,154,207,362)(35,155,208,363)(36,156,209,364)(37,157,210,337)(38,158,211,338)(39,159,212,339)(40,160,213,340)(41,161,214,341)(42,162,215,342)(43,163,216,343)(44,164,217,344)(45,165,218,345)(46,166,219,346)(47,167,220,347)(48,168,221,348)(49,141,222,349)(50,142,223,350)(51,143,224,351)(52,144,197,352)(53,145,198,353)(54,146,199,354)(55,147,200,355)(56,148,201,356)(169,309,247,375)(170,310,248,376)(171,311,249,377)(172,312,250,378)(173,313,251,379)(174,314,252,380)(175,315,225,381)(176,316,226,382)(177,317,227,383)(178,318,228,384)(179,319,229,385)(180,320,230,386)(181,321,231,387)(182,322,232,388)(183,323,233,389)(184,324,234,390)(185,325,235,391)(186,326,236,392)(187,327,237,365)(188,328,238,366)(189,329,239,367)(190,330,240,368)(191,331,241,369)(192,332,242,370)(193,333,243,371)(194,334,244,372)(195,335,245,373)(196,336,246,374)(253,431,415,286)(254,432,416,287)(255,433,417,288)(256,434,418,289)(257,435,419,290)(258,436,420,291)(259,437,393,292)(260,438,394,293)(261,439,395,294)(262,440,396,295)(263,441,397,296)(264,442,398,297)(265,443,399,298)(266,444,400,299)(267,445,401,300)(268,446,402,301)(269,447,403,302)(270,448,404,303)(271,421,405,304)(272,422,406,305)(273,423,407,306)(274,424,408,307)(275,425,409,308)(276,426,410,281)(277,427,411,282)(278,428,412,283)(279,429,413,284)(280,430,414,285), (1,44,83,217)(2,45,84,218)(3,46,57,219)(4,47,58,220)(5,48,59,221)(6,49,60,222)(7,50,61,223)(8,51,62,224)(9,52,63,197)(10,53,64,198)(11,54,65,199)(12,55,66,200)(13,56,67,201)(14,29,68,202)(15,30,69,203)(16,31,70,204)(17,32,71,205)(18,33,72,206)(19,34,73,207)(20,35,74,208)(21,36,75,209)(22,37,76,210)(23,38,77,211)(24,39,78,212)(25,40,79,213)(26,41,80,214)(27,42,81,215)(28,43,82,216)(85,153,130,361)(86,154,131,362)(87,155,132,363)(88,156,133,364)(89,157,134,337)(90,158,135,338)(91,159,136,339)(92,160,137,340)(93,161,138,341)(94,162,139,342)(95,163,140,343)(96,164,113,344)(97,165,114,345)(98,166,115,346)(99,167,116,347)(100,168,117,348)(101,141,118,349)(102,142,119,350)(103,143,120,351)(104,144,121,352)(105,145,122,353)(106,146,123,354)(107,147,124,355)(108,148,125,356)(109,149,126,357)(110,150,127,358)(111,151,128,359)(112,152,129,360)(169,416,247,254)(170,417,248,255)(171,418,249,256)(172,419,250,257)(173,420,251,258)(174,393,252,259)(175,394,225,260)(176,395,226,261)(177,396,227,262)(178,397,228,263)(179,398,229,264)(180,399,230,265)(181,400,231,266)(182,401,232,267)(183,402,233,268)(184,403,234,269)(185,404,235,270)(186,405,236,271)(187,406,237,272)(188,407,238,273)(189,408,239,274)(190,409,240,275)(191,410,241,276)(192,411,242,277)(193,412,243,278)(194,413,244,279)(195,414,245,280)(196,415,246,253)(281,331,426,369)(282,332,427,370)(283,333,428,371)(284,334,429,372)(285,335,430,373)(286,336,431,374)(287,309,432,375)(288,310,433,376)(289,311,434,377)(290,312,435,378)(291,313,436,379)(292,314,437,380)(293,315,438,381)(294,316,439,382)(295,317,440,383)(296,318,441,384)(297,319,442,385)(298,320,443,386)(299,321,444,387)(300,322,445,388)(301,323,446,389)(302,324,447,390)(303,325,448,391)(304,326,421,392)(305,327,422,365)(306,328,423,366)(307,329,424,367)(308,330,425,368), (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), (1,432,15,446)(2,431,16,445)(3,430,17,444)(4,429,18,443)(5,428,19,442)(6,427,20,441)(7,426,21,440)(8,425,22,439)(9,424,23,438)(10,423,24,437)(11,422,25,436)(12,421,26,435)(13,448,27,434)(14,447,28,433)(29,324,43,310)(30,323,44,309)(31,322,45,336)(32,321,46,335)(33,320,47,334)(34,319,48,333)(35,318,49,332)(36,317,50,331)(37,316,51,330)(38,315,52,329)(39,314,53,328)(40,313,54,327)(41,312,55,326)(42,311,56,325)(57,285,71,299)(58,284,72,298)(59,283,73,297)(60,282,74,296)(61,281,75,295)(62,308,76,294)(63,307,77,293)(64,306,78,292)(65,305,79,291)(66,304,80,290)(67,303,81,289)(68,302,82,288)(69,301,83,287)(70,300,84,286)(85,265,99,279)(86,264,100,278)(87,263,101,277)(88,262,102,276)(89,261,103,275)(90,260,104,274)(91,259,105,273)(92,258,106,272)(93,257,107,271)(94,256,108,270)(95,255,109,269)(96,254,110,268)(97,253,111,267)(98,280,112,266)(113,416,127,402)(114,415,128,401)(115,414,129,400)(116,413,130,399)(117,412,131,398)(118,411,132,397)(119,410,133,396)(120,409,134,395)(121,408,135,394)(122,407,136,393)(123,406,137,420)(124,405,138,419)(125,404,139,418)(126,403,140,417)(141,242,155,228)(142,241,156,227)(143,240,157,226)(144,239,158,225)(145,238,159,252)(146,237,160,251)(147,236,161,250)(148,235,162,249)(149,234,163,248)(150,233,164,247)(151,232,165,246)(152,231,166,245)(153,230,167,244)(154,229,168,243)(169,358,183,344)(170,357,184,343)(171,356,185,342)(172,355,186,341)(173,354,187,340)(174,353,188,339)(175,352,189,338)(176,351,190,337)(177,350,191,364)(178,349,192,363)(179,348,193,362)(180,347,194,361)(181,346,195,360)(182,345,196,359)(197,367,211,381)(198,366,212,380)(199,365,213,379)(200,392,214,378)(201,391,215,377)(202,390,216,376)(203,389,217,375)(204,388,218,374)(205,387,219,373)(206,386,220,372)(207,385,221,371)(208,384,222,370)(209,383,223,369)(210,382,224,368)>;
G:=Group( (1,113,83,96)(2,114,84,97)(3,115,57,98)(4,116,58,99)(5,117,59,100)(6,118,60,101)(7,119,61,102)(8,120,62,103)(9,121,63,104)(10,122,64,105)(11,123,65,106)(12,124,66,107)(13,125,67,108)(14,126,68,109)(15,127,69,110)(16,128,70,111)(17,129,71,112)(18,130,72,85)(19,131,73,86)(20,132,74,87)(21,133,75,88)(22,134,76,89)(23,135,77,90)(24,136,78,91)(25,137,79,92)(26,138,80,93)(27,139,81,94)(28,140,82,95)(29,149,202,357)(30,150,203,358)(31,151,204,359)(32,152,205,360)(33,153,206,361)(34,154,207,362)(35,155,208,363)(36,156,209,364)(37,157,210,337)(38,158,211,338)(39,159,212,339)(40,160,213,340)(41,161,214,341)(42,162,215,342)(43,163,216,343)(44,164,217,344)(45,165,218,345)(46,166,219,346)(47,167,220,347)(48,168,221,348)(49,141,222,349)(50,142,223,350)(51,143,224,351)(52,144,197,352)(53,145,198,353)(54,146,199,354)(55,147,200,355)(56,148,201,356)(169,309,247,375)(170,310,248,376)(171,311,249,377)(172,312,250,378)(173,313,251,379)(174,314,252,380)(175,315,225,381)(176,316,226,382)(177,317,227,383)(178,318,228,384)(179,319,229,385)(180,320,230,386)(181,321,231,387)(182,322,232,388)(183,323,233,389)(184,324,234,390)(185,325,235,391)(186,326,236,392)(187,327,237,365)(188,328,238,366)(189,329,239,367)(190,330,240,368)(191,331,241,369)(192,332,242,370)(193,333,243,371)(194,334,244,372)(195,335,245,373)(196,336,246,374)(253,431,415,286)(254,432,416,287)(255,433,417,288)(256,434,418,289)(257,435,419,290)(258,436,420,291)(259,437,393,292)(260,438,394,293)(261,439,395,294)(262,440,396,295)(263,441,397,296)(264,442,398,297)(265,443,399,298)(266,444,400,299)(267,445,401,300)(268,446,402,301)(269,447,403,302)(270,448,404,303)(271,421,405,304)(272,422,406,305)(273,423,407,306)(274,424,408,307)(275,425,409,308)(276,426,410,281)(277,427,411,282)(278,428,412,283)(279,429,413,284)(280,430,414,285), (1,44,83,217)(2,45,84,218)(3,46,57,219)(4,47,58,220)(5,48,59,221)(6,49,60,222)(7,50,61,223)(8,51,62,224)(9,52,63,197)(10,53,64,198)(11,54,65,199)(12,55,66,200)(13,56,67,201)(14,29,68,202)(15,30,69,203)(16,31,70,204)(17,32,71,205)(18,33,72,206)(19,34,73,207)(20,35,74,208)(21,36,75,209)(22,37,76,210)(23,38,77,211)(24,39,78,212)(25,40,79,213)(26,41,80,214)(27,42,81,215)(28,43,82,216)(85,153,130,361)(86,154,131,362)(87,155,132,363)(88,156,133,364)(89,157,134,337)(90,158,135,338)(91,159,136,339)(92,160,137,340)(93,161,138,341)(94,162,139,342)(95,163,140,343)(96,164,113,344)(97,165,114,345)(98,166,115,346)(99,167,116,347)(100,168,117,348)(101,141,118,349)(102,142,119,350)(103,143,120,351)(104,144,121,352)(105,145,122,353)(106,146,123,354)(107,147,124,355)(108,148,125,356)(109,149,126,357)(110,150,127,358)(111,151,128,359)(112,152,129,360)(169,416,247,254)(170,417,248,255)(171,418,249,256)(172,419,250,257)(173,420,251,258)(174,393,252,259)(175,394,225,260)(176,395,226,261)(177,396,227,262)(178,397,228,263)(179,398,229,264)(180,399,230,265)(181,400,231,266)(182,401,232,267)(183,402,233,268)(184,403,234,269)(185,404,235,270)(186,405,236,271)(187,406,237,272)(188,407,238,273)(189,408,239,274)(190,409,240,275)(191,410,241,276)(192,411,242,277)(193,412,243,278)(194,413,244,279)(195,414,245,280)(196,415,246,253)(281,331,426,369)(282,332,427,370)(283,333,428,371)(284,334,429,372)(285,335,430,373)(286,336,431,374)(287,309,432,375)(288,310,433,376)(289,311,434,377)(290,312,435,378)(291,313,436,379)(292,314,437,380)(293,315,438,381)(294,316,439,382)(295,317,440,383)(296,318,441,384)(297,319,442,385)(298,320,443,386)(299,321,444,387)(300,322,445,388)(301,323,446,389)(302,324,447,390)(303,325,448,391)(304,326,421,392)(305,327,422,365)(306,328,423,366)(307,329,424,367)(308,330,425,368), (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), (1,432,15,446)(2,431,16,445)(3,430,17,444)(4,429,18,443)(5,428,19,442)(6,427,20,441)(7,426,21,440)(8,425,22,439)(9,424,23,438)(10,423,24,437)(11,422,25,436)(12,421,26,435)(13,448,27,434)(14,447,28,433)(29,324,43,310)(30,323,44,309)(31,322,45,336)(32,321,46,335)(33,320,47,334)(34,319,48,333)(35,318,49,332)(36,317,50,331)(37,316,51,330)(38,315,52,329)(39,314,53,328)(40,313,54,327)(41,312,55,326)(42,311,56,325)(57,285,71,299)(58,284,72,298)(59,283,73,297)(60,282,74,296)(61,281,75,295)(62,308,76,294)(63,307,77,293)(64,306,78,292)(65,305,79,291)(66,304,80,290)(67,303,81,289)(68,302,82,288)(69,301,83,287)(70,300,84,286)(85,265,99,279)(86,264,100,278)(87,263,101,277)(88,262,102,276)(89,261,103,275)(90,260,104,274)(91,259,105,273)(92,258,106,272)(93,257,107,271)(94,256,108,270)(95,255,109,269)(96,254,110,268)(97,253,111,267)(98,280,112,266)(113,416,127,402)(114,415,128,401)(115,414,129,400)(116,413,130,399)(117,412,131,398)(118,411,132,397)(119,410,133,396)(120,409,134,395)(121,408,135,394)(122,407,136,393)(123,406,137,420)(124,405,138,419)(125,404,139,418)(126,403,140,417)(141,242,155,228)(142,241,156,227)(143,240,157,226)(144,239,158,225)(145,238,159,252)(146,237,160,251)(147,236,161,250)(148,235,162,249)(149,234,163,248)(150,233,164,247)(151,232,165,246)(152,231,166,245)(153,230,167,244)(154,229,168,243)(169,358,183,344)(170,357,184,343)(171,356,185,342)(172,355,186,341)(173,354,187,340)(174,353,188,339)(175,352,189,338)(176,351,190,337)(177,350,191,364)(178,349,192,363)(179,348,193,362)(180,347,194,361)(181,346,195,360)(182,345,196,359)(197,367,211,381)(198,366,212,380)(199,365,213,379)(200,392,214,378)(201,391,215,377)(202,390,216,376)(203,389,217,375)(204,388,218,374)(205,387,219,373)(206,386,220,372)(207,385,221,371)(208,384,222,370)(209,383,223,369)(210,382,224,368) );
G=PermutationGroup([(1,113,83,96),(2,114,84,97),(3,115,57,98),(4,116,58,99),(5,117,59,100),(6,118,60,101),(7,119,61,102),(8,120,62,103),(9,121,63,104),(10,122,64,105),(11,123,65,106),(12,124,66,107),(13,125,67,108),(14,126,68,109),(15,127,69,110),(16,128,70,111),(17,129,71,112),(18,130,72,85),(19,131,73,86),(20,132,74,87),(21,133,75,88),(22,134,76,89),(23,135,77,90),(24,136,78,91),(25,137,79,92),(26,138,80,93),(27,139,81,94),(28,140,82,95),(29,149,202,357),(30,150,203,358),(31,151,204,359),(32,152,205,360),(33,153,206,361),(34,154,207,362),(35,155,208,363),(36,156,209,364),(37,157,210,337),(38,158,211,338),(39,159,212,339),(40,160,213,340),(41,161,214,341),(42,162,215,342),(43,163,216,343),(44,164,217,344),(45,165,218,345),(46,166,219,346),(47,167,220,347),(48,168,221,348),(49,141,222,349),(50,142,223,350),(51,143,224,351),(52,144,197,352),(53,145,198,353),(54,146,199,354),(55,147,200,355),(56,148,201,356),(169,309,247,375),(170,310,248,376),(171,311,249,377),(172,312,250,378),(173,313,251,379),(174,314,252,380),(175,315,225,381),(176,316,226,382),(177,317,227,383),(178,318,228,384),(179,319,229,385),(180,320,230,386),(181,321,231,387),(182,322,232,388),(183,323,233,389),(184,324,234,390),(185,325,235,391),(186,326,236,392),(187,327,237,365),(188,328,238,366),(189,329,239,367),(190,330,240,368),(191,331,241,369),(192,332,242,370),(193,333,243,371),(194,334,244,372),(195,335,245,373),(196,336,246,374),(253,431,415,286),(254,432,416,287),(255,433,417,288),(256,434,418,289),(257,435,419,290),(258,436,420,291),(259,437,393,292),(260,438,394,293),(261,439,395,294),(262,440,396,295),(263,441,397,296),(264,442,398,297),(265,443,399,298),(266,444,400,299),(267,445,401,300),(268,446,402,301),(269,447,403,302),(270,448,404,303),(271,421,405,304),(272,422,406,305),(273,423,407,306),(274,424,408,307),(275,425,409,308),(276,426,410,281),(277,427,411,282),(278,428,412,283),(279,429,413,284),(280,430,414,285)], [(1,44,83,217),(2,45,84,218),(3,46,57,219),(4,47,58,220),(5,48,59,221),(6,49,60,222),(7,50,61,223),(8,51,62,224),(9,52,63,197),(10,53,64,198),(11,54,65,199),(12,55,66,200),(13,56,67,201),(14,29,68,202),(15,30,69,203),(16,31,70,204),(17,32,71,205),(18,33,72,206),(19,34,73,207),(20,35,74,208),(21,36,75,209),(22,37,76,210),(23,38,77,211),(24,39,78,212),(25,40,79,213),(26,41,80,214),(27,42,81,215),(28,43,82,216),(85,153,130,361),(86,154,131,362),(87,155,132,363),(88,156,133,364),(89,157,134,337),(90,158,135,338),(91,159,136,339),(92,160,137,340),(93,161,138,341),(94,162,139,342),(95,163,140,343),(96,164,113,344),(97,165,114,345),(98,166,115,346),(99,167,116,347),(100,168,117,348),(101,141,118,349),(102,142,119,350),(103,143,120,351),(104,144,121,352),(105,145,122,353),(106,146,123,354),(107,147,124,355),(108,148,125,356),(109,149,126,357),(110,150,127,358),(111,151,128,359),(112,152,129,360),(169,416,247,254),(170,417,248,255),(171,418,249,256),(172,419,250,257),(173,420,251,258),(174,393,252,259),(175,394,225,260),(176,395,226,261),(177,396,227,262),(178,397,228,263),(179,398,229,264),(180,399,230,265),(181,400,231,266),(182,401,232,267),(183,402,233,268),(184,403,234,269),(185,404,235,270),(186,405,236,271),(187,406,237,272),(188,407,238,273),(189,408,239,274),(190,409,240,275),(191,410,241,276),(192,411,242,277),(193,412,243,278),(194,413,244,279),(195,414,245,280),(196,415,246,253),(281,331,426,369),(282,332,427,370),(283,333,428,371),(284,334,429,372),(285,335,430,373),(286,336,431,374),(287,309,432,375),(288,310,433,376),(289,311,434,377),(290,312,435,378),(291,313,436,379),(292,314,437,380),(293,315,438,381),(294,316,439,382),(295,317,440,383),(296,318,441,384),(297,319,442,385),(298,320,443,386),(299,321,444,387),(300,322,445,388),(301,323,446,389),(302,324,447,390),(303,325,448,391),(304,326,421,392),(305,327,422,365),(306,328,423,366),(307,329,424,367),(308,330,425,368)], [(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)], [(1,432,15,446),(2,431,16,445),(3,430,17,444),(4,429,18,443),(5,428,19,442),(6,427,20,441),(7,426,21,440),(8,425,22,439),(9,424,23,438),(10,423,24,437),(11,422,25,436),(12,421,26,435),(13,448,27,434),(14,447,28,433),(29,324,43,310),(30,323,44,309),(31,322,45,336),(32,321,46,335),(33,320,47,334),(34,319,48,333),(35,318,49,332),(36,317,50,331),(37,316,51,330),(38,315,52,329),(39,314,53,328),(40,313,54,327),(41,312,55,326),(42,311,56,325),(57,285,71,299),(58,284,72,298),(59,283,73,297),(60,282,74,296),(61,281,75,295),(62,308,76,294),(63,307,77,293),(64,306,78,292),(65,305,79,291),(66,304,80,290),(67,303,81,289),(68,302,82,288),(69,301,83,287),(70,300,84,286),(85,265,99,279),(86,264,100,278),(87,263,101,277),(88,262,102,276),(89,261,103,275),(90,260,104,274),(91,259,105,273),(92,258,106,272),(93,257,107,271),(94,256,108,270),(95,255,109,269),(96,254,110,268),(97,253,111,267),(98,280,112,266),(113,416,127,402),(114,415,128,401),(115,414,129,400),(116,413,130,399),(117,412,131,398),(118,411,132,397),(119,410,133,396),(120,409,134,395),(121,408,135,394),(122,407,136,393),(123,406,137,420),(124,405,138,419),(125,404,139,418),(126,403,140,417),(141,242,155,228),(142,241,156,227),(143,240,157,226),(144,239,158,225),(145,238,159,252),(146,237,160,251),(147,236,161,250),(148,235,162,249),(149,234,163,248),(150,233,164,247),(151,232,165,246),(152,231,166,245),(153,230,167,244),(154,229,168,243),(169,358,183,344),(170,357,184,343),(171,356,185,342),(172,355,186,341),(173,354,187,340),(174,353,188,339),(175,352,189,338),(176,351,190,337),(177,350,191,364),(178,349,192,363),(179,348,193,362),(180,347,194,361),(181,346,195,360),(182,345,196,359),(197,367,211,381),(198,366,212,380),(199,365,213,379),(200,392,214,378),(201,391,215,377),(202,390,216,376),(203,389,217,375),(204,388,218,374),(205,387,219,373),(206,386,220,372),(207,385,221,371),(208,384,222,370),(209,383,223,369),(210,382,224,368)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 5 |
| 0 | 0 | 0 | 0 | 17 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 26 |
| 0 | 0 | 0 | 0 | 21 | 27 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 3 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 5 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 25 | 0 | 0 | 0 | 0 |
| 20 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 27 |
| 0 | 0 | 0 | 0 | 28 | 12 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,17,0,0,0,0,5,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,2,21,0,0,0,0,26,27],[7,3,0,0,0,0,0,25,0,0,0,0,0,0,20,1,0,0,0,0,5,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,20,0,0,0,0,25,20,0,0,0,0,0,0,0,21,0,0,0,0,11,0,0,0,0,0,0,0,17,28,0,0,0,0,27,12] >;
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 | D7 | C4○D4 | D14 | D14 | D14 | Dic14 | 2- (1+4) | Q8⋊2D7 | D4.10D14 |
| kernel | Q8⋊6Dic14 | C4×Dic14 | C28⋊2Q8 | C4.Dic14 | Q8×Dic7 | Q8×C28 | C7×Q8 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | Q8 | C14 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_6Dic_{14} % in TeX
G:=Group("Q8:6Dic14"); // GroupNames label
G:=SmallGroup(448,1023);
// by ID
G=gap.SmallGroup(448,1023);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,387,184,1571,192,18822]);
// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
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