direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C2×C28, C2.5(C23×C28), (C2×C42).17C14, C4.17(C22×C28), C42.86(C2×C14), C14.57(C23×C4), C22.17(Q8×C14), C14.56(C22×Q8), (C2×C14).336C24, C28.162(C22×C4), (C4×C28).370C22, (C2×C28).708C23, C22.9(C23×C14), (C22×Q8).10C14, C23.68(C22×C14), C22.26(C22×C28), (Q8×C14).282C22, (C22×C14).468C23, (C22×C28).594C22, C2.2(Q8×C2×C14), (C2×C4×C28).40C2, C2.3(C14×C4○D4), (C2×C4⋊C4).22C14, (C14×C4⋊C4).51C2, (Q8×C2×C14).20C2, C4⋊C4.80(C2×C14), (C2×C4).52(C2×C28), (C2×C28).273(C2×C4), C14.222(C2×C4○D4), (C2×Q8).70(C2×C14), (C2×C14).115(C2×Q8), C22.28(C7×C4○D4), (C7×C4⋊C4).405C22, (C22×C4).98(C2×C14), (C2×C4).55(C22×C14), (C2×C14).228(C4○D4), (C2×C14).246(C22×C4), SmallGroup(448,1299)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C2×C28
G = < a,b,c,d | a2=b28=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 322 in 298 conjugacy classes, 274 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C2×C28, C2×C28, C7×Q8, C22×C14, C2×C4×Q8, C4×C28, C7×C4⋊C4, C22×C28, C22×C28, Q8×C14, C2×C4×C28, C14×C4⋊C4, Q8×C28, Q8×C2×C14, Q8×C2×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, Q8, C23, C14, C22×C4, C2×Q8, C4○D4, C24, C28, C2×C14, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C2×C28, C7×Q8, C22×C14, C2×C4×Q8, C22×C28, Q8×C14, C7×C4○D4, C23×C14, Q8×C28, C23×C28, Q8×C2×C14, C14×C4○D4, Q8×C2×C28
►Regular action on 448 points
Generators in S448
(1 422)(2 423)(3 424)(4 425)(5 426)(6 427)(7 428)(8 429)(9 430)(10 431)(11 432)(12 433)(13 434)(14 435)(15 436)(16 437)(17 438)(18 439)(19 440)(20 441)(21 442)(22 443)(23 444)(24 445)(25 446)(26 447)(27 448)(28 421)(29 308)(30 281)(31 282)(32 283)(33 284)(34 285)(35 286)(36 287)(37 288)(38 289)(39 290)(40 291)(41 292)(42 293)(43 294)(44 295)(45 296)(46 297)(47 298)(48 299)(49 300)(50 301)(51 302)(52 303)(53 304)(54 305)(55 306)(56 307)(57 353)(58 354)(59 355)(60 356)(61 357)(62 358)(63 359)(64 360)(65 361)(66 362)(67 363)(68 364)(69 337)(70 338)(71 339)(72 340)(73 341)(74 342)(75 343)(76 344)(77 345)(78 346)(79 347)(80 348)(81 349)(82 350)(83 351)(84 352)(85 217)(86 218)(87 219)(88 220)(89 221)(90 222)(91 223)(92 224)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(105 209)(106 210)(107 211)(108 212)(109 213)(110 214)(111 215)(112 216)(113 252)(114 225)(115 226)(116 227)(117 228)(118 229)(119 230)(120 231)(121 232)(122 233)(123 234)(124 235)(125 236)(126 237)(127 238)(128 239)(129 240)(130 241)(131 242)(132 243)(133 244)(134 245)(135 246)(136 247)(137 248)(138 249)(139 250)(140 251)(141 389)(142 390)(143 391)(144 392)(145 365)(146 366)(147 367)(148 368)(149 369)(150 370)(151 371)(152 372)(153 373)(154 374)(155 375)(156 376)(157 377)(158 378)(159 379)(160 380)(161 381)(162 382)(163 383)(164 384)(165 385)(166 386)(167 387)(168 388)(169 270)(170 271)(171 272)(172 273)(173 274)(174 275)(175 276)(176 277)(177 278)(178 279)(179 280)(180 253)(181 254)(182 255)(183 256)(184 257)(185 258)(186 259)(187 260)(188 261)(189 262)(190 263)(191 264)(192 265)(193 266)(194 267)(195 268)(196 269)(309 397)(310 398)(311 399)(312 400)(313 401)(314 402)(315 403)(316 404)(317 405)(318 406)(319 407)(320 408)(321 409)(322 410)(323 411)(324 412)(325 413)(326 414)(327 415)(328 416)(329 417)(330 418)(331 419)(332 420)(333 393)(334 394)(335 395)(336 396)
(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)
(1 215 372 39)(2 216 373 40)(3 217 374 41)(4 218 375 42)(5 219 376 43)(6 220 377 44)(7 221 378 45)(8 222 379 46)(9 223 380 47)(10 224 381 48)(11 197 382 49)(12 198 383 50)(13 199 384 51)(14 200 385 52)(15 201 386 53)(16 202 387 54)(17 203 388 55)(18 204 389 56)(19 205 390 29)(20 206 391 30)(21 207 392 31)(22 208 365 32)(23 209 366 33)(24 210 367 34)(25 211 368 35)(26 212 369 36)(27 213 370 37)(28 214 371 38)(57 268 133 323)(58 269 134 324)(59 270 135 325)(60 271 136 326)(61 272 137 327)(62 273 138 328)(63 274 139 329)(64 275 140 330)(65 276 113 331)(66 277 114 332)(67 278 115 333)(68 279 116 334)(69 280 117 335)(70 253 118 336)(71 254 119 309)(72 255 120 310)(73 256 121 311)(74 257 122 312)(75 258 123 313)(76 259 124 314)(77 260 125 315)(78 261 126 316)(79 262 127 317)(80 263 128 318)(81 264 129 319)(82 265 130 320)(83 266 131 321)(84 267 132 322)(85 154 292 424)(86 155 293 425)(87 156 294 426)(88 157 295 427)(89 158 296 428)(90 159 297 429)(91 160 298 430)(92 161 299 431)(93 162 300 432)(94 163 301 433)(95 164 302 434)(96 165 303 435)(97 166 304 436)(98 167 305 437)(99 168 306 438)(100 141 307 439)(101 142 308 440)(102 143 281 441)(103 144 282 442)(104 145 283 443)(105 146 284 444)(106 147 285 445)(107 148 286 446)(108 149 287 447)(109 150 288 448)(110 151 289 421)(111 152 290 422)(112 153 291 423)(169 246 413 355)(170 247 414 356)(171 248 415 357)(172 249 416 358)(173 250 417 359)(174 251 418 360)(175 252 419 361)(176 225 420 362)(177 226 393 363)(178 227 394 364)(179 228 395 337)(180 229 396 338)(181 230 397 339)(182 231 398 340)(183 232 399 341)(184 233 400 342)(185 234 401 343)(186 235 402 344)(187 236 403 345)(188 237 404 346)(189 238 405 347)(190 239 406 348)(191 240 407 349)(192 241 408 350)(193 242 409 351)(194 243 410 352)(195 244 411 353)(196 245 412 354)
(1 311 372 256)(2 312 373 257)(3 313 374 258)(4 314 375 259)(5 315 376 260)(6 316 377 261)(7 317 378 262)(8 318 379 263)(9 319 380 264)(10 320 381 265)(11 321 382 266)(12 322 383 267)(13 323 384 268)(14 324 385 269)(15 325 386 270)(16 326 387 271)(17 327 388 272)(18 328 389 273)(19 329 390 274)(20 330 391 275)(21 331 392 276)(22 332 365 277)(23 333 366 278)(24 334 367 279)(25 335 368 280)(26 336 369 253)(27 309 370 254)(28 310 371 255)(29 63 205 139)(30 64 206 140)(31 65 207 113)(32 66 208 114)(33 67 209 115)(34 68 210 116)(35 69 211 117)(36 70 212 118)(37 71 213 119)(38 72 214 120)(39 73 215 121)(40 74 216 122)(41 75 217 123)(42 76 218 124)(43 77 219 125)(44 78 220 126)(45 79 221 127)(46 80 222 128)(47 81 223 129)(48 82 224 130)(49 83 197 131)(50 84 198 132)(51 57 199 133)(52 58 200 134)(53 59 201 135)(54 60 202 136)(55 61 203 137)(56 62 204 138)(85 234 292 343)(86 235 293 344)(87 236 294 345)(88 237 295 346)(89 238 296 347)(90 239 297 348)(91 240 298 349)(92 241 299 350)(93 242 300 351)(94 243 301 352)(95 244 302 353)(96 245 303 354)(97 246 304 355)(98 247 305 356)(99 248 306 357)(100 249 307 358)(101 250 308 359)(102 251 281 360)(103 252 282 361)(104 225 283 362)(105 226 284 363)(106 227 285 364)(107 228 286 337)(108 229 287 338)(109 230 288 339)(110 231 289 340)(111 232 290 341)(112 233 291 342)(141 172 439 416)(142 173 440 417)(143 174 441 418)(144 175 442 419)(145 176 443 420)(146 177 444 393)(147 178 445 394)(148 179 446 395)(149 180 447 396)(150 181 448 397)(151 182 421 398)(152 183 422 399)(153 184 423 400)(154 185 424 401)(155 186 425 402)(156 187 426 403)(157 188 427 404)(158 189 428 405)(159 190 429 406)(160 191 430 407)(161 192 431 408)(162 193 432 409)(163 194 433 410)(164 195 434 411)(165 196 435 412)(166 169 436 413)(167 170 437 414)(168 171 438 415)
G:=sub<Sym(448)| (1,422)(2,423)(3,424)(4,425)(5,426)(6,427)(7,428)(8,429)(9,430)(10,431)(11,432)(12,433)(13,434)(14,435)(15,436)(16,437)(17,438)(18,439)(19,440)(20,441)(21,442)(22,443)(23,444)(24,445)(25,446)(26,447)(27,448)(28,421)(29,308)(30,281)(31,282)(32,283)(33,284)(34,285)(35,286)(36,287)(37,288)(38,289)(39,290)(40,291)(41,292)(42,293)(43,294)(44,295)(45,296)(46,297)(47,298)(48,299)(49,300)(50,301)(51,302)(52,303)(53,304)(54,305)(55,306)(56,307)(57,353)(58,354)(59,355)(60,356)(61,357)(62,358)(63,359)(64,360)(65,361)(66,362)(67,363)(68,364)(69,337)(70,338)(71,339)(72,340)(73,341)(74,342)(75,343)(76,344)(77,345)(78,346)(79,347)(80,348)(81,349)(82,350)(83,351)(84,352)(85,217)(86,218)(87,219)(88,220)(89,221)(90,222)(91,223)(92,224)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208)(105,209)(106,210)(107,211)(108,212)(109,213)(110,214)(111,215)(112,216)(113,252)(114,225)(115,226)(116,227)(117,228)(118,229)(119,230)(120,231)(121,232)(122,233)(123,234)(124,235)(125,236)(126,237)(127,238)(128,239)(129,240)(130,241)(131,242)(132,243)(133,244)(134,245)(135,246)(136,247)(137,248)(138,249)(139,250)(140,251)(141,389)(142,390)(143,391)(144,392)(145,365)(146,366)(147,367)(148,368)(149,369)(150,370)(151,371)(152,372)(153,373)(154,374)(155,375)(156,376)(157,377)(158,378)(159,379)(160,380)(161,381)(162,382)(163,383)(164,384)(165,385)(166,386)(167,387)(168,388)(169,270)(170,271)(171,272)(172,273)(173,274)(174,275)(175,276)(176,277)(177,278)(178,279)(179,280)(180,253)(181,254)(182,255)(183,256)(184,257)(185,258)(186,259)(187,260)(188,261)(189,262)(190,263)(191,264)(192,265)(193,266)(194,267)(195,268)(196,269)(309,397)(310,398)(311,399)(312,400)(313,401)(314,402)(315,403)(316,404)(317,405)(318,406)(319,407)(320,408)(321,409)(322,410)(323,411)(324,412)(325,413)(326,414)(327,415)(328,416)(329,417)(330,418)(331,419)(332,420)(333,393)(334,394)(335,395)(336,396), (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), (1,215,372,39)(2,216,373,40)(3,217,374,41)(4,218,375,42)(5,219,376,43)(6,220,377,44)(7,221,378,45)(8,222,379,46)(9,223,380,47)(10,224,381,48)(11,197,382,49)(12,198,383,50)(13,199,384,51)(14,200,385,52)(15,201,386,53)(16,202,387,54)(17,203,388,55)(18,204,389,56)(19,205,390,29)(20,206,391,30)(21,207,392,31)(22,208,365,32)(23,209,366,33)(24,210,367,34)(25,211,368,35)(26,212,369,36)(27,213,370,37)(28,214,371,38)(57,268,133,323)(58,269,134,324)(59,270,135,325)(60,271,136,326)(61,272,137,327)(62,273,138,328)(63,274,139,329)(64,275,140,330)(65,276,113,331)(66,277,114,332)(67,278,115,333)(68,279,116,334)(69,280,117,335)(70,253,118,336)(71,254,119,309)(72,255,120,310)(73,256,121,311)(74,257,122,312)(75,258,123,313)(76,259,124,314)(77,260,125,315)(78,261,126,316)(79,262,127,317)(80,263,128,318)(81,264,129,319)(82,265,130,320)(83,266,131,321)(84,267,132,322)(85,154,292,424)(86,155,293,425)(87,156,294,426)(88,157,295,427)(89,158,296,428)(90,159,297,429)(91,160,298,430)(92,161,299,431)(93,162,300,432)(94,163,301,433)(95,164,302,434)(96,165,303,435)(97,166,304,436)(98,167,305,437)(99,168,306,438)(100,141,307,439)(101,142,308,440)(102,143,281,441)(103,144,282,442)(104,145,283,443)(105,146,284,444)(106,147,285,445)(107,148,286,446)(108,149,287,447)(109,150,288,448)(110,151,289,421)(111,152,290,422)(112,153,291,423)(169,246,413,355)(170,247,414,356)(171,248,415,357)(172,249,416,358)(173,250,417,359)(174,251,418,360)(175,252,419,361)(176,225,420,362)(177,226,393,363)(178,227,394,364)(179,228,395,337)(180,229,396,338)(181,230,397,339)(182,231,398,340)(183,232,399,341)(184,233,400,342)(185,234,401,343)(186,235,402,344)(187,236,403,345)(188,237,404,346)(189,238,405,347)(190,239,406,348)(191,240,407,349)(192,241,408,350)(193,242,409,351)(194,243,410,352)(195,244,411,353)(196,245,412,354), (1,311,372,256)(2,312,373,257)(3,313,374,258)(4,314,375,259)(5,315,376,260)(6,316,377,261)(7,317,378,262)(8,318,379,263)(9,319,380,264)(10,320,381,265)(11,321,382,266)(12,322,383,267)(13,323,384,268)(14,324,385,269)(15,325,386,270)(16,326,387,271)(17,327,388,272)(18,328,389,273)(19,329,390,274)(20,330,391,275)(21,331,392,276)(22,332,365,277)(23,333,366,278)(24,334,367,279)(25,335,368,280)(26,336,369,253)(27,309,370,254)(28,310,371,255)(29,63,205,139)(30,64,206,140)(31,65,207,113)(32,66,208,114)(33,67,209,115)(34,68,210,116)(35,69,211,117)(36,70,212,118)(37,71,213,119)(38,72,214,120)(39,73,215,121)(40,74,216,122)(41,75,217,123)(42,76,218,124)(43,77,219,125)(44,78,220,126)(45,79,221,127)(46,80,222,128)(47,81,223,129)(48,82,224,130)(49,83,197,131)(50,84,198,132)(51,57,199,133)(52,58,200,134)(53,59,201,135)(54,60,202,136)(55,61,203,137)(56,62,204,138)(85,234,292,343)(86,235,293,344)(87,236,294,345)(88,237,295,346)(89,238,296,347)(90,239,297,348)(91,240,298,349)(92,241,299,350)(93,242,300,351)(94,243,301,352)(95,244,302,353)(96,245,303,354)(97,246,304,355)(98,247,305,356)(99,248,306,357)(100,249,307,358)(101,250,308,359)(102,251,281,360)(103,252,282,361)(104,225,283,362)(105,226,284,363)(106,227,285,364)(107,228,286,337)(108,229,287,338)(109,230,288,339)(110,231,289,340)(111,232,290,341)(112,233,291,342)(141,172,439,416)(142,173,440,417)(143,174,441,418)(144,175,442,419)(145,176,443,420)(146,177,444,393)(147,178,445,394)(148,179,446,395)(149,180,447,396)(150,181,448,397)(151,182,421,398)(152,183,422,399)(153,184,423,400)(154,185,424,401)(155,186,425,402)(156,187,426,403)(157,188,427,404)(158,189,428,405)(159,190,429,406)(160,191,430,407)(161,192,431,408)(162,193,432,409)(163,194,433,410)(164,195,434,411)(165,196,435,412)(166,169,436,413)(167,170,437,414)(168,171,438,415)>;
G:=Group( (1,422)(2,423)(3,424)(4,425)(5,426)(6,427)(7,428)(8,429)(9,430)(10,431)(11,432)(12,433)(13,434)(14,435)(15,436)(16,437)(17,438)(18,439)(19,440)(20,441)(21,442)(22,443)(23,444)(24,445)(25,446)(26,447)(27,448)(28,421)(29,308)(30,281)(31,282)(32,283)(33,284)(34,285)(35,286)(36,287)(37,288)(38,289)(39,290)(40,291)(41,292)(42,293)(43,294)(44,295)(45,296)(46,297)(47,298)(48,299)(49,300)(50,301)(51,302)(52,303)(53,304)(54,305)(55,306)(56,307)(57,353)(58,354)(59,355)(60,356)(61,357)(62,358)(63,359)(64,360)(65,361)(66,362)(67,363)(68,364)(69,337)(70,338)(71,339)(72,340)(73,341)(74,342)(75,343)(76,344)(77,345)(78,346)(79,347)(80,348)(81,349)(82,350)(83,351)(84,352)(85,217)(86,218)(87,219)(88,220)(89,221)(90,222)(91,223)(92,224)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208)(105,209)(106,210)(107,211)(108,212)(109,213)(110,214)(111,215)(112,216)(113,252)(114,225)(115,226)(116,227)(117,228)(118,229)(119,230)(120,231)(121,232)(122,233)(123,234)(124,235)(125,236)(126,237)(127,238)(128,239)(129,240)(130,241)(131,242)(132,243)(133,244)(134,245)(135,246)(136,247)(137,248)(138,249)(139,250)(140,251)(141,389)(142,390)(143,391)(144,392)(145,365)(146,366)(147,367)(148,368)(149,369)(150,370)(151,371)(152,372)(153,373)(154,374)(155,375)(156,376)(157,377)(158,378)(159,379)(160,380)(161,381)(162,382)(163,383)(164,384)(165,385)(166,386)(167,387)(168,388)(169,270)(170,271)(171,272)(172,273)(173,274)(174,275)(175,276)(176,277)(177,278)(178,279)(179,280)(180,253)(181,254)(182,255)(183,256)(184,257)(185,258)(186,259)(187,260)(188,261)(189,262)(190,263)(191,264)(192,265)(193,266)(194,267)(195,268)(196,269)(309,397)(310,398)(311,399)(312,400)(313,401)(314,402)(315,403)(316,404)(317,405)(318,406)(319,407)(320,408)(321,409)(322,410)(323,411)(324,412)(325,413)(326,414)(327,415)(328,416)(329,417)(330,418)(331,419)(332,420)(333,393)(334,394)(335,395)(336,396), (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), (1,215,372,39)(2,216,373,40)(3,217,374,41)(4,218,375,42)(5,219,376,43)(6,220,377,44)(7,221,378,45)(8,222,379,46)(9,223,380,47)(10,224,381,48)(11,197,382,49)(12,198,383,50)(13,199,384,51)(14,200,385,52)(15,201,386,53)(16,202,387,54)(17,203,388,55)(18,204,389,56)(19,205,390,29)(20,206,391,30)(21,207,392,31)(22,208,365,32)(23,209,366,33)(24,210,367,34)(25,211,368,35)(26,212,369,36)(27,213,370,37)(28,214,371,38)(57,268,133,323)(58,269,134,324)(59,270,135,325)(60,271,136,326)(61,272,137,327)(62,273,138,328)(63,274,139,329)(64,275,140,330)(65,276,113,331)(66,277,114,332)(67,278,115,333)(68,279,116,334)(69,280,117,335)(70,253,118,336)(71,254,119,309)(72,255,120,310)(73,256,121,311)(74,257,122,312)(75,258,123,313)(76,259,124,314)(77,260,125,315)(78,261,126,316)(79,262,127,317)(80,263,128,318)(81,264,129,319)(82,265,130,320)(83,266,131,321)(84,267,132,322)(85,154,292,424)(86,155,293,425)(87,156,294,426)(88,157,295,427)(89,158,296,428)(90,159,297,429)(91,160,298,430)(92,161,299,431)(93,162,300,432)(94,163,301,433)(95,164,302,434)(96,165,303,435)(97,166,304,436)(98,167,305,437)(99,168,306,438)(100,141,307,439)(101,142,308,440)(102,143,281,441)(103,144,282,442)(104,145,283,443)(105,146,284,444)(106,147,285,445)(107,148,286,446)(108,149,287,447)(109,150,288,448)(110,151,289,421)(111,152,290,422)(112,153,291,423)(169,246,413,355)(170,247,414,356)(171,248,415,357)(172,249,416,358)(173,250,417,359)(174,251,418,360)(175,252,419,361)(176,225,420,362)(177,226,393,363)(178,227,394,364)(179,228,395,337)(180,229,396,338)(181,230,397,339)(182,231,398,340)(183,232,399,341)(184,233,400,342)(185,234,401,343)(186,235,402,344)(187,236,403,345)(188,237,404,346)(189,238,405,347)(190,239,406,348)(191,240,407,349)(192,241,408,350)(193,242,409,351)(194,243,410,352)(195,244,411,353)(196,245,412,354), (1,311,372,256)(2,312,373,257)(3,313,374,258)(4,314,375,259)(5,315,376,260)(6,316,377,261)(7,317,378,262)(8,318,379,263)(9,319,380,264)(10,320,381,265)(11,321,382,266)(12,322,383,267)(13,323,384,268)(14,324,385,269)(15,325,386,270)(16,326,387,271)(17,327,388,272)(18,328,389,273)(19,329,390,274)(20,330,391,275)(21,331,392,276)(22,332,365,277)(23,333,366,278)(24,334,367,279)(25,335,368,280)(26,336,369,253)(27,309,370,254)(28,310,371,255)(29,63,205,139)(30,64,206,140)(31,65,207,113)(32,66,208,114)(33,67,209,115)(34,68,210,116)(35,69,211,117)(36,70,212,118)(37,71,213,119)(38,72,214,120)(39,73,215,121)(40,74,216,122)(41,75,217,123)(42,76,218,124)(43,77,219,125)(44,78,220,126)(45,79,221,127)(46,80,222,128)(47,81,223,129)(48,82,224,130)(49,83,197,131)(50,84,198,132)(51,57,199,133)(52,58,200,134)(53,59,201,135)(54,60,202,136)(55,61,203,137)(56,62,204,138)(85,234,292,343)(86,235,293,344)(87,236,294,345)(88,237,295,346)(89,238,296,347)(90,239,297,348)(91,240,298,349)(92,241,299,350)(93,242,300,351)(94,243,301,352)(95,244,302,353)(96,245,303,354)(97,246,304,355)(98,247,305,356)(99,248,306,357)(100,249,307,358)(101,250,308,359)(102,251,281,360)(103,252,282,361)(104,225,283,362)(105,226,284,363)(106,227,285,364)(107,228,286,337)(108,229,287,338)(109,230,288,339)(110,231,289,340)(111,232,290,341)(112,233,291,342)(141,172,439,416)(142,173,440,417)(143,174,441,418)(144,175,442,419)(145,176,443,420)(146,177,444,393)(147,178,445,394)(148,179,446,395)(149,180,447,396)(150,181,448,397)(151,182,421,398)(152,183,422,399)(153,184,423,400)(154,185,424,401)(155,186,425,402)(156,187,426,403)(157,188,427,404)(158,189,428,405)(159,190,429,406)(160,191,430,407)(161,192,431,408)(162,193,432,409)(163,194,433,410)(164,195,434,411)(165,196,435,412)(166,169,436,413)(167,170,437,414)(168,171,438,415) );
G=PermutationGroup([[(1,422),(2,423),(3,424),(4,425),(5,426),(6,427),(7,428),(8,429),(9,430),(10,431),(11,432),(12,433),(13,434),(14,435),(15,436),(16,437),(17,438),(18,439),(19,440),(20,441),(21,442),(22,443),(23,444),(24,445),(25,446),(26,447),(27,448),(28,421),(29,308),(30,281),(31,282),(32,283),(33,284),(34,285),(35,286),(36,287),(37,288),(38,289),(39,290),(40,291),(41,292),(42,293),(43,294),(44,295),(45,296),(46,297),(47,298),(48,299),(49,300),(50,301),(51,302),(52,303),(53,304),(54,305),(55,306),(56,307),(57,353),(58,354),(59,355),(60,356),(61,357),(62,358),(63,359),(64,360),(65,361),(66,362),(67,363),(68,364),(69,337),(70,338),(71,339),(72,340),(73,341),(74,342),(75,343),(76,344),(77,345),(78,346),(79,347),(80,348),(81,349),(82,350),(83,351),(84,352),(85,217),(86,218),(87,219),(88,220),(89,221),(90,222),(91,223),(92,224),(93,197),(94,198),(95,199),(96,200),(97,201),(98,202),(99,203),(100,204),(101,205),(102,206),(103,207),(104,208),(105,209),(106,210),(107,211),(108,212),(109,213),(110,214),(111,215),(112,216),(113,252),(114,225),(115,226),(116,227),(117,228),(118,229),(119,230),(120,231),(121,232),(122,233),(123,234),(124,235),(125,236),(126,237),(127,238),(128,239),(129,240),(130,241),(131,242),(132,243),(133,244),(134,245),(135,246),(136,247),(137,248),(138,249),(139,250),(140,251),(141,389),(142,390),(143,391),(144,392),(145,365),(146,366),(147,367),(148,368),(149,369),(150,370),(151,371),(152,372),(153,373),(154,374),(155,375),(156,376),(157,377),(158,378),(159,379),(160,380),(161,381),(162,382),(163,383),(164,384),(165,385),(166,386),(167,387),(168,388),(169,270),(170,271),(171,272),(172,273),(173,274),(174,275),(175,276),(176,277),(177,278),(178,279),(179,280),(180,253),(181,254),(182,255),(183,256),(184,257),(185,258),(186,259),(187,260),(188,261),(189,262),(190,263),(191,264),(192,265),(193,266),(194,267),(195,268),(196,269),(309,397),(310,398),(311,399),(312,400),(313,401),(314,402),(315,403),(316,404),(317,405),(318,406),(319,407),(320,408),(321,409),(322,410),(323,411),(324,412),(325,413),(326,414),(327,415),(328,416),(329,417),(330,418),(331,419),(332,420),(333,393),(334,394),(335,395),(336,396)], [(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)], [(1,215,372,39),(2,216,373,40),(3,217,374,41),(4,218,375,42),(5,219,376,43),(6,220,377,44),(7,221,378,45),(8,222,379,46),(9,223,380,47),(10,224,381,48),(11,197,382,49),(12,198,383,50),(13,199,384,51),(14,200,385,52),(15,201,386,53),(16,202,387,54),(17,203,388,55),(18,204,389,56),(19,205,390,29),(20,206,391,30),(21,207,392,31),(22,208,365,32),(23,209,366,33),(24,210,367,34),(25,211,368,35),(26,212,369,36),(27,213,370,37),(28,214,371,38),(57,268,133,323),(58,269,134,324),(59,270,135,325),(60,271,136,326),(61,272,137,327),(62,273,138,328),(63,274,139,329),(64,275,140,330),(65,276,113,331),(66,277,114,332),(67,278,115,333),(68,279,116,334),(69,280,117,335),(70,253,118,336),(71,254,119,309),(72,255,120,310),(73,256,121,311),(74,257,122,312),(75,258,123,313),(76,259,124,314),(77,260,125,315),(78,261,126,316),(79,262,127,317),(80,263,128,318),(81,264,129,319),(82,265,130,320),(83,266,131,321),(84,267,132,322),(85,154,292,424),(86,155,293,425),(87,156,294,426),(88,157,295,427),(89,158,296,428),(90,159,297,429),(91,160,298,430),(92,161,299,431),(93,162,300,432),(94,163,301,433),(95,164,302,434),(96,165,303,435),(97,166,304,436),(98,167,305,437),(99,168,306,438),(100,141,307,439),(101,142,308,440),(102,143,281,441),(103,144,282,442),(104,145,283,443),(105,146,284,444),(106,147,285,445),(107,148,286,446),(108,149,287,447),(109,150,288,448),(110,151,289,421),(111,152,290,422),(112,153,291,423),(169,246,413,355),(170,247,414,356),(171,248,415,357),(172,249,416,358),(173,250,417,359),(174,251,418,360),(175,252,419,361),(176,225,420,362),(177,226,393,363),(178,227,394,364),(179,228,395,337),(180,229,396,338),(181,230,397,339),(182,231,398,340),(183,232,399,341),(184,233,400,342),(185,234,401,343),(186,235,402,344),(187,236,403,345),(188,237,404,346),(189,238,405,347),(190,239,406,348),(191,240,407,349),(192,241,408,350),(193,242,409,351),(194,243,410,352),(195,244,411,353),(196,245,412,354)], [(1,311,372,256),(2,312,373,257),(3,313,374,258),(4,314,375,259),(5,315,376,260),(6,316,377,261),(7,317,378,262),(8,318,379,263),(9,319,380,264),(10,320,381,265),(11,321,382,266),(12,322,383,267),(13,323,384,268),(14,324,385,269),(15,325,386,270),(16,326,387,271),(17,327,388,272),(18,328,389,273),(19,329,390,274),(20,330,391,275),(21,331,392,276),(22,332,365,277),(23,333,366,278),(24,334,367,279),(25,335,368,280),(26,336,369,253),(27,309,370,254),(28,310,371,255),(29,63,205,139),(30,64,206,140),(31,65,207,113),(32,66,208,114),(33,67,209,115),(34,68,210,116),(35,69,211,117),(36,70,212,118),(37,71,213,119),(38,72,214,120),(39,73,215,121),(40,74,216,122),(41,75,217,123),(42,76,218,124),(43,77,219,125),(44,78,220,126),(45,79,221,127),(46,80,222,128),(47,81,223,129),(48,82,224,130),(49,83,197,131),(50,84,198,132),(51,57,199,133),(52,58,200,134),(53,59,201,135),(54,60,202,136),(55,61,203,137),(56,62,204,138),(85,234,292,343),(86,235,293,344),(87,236,294,345),(88,237,295,346),(89,238,296,347),(90,239,297,348),(91,240,298,349),(92,241,299,350),(93,242,300,351),(94,243,301,352),(95,244,302,353),(96,245,303,354),(97,246,304,355),(98,247,305,356),(99,248,306,357),(100,249,307,358),(101,250,308,359),(102,251,281,360),(103,252,282,361),(104,225,283,362),(105,226,284,363),(106,227,285,364),(107,228,286,337),(108,229,287,338),(109,230,288,339),(110,231,289,340),(111,232,290,341),(112,233,291,342),(141,172,439,416),(142,173,440,417),(143,174,441,418),(144,175,442,419),(145,176,443,420),(146,177,444,393),(147,178,445,394),(148,179,446,395),(149,180,447,396),(150,181,448,397),(151,182,421,398),(152,183,422,399),(153,184,423,400),(154,185,424,401),(155,186,425,402),(156,187,426,403),(157,188,427,404),(158,189,428,405),(159,190,429,406),(160,191,430,407),(161,192,431,408),(162,193,432,409),(163,194,433,410),(164,195,434,411),(165,196,435,412),(166,169,436,413),(167,170,437,414),(168,171,438,415)]])
280 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 7A | ··· | 7F | 14A | ··· | 14AP | 28A | ··· | 28AV | 28AW | ··· | 28GJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | Q8 | C4○D4 | C7×Q8 | C7×C4○D4 |
| kernel | Q8×C2×C28 | C2×C4×C28 | C14×C4⋊C4 | Q8×C28 | Q8×C2×C14 | Q8×C14 | C2×C4×Q8 | C2×C42 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×Q8 | C2×C28 | C2×C14 | C2×C4 | C22 |
| # reps | 1 | 3 | 3 | 8 | 1 | 16 | 6 | 18 | 18 | 48 | 6 | 96 | 4 | 4 | 24 | 24 |
Matrix representation of Q8×C2×C28 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 9 | 11 |
| 0 | 0 | 11 | 20 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,28,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0],[28,0,0,0,0,28,0,0,0,0,9,11,0,0,11,20] >;
Q8×C2×C28 in GAP, Magma, Sage, TeX
Q_8\times C_2\times C_{28} % in TeX
G:=Group("Q8xC2xC28"); // GroupNames label
G:=SmallGroup(448,1299);
// by ID
G=gap.SmallGroup(448,1299);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,792,1780]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations