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