metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).287D4, C14.26(C4⋊Q8), (C2×Dic7).5Q8, C22.46(Q8×D7), C2.8(C28⋊2D4), (C2×Dic7).59D4, (C22×C4).41D14, C22.245(D4×D7), C14.89(C4⋊D4), C14.45(C22⋊Q8), C2.21(D14⋊Q8), C2.6(Dic7⋊Q8), C2.11(Dic7.Q8), (C22×C28).29C22, C14.19(C42.C2), C23.375(C22×D7), C22.103(C4○D28), C22.99(D4⋊2D7), C14.C42.18C2, (C22×C14).344C23, C7⋊3(C23.81C23), C14.62(C22.D4), (C22×Dic7).53C22, C2.12(C23.23D14), (C2×C4⋊C4).17D7, (C14×C4⋊C4).31C2, (C2×C14).79(C2×Q8), (C2×C14).448(C2×D4), (C2×C4).36(C7⋊D4), (C2×Dic7⋊C4).14C2, C22.135(C2×C7⋊D4), (C2×C14).154(C4○D4), SmallGroup(448,514)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28).287D4
G = < a,b,c,d | a2=b28=c4=1, d2=b14, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab13, dcd-1=ab14c-1 >
Subgroups: 612 in 150 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.81C23, Dic7⋊C4, C7×C4⋊C4, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C14.C42, C2×Dic7⋊C4, C2×Dic7⋊C4, C14×C4⋊C4, (C2×C28).287D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C7⋊D4, C22×D7, C23.81C23, C4○D28, D4×D7, D4⋊2D7, Q8×D7, C2×C7⋊D4, Dic7.Q8, D14⋊Q8, C23.23D14, C28⋊2D4, Dic7⋊Q8, (C2×C28).287D4
►Regular action on 448 points
Generators in S448
(1 162)(2 163)(3 164)(4 165)(5 166)(6 167)(7 168)(8 141)(9 142)(10 143)(11 144)(12 145)(13 146)(14 147)(15 148)(16 149)(17 150)(18 151)(19 152)(20 153)(21 154)(22 155)(23 156)(24 157)(25 158)(26 159)(27 160)(28 161)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 99)(57 384)(58 385)(59 386)(60 387)(61 388)(62 389)(63 390)(64 391)(65 392)(66 365)(67 366)(68 367)(69 368)(70 369)(71 370)(72 371)(73 372)(74 373)(75 374)(76 375)(77 376)(78 377)(79 378)(80 379)(81 380)(82 381)(83 382)(84 383)(113 397)(114 398)(115 399)(116 400)(117 401)(118 402)(119 403)(120 404)(121 405)(122 406)(123 407)(124 408)(125 409)(126 410)(127 411)(128 412)(129 413)(130 414)(131 415)(132 416)(133 417)(134 418)(135 419)(136 420)(137 393)(138 394)(139 395)(140 396)(169 210)(170 211)(171 212)(172 213)(173 214)(174 215)(175 216)(176 217)(177 218)(178 219)(179 220)(180 221)(181 222)(182 223)(183 224)(184 197)(185 198)(186 199)(187 200)(188 201)(189 202)(190 203)(191 204)(192 205)(193 206)(194 207)(195 208)(196 209)(225 311)(226 312)(227 313)(228 314)(229 315)(230 316)(231 317)(232 318)(233 319)(234 320)(235 321)(236 322)(237 323)(238 324)(239 325)(240 326)(241 327)(242 328)(243 329)(244 330)(245 331)(246 332)(247 333)(248 334)(249 335)(250 336)(251 309)(252 310)(253 337)(254 338)(255 339)(256 340)(257 341)(258 342)(259 343)(260 344)(261 345)(262 346)(263 347)(264 348)(265 349)(266 350)(267 351)(268 352)(269 353)(270 354)(271 355)(272 356)(273 357)(274 358)(275 359)(276 360)(277 361)(278 362)(279 363)(280 364)(281 437)(282 438)(283 439)(284 440)(285 441)(286 442)(287 443)(288 444)(289 445)(290 446)(291 447)(292 448)(293 421)(294 422)(295 423)(296 424)(297 425)(298 426)(299 427)(300 428)(301 429)(302 430)(303 431)(304 432)(305 433)(306 434)(307 435)(308 436)
(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)
(1 322 102 195)(2 235 103 207)(3 320 104 193)(4 233 105 205)(5 318 106 191)(6 231 107 203)(7 316 108 189)(8 229 109 201)(9 314 110 187)(10 227 111 199)(11 312 112 185)(12 225 85 197)(13 310 86 183)(14 251 87 223)(15 336 88 181)(16 249 89 221)(17 334 90 179)(18 247 91 219)(19 332 92 177)(20 245 93 217)(21 330 94 175)(22 243 95 215)(23 328 96 173)(24 241 97 213)(25 326 98 171)(26 239 99 211)(27 324 100 169)(28 237 101 209)(29 210 160 238)(30 196 161 323)(31 208 162 236)(32 194 163 321)(33 206 164 234)(34 192 165 319)(35 204 166 232)(36 190 167 317)(37 202 168 230)(38 188 141 315)(39 200 142 228)(40 186 143 313)(41 198 144 226)(42 184 145 311)(43 224 146 252)(44 182 147 309)(45 222 148 250)(46 180 149 335)(47 220 150 248)(48 178 151 333)(49 218 152 246)(50 176 153 331)(51 216 154 244)(52 174 155 329)(53 214 156 242)(54 172 157 327)(55 212 158 240)(56 170 159 325)(57 424 338 118)(58 295 339 401)(59 422 340 116)(60 293 341 399)(61 448 342 114)(62 291 343 397)(63 446 344 140)(64 289 345 395)(65 444 346 138)(66 287 347 393)(67 442 348 136)(68 285 349 419)(69 440 350 134)(70 283 351 417)(71 438 352 132)(72 281 353 415)(73 436 354 130)(74 307 355 413)(75 434 356 128)(76 305 357 411)(77 432 358 126)(78 303 359 409)(79 430 360 124)(80 301 361 407)(81 428 362 122)(82 299 363 405)(83 426 364 120)(84 297 337 403)(113 389 447 259)(115 387 421 257)(117 385 423 255)(119 383 425 253)(121 381 427 279)(123 379 429 277)(125 377 431 275)(127 375 433 273)(129 373 435 271)(131 371 437 269)(133 369 439 267)(135 367 441 265)(137 365 443 263)(139 391 445 261)(254 402 384 296)(256 400 386 294)(258 398 388 292)(260 396 390 290)(262 394 392 288)(264 420 366 286)(266 418 368 284)(268 416 370 282)(270 414 372 308)(272 412 374 306)(274 410 376 304)(276 408 378 302)(278 406 380 300)(280 404 382 298)
(1 413 15 399)(2 114 16 128)(3 411 17 397)(4 140 18 126)(5 409 19 395)(6 138 20 124)(7 407 21 393)(8 136 22 122)(9 405 23 419)(10 134 24 120)(11 403 25 417)(12 132 26 118)(13 401 27 415)(14 130 28 116)(29 437 43 423)(30 294 44 308)(31 435 45 421)(32 292 46 306)(33 433 47 447)(34 290 48 304)(35 431 49 445)(36 288 50 302)(37 429 51 443)(38 286 52 300)(39 427 53 441)(40 284 54 298)(41 425 55 439)(42 282 56 296)(57 170 71 184)(58 224 72 210)(59 196 73 182)(60 222 74 208)(61 194 75 180)(62 220 76 206)(63 192 77 178)(64 218 78 204)(65 190 79 176)(66 216 80 202)(67 188 81 174)(68 214 82 200)(69 186 83 172)(70 212 84 198)(85 438 99 424)(86 295 100 281)(87 436 101 422)(88 293 102 307)(89 434 103 448)(90 291 104 305)(91 432 105 446)(92 289 106 303)(93 430 107 444)(94 287 108 301)(95 428 109 442)(96 285 110 299)(97 426 111 440)(98 283 112 297)(113 164 127 150)(115 162 129 148)(117 160 131 146)(119 158 133 144)(121 156 135 142)(123 154 137 168)(125 152 139 166)(141 420 155 406)(143 418 157 404)(145 416 159 402)(147 414 161 400)(149 412 163 398)(151 410 165 396)(153 408 167 394)(169 385 183 371)(171 383 185 369)(173 381 187 367)(175 379 189 365)(177 377 191 391)(179 375 193 389)(181 373 195 387)(197 384 211 370)(199 382 213 368)(201 380 215 366)(203 378 217 392)(205 376 219 390)(207 374 221 388)(209 372 223 386)(225 254 239 268)(226 351 240 337)(227 280 241 266)(228 349 242 363)(229 278 243 264)(230 347 244 361)(231 276 245 262)(232 345 246 359)(233 274 247 260)(234 343 248 357)(235 272 249 258)(236 341 250 355)(237 270 251 256)(238 339 252 353)(253 312 267 326)(255 310 269 324)(257 336 271 322)(259 334 273 320)(261 332 275 318)(263 330 277 316)(265 328 279 314)(309 340 323 354)(311 338 325 352)(313 364 327 350)(315 362 329 348)(317 360 331 346)(319 358 333 344)(321 356 335 342)
G:=sub<Sym(448)| (1,162)(2,163)(3,164)(4,165)(5,166)(6,167)(7,168)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,149)(17,150)(18,151)(19,152)(20,153)(21,154)(22,155)(23,156)(24,157)(25,158)(26,159)(27,160)(28,161)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,99)(57,384)(58,385)(59,386)(60,387)(61,388)(62,389)(63,390)(64,391)(65,392)(66,365)(67,366)(68,367)(69,368)(70,369)(71,370)(72,371)(73,372)(74,373)(75,374)(76,375)(77,376)(78,377)(79,378)(80,379)(81,380)(82,381)(83,382)(84,383)(113,397)(114,398)(115,399)(116,400)(117,401)(118,402)(119,403)(120,404)(121,405)(122,406)(123,407)(124,408)(125,409)(126,410)(127,411)(128,412)(129,413)(130,414)(131,415)(132,416)(133,417)(134,418)(135,419)(136,420)(137,393)(138,394)(139,395)(140,396)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)(179,220)(180,221)(181,222)(182,223)(183,224)(184,197)(185,198)(186,199)(187,200)(188,201)(189,202)(190,203)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(225,311)(226,312)(227,313)(228,314)(229,315)(230,316)(231,317)(232,318)(233,319)(234,320)(235,321)(236,322)(237,323)(238,324)(239,325)(240,326)(241,327)(242,328)(243,329)(244,330)(245,331)(246,332)(247,333)(248,334)(249,335)(250,336)(251,309)(252,310)(253,337)(254,338)(255,339)(256,340)(257,341)(258,342)(259,343)(260,344)(261,345)(262,346)(263,347)(264,348)(265,349)(266,350)(267,351)(268,352)(269,353)(270,354)(271,355)(272,356)(273,357)(274,358)(275,359)(276,360)(277,361)(278,362)(279,363)(280,364)(281,437)(282,438)(283,439)(284,440)(285,441)(286,442)(287,443)(288,444)(289,445)(290,446)(291,447)(292,448)(293,421)(294,422)(295,423)(296,424)(297,425)(298,426)(299,427)(300,428)(301,429)(302,430)(303,431)(304,432)(305,433)(306,434)(307,435)(308,436), (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), (1,322,102,195)(2,235,103,207)(3,320,104,193)(4,233,105,205)(5,318,106,191)(6,231,107,203)(7,316,108,189)(8,229,109,201)(9,314,110,187)(10,227,111,199)(11,312,112,185)(12,225,85,197)(13,310,86,183)(14,251,87,223)(15,336,88,181)(16,249,89,221)(17,334,90,179)(18,247,91,219)(19,332,92,177)(20,245,93,217)(21,330,94,175)(22,243,95,215)(23,328,96,173)(24,241,97,213)(25,326,98,171)(26,239,99,211)(27,324,100,169)(28,237,101,209)(29,210,160,238)(30,196,161,323)(31,208,162,236)(32,194,163,321)(33,206,164,234)(34,192,165,319)(35,204,166,232)(36,190,167,317)(37,202,168,230)(38,188,141,315)(39,200,142,228)(40,186,143,313)(41,198,144,226)(42,184,145,311)(43,224,146,252)(44,182,147,309)(45,222,148,250)(46,180,149,335)(47,220,150,248)(48,178,151,333)(49,218,152,246)(50,176,153,331)(51,216,154,244)(52,174,155,329)(53,214,156,242)(54,172,157,327)(55,212,158,240)(56,170,159,325)(57,424,338,118)(58,295,339,401)(59,422,340,116)(60,293,341,399)(61,448,342,114)(62,291,343,397)(63,446,344,140)(64,289,345,395)(65,444,346,138)(66,287,347,393)(67,442,348,136)(68,285,349,419)(69,440,350,134)(70,283,351,417)(71,438,352,132)(72,281,353,415)(73,436,354,130)(74,307,355,413)(75,434,356,128)(76,305,357,411)(77,432,358,126)(78,303,359,409)(79,430,360,124)(80,301,361,407)(81,428,362,122)(82,299,363,405)(83,426,364,120)(84,297,337,403)(113,389,447,259)(115,387,421,257)(117,385,423,255)(119,383,425,253)(121,381,427,279)(123,379,429,277)(125,377,431,275)(127,375,433,273)(129,373,435,271)(131,371,437,269)(133,369,439,267)(135,367,441,265)(137,365,443,263)(139,391,445,261)(254,402,384,296)(256,400,386,294)(258,398,388,292)(260,396,390,290)(262,394,392,288)(264,420,366,286)(266,418,368,284)(268,416,370,282)(270,414,372,308)(272,412,374,306)(274,410,376,304)(276,408,378,302)(278,406,380,300)(280,404,382,298), (1,413,15,399)(2,114,16,128)(3,411,17,397)(4,140,18,126)(5,409,19,395)(6,138,20,124)(7,407,21,393)(8,136,22,122)(9,405,23,419)(10,134,24,120)(11,403,25,417)(12,132,26,118)(13,401,27,415)(14,130,28,116)(29,437,43,423)(30,294,44,308)(31,435,45,421)(32,292,46,306)(33,433,47,447)(34,290,48,304)(35,431,49,445)(36,288,50,302)(37,429,51,443)(38,286,52,300)(39,427,53,441)(40,284,54,298)(41,425,55,439)(42,282,56,296)(57,170,71,184)(58,224,72,210)(59,196,73,182)(60,222,74,208)(61,194,75,180)(62,220,76,206)(63,192,77,178)(64,218,78,204)(65,190,79,176)(66,216,80,202)(67,188,81,174)(68,214,82,200)(69,186,83,172)(70,212,84,198)(85,438,99,424)(86,295,100,281)(87,436,101,422)(88,293,102,307)(89,434,103,448)(90,291,104,305)(91,432,105,446)(92,289,106,303)(93,430,107,444)(94,287,108,301)(95,428,109,442)(96,285,110,299)(97,426,111,440)(98,283,112,297)(113,164,127,150)(115,162,129,148)(117,160,131,146)(119,158,133,144)(121,156,135,142)(123,154,137,168)(125,152,139,166)(141,420,155,406)(143,418,157,404)(145,416,159,402)(147,414,161,400)(149,412,163,398)(151,410,165,396)(153,408,167,394)(169,385,183,371)(171,383,185,369)(173,381,187,367)(175,379,189,365)(177,377,191,391)(179,375,193,389)(181,373,195,387)(197,384,211,370)(199,382,213,368)(201,380,215,366)(203,378,217,392)(205,376,219,390)(207,374,221,388)(209,372,223,386)(225,254,239,268)(226,351,240,337)(227,280,241,266)(228,349,242,363)(229,278,243,264)(230,347,244,361)(231,276,245,262)(232,345,246,359)(233,274,247,260)(234,343,248,357)(235,272,249,258)(236,341,250,355)(237,270,251,256)(238,339,252,353)(253,312,267,326)(255,310,269,324)(257,336,271,322)(259,334,273,320)(261,332,275,318)(263,330,277,316)(265,328,279,314)(309,340,323,354)(311,338,325,352)(313,364,327,350)(315,362,329,348)(317,360,331,346)(319,358,333,344)(321,356,335,342)>;
G:=Group( (1,162)(2,163)(3,164)(4,165)(5,166)(6,167)(7,168)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,149)(17,150)(18,151)(19,152)(20,153)(21,154)(22,155)(23,156)(24,157)(25,158)(26,159)(27,160)(28,161)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,99)(57,384)(58,385)(59,386)(60,387)(61,388)(62,389)(63,390)(64,391)(65,392)(66,365)(67,366)(68,367)(69,368)(70,369)(71,370)(72,371)(73,372)(74,373)(75,374)(76,375)(77,376)(78,377)(79,378)(80,379)(81,380)(82,381)(83,382)(84,383)(113,397)(114,398)(115,399)(116,400)(117,401)(118,402)(119,403)(120,404)(121,405)(122,406)(123,407)(124,408)(125,409)(126,410)(127,411)(128,412)(129,413)(130,414)(131,415)(132,416)(133,417)(134,418)(135,419)(136,420)(137,393)(138,394)(139,395)(140,396)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)(179,220)(180,221)(181,222)(182,223)(183,224)(184,197)(185,198)(186,199)(187,200)(188,201)(189,202)(190,203)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(225,311)(226,312)(227,313)(228,314)(229,315)(230,316)(231,317)(232,318)(233,319)(234,320)(235,321)(236,322)(237,323)(238,324)(239,325)(240,326)(241,327)(242,328)(243,329)(244,330)(245,331)(246,332)(247,333)(248,334)(249,335)(250,336)(251,309)(252,310)(253,337)(254,338)(255,339)(256,340)(257,341)(258,342)(259,343)(260,344)(261,345)(262,346)(263,347)(264,348)(265,349)(266,350)(267,351)(268,352)(269,353)(270,354)(271,355)(272,356)(273,357)(274,358)(275,359)(276,360)(277,361)(278,362)(279,363)(280,364)(281,437)(282,438)(283,439)(284,440)(285,441)(286,442)(287,443)(288,444)(289,445)(290,446)(291,447)(292,448)(293,421)(294,422)(295,423)(296,424)(297,425)(298,426)(299,427)(300,428)(301,429)(302,430)(303,431)(304,432)(305,433)(306,434)(307,435)(308,436), (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), (1,322,102,195)(2,235,103,207)(3,320,104,193)(4,233,105,205)(5,318,106,191)(6,231,107,203)(7,316,108,189)(8,229,109,201)(9,314,110,187)(10,227,111,199)(11,312,112,185)(12,225,85,197)(13,310,86,183)(14,251,87,223)(15,336,88,181)(16,249,89,221)(17,334,90,179)(18,247,91,219)(19,332,92,177)(20,245,93,217)(21,330,94,175)(22,243,95,215)(23,328,96,173)(24,241,97,213)(25,326,98,171)(26,239,99,211)(27,324,100,169)(28,237,101,209)(29,210,160,238)(30,196,161,323)(31,208,162,236)(32,194,163,321)(33,206,164,234)(34,192,165,319)(35,204,166,232)(36,190,167,317)(37,202,168,230)(38,188,141,315)(39,200,142,228)(40,186,143,313)(41,198,144,226)(42,184,145,311)(43,224,146,252)(44,182,147,309)(45,222,148,250)(46,180,149,335)(47,220,150,248)(48,178,151,333)(49,218,152,246)(50,176,153,331)(51,216,154,244)(52,174,155,329)(53,214,156,242)(54,172,157,327)(55,212,158,240)(56,170,159,325)(57,424,338,118)(58,295,339,401)(59,422,340,116)(60,293,341,399)(61,448,342,114)(62,291,343,397)(63,446,344,140)(64,289,345,395)(65,444,346,138)(66,287,347,393)(67,442,348,136)(68,285,349,419)(69,440,350,134)(70,283,351,417)(71,438,352,132)(72,281,353,415)(73,436,354,130)(74,307,355,413)(75,434,356,128)(76,305,357,411)(77,432,358,126)(78,303,359,409)(79,430,360,124)(80,301,361,407)(81,428,362,122)(82,299,363,405)(83,426,364,120)(84,297,337,403)(113,389,447,259)(115,387,421,257)(117,385,423,255)(119,383,425,253)(121,381,427,279)(123,379,429,277)(125,377,431,275)(127,375,433,273)(129,373,435,271)(131,371,437,269)(133,369,439,267)(135,367,441,265)(137,365,443,263)(139,391,445,261)(254,402,384,296)(256,400,386,294)(258,398,388,292)(260,396,390,290)(262,394,392,288)(264,420,366,286)(266,418,368,284)(268,416,370,282)(270,414,372,308)(272,412,374,306)(274,410,376,304)(276,408,378,302)(278,406,380,300)(280,404,382,298), (1,413,15,399)(2,114,16,128)(3,411,17,397)(4,140,18,126)(5,409,19,395)(6,138,20,124)(7,407,21,393)(8,136,22,122)(9,405,23,419)(10,134,24,120)(11,403,25,417)(12,132,26,118)(13,401,27,415)(14,130,28,116)(29,437,43,423)(30,294,44,308)(31,435,45,421)(32,292,46,306)(33,433,47,447)(34,290,48,304)(35,431,49,445)(36,288,50,302)(37,429,51,443)(38,286,52,300)(39,427,53,441)(40,284,54,298)(41,425,55,439)(42,282,56,296)(57,170,71,184)(58,224,72,210)(59,196,73,182)(60,222,74,208)(61,194,75,180)(62,220,76,206)(63,192,77,178)(64,218,78,204)(65,190,79,176)(66,216,80,202)(67,188,81,174)(68,214,82,200)(69,186,83,172)(70,212,84,198)(85,438,99,424)(86,295,100,281)(87,436,101,422)(88,293,102,307)(89,434,103,448)(90,291,104,305)(91,432,105,446)(92,289,106,303)(93,430,107,444)(94,287,108,301)(95,428,109,442)(96,285,110,299)(97,426,111,440)(98,283,112,297)(113,164,127,150)(115,162,129,148)(117,160,131,146)(119,158,133,144)(121,156,135,142)(123,154,137,168)(125,152,139,166)(141,420,155,406)(143,418,157,404)(145,416,159,402)(147,414,161,400)(149,412,163,398)(151,410,165,396)(153,408,167,394)(169,385,183,371)(171,383,185,369)(173,381,187,367)(175,379,189,365)(177,377,191,391)(179,375,193,389)(181,373,195,387)(197,384,211,370)(199,382,213,368)(201,380,215,366)(203,378,217,392)(205,376,219,390)(207,374,221,388)(209,372,223,386)(225,254,239,268)(226,351,240,337)(227,280,241,266)(228,349,242,363)(229,278,243,264)(230,347,244,361)(231,276,245,262)(232,345,246,359)(233,274,247,260)(234,343,248,357)(235,272,249,258)(236,341,250,355)(237,270,251,256)(238,339,252,353)(253,312,267,326)(255,310,269,324)(257,336,271,322)(259,334,273,320)(261,332,275,318)(263,330,277,316)(265,328,279,314)(309,340,323,354)(311,338,325,352)(313,364,327,350)(315,362,329,348)(317,360,331,346)(319,358,333,344)(321,356,335,342) );
G=PermutationGroup([[(1,162),(2,163),(3,164),(4,165),(5,166),(6,167),(7,168),(8,141),(9,142),(10,143),(11,144),(12,145),(13,146),(14,147),(15,148),(16,149),(17,150),(18,151),(19,152),(20,153),(21,154),(22,155),(23,156),(24,157),(25,158),(26,159),(27,160),(28,161),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,96),(54,97),(55,98),(56,99),(57,384),(58,385),(59,386),(60,387),(61,388),(62,389),(63,390),(64,391),(65,392),(66,365),(67,366),(68,367),(69,368),(70,369),(71,370),(72,371),(73,372),(74,373),(75,374),(76,375),(77,376),(78,377),(79,378),(80,379),(81,380),(82,381),(83,382),(84,383),(113,397),(114,398),(115,399),(116,400),(117,401),(118,402),(119,403),(120,404),(121,405),(122,406),(123,407),(124,408),(125,409),(126,410),(127,411),(128,412),(129,413),(130,414),(131,415),(132,416),(133,417),(134,418),(135,419),(136,420),(137,393),(138,394),(139,395),(140,396),(169,210),(170,211),(171,212),(172,213),(173,214),(174,215),(175,216),(176,217),(177,218),(178,219),(179,220),(180,221),(181,222),(182,223),(183,224),(184,197),(185,198),(186,199),(187,200),(188,201),(189,202),(190,203),(191,204),(192,205),(193,206),(194,207),(195,208),(196,209),(225,311),(226,312),(227,313),(228,314),(229,315),(230,316),(231,317),(232,318),(233,319),(234,320),(235,321),(236,322),(237,323),(238,324),(239,325),(240,326),(241,327),(242,328),(243,329),(244,330),(245,331),(246,332),(247,333),(248,334),(249,335),(250,336),(251,309),(252,310),(253,337),(254,338),(255,339),(256,340),(257,341),(258,342),(259,343),(260,344),(261,345),(262,346),(263,347),(264,348),(265,349),(266,350),(267,351),(268,352),(269,353),(270,354),(271,355),(272,356),(273,357),(274,358),(275,359),(276,360),(277,361),(278,362),(279,363),(280,364),(281,437),(282,438),(283,439),(284,440),(285,441),(286,442),(287,443),(288,444),(289,445),(290,446),(291,447),(292,448),(293,421),(294,422),(295,423),(296,424),(297,425),(298,426),(299,427),(300,428),(301,429),(302,430),(303,431),(304,432),(305,433),(306,434),(307,435),(308,436)], [(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)], [(1,322,102,195),(2,235,103,207),(3,320,104,193),(4,233,105,205),(5,318,106,191),(6,231,107,203),(7,316,108,189),(8,229,109,201),(9,314,110,187),(10,227,111,199),(11,312,112,185),(12,225,85,197),(13,310,86,183),(14,251,87,223),(15,336,88,181),(16,249,89,221),(17,334,90,179),(18,247,91,219),(19,332,92,177),(20,245,93,217),(21,330,94,175),(22,243,95,215),(23,328,96,173),(24,241,97,213),(25,326,98,171),(26,239,99,211),(27,324,100,169),(28,237,101,209),(29,210,160,238),(30,196,161,323),(31,208,162,236),(32,194,163,321),(33,206,164,234),(34,192,165,319),(35,204,166,232),(36,190,167,317),(37,202,168,230),(38,188,141,315),(39,200,142,228),(40,186,143,313),(41,198,144,226),(42,184,145,311),(43,224,146,252),(44,182,147,309),(45,222,148,250),(46,180,149,335),(47,220,150,248),(48,178,151,333),(49,218,152,246),(50,176,153,331),(51,216,154,244),(52,174,155,329),(53,214,156,242),(54,172,157,327),(55,212,158,240),(56,170,159,325),(57,424,338,118),(58,295,339,401),(59,422,340,116),(60,293,341,399),(61,448,342,114),(62,291,343,397),(63,446,344,140),(64,289,345,395),(65,444,346,138),(66,287,347,393),(67,442,348,136),(68,285,349,419),(69,440,350,134),(70,283,351,417),(71,438,352,132),(72,281,353,415),(73,436,354,130),(74,307,355,413),(75,434,356,128),(76,305,357,411),(77,432,358,126),(78,303,359,409),(79,430,360,124),(80,301,361,407),(81,428,362,122),(82,299,363,405),(83,426,364,120),(84,297,337,403),(113,389,447,259),(115,387,421,257),(117,385,423,255),(119,383,425,253),(121,381,427,279),(123,379,429,277),(125,377,431,275),(127,375,433,273),(129,373,435,271),(131,371,437,269),(133,369,439,267),(135,367,441,265),(137,365,443,263),(139,391,445,261),(254,402,384,296),(256,400,386,294),(258,398,388,292),(260,396,390,290),(262,394,392,288),(264,420,366,286),(266,418,368,284),(268,416,370,282),(270,414,372,308),(272,412,374,306),(274,410,376,304),(276,408,378,302),(278,406,380,300),(280,404,382,298)], [(1,413,15,399),(2,114,16,128),(3,411,17,397),(4,140,18,126),(5,409,19,395),(6,138,20,124),(7,407,21,393),(8,136,22,122),(9,405,23,419),(10,134,24,120),(11,403,25,417),(12,132,26,118),(13,401,27,415),(14,130,28,116),(29,437,43,423),(30,294,44,308),(31,435,45,421),(32,292,46,306),(33,433,47,447),(34,290,48,304),(35,431,49,445),(36,288,50,302),(37,429,51,443),(38,286,52,300),(39,427,53,441),(40,284,54,298),(41,425,55,439),(42,282,56,296),(57,170,71,184),(58,224,72,210),(59,196,73,182),(60,222,74,208),(61,194,75,180),(62,220,76,206),(63,192,77,178),(64,218,78,204),(65,190,79,176),(66,216,80,202),(67,188,81,174),(68,214,82,200),(69,186,83,172),(70,212,84,198),(85,438,99,424),(86,295,100,281),(87,436,101,422),(88,293,102,307),(89,434,103,448),(90,291,104,305),(91,432,105,446),(92,289,106,303),(93,430,107,444),(94,287,108,301),(95,428,109,442),(96,285,110,299),(97,426,111,440),(98,283,112,297),(113,164,127,150),(115,162,129,148),(117,160,131,146),(119,158,133,144),(121,156,135,142),(123,154,137,168),(125,152,139,166),(141,420,155,406),(143,418,157,404),(145,416,159,402),(147,414,161,400),(149,412,163,398),(151,410,165,396),(153,408,167,394),(169,385,183,371),(171,383,185,369),(173,381,187,367),(175,379,189,365),(177,377,191,391),(179,375,193,389),(181,373,195,387),(197,384,211,370),(199,382,213,368),(201,380,215,366),(203,378,217,392),(205,376,219,390),(207,374,221,388),(209,372,223,386),(225,254,239,268),(226,351,240,337),(227,280,241,266),(228,349,242,363),(229,278,243,264),(230,347,244,361),(231,276,245,262),(232,345,246,359),(233,274,247,260),(234,343,248,357),(235,272,249,258),(236,341,250,355),(237,270,251,256),(238,339,252,353),(253,312,267,326),(255,310,269,324),(257,336,271,322),(259,334,273,320),(261,332,275,318),(263,330,277,316),(265,328,279,314),(309,340,323,354),(311,338,325,352),(313,364,327,350),(315,362,329,348),(317,360,331,346),(319,358,333,344),(321,356,335,342)]])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | D4 | Q8 | D4 | D7 | C4○D4 | D14 | C7⋊D4 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 |
| kernel | (C2×C28).287D4 | C14.C42 | C2×Dic7⋊C4 | C14×C4⋊C4 | C2×Dic7 | C2×Dic7 | C2×C28 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 4 | 2 | 3 | 6 | 9 | 12 | 24 | 3 | 3 | 6 |
Matrix representation of (C2×C28).287D4 ►in GL6(𝔽29)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 19 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 2 | 0 | 0 |
| 0 | 0 | 27 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 23 |
| 0 | 0 | 0 | 0 | 18 | 7 |
| 27 | 28 | 0 | 0 | 0 | 0 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 25 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 24 | 12 | 0 | 0 | 0 | 0 |
| 27 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 17 | 0 | 0 |
| 0 | 0 | 24 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 6 |
| 0 | 0 | 0 | 0 | 11 | 22 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,19,0,0,0,0,2,22,0,0,0,0,0,0,24,27,0,0,0,0,2,1,0,0,0,0,0,0,22,18,0,0,0,0,23,7],[27,5,0,0,0,0,28,2,0,0,0,0,0,0,21,26,0,0,0,0,21,8,0,0,0,0,0,0,28,0,0,0,0,0,25,1],[24,27,0,0,0,0,12,5,0,0,0,0,0,0,12,24,0,0,0,0,17,17,0,0,0,0,0,0,7,11,0,0,0,0,6,22] >;
(C2×C28).287D4 in GAP, Magma, Sage, TeX
(C_2\times C_{28})._{287}D_4 % in TeX
G:=Group("(C2xC28).287D4"); // GroupNames label
G:=SmallGroup(448,514);
// by ID
G=gap.SmallGroup(448,514);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,232,254,387,100,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=b^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=a*b^13,d*c*d^-1=a*b^14*c^-1>;
// generators/relations