metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).28D4, (C2×C4).17D28, (C2×Dic7).3Q8, C22.41(Q8×D7), (C22×C4).12D14, C22.78(C2×D28), C2.7(D14⋊2Q8), C2.5(C4.D28), C14.1(C4.4D4), C14.23(C22⋊Q8), (C22×C28).9C22, C22.86(C4○D28), C2.10(Dic7.Q8), C14.C42.7C2, C14.13(C42.C2), C2.C42.14D7, C23.355(C22×D7), C14.10(C42⋊2C2), C22.84(D4⋊2D7), (C22×C14).284C23, C7⋊2(C23.83C23), C2.6(C22.D28), C2.9(C23.D14), C14.6(C22.D4), (C22×Dic7).9C22, (C2×C14).94(C2×D4), (C2×C4⋊Dic7).8C2, (C2×C14).65(C2×Q8), (C2×Dic7⋊C4).19C2, (C2×C14).128(C4○D4), (C7×C2.C42).10C2, SmallGroup(448,193)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28).28D4
G = < a,b,c,d | a2=b4=c28=1, d2=ab2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=ab-1, dcd-1=b2c-1 >
Subgroups: 572 in 134 conjugacy classes, 57 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.C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.83C23, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C7×C2.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, (C2×C28).28D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, D28, C22×D7, C23.83C23, C2×D28, C4○D28, D4⋊2D7, Q8×D7, C4.D28, C23.D14, C22.D28, Dic7.Q8, D14⋊2Q8, (C2×C28).28D4
►Regular action on 448 points
Generators in S448
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 201)(30 202)(31 203)(32 204)(33 205)(34 206)(35 207)(36 208)(37 209)(38 210)(39 211)(40 212)(41 213)(42 214)(43 215)(44 216)(45 217)(46 218)(47 219)(48 220)(49 221)(50 222)(51 223)(52 224)(53 197)(54 198)(55 199)(56 200)(57 311)(58 312)(59 313)(60 314)(61 315)(62 316)(63 317)(64 318)(65 319)(66 320)(67 321)(68 322)(69 323)(70 324)(71 325)(72 326)(73 327)(74 328)(75 329)(76 330)(77 331)(78 332)(79 333)(80 334)(81 335)(82 336)(83 309)(84 310)(113 168)(114 141)(115 142)(116 143)(117 144)(118 145)(119 146)(120 147)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 161)(135 162)(136 163)(137 164)(138 165)(139 166)(140 167)(169 370)(170 371)(171 372)(172 373)(173 374)(174 375)(175 376)(176 377)(177 378)(178 379)(179 380)(180 381)(181 382)(182 383)(183 384)(184 385)(185 386)(186 387)(187 388)(188 389)(189 390)(190 391)(191 392)(192 365)(193 366)(194 367)(195 368)(196 369)(225 447)(226 448)(227 421)(228 422)(229 423)(230 424)(231 425)(232 426)(233 427)(234 428)(235 429)(236 430)(237 431)(238 432)(239 433)(240 434)(241 435)(242 436)(243 437)(244 438)(245 439)(246 440)(247 441)(248 442)(249 443)(250 444)(251 445)(252 446)(253 283)(254 284)(255 285)(256 286)(257 287)(258 288)(259 289)(260 290)(261 291)(262 292)(263 293)(264 294)(265 295)(266 296)(267 297)(268 298)(269 299)(270 300)(271 301)(272 302)(273 303)(274 304)(275 305)(276 306)(277 307)(278 308)(279 281)(280 282)(337 395)(338 396)(339 397)(340 398)(341 399)(342 400)(343 401)(344 402)(345 403)(346 404)(347 405)(348 406)(349 407)(350 408)(351 409)(352 410)(353 411)(354 412)(355 413)(356 414)(357 415)(358 416)(359 417)(360 418)(361 419)(362 420)(363 393)(364 394)
(1 146 44 388)(2 120 45 188)(3 148 46 390)(4 122 47 190)(5 150 48 392)(6 124 49 192)(7 152 50 366)(8 126 51 194)(9 154 52 368)(10 128 53 196)(11 156 54 370)(12 130 55 170)(13 158 56 372)(14 132 29 172)(15 160 30 374)(16 134 31 174)(17 162 32 376)(18 136 33 176)(19 164 34 378)(20 138 35 178)(21 166 36 380)(22 140 37 180)(23 168 38 382)(24 114 39 182)(25 142 40 384)(26 116 41 184)(27 144 42 386)(28 118 43 186)(57 435 270 418)(58 242 271 361)(59 437 272 420)(60 244 273 363)(61 439 274 394)(62 246 275 337)(63 441 276 396)(64 248 277 339)(65 443 278 398)(66 250 279 341)(67 445 280 400)(68 252 253 343)(69 447 254 402)(70 226 255 345)(71 421 256 404)(72 228 257 347)(73 423 258 406)(74 230 259 349)(75 425 260 408)(76 232 261 351)(77 427 262 410)(78 234 263 353)(79 429 264 412)(80 236 265 355)(81 431 266 414)(82 238 267 357)(83 433 268 416)(84 240 269 359)(85 119 216 187)(86 147 217 389)(87 121 218 189)(88 149 219 391)(89 123 220 191)(90 151 221 365)(91 125 222 193)(92 153 223 367)(93 127 224 195)(94 155 197 369)(95 129 198 169)(96 157 199 371)(97 131 200 171)(98 159 201 373)(99 133 202 173)(100 161 203 375)(101 135 204 175)(102 163 205 377)(103 137 206 177)(104 165 207 379)(105 139 208 179)(106 167 209 381)(107 113 210 181)(108 141 211 383)(109 115 212 183)(110 143 213 385)(111 117 214 185)(112 145 215 387)(225 284 344 323)(227 286 346 325)(229 288 348 327)(231 290 350 329)(233 292 352 331)(235 294 354 333)(237 296 356 335)(239 298 358 309)(241 300 360 311)(243 302 362 313)(245 304 364 315)(247 306 338 317)(249 308 340 319)(251 282 342 321)(281 399 320 444)(283 401 322 446)(285 403 324 448)(287 405 326 422)(289 407 328 424)(291 409 330 426)(293 411 332 428)(295 413 334 430)(297 415 336 432)(299 417 310 434)(301 419 312 436)(303 393 314 438)(305 395 316 440)(307 397 318 442)
(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)
(1 444 216 341)(2 398 217 249)(3 442 218 339)(4 396 219 247)(5 440 220 337)(6 394 221 245)(7 438 222 363)(8 420 223 243)(9 436 224 361)(10 418 197 241)(11 434 198 359)(12 416 199 239)(13 432 200 357)(14 414 201 237)(15 430 202 355)(16 412 203 235)(17 428 204 353)(18 410 205 233)(19 426 206 351)(20 408 207 231)(21 424 208 349)(22 406 209 229)(23 422 210 347)(24 404 211 227)(25 448 212 345)(26 402 213 225)(27 446 214 343)(28 400 215 251)(29 431 98 356)(30 413 99 236)(31 429 100 354)(32 411 101 234)(33 427 102 352)(34 409 103 232)(35 425 104 350)(36 407 105 230)(37 423 106 348)(38 405 107 228)(39 421 108 346)(40 403 109 226)(41 447 110 344)(42 401 111 252)(43 445 112 342)(44 399 85 250)(45 443 86 340)(46 397 87 248)(47 441 88 338)(48 395 89 246)(49 439 90 364)(50 393 91 244)(51 437 92 362)(52 419 93 242)(53 435 94 360)(54 417 95 240)(55 433 96 358)(56 415 97 238)(57 128 300 369)(58 195 301 154)(59 126 302 367)(60 193 303 152)(61 124 304 365)(62 191 305 150)(63 122 306 391)(64 189 307 148)(65 120 308 389)(66 187 281 146)(67 118 282 387)(68 185 283 144)(69 116 284 385)(70 183 285 142)(71 114 286 383)(72 181 287 168)(73 140 288 381)(74 179 289 166)(75 138 290 379)(76 177 291 164)(77 136 292 377)(78 175 293 162)(79 134 294 375)(80 173 295 160)(81 132 296 373)(82 171 297 158)(83 130 298 371)(84 169 299 156)(113 326 382 257)(115 324 384 255)(117 322 386 253)(119 320 388 279)(121 318 390 277)(123 316 392 275)(125 314 366 273)(127 312 368 271)(129 310 370 269)(131 336 372 267)(133 334 374 265)(135 332 376 263)(137 330 378 261)(139 328 380 259)(141 256 182 325)(143 254 184 323)(145 280 186 321)(147 278 188 319)(149 276 190 317)(151 274 192 315)(153 272 194 313)(155 270 196 311)(157 268 170 309)(159 266 172 335)(161 264 174 333)(163 262 176 331)(165 260 178 329)(167 258 180 327)
G:=sub<Sym(448)| (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,201)(30,202)(31,203)(32,204)(33,205)(34,206)(35,207)(36,208)(37,209)(38,210)(39,211)(40,212)(41,213)(42,214)(43,215)(44,216)(45,217)(46,218)(47,219)(48,220)(49,221)(50,222)(51,223)(52,224)(53,197)(54,198)(55,199)(56,200)(57,311)(58,312)(59,313)(60,314)(61,315)(62,316)(63,317)(64,318)(65,319)(66,320)(67,321)(68,322)(69,323)(70,324)(71,325)(72,326)(73,327)(74,328)(75,329)(76,330)(77,331)(78,332)(79,333)(80,334)(81,335)(82,336)(83,309)(84,310)(113,168)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(138,165)(139,166)(140,167)(169,370)(170,371)(171,372)(172,373)(173,374)(174,375)(175,376)(176,377)(177,378)(178,379)(179,380)(180,381)(181,382)(182,383)(183,384)(184,385)(185,386)(186,387)(187,388)(188,389)(189,390)(190,391)(191,392)(192,365)(193,366)(194,367)(195,368)(196,369)(225,447)(226,448)(227,421)(228,422)(229,423)(230,424)(231,425)(232,426)(233,427)(234,428)(235,429)(236,430)(237,431)(238,432)(239,433)(240,434)(241,435)(242,436)(243,437)(244,438)(245,439)(246,440)(247,441)(248,442)(249,443)(250,444)(251,445)(252,446)(253,283)(254,284)(255,285)(256,286)(257,287)(258,288)(259,289)(260,290)(261,291)(262,292)(263,293)(264,294)(265,295)(266,296)(267,297)(268,298)(269,299)(270,300)(271,301)(272,302)(273,303)(274,304)(275,305)(276,306)(277,307)(278,308)(279,281)(280,282)(337,395)(338,396)(339,397)(340,398)(341,399)(342,400)(343,401)(344,402)(345,403)(346,404)(347,405)(348,406)(349,407)(350,408)(351,409)(352,410)(353,411)(354,412)(355,413)(356,414)(357,415)(358,416)(359,417)(360,418)(361,419)(362,420)(363,393)(364,394), (1,146,44,388)(2,120,45,188)(3,148,46,390)(4,122,47,190)(5,150,48,392)(6,124,49,192)(7,152,50,366)(8,126,51,194)(9,154,52,368)(10,128,53,196)(11,156,54,370)(12,130,55,170)(13,158,56,372)(14,132,29,172)(15,160,30,374)(16,134,31,174)(17,162,32,376)(18,136,33,176)(19,164,34,378)(20,138,35,178)(21,166,36,380)(22,140,37,180)(23,168,38,382)(24,114,39,182)(25,142,40,384)(26,116,41,184)(27,144,42,386)(28,118,43,186)(57,435,270,418)(58,242,271,361)(59,437,272,420)(60,244,273,363)(61,439,274,394)(62,246,275,337)(63,441,276,396)(64,248,277,339)(65,443,278,398)(66,250,279,341)(67,445,280,400)(68,252,253,343)(69,447,254,402)(70,226,255,345)(71,421,256,404)(72,228,257,347)(73,423,258,406)(74,230,259,349)(75,425,260,408)(76,232,261,351)(77,427,262,410)(78,234,263,353)(79,429,264,412)(80,236,265,355)(81,431,266,414)(82,238,267,357)(83,433,268,416)(84,240,269,359)(85,119,216,187)(86,147,217,389)(87,121,218,189)(88,149,219,391)(89,123,220,191)(90,151,221,365)(91,125,222,193)(92,153,223,367)(93,127,224,195)(94,155,197,369)(95,129,198,169)(96,157,199,371)(97,131,200,171)(98,159,201,373)(99,133,202,173)(100,161,203,375)(101,135,204,175)(102,163,205,377)(103,137,206,177)(104,165,207,379)(105,139,208,179)(106,167,209,381)(107,113,210,181)(108,141,211,383)(109,115,212,183)(110,143,213,385)(111,117,214,185)(112,145,215,387)(225,284,344,323)(227,286,346,325)(229,288,348,327)(231,290,350,329)(233,292,352,331)(235,294,354,333)(237,296,356,335)(239,298,358,309)(241,300,360,311)(243,302,362,313)(245,304,364,315)(247,306,338,317)(249,308,340,319)(251,282,342,321)(281,399,320,444)(283,401,322,446)(285,403,324,448)(287,405,326,422)(289,407,328,424)(291,409,330,426)(293,411,332,428)(295,413,334,430)(297,415,336,432)(299,417,310,434)(301,419,312,436)(303,393,314,438)(305,395,316,440)(307,397,318,442), (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), (1,444,216,341)(2,398,217,249)(3,442,218,339)(4,396,219,247)(5,440,220,337)(6,394,221,245)(7,438,222,363)(8,420,223,243)(9,436,224,361)(10,418,197,241)(11,434,198,359)(12,416,199,239)(13,432,200,357)(14,414,201,237)(15,430,202,355)(16,412,203,235)(17,428,204,353)(18,410,205,233)(19,426,206,351)(20,408,207,231)(21,424,208,349)(22,406,209,229)(23,422,210,347)(24,404,211,227)(25,448,212,345)(26,402,213,225)(27,446,214,343)(28,400,215,251)(29,431,98,356)(30,413,99,236)(31,429,100,354)(32,411,101,234)(33,427,102,352)(34,409,103,232)(35,425,104,350)(36,407,105,230)(37,423,106,348)(38,405,107,228)(39,421,108,346)(40,403,109,226)(41,447,110,344)(42,401,111,252)(43,445,112,342)(44,399,85,250)(45,443,86,340)(46,397,87,248)(47,441,88,338)(48,395,89,246)(49,439,90,364)(50,393,91,244)(51,437,92,362)(52,419,93,242)(53,435,94,360)(54,417,95,240)(55,433,96,358)(56,415,97,238)(57,128,300,369)(58,195,301,154)(59,126,302,367)(60,193,303,152)(61,124,304,365)(62,191,305,150)(63,122,306,391)(64,189,307,148)(65,120,308,389)(66,187,281,146)(67,118,282,387)(68,185,283,144)(69,116,284,385)(70,183,285,142)(71,114,286,383)(72,181,287,168)(73,140,288,381)(74,179,289,166)(75,138,290,379)(76,177,291,164)(77,136,292,377)(78,175,293,162)(79,134,294,375)(80,173,295,160)(81,132,296,373)(82,171,297,158)(83,130,298,371)(84,169,299,156)(113,326,382,257)(115,324,384,255)(117,322,386,253)(119,320,388,279)(121,318,390,277)(123,316,392,275)(125,314,366,273)(127,312,368,271)(129,310,370,269)(131,336,372,267)(133,334,374,265)(135,332,376,263)(137,330,378,261)(139,328,380,259)(141,256,182,325)(143,254,184,323)(145,280,186,321)(147,278,188,319)(149,276,190,317)(151,274,192,315)(153,272,194,313)(155,270,196,311)(157,268,170,309)(159,266,172,335)(161,264,174,333)(163,262,176,331)(165,260,178,329)(167,258,180,327)>;
G:=Group( (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,201)(30,202)(31,203)(32,204)(33,205)(34,206)(35,207)(36,208)(37,209)(38,210)(39,211)(40,212)(41,213)(42,214)(43,215)(44,216)(45,217)(46,218)(47,219)(48,220)(49,221)(50,222)(51,223)(52,224)(53,197)(54,198)(55,199)(56,200)(57,311)(58,312)(59,313)(60,314)(61,315)(62,316)(63,317)(64,318)(65,319)(66,320)(67,321)(68,322)(69,323)(70,324)(71,325)(72,326)(73,327)(74,328)(75,329)(76,330)(77,331)(78,332)(79,333)(80,334)(81,335)(82,336)(83,309)(84,310)(113,168)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(138,165)(139,166)(140,167)(169,370)(170,371)(171,372)(172,373)(173,374)(174,375)(175,376)(176,377)(177,378)(178,379)(179,380)(180,381)(181,382)(182,383)(183,384)(184,385)(185,386)(186,387)(187,388)(188,389)(189,390)(190,391)(191,392)(192,365)(193,366)(194,367)(195,368)(196,369)(225,447)(226,448)(227,421)(228,422)(229,423)(230,424)(231,425)(232,426)(233,427)(234,428)(235,429)(236,430)(237,431)(238,432)(239,433)(240,434)(241,435)(242,436)(243,437)(244,438)(245,439)(246,440)(247,441)(248,442)(249,443)(250,444)(251,445)(252,446)(253,283)(254,284)(255,285)(256,286)(257,287)(258,288)(259,289)(260,290)(261,291)(262,292)(263,293)(264,294)(265,295)(266,296)(267,297)(268,298)(269,299)(270,300)(271,301)(272,302)(273,303)(274,304)(275,305)(276,306)(277,307)(278,308)(279,281)(280,282)(337,395)(338,396)(339,397)(340,398)(341,399)(342,400)(343,401)(344,402)(345,403)(346,404)(347,405)(348,406)(349,407)(350,408)(351,409)(352,410)(353,411)(354,412)(355,413)(356,414)(357,415)(358,416)(359,417)(360,418)(361,419)(362,420)(363,393)(364,394), (1,146,44,388)(2,120,45,188)(3,148,46,390)(4,122,47,190)(5,150,48,392)(6,124,49,192)(7,152,50,366)(8,126,51,194)(9,154,52,368)(10,128,53,196)(11,156,54,370)(12,130,55,170)(13,158,56,372)(14,132,29,172)(15,160,30,374)(16,134,31,174)(17,162,32,376)(18,136,33,176)(19,164,34,378)(20,138,35,178)(21,166,36,380)(22,140,37,180)(23,168,38,382)(24,114,39,182)(25,142,40,384)(26,116,41,184)(27,144,42,386)(28,118,43,186)(57,435,270,418)(58,242,271,361)(59,437,272,420)(60,244,273,363)(61,439,274,394)(62,246,275,337)(63,441,276,396)(64,248,277,339)(65,443,278,398)(66,250,279,341)(67,445,280,400)(68,252,253,343)(69,447,254,402)(70,226,255,345)(71,421,256,404)(72,228,257,347)(73,423,258,406)(74,230,259,349)(75,425,260,408)(76,232,261,351)(77,427,262,410)(78,234,263,353)(79,429,264,412)(80,236,265,355)(81,431,266,414)(82,238,267,357)(83,433,268,416)(84,240,269,359)(85,119,216,187)(86,147,217,389)(87,121,218,189)(88,149,219,391)(89,123,220,191)(90,151,221,365)(91,125,222,193)(92,153,223,367)(93,127,224,195)(94,155,197,369)(95,129,198,169)(96,157,199,371)(97,131,200,171)(98,159,201,373)(99,133,202,173)(100,161,203,375)(101,135,204,175)(102,163,205,377)(103,137,206,177)(104,165,207,379)(105,139,208,179)(106,167,209,381)(107,113,210,181)(108,141,211,383)(109,115,212,183)(110,143,213,385)(111,117,214,185)(112,145,215,387)(225,284,344,323)(227,286,346,325)(229,288,348,327)(231,290,350,329)(233,292,352,331)(235,294,354,333)(237,296,356,335)(239,298,358,309)(241,300,360,311)(243,302,362,313)(245,304,364,315)(247,306,338,317)(249,308,340,319)(251,282,342,321)(281,399,320,444)(283,401,322,446)(285,403,324,448)(287,405,326,422)(289,407,328,424)(291,409,330,426)(293,411,332,428)(295,413,334,430)(297,415,336,432)(299,417,310,434)(301,419,312,436)(303,393,314,438)(305,395,316,440)(307,397,318,442), (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), (1,444,216,341)(2,398,217,249)(3,442,218,339)(4,396,219,247)(5,440,220,337)(6,394,221,245)(7,438,222,363)(8,420,223,243)(9,436,224,361)(10,418,197,241)(11,434,198,359)(12,416,199,239)(13,432,200,357)(14,414,201,237)(15,430,202,355)(16,412,203,235)(17,428,204,353)(18,410,205,233)(19,426,206,351)(20,408,207,231)(21,424,208,349)(22,406,209,229)(23,422,210,347)(24,404,211,227)(25,448,212,345)(26,402,213,225)(27,446,214,343)(28,400,215,251)(29,431,98,356)(30,413,99,236)(31,429,100,354)(32,411,101,234)(33,427,102,352)(34,409,103,232)(35,425,104,350)(36,407,105,230)(37,423,106,348)(38,405,107,228)(39,421,108,346)(40,403,109,226)(41,447,110,344)(42,401,111,252)(43,445,112,342)(44,399,85,250)(45,443,86,340)(46,397,87,248)(47,441,88,338)(48,395,89,246)(49,439,90,364)(50,393,91,244)(51,437,92,362)(52,419,93,242)(53,435,94,360)(54,417,95,240)(55,433,96,358)(56,415,97,238)(57,128,300,369)(58,195,301,154)(59,126,302,367)(60,193,303,152)(61,124,304,365)(62,191,305,150)(63,122,306,391)(64,189,307,148)(65,120,308,389)(66,187,281,146)(67,118,282,387)(68,185,283,144)(69,116,284,385)(70,183,285,142)(71,114,286,383)(72,181,287,168)(73,140,288,381)(74,179,289,166)(75,138,290,379)(76,177,291,164)(77,136,292,377)(78,175,293,162)(79,134,294,375)(80,173,295,160)(81,132,296,373)(82,171,297,158)(83,130,298,371)(84,169,299,156)(113,326,382,257)(115,324,384,255)(117,322,386,253)(119,320,388,279)(121,318,390,277)(123,316,392,275)(125,314,366,273)(127,312,368,271)(129,310,370,269)(131,336,372,267)(133,334,374,265)(135,332,376,263)(137,330,378,261)(139,328,380,259)(141,256,182,325)(143,254,184,323)(145,280,186,321)(147,278,188,319)(149,276,190,317)(151,274,192,315)(153,272,194,313)(155,270,196,311)(157,268,170,309)(159,266,172,335)(161,264,174,333)(163,262,176,331)(165,260,178,329)(167,258,180,327) );
G=PermutationGroup([[(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,201),(30,202),(31,203),(32,204),(33,205),(34,206),(35,207),(36,208),(37,209),(38,210),(39,211),(40,212),(41,213),(42,214),(43,215),(44,216),(45,217),(46,218),(47,219),(48,220),(49,221),(50,222),(51,223),(52,224),(53,197),(54,198),(55,199),(56,200),(57,311),(58,312),(59,313),(60,314),(61,315),(62,316),(63,317),(64,318),(65,319),(66,320),(67,321),(68,322),(69,323),(70,324),(71,325),(72,326),(73,327),(74,328),(75,329),(76,330),(77,331),(78,332),(79,333),(80,334),(81,335),(82,336),(83,309),(84,310),(113,168),(114,141),(115,142),(116,143),(117,144),(118,145),(119,146),(120,147),(121,148),(122,149),(123,150),(124,151),(125,152),(126,153),(127,154),(128,155),(129,156),(130,157),(131,158),(132,159),(133,160),(134,161),(135,162),(136,163),(137,164),(138,165),(139,166),(140,167),(169,370),(170,371),(171,372),(172,373),(173,374),(174,375),(175,376),(176,377),(177,378),(178,379),(179,380),(180,381),(181,382),(182,383),(183,384),(184,385),(185,386),(186,387),(187,388),(188,389),(189,390),(190,391),(191,392),(192,365),(193,366),(194,367),(195,368),(196,369),(225,447),(226,448),(227,421),(228,422),(229,423),(230,424),(231,425),(232,426),(233,427),(234,428),(235,429),(236,430),(237,431),(238,432),(239,433),(240,434),(241,435),(242,436),(243,437),(244,438),(245,439),(246,440),(247,441),(248,442),(249,443),(250,444),(251,445),(252,446),(253,283),(254,284),(255,285),(256,286),(257,287),(258,288),(259,289),(260,290),(261,291),(262,292),(263,293),(264,294),(265,295),(266,296),(267,297),(268,298),(269,299),(270,300),(271,301),(272,302),(273,303),(274,304),(275,305),(276,306),(277,307),(278,308),(279,281),(280,282),(337,395),(338,396),(339,397),(340,398),(341,399),(342,400),(343,401),(344,402),(345,403),(346,404),(347,405),(348,406),(349,407),(350,408),(351,409),(352,410),(353,411),(354,412),(355,413),(356,414),(357,415),(358,416),(359,417),(360,418),(361,419),(362,420),(363,393),(364,394)], [(1,146,44,388),(2,120,45,188),(3,148,46,390),(4,122,47,190),(5,150,48,392),(6,124,49,192),(7,152,50,366),(8,126,51,194),(9,154,52,368),(10,128,53,196),(11,156,54,370),(12,130,55,170),(13,158,56,372),(14,132,29,172),(15,160,30,374),(16,134,31,174),(17,162,32,376),(18,136,33,176),(19,164,34,378),(20,138,35,178),(21,166,36,380),(22,140,37,180),(23,168,38,382),(24,114,39,182),(25,142,40,384),(26,116,41,184),(27,144,42,386),(28,118,43,186),(57,435,270,418),(58,242,271,361),(59,437,272,420),(60,244,273,363),(61,439,274,394),(62,246,275,337),(63,441,276,396),(64,248,277,339),(65,443,278,398),(66,250,279,341),(67,445,280,400),(68,252,253,343),(69,447,254,402),(70,226,255,345),(71,421,256,404),(72,228,257,347),(73,423,258,406),(74,230,259,349),(75,425,260,408),(76,232,261,351),(77,427,262,410),(78,234,263,353),(79,429,264,412),(80,236,265,355),(81,431,266,414),(82,238,267,357),(83,433,268,416),(84,240,269,359),(85,119,216,187),(86,147,217,389),(87,121,218,189),(88,149,219,391),(89,123,220,191),(90,151,221,365),(91,125,222,193),(92,153,223,367),(93,127,224,195),(94,155,197,369),(95,129,198,169),(96,157,199,371),(97,131,200,171),(98,159,201,373),(99,133,202,173),(100,161,203,375),(101,135,204,175),(102,163,205,377),(103,137,206,177),(104,165,207,379),(105,139,208,179),(106,167,209,381),(107,113,210,181),(108,141,211,383),(109,115,212,183),(110,143,213,385),(111,117,214,185),(112,145,215,387),(225,284,344,323),(227,286,346,325),(229,288,348,327),(231,290,350,329),(233,292,352,331),(235,294,354,333),(237,296,356,335),(239,298,358,309),(241,300,360,311),(243,302,362,313),(245,304,364,315),(247,306,338,317),(249,308,340,319),(251,282,342,321),(281,399,320,444),(283,401,322,446),(285,403,324,448),(287,405,326,422),(289,407,328,424),(291,409,330,426),(293,411,332,428),(295,413,334,430),(297,415,336,432),(299,417,310,434),(301,419,312,436),(303,393,314,438),(305,395,316,440),(307,397,318,442)], [(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)], [(1,444,216,341),(2,398,217,249),(3,442,218,339),(4,396,219,247),(5,440,220,337),(6,394,221,245),(7,438,222,363),(8,420,223,243),(9,436,224,361),(10,418,197,241),(11,434,198,359),(12,416,199,239),(13,432,200,357),(14,414,201,237),(15,430,202,355),(16,412,203,235),(17,428,204,353),(18,410,205,233),(19,426,206,351),(20,408,207,231),(21,424,208,349),(22,406,209,229),(23,422,210,347),(24,404,211,227),(25,448,212,345),(26,402,213,225),(27,446,214,343),(28,400,215,251),(29,431,98,356),(30,413,99,236),(31,429,100,354),(32,411,101,234),(33,427,102,352),(34,409,103,232),(35,425,104,350),(36,407,105,230),(37,423,106,348),(38,405,107,228),(39,421,108,346),(40,403,109,226),(41,447,110,344),(42,401,111,252),(43,445,112,342),(44,399,85,250),(45,443,86,340),(46,397,87,248),(47,441,88,338),(48,395,89,246),(49,439,90,364),(50,393,91,244),(51,437,92,362),(52,419,93,242),(53,435,94,360),(54,417,95,240),(55,433,96,358),(56,415,97,238),(57,128,300,369),(58,195,301,154),(59,126,302,367),(60,193,303,152),(61,124,304,365),(62,191,305,150),(63,122,306,391),(64,189,307,148),(65,120,308,389),(66,187,281,146),(67,118,282,387),(68,185,283,144),(69,116,284,385),(70,183,285,142),(71,114,286,383),(72,181,287,168),(73,140,288,381),(74,179,289,166),(75,138,290,379),(76,177,291,164),(77,136,292,377),(78,175,293,162),(79,134,294,375),(80,173,295,160),(81,132,296,373),(82,171,297,158),(83,130,298,371),(84,169,299,156),(113,326,382,257),(115,324,384,255),(117,322,386,253),(119,320,388,279),(121,318,390,277),(123,316,392,275),(125,314,366,273),(127,312,368,271),(129,310,370,269),(131,336,372,267),(133,334,374,265),(135,332,376,263),(137,330,378,261),(139,328,380,259),(141,256,182,325),(143,254,184,323),(145,280,186,321),(147,278,188,319),(149,276,190,317),(151,274,192,315),(153,272,194,313),(155,270,196,311),(157,268,170,309),(159,266,172,335),(161,264,174,333),(163,262,176,331),(165,260,178,329),(167,258,180,327)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | C4○D4 | D14 | D28 | C4○D28 | D4⋊2D7 | Q8×D7 |
| kernel | (C2×C28).28D4 | C14.C42 | C7×C2.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×Dic7 | C2×C28 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 3 | 10 | 9 | 12 | 24 | 9 | 3 |
Matrix representation of (C2×C28).28D4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 0 | 0 | 0 | 0 | 18 | 2 |
| 1 | 27 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 5 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 7 |
| 0 | 0 | 0 | 0 | 22 | 12 |
| 17 | 24 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 3 | 0 | 0 |
| 0 | 0 | 18 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 12 |
| 0 | 0 | 0 | 0 | 20 | 7 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,5,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,7,4,0,0,0,0,5,3,0,0,0,0,0,0,20,22,0,0,0,0,7,12],[17,0,0,0,0,0,24,12,0,0,0,0,0,0,18,18,0,0,0,0,3,11,0,0,0,0,0,0,22,20,0,0,0,0,12,7] >;
(C2×C28).28D4 in GAP, Magma, Sage, TeX
(C_2\times C_{28})._{28}D_4 % in TeX
G:=Group("(C2xC28).28D4"); // GroupNames label
G:=SmallGroup(448,193);
// by ID
G=gap.SmallGroup(448,193);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,64,254,387,268,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^28=1,d^2=a*b^2,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=a*b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations