metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic28⋊3C4, C42.22D14, C8.7(C4×D7), C56.4(C2×C4), C14.13(C4×D4), C2.16(C4×D28), (C2×C8).57D14, C8⋊C4.2D7, C8⋊Dic7.3C2, C7⋊1(Q16⋊C4), (C2×C28).238D4, (C2×C4).116D28, (C2×C56).58C22, (C4×C28).16C22, (C4×Dic14).5C2, (C2×Dic28).7C2, C22.32(C2×D28), C28.226(C4○D4), C4.110(C4○D28), (C2×C28).737C23, C28.107(C22×C4), Dic14.14(C2×C4), C2.2(C8.D14), C14.6(C8.C22), C28.44D4.16C2, C4⋊Dic7.267C22, (C2×Dic14).209C22, C4.65(C2×C4×D7), (C7×C8⋊C4).2C2, (C2×C14).120(C2×D4), (C2×C4).681(C22×D7), SmallGroup(448,250)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic28⋊C4
G = < a,b,c | a56=c4=1, b2=a28, bab-1=a-1, cac-1=a29, bc=cb >
Subgroups: 484 in 108 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C56, C56, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, Q16⋊C4, Dic28, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28.44D4, C8⋊Dic7, C7×C8⋊C4, C4×Dic14, C2×Dic28, Dic28⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8.C22, C4×D7, D28, C22×D7, Q16⋊C4, C2×C4×D7, C2×D28, C4○D28, C4×D28, C8.D14, Dic28⋊C4
►Regular action on 448 points
Generators in S448
(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)
(1 386 29 358)(2 385 30 357)(3 384 31 356)(4 383 32 355)(5 382 33 354)(6 381 34 353)(7 380 35 352)(8 379 36 351)(9 378 37 350)(10 377 38 349)(11 376 39 348)(12 375 40 347)(13 374 41 346)(14 373 42 345)(15 372 43 344)(16 371 44 343)(17 370 45 342)(18 369 46 341)(19 368 47 340)(20 367 48 339)(21 366 49 338)(22 365 50 337)(23 364 51 392)(24 363 52 391)(25 362 53 390)(26 361 54 389)(27 360 55 388)(28 359 56 387)(57 304 85 332)(58 303 86 331)(59 302 87 330)(60 301 88 329)(61 300 89 328)(62 299 90 327)(63 298 91 326)(64 297 92 325)(65 296 93 324)(66 295 94 323)(67 294 95 322)(68 293 96 321)(69 292 97 320)(70 291 98 319)(71 290 99 318)(72 289 100 317)(73 288 101 316)(74 287 102 315)(75 286 103 314)(76 285 104 313)(77 284 105 312)(78 283 106 311)(79 282 107 310)(80 281 108 309)(81 336 109 308)(82 335 110 307)(83 334 111 306)(84 333 112 305)(113 410 141 438)(114 409 142 437)(115 408 143 436)(116 407 144 435)(117 406 145 434)(118 405 146 433)(119 404 147 432)(120 403 148 431)(121 402 149 430)(122 401 150 429)(123 400 151 428)(124 399 152 427)(125 398 153 426)(126 397 154 425)(127 396 155 424)(128 395 156 423)(129 394 157 422)(130 393 158 421)(131 448 159 420)(132 447 160 419)(133 446 161 418)(134 445 162 417)(135 444 163 416)(136 443 164 415)(137 442 165 414)(138 441 166 413)(139 440 167 412)(140 439 168 411)(169 245 197 273)(170 244 198 272)(171 243 199 271)(172 242 200 270)(173 241 201 269)(174 240 202 268)(175 239 203 267)(176 238 204 266)(177 237 205 265)(178 236 206 264)(179 235 207 263)(180 234 208 262)(181 233 209 261)(182 232 210 260)(183 231 211 259)(184 230 212 258)(185 229 213 257)(186 228 214 256)(187 227 215 255)(188 226 216 254)(189 225 217 253)(190 280 218 252)(191 279 219 251)(192 278 220 250)(193 277 221 249)(194 276 222 248)(195 275 223 247)(196 274 224 246)
(1 93 149 260)(2 66 150 233)(3 95 151 262)(4 68 152 235)(5 97 153 264)(6 70 154 237)(7 99 155 266)(8 72 156 239)(9 101 157 268)(10 74 158 241)(11 103 159 270)(12 76 160 243)(13 105 161 272)(14 78 162 245)(15 107 163 274)(16 80 164 247)(17 109 165 276)(18 82 166 249)(19 111 167 278)(20 84 168 251)(21 57 113 280)(22 86 114 253)(23 59 115 226)(24 88 116 255)(25 61 117 228)(26 90 118 257)(27 63 119 230)(28 92 120 259)(29 65 121 232)(30 94 122 261)(31 67 123 234)(32 96 124 263)(33 69 125 236)(34 98 126 265)(35 71 127 238)(36 100 128 267)(37 73 129 240)(38 102 130 269)(39 75 131 242)(40 104 132 271)(41 77 133 244)(42 106 134 273)(43 79 135 246)(44 108 136 275)(45 81 137 248)(46 110 138 277)(47 83 139 250)(48 112 140 279)(49 85 141 252)(50 58 142 225)(51 87 143 254)(52 60 144 227)(53 89 145 256)(54 62 146 229)(55 91 147 258)(56 64 148 231)(169 345 311 445)(170 374 312 418)(171 347 313 447)(172 376 314 420)(173 349 315 393)(174 378 316 422)(175 351 317 395)(176 380 318 424)(177 353 319 397)(178 382 320 426)(179 355 321 399)(180 384 322 428)(181 357 323 401)(182 386 324 430)(183 359 325 403)(184 388 326 432)(185 361 327 405)(186 390 328 434)(187 363 329 407)(188 392 330 436)(189 365 331 409)(190 338 332 438)(191 367 333 411)(192 340 334 440)(193 369 335 413)(194 342 336 442)(195 371 281 415)(196 344 282 444)(197 373 283 417)(198 346 284 446)(199 375 285 419)(200 348 286 448)(201 377 287 421)(202 350 288 394)(203 379 289 423)(204 352 290 396)(205 381 291 425)(206 354 292 398)(207 383 293 427)(208 356 294 400)(209 385 295 429)(210 358 296 402)(211 387 297 431)(212 360 298 404)(213 389 299 433)(214 362 300 406)(215 391 301 435)(216 364 302 408)(217 337 303 437)(218 366 304 410)(219 339 305 439)(220 368 306 412)(221 341 307 441)(222 370 308 414)(223 343 309 443)(224 372 310 416)
G:=sub<Sym(448)| (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), (1,386,29,358)(2,385,30,357)(3,384,31,356)(4,383,32,355)(5,382,33,354)(6,381,34,353)(7,380,35,352)(8,379,36,351)(9,378,37,350)(10,377,38,349)(11,376,39,348)(12,375,40,347)(13,374,41,346)(14,373,42,345)(15,372,43,344)(16,371,44,343)(17,370,45,342)(18,369,46,341)(19,368,47,340)(20,367,48,339)(21,366,49,338)(22,365,50,337)(23,364,51,392)(24,363,52,391)(25,362,53,390)(26,361,54,389)(27,360,55,388)(28,359,56,387)(57,304,85,332)(58,303,86,331)(59,302,87,330)(60,301,88,329)(61,300,89,328)(62,299,90,327)(63,298,91,326)(64,297,92,325)(65,296,93,324)(66,295,94,323)(67,294,95,322)(68,293,96,321)(69,292,97,320)(70,291,98,319)(71,290,99,318)(72,289,100,317)(73,288,101,316)(74,287,102,315)(75,286,103,314)(76,285,104,313)(77,284,105,312)(78,283,106,311)(79,282,107,310)(80,281,108,309)(81,336,109,308)(82,335,110,307)(83,334,111,306)(84,333,112,305)(113,410,141,438)(114,409,142,437)(115,408,143,436)(116,407,144,435)(117,406,145,434)(118,405,146,433)(119,404,147,432)(120,403,148,431)(121,402,149,430)(122,401,150,429)(123,400,151,428)(124,399,152,427)(125,398,153,426)(126,397,154,425)(127,396,155,424)(128,395,156,423)(129,394,157,422)(130,393,158,421)(131,448,159,420)(132,447,160,419)(133,446,161,418)(134,445,162,417)(135,444,163,416)(136,443,164,415)(137,442,165,414)(138,441,166,413)(139,440,167,412)(140,439,168,411)(169,245,197,273)(170,244,198,272)(171,243,199,271)(172,242,200,270)(173,241,201,269)(174,240,202,268)(175,239,203,267)(176,238,204,266)(177,237,205,265)(178,236,206,264)(179,235,207,263)(180,234,208,262)(181,233,209,261)(182,232,210,260)(183,231,211,259)(184,230,212,258)(185,229,213,257)(186,228,214,256)(187,227,215,255)(188,226,216,254)(189,225,217,253)(190,280,218,252)(191,279,219,251)(192,278,220,250)(193,277,221,249)(194,276,222,248)(195,275,223,247)(196,274,224,246), (1,93,149,260)(2,66,150,233)(3,95,151,262)(4,68,152,235)(5,97,153,264)(6,70,154,237)(7,99,155,266)(8,72,156,239)(9,101,157,268)(10,74,158,241)(11,103,159,270)(12,76,160,243)(13,105,161,272)(14,78,162,245)(15,107,163,274)(16,80,164,247)(17,109,165,276)(18,82,166,249)(19,111,167,278)(20,84,168,251)(21,57,113,280)(22,86,114,253)(23,59,115,226)(24,88,116,255)(25,61,117,228)(26,90,118,257)(27,63,119,230)(28,92,120,259)(29,65,121,232)(30,94,122,261)(31,67,123,234)(32,96,124,263)(33,69,125,236)(34,98,126,265)(35,71,127,238)(36,100,128,267)(37,73,129,240)(38,102,130,269)(39,75,131,242)(40,104,132,271)(41,77,133,244)(42,106,134,273)(43,79,135,246)(44,108,136,275)(45,81,137,248)(46,110,138,277)(47,83,139,250)(48,112,140,279)(49,85,141,252)(50,58,142,225)(51,87,143,254)(52,60,144,227)(53,89,145,256)(54,62,146,229)(55,91,147,258)(56,64,148,231)(169,345,311,445)(170,374,312,418)(171,347,313,447)(172,376,314,420)(173,349,315,393)(174,378,316,422)(175,351,317,395)(176,380,318,424)(177,353,319,397)(178,382,320,426)(179,355,321,399)(180,384,322,428)(181,357,323,401)(182,386,324,430)(183,359,325,403)(184,388,326,432)(185,361,327,405)(186,390,328,434)(187,363,329,407)(188,392,330,436)(189,365,331,409)(190,338,332,438)(191,367,333,411)(192,340,334,440)(193,369,335,413)(194,342,336,442)(195,371,281,415)(196,344,282,444)(197,373,283,417)(198,346,284,446)(199,375,285,419)(200,348,286,448)(201,377,287,421)(202,350,288,394)(203,379,289,423)(204,352,290,396)(205,381,291,425)(206,354,292,398)(207,383,293,427)(208,356,294,400)(209,385,295,429)(210,358,296,402)(211,387,297,431)(212,360,298,404)(213,389,299,433)(214,362,300,406)(215,391,301,435)(216,364,302,408)(217,337,303,437)(218,366,304,410)(219,339,305,439)(220,368,306,412)(221,341,307,441)(222,370,308,414)(223,343,309,443)(224,372,310,416)>;
G:=Group( (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), (1,386,29,358)(2,385,30,357)(3,384,31,356)(4,383,32,355)(5,382,33,354)(6,381,34,353)(7,380,35,352)(8,379,36,351)(9,378,37,350)(10,377,38,349)(11,376,39,348)(12,375,40,347)(13,374,41,346)(14,373,42,345)(15,372,43,344)(16,371,44,343)(17,370,45,342)(18,369,46,341)(19,368,47,340)(20,367,48,339)(21,366,49,338)(22,365,50,337)(23,364,51,392)(24,363,52,391)(25,362,53,390)(26,361,54,389)(27,360,55,388)(28,359,56,387)(57,304,85,332)(58,303,86,331)(59,302,87,330)(60,301,88,329)(61,300,89,328)(62,299,90,327)(63,298,91,326)(64,297,92,325)(65,296,93,324)(66,295,94,323)(67,294,95,322)(68,293,96,321)(69,292,97,320)(70,291,98,319)(71,290,99,318)(72,289,100,317)(73,288,101,316)(74,287,102,315)(75,286,103,314)(76,285,104,313)(77,284,105,312)(78,283,106,311)(79,282,107,310)(80,281,108,309)(81,336,109,308)(82,335,110,307)(83,334,111,306)(84,333,112,305)(113,410,141,438)(114,409,142,437)(115,408,143,436)(116,407,144,435)(117,406,145,434)(118,405,146,433)(119,404,147,432)(120,403,148,431)(121,402,149,430)(122,401,150,429)(123,400,151,428)(124,399,152,427)(125,398,153,426)(126,397,154,425)(127,396,155,424)(128,395,156,423)(129,394,157,422)(130,393,158,421)(131,448,159,420)(132,447,160,419)(133,446,161,418)(134,445,162,417)(135,444,163,416)(136,443,164,415)(137,442,165,414)(138,441,166,413)(139,440,167,412)(140,439,168,411)(169,245,197,273)(170,244,198,272)(171,243,199,271)(172,242,200,270)(173,241,201,269)(174,240,202,268)(175,239,203,267)(176,238,204,266)(177,237,205,265)(178,236,206,264)(179,235,207,263)(180,234,208,262)(181,233,209,261)(182,232,210,260)(183,231,211,259)(184,230,212,258)(185,229,213,257)(186,228,214,256)(187,227,215,255)(188,226,216,254)(189,225,217,253)(190,280,218,252)(191,279,219,251)(192,278,220,250)(193,277,221,249)(194,276,222,248)(195,275,223,247)(196,274,224,246), (1,93,149,260)(2,66,150,233)(3,95,151,262)(4,68,152,235)(5,97,153,264)(6,70,154,237)(7,99,155,266)(8,72,156,239)(9,101,157,268)(10,74,158,241)(11,103,159,270)(12,76,160,243)(13,105,161,272)(14,78,162,245)(15,107,163,274)(16,80,164,247)(17,109,165,276)(18,82,166,249)(19,111,167,278)(20,84,168,251)(21,57,113,280)(22,86,114,253)(23,59,115,226)(24,88,116,255)(25,61,117,228)(26,90,118,257)(27,63,119,230)(28,92,120,259)(29,65,121,232)(30,94,122,261)(31,67,123,234)(32,96,124,263)(33,69,125,236)(34,98,126,265)(35,71,127,238)(36,100,128,267)(37,73,129,240)(38,102,130,269)(39,75,131,242)(40,104,132,271)(41,77,133,244)(42,106,134,273)(43,79,135,246)(44,108,136,275)(45,81,137,248)(46,110,138,277)(47,83,139,250)(48,112,140,279)(49,85,141,252)(50,58,142,225)(51,87,143,254)(52,60,144,227)(53,89,145,256)(54,62,146,229)(55,91,147,258)(56,64,148,231)(169,345,311,445)(170,374,312,418)(171,347,313,447)(172,376,314,420)(173,349,315,393)(174,378,316,422)(175,351,317,395)(176,380,318,424)(177,353,319,397)(178,382,320,426)(179,355,321,399)(180,384,322,428)(181,357,323,401)(182,386,324,430)(183,359,325,403)(184,388,326,432)(185,361,327,405)(186,390,328,434)(187,363,329,407)(188,392,330,436)(189,365,331,409)(190,338,332,438)(191,367,333,411)(192,340,334,440)(193,369,335,413)(194,342,336,442)(195,371,281,415)(196,344,282,444)(197,373,283,417)(198,346,284,446)(199,375,285,419)(200,348,286,448)(201,377,287,421)(202,350,288,394)(203,379,289,423)(204,352,290,396)(205,381,291,425)(206,354,292,398)(207,383,293,427)(208,356,294,400)(209,385,295,429)(210,358,296,402)(211,387,297,431)(212,360,298,404)(213,389,299,433)(214,362,300,406)(215,391,301,435)(216,364,302,408)(217,337,303,437)(218,366,304,410)(219,339,305,439)(220,368,306,412)(221,341,307,441)(222,370,308,414)(223,343,309,443)(224,372,310,416) );
G=PermutationGroup([[(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)], [(1,386,29,358),(2,385,30,357),(3,384,31,356),(4,383,32,355),(5,382,33,354),(6,381,34,353),(7,380,35,352),(8,379,36,351),(9,378,37,350),(10,377,38,349),(11,376,39,348),(12,375,40,347),(13,374,41,346),(14,373,42,345),(15,372,43,344),(16,371,44,343),(17,370,45,342),(18,369,46,341),(19,368,47,340),(20,367,48,339),(21,366,49,338),(22,365,50,337),(23,364,51,392),(24,363,52,391),(25,362,53,390),(26,361,54,389),(27,360,55,388),(28,359,56,387),(57,304,85,332),(58,303,86,331),(59,302,87,330),(60,301,88,329),(61,300,89,328),(62,299,90,327),(63,298,91,326),(64,297,92,325),(65,296,93,324),(66,295,94,323),(67,294,95,322),(68,293,96,321),(69,292,97,320),(70,291,98,319),(71,290,99,318),(72,289,100,317),(73,288,101,316),(74,287,102,315),(75,286,103,314),(76,285,104,313),(77,284,105,312),(78,283,106,311),(79,282,107,310),(80,281,108,309),(81,336,109,308),(82,335,110,307),(83,334,111,306),(84,333,112,305),(113,410,141,438),(114,409,142,437),(115,408,143,436),(116,407,144,435),(117,406,145,434),(118,405,146,433),(119,404,147,432),(120,403,148,431),(121,402,149,430),(122,401,150,429),(123,400,151,428),(124,399,152,427),(125,398,153,426),(126,397,154,425),(127,396,155,424),(128,395,156,423),(129,394,157,422),(130,393,158,421),(131,448,159,420),(132,447,160,419),(133,446,161,418),(134,445,162,417),(135,444,163,416),(136,443,164,415),(137,442,165,414),(138,441,166,413),(139,440,167,412),(140,439,168,411),(169,245,197,273),(170,244,198,272),(171,243,199,271),(172,242,200,270),(173,241,201,269),(174,240,202,268),(175,239,203,267),(176,238,204,266),(177,237,205,265),(178,236,206,264),(179,235,207,263),(180,234,208,262),(181,233,209,261),(182,232,210,260),(183,231,211,259),(184,230,212,258),(185,229,213,257),(186,228,214,256),(187,227,215,255),(188,226,216,254),(189,225,217,253),(190,280,218,252),(191,279,219,251),(192,278,220,250),(193,277,221,249),(194,276,222,248),(195,275,223,247),(196,274,224,246)], [(1,93,149,260),(2,66,150,233),(3,95,151,262),(4,68,152,235),(5,97,153,264),(6,70,154,237),(7,99,155,266),(8,72,156,239),(9,101,157,268),(10,74,158,241),(11,103,159,270),(12,76,160,243),(13,105,161,272),(14,78,162,245),(15,107,163,274),(16,80,164,247),(17,109,165,276),(18,82,166,249),(19,111,167,278),(20,84,168,251),(21,57,113,280),(22,86,114,253),(23,59,115,226),(24,88,116,255),(25,61,117,228),(26,90,118,257),(27,63,119,230),(28,92,120,259),(29,65,121,232),(30,94,122,261),(31,67,123,234),(32,96,124,263),(33,69,125,236),(34,98,126,265),(35,71,127,238),(36,100,128,267),(37,73,129,240),(38,102,130,269),(39,75,131,242),(40,104,132,271),(41,77,133,244),(42,106,134,273),(43,79,135,246),(44,108,136,275),(45,81,137,248),(46,110,138,277),(47,83,139,250),(48,112,140,279),(49,85,141,252),(50,58,142,225),(51,87,143,254),(52,60,144,227),(53,89,145,256),(54,62,146,229),(55,91,147,258),(56,64,148,231),(169,345,311,445),(170,374,312,418),(171,347,313,447),(172,376,314,420),(173,349,315,393),(174,378,316,422),(175,351,317,395),(176,380,318,424),(177,353,319,397),(178,382,320,426),(179,355,321,399),(180,384,322,428),(181,357,323,401),(182,386,324,430),(183,359,325,403),(184,388,326,432),(185,361,327,405),(186,390,328,434),(187,363,329,407),(188,392,330,436),(189,365,331,409),(190,338,332,438),(191,367,333,411),(192,340,334,440),(193,369,335,413),(194,342,336,442),(195,371,281,415),(196,344,282,444),(197,373,283,417),(198,346,284,446),(199,375,285,419),(200,348,286,448),(201,377,287,421),(202,350,288,394),(203,379,289,423),(204,352,290,396),(205,381,291,425),(206,354,292,398),(207,383,293,427),(208,356,294,400),(209,385,295,429),(210,358,296,402),(211,387,297,431),(212,360,298,404),(213,389,299,433),(214,362,300,406),(215,391,301,435),(216,364,302,408),(217,337,303,437),(218,366,304,410),(219,339,305,439),(220,368,306,412),(221,341,307,441),(222,370,308,414),(223,343,309,443),(224,372,310,416)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | D28 | C4○D28 | C8.C22 | C8.D14 |
| kernel | Dic28⋊C4 | C28.44D4 | C8⋊Dic7 | C7×C8⋊C4 | C4×Dic14 | C2×Dic28 | Dic28 | C2×C28 | C8⋊C4 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 3 | 2 | 3 | 6 | 12 | 12 | 12 | 2 | 12 |
Matrix representation of Dic28⋊C4 ►in GL6(𝔽113)
| 85 | 112 | 0 | 0 | 0 | 0 |
| 107 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 60 | 102 | 47 |
| 0 | 0 | 106 | 35 | 80 | 57 |
| 0 | 0 | 3 | 53 | 47 | 53 |
| 0 | 0 | 93 | 34 | 60 | 53 |
| 83 | 98 | 0 | 0 | 0 | 0 |
| 75 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 71 | 19 | 103 | 90 |
| 0 | 0 | 32 | 42 | 1 | 29 |
| 0 | 0 | 0 | 0 | 14 | 37 |
| 0 | 0 | 0 | 0 | 13 | 99 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 70 |
| 0 | 0 | 0 | 112 | 43 | 58 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(113))| [85,107,0,0,0,0,112,28,0,0,0,0,0,0,91,106,3,93,0,0,60,35,53,34,0,0,102,80,47,60,0,0,47,57,53,53],[83,75,0,0,0,0,98,30,0,0,0,0,0,0,71,32,0,0,0,0,19,42,0,0,0,0,103,1,14,13,0,0,90,29,37,99],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,43,1,0,0,0,70,58,0,1] >;
Dic28⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{28}\rtimes C_4 % in TeX
G:=Group("Dic28:C4"); // GroupNames label
G:=SmallGroup(448,250);
// by ID
G=gap.SmallGroup(448,250);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,344,387,58,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=c^4=1,b^2=a^28,b*a*b^-1=a^-1,c*a*c^-1=a^29,b*c=c*b>;
// generators/relations