metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic28⋊6C4, Dic7⋊5Q16, C7⋊3(C4×Q16), C8.15(C4×D7), C2.3(D7×Q16), C56.16(C2×C4), C14.53(C4×D4), C2.D8.9D7, C4⋊C4.167D14, (C2×C8).226D14, C14.20(C2×Q16), (C8×Dic7).1C2, C22.88(D4×D7), C28.36(C4○D4), C14.26(C4○D8), C2.3(D8⋊3D7), C28.47(C22×C4), (C2×C56).78C22, C4.8(Q8⋊2D7), (C2×Dic28).9C2, Dic14.9(C2×C4), C14.Q16.7C2, (C2×C28).289C23, Dic7⋊3Q8.7C2, (C2×Dic7).208D4, C2.13(D28⋊C4), (C4×Dic7).232C22, (C2×Dic14).86C22, C4.44(C2×C4×D7), (C7×C2.D8).4C2, (C2×C14).294(C2×D4), (C7×C4⋊C4).82C22, (C2×C7⋊C8).230C22, (C2×C4).392(C22×D7), SmallGroup(448,407)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic28⋊6C4
G = < a,b,c | a56=c4=1, b2=a28, bab-1=a-1, cac-1=a15, bc=cb >
Subgroups: 460 in 110 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C4×Q16, Dic28, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C14.Q16, C8×Dic7, C7×C2.D8, Dic7⋊3Q8, C2×Dic28, Dic28⋊6C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, Q16, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×Q16, C4○D8, C4×D7, C22×D7, C4×Q16, C2×C4×D7, D4×D7, Q8⋊2D7, D28⋊C4, D8⋊3D7, D7×Q16, Dic28⋊6C4
►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)
(1 215 29 187)(2 214 30 186)(3 213 31 185)(4 212 32 184)(5 211 33 183)(6 210 34 182)(7 209 35 181)(8 208 36 180)(9 207 37 179)(10 206 38 178)(11 205 39 177)(12 204 40 176)(13 203 41 175)(14 202 42 174)(15 201 43 173)(16 200 44 172)(17 199 45 171)(18 198 46 170)(19 197 47 169)(20 196 48 224)(21 195 49 223)(22 194 50 222)(23 193 51 221)(24 192 52 220)(25 191 53 219)(26 190 54 218)(27 189 55 217)(28 188 56 216)(57 258 85 230)(58 257 86 229)(59 256 87 228)(60 255 88 227)(61 254 89 226)(62 253 90 225)(63 252 91 280)(64 251 92 279)(65 250 93 278)(66 249 94 277)(67 248 95 276)(68 247 96 275)(69 246 97 274)(70 245 98 273)(71 244 99 272)(72 243 100 271)(73 242 101 270)(74 241 102 269)(75 240 103 268)(76 239 104 267)(77 238 105 266)(78 237 106 265)(79 236 107 264)(80 235 108 263)(81 234 109 262)(82 233 110 261)(83 232 111 260)(84 231 112 259)(113 422 141 394)(114 421 142 393)(115 420 143 448)(116 419 144 447)(117 418 145 446)(118 417 146 445)(119 416 147 444)(120 415 148 443)(121 414 149 442)(122 413 150 441)(123 412 151 440)(124 411 152 439)(125 410 153 438)(126 409 154 437)(127 408 155 436)(128 407 156 435)(129 406 157 434)(130 405 158 433)(131 404 159 432)(132 403 160 431)(133 402 161 430)(134 401 162 429)(135 400 163 428)(136 399 164 427)(137 398 165 426)(138 397 166 425)(139 396 167 424)(140 395 168 423)(281 385 309 357)(282 384 310 356)(283 383 311 355)(284 382 312 354)(285 381 313 353)(286 380 314 352)(287 379 315 351)(288 378 316 350)(289 377 317 349)(290 376 318 348)(291 375 319 347)(292 374 320 346)(293 373 321 345)(294 372 322 344)(295 371 323 343)(296 370 324 342)(297 369 325 341)(298 368 326 340)(299 367 327 339)(300 366 328 338)(301 365 329 337)(302 364 330 392)(303 363 331 391)(304 362 332 390)(305 361 333 389)(306 360 334 388)(307 359 335 387)(308 358 336 386)
(1 389 427 67)(2 348 428 82)(3 363 429 97)(4 378 430 112)(5 337 431 71)(6 352 432 86)(7 367 433 101)(8 382 434 60)(9 341 435 75)(10 356 436 90)(11 371 437 105)(12 386 438 64)(13 345 439 79)(14 360 440 94)(15 375 441 109)(16 390 442 68)(17 349 443 83)(18 364 444 98)(19 379 445 57)(20 338 446 72)(21 353 447 87)(22 368 448 102)(23 383 393 61)(24 342 394 76)(25 357 395 91)(26 372 396 106)(27 387 397 65)(28 346 398 80)(29 361 399 95)(30 376 400 110)(31 391 401 69)(32 350 402 84)(33 365 403 99)(34 380 404 58)(35 339 405 73)(36 354 406 88)(37 369 407 103)(38 384 408 62)(39 343 409 77)(40 358 410 92)(41 373 411 107)(42 388 412 66)(43 347 413 81)(44 362 414 96)(45 377 415 111)(46 392 416 70)(47 351 417 85)(48 366 418 100)(49 381 419 59)(50 340 420 74)(51 355 421 89)(52 370 422 104)(53 385 423 63)(54 344 424 78)(55 359 425 93)(56 374 426 108)(113 239 192 296)(114 254 193 311)(115 269 194 326)(116 228 195 285)(117 243 196 300)(118 258 197 315)(119 273 198 330)(120 232 199 289)(121 247 200 304)(122 262 201 319)(123 277 202 334)(124 236 203 293)(125 251 204 308)(126 266 205 323)(127 225 206 282)(128 240 207 297)(129 255 208 312)(130 270 209 327)(131 229 210 286)(132 244 211 301)(133 259 212 316)(134 274 213 331)(135 233 214 290)(136 248 215 305)(137 263 216 320)(138 278 217 335)(139 237 218 294)(140 252 219 309)(141 267 220 324)(142 226 221 283)(143 241 222 298)(144 256 223 313)(145 271 224 328)(146 230 169 287)(147 245 170 302)(148 260 171 317)(149 275 172 332)(150 234 173 291)(151 249 174 306)(152 264 175 321)(153 279 176 336)(154 238 177 295)(155 253 178 310)(156 268 179 325)(157 227 180 284)(158 242 181 299)(159 257 182 314)(160 272 183 329)(161 231 184 288)(162 246 185 303)(163 261 186 318)(164 276 187 333)(165 235 188 292)(166 250 189 307)(167 265 190 322)(168 280 191 281)
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), (1,215,29,187)(2,214,30,186)(3,213,31,185)(4,212,32,184)(5,211,33,183)(6,210,34,182)(7,209,35,181)(8,208,36,180)(9,207,37,179)(10,206,38,178)(11,205,39,177)(12,204,40,176)(13,203,41,175)(14,202,42,174)(15,201,43,173)(16,200,44,172)(17,199,45,171)(18,198,46,170)(19,197,47,169)(20,196,48,224)(21,195,49,223)(22,194,50,222)(23,193,51,221)(24,192,52,220)(25,191,53,219)(26,190,54,218)(27,189,55,217)(28,188,56,216)(57,258,85,230)(58,257,86,229)(59,256,87,228)(60,255,88,227)(61,254,89,226)(62,253,90,225)(63,252,91,280)(64,251,92,279)(65,250,93,278)(66,249,94,277)(67,248,95,276)(68,247,96,275)(69,246,97,274)(70,245,98,273)(71,244,99,272)(72,243,100,271)(73,242,101,270)(74,241,102,269)(75,240,103,268)(76,239,104,267)(77,238,105,266)(78,237,106,265)(79,236,107,264)(80,235,108,263)(81,234,109,262)(82,233,110,261)(83,232,111,260)(84,231,112,259)(113,422,141,394)(114,421,142,393)(115,420,143,448)(116,419,144,447)(117,418,145,446)(118,417,146,445)(119,416,147,444)(120,415,148,443)(121,414,149,442)(122,413,150,441)(123,412,151,440)(124,411,152,439)(125,410,153,438)(126,409,154,437)(127,408,155,436)(128,407,156,435)(129,406,157,434)(130,405,158,433)(131,404,159,432)(132,403,160,431)(133,402,161,430)(134,401,162,429)(135,400,163,428)(136,399,164,427)(137,398,165,426)(138,397,166,425)(139,396,167,424)(140,395,168,423)(281,385,309,357)(282,384,310,356)(283,383,311,355)(284,382,312,354)(285,381,313,353)(286,380,314,352)(287,379,315,351)(288,378,316,350)(289,377,317,349)(290,376,318,348)(291,375,319,347)(292,374,320,346)(293,373,321,345)(294,372,322,344)(295,371,323,343)(296,370,324,342)(297,369,325,341)(298,368,326,340)(299,367,327,339)(300,366,328,338)(301,365,329,337)(302,364,330,392)(303,363,331,391)(304,362,332,390)(305,361,333,389)(306,360,334,388)(307,359,335,387)(308,358,336,386), (1,389,427,67)(2,348,428,82)(3,363,429,97)(4,378,430,112)(5,337,431,71)(6,352,432,86)(7,367,433,101)(8,382,434,60)(9,341,435,75)(10,356,436,90)(11,371,437,105)(12,386,438,64)(13,345,439,79)(14,360,440,94)(15,375,441,109)(16,390,442,68)(17,349,443,83)(18,364,444,98)(19,379,445,57)(20,338,446,72)(21,353,447,87)(22,368,448,102)(23,383,393,61)(24,342,394,76)(25,357,395,91)(26,372,396,106)(27,387,397,65)(28,346,398,80)(29,361,399,95)(30,376,400,110)(31,391,401,69)(32,350,402,84)(33,365,403,99)(34,380,404,58)(35,339,405,73)(36,354,406,88)(37,369,407,103)(38,384,408,62)(39,343,409,77)(40,358,410,92)(41,373,411,107)(42,388,412,66)(43,347,413,81)(44,362,414,96)(45,377,415,111)(46,392,416,70)(47,351,417,85)(48,366,418,100)(49,381,419,59)(50,340,420,74)(51,355,421,89)(52,370,422,104)(53,385,423,63)(54,344,424,78)(55,359,425,93)(56,374,426,108)(113,239,192,296)(114,254,193,311)(115,269,194,326)(116,228,195,285)(117,243,196,300)(118,258,197,315)(119,273,198,330)(120,232,199,289)(121,247,200,304)(122,262,201,319)(123,277,202,334)(124,236,203,293)(125,251,204,308)(126,266,205,323)(127,225,206,282)(128,240,207,297)(129,255,208,312)(130,270,209,327)(131,229,210,286)(132,244,211,301)(133,259,212,316)(134,274,213,331)(135,233,214,290)(136,248,215,305)(137,263,216,320)(138,278,217,335)(139,237,218,294)(140,252,219,309)(141,267,220,324)(142,226,221,283)(143,241,222,298)(144,256,223,313)(145,271,224,328)(146,230,169,287)(147,245,170,302)(148,260,171,317)(149,275,172,332)(150,234,173,291)(151,249,174,306)(152,264,175,321)(153,279,176,336)(154,238,177,295)(155,253,178,310)(156,268,179,325)(157,227,180,284)(158,242,181,299)(159,257,182,314)(160,272,183,329)(161,231,184,288)(162,246,185,303)(163,261,186,318)(164,276,187,333)(165,235,188,292)(166,250,189,307)(167,265,190,322)(168,280,191,281)>;
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), (1,215,29,187)(2,214,30,186)(3,213,31,185)(4,212,32,184)(5,211,33,183)(6,210,34,182)(7,209,35,181)(8,208,36,180)(9,207,37,179)(10,206,38,178)(11,205,39,177)(12,204,40,176)(13,203,41,175)(14,202,42,174)(15,201,43,173)(16,200,44,172)(17,199,45,171)(18,198,46,170)(19,197,47,169)(20,196,48,224)(21,195,49,223)(22,194,50,222)(23,193,51,221)(24,192,52,220)(25,191,53,219)(26,190,54,218)(27,189,55,217)(28,188,56,216)(57,258,85,230)(58,257,86,229)(59,256,87,228)(60,255,88,227)(61,254,89,226)(62,253,90,225)(63,252,91,280)(64,251,92,279)(65,250,93,278)(66,249,94,277)(67,248,95,276)(68,247,96,275)(69,246,97,274)(70,245,98,273)(71,244,99,272)(72,243,100,271)(73,242,101,270)(74,241,102,269)(75,240,103,268)(76,239,104,267)(77,238,105,266)(78,237,106,265)(79,236,107,264)(80,235,108,263)(81,234,109,262)(82,233,110,261)(83,232,111,260)(84,231,112,259)(113,422,141,394)(114,421,142,393)(115,420,143,448)(116,419,144,447)(117,418,145,446)(118,417,146,445)(119,416,147,444)(120,415,148,443)(121,414,149,442)(122,413,150,441)(123,412,151,440)(124,411,152,439)(125,410,153,438)(126,409,154,437)(127,408,155,436)(128,407,156,435)(129,406,157,434)(130,405,158,433)(131,404,159,432)(132,403,160,431)(133,402,161,430)(134,401,162,429)(135,400,163,428)(136,399,164,427)(137,398,165,426)(138,397,166,425)(139,396,167,424)(140,395,168,423)(281,385,309,357)(282,384,310,356)(283,383,311,355)(284,382,312,354)(285,381,313,353)(286,380,314,352)(287,379,315,351)(288,378,316,350)(289,377,317,349)(290,376,318,348)(291,375,319,347)(292,374,320,346)(293,373,321,345)(294,372,322,344)(295,371,323,343)(296,370,324,342)(297,369,325,341)(298,368,326,340)(299,367,327,339)(300,366,328,338)(301,365,329,337)(302,364,330,392)(303,363,331,391)(304,362,332,390)(305,361,333,389)(306,360,334,388)(307,359,335,387)(308,358,336,386), (1,389,427,67)(2,348,428,82)(3,363,429,97)(4,378,430,112)(5,337,431,71)(6,352,432,86)(7,367,433,101)(8,382,434,60)(9,341,435,75)(10,356,436,90)(11,371,437,105)(12,386,438,64)(13,345,439,79)(14,360,440,94)(15,375,441,109)(16,390,442,68)(17,349,443,83)(18,364,444,98)(19,379,445,57)(20,338,446,72)(21,353,447,87)(22,368,448,102)(23,383,393,61)(24,342,394,76)(25,357,395,91)(26,372,396,106)(27,387,397,65)(28,346,398,80)(29,361,399,95)(30,376,400,110)(31,391,401,69)(32,350,402,84)(33,365,403,99)(34,380,404,58)(35,339,405,73)(36,354,406,88)(37,369,407,103)(38,384,408,62)(39,343,409,77)(40,358,410,92)(41,373,411,107)(42,388,412,66)(43,347,413,81)(44,362,414,96)(45,377,415,111)(46,392,416,70)(47,351,417,85)(48,366,418,100)(49,381,419,59)(50,340,420,74)(51,355,421,89)(52,370,422,104)(53,385,423,63)(54,344,424,78)(55,359,425,93)(56,374,426,108)(113,239,192,296)(114,254,193,311)(115,269,194,326)(116,228,195,285)(117,243,196,300)(118,258,197,315)(119,273,198,330)(120,232,199,289)(121,247,200,304)(122,262,201,319)(123,277,202,334)(124,236,203,293)(125,251,204,308)(126,266,205,323)(127,225,206,282)(128,240,207,297)(129,255,208,312)(130,270,209,327)(131,229,210,286)(132,244,211,301)(133,259,212,316)(134,274,213,331)(135,233,214,290)(136,248,215,305)(137,263,216,320)(138,278,217,335)(139,237,218,294)(140,252,219,309)(141,267,220,324)(142,226,221,283)(143,241,222,298)(144,256,223,313)(145,271,224,328)(146,230,169,287)(147,245,170,302)(148,260,171,317)(149,275,172,332)(150,234,173,291)(151,249,174,306)(152,264,175,321)(153,279,176,336)(154,238,177,295)(155,253,178,310)(156,268,179,325)(157,227,180,284)(158,242,181,299)(159,257,182,314)(160,272,183,329)(161,231,184,288)(162,246,185,303)(163,261,186,318)(164,276,187,333)(165,235,188,292)(166,250,189,307)(167,265,190,322)(168,280,191,281) );
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)], [(1,215,29,187),(2,214,30,186),(3,213,31,185),(4,212,32,184),(5,211,33,183),(6,210,34,182),(7,209,35,181),(8,208,36,180),(9,207,37,179),(10,206,38,178),(11,205,39,177),(12,204,40,176),(13,203,41,175),(14,202,42,174),(15,201,43,173),(16,200,44,172),(17,199,45,171),(18,198,46,170),(19,197,47,169),(20,196,48,224),(21,195,49,223),(22,194,50,222),(23,193,51,221),(24,192,52,220),(25,191,53,219),(26,190,54,218),(27,189,55,217),(28,188,56,216),(57,258,85,230),(58,257,86,229),(59,256,87,228),(60,255,88,227),(61,254,89,226),(62,253,90,225),(63,252,91,280),(64,251,92,279),(65,250,93,278),(66,249,94,277),(67,248,95,276),(68,247,96,275),(69,246,97,274),(70,245,98,273),(71,244,99,272),(72,243,100,271),(73,242,101,270),(74,241,102,269),(75,240,103,268),(76,239,104,267),(77,238,105,266),(78,237,106,265),(79,236,107,264),(80,235,108,263),(81,234,109,262),(82,233,110,261),(83,232,111,260),(84,231,112,259),(113,422,141,394),(114,421,142,393),(115,420,143,448),(116,419,144,447),(117,418,145,446),(118,417,146,445),(119,416,147,444),(120,415,148,443),(121,414,149,442),(122,413,150,441),(123,412,151,440),(124,411,152,439),(125,410,153,438),(126,409,154,437),(127,408,155,436),(128,407,156,435),(129,406,157,434),(130,405,158,433),(131,404,159,432),(132,403,160,431),(133,402,161,430),(134,401,162,429),(135,400,163,428),(136,399,164,427),(137,398,165,426),(138,397,166,425),(139,396,167,424),(140,395,168,423),(281,385,309,357),(282,384,310,356),(283,383,311,355),(284,382,312,354),(285,381,313,353),(286,380,314,352),(287,379,315,351),(288,378,316,350),(289,377,317,349),(290,376,318,348),(291,375,319,347),(292,374,320,346),(293,373,321,345),(294,372,322,344),(295,371,323,343),(296,370,324,342),(297,369,325,341),(298,368,326,340),(299,367,327,339),(300,366,328,338),(301,365,329,337),(302,364,330,392),(303,363,331,391),(304,362,332,390),(305,361,333,389),(306,360,334,388),(307,359,335,387),(308,358,336,386)], [(1,389,427,67),(2,348,428,82),(3,363,429,97),(4,378,430,112),(5,337,431,71),(6,352,432,86),(7,367,433,101),(8,382,434,60),(9,341,435,75),(10,356,436,90),(11,371,437,105),(12,386,438,64),(13,345,439,79),(14,360,440,94),(15,375,441,109),(16,390,442,68),(17,349,443,83),(18,364,444,98),(19,379,445,57),(20,338,446,72),(21,353,447,87),(22,368,448,102),(23,383,393,61),(24,342,394,76),(25,357,395,91),(26,372,396,106),(27,387,397,65),(28,346,398,80),(29,361,399,95),(30,376,400,110),(31,391,401,69),(32,350,402,84),(33,365,403,99),(34,380,404,58),(35,339,405,73),(36,354,406,88),(37,369,407,103),(38,384,408,62),(39,343,409,77),(40,358,410,92),(41,373,411,107),(42,388,412,66),(43,347,413,81),(44,362,414,96),(45,377,415,111),(46,392,416,70),(47,351,417,85),(48,366,418,100),(49,381,419,59),(50,340,420,74),(51,355,421,89),(52,370,422,104),(53,385,423,63),(54,344,424,78),(55,359,425,93),(56,374,426,108),(113,239,192,296),(114,254,193,311),(115,269,194,326),(116,228,195,285),(117,243,196,300),(118,258,197,315),(119,273,198,330),(120,232,199,289),(121,247,200,304),(122,262,201,319),(123,277,202,334),(124,236,203,293),(125,251,204,308),(126,266,205,323),(127,225,206,282),(128,240,207,297),(129,255,208,312),(130,270,209,327),(131,229,210,286),(132,244,211,301),(133,259,212,316),(134,274,213,331),(135,233,214,290),(136,248,215,305),(137,263,216,320),(138,278,217,335),(139,237,218,294),(140,252,219,309),(141,267,220,324),(142,226,221,283),(143,241,222,298),(144,256,223,313),(145,271,224,328),(146,230,169,287),(147,245,170,302),(148,260,171,317),(149,275,172,332),(150,234,173,291),(151,249,174,306),(152,264,175,321),(153,279,176,336),(154,238,177,295),(155,253,178,310),(156,268,179,325),(157,227,180,284),(158,242,181,299),(159,257,182,314),(160,272,183,329),(161,231,184,288),(162,246,185,303),(163,261,186,318),(164,276,187,333),(165,235,188,292),(166,250,189,307),(167,265,190,322),(168,280,191,281)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | C4○D8 | C4×D7 | Q8⋊2D7 | D4×D7 | D8⋊3D7 | D7×Q16 |
| kernel | Dic28⋊6C4 | C14.Q16 | C8×Dic7 | C7×C2.D8 | Dic7⋊3Q8 | C2×Dic28 | Dic28 | C2×Dic7 | C2.D8 | Dic7 | C28 | C4⋊C4 | C2×C8 | C14 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 3 | 4 | 2 | 6 | 3 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of Dic28⋊6C4 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 24 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 62 | 18 |
| 36 | 17 | 0 | 0 |
| 90 | 77 | 0 | 0 |
| 0 | 0 | 41 | 63 |
| 0 | 0 | 2 | 72 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 50 | 41 |
| 0 | 0 | 30 | 63 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,24,0,0,0,0,44,62,0,0,0,18],[36,90,0,0,17,77,0,0,0,0,41,2,0,0,63,72],[98,0,0,0,0,98,0,0,0,0,50,30,0,0,41,63] >;
Dic28⋊6C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{28}\rtimes_6C_4 % in TeX
G:=Group("Dic28:6C4"); // GroupNames label
G:=SmallGroup(448,407);
// by ID
G=gap.SmallGroup(448,407);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,135,268,570,297,136,18822]);
// 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^15,b*c=c*b>;
// generators/relations