metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic28⋊9C4, C8.9(C4×D7), C56.24(C2×C4), C4.Q8.2D7, C14.50(C4×D4), (C2×C8).58D14, C56⋊C4.1C2, C4⋊C4.159D14, C7⋊3(Q16⋊C4), C22.82(D4×D7), C28.26(C4○D4), C28.41(C22×C4), C4.2(Q8⋊2D7), Dic14.8(C2×C4), C14.Q16.4C2, (C2×C28).269C23, (C2×C56).107C22, Dic7⋊3Q8.4C2, (C2×Dic7).160D4, (C2×Dic28).13C2, C2.5(SD16⋊D7), C2.10(D28⋊C4), C14.37(C8.C22), (C4×Dic7).27C22, (C2×Dic14).79C22, C4.41(C2×C4×D7), (C7×C4.Q8).2C2, (C2×C7⋊C8).51C22, (C2×C14).274(C2×D4), (C7×C4⋊C4).62C22, (C2×C4).372(C22×D7), SmallGroup(448,387)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic28⋊9C4
G = < a,b,c | a56=c4=1, b2=a28, bab-1=a-1, cac-1=a43, cbc-1=a28b >
Subgroups: 460 in 108 conjugacy classes, 49 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, Q16⋊C4, Dic28, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C14.Q16, C56⋊C4, C7×C4.Q8, Dic7⋊3Q8, C2×Dic28, Dic28⋊9C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8.C22, C4×D7, C22×D7, Q16⋊C4, C2×C4×D7, D4×D7, Q8⋊2D7, D28⋊C4, SD16⋊D7, Dic28⋊9C4
►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)
(1 431 29 403)(2 430 30 402)(3 429 31 401)(4 428 32 400)(5 427 33 399)(6 426 34 398)(7 425 35 397)(8 424 36 396)(9 423 37 395)(10 422 38 394)(11 421 39 393)(12 420 40 448)(13 419 41 447)(14 418 42 446)(15 417 43 445)(16 416 44 444)(17 415 45 443)(18 414 46 442)(19 413 47 441)(20 412 48 440)(21 411 49 439)(22 410 50 438)(23 409 51 437)(24 408 52 436)(25 407 53 435)(26 406 54 434)(27 405 55 433)(28 404 56 432)(57 162 85 134)(58 161 86 133)(59 160 87 132)(60 159 88 131)(61 158 89 130)(62 157 90 129)(63 156 91 128)(64 155 92 127)(65 154 93 126)(66 153 94 125)(67 152 95 124)(68 151 96 123)(69 150 97 122)(70 149 98 121)(71 148 99 120)(72 147 100 119)(73 146 101 118)(74 145 102 117)(75 144 103 116)(76 143 104 115)(77 142 105 114)(78 141 106 113)(79 140 107 168)(80 139 108 167)(81 138 109 166)(82 137 110 165)(83 136 111 164)(84 135 112 163)(169 263 197 235)(170 262 198 234)(171 261 199 233)(172 260 200 232)(173 259 201 231)(174 258 202 230)(175 257 203 229)(176 256 204 228)(177 255 205 227)(178 254 206 226)(179 253 207 225)(180 252 208 280)(181 251 209 279)(182 250 210 278)(183 249 211 277)(184 248 212 276)(185 247 213 275)(186 246 214 274)(187 245 215 273)(188 244 216 272)(189 243 217 271)(190 242 218 270)(191 241 219 269)(192 240 220 268)(193 239 221 267)(194 238 222 266)(195 237 223 265)(196 236 224 264)(281 392 309 364)(282 391 310 363)(283 390 311 362)(284 389 312 361)(285 388 313 360)(286 387 314 359)(287 386 315 358)(288 385 316 357)(289 384 317 356)(290 383 318 355)(291 382 319 354)(292 381 320 353)(293 380 321 352)(294 379 322 351)(295 378 323 350)(296 377 324 349)(297 376 325 348)(298 375 326 347)(299 374 327 346)(300 373 328 345)(301 372 329 344)(302 371 330 343)(303 370 331 342)(304 369 332 341)(305 368 333 340)(306 367 334 339)(307 366 335 338)(308 365 336 337)
(1 295 124 226)(2 282 125 269)(3 325 126 256)(4 312 127 243)(5 299 128 230)(6 286 129 273)(7 329 130 260)(8 316 131 247)(9 303 132 234)(10 290 133 277)(11 333 134 264)(12 320 135 251)(13 307 136 238)(14 294 137 225)(15 281 138 268)(16 324 139 255)(17 311 140 242)(18 298 141 229)(19 285 142 272)(20 328 143 259)(21 315 144 246)(22 302 145 233)(23 289 146 276)(24 332 147 263)(25 319 148 250)(26 306 149 237)(27 293 150 280)(28 336 151 267)(29 323 152 254)(30 310 153 241)(31 297 154 228)(32 284 155 271)(33 327 156 258)(34 314 157 245)(35 301 158 232)(36 288 159 275)(37 331 160 262)(38 318 161 249)(39 305 162 236)(40 292 163 279)(41 335 164 266)(42 322 165 253)(43 309 166 240)(44 296 167 227)(45 283 168 270)(46 326 113 257)(47 313 114 244)(48 300 115 231)(49 287 116 274)(50 330 117 261)(51 317 118 248)(52 304 119 235)(53 291 120 278)(54 334 121 265)(55 321 122 252)(56 308 123 239)(57 224 421 368)(58 211 422 355)(59 198 423 342)(60 185 424 385)(61 172 425 372)(62 215 426 359)(63 202 427 346)(64 189 428 389)(65 176 429 376)(66 219 430 363)(67 206 431 350)(68 193 432 337)(69 180 433 380)(70 223 434 367)(71 210 435 354)(72 197 436 341)(73 184 437 384)(74 171 438 371)(75 214 439 358)(76 201 440 345)(77 188 441 388)(78 175 442 375)(79 218 443 362)(80 205 444 349)(81 192 445 392)(82 179 446 379)(83 222 447 366)(84 209 448 353)(85 196 393 340)(86 183 394 383)(87 170 395 370)(88 213 396 357)(89 200 397 344)(90 187 398 387)(91 174 399 374)(92 217 400 361)(93 204 401 348)(94 191 402 391)(95 178 403 378)(96 221 404 365)(97 208 405 352)(98 195 406 339)(99 182 407 382)(100 169 408 369)(101 212 409 356)(102 199 410 343)(103 186 411 386)(104 173 412 373)(105 216 413 360)(106 203 414 347)(107 190 415 390)(108 177 416 377)(109 220 417 364)(110 207 418 351)(111 194 419 338)(112 181 420 381)
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), (1,431,29,403)(2,430,30,402)(3,429,31,401)(4,428,32,400)(5,427,33,399)(6,426,34,398)(7,425,35,397)(8,424,36,396)(9,423,37,395)(10,422,38,394)(11,421,39,393)(12,420,40,448)(13,419,41,447)(14,418,42,446)(15,417,43,445)(16,416,44,444)(17,415,45,443)(18,414,46,442)(19,413,47,441)(20,412,48,440)(21,411,49,439)(22,410,50,438)(23,409,51,437)(24,408,52,436)(25,407,53,435)(26,406,54,434)(27,405,55,433)(28,404,56,432)(57,162,85,134)(58,161,86,133)(59,160,87,132)(60,159,88,131)(61,158,89,130)(62,157,90,129)(63,156,91,128)(64,155,92,127)(65,154,93,126)(66,153,94,125)(67,152,95,124)(68,151,96,123)(69,150,97,122)(70,149,98,121)(71,148,99,120)(72,147,100,119)(73,146,101,118)(74,145,102,117)(75,144,103,116)(76,143,104,115)(77,142,105,114)(78,141,106,113)(79,140,107,168)(80,139,108,167)(81,138,109,166)(82,137,110,165)(83,136,111,164)(84,135,112,163)(169,263,197,235)(170,262,198,234)(171,261,199,233)(172,260,200,232)(173,259,201,231)(174,258,202,230)(175,257,203,229)(176,256,204,228)(177,255,205,227)(178,254,206,226)(179,253,207,225)(180,252,208,280)(181,251,209,279)(182,250,210,278)(183,249,211,277)(184,248,212,276)(185,247,213,275)(186,246,214,274)(187,245,215,273)(188,244,216,272)(189,243,217,271)(190,242,218,270)(191,241,219,269)(192,240,220,268)(193,239,221,267)(194,238,222,266)(195,237,223,265)(196,236,224,264)(281,392,309,364)(282,391,310,363)(283,390,311,362)(284,389,312,361)(285,388,313,360)(286,387,314,359)(287,386,315,358)(288,385,316,357)(289,384,317,356)(290,383,318,355)(291,382,319,354)(292,381,320,353)(293,380,321,352)(294,379,322,351)(295,378,323,350)(296,377,324,349)(297,376,325,348)(298,375,326,347)(299,374,327,346)(300,373,328,345)(301,372,329,344)(302,371,330,343)(303,370,331,342)(304,369,332,341)(305,368,333,340)(306,367,334,339)(307,366,335,338)(308,365,336,337), (1,295,124,226)(2,282,125,269)(3,325,126,256)(4,312,127,243)(5,299,128,230)(6,286,129,273)(7,329,130,260)(8,316,131,247)(9,303,132,234)(10,290,133,277)(11,333,134,264)(12,320,135,251)(13,307,136,238)(14,294,137,225)(15,281,138,268)(16,324,139,255)(17,311,140,242)(18,298,141,229)(19,285,142,272)(20,328,143,259)(21,315,144,246)(22,302,145,233)(23,289,146,276)(24,332,147,263)(25,319,148,250)(26,306,149,237)(27,293,150,280)(28,336,151,267)(29,323,152,254)(30,310,153,241)(31,297,154,228)(32,284,155,271)(33,327,156,258)(34,314,157,245)(35,301,158,232)(36,288,159,275)(37,331,160,262)(38,318,161,249)(39,305,162,236)(40,292,163,279)(41,335,164,266)(42,322,165,253)(43,309,166,240)(44,296,167,227)(45,283,168,270)(46,326,113,257)(47,313,114,244)(48,300,115,231)(49,287,116,274)(50,330,117,261)(51,317,118,248)(52,304,119,235)(53,291,120,278)(54,334,121,265)(55,321,122,252)(56,308,123,239)(57,224,421,368)(58,211,422,355)(59,198,423,342)(60,185,424,385)(61,172,425,372)(62,215,426,359)(63,202,427,346)(64,189,428,389)(65,176,429,376)(66,219,430,363)(67,206,431,350)(68,193,432,337)(69,180,433,380)(70,223,434,367)(71,210,435,354)(72,197,436,341)(73,184,437,384)(74,171,438,371)(75,214,439,358)(76,201,440,345)(77,188,441,388)(78,175,442,375)(79,218,443,362)(80,205,444,349)(81,192,445,392)(82,179,446,379)(83,222,447,366)(84,209,448,353)(85,196,393,340)(86,183,394,383)(87,170,395,370)(88,213,396,357)(89,200,397,344)(90,187,398,387)(91,174,399,374)(92,217,400,361)(93,204,401,348)(94,191,402,391)(95,178,403,378)(96,221,404,365)(97,208,405,352)(98,195,406,339)(99,182,407,382)(100,169,408,369)(101,212,409,356)(102,199,410,343)(103,186,411,386)(104,173,412,373)(105,216,413,360)(106,203,414,347)(107,190,415,390)(108,177,416,377)(109,220,417,364)(110,207,418,351)(111,194,419,338)(112,181,420,381)>;
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), (1,431,29,403)(2,430,30,402)(3,429,31,401)(4,428,32,400)(5,427,33,399)(6,426,34,398)(7,425,35,397)(8,424,36,396)(9,423,37,395)(10,422,38,394)(11,421,39,393)(12,420,40,448)(13,419,41,447)(14,418,42,446)(15,417,43,445)(16,416,44,444)(17,415,45,443)(18,414,46,442)(19,413,47,441)(20,412,48,440)(21,411,49,439)(22,410,50,438)(23,409,51,437)(24,408,52,436)(25,407,53,435)(26,406,54,434)(27,405,55,433)(28,404,56,432)(57,162,85,134)(58,161,86,133)(59,160,87,132)(60,159,88,131)(61,158,89,130)(62,157,90,129)(63,156,91,128)(64,155,92,127)(65,154,93,126)(66,153,94,125)(67,152,95,124)(68,151,96,123)(69,150,97,122)(70,149,98,121)(71,148,99,120)(72,147,100,119)(73,146,101,118)(74,145,102,117)(75,144,103,116)(76,143,104,115)(77,142,105,114)(78,141,106,113)(79,140,107,168)(80,139,108,167)(81,138,109,166)(82,137,110,165)(83,136,111,164)(84,135,112,163)(169,263,197,235)(170,262,198,234)(171,261,199,233)(172,260,200,232)(173,259,201,231)(174,258,202,230)(175,257,203,229)(176,256,204,228)(177,255,205,227)(178,254,206,226)(179,253,207,225)(180,252,208,280)(181,251,209,279)(182,250,210,278)(183,249,211,277)(184,248,212,276)(185,247,213,275)(186,246,214,274)(187,245,215,273)(188,244,216,272)(189,243,217,271)(190,242,218,270)(191,241,219,269)(192,240,220,268)(193,239,221,267)(194,238,222,266)(195,237,223,265)(196,236,224,264)(281,392,309,364)(282,391,310,363)(283,390,311,362)(284,389,312,361)(285,388,313,360)(286,387,314,359)(287,386,315,358)(288,385,316,357)(289,384,317,356)(290,383,318,355)(291,382,319,354)(292,381,320,353)(293,380,321,352)(294,379,322,351)(295,378,323,350)(296,377,324,349)(297,376,325,348)(298,375,326,347)(299,374,327,346)(300,373,328,345)(301,372,329,344)(302,371,330,343)(303,370,331,342)(304,369,332,341)(305,368,333,340)(306,367,334,339)(307,366,335,338)(308,365,336,337), (1,295,124,226)(2,282,125,269)(3,325,126,256)(4,312,127,243)(5,299,128,230)(6,286,129,273)(7,329,130,260)(8,316,131,247)(9,303,132,234)(10,290,133,277)(11,333,134,264)(12,320,135,251)(13,307,136,238)(14,294,137,225)(15,281,138,268)(16,324,139,255)(17,311,140,242)(18,298,141,229)(19,285,142,272)(20,328,143,259)(21,315,144,246)(22,302,145,233)(23,289,146,276)(24,332,147,263)(25,319,148,250)(26,306,149,237)(27,293,150,280)(28,336,151,267)(29,323,152,254)(30,310,153,241)(31,297,154,228)(32,284,155,271)(33,327,156,258)(34,314,157,245)(35,301,158,232)(36,288,159,275)(37,331,160,262)(38,318,161,249)(39,305,162,236)(40,292,163,279)(41,335,164,266)(42,322,165,253)(43,309,166,240)(44,296,167,227)(45,283,168,270)(46,326,113,257)(47,313,114,244)(48,300,115,231)(49,287,116,274)(50,330,117,261)(51,317,118,248)(52,304,119,235)(53,291,120,278)(54,334,121,265)(55,321,122,252)(56,308,123,239)(57,224,421,368)(58,211,422,355)(59,198,423,342)(60,185,424,385)(61,172,425,372)(62,215,426,359)(63,202,427,346)(64,189,428,389)(65,176,429,376)(66,219,430,363)(67,206,431,350)(68,193,432,337)(69,180,433,380)(70,223,434,367)(71,210,435,354)(72,197,436,341)(73,184,437,384)(74,171,438,371)(75,214,439,358)(76,201,440,345)(77,188,441,388)(78,175,442,375)(79,218,443,362)(80,205,444,349)(81,192,445,392)(82,179,446,379)(83,222,447,366)(84,209,448,353)(85,196,393,340)(86,183,394,383)(87,170,395,370)(88,213,396,357)(89,200,397,344)(90,187,398,387)(91,174,399,374)(92,217,400,361)(93,204,401,348)(94,191,402,391)(95,178,403,378)(96,221,404,365)(97,208,405,352)(98,195,406,339)(99,182,407,382)(100,169,408,369)(101,212,409,356)(102,199,410,343)(103,186,411,386)(104,173,412,373)(105,216,413,360)(106,203,414,347)(107,190,415,390)(108,177,416,377)(109,220,417,364)(110,207,418,351)(111,194,419,338)(112,181,420,381) );
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)], [(1,431,29,403),(2,430,30,402),(3,429,31,401),(4,428,32,400),(5,427,33,399),(6,426,34,398),(7,425,35,397),(8,424,36,396),(9,423,37,395),(10,422,38,394),(11,421,39,393),(12,420,40,448),(13,419,41,447),(14,418,42,446),(15,417,43,445),(16,416,44,444),(17,415,45,443),(18,414,46,442),(19,413,47,441),(20,412,48,440),(21,411,49,439),(22,410,50,438),(23,409,51,437),(24,408,52,436),(25,407,53,435),(26,406,54,434),(27,405,55,433),(28,404,56,432),(57,162,85,134),(58,161,86,133),(59,160,87,132),(60,159,88,131),(61,158,89,130),(62,157,90,129),(63,156,91,128),(64,155,92,127),(65,154,93,126),(66,153,94,125),(67,152,95,124),(68,151,96,123),(69,150,97,122),(70,149,98,121),(71,148,99,120),(72,147,100,119),(73,146,101,118),(74,145,102,117),(75,144,103,116),(76,143,104,115),(77,142,105,114),(78,141,106,113),(79,140,107,168),(80,139,108,167),(81,138,109,166),(82,137,110,165),(83,136,111,164),(84,135,112,163),(169,263,197,235),(170,262,198,234),(171,261,199,233),(172,260,200,232),(173,259,201,231),(174,258,202,230),(175,257,203,229),(176,256,204,228),(177,255,205,227),(178,254,206,226),(179,253,207,225),(180,252,208,280),(181,251,209,279),(182,250,210,278),(183,249,211,277),(184,248,212,276),(185,247,213,275),(186,246,214,274),(187,245,215,273),(188,244,216,272),(189,243,217,271),(190,242,218,270),(191,241,219,269),(192,240,220,268),(193,239,221,267),(194,238,222,266),(195,237,223,265),(196,236,224,264),(281,392,309,364),(282,391,310,363),(283,390,311,362),(284,389,312,361),(285,388,313,360),(286,387,314,359),(287,386,315,358),(288,385,316,357),(289,384,317,356),(290,383,318,355),(291,382,319,354),(292,381,320,353),(293,380,321,352),(294,379,322,351),(295,378,323,350),(296,377,324,349),(297,376,325,348),(298,375,326,347),(299,374,327,346),(300,373,328,345),(301,372,329,344),(302,371,330,343),(303,370,331,342),(304,369,332,341),(305,368,333,340),(306,367,334,339),(307,366,335,338),(308,365,336,337)], [(1,295,124,226),(2,282,125,269),(3,325,126,256),(4,312,127,243),(5,299,128,230),(6,286,129,273),(7,329,130,260),(8,316,131,247),(9,303,132,234),(10,290,133,277),(11,333,134,264),(12,320,135,251),(13,307,136,238),(14,294,137,225),(15,281,138,268),(16,324,139,255),(17,311,140,242),(18,298,141,229),(19,285,142,272),(20,328,143,259),(21,315,144,246),(22,302,145,233),(23,289,146,276),(24,332,147,263),(25,319,148,250),(26,306,149,237),(27,293,150,280),(28,336,151,267),(29,323,152,254),(30,310,153,241),(31,297,154,228),(32,284,155,271),(33,327,156,258),(34,314,157,245),(35,301,158,232),(36,288,159,275),(37,331,160,262),(38,318,161,249),(39,305,162,236),(40,292,163,279),(41,335,164,266),(42,322,165,253),(43,309,166,240),(44,296,167,227),(45,283,168,270),(46,326,113,257),(47,313,114,244),(48,300,115,231),(49,287,116,274),(50,330,117,261),(51,317,118,248),(52,304,119,235),(53,291,120,278),(54,334,121,265),(55,321,122,252),(56,308,123,239),(57,224,421,368),(58,211,422,355),(59,198,423,342),(60,185,424,385),(61,172,425,372),(62,215,426,359),(63,202,427,346),(64,189,428,389),(65,176,429,376),(66,219,430,363),(67,206,431,350),(68,193,432,337),(69,180,433,380),(70,223,434,367),(71,210,435,354),(72,197,436,341),(73,184,437,384),(74,171,438,371),(75,214,439,358),(76,201,440,345),(77,188,441,388),(78,175,442,375),(79,218,443,362),(80,205,444,349),(81,192,445,392),(82,179,446,379),(83,222,447,366),(84,209,448,353),(85,196,393,340),(86,183,394,383),(87,170,395,370),(88,213,396,357),(89,200,397,344),(90,187,398,387),(91,174,399,374),(92,217,400,361),(93,204,401,348),(94,191,402,391),(95,178,403,378),(96,221,404,365),(97,208,405,352),(98,195,406,339),(99,182,407,382),(100,169,408,369),(101,212,409,356),(102,199,410,343),(103,186,411,386),(104,173,412,373),(105,216,413,360),(106,203,414,347),(107,190,415,390),(108,177,416,377),(109,220,417,364),(110,207,418,351),(111,194,419,338),(112,181,420,381)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 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 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | C8.C22 | Q8⋊2D7 | D4×D7 | SD16⋊D7 |
| kernel | Dic28⋊9C4 | C14.Q16 | C56⋊C4 | C7×C4.Q8 | Dic7⋊3Q8 | C2×Dic28 | Dic28 | C2×Dic7 | C4.Q8 | C28 | C4⋊C4 | C2×C8 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 3 | 2 | 6 | 3 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of Dic28⋊9C4 ►in GL6(𝔽113)
| 112 | 1 | 0 | 0 | 0 | 0 |
| 87 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 58 |
| 0 | 0 | 0 | 0 | 98 | 41 |
| 0 | 0 | 79 | 18 | 17 | 93 |
| 0 | 0 | 24 | 81 | 20 | 63 |
| 100 | 36 | 0 | 0 | 0 | 0 |
| 33 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 18 | 44 | 41 |
| 0 | 0 | 56 | 65 | 71 | 31 |
| 0 | 0 | 23 | 68 | 50 | 20 |
| 0 | 0 | 2 | 29 | 18 | 63 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 19 | 59 | 4 |
| 0 | 0 | 103 | 13 | 24 | 72 |
| 0 | 0 | 2 | 88 | 77 | 94 |
| 0 | 0 | 99 | 62 | 19 | 19 |
G:=sub<GL(6,GF(113))| [112,87,0,0,0,0,1,25,0,0,0,0,0,0,0,0,79,24,0,0,0,0,18,81,0,0,22,98,17,20,0,0,58,41,93,63],[100,33,0,0,0,0,36,13,0,0,0,0,0,0,48,56,23,2,0,0,18,65,68,29,0,0,44,71,50,18,0,0,41,31,20,63],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,4,103,2,99,0,0,19,13,88,62,0,0,59,24,77,19,0,0,4,72,94,19] >;
Dic28⋊9C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{28}\rtimes_9C_4 % in TeX
G:=Group("Dic28:9C4"); // GroupNames label
G:=SmallGroup(448,387);
// by ID
G=gap.SmallGroup(448,387);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,758,135,100,570,297,136,18822]);
// 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^43,c*b*c^-1=a^28*b>;
// generators/relations