metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.1Q16, C4.5Dic28, C42.3D14, C28.35SD16, C4⋊C8.3D7, C28⋊C8.8C2, (C2×C4).121D28, (C2×C28).463D4, C4.9(C56⋊C2), C4.9(D4.D7), C28⋊2Q8.7C2, C7⋊1(C4.6Q16), C4.9(C7⋊Q16), (C4×C28).39C22, (C2×Dic14).2C4, C14.7(Q8⋊C4), C14.3(C4.D4), C2.4(C14.Q16), C22.60(D14⋊C4), C2.4(C28.46D4), C2.4(C28.44D4), (C7×C4⋊C8).3C2, (C2×C4).14(C4×D7), (C2×C28).26(C2×C4), (C2×C4).227(C7⋊D4), (C2×C14).43(C22⋊C4), SmallGroup(448,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic28
G = < a,b,c | a4=b56=1, c2=ab28, bab-1=a-1, ac=ca, cbc-1=a-1b-1 >
Subgroups: 324 in 64 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C4⋊C8, C4⋊C8, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C4.6Q16, C2×C7⋊C8, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28⋊C8, C7×C4⋊C8, C28⋊2Q8, C4.Dic28
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, SD16, Q16, D14, C4.D4, Q8⋊C4, C4×D7, D28, C7⋊D4, C4.6Q16, C56⋊C2, Dic28, D14⋊C4, D4.D7, C7⋊Q16, C14.Q16, C28.44D4, C28.46D4, C4.Dic28
►Regular action on 448 points
Generators in S448
(1 163 74 434)(2 435 75 164)(3 165 76 436)(4 437 77 166)(5 167 78 438)(6 439 79 168)(7 113 80 440)(8 441 81 114)(9 115 82 442)(10 443 83 116)(11 117 84 444)(12 445 85 118)(13 119 86 446)(14 447 87 120)(15 121 88 448)(16 393 89 122)(17 123 90 394)(18 395 91 124)(19 125 92 396)(20 397 93 126)(21 127 94 398)(22 399 95 128)(23 129 96 400)(24 401 97 130)(25 131 98 402)(26 403 99 132)(27 133 100 404)(28 405 101 134)(29 135 102 406)(30 407 103 136)(31 137 104 408)(32 409 105 138)(33 139 106 410)(34 411 107 140)(35 141 108 412)(36 413 109 142)(37 143 110 414)(38 415 111 144)(39 145 112 416)(40 417 57 146)(41 147 58 418)(42 419 59 148)(43 149 60 420)(44 421 61 150)(45 151 62 422)(46 423 63 152)(47 153 64 424)(48 425 65 154)(49 155 66 426)(50 427 67 156)(51 157 68 428)(52 429 69 158)(53 159 70 430)(54 431 71 160)(55 161 72 432)(56 433 73 162)(169 249 302 339)(170 340 303 250)(171 251 304 341)(172 342 305 252)(173 253 306 343)(174 344 307 254)(175 255 308 345)(176 346 309 256)(177 257 310 347)(178 348 311 258)(179 259 312 349)(180 350 313 260)(181 261 314 351)(182 352 315 262)(183 263 316 353)(184 354 317 264)(185 265 318 355)(186 356 319 266)(187 267 320 357)(188 358 321 268)(189 269 322 359)(190 360 323 270)(191 271 324 361)(192 362 325 272)(193 273 326 363)(194 364 327 274)(195 275 328 365)(196 366 329 276)(197 277 330 367)(198 368 331 278)(199 279 332 369)(200 370 333 280)(201 225 334 371)(202 372 335 226)(203 227 336 373)(204 374 281 228)(205 229 282 375)(206 376 283 230)(207 231 284 377)(208 378 285 232)(209 233 286 379)(210 380 287 234)(211 235 288 381)(212 382 289 236)(213 237 290 383)(214 384 291 238)(215 239 292 385)(216 386 293 240)(217 241 294 387)(218 388 295 242)(219 243 296 389)(220 390 297 244)(221 245 298 391)(222 392 299 246)(223 247 300 337)(224 338 301 248)
(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)
(1 248 135 196 74 338 406 329)(2 223 407 275 75 300 136 365)(3 246 137 194 76 392 408 327)(4 221 409 273 77 298 138 363)(5 244 139 192 78 390 410 325)(6 219 411 271 79 296 140 361)(7 242 141 190 80 388 412 323)(8 217 413 269 81 294 142 359)(9 240 143 188 82 386 414 321)(10 215 415 267 83 292 144 357)(11 238 145 186 84 384 416 319)(12 213 417 265 85 290 146 355)(13 236 147 184 86 382 418 317)(14 211 419 263 87 288 148 353)(15 234 149 182 88 380 420 315)(16 209 421 261 89 286 150 351)(17 232 151 180 90 378 422 313)(18 207 423 259 91 284 152 349)(19 230 153 178 92 376 424 311)(20 205 425 257 93 282 154 347)(21 228 155 176 94 374 426 309)(22 203 427 255 95 336 156 345)(23 226 157 174 96 372 428 307)(24 201 429 253 97 334 158 343)(25 280 159 172 98 370 430 305)(26 199 431 251 99 332 160 341)(27 278 161 170 100 368 432 303)(28 197 433 249 101 330 162 339)(29 276 163 224 102 366 434 301)(30 195 435 247 103 328 164 337)(31 274 165 222 104 364 436 299)(32 193 437 245 105 326 166 391)(33 272 167 220 106 362 438 297)(34 191 439 243 107 324 168 389)(35 270 113 218 108 360 440 295)(36 189 441 241 109 322 114 387)(37 268 115 216 110 358 442 293)(38 187 443 239 111 320 116 385)(39 266 117 214 112 356 444 291)(40 185 445 237 57 318 118 383)(41 264 119 212 58 354 446 289)(42 183 447 235 59 316 120 381)(43 262 121 210 60 352 448 287)(44 181 393 233 61 314 122 379)(45 260 123 208 62 350 394 285)(46 179 395 231 63 312 124 377)(47 258 125 206 64 348 396 283)(48 177 397 229 65 310 126 375)(49 256 127 204 66 346 398 281)(50 175 399 227 67 308 128 373)(51 254 129 202 68 344 400 335)(52 173 401 225 69 306 130 371)(53 252 131 200 70 342 402 333)(54 171 403 279 71 304 132 369)(55 250 133 198 72 340 404 331)(56 169 405 277 73 302 134 367)
G:=sub<Sym(448)| (1,163,74,434)(2,435,75,164)(3,165,76,436)(4,437,77,166)(5,167,78,438)(6,439,79,168)(7,113,80,440)(8,441,81,114)(9,115,82,442)(10,443,83,116)(11,117,84,444)(12,445,85,118)(13,119,86,446)(14,447,87,120)(15,121,88,448)(16,393,89,122)(17,123,90,394)(18,395,91,124)(19,125,92,396)(20,397,93,126)(21,127,94,398)(22,399,95,128)(23,129,96,400)(24,401,97,130)(25,131,98,402)(26,403,99,132)(27,133,100,404)(28,405,101,134)(29,135,102,406)(30,407,103,136)(31,137,104,408)(32,409,105,138)(33,139,106,410)(34,411,107,140)(35,141,108,412)(36,413,109,142)(37,143,110,414)(38,415,111,144)(39,145,112,416)(40,417,57,146)(41,147,58,418)(42,419,59,148)(43,149,60,420)(44,421,61,150)(45,151,62,422)(46,423,63,152)(47,153,64,424)(48,425,65,154)(49,155,66,426)(50,427,67,156)(51,157,68,428)(52,429,69,158)(53,159,70,430)(54,431,71,160)(55,161,72,432)(56,433,73,162)(169,249,302,339)(170,340,303,250)(171,251,304,341)(172,342,305,252)(173,253,306,343)(174,344,307,254)(175,255,308,345)(176,346,309,256)(177,257,310,347)(178,348,311,258)(179,259,312,349)(180,350,313,260)(181,261,314,351)(182,352,315,262)(183,263,316,353)(184,354,317,264)(185,265,318,355)(186,356,319,266)(187,267,320,357)(188,358,321,268)(189,269,322,359)(190,360,323,270)(191,271,324,361)(192,362,325,272)(193,273,326,363)(194,364,327,274)(195,275,328,365)(196,366,329,276)(197,277,330,367)(198,368,331,278)(199,279,332,369)(200,370,333,280)(201,225,334,371)(202,372,335,226)(203,227,336,373)(204,374,281,228)(205,229,282,375)(206,376,283,230)(207,231,284,377)(208,378,285,232)(209,233,286,379)(210,380,287,234)(211,235,288,381)(212,382,289,236)(213,237,290,383)(214,384,291,238)(215,239,292,385)(216,386,293,240)(217,241,294,387)(218,388,295,242)(219,243,296,389)(220,390,297,244)(221,245,298,391)(222,392,299,246)(223,247,300,337)(224,338,301,248), (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), (1,248,135,196,74,338,406,329)(2,223,407,275,75,300,136,365)(3,246,137,194,76,392,408,327)(4,221,409,273,77,298,138,363)(5,244,139,192,78,390,410,325)(6,219,411,271,79,296,140,361)(7,242,141,190,80,388,412,323)(8,217,413,269,81,294,142,359)(9,240,143,188,82,386,414,321)(10,215,415,267,83,292,144,357)(11,238,145,186,84,384,416,319)(12,213,417,265,85,290,146,355)(13,236,147,184,86,382,418,317)(14,211,419,263,87,288,148,353)(15,234,149,182,88,380,420,315)(16,209,421,261,89,286,150,351)(17,232,151,180,90,378,422,313)(18,207,423,259,91,284,152,349)(19,230,153,178,92,376,424,311)(20,205,425,257,93,282,154,347)(21,228,155,176,94,374,426,309)(22,203,427,255,95,336,156,345)(23,226,157,174,96,372,428,307)(24,201,429,253,97,334,158,343)(25,280,159,172,98,370,430,305)(26,199,431,251,99,332,160,341)(27,278,161,170,100,368,432,303)(28,197,433,249,101,330,162,339)(29,276,163,224,102,366,434,301)(30,195,435,247,103,328,164,337)(31,274,165,222,104,364,436,299)(32,193,437,245,105,326,166,391)(33,272,167,220,106,362,438,297)(34,191,439,243,107,324,168,389)(35,270,113,218,108,360,440,295)(36,189,441,241,109,322,114,387)(37,268,115,216,110,358,442,293)(38,187,443,239,111,320,116,385)(39,266,117,214,112,356,444,291)(40,185,445,237,57,318,118,383)(41,264,119,212,58,354,446,289)(42,183,447,235,59,316,120,381)(43,262,121,210,60,352,448,287)(44,181,393,233,61,314,122,379)(45,260,123,208,62,350,394,285)(46,179,395,231,63,312,124,377)(47,258,125,206,64,348,396,283)(48,177,397,229,65,310,126,375)(49,256,127,204,66,346,398,281)(50,175,399,227,67,308,128,373)(51,254,129,202,68,344,400,335)(52,173,401,225,69,306,130,371)(53,252,131,200,70,342,402,333)(54,171,403,279,71,304,132,369)(55,250,133,198,72,340,404,331)(56,169,405,277,73,302,134,367)>;
G:=Group( (1,163,74,434)(2,435,75,164)(3,165,76,436)(4,437,77,166)(5,167,78,438)(6,439,79,168)(7,113,80,440)(8,441,81,114)(9,115,82,442)(10,443,83,116)(11,117,84,444)(12,445,85,118)(13,119,86,446)(14,447,87,120)(15,121,88,448)(16,393,89,122)(17,123,90,394)(18,395,91,124)(19,125,92,396)(20,397,93,126)(21,127,94,398)(22,399,95,128)(23,129,96,400)(24,401,97,130)(25,131,98,402)(26,403,99,132)(27,133,100,404)(28,405,101,134)(29,135,102,406)(30,407,103,136)(31,137,104,408)(32,409,105,138)(33,139,106,410)(34,411,107,140)(35,141,108,412)(36,413,109,142)(37,143,110,414)(38,415,111,144)(39,145,112,416)(40,417,57,146)(41,147,58,418)(42,419,59,148)(43,149,60,420)(44,421,61,150)(45,151,62,422)(46,423,63,152)(47,153,64,424)(48,425,65,154)(49,155,66,426)(50,427,67,156)(51,157,68,428)(52,429,69,158)(53,159,70,430)(54,431,71,160)(55,161,72,432)(56,433,73,162)(169,249,302,339)(170,340,303,250)(171,251,304,341)(172,342,305,252)(173,253,306,343)(174,344,307,254)(175,255,308,345)(176,346,309,256)(177,257,310,347)(178,348,311,258)(179,259,312,349)(180,350,313,260)(181,261,314,351)(182,352,315,262)(183,263,316,353)(184,354,317,264)(185,265,318,355)(186,356,319,266)(187,267,320,357)(188,358,321,268)(189,269,322,359)(190,360,323,270)(191,271,324,361)(192,362,325,272)(193,273,326,363)(194,364,327,274)(195,275,328,365)(196,366,329,276)(197,277,330,367)(198,368,331,278)(199,279,332,369)(200,370,333,280)(201,225,334,371)(202,372,335,226)(203,227,336,373)(204,374,281,228)(205,229,282,375)(206,376,283,230)(207,231,284,377)(208,378,285,232)(209,233,286,379)(210,380,287,234)(211,235,288,381)(212,382,289,236)(213,237,290,383)(214,384,291,238)(215,239,292,385)(216,386,293,240)(217,241,294,387)(218,388,295,242)(219,243,296,389)(220,390,297,244)(221,245,298,391)(222,392,299,246)(223,247,300,337)(224,338,301,248), (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), (1,248,135,196,74,338,406,329)(2,223,407,275,75,300,136,365)(3,246,137,194,76,392,408,327)(4,221,409,273,77,298,138,363)(5,244,139,192,78,390,410,325)(6,219,411,271,79,296,140,361)(7,242,141,190,80,388,412,323)(8,217,413,269,81,294,142,359)(9,240,143,188,82,386,414,321)(10,215,415,267,83,292,144,357)(11,238,145,186,84,384,416,319)(12,213,417,265,85,290,146,355)(13,236,147,184,86,382,418,317)(14,211,419,263,87,288,148,353)(15,234,149,182,88,380,420,315)(16,209,421,261,89,286,150,351)(17,232,151,180,90,378,422,313)(18,207,423,259,91,284,152,349)(19,230,153,178,92,376,424,311)(20,205,425,257,93,282,154,347)(21,228,155,176,94,374,426,309)(22,203,427,255,95,336,156,345)(23,226,157,174,96,372,428,307)(24,201,429,253,97,334,158,343)(25,280,159,172,98,370,430,305)(26,199,431,251,99,332,160,341)(27,278,161,170,100,368,432,303)(28,197,433,249,101,330,162,339)(29,276,163,224,102,366,434,301)(30,195,435,247,103,328,164,337)(31,274,165,222,104,364,436,299)(32,193,437,245,105,326,166,391)(33,272,167,220,106,362,438,297)(34,191,439,243,107,324,168,389)(35,270,113,218,108,360,440,295)(36,189,441,241,109,322,114,387)(37,268,115,216,110,358,442,293)(38,187,443,239,111,320,116,385)(39,266,117,214,112,356,444,291)(40,185,445,237,57,318,118,383)(41,264,119,212,58,354,446,289)(42,183,447,235,59,316,120,381)(43,262,121,210,60,352,448,287)(44,181,393,233,61,314,122,379)(45,260,123,208,62,350,394,285)(46,179,395,231,63,312,124,377)(47,258,125,206,64,348,396,283)(48,177,397,229,65,310,126,375)(49,256,127,204,66,346,398,281)(50,175,399,227,67,308,128,373)(51,254,129,202,68,344,400,335)(52,173,401,225,69,306,130,371)(53,252,131,200,70,342,402,333)(54,171,403,279,71,304,132,369)(55,250,133,198,72,340,404,331)(56,169,405,277,73,302,134,367) );
G=PermutationGroup([[(1,163,74,434),(2,435,75,164),(3,165,76,436),(4,437,77,166),(5,167,78,438),(6,439,79,168),(7,113,80,440),(8,441,81,114),(9,115,82,442),(10,443,83,116),(11,117,84,444),(12,445,85,118),(13,119,86,446),(14,447,87,120),(15,121,88,448),(16,393,89,122),(17,123,90,394),(18,395,91,124),(19,125,92,396),(20,397,93,126),(21,127,94,398),(22,399,95,128),(23,129,96,400),(24,401,97,130),(25,131,98,402),(26,403,99,132),(27,133,100,404),(28,405,101,134),(29,135,102,406),(30,407,103,136),(31,137,104,408),(32,409,105,138),(33,139,106,410),(34,411,107,140),(35,141,108,412),(36,413,109,142),(37,143,110,414),(38,415,111,144),(39,145,112,416),(40,417,57,146),(41,147,58,418),(42,419,59,148),(43,149,60,420),(44,421,61,150),(45,151,62,422),(46,423,63,152),(47,153,64,424),(48,425,65,154),(49,155,66,426),(50,427,67,156),(51,157,68,428),(52,429,69,158),(53,159,70,430),(54,431,71,160),(55,161,72,432),(56,433,73,162),(169,249,302,339),(170,340,303,250),(171,251,304,341),(172,342,305,252),(173,253,306,343),(174,344,307,254),(175,255,308,345),(176,346,309,256),(177,257,310,347),(178,348,311,258),(179,259,312,349),(180,350,313,260),(181,261,314,351),(182,352,315,262),(183,263,316,353),(184,354,317,264),(185,265,318,355),(186,356,319,266),(187,267,320,357),(188,358,321,268),(189,269,322,359),(190,360,323,270),(191,271,324,361),(192,362,325,272),(193,273,326,363),(194,364,327,274),(195,275,328,365),(196,366,329,276),(197,277,330,367),(198,368,331,278),(199,279,332,369),(200,370,333,280),(201,225,334,371),(202,372,335,226),(203,227,336,373),(204,374,281,228),(205,229,282,375),(206,376,283,230),(207,231,284,377),(208,378,285,232),(209,233,286,379),(210,380,287,234),(211,235,288,381),(212,382,289,236),(213,237,290,383),(214,384,291,238),(215,239,292,385),(216,386,293,240),(217,241,294,387),(218,388,295,242),(219,243,296,389),(220,390,297,244),(221,245,298,391),(222,392,299,246),(223,247,300,337),(224,338,301,248)], [(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)], [(1,248,135,196,74,338,406,329),(2,223,407,275,75,300,136,365),(3,246,137,194,76,392,408,327),(4,221,409,273,77,298,138,363),(5,244,139,192,78,390,410,325),(6,219,411,271,79,296,140,361),(7,242,141,190,80,388,412,323),(8,217,413,269,81,294,142,359),(9,240,143,188,82,386,414,321),(10,215,415,267,83,292,144,357),(11,238,145,186,84,384,416,319),(12,213,417,265,85,290,146,355),(13,236,147,184,86,382,418,317),(14,211,419,263,87,288,148,353),(15,234,149,182,88,380,420,315),(16,209,421,261,89,286,150,351),(17,232,151,180,90,378,422,313),(18,207,423,259,91,284,152,349),(19,230,153,178,92,376,424,311),(20,205,425,257,93,282,154,347),(21,228,155,176,94,374,426,309),(22,203,427,255,95,336,156,345),(23,226,157,174,96,372,428,307),(24,201,429,253,97,334,158,343),(25,280,159,172,98,370,430,305),(26,199,431,251,99,332,160,341),(27,278,161,170,100,368,432,303),(28,197,433,249,101,330,162,339),(29,276,163,224,102,366,434,301),(30,195,435,247,103,328,164,337),(31,274,165,222,104,364,436,299),(32,193,437,245,105,326,166,391),(33,272,167,220,106,362,438,297),(34,191,439,243,107,324,168,389),(35,270,113,218,108,360,440,295),(36,189,441,241,109,322,114,387),(37,268,115,216,110,358,442,293),(38,187,443,239,111,320,116,385),(39,266,117,214,112,356,444,291),(40,185,445,237,57,318,118,383),(41,264,119,212,58,354,446,289),(42,183,447,235,59,316,120,381),(43,262,121,210,60,352,448,287),(44,181,393,233,61,314,122,379),(45,260,123,208,62,350,394,285),(46,179,395,231,63,312,124,377),(47,258,125,206,64,348,396,283),(48,177,397,229,65,310,126,375),(49,256,127,204,66,346,398,281),(50,175,399,227,67,308,128,373),(51,254,129,202,68,344,400,335),(52,173,401,225,69,306,130,371),(53,252,131,200,70,342,402,333),(54,171,403,279,71,304,132,369),(55,250,133,198,72,340,404,331),(56,169,405,277,73,302,134,367)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D7 | SD16 | Q16 | D14 | C4×D7 | D28 | C7⋊D4 | C56⋊C2 | Dic28 | C4.D4 | D4.D7 | C7⋊Q16 | C28.46D4 |
| kernel | C4.Dic28 | C28⋊C8 | C7×C4⋊C8 | C28⋊2Q8 | C2×Dic14 | C2×C28 | C4⋊C8 | C28 | C28 | C42 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 4 | 4 | 3 | 6 | 6 | 6 | 12 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of C4.Dic28 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 60 | 91 | 0 | 0 | 0 | 0 |
| 87 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 44 | 0 | 0 |
| 0 | 0 | 104 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 103 | 50 |
| 0 | 0 | 0 | 0 | 50 | 10 |
| 104 | 100 | 0 | 0 | 0 | 0 |
| 15 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 77 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 0 | 13 | 13 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[60,87,0,0,0,0,91,34,0,0,0,0,0,0,12,104,0,0,0,0,44,14,0,0,0,0,0,0,103,50,0,0,0,0,50,10],[104,15,0,0,0,0,100,9,0,0,0,0,0,0,112,77,0,0,0,0,0,1,0,0,0,0,0,0,13,13,0,0,0,0,100,13] >;
C4.Dic28 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_{28} % in TeX
G:=Group("C4.Dic28"); // GroupNames label
G:=SmallGroup(448,38);
// by ID
G=gap.SmallGroup(448,38);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,85,92,422,387,268,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^56=1,c^2=a*b^28,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^-1>;
// generators/relations