metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊1Q16, C4⋊2Dic28, C42.40D14, Dic14.19D4, C4⋊C8.7D7, C4.134(D4×D7), (C2×C8).23D14, C7⋊2(C4⋊2Q16), C14.7(C2×Q16), (C2×C4).136D28, (C2×C28).125D4, C28.343(C2×D4), C2.9(C2×Dic28), (C4×C28).75C22, (C2×C56).27C22, C28⋊2Q8.10C2, (C2×Dic28).4C2, C28.332(C4○D4), C14.42(C4⋊D4), C2.15(C4⋊D28), (C2×C28).759C23, C4.48(Q8⋊2D7), (C4×Dic14).11C2, C28.44D4.3C2, C22.122(C2×D28), C2.20(C8.D14), C14.17(C8.C22), C4⋊Dic7.277C22, (C2×Dic14).16C22, (C7×C4⋊C8).12C2, (C2×C14).142(C2×D4), (C2×C4).704(C22×D7), SmallGroup(448,383)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic28
G = < a,b,c | a4=b56=1, c2=b28, bab-1=a-1, ac=ca, cbc-1=b-1 >
Subgroups: 548 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, Dic7, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C56, Dic14, Dic14, C2×Dic7, C2×C28, C4⋊2Q16, Dic28, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C2×Dic14, C28.44D4, C7×C4⋊C8, C4×Dic14, C28⋊2Q8, C2×Dic28, C4⋊Dic28
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C4○D4, D14, C4⋊D4, C2×Q16, C8.C22, D28, C22×D7, C4⋊2Q16, Dic28, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, C2×Dic28, C8.D14, C4⋊Dic28
►Regular action on 448 points
Generators in S448
(1 169 428 227)(2 228 429 170)(3 171 430 229)(4 230 431 172)(5 173 432 231)(6 232 433 174)(7 175 434 233)(8 234 435 176)(9 177 436 235)(10 236 437 178)(11 179 438 237)(12 238 439 180)(13 181 440 239)(14 240 441 182)(15 183 442 241)(16 242 443 184)(17 185 444 243)(18 244 445 186)(19 187 446 245)(20 246 447 188)(21 189 448 247)(22 248 393 190)(23 191 394 249)(24 250 395 192)(25 193 396 251)(26 252 397 194)(27 195 398 253)(28 254 399 196)(29 197 400 255)(30 256 401 198)(31 199 402 257)(32 258 403 200)(33 201 404 259)(34 260 405 202)(35 203 406 261)(36 262 407 204)(37 205 408 263)(38 264 409 206)(39 207 410 265)(40 266 411 208)(41 209 412 267)(42 268 413 210)(43 211 414 269)(44 270 415 212)(45 213 416 271)(46 272 417 214)(47 215 418 273)(48 274 419 216)(49 217 420 275)(50 276 421 218)(51 219 422 277)(52 278 423 220)(53 221 424 279)(54 280 425 222)(55 223 426 225)(56 226 427 224)(57 317 390 163)(58 164 391 318)(59 319 392 165)(60 166 337 320)(61 321 338 167)(62 168 339 322)(63 323 340 113)(64 114 341 324)(65 325 342 115)(66 116 343 326)(67 327 344 117)(68 118 345 328)(69 329 346 119)(70 120 347 330)(71 331 348 121)(72 122 349 332)(73 333 350 123)(74 124 351 334)(75 335 352 125)(76 126 353 336)(77 281 354 127)(78 128 355 282)(79 283 356 129)(80 130 357 284)(81 285 358 131)(82 132 359 286)(83 287 360 133)(84 134 361 288)(85 289 362 135)(86 136 363 290)(87 291 364 137)(88 138 365 292)(89 293 366 139)(90 140 367 294)(91 295 368 141)(92 142 369 296)(93 297 370 143)(94 144 371 298)(95 299 372 145)(96 146 373 300)(97 301 374 147)(98 148 375 302)(99 303 376 149)(100 150 377 304)(101 305 378 151)(102 152 379 306)(103 307 380 153)(104 154 381 308)(105 309 382 155)(106 156 383 310)(107 311 384 157)(108 158 385 312)(109 313 386 159)(110 160 387 314)(111 315 388 161)(112 162 389 316)
(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)
(1 379 29 351)(2 378 30 350)(3 377 31 349)(4 376 32 348)(5 375 33 347)(6 374 34 346)(7 373 35 345)(8 372 36 344)(9 371 37 343)(10 370 38 342)(11 369 39 341)(12 368 40 340)(13 367 41 339)(14 366 42 338)(15 365 43 337)(16 364 44 392)(17 363 45 391)(18 362 46 390)(19 361 47 389)(20 360 48 388)(21 359 49 387)(22 358 50 386)(23 357 51 385)(24 356 52 384)(25 355 53 383)(26 354 54 382)(27 353 55 381)(28 352 56 380)(57 445 85 417)(58 444 86 416)(59 443 87 415)(60 442 88 414)(61 441 89 413)(62 440 90 412)(63 439 91 411)(64 438 92 410)(65 437 93 409)(66 436 94 408)(67 435 95 407)(68 434 96 406)(69 433 97 405)(70 432 98 404)(71 431 99 403)(72 430 100 402)(73 429 101 401)(74 428 102 400)(75 427 103 399)(76 426 104 398)(77 425 105 397)(78 424 106 396)(79 423 107 395)(80 422 108 394)(81 421 109 393)(82 420 110 448)(83 419 111 447)(84 418 112 446)(113 238 141 266)(114 237 142 265)(115 236 143 264)(116 235 144 263)(117 234 145 262)(118 233 146 261)(119 232 147 260)(120 231 148 259)(121 230 149 258)(122 229 150 257)(123 228 151 256)(124 227 152 255)(125 226 153 254)(126 225 154 253)(127 280 155 252)(128 279 156 251)(129 278 157 250)(130 277 158 249)(131 276 159 248)(132 275 160 247)(133 274 161 246)(134 273 162 245)(135 272 163 244)(136 271 164 243)(137 270 165 242)(138 269 166 241)(139 268 167 240)(140 267 168 239)(169 306 197 334)(170 305 198 333)(171 304 199 332)(172 303 200 331)(173 302 201 330)(174 301 202 329)(175 300 203 328)(176 299 204 327)(177 298 205 326)(178 297 206 325)(179 296 207 324)(180 295 208 323)(181 294 209 322)(182 293 210 321)(183 292 211 320)(184 291 212 319)(185 290 213 318)(186 289 214 317)(187 288 215 316)(188 287 216 315)(189 286 217 314)(190 285 218 313)(191 284 219 312)(192 283 220 311)(193 282 221 310)(194 281 222 309)(195 336 223 308)(196 335 224 307)
G:=sub<Sym(448)| (1,169,428,227)(2,228,429,170)(3,171,430,229)(4,230,431,172)(5,173,432,231)(6,232,433,174)(7,175,434,233)(8,234,435,176)(9,177,436,235)(10,236,437,178)(11,179,438,237)(12,238,439,180)(13,181,440,239)(14,240,441,182)(15,183,442,241)(16,242,443,184)(17,185,444,243)(18,244,445,186)(19,187,446,245)(20,246,447,188)(21,189,448,247)(22,248,393,190)(23,191,394,249)(24,250,395,192)(25,193,396,251)(26,252,397,194)(27,195,398,253)(28,254,399,196)(29,197,400,255)(30,256,401,198)(31,199,402,257)(32,258,403,200)(33,201,404,259)(34,260,405,202)(35,203,406,261)(36,262,407,204)(37,205,408,263)(38,264,409,206)(39,207,410,265)(40,266,411,208)(41,209,412,267)(42,268,413,210)(43,211,414,269)(44,270,415,212)(45,213,416,271)(46,272,417,214)(47,215,418,273)(48,274,419,216)(49,217,420,275)(50,276,421,218)(51,219,422,277)(52,278,423,220)(53,221,424,279)(54,280,425,222)(55,223,426,225)(56,226,427,224)(57,317,390,163)(58,164,391,318)(59,319,392,165)(60,166,337,320)(61,321,338,167)(62,168,339,322)(63,323,340,113)(64,114,341,324)(65,325,342,115)(66,116,343,326)(67,327,344,117)(68,118,345,328)(69,329,346,119)(70,120,347,330)(71,331,348,121)(72,122,349,332)(73,333,350,123)(74,124,351,334)(75,335,352,125)(76,126,353,336)(77,281,354,127)(78,128,355,282)(79,283,356,129)(80,130,357,284)(81,285,358,131)(82,132,359,286)(83,287,360,133)(84,134,361,288)(85,289,362,135)(86,136,363,290)(87,291,364,137)(88,138,365,292)(89,293,366,139)(90,140,367,294)(91,295,368,141)(92,142,369,296)(93,297,370,143)(94,144,371,298)(95,299,372,145)(96,146,373,300)(97,301,374,147)(98,148,375,302)(99,303,376,149)(100,150,377,304)(101,305,378,151)(102,152,379,306)(103,307,380,153)(104,154,381,308)(105,309,382,155)(106,156,383,310)(107,311,384,157)(108,158,385,312)(109,313,386,159)(110,160,387,314)(111,315,388,161)(112,162,389,316), (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), (1,379,29,351)(2,378,30,350)(3,377,31,349)(4,376,32,348)(5,375,33,347)(6,374,34,346)(7,373,35,345)(8,372,36,344)(9,371,37,343)(10,370,38,342)(11,369,39,341)(12,368,40,340)(13,367,41,339)(14,366,42,338)(15,365,43,337)(16,364,44,392)(17,363,45,391)(18,362,46,390)(19,361,47,389)(20,360,48,388)(21,359,49,387)(22,358,50,386)(23,357,51,385)(24,356,52,384)(25,355,53,383)(26,354,54,382)(27,353,55,381)(28,352,56,380)(57,445,85,417)(58,444,86,416)(59,443,87,415)(60,442,88,414)(61,441,89,413)(62,440,90,412)(63,439,91,411)(64,438,92,410)(65,437,93,409)(66,436,94,408)(67,435,95,407)(68,434,96,406)(69,433,97,405)(70,432,98,404)(71,431,99,403)(72,430,100,402)(73,429,101,401)(74,428,102,400)(75,427,103,399)(76,426,104,398)(77,425,105,397)(78,424,106,396)(79,423,107,395)(80,422,108,394)(81,421,109,393)(82,420,110,448)(83,419,111,447)(84,418,112,446)(113,238,141,266)(114,237,142,265)(115,236,143,264)(116,235,144,263)(117,234,145,262)(118,233,146,261)(119,232,147,260)(120,231,148,259)(121,230,149,258)(122,229,150,257)(123,228,151,256)(124,227,152,255)(125,226,153,254)(126,225,154,253)(127,280,155,252)(128,279,156,251)(129,278,157,250)(130,277,158,249)(131,276,159,248)(132,275,160,247)(133,274,161,246)(134,273,162,245)(135,272,163,244)(136,271,164,243)(137,270,165,242)(138,269,166,241)(139,268,167,240)(140,267,168,239)(169,306,197,334)(170,305,198,333)(171,304,199,332)(172,303,200,331)(173,302,201,330)(174,301,202,329)(175,300,203,328)(176,299,204,327)(177,298,205,326)(178,297,206,325)(179,296,207,324)(180,295,208,323)(181,294,209,322)(182,293,210,321)(183,292,211,320)(184,291,212,319)(185,290,213,318)(186,289,214,317)(187,288,215,316)(188,287,216,315)(189,286,217,314)(190,285,218,313)(191,284,219,312)(192,283,220,311)(193,282,221,310)(194,281,222,309)(195,336,223,308)(196,335,224,307)>;
G:=Group( (1,169,428,227)(2,228,429,170)(3,171,430,229)(4,230,431,172)(5,173,432,231)(6,232,433,174)(7,175,434,233)(8,234,435,176)(9,177,436,235)(10,236,437,178)(11,179,438,237)(12,238,439,180)(13,181,440,239)(14,240,441,182)(15,183,442,241)(16,242,443,184)(17,185,444,243)(18,244,445,186)(19,187,446,245)(20,246,447,188)(21,189,448,247)(22,248,393,190)(23,191,394,249)(24,250,395,192)(25,193,396,251)(26,252,397,194)(27,195,398,253)(28,254,399,196)(29,197,400,255)(30,256,401,198)(31,199,402,257)(32,258,403,200)(33,201,404,259)(34,260,405,202)(35,203,406,261)(36,262,407,204)(37,205,408,263)(38,264,409,206)(39,207,410,265)(40,266,411,208)(41,209,412,267)(42,268,413,210)(43,211,414,269)(44,270,415,212)(45,213,416,271)(46,272,417,214)(47,215,418,273)(48,274,419,216)(49,217,420,275)(50,276,421,218)(51,219,422,277)(52,278,423,220)(53,221,424,279)(54,280,425,222)(55,223,426,225)(56,226,427,224)(57,317,390,163)(58,164,391,318)(59,319,392,165)(60,166,337,320)(61,321,338,167)(62,168,339,322)(63,323,340,113)(64,114,341,324)(65,325,342,115)(66,116,343,326)(67,327,344,117)(68,118,345,328)(69,329,346,119)(70,120,347,330)(71,331,348,121)(72,122,349,332)(73,333,350,123)(74,124,351,334)(75,335,352,125)(76,126,353,336)(77,281,354,127)(78,128,355,282)(79,283,356,129)(80,130,357,284)(81,285,358,131)(82,132,359,286)(83,287,360,133)(84,134,361,288)(85,289,362,135)(86,136,363,290)(87,291,364,137)(88,138,365,292)(89,293,366,139)(90,140,367,294)(91,295,368,141)(92,142,369,296)(93,297,370,143)(94,144,371,298)(95,299,372,145)(96,146,373,300)(97,301,374,147)(98,148,375,302)(99,303,376,149)(100,150,377,304)(101,305,378,151)(102,152,379,306)(103,307,380,153)(104,154,381,308)(105,309,382,155)(106,156,383,310)(107,311,384,157)(108,158,385,312)(109,313,386,159)(110,160,387,314)(111,315,388,161)(112,162,389,316), (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), (1,379,29,351)(2,378,30,350)(3,377,31,349)(4,376,32,348)(5,375,33,347)(6,374,34,346)(7,373,35,345)(8,372,36,344)(9,371,37,343)(10,370,38,342)(11,369,39,341)(12,368,40,340)(13,367,41,339)(14,366,42,338)(15,365,43,337)(16,364,44,392)(17,363,45,391)(18,362,46,390)(19,361,47,389)(20,360,48,388)(21,359,49,387)(22,358,50,386)(23,357,51,385)(24,356,52,384)(25,355,53,383)(26,354,54,382)(27,353,55,381)(28,352,56,380)(57,445,85,417)(58,444,86,416)(59,443,87,415)(60,442,88,414)(61,441,89,413)(62,440,90,412)(63,439,91,411)(64,438,92,410)(65,437,93,409)(66,436,94,408)(67,435,95,407)(68,434,96,406)(69,433,97,405)(70,432,98,404)(71,431,99,403)(72,430,100,402)(73,429,101,401)(74,428,102,400)(75,427,103,399)(76,426,104,398)(77,425,105,397)(78,424,106,396)(79,423,107,395)(80,422,108,394)(81,421,109,393)(82,420,110,448)(83,419,111,447)(84,418,112,446)(113,238,141,266)(114,237,142,265)(115,236,143,264)(116,235,144,263)(117,234,145,262)(118,233,146,261)(119,232,147,260)(120,231,148,259)(121,230,149,258)(122,229,150,257)(123,228,151,256)(124,227,152,255)(125,226,153,254)(126,225,154,253)(127,280,155,252)(128,279,156,251)(129,278,157,250)(130,277,158,249)(131,276,159,248)(132,275,160,247)(133,274,161,246)(134,273,162,245)(135,272,163,244)(136,271,164,243)(137,270,165,242)(138,269,166,241)(139,268,167,240)(140,267,168,239)(169,306,197,334)(170,305,198,333)(171,304,199,332)(172,303,200,331)(173,302,201,330)(174,301,202,329)(175,300,203,328)(176,299,204,327)(177,298,205,326)(178,297,206,325)(179,296,207,324)(180,295,208,323)(181,294,209,322)(182,293,210,321)(183,292,211,320)(184,291,212,319)(185,290,213,318)(186,289,214,317)(187,288,215,316)(188,287,216,315)(189,286,217,314)(190,285,218,313)(191,284,219,312)(192,283,220,311)(193,282,221,310)(194,281,222,309)(195,336,223,308)(196,335,224,307) );
G=PermutationGroup([[(1,169,428,227),(2,228,429,170),(3,171,430,229),(4,230,431,172),(5,173,432,231),(6,232,433,174),(7,175,434,233),(8,234,435,176),(9,177,436,235),(10,236,437,178),(11,179,438,237),(12,238,439,180),(13,181,440,239),(14,240,441,182),(15,183,442,241),(16,242,443,184),(17,185,444,243),(18,244,445,186),(19,187,446,245),(20,246,447,188),(21,189,448,247),(22,248,393,190),(23,191,394,249),(24,250,395,192),(25,193,396,251),(26,252,397,194),(27,195,398,253),(28,254,399,196),(29,197,400,255),(30,256,401,198),(31,199,402,257),(32,258,403,200),(33,201,404,259),(34,260,405,202),(35,203,406,261),(36,262,407,204),(37,205,408,263),(38,264,409,206),(39,207,410,265),(40,266,411,208),(41,209,412,267),(42,268,413,210),(43,211,414,269),(44,270,415,212),(45,213,416,271),(46,272,417,214),(47,215,418,273),(48,274,419,216),(49,217,420,275),(50,276,421,218),(51,219,422,277),(52,278,423,220),(53,221,424,279),(54,280,425,222),(55,223,426,225),(56,226,427,224),(57,317,390,163),(58,164,391,318),(59,319,392,165),(60,166,337,320),(61,321,338,167),(62,168,339,322),(63,323,340,113),(64,114,341,324),(65,325,342,115),(66,116,343,326),(67,327,344,117),(68,118,345,328),(69,329,346,119),(70,120,347,330),(71,331,348,121),(72,122,349,332),(73,333,350,123),(74,124,351,334),(75,335,352,125),(76,126,353,336),(77,281,354,127),(78,128,355,282),(79,283,356,129),(80,130,357,284),(81,285,358,131),(82,132,359,286),(83,287,360,133),(84,134,361,288),(85,289,362,135),(86,136,363,290),(87,291,364,137),(88,138,365,292),(89,293,366,139),(90,140,367,294),(91,295,368,141),(92,142,369,296),(93,297,370,143),(94,144,371,298),(95,299,372,145),(96,146,373,300),(97,301,374,147),(98,148,375,302),(99,303,376,149),(100,150,377,304),(101,305,378,151),(102,152,379,306),(103,307,380,153),(104,154,381,308),(105,309,382,155),(106,156,383,310),(107,311,384,157),(108,158,385,312),(109,313,386,159),(110,160,387,314),(111,315,388,161),(112,162,389,316)], [(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)], [(1,379,29,351),(2,378,30,350),(3,377,31,349),(4,376,32,348),(5,375,33,347),(6,374,34,346),(7,373,35,345),(8,372,36,344),(9,371,37,343),(10,370,38,342),(11,369,39,341),(12,368,40,340),(13,367,41,339),(14,366,42,338),(15,365,43,337),(16,364,44,392),(17,363,45,391),(18,362,46,390),(19,361,47,389),(20,360,48,388),(21,359,49,387),(22,358,50,386),(23,357,51,385),(24,356,52,384),(25,355,53,383),(26,354,54,382),(27,353,55,381),(28,352,56,380),(57,445,85,417),(58,444,86,416),(59,443,87,415),(60,442,88,414),(61,441,89,413),(62,440,90,412),(63,439,91,411),(64,438,92,410),(65,437,93,409),(66,436,94,408),(67,435,95,407),(68,434,96,406),(69,433,97,405),(70,432,98,404),(71,431,99,403),(72,430,100,402),(73,429,101,401),(74,428,102,400),(75,427,103,399),(76,426,104,398),(77,425,105,397),(78,424,106,396),(79,423,107,395),(80,422,108,394),(81,421,109,393),(82,420,110,448),(83,419,111,447),(84,418,112,446),(113,238,141,266),(114,237,142,265),(115,236,143,264),(116,235,144,263),(117,234,145,262),(118,233,146,261),(119,232,147,260),(120,231,148,259),(121,230,149,258),(122,229,150,257),(123,228,151,256),(124,227,152,255),(125,226,153,254),(126,225,154,253),(127,280,155,252),(128,279,156,251),(129,278,157,250),(130,277,158,249),(131,276,159,248),(132,275,160,247),(133,274,161,246),(134,273,162,245),(135,272,163,244),(136,271,164,243),(137,270,165,242),(138,269,166,241),(139,268,167,240),(140,267,168,239),(169,306,197,334),(170,305,198,333),(171,304,199,332),(172,303,200,331),(173,302,201,330),(174,301,202,329),(175,300,203,328),(176,299,204,327),(177,298,205,326),(178,297,206,325),(179,296,207,324),(180,295,208,323),(181,294,209,322),(182,293,210,321),(183,292,211,320),(184,291,212,319),(185,290,213,318),(186,289,214,317),(187,288,215,316),(188,287,216,315),(189,286,217,314),(190,285,218,313),(191,284,219,312),(192,283,220,311),(193,282,221,310),(194,281,222,309),(195,336,223,308),(196,335,224,307)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | D28 | Dic28 | C8.C22 | D4×D7 | Q8⋊2D7 | C8.D14 |
| kernel | C4⋊Dic28 | C28.44D4 | C7×C4⋊C8 | C4×Dic14 | C28⋊2Q8 | C2×Dic28 | Dic14 | C2×C28 | C4⋊C8 | C28 | C28 | C42 | C2×C8 | C2×C4 | C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 24 | 1 | 3 | 3 | 6 |
Matrix representation of C4⋊Dic28 ►in GL4(𝔽113) generated by
| 13 | 13 | 0 | 0 |
| 13 | 100 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 92 | 91 |
| 0 | 0 | 87 | 5 |
| 82 | 82 | 0 | 0 |
| 82 | 31 | 0 | 0 |
| 0 | 0 | 55 | 38 |
| 0 | 0 | 75 | 58 |
G:=sub<GL(4,GF(113))| [13,13,0,0,13,100,0,0,0,0,1,0,0,0,0,1],[0,112,0,0,1,0,0,0,0,0,92,87,0,0,91,5],[82,82,0,0,82,31,0,0,0,0,55,75,0,0,38,58] >;
C4⋊Dic28 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_{28} % in TeX
G:=Group("C4:Dic28"); // GroupNames label
G:=SmallGroup(448,383);
// by ID
G=gap.SmallGroup(448,383);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,254,219,226,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^56=1,c^2=b^28,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations