direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Dic28, C28⋊5Q16, C42.265D14, C7⋊1(C4×Q16), (C4×C8).8D7, C8.21(C4×D7), C14.8(C4×D4), C56.51(C2×C4), (C4×C56).10C2, (C2×C4).64D28, C2.11(C4×D28), C14.3(C2×Q16), C14.6(C4○D8), (C2×C8).289D14, (C2×C28).354D4, C56⋊1C4.18C2, C2.1(C2×Dic28), (C4×Dic14).3C2, C22.29(C2×D28), C4.104(C4○D28), C28.220(C4○D4), C2.3(D56⋊7C2), (C4×C28).327C22, C28.103(C22×C4), (C2×C56).349C22, (C2×C28).727C23, (C2×Dic28).14C2, Dic14.12(C2×C4), C28.44D4.17C2, C4⋊Dic7.264C22, (C2×Dic14).207C22, C4.61(C2×C4×D7), (C2×C14).110(C2×D4), (C2×C4).670(C22×D7), SmallGroup(448,232)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Dic28
G = < a,b,c | a4=b56=1, c2=b28, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 484 in 110 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, Dic7, C28, C28, C28, C2×C14, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C56, C56, Dic14, Dic14, C2×Dic7, C2×C28, C4×Q16, Dic28, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28.44D4, C56⋊1C4, C4×C56, C4×Dic14, C2×Dic28, C4×Dic28
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, D28, C22×D7, C4×Q16, Dic28, C2×C4×D7, C2×D28, C4○D28, C4×D28, D56⋊7C2, C2×Dic28, C4×Dic28
►Regular action on 448 points
Generators in S448
(1 430 96 255)(2 431 97 256)(3 432 98 257)(4 433 99 258)(5 434 100 259)(6 435 101 260)(7 436 102 261)(8 437 103 262)(9 438 104 263)(10 439 105 264)(11 440 106 265)(12 441 107 266)(13 442 108 267)(14 443 109 268)(15 444 110 269)(16 445 111 270)(17 446 112 271)(18 447 57 272)(19 448 58 273)(20 393 59 274)(21 394 60 275)(22 395 61 276)(23 396 62 277)(24 397 63 278)(25 398 64 279)(26 399 65 280)(27 400 66 225)(28 401 67 226)(29 402 68 227)(30 403 69 228)(31 404 70 229)(32 405 71 230)(33 406 72 231)(34 407 73 232)(35 408 74 233)(36 409 75 234)(37 410 76 235)(38 411 77 236)(39 412 78 237)(40 413 79 238)(41 414 80 239)(42 415 81 240)(43 416 82 241)(44 417 83 242)(45 418 84 243)(46 419 85 244)(47 420 86 245)(48 421 87 246)(49 422 88 247)(50 423 89 248)(51 424 90 249)(52 425 91 250)(53 426 92 251)(54 427 93 252)(55 428 94 253)(56 429 95 254)(113 376 285 222)(114 377 286 223)(115 378 287 224)(116 379 288 169)(117 380 289 170)(118 381 290 171)(119 382 291 172)(120 383 292 173)(121 384 293 174)(122 385 294 175)(123 386 295 176)(124 387 296 177)(125 388 297 178)(126 389 298 179)(127 390 299 180)(128 391 300 181)(129 392 301 182)(130 337 302 183)(131 338 303 184)(132 339 304 185)(133 340 305 186)(134 341 306 187)(135 342 307 188)(136 343 308 189)(137 344 309 190)(138 345 310 191)(139 346 311 192)(140 347 312 193)(141 348 313 194)(142 349 314 195)(143 350 315 196)(144 351 316 197)(145 352 317 198)(146 353 318 199)(147 354 319 200)(148 355 320 201)(149 356 321 202)(150 357 322 203)(151 358 323 204)(152 359 324 205)(153 360 325 206)(154 361 326 207)(155 362 327 208)(156 363 328 209)(157 364 329 210)(158 365 330 211)(159 366 331 212)(160 367 332 213)(161 368 333 214)(162 369 334 215)(163 370 335 216)(164 371 336 217)(165 372 281 218)(166 373 282 219)(167 374 283 220)(168 375 284 221)
(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)
(1 331 29 303)(2 330 30 302)(3 329 31 301)(4 328 32 300)(5 327 33 299)(6 326 34 298)(7 325 35 297)(8 324 36 296)(9 323 37 295)(10 322 38 294)(11 321 39 293)(12 320 40 292)(13 319 41 291)(14 318 42 290)(15 317 43 289)(16 316 44 288)(17 315 45 287)(18 314 46 286)(19 313 47 285)(20 312 48 284)(21 311 49 283)(22 310 50 282)(23 309 51 281)(24 308 52 336)(25 307 53 335)(26 306 54 334)(27 305 55 333)(28 304 56 332)(57 142 85 114)(58 141 86 113)(59 140 87 168)(60 139 88 167)(61 138 89 166)(62 137 90 165)(63 136 91 164)(64 135 92 163)(65 134 93 162)(66 133 94 161)(67 132 95 160)(68 131 96 159)(69 130 97 158)(70 129 98 157)(71 128 99 156)(72 127 100 155)(73 126 101 154)(74 125 102 153)(75 124 103 152)(76 123 104 151)(77 122 105 150)(78 121 106 149)(79 120 107 148)(80 119 108 147)(81 118 109 146)(82 117 110 145)(83 116 111 144)(84 115 112 143)(169 445 197 417)(170 444 198 416)(171 443 199 415)(172 442 200 414)(173 441 201 413)(174 440 202 412)(175 439 203 411)(176 438 204 410)(177 437 205 409)(178 436 206 408)(179 435 207 407)(180 434 208 406)(181 433 209 405)(182 432 210 404)(183 431 211 403)(184 430 212 402)(185 429 213 401)(186 428 214 400)(187 427 215 399)(188 426 216 398)(189 425 217 397)(190 424 218 396)(191 423 219 395)(192 422 220 394)(193 421 221 393)(194 420 222 448)(195 419 223 447)(196 418 224 446)(225 340 253 368)(226 339 254 367)(227 338 255 366)(228 337 256 365)(229 392 257 364)(230 391 258 363)(231 390 259 362)(232 389 260 361)(233 388 261 360)(234 387 262 359)(235 386 263 358)(236 385 264 357)(237 384 265 356)(238 383 266 355)(239 382 267 354)(240 381 268 353)(241 380 269 352)(242 379 270 351)(243 378 271 350)(244 377 272 349)(245 376 273 348)(246 375 274 347)(247 374 275 346)(248 373 276 345)(249 372 277 344)(250 371 278 343)(251 370 279 342)(252 369 280 341)
G:=sub<Sym(448)| (1,430,96,255)(2,431,97,256)(3,432,98,257)(4,433,99,258)(5,434,100,259)(6,435,101,260)(7,436,102,261)(8,437,103,262)(9,438,104,263)(10,439,105,264)(11,440,106,265)(12,441,107,266)(13,442,108,267)(14,443,109,268)(15,444,110,269)(16,445,111,270)(17,446,112,271)(18,447,57,272)(19,448,58,273)(20,393,59,274)(21,394,60,275)(22,395,61,276)(23,396,62,277)(24,397,63,278)(25,398,64,279)(26,399,65,280)(27,400,66,225)(28,401,67,226)(29,402,68,227)(30,403,69,228)(31,404,70,229)(32,405,71,230)(33,406,72,231)(34,407,73,232)(35,408,74,233)(36,409,75,234)(37,410,76,235)(38,411,77,236)(39,412,78,237)(40,413,79,238)(41,414,80,239)(42,415,81,240)(43,416,82,241)(44,417,83,242)(45,418,84,243)(46,419,85,244)(47,420,86,245)(48,421,87,246)(49,422,88,247)(50,423,89,248)(51,424,90,249)(52,425,91,250)(53,426,92,251)(54,427,93,252)(55,428,94,253)(56,429,95,254)(113,376,285,222)(114,377,286,223)(115,378,287,224)(116,379,288,169)(117,380,289,170)(118,381,290,171)(119,382,291,172)(120,383,292,173)(121,384,293,174)(122,385,294,175)(123,386,295,176)(124,387,296,177)(125,388,297,178)(126,389,298,179)(127,390,299,180)(128,391,300,181)(129,392,301,182)(130,337,302,183)(131,338,303,184)(132,339,304,185)(133,340,305,186)(134,341,306,187)(135,342,307,188)(136,343,308,189)(137,344,309,190)(138,345,310,191)(139,346,311,192)(140,347,312,193)(141,348,313,194)(142,349,314,195)(143,350,315,196)(144,351,316,197)(145,352,317,198)(146,353,318,199)(147,354,319,200)(148,355,320,201)(149,356,321,202)(150,357,322,203)(151,358,323,204)(152,359,324,205)(153,360,325,206)(154,361,326,207)(155,362,327,208)(156,363,328,209)(157,364,329,210)(158,365,330,211)(159,366,331,212)(160,367,332,213)(161,368,333,214)(162,369,334,215)(163,370,335,216)(164,371,336,217)(165,372,281,218)(166,373,282,219)(167,374,283,220)(168,375,284,221), (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), (1,331,29,303)(2,330,30,302)(3,329,31,301)(4,328,32,300)(5,327,33,299)(6,326,34,298)(7,325,35,297)(8,324,36,296)(9,323,37,295)(10,322,38,294)(11,321,39,293)(12,320,40,292)(13,319,41,291)(14,318,42,290)(15,317,43,289)(16,316,44,288)(17,315,45,287)(18,314,46,286)(19,313,47,285)(20,312,48,284)(21,311,49,283)(22,310,50,282)(23,309,51,281)(24,308,52,336)(25,307,53,335)(26,306,54,334)(27,305,55,333)(28,304,56,332)(57,142,85,114)(58,141,86,113)(59,140,87,168)(60,139,88,167)(61,138,89,166)(62,137,90,165)(63,136,91,164)(64,135,92,163)(65,134,93,162)(66,133,94,161)(67,132,95,160)(68,131,96,159)(69,130,97,158)(70,129,98,157)(71,128,99,156)(72,127,100,155)(73,126,101,154)(74,125,102,153)(75,124,103,152)(76,123,104,151)(77,122,105,150)(78,121,106,149)(79,120,107,148)(80,119,108,147)(81,118,109,146)(82,117,110,145)(83,116,111,144)(84,115,112,143)(169,445,197,417)(170,444,198,416)(171,443,199,415)(172,442,200,414)(173,441,201,413)(174,440,202,412)(175,439,203,411)(176,438,204,410)(177,437,205,409)(178,436,206,408)(179,435,207,407)(180,434,208,406)(181,433,209,405)(182,432,210,404)(183,431,211,403)(184,430,212,402)(185,429,213,401)(186,428,214,400)(187,427,215,399)(188,426,216,398)(189,425,217,397)(190,424,218,396)(191,423,219,395)(192,422,220,394)(193,421,221,393)(194,420,222,448)(195,419,223,447)(196,418,224,446)(225,340,253,368)(226,339,254,367)(227,338,255,366)(228,337,256,365)(229,392,257,364)(230,391,258,363)(231,390,259,362)(232,389,260,361)(233,388,261,360)(234,387,262,359)(235,386,263,358)(236,385,264,357)(237,384,265,356)(238,383,266,355)(239,382,267,354)(240,381,268,353)(241,380,269,352)(242,379,270,351)(243,378,271,350)(244,377,272,349)(245,376,273,348)(246,375,274,347)(247,374,275,346)(248,373,276,345)(249,372,277,344)(250,371,278,343)(251,370,279,342)(252,369,280,341)>;
G:=Group( (1,430,96,255)(2,431,97,256)(3,432,98,257)(4,433,99,258)(5,434,100,259)(6,435,101,260)(7,436,102,261)(8,437,103,262)(9,438,104,263)(10,439,105,264)(11,440,106,265)(12,441,107,266)(13,442,108,267)(14,443,109,268)(15,444,110,269)(16,445,111,270)(17,446,112,271)(18,447,57,272)(19,448,58,273)(20,393,59,274)(21,394,60,275)(22,395,61,276)(23,396,62,277)(24,397,63,278)(25,398,64,279)(26,399,65,280)(27,400,66,225)(28,401,67,226)(29,402,68,227)(30,403,69,228)(31,404,70,229)(32,405,71,230)(33,406,72,231)(34,407,73,232)(35,408,74,233)(36,409,75,234)(37,410,76,235)(38,411,77,236)(39,412,78,237)(40,413,79,238)(41,414,80,239)(42,415,81,240)(43,416,82,241)(44,417,83,242)(45,418,84,243)(46,419,85,244)(47,420,86,245)(48,421,87,246)(49,422,88,247)(50,423,89,248)(51,424,90,249)(52,425,91,250)(53,426,92,251)(54,427,93,252)(55,428,94,253)(56,429,95,254)(113,376,285,222)(114,377,286,223)(115,378,287,224)(116,379,288,169)(117,380,289,170)(118,381,290,171)(119,382,291,172)(120,383,292,173)(121,384,293,174)(122,385,294,175)(123,386,295,176)(124,387,296,177)(125,388,297,178)(126,389,298,179)(127,390,299,180)(128,391,300,181)(129,392,301,182)(130,337,302,183)(131,338,303,184)(132,339,304,185)(133,340,305,186)(134,341,306,187)(135,342,307,188)(136,343,308,189)(137,344,309,190)(138,345,310,191)(139,346,311,192)(140,347,312,193)(141,348,313,194)(142,349,314,195)(143,350,315,196)(144,351,316,197)(145,352,317,198)(146,353,318,199)(147,354,319,200)(148,355,320,201)(149,356,321,202)(150,357,322,203)(151,358,323,204)(152,359,324,205)(153,360,325,206)(154,361,326,207)(155,362,327,208)(156,363,328,209)(157,364,329,210)(158,365,330,211)(159,366,331,212)(160,367,332,213)(161,368,333,214)(162,369,334,215)(163,370,335,216)(164,371,336,217)(165,372,281,218)(166,373,282,219)(167,374,283,220)(168,375,284,221), (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), (1,331,29,303)(2,330,30,302)(3,329,31,301)(4,328,32,300)(5,327,33,299)(6,326,34,298)(7,325,35,297)(8,324,36,296)(9,323,37,295)(10,322,38,294)(11,321,39,293)(12,320,40,292)(13,319,41,291)(14,318,42,290)(15,317,43,289)(16,316,44,288)(17,315,45,287)(18,314,46,286)(19,313,47,285)(20,312,48,284)(21,311,49,283)(22,310,50,282)(23,309,51,281)(24,308,52,336)(25,307,53,335)(26,306,54,334)(27,305,55,333)(28,304,56,332)(57,142,85,114)(58,141,86,113)(59,140,87,168)(60,139,88,167)(61,138,89,166)(62,137,90,165)(63,136,91,164)(64,135,92,163)(65,134,93,162)(66,133,94,161)(67,132,95,160)(68,131,96,159)(69,130,97,158)(70,129,98,157)(71,128,99,156)(72,127,100,155)(73,126,101,154)(74,125,102,153)(75,124,103,152)(76,123,104,151)(77,122,105,150)(78,121,106,149)(79,120,107,148)(80,119,108,147)(81,118,109,146)(82,117,110,145)(83,116,111,144)(84,115,112,143)(169,445,197,417)(170,444,198,416)(171,443,199,415)(172,442,200,414)(173,441,201,413)(174,440,202,412)(175,439,203,411)(176,438,204,410)(177,437,205,409)(178,436,206,408)(179,435,207,407)(180,434,208,406)(181,433,209,405)(182,432,210,404)(183,431,211,403)(184,430,212,402)(185,429,213,401)(186,428,214,400)(187,427,215,399)(188,426,216,398)(189,425,217,397)(190,424,218,396)(191,423,219,395)(192,422,220,394)(193,421,221,393)(194,420,222,448)(195,419,223,447)(196,418,224,446)(225,340,253,368)(226,339,254,367)(227,338,255,366)(228,337,256,365)(229,392,257,364)(230,391,258,363)(231,390,259,362)(232,389,260,361)(233,388,261,360)(234,387,262,359)(235,386,263,358)(236,385,264,357)(237,384,265,356)(238,383,266,355)(239,382,267,354)(240,381,268,353)(241,380,269,352)(242,379,270,351)(243,378,271,350)(244,377,272,349)(245,376,273,348)(246,375,274,347)(247,374,275,346)(248,373,276,345)(249,372,277,344)(250,371,278,343)(251,370,279,342)(252,369,280,341) );
G=PermutationGroup([[(1,430,96,255),(2,431,97,256),(3,432,98,257),(4,433,99,258),(5,434,100,259),(6,435,101,260),(7,436,102,261),(8,437,103,262),(9,438,104,263),(10,439,105,264),(11,440,106,265),(12,441,107,266),(13,442,108,267),(14,443,109,268),(15,444,110,269),(16,445,111,270),(17,446,112,271),(18,447,57,272),(19,448,58,273),(20,393,59,274),(21,394,60,275),(22,395,61,276),(23,396,62,277),(24,397,63,278),(25,398,64,279),(26,399,65,280),(27,400,66,225),(28,401,67,226),(29,402,68,227),(30,403,69,228),(31,404,70,229),(32,405,71,230),(33,406,72,231),(34,407,73,232),(35,408,74,233),(36,409,75,234),(37,410,76,235),(38,411,77,236),(39,412,78,237),(40,413,79,238),(41,414,80,239),(42,415,81,240),(43,416,82,241),(44,417,83,242),(45,418,84,243),(46,419,85,244),(47,420,86,245),(48,421,87,246),(49,422,88,247),(50,423,89,248),(51,424,90,249),(52,425,91,250),(53,426,92,251),(54,427,93,252),(55,428,94,253),(56,429,95,254),(113,376,285,222),(114,377,286,223),(115,378,287,224),(116,379,288,169),(117,380,289,170),(118,381,290,171),(119,382,291,172),(120,383,292,173),(121,384,293,174),(122,385,294,175),(123,386,295,176),(124,387,296,177),(125,388,297,178),(126,389,298,179),(127,390,299,180),(128,391,300,181),(129,392,301,182),(130,337,302,183),(131,338,303,184),(132,339,304,185),(133,340,305,186),(134,341,306,187),(135,342,307,188),(136,343,308,189),(137,344,309,190),(138,345,310,191),(139,346,311,192),(140,347,312,193),(141,348,313,194),(142,349,314,195),(143,350,315,196),(144,351,316,197),(145,352,317,198),(146,353,318,199),(147,354,319,200),(148,355,320,201),(149,356,321,202),(150,357,322,203),(151,358,323,204),(152,359,324,205),(153,360,325,206),(154,361,326,207),(155,362,327,208),(156,363,328,209),(157,364,329,210),(158,365,330,211),(159,366,331,212),(160,367,332,213),(161,368,333,214),(162,369,334,215),(163,370,335,216),(164,371,336,217),(165,372,281,218),(166,373,282,219),(167,374,283,220),(168,375,284,221)], [(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)], [(1,331,29,303),(2,330,30,302),(3,329,31,301),(4,328,32,300),(5,327,33,299),(6,326,34,298),(7,325,35,297),(8,324,36,296),(9,323,37,295),(10,322,38,294),(11,321,39,293),(12,320,40,292),(13,319,41,291),(14,318,42,290),(15,317,43,289),(16,316,44,288),(17,315,45,287),(18,314,46,286),(19,313,47,285),(20,312,48,284),(21,311,49,283),(22,310,50,282),(23,309,51,281),(24,308,52,336),(25,307,53,335),(26,306,54,334),(27,305,55,333),(28,304,56,332),(57,142,85,114),(58,141,86,113),(59,140,87,168),(60,139,88,167),(61,138,89,166),(62,137,90,165),(63,136,91,164),(64,135,92,163),(65,134,93,162),(66,133,94,161),(67,132,95,160),(68,131,96,159),(69,130,97,158),(70,129,98,157),(71,128,99,156),(72,127,100,155),(73,126,101,154),(74,125,102,153),(75,124,103,152),(76,123,104,151),(77,122,105,150),(78,121,106,149),(79,120,107,148),(80,119,108,147),(81,118,109,146),(82,117,110,145),(83,116,111,144),(84,115,112,143),(169,445,197,417),(170,444,198,416),(171,443,199,415),(172,442,200,414),(173,441,201,413),(174,440,202,412),(175,439,203,411),(176,438,204,410),(177,437,205,409),(178,436,206,408),(179,435,207,407),(180,434,208,406),(181,433,209,405),(182,432,210,404),(183,431,211,403),(184,430,212,402),(185,429,213,401),(186,428,214,400),(187,427,215,399),(188,426,216,398),(189,425,217,397),(190,424,218,396),(191,423,219,395),(192,422,220,394),(193,421,221,393),(194,420,222,448),(195,419,223,447),(196,418,224,446),(225,340,253,368),(226,339,254,367),(227,338,255,366),(228,337,256,365),(229,392,257,364),(230,391,258,363),(231,390,259,362),(232,389,260,361),(233,388,261,360),(234,387,262,359),(235,386,263,358),(236,385,264,357),(237,384,265,356),(238,383,266,355),(239,382,267,354),(240,381,268,353),(241,380,269,352),(242,379,270,351),(243,378,271,350),(244,377,272,349),(245,376,273,348),(246,375,274,347),(247,374,275,346),(248,373,276,345),(249,372,277,344),(250,371,278,343),(251,370,279,342),(252,369,280,341)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | C4○D8 | C4×D7 | D28 | Dic28 | C4○D28 | D56⋊7C2 |
| kernel | C4×Dic28 | C28.44D4 | C56⋊1C4 | C4×C56 | C4×Dic14 | C2×Dic28 | Dic28 | C2×C28 | C4×C8 | C28 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 3 | 4 | 2 | 3 | 6 | 4 | 12 | 12 | 24 | 12 | 24 |
Matrix representation of C4×Dic28 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 112 | 0 | 0 |
| 0 | 43 | 84 |
| 0 | 29 | 12 |
| 1 | 0 | 0 |
| 0 | 112 | 76 |
| 0 | 110 | 1 |
G:=sub<GL(3,GF(113))| [98,0,0,0,1,0,0,0,1],[112,0,0,0,43,29,0,84,12],[1,0,0,0,112,110,0,76,1] >;
C4×Dic28 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{28} % in TeX
G:=Group("C4xDic28"); // GroupNames label
G:=SmallGroup(448,232);
// by ID
G=gap.SmallGroup(448,232);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,344,58,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^56=1,c^2=b^28,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations