metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.1C42, C4⋊C4⋊3Dic7, C28.6(C4⋊C4), (C2×C28).4Q8, C4⋊Dic7⋊10C4, (C2×C14).35D8, C4.1(C4×Dic7), (C2×C28).102D4, (C2×C4).125D28, C4.1(C4⋊Dic7), (C2×C14).14Q16, C14.5(C2.D8), C14.4(C4.Q8), C4.31(D14⋊C4), C28.7(C22⋊C4), C4.6(Dic7⋊C4), C7⋊1(C22.4Q16), (C2×C14).38SD16, (C2×C4).23Dic14, C22.9(Q8⋊D7), C2.2(C14.D8), C22.16(D4⋊D7), C14.4(Q8⋊C4), C2.1(D4⋊Dic7), C2.2(C14.Q16), C2.1(Q8⋊Dic7), C2.2(C28.Q8), (C22×C14).178D4, (C22×C4).323D14, C23.94(C7⋊D4), C22.9(D4.D7), C2.2(C4.Dic14), C14.21(D4⋊C4), C22.6(C7⋊Q16), C22.38(D14⋊C4), C14.5(C2.C42), C2.6(C14.C42), C22.20(Dic7⋊C4), (C22×C28).118C22, C22.26(C23.D7), (C2×C7⋊C8)⋊4C4, (C7×C4⋊C4)⋊5C4, (C2×C4⋊C4).2D7, (C14×C4⋊C4).1C2, (C22×C7⋊C8).1C2, (C2×C28).54(C2×C4), (C2×C4).138(C4×D7), (C2×C14).35(C4⋊C4), (C2×C4⋊Dic7).27C2, (C2×C4).35(C2×Dic7), (C2×C4).174(C7⋊D4), (C2×C14).87(C22⋊C4), SmallGroup(448,86)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.C42
G = < a,b,c | a28=c4=1, b4=a14, bab-1=a13, cac-1=a15, cbc-1=a7b >
Subgroups: 420 in 114 conjugacy classes, 67 normal (59 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C23, C14, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C22×C14, C22.4Q16, C2×C7⋊C8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, C22×C28, C22×C7⋊C8, C2×C4⋊Dic7, C14×C4⋊C4, C28.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, Dic7, D14, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C22.4Q16, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C23.D7, C28.Q8, C4.Dic14, C14.D8, C14.Q16, C14.C42, D4⋊Dic7, Q8⋊Dic7, C28.C42
►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)
(1 400 135 99 15 414 121 85)(2 413 136 112 16 399 122 98)(3 398 137 97 17 412 123 111)(4 411 138 110 18 397 124 96)(5 396 139 95 19 410 125 109)(6 409 140 108 20 395 126 94)(7 394 113 93 21 408 127 107)(8 407 114 106 22 393 128 92)(9 420 115 91 23 406 129 105)(10 405 116 104 24 419 130 90)(11 418 117 89 25 404 131 103)(12 403 118 102 26 417 132 88)(13 416 119 87 27 402 133 101)(14 401 120 100 28 415 134 86)(29 273 217 143 43 259 203 157)(30 258 218 156 44 272 204 142)(31 271 219 141 45 257 205 155)(32 256 220 154 46 270 206 168)(33 269 221 167 47 255 207 153)(34 254 222 152 48 268 208 166)(35 267 223 165 49 253 209 151)(36 280 224 150 50 266 210 164)(37 265 197 163 51 279 211 149)(38 278 198 148 52 264 212 162)(39 263 199 161 53 277 213 147)(40 276 200 146 54 262 214 160)(41 261 201 159 55 275 215 145)(42 274 202 144 56 260 216 158)(57 390 183 348 71 376 169 362)(58 375 184 361 72 389 170 347)(59 388 185 346 73 374 171 360)(60 373 186 359 74 387 172 345)(61 386 187 344 75 372 173 358)(62 371 188 357 76 385 174 343)(63 384 189 342 77 370 175 356)(64 369 190 355 78 383 176 341)(65 382 191 340 79 368 177 354)(66 367 192 353 80 381 178 339)(67 380 193 338 81 366 179 352)(68 365 194 351 82 379 180 337)(69 378 195 364 83 392 181 350)(70 391 196 349 84 377 182 363)(225 321 443 307 239 335 429 293)(226 334 444 292 240 320 430 306)(227 319 445 305 241 333 431 291)(228 332 446 290 242 318 432 304)(229 317 447 303 243 331 433 289)(230 330 448 288 244 316 434 302)(231 315 421 301 245 329 435 287)(232 328 422 286 246 314 436 300)(233 313 423 299 247 327 437 285)(234 326 424 284 248 312 438 298)(235 311 425 297 249 325 439 283)(236 324 426 282 250 310 440 296)(237 309 427 295 251 323 441 281)(238 322 428 308 252 336 442 294)
(1 41 248 177)(2 56 249 192)(3 43 250 179)(4 30 251 194)(5 45 252 181)(6 32 225 196)(7 47 226 183)(8 34 227 170)(9 49 228 185)(10 36 229 172)(11 51 230 187)(12 38 231 174)(13 53 232 189)(14 40 233 176)(15 55 234 191)(16 42 235 178)(17 29 236 193)(18 44 237 180)(19 31 238 195)(20 46 239 182)(21 33 240 169)(22 48 241 184)(23 35 242 171)(24 50 243 186)(25 37 244 173)(26 52 245 188)(27 39 246 175)(28 54 247 190)(57 113 221 444)(58 128 222 431)(59 115 223 446)(60 130 224 433)(61 117 197 448)(62 132 198 435)(63 119 199 422)(64 134 200 437)(65 121 201 424)(66 136 202 439)(67 123 203 426)(68 138 204 441)(69 125 205 428)(70 140 206 443)(71 127 207 430)(72 114 208 445)(73 129 209 432)(74 116 210 447)(75 131 211 434)(76 118 212 421)(77 133 213 436)(78 120 214 423)(79 135 215 438)(80 122 216 425)(81 137 217 440)(82 124 218 427)(83 139 219 442)(84 126 220 429)(85 166 284 389)(86 153 285 376)(87 168 286 391)(88 155 287 378)(89 142 288 365)(90 157 289 380)(91 144 290 367)(92 159 291 382)(93 146 292 369)(94 161 293 384)(95 148 294 371)(96 163 295 386)(97 150 296 373)(98 165 297 388)(99 152 298 375)(100 167 299 390)(101 154 300 377)(102 141 301 392)(103 156 302 379)(104 143 303 366)(105 158 304 381)(106 145 305 368)(107 160 306 383)(108 147 307 370)(109 162 308 385)(110 149 281 372)(111 164 282 387)(112 151 283 374)(253 311 346 399)(254 326 347 414)(255 313 348 401)(256 328 349 416)(257 315 350 403)(258 330 351 418)(259 317 352 405)(260 332 353 420)(261 319 354 407)(262 334 355 394)(263 321 356 409)(264 336 357 396)(265 323 358 411)(266 310 359 398)(267 325 360 413)(268 312 361 400)(269 327 362 415)(270 314 363 402)(271 329 364 417)(272 316 337 404)(273 331 338 419)(274 318 339 406)(275 333 340 393)(276 320 341 408)(277 335 342 395)(278 322 343 410)(279 309 344 397)(280 324 345 412)
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), (1,400,135,99,15,414,121,85)(2,413,136,112,16,399,122,98)(3,398,137,97,17,412,123,111)(4,411,138,110,18,397,124,96)(5,396,139,95,19,410,125,109)(6,409,140,108,20,395,126,94)(7,394,113,93,21,408,127,107)(8,407,114,106,22,393,128,92)(9,420,115,91,23,406,129,105)(10,405,116,104,24,419,130,90)(11,418,117,89,25,404,131,103)(12,403,118,102,26,417,132,88)(13,416,119,87,27,402,133,101)(14,401,120,100,28,415,134,86)(29,273,217,143,43,259,203,157)(30,258,218,156,44,272,204,142)(31,271,219,141,45,257,205,155)(32,256,220,154,46,270,206,168)(33,269,221,167,47,255,207,153)(34,254,222,152,48,268,208,166)(35,267,223,165,49,253,209,151)(36,280,224,150,50,266,210,164)(37,265,197,163,51,279,211,149)(38,278,198,148,52,264,212,162)(39,263,199,161,53,277,213,147)(40,276,200,146,54,262,214,160)(41,261,201,159,55,275,215,145)(42,274,202,144,56,260,216,158)(57,390,183,348,71,376,169,362)(58,375,184,361,72,389,170,347)(59,388,185,346,73,374,171,360)(60,373,186,359,74,387,172,345)(61,386,187,344,75,372,173,358)(62,371,188,357,76,385,174,343)(63,384,189,342,77,370,175,356)(64,369,190,355,78,383,176,341)(65,382,191,340,79,368,177,354)(66,367,192,353,80,381,178,339)(67,380,193,338,81,366,179,352)(68,365,194,351,82,379,180,337)(69,378,195,364,83,392,181,350)(70,391,196,349,84,377,182,363)(225,321,443,307,239,335,429,293)(226,334,444,292,240,320,430,306)(227,319,445,305,241,333,431,291)(228,332,446,290,242,318,432,304)(229,317,447,303,243,331,433,289)(230,330,448,288,244,316,434,302)(231,315,421,301,245,329,435,287)(232,328,422,286,246,314,436,300)(233,313,423,299,247,327,437,285)(234,326,424,284,248,312,438,298)(235,311,425,297,249,325,439,283)(236,324,426,282,250,310,440,296)(237,309,427,295,251,323,441,281)(238,322,428,308,252,336,442,294), (1,41,248,177)(2,56,249,192)(3,43,250,179)(4,30,251,194)(5,45,252,181)(6,32,225,196)(7,47,226,183)(8,34,227,170)(9,49,228,185)(10,36,229,172)(11,51,230,187)(12,38,231,174)(13,53,232,189)(14,40,233,176)(15,55,234,191)(16,42,235,178)(17,29,236,193)(18,44,237,180)(19,31,238,195)(20,46,239,182)(21,33,240,169)(22,48,241,184)(23,35,242,171)(24,50,243,186)(25,37,244,173)(26,52,245,188)(27,39,246,175)(28,54,247,190)(57,113,221,444)(58,128,222,431)(59,115,223,446)(60,130,224,433)(61,117,197,448)(62,132,198,435)(63,119,199,422)(64,134,200,437)(65,121,201,424)(66,136,202,439)(67,123,203,426)(68,138,204,441)(69,125,205,428)(70,140,206,443)(71,127,207,430)(72,114,208,445)(73,129,209,432)(74,116,210,447)(75,131,211,434)(76,118,212,421)(77,133,213,436)(78,120,214,423)(79,135,215,438)(80,122,216,425)(81,137,217,440)(82,124,218,427)(83,139,219,442)(84,126,220,429)(85,166,284,389)(86,153,285,376)(87,168,286,391)(88,155,287,378)(89,142,288,365)(90,157,289,380)(91,144,290,367)(92,159,291,382)(93,146,292,369)(94,161,293,384)(95,148,294,371)(96,163,295,386)(97,150,296,373)(98,165,297,388)(99,152,298,375)(100,167,299,390)(101,154,300,377)(102,141,301,392)(103,156,302,379)(104,143,303,366)(105,158,304,381)(106,145,305,368)(107,160,306,383)(108,147,307,370)(109,162,308,385)(110,149,281,372)(111,164,282,387)(112,151,283,374)(253,311,346,399)(254,326,347,414)(255,313,348,401)(256,328,349,416)(257,315,350,403)(258,330,351,418)(259,317,352,405)(260,332,353,420)(261,319,354,407)(262,334,355,394)(263,321,356,409)(264,336,357,396)(265,323,358,411)(266,310,359,398)(267,325,360,413)(268,312,361,400)(269,327,362,415)(270,314,363,402)(271,329,364,417)(272,316,337,404)(273,331,338,419)(274,318,339,406)(275,333,340,393)(276,320,341,408)(277,335,342,395)(278,322,343,410)(279,309,344,397)(280,324,345,412)>;
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), (1,400,135,99,15,414,121,85)(2,413,136,112,16,399,122,98)(3,398,137,97,17,412,123,111)(4,411,138,110,18,397,124,96)(5,396,139,95,19,410,125,109)(6,409,140,108,20,395,126,94)(7,394,113,93,21,408,127,107)(8,407,114,106,22,393,128,92)(9,420,115,91,23,406,129,105)(10,405,116,104,24,419,130,90)(11,418,117,89,25,404,131,103)(12,403,118,102,26,417,132,88)(13,416,119,87,27,402,133,101)(14,401,120,100,28,415,134,86)(29,273,217,143,43,259,203,157)(30,258,218,156,44,272,204,142)(31,271,219,141,45,257,205,155)(32,256,220,154,46,270,206,168)(33,269,221,167,47,255,207,153)(34,254,222,152,48,268,208,166)(35,267,223,165,49,253,209,151)(36,280,224,150,50,266,210,164)(37,265,197,163,51,279,211,149)(38,278,198,148,52,264,212,162)(39,263,199,161,53,277,213,147)(40,276,200,146,54,262,214,160)(41,261,201,159,55,275,215,145)(42,274,202,144,56,260,216,158)(57,390,183,348,71,376,169,362)(58,375,184,361,72,389,170,347)(59,388,185,346,73,374,171,360)(60,373,186,359,74,387,172,345)(61,386,187,344,75,372,173,358)(62,371,188,357,76,385,174,343)(63,384,189,342,77,370,175,356)(64,369,190,355,78,383,176,341)(65,382,191,340,79,368,177,354)(66,367,192,353,80,381,178,339)(67,380,193,338,81,366,179,352)(68,365,194,351,82,379,180,337)(69,378,195,364,83,392,181,350)(70,391,196,349,84,377,182,363)(225,321,443,307,239,335,429,293)(226,334,444,292,240,320,430,306)(227,319,445,305,241,333,431,291)(228,332,446,290,242,318,432,304)(229,317,447,303,243,331,433,289)(230,330,448,288,244,316,434,302)(231,315,421,301,245,329,435,287)(232,328,422,286,246,314,436,300)(233,313,423,299,247,327,437,285)(234,326,424,284,248,312,438,298)(235,311,425,297,249,325,439,283)(236,324,426,282,250,310,440,296)(237,309,427,295,251,323,441,281)(238,322,428,308,252,336,442,294), (1,41,248,177)(2,56,249,192)(3,43,250,179)(4,30,251,194)(5,45,252,181)(6,32,225,196)(7,47,226,183)(8,34,227,170)(9,49,228,185)(10,36,229,172)(11,51,230,187)(12,38,231,174)(13,53,232,189)(14,40,233,176)(15,55,234,191)(16,42,235,178)(17,29,236,193)(18,44,237,180)(19,31,238,195)(20,46,239,182)(21,33,240,169)(22,48,241,184)(23,35,242,171)(24,50,243,186)(25,37,244,173)(26,52,245,188)(27,39,246,175)(28,54,247,190)(57,113,221,444)(58,128,222,431)(59,115,223,446)(60,130,224,433)(61,117,197,448)(62,132,198,435)(63,119,199,422)(64,134,200,437)(65,121,201,424)(66,136,202,439)(67,123,203,426)(68,138,204,441)(69,125,205,428)(70,140,206,443)(71,127,207,430)(72,114,208,445)(73,129,209,432)(74,116,210,447)(75,131,211,434)(76,118,212,421)(77,133,213,436)(78,120,214,423)(79,135,215,438)(80,122,216,425)(81,137,217,440)(82,124,218,427)(83,139,219,442)(84,126,220,429)(85,166,284,389)(86,153,285,376)(87,168,286,391)(88,155,287,378)(89,142,288,365)(90,157,289,380)(91,144,290,367)(92,159,291,382)(93,146,292,369)(94,161,293,384)(95,148,294,371)(96,163,295,386)(97,150,296,373)(98,165,297,388)(99,152,298,375)(100,167,299,390)(101,154,300,377)(102,141,301,392)(103,156,302,379)(104,143,303,366)(105,158,304,381)(106,145,305,368)(107,160,306,383)(108,147,307,370)(109,162,308,385)(110,149,281,372)(111,164,282,387)(112,151,283,374)(253,311,346,399)(254,326,347,414)(255,313,348,401)(256,328,349,416)(257,315,350,403)(258,330,351,418)(259,317,352,405)(260,332,353,420)(261,319,354,407)(262,334,355,394)(263,321,356,409)(264,336,357,396)(265,323,358,411)(266,310,359,398)(267,325,360,413)(268,312,361,400)(269,327,362,415)(270,314,363,402)(271,329,364,417)(272,316,337,404)(273,331,338,419)(274,318,339,406)(275,333,340,393)(276,320,341,408)(277,335,342,395)(278,322,343,410)(279,309,344,397)(280,324,345,412) );
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)], [(1,400,135,99,15,414,121,85),(2,413,136,112,16,399,122,98),(3,398,137,97,17,412,123,111),(4,411,138,110,18,397,124,96),(5,396,139,95,19,410,125,109),(6,409,140,108,20,395,126,94),(7,394,113,93,21,408,127,107),(8,407,114,106,22,393,128,92),(9,420,115,91,23,406,129,105),(10,405,116,104,24,419,130,90),(11,418,117,89,25,404,131,103),(12,403,118,102,26,417,132,88),(13,416,119,87,27,402,133,101),(14,401,120,100,28,415,134,86),(29,273,217,143,43,259,203,157),(30,258,218,156,44,272,204,142),(31,271,219,141,45,257,205,155),(32,256,220,154,46,270,206,168),(33,269,221,167,47,255,207,153),(34,254,222,152,48,268,208,166),(35,267,223,165,49,253,209,151),(36,280,224,150,50,266,210,164),(37,265,197,163,51,279,211,149),(38,278,198,148,52,264,212,162),(39,263,199,161,53,277,213,147),(40,276,200,146,54,262,214,160),(41,261,201,159,55,275,215,145),(42,274,202,144,56,260,216,158),(57,390,183,348,71,376,169,362),(58,375,184,361,72,389,170,347),(59,388,185,346,73,374,171,360),(60,373,186,359,74,387,172,345),(61,386,187,344,75,372,173,358),(62,371,188,357,76,385,174,343),(63,384,189,342,77,370,175,356),(64,369,190,355,78,383,176,341),(65,382,191,340,79,368,177,354),(66,367,192,353,80,381,178,339),(67,380,193,338,81,366,179,352),(68,365,194,351,82,379,180,337),(69,378,195,364,83,392,181,350),(70,391,196,349,84,377,182,363),(225,321,443,307,239,335,429,293),(226,334,444,292,240,320,430,306),(227,319,445,305,241,333,431,291),(228,332,446,290,242,318,432,304),(229,317,447,303,243,331,433,289),(230,330,448,288,244,316,434,302),(231,315,421,301,245,329,435,287),(232,328,422,286,246,314,436,300),(233,313,423,299,247,327,437,285),(234,326,424,284,248,312,438,298),(235,311,425,297,249,325,439,283),(236,324,426,282,250,310,440,296),(237,309,427,295,251,323,441,281),(238,322,428,308,252,336,442,294)], [(1,41,248,177),(2,56,249,192),(3,43,250,179),(4,30,251,194),(5,45,252,181),(6,32,225,196),(7,47,226,183),(8,34,227,170),(9,49,228,185),(10,36,229,172),(11,51,230,187),(12,38,231,174),(13,53,232,189),(14,40,233,176),(15,55,234,191),(16,42,235,178),(17,29,236,193),(18,44,237,180),(19,31,238,195),(20,46,239,182),(21,33,240,169),(22,48,241,184),(23,35,242,171),(24,50,243,186),(25,37,244,173),(26,52,245,188),(27,39,246,175),(28,54,247,190),(57,113,221,444),(58,128,222,431),(59,115,223,446),(60,130,224,433),(61,117,197,448),(62,132,198,435),(63,119,199,422),(64,134,200,437),(65,121,201,424),(66,136,202,439),(67,123,203,426),(68,138,204,441),(69,125,205,428),(70,140,206,443),(71,127,207,430),(72,114,208,445),(73,129,209,432),(74,116,210,447),(75,131,211,434),(76,118,212,421),(77,133,213,436),(78,120,214,423),(79,135,215,438),(80,122,216,425),(81,137,217,440),(82,124,218,427),(83,139,219,442),(84,126,220,429),(85,166,284,389),(86,153,285,376),(87,168,286,391),(88,155,287,378),(89,142,288,365),(90,157,289,380),(91,144,290,367),(92,159,291,382),(93,146,292,369),(94,161,293,384),(95,148,294,371),(96,163,295,386),(97,150,296,373),(98,165,297,388),(99,152,298,375),(100,167,299,390),(101,154,300,377),(102,141,301,392),(103,156,302,379),(104,143,303,366),(105,158,304,381),(106,145,305,368),(107,160,306,383),(108,147,307,370),(109,162,308,385),(110,149,281,372),(111,164,282,387),(112,151,283,374),(253,311,346,399),(254,326,347,414),(255,313,348,401),(256,328,349,416),(257,315,350,403),(258,330,351,418),(259,317,352,405),(260,332,353,420),(261,319,354,407),(262,334,355,394),(263,321,356,409),(264,336,357,396),(265,323,358,411),(266,310,359,398),(267,325,360,413),(268,312,361,400),(269,327,362,415),(270,314,363,402),(271,329,364,417),(272,316,337,404),(273,331,338,419),(274,318,339,406),(275,333,340,393),(276,320,341,408),(277,335,342,395),(278,322,343,410),(279,309,344,397),(280,324,345,412)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | - | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D7 | D8 | SD16 | Q16 | Dic7 | D14 | Dic14 | C4×D7 | D28 | C7⋊D4 | C7⋊D4 | D4⋊D7 | D4.D7 | Q8⋊D7 | C7⋊Q16 |
| kernel | C28.C42 | C22×C7⋊C8 | C2×C4⋊Dic7 | C14×C4⋊C4 | C2×C7⋊C8 | C4⋊Dic7 | C7×C4⋊C4 | C2×C28 | C2×C28 | C22×C14 | C2×C4⋊C4 | C2×C14 | C2×C14 | C2×C14 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 3 | 2 | 4 | 2 | 6 | 3 | 6 | 12 | 6 | 6 | 6 | 3 | 3 | 3 | 3 |
Matrix representation of C28.C42 ►in GL6(𝔽113)
| 25 | 89 | 0 | 0 | 0 | 0 |
| 59 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 89 | 0 | 0 |
| 0 | 0 | 24 | 103 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 2 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 13 | 77 | 0 | 0 | 0 | 0 |
| 80 | 100 | 0 | 0 | 0 | 0 |
| 0 | 0 | 107 | 64 | 0 | 0 |
| 0 | 0 | 33 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 26 |
| 0 | 0 | 0 | 0 | 100 | 26 |
| 22 | 7 | 0 | 0 | 0 | 0 |
| 44 | 91 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 105 | 0 | 0 |
| 0 | 0 | 8 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 74 |
| 0 | 0 | 0 | 0 | 72 | 78 |
G:=sub<GL(6,GF(113))| [25,59,0,0,0,0,89,79,0,0,0,0,0,0,1,24,0,0,0,0,89,103,0,0,0,0,0,0,112,112,0,0,0,0,2,1],[13,80,0,0,0,0,77,100,0,0,0,0,0,0,107,33,0,0,0,0,64,6,0,0,0,0,0,0,0,100,0,0,0,0,26,26],[22,44,0,0,0,0,7,91,0,0,0,0,0,0,96,8,0,0,0,0,105,17,0,0,0,0,0,0,35,72,0,0,0,0,74,78] >;
C28.C42 in GAP, Magma, Sage, TeX
C_{28}.C_4^2 % in TeX
G:=Group("C28.C4^2"); // GroupNames label
G:=SmallGroup(448,86);
// by ID
G=gap.SmallGroup(448,86);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,365,36,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=c^4=1,b^4=a^14,b*a*b^-1=a^13,c*a*c^-1=a^15,c*b*c^-1=a^7*b>;
// generators/relations