metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.2D8, C4.10D56, C28.22Q16, C28.1SD16, C42.6D14, C4⋊C8.6D7, C4⋊Dic7.3C4, (C2×C28).466D4, (C2×C4).124D28, C4.7(C56⋊C2), C7⋊2(C4.10D8), C28⋊2Q8.9C2, C28⋊C8.10C2, C4.10(D4.D7), (C4×C28).44C22, C4.10(C7⋊Q16), C2.5(C2.D56), C14.3(Q8⋊C4), C2.5(C14.Q16), C14.13(D4⋊C4), C22.63(D14⋊C4), C14.4(C4.10D4), C2.5(C4.12D28), (C7×C4⋊C8).6C2, (C2×C4).17(C4×D7), (C2×C28).29(C2×C4), (C2×C4).230(C7⋊D4), (C2×C14).48(C22⋊C4), SmallGroup(448,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.2D8
G = < a,b,c | a28=b8=1, c2=a7, bab-1=a15, cac-1=a13, cbc-1=a21b-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, C4.10D8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28⋊C8, C7×C4⋊C8, C28⋊2Q8, C28.2D8
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Q16, D14, C4.10D4, D4⋊C4, Q8⋊C4, C4×D7, D28, C7⋊D4, C4.10D8, C56⋊C2, D56, D14⋊C4, D4.D7, C7⋊Q16, C14.Q16, C2.D56, C4.12D28, C28.2D8
►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 33 418 376 164 181 60 326)(2 48 419 391 165 196 61 313)(3 35 420 378 166 183 62 328)(4 50 393 365 167 170 63 315)(5 37 394 380 168 185 64 330)(6 52 395 367 141 172 65 317)(7 39 396 382 142 187 66 332)(8 54 397 369 143 174 67 319)(9 41 398 384 144 189 68 334)(10 56 399 371 145 176 69 321)(11 43 400 386 146 191 70 336)(12 30 401 373 147 178 71 323)(13 45 402 388 148 193 72 310)(14 32 403 375 149 180 73 325)(15 47 404 390 150 195 74 312)(16 34 405 377 151 182 75 327)(17 49 406 392 152 169 76 314)(18 36 407 379 153 184 77 329)(19 51 408 366 154 171 78 316)(20 38 409 381 155 186 79 331)(21 53 410 368 156 173 80 318)(22 40 411 383 157 188 81 333)(23 55 412 370 158 175 82 320)(24 42 413 385 159 190 83 335)(25 29 414 372 160 177 84 322)(26 44 415 387 161 192 57 309)(27 31 416 374 162 179 58 324)(28 46 417 389 163 194 59 311)(85 251 199 337 432 126 278 282)(86 238 200 352 433 113 279 297)(87 225 201 339 434 128 280 284)(88 240 202 354 435 115 253 299)(89 227 203 341 436 130 254 286)(90 242 204 356 437 117 255 301)(91 229 205 343 438 132 256 288)(92 244 206 358 439 119 257 303)(93 231 207 345 440 134 258 290)(94 246 208 360 441 121 259 305)(95 233 209 347 442 136 260 292)(96 248 210 362 443 123 261 307)(97 235 211 349 444 138 262 294)(98 250 212 364 445 125 263 281)(99 237 213 351 446 140 264 296)(100 252 214 338 447 127 265 283)(101 239 215 353 448 114 266 298)(102 226 216 340 421 129 267 285)(103 241 217 355 422 116 268 300)(104 228 218 342 423 131 269 287)(105 243 219 357 424 118 270 302)(106 230 220 344 425 133 271 289)(107 245 221 359 426 120 272 304)(108 232 222 346 427 135 273 291)(109 247 223 361 428 122 274 306)(110 234 224 348 429 137 275 293)(111 249 197 363 430 124 276 308)(112 236 198 350 431 139 277 295)
(1 103 8 110 15 89 22 96)(2 88 9 95 16 102 23 109)(3 101 10 108 17 87 24 94)(4 86 11 93 18 100 25 107)(5 99 12 106 19 85 26 92)(6 112 13 91 20 98 27 105)(7 97 14 104 21 111 28 90)(29 297 36 304 43 283 50 290)(30 282 37 289 44 296 51 303)(31 295 38 302 45 281 52 288)(32 308 39 287 46 294 53 301)(33 293 40 300 47 307 54 286)(34 306 41 285 48 292 55 299)(35 291 42 298 49 305 56 284)(57 206 64 213 71 220 78 199)(58 219 65 198 72 205 79 212)(59 204 66 211 73 218 80 197)(60 217 67 224 74 203 81 210)(61 202 68 209 75 216 82 223)(62 215 69 222 76 201 83 208)(63 200 70 207 77 214 84 221)(113 379 120 386 127 365 134 372)(114 392 121 371 128 378 135 385)(115 377 122 384 129 391 136 370)(116 390 123 369 130 376 137 383)(117 375 124 382 131 389 138 368)(118 388 125 367 132 374 139 381)(119 373 126 380 133 387 140 366)(141 431 148 438 155 445 162 424)(142 444 149 423 156 430 163 437)(143 429 150 436 157 443 164 422)(144 442 151 421 158 428 165 435)(145 427 152 434 159 441 166 448)(146 440 153 447 160 426 167 433)(147 425 154 432 161 439 168 446)(169 360 176 339 183 346 190 353)(170 345 177 352 184 359 191 338)(171 358 178 337 185 344 192 351)(172 343 179 350 186 357 193 364)(173 356 180 363 187 342 194 349)(174 341 181 348 188 355 195 362)(175 354 182 361 189 340 196 347)(225 328 232 335 239 314 246 321)(226 313 233 320 240 327 247 334)(227 326 234 333 241 312 248 319)(228 311 235 318 242 325 249 332)(229 324 236 331 243 310 250 317)(230 309 237 316 244 323 251 330)(231 322 238 329 245 336 252 315)(253 398 260 405 267 412 274 419)(254 411 261 418 268 397 275 404)(255 396 262 403 269 410 276 417)(256 409 263 416 270 395 277 402)(257 394 264 401 271 408 278 415)(258 407 265 414 272 393 279 400)(259 420 266 399 273 406 280 413)
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), (1,33,418,376,164,181,60,326)(2,48,419,391,165,196,61,313)(3,35,420,378,166,183,62,328)(4,50,393,365,167,170,63,315)(5,37,394,380,168,185,64,330)(6,52,395,367,141,172,65,317)(7,39,396,382,142,187,66,332)(8,54,397,369,143,174,67,319)(9,41,398,384,144,189,68,334)(10,56,399,371,145,176,69,321)(11,43,400,386,146,191,70,336)(12,30,401,373,147,178,71,323)(13,45,402,388,148,193,72,310)(14,32,403,375,149,180,73,325)(15,47,404,390,150,195,74,312)(16,34,405,377,151,182,75,327)(17,49,406,392,152,169,76,314)(18,36,407,379,153,184,77,329)(19,51,408,366,154,171,78,316)(20,38,409,381,155,186,79,331)(21,53,410,368,156,173,80,318)(22,40,411,383,157,188,81,333)(23,55,412,370,158,175,82,320)(24,42,413,385,159,190,83,335)(25,29,414,372,160,177,84,322)(26,44,415,387,161,192,57,309)(27,31,416,374,162,179,58,324)(28,46,417,389,163,194,59,311)(85,251,199,337,432,126,278,282)(86,238,200,352,433,113,279,297)(87,225,201,339,434,128,280,284)(88,240,202,354,435,115,253,299)(89,227,203,341,436,130,254,286)(90,242,204,356,437,117,255,301)(91,229,205,343,438,132,256,288)(92,244,206,358,439,119,257,303)(93,231,207,345,440,134,258,290)(94,246,208,360,441,121,259,305)(95,233,209,347,442,136,260,292)(96,248,210,362,443,123,261,307)(97,235,211,349,444,138,262,294)(98,250,212,364,445,125,263,281)(99,237,213,351,446,140,264,296)(100,252,214,338,447,127,265,283)(101,239,215,353,448,114,266,298)(102,226,216,340,421,129,267,285)(103,241,217,355,422,116,268,300)(104,228,218,342,423,131,269,287)(105,243,219,357,424,118,270,302)(106,230,220,344,425,133,271,289)(107,245,221,359,426,120,272,304)(108,232,222,346,427,135,273,291)(109,247,223,361,428,122,274,306)(110,234,224,348,429,137,275,293)(111,249,197,363,430,124,276,308)(112,236,198,350,431,139,277,295), (1,103,8,110,15,89,22,96)(2,88,9,95,16,102,23,109)(3,101,10,108,17,87,24,94)(4,86,11,93,18,100,25,107)(5,99,12,106,19,85,26,92)(6,112,13,91,20,98,27,105)(7,97,14,104,21,111,28,90)(29,297,36,304,43,283,50,290)(30,282,37,289,44,296,51,303)(31,295,38,302,45,281,52,288)(32,308,39,287,46,294,53,301)(33,293,40,300,47,307,54,286)(34,306,41,285,48,292,55,299)(35,291,42,298,49,305,56,284)(57,206,64,213,71,220,78,199)(58,219,65,198,72,205,79,212)(59,204,66,211,73,218,80,197)(60,217,67,224,74,203,81,210)(61,202,68,209,75,216,82,223)(62,215,69,222,76,201,83,208)(63,200,70,207,77,214,84,221)(113,379,120,386,127,365,134,372)(114,392,121,371,128,378,135,385)(115,377,122,384,129,391,136,370)(116,390,123,369,130,376,137,383)(117,375,124,382,131,389,138,368)(118,388,125,367,132,374,139,381)(119,373,126,380,133,387,140,366)(141,431,148,438,155,445,162,424)(142,444,149,423,156,430,163,437)(143,429,150,436,157,443,164,422)(144,442,151,421,158,428,165,435)(145,427,152,434,159,441,166,448)(146,440,153,447,160,426,167,433)(147,425,154,432,161,439,168,446)(169,360,176,339,183,346,190,353)(170,345,177,352,184,359,191,338)(171,358,178,337,185,344,192,351)(172,343,179,350,186,357,193,364)(173,356,180,363,187,342,194,349)(174,341,181,348,188,355,195,362)(175,354,182,361,189,340,196,347)(225,328,232,335,239,314,246,321)(226,313,233,320,240,327,247,334)(227,326,234,333,241,312,248,319)(228,311,235,318,242,325,249,332)(229,324,236,331,243,310,250,317)(230,309,237,316,244,323,251,330)(231,322,238,329,245,336,252,315)(253,398,260,405,267,412,274,419)(254,411,261,418,268,397,275,404)(255,396,262,403,269,410,276,417)(256,409,263,416,270,395,277,402)(257,394,264,401,271,408,278,415)(258,407,265,414,272,393,279,400)(259,420,266,399,273,406,280,413)>;
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), (1,33,418,376,164,181,60,326)(2,48,419,391,165,196,61,313)(3,35,420,378,166,183,62,328)(4,50,393,365,167,170,63,315)(5,37,394,380,168,185,64,330)(6,52,395,367,141,172,65,317)(7,39,396,382,142,187,66,332)(8,54,397,369,143,174,67,319)(9,41,398,384,144,189,68,334)(10,56,399,371,145,176,69,321)(11,43,400,386,146,191,70,336)(12,30,401,373,147,178,71,323)(13,45,402,388,148,193,72,310)(14,32,403,375,149,180,73,325)(15,47,404,390,150,195,74,312)(16,34,405,377,151,182,75,327)(17,49,406,392,152,169,76,314)(18,36,407,379,153,184,77,329)(19,51,408,366,154,171,78,316)(20,38,409,381,155,186,79,331)(21,53,410,368,156,173,80,318)(22,40,411,383,157,188,81,333)(23,55,412,370,158,175,82,320)(24,42,413,385,159,190,83,335)(25,29,414,372,160,177,84,322)(26,44,415,387,161,192,57,309)(27,31,416,374,162,179,58,324)(28,46,417,389,163,194,59,311)(85,251,199,337,432,126,278,282)(86,238,200,352,433,113,279,297)(87,225,201,339,434,128,280,284)(88,240,202,354,435,115,253,299)(89,227,203,341,436,130,254,286)(90,242,204,356,437,117,255,301)(91,229,205,343,438,132,256,288)(92,244,206,358,439,119,257,303)(93,231,207,345,440,134,258,290)(94,246,208,360,441,121,259,305)(95,233,209,347,442,136,260,292)(96,248,210,362,443,123,261,307)(97,235,211,349,444,138,262,294)(98,250,212,364,445,125,263,281)(99,237,213,351,446,140,264,296)(100,252,214,338,447,127,265,283)(101,239,215,353,448,114,266,298)(102,226,216,340,421,129,267,285)(103,241,217,355,422,116,268,300)(104,228,218,342,423,131,269,287)(105,243,219,357,424,118,270,302)(106,230,220,344,425,133,271,289)(107,245,221,359,426,120,272,304)(108,232,222,346,427,135,273,291)(109,247,223,361,428,122,274,306)(110,234,224,348,429,137,275,293)(111,249,197,363,430,124,276,308)(112,236,198,350,431,139,277,295), (1,103,8,110,15,89,22,96)(2,88,9,95,16,102,23,109)(3,101,10,108,17,87,24,94)(4,86,11,93,18,100,25,107)(5,99,12,106,19,85,26,92)(6,112,13,91,20,98,27,105)(7,97,14,104,21,111,28,90)(29,297,36,304,43,283,50,290)(30,282,37,289,44,296,51,303)(31,295,38,302,45,281,52,288)(32,308,39,287,46,294,53,301)(33,293,40,300,47,307,54,286)(34,306,41,285,48,292,55,299)(35,291,42,298,49,305,56,284)(57,206,64,213,71,220,78,199)(58,219,65,198,72,205,79,212)(59,204,66,211,73,218,80,197)(60,217,67,224,74,203,81,210)(61,202,68,209,75,216,82,223)(62,215,69,222,76,201,83,208)(63,200,70,207,77,214,84,221)(113,379,120,386,127,365,134,372)(114,392,121,371,128,378,135,385)(115,377,122,384,129,391,136,370)(116,390,123,369,130,376,137,383)(117,375,124,382,131,389,138,368)(118,388,125,367,132,374,139,381)(119,373,126,380,133,387,140,366)(141,431,148,438,155,445,162,424)(142,444,149,423,156,430,163,437)(143,429,150,436,157,443,164,422)(144,442,151,421,158,428,165,435)(145,427,152,434,159,441,166,448)(146,440,153,447,160,426,167,433)(147,425,154,432,161,439,168,446)(169,360,176,339,183,346,190,353)(170,345,177,352,184,359,191,338)(171,358,178,337,185,344,192,351)(172,343,179,350,186,357,193,364)(173,356,180,363,187,342,194,349)(174,341,181,348,188,355,195,362)(175,354,182,361,189,340,196,347)(225,328,232,335,239,314,246,321)(226,313,233,320,240,327,247,334)(227,326,234,333,241,312,248,319)(228,311,235,318,242,325,249,332)(229,324,236,331,243,310,250,317)(230,309,237,316,244,323,251,330)(231,322,238,329,245,336,252,315)(253,398,260,405,267,412,274,419)(254,411,261,418,268,397,275,404)(255,396,262,403,269,410,276,417)(256,409,263,416,270,395,277,402)(257,394,264,401,271,408,278,415)(258,407,265,414,272,393,279,400)(259,420,266,399,273,406,280,413) );
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)], [(1,33,418,376,164,181,60,326),(2,48,419,391,165,196,61,313),(3,35,420,378,166,183,62,328),(4,50,393,365,167,170,63,315),(5,37,394,380,168,185,64,330),(6,52,395,367,141,172,65,317),(7,39,396,382,142,187,66,332),(8,54,397,369,143,174,67,319),(9,41,398,384,144,189,68,334),(10,56,399,371,145,176,69,321),(11,43,400,386,146,191,70,336),(12,30,401,373,147,178,71,323),(13,45,402,388,148,193,72,310),(14,32,403,375,149,180,73,325),(15,47,404,390,150,195,74,312),(16,34,405,377,151,182,75,327),(17,49,406,392,152,169,76,314),(18,36,407,379,153,184,77,329),(19,51,408,366,154,171,78,316),(20,38,409,381,155,186,79,331),(21,53,410,368,156,173,80,318),(22,40,411,383,157,188,81,333),(23,55,412,370,158,175,82,320),(24,42,413,385,159,190,83,335),(25,29,414,372,160,177,84,322),(26,44,415,387,161,192,57,309),(27,31,416,374,162,179,58,324),(28,46,417,389,163,194,59,311),(85,251,199,337,432,126,278,282),(86,238,200,352,433,113,279,297),(87,225,201,339,434,128,280,284),(88,240,202,354,435,115,253,299),(89,227,203,341,436,130,254,286),(90,242,204,356,437,117,255,301),(91,229,205,343,438,132,256,288),(92,244,206,358,439,119,257,303),(93,231,207,345,440,134,258,290),(94,246,208,360,441,121,259,305),(95,233,209,347,442,136,260,292),(96,248,210,362,443,123,261,307),(97,235,211,349,444,138,262,294),(98,250,212,364,445,125,263,281),(99,237,213,351,446,140,264,296),(100,252,214,338,447,127,265,283),(101,239,215,353,448,114,266,298),(102,226,216,340,421,129,267,285),(103,241,217,355,422,116,268,300),(104,228,218,342,423,131,269,287),(105,243,219,357,424,118,270,302),(106,230,220,344,425,133,271,289),(107,245,221,359,426,120,272,304),(108,232,222,346,427,135,273,291),(109,247,223,361,428,122,274,306),(110,234,224,348,429,137,275,293),(111,249,197,363,430,124,276,308),(112,236,198,350,431,139,277,295)], [(1,103,8,110,15,89,22,96),(2,88,9,95,16,102,23,109),(3,101,10,108,17,87,24,94),(4,86,11,93,18,100,25,107),(5,99,12,106,19,85,26,92),(6,112,13,91,20,98,27,105),(7,97,14,104,21,111,28,90),(29,297,36,304,43,283,50,290),(30,282,37,289,44,296,51,303),(31,295,38,302,45,281,52,288),(32,308,39,287,46,294,53,301),(33,293,40,300,47,307,54,286),(34,306,41,285,48,292,55,299),(35,291,42,298,49,305,56,284),(57,206,64,213,71,220,78,199),(58,219,65,198,72,205,79,212),(59,204,66,211,73,218,80,197),(60,217,67,224,74,203,81,210),(61,202,68,209,75,216,82,223),(62,215,69,222,76,201,83,208),(63,200,70,207,77,214,84,221),(113,379,120,386,127,365,134,372),(114,392,121,371,128,378,135,385),(115,377,122,384,129,391,136,370),(116,390,123,369,130,376,137,383),(117,375,124,382,131,389,138,368),(118,388,125,367,132,374,139,381),(119,373,126,380,133,387,140,366),(141,431,148,438,155,445,162,424),(142,444,149,423,156,430,163,437),(143,429,150,436,157,443,164,422),(144,442,151,421,158,428,165,435),(145,427,152,434,159,441,166,448),(146,440,153,447,160,426,167,433),(147,425,154,432,161,439,168,446),(169,360,176,339,183,346,190,353),(170,345,177,352,184,359,191,338),(171,358,178,337,185,344,192,351),(172,343,179,350,186,357,193,364),(173,356,180,363,187,342,194,349),(174,341,181,348,188,355,195,362),(175,354,182,361,189,340,196,347),(225,328,232,335,239,314,246,321),(226,313,233,320,240,327,247,334),(227,326,234,333,241,312,248,319),(228,311,235,318,242,325,249,332),(229,324,236,331,243,310,250,317),(230,309,237,316,244,323,251,330),(231,322,238,329,245,336,252,315),(253,398,260,405,267,412,274,419),(254,411,261,418,268,397,275,404),(255,396,262,403,269,410,276,417),(256,409,263,416,270,395,277,402),(257,394,264,401,271,408,278,415),(258,407,265,414,272,393,279,400),(259,420,266,399,273,406,280,413)]])
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 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D7 | D8 | SD16 | Q16 | D14 | C4×D7 | D28 | C7⋊D4 | C56⋊C2 | D56 | C4.10D4 | D4.D7 | C7⋊Q16 | C4.12D28 |
| kernel | C28.2D8 | C28⋊C8 | C7×C4⋊C8 | C28⋊2Q8 | C4⋊Dic7 | C2×C28 | C4⋊C8 | C28 | C28 | C28 | C42 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 2 | 4 | 2 | 3 | 6 | 6 | 6 | 12 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of C28.2D8 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 104 | 0 | 0 |
| 0 | 0 | 88 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 98 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 15 | 87 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 108 | 0 | 0 |
| 0 | 0 | 74 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 87 | 37 |
| 0 | 0 | 0 | 0 | 52 | 26 |
| 110 | 66 | 0 | 0 | 0 | 0 |
| 106 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 80 | 32 | 0 | 0 |
| 0 | 0 | 86 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 31 |
| 0 | 0 | 0 | 0 | 0 | 95 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,88,0,0,0,0,104,34,0,0,0,0,0,0,98,0,0,0,0,0,98,15],[0,15,0,0,0,0,98,87,0,0,0,0,0,0,91,74,0,0,0,0,108,22,0,0,0,0,0,0,87,52,0,0,0,0,37,26],[110,106,0,0,0,0,66,3,0,0,0,0,0,0,80,86,0,0,0,0,32,33,0,0,0,0,0,0,44,0,0,0,0,0,31,95] >;
C28.2D8 in GAP, Magma, Sage, TeX
C_{28}._2D_8 % in TeX
G:=Group("C28.2D8"); // GroupNames label
G:=SmallGroup(448,43);
// by ID
G=gap.SmallGroup(448,43);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,141,36,1094,268,1123,346,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^7,b*a*b^-1=a^15,c*a*c^-1=a^13,c*b*c^-1=a^21*b^-1>;
// generators/relations