metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.17D8, C28.8Q16, C42.220D14, C7⋊C8⋊7Q8, C4⋊Q8.4D7, C7⋊3(C8⋊2Q8), C4.34(Q8×D7), C4⋊C4.79D14, C4.6(D4⋊D7), C14.59(C2×D8), C28.35(C2×Q8), (C2×C28).151D4, C14.30(C4⋊Q8), C14.40(C2×Q16), C4.3(C7⋊Q16), C28⋊2Q8.18C2, (C4×C28).125C22, (C2×C28).396C23, C28.Q8.15C2, C4⋊Dic7.156C22, C2.10(Dic7⋊Q8), (C4×C7⋊C8).11C2, (C7×C4⋊Q8).4C2, C2.14(C2×D4⋊D7), C2.11(C2×C7⋊Q16), (C2×C14).527(C2×D4), (C2×C7⋊C8).259C22, (C2×C4).133(C7⋊D4), (C7×C4⋊C4).126C22, (C2×C4).493(C22×D7), C22.199(C2×C7⋊D4), SmallGroup(448,612)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.17D8
G = < a,b,c | a28=b8=1, c2=a14, bab-1=a13, cac-1=a-1, cbc-1=b-1 >
Subgroups: 396 in 98 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C4×C8, C2.D8, C4⋊Q8, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C8⋊2Q8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C4×C7⋊C8, C28.Q8, C28⋊2Q8, C7×C4⋊Q8, C28.17D8
Quotients: C1, C2, C22, D4, Q8, C23, D7, D8, Q16, C2×D4, C2×Q8, D14, C4⋊Q8, C2×D8, C2×Q16, C7⋊D4, C22×D7, C8⋊2Q8, D4⋊D7, C7⋊Q16, Q8×D7, C2×C7⋊D4, C2×D4⋊D7, C2×C7⋊Q16, Dic7⋊Q8, C28.17D8
►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)
(1 225 52 106 414 323 155 424)(2 238 53 91 415 336 156 437)(3 251 54 104 416 321 157 422)(4 236 55 89 417 334 158 435)(5 249 56 102 418 319 159 448)(6 234 29 87 419 332 160 433)(7 247 30 100 420 317 161 446)(8 232 31 85 393 330 162 431)(9 245 32 98 394 315 163 444)(10 230 33 111 395 328 164 429)(11 243 34 96 396 313 165 442)(12 228 35 109 397 326 166 427)(13 241 36 94 398 311 167 440)(14 226 37 107 399 324 168 425)(15 239 38 92 400 309 141 438)(16 252 39 105 401 322 142 423)(17 237 40 90 402 335 143 436)(18 250 41 103 403 320 144 421)(19 235 42 88 404 333 145 434)(20 248 43 101 405 318 146 447)(21 233 44 86 406 331 147 432)(22 246 45 99 407 316 148 445)(23 231 46 112 408 329 149 430)(24 244 47 97 409 314 150 443)(25 229 48 110 410 327 151 428)(26 242 49 95 411 312 152 441)(27 227 50 108 412 325 153 426)(28 240 51 93 413 310 154 439)(57 372 209 359 301 279 175 137)(58 385 210 344 302 264 176 122)(59 370 211 357 303 277 177 135)(60 383 212 342 304 262 178 120)(61 368 213 355 305 275 179 133)(62 381 214 340 306 260 180 118)(63 366 215 353 307 273 181 131)(64 379 216 338 308 258 182 116)(65 392 217 351 281 271 183 129)(66 377 218 364 282 256 184 114)(67 390 219 349 283 269 185 127)(68 375 220 362 284 254 186 140)(69 388 221 347 285 267 187 125)(70 373 222 360 286 280 188 138)(71 386 223 345 287 265 189 123)(72 371 224 358 288 278 190 136)(73 384 197 343 289 263 191 121)(74 369 198 356 290 276 192 134)(75 382 199 341 291 261 193 119)(76 367 200 354 292 274 194 132)(77 380 201 339 293 259 195 117)(78 365 202 352 294 272 196 130)(79 378 203 337 295 257 169 115)(80 391 204 350 296 270 170 128)(81 376 205 363 297 255 171 113)(82 389 206 348 298 268 172 126)(83 374 207 361 299 253 173 139)(84 387 208 346 300 266 174 124)
(1 62 15 76)(2 61 16 75)(3 60 17 74)(4 59 18 73)(5 58 19 72)(6 57 20 71)(7 84 21 70)(8 83 22 69)(9 82 23 68)(10 81 24 67)(11 80 25 66)(12 79 26 65)(13 78 27 64)(14 77 28 63)(29 175 43 189)(30 174 44 188)(31 173 45 187)(32 172 46 186)(33 171 47 185)(34 170 48 184)(35 169 49 183)(36 196 50 182)(37 195 51 181)(38 194 52 180)(39 193 53 179)(40 192 54 178)(41 191 55 177)(42 190 56 176)(85 253 99 267)(86 280 100 266)(87 279 101 265)(88 278 102 264)(89 277 103 263)(90 276 104 262)(91 275 105 261)(92 274 106 260)(93 273 107 259)(94 272 108 258)(95 271 109 257)(96 270 110 256)(97 269 111 255)(98 268 112 254)(113 244 127 230)(114 243 128 229)(115 242 129 228)(116 241 130 227)(117 240 131 226)(118 239 132 225)(119 238 133 252)(120 237 134 251)(121 236 135 250)(122 235 136 249)(123 234 137 248)(124 233 138 247)(125 232 139 246)(126 231 140 245)(141 200 155 214)(142 199 156 213)(143 198 157 212)(144 197 158 211)(145 224 159 210)(146 223 160 209)(147 222 161 208)(148 221 162 207)(149 220 163 206)(150 219 164 205)(151 218 165 204)(152 217 166 203)(153 216 167 202)(154 215 168 201)(281 397 295 411)(282 396 296 410)(283 395 297 409)(284 394 298 408)(285 393 299 407)(286 420 300 406)(287 419 301 405)(288 418 302 404)(289 417 303 403)(290 416 304 402)(291 415 305 401)(292 414 306 400)(293 413 307 399)(294 412 308 398)(309 354 323 340)(310 353 324 339)(311 352 325 338)(312 351 326 337)(313 350 327 364)(314 349 328 363)(315 348 329 362)(316 347 330 361)(317 346 331 360)(318 345 332 359)(319 344 333 358)(320 343 334 357)(321 342 335 356)(322 341 336 355)(365 426 379 440)(366 425 380 439)(367 424 381 438)(368 423 382 437)(369 422 383 436)(370 421 384 435)(371 448 385 434)(372 447 386 433)(373 446 387 432)(374 445 388 431)(375 444 389 430)(376 443 390 429)(377 442 391 428)(378 441 392 427)
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), (1,225,52,106,414,323,155,424)(2,238,53,91,415,336,156,437)(3,251,54,104,416,321,157,422)(4,236,55,89,417,334,158,435)(5,249,56,102,418,319,159,448)(6,234,29,87,419,332,160,433)(7,247,30,100,420,317,161,446)(8,232,31,85,393,330,162,431)(9,245,32,98,394,315,163,444)(10,230,33,111,395,328,164,429)(11,243,34,96,396,313,165,442)(12,228,35,109,397,326,166,427)(13,241,36,94,398,311,167,440)(14,226,37,107,399,324,168,425)(15,239,38,92,400,309,141,438)(16,252,39,105,401,322,142,423)(17,237,40,90,402,335,143,436)(18,250,41,103,403,320,144,421)(19,235,42,88,404,333,145,434)(20,248,43,101,405,318,146,447)(21,233,44,86,406,331,147,432)(22,246,45,99,407,316,148,445)(23,231,46,112,408,329,149,430)(24,244,47,97,409,314,150,443)(25,229,48,110,410,327,151,428)(26,242,49,95,411,312,152,441)(27,227,50,108,412,325,153,426)(28,240,51,93,413,310,154,439)(57,372,209,359,301,279,175,137)(58,385,210,344,302,264,176,122)(59,370,211,357,303,277,177,135)(60,383,212,342,304,262,178,120)(61,368,213,355,305,275,179,133)(62,381,214,340,306,260,180,118)(63,366,215,353,307,273,181,131)(64,379,216,338,308,258,182,116)(65,392,217,351,281,271,183,129)(66,377,218,364,282,256,184,114)(67,390,219,349,283,269,185,127)(68,375,220,362,284,254,186,140)(69,388,221,347,285,267,187,125)(70,373,222,360,286,280,188,138)(71,386,223,345,287,265,189,123)(72,371,224,358,288,278,190,136)(73,384,197,343,289,263,191,121)(74,369,198,356,290,276,192,134)(75,382,199,341,291,261,193,119)(76,367,200,354,292,274,194,132)(77,380,201,339,293,259,195,117)(78,365,202,352,294,272,196,130)(79,378,203,337,295,257,169,115)(80,391,204,350,296,270,170,128)(81,376,205,363,297,255,171,113)(82,389,206,348,298,268,172,126)(83,374,207,361,299,253,173,139)(84,387,208,346,300,266,174,124), (1,62,15,76)(2,61,16,75)(3,60,17,74)(4,59,18,73)(5,58,19,72)(6,57,20,71)(7,84,21,70)(8,83,22,69)(9,82,23,68)(10,81,24,67)(11,80,25,66)(12,79,26,65)(13,78,27,64)(14,77,28,63)(29,175,43,189)(30,174,44,188)(31,173,45,187)(32,172,46,186)(33,171,47,185)(34,170,48,184)(35,169,49,183)(36,196,50,182)(37,195,51,181)(38,194,52,180)(39,193,53,179)(40,192,54,178)(41,191,55,177)(42,190,56,176)(85,253,99,267)(86,280,100,266)(87,279,101,265)(88,278,102,264)(89,277,103,263)(90,276,104,262)(91,275,105,261)(92,274,106,260)(93,273,107,259)(94,272,108,258)(95,271,109,257)(96,270,110,256)(97,269,111,255)(98,268,112,254)(113,244,127,230)(114,243,128,229)(115,242,129,228)(116,241,130,227)(117,240,131,226)(118,239,132,225)(119,238,133,252)(120,237,134,251)(121,236,135,250)(122,235,136,249)(123,234,137,248)(124,233,138,247)(125,232,139,246)(126,231,140,245)(141,200,155,214)(142,199,156,213)(143,198,157,212)(144,197,158,211)(145,224,159,210)(146,223,160,209)(147,222,161,208)(148,221,162,207)(149,220,163,206)(150,219,164,205)(151,218,165,204)(152,217,166,203)(153,216,167,202)(154,215,168,201)(281,397,295,411)(282,396,296,410)(283,395,297,409)(284,394,298,408)(285,393,299,407)(286,420,300,406)(287,419,301,405)(288,418,302,404)(289,417,303,403)(290,416,304,402)(291,415,305,401)(292,414,306,400)(293,413,307,399)(294,412,308,398)(309,354,323,340)(310,353,324,339)(311,352,325,338)(312,351,326,337)(313,350,327,364)(314,349,328,363)(315,348,329,362)(316,347,330,361)(317,346,331,360)(318,345,332,359)(319,344,333,358)(320,343,334,357)(321,342,335,356)(322,341,336,355)(365,426,379,440)(366,425,380,439)(367,424,381,438)(368,423,382,437)(369,422,383,436)(370,421,384,435)(371,448,385,434)(372,447,386,433)(373,446,387,432)(374,445,388,431)(375,444,389,430)(376,443,390,429)(377,442,391,428)(378,441,392,427)>;
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), (1,225,52,106,414,323,155,424)(2,238,53,91,415,336,156,437)(3,251,54,104,416,321,157,422)(4,236,55,89,417,334,158,435)(5,249,56,102,418,319,159,448)(6,234,29,87,419,332,160,433)(7,247,30,100,420,317,161,446)(8,232,31,85,393,330,162,431)(9,245,32,98,394,315,163,444)(10,230,33,111,395,328,164,429)(11,243,34,96,396,313,165,442)(12,228,35,109,397,326,166,427)(13,241,36,94,398,311,167,440)(14,226,37,107,399,324,168,425)(15,239,38,92,400,309,141,438)(16,252,39,105,401,322,142,423)(17,237,40,90,402,335,143,436)(18,250,41,103,403,320,144,421)(19,235,42,88,404,333,145,434)(20,248,43,101,405,318,146,447)(21,233,44,86,406,331,147,432)(22,246,45,99,407,316,148,445)(23,231,46,112,408,329,149,430)(24,244,47,97,409,314,150,443)(25,229,48,110,410,327,151,428)(26,242,49,95,411,312,152,441)(27,227,50,108,412,325,153,426)(28,240,51,93,413,310,154,439)(57,372,209,359,301,279,175,137)(58,385,210,344,302,264,176,122)(59,370,211,357,303,277,177,135)(60,383,212,342,304,262,178,120)(61,368,213,355,305,275,179,133)(62,381,214,340,306,260,180,118)(63,366,215,353,307,273,181,131)(64,379,216,338,308,258,182,116)(65,392,217,351,281,271,183,129)(66,377,218,364,282,256,184,114)(67,390,219,349,283,269,185,127)(68,375,220,362,284,254,186,140)(69,388,221,347,285,267,187,125)(70,373,222,360,286,280,188,138)(71,386,223,345,287,265,189,123)(72,371,224,358,288,278,190,136)(73,384,197,343,289,263,191,121)(74,369,198,356,290,276,192,134)(75,382,199,341,291,261,193,119)(76,367,200,354,292,274,194,132)(77,380,201,339,293,259,195,117)(78,365,202,352,294,272,196,130)(79,378,203,337,295,257,169,115)(80,391,204,350,296,270,170,128)(81,376,205,363,297,255,171,113)(82,389,206,348,298,268,172,126)(83,374,207,361,299,253,173,139)(84,387,208,346,300,266,174,124), (1,62,15,76)(2,61,16,75)(3,60,17,74)(4,59,18,73)(5,58,19,72)(6,57,20,71)(7,84,21,70)(8,83,22,69)(9,82,23,68)(10,81,24,67)(11,80,25,66)(12,79,26,65)(13,78,27,64)(14,77,28,63)(29,175,43,189)(30,174,44,188)(31,173,45,187)(32,172,46,186)(33,171,47,185)(34,170,48,184)(35,169,49,183)(36,196,50,182)(37,195,51,181)(38,194,52,180)(39,193,53,179)(40,192,54,178)(41,191,55,177)(42,190,56,176)(85,253,99,267)(86,280,100,266)(87,279,101,265)(88,278,102,264)(89,277,103,263)(90,276,104,262)(91,275,105,261)(92,274,106,260)(93,273,107,259)(94,272,108,258)(95,271,109,257)(96,270,110,256)(97,269,111,255)(98,268,112,254)(113,244,127,230)(114,243,128,229)(115,242,129,228)(116,241,130,227)(117,240,131,226)(118,239,132,225)(119,238,133,252)(120,237,134,251)(121,236,135,250)(122,235,136,249)(123,234,137,248)(124,233,138,247)(125,232,139,246)(126,231,140,245)(141,200,155,214)(142,199,156,213)(143,198,157,212)(144,197,158,211)(145,224,159,210)(146,223,160,209)(147,222,161,208)(148,221,162,207)(149,220,163,206)(150,219,164,205)(151,218,165,204)(152,217,166,203)(153,216,167,202)(154,215,168,201)(281,397,295,411)(282,396,296,410)(283,395,297,409)(284,394,298,408)(285,393,299,407)(286,420,300,406)(287,419,301,405)(288,418,302,404)(289,417,303,403)(290,416,304,402)(291,415,305,401)(292,414,306,400)(293,413,307,399)(294,412,308,398)(309,354,323,340)(310,353,324,339)(311,352,325,338)(312,351,326,337)(313,350,327,364)(314,349,328,363)(315,348,329,362)(316,347,330,361)(317,346,331,360)(318,345,332,359)(319,344,333,358)(320,343,334,357)(321,342,335,356)(322,341,336,355)(365,426,379,440)(366,425,380,439)(367,424,381,438)(368,423,382,437)(369,422,383,436)(370,421,384,435)(371,448,385,434)(372,447,386,433)(373,446,387,432)(374,445,388,431)(375,444,389,430)(376,443,390,429)(377,442,391,428)(378,441,392,427) );
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)], [(1,225,52,106,414,323,155,424),(2,238,53,91,415,336,156,437),(3,251,54,104,416,321,157,422),(4,236,55,89,417,334,158,435),(5,249,56,102,418,319,159,448),(6,234,29,87,419,332,160,433),(7,247,30,100,420,317,161,446),(8,232,31,85,393,330,162,431),(9,245,32,98,394,315,163,444),(10,230,33,111,395,328,164,429),(11,243,34,96,396,313,165,442),(12,228,35,109,397,326,166,427),(13,241,36,94,398,311,167,440),(14,226,37,107,399,324,168,425),(15,239,38,92,400,309,141,438),(16,252,39,105,401,322,142,423),(17,237,40,90,402,335,143,436),(18,250,41,103,403,320,144,421),(19,235,42,88,404,333,145,434),(20,248,43,101,405,318,146,447),(21,233,44,86,406,331,147,432),(22,246,45,99,407,316,148,445),(23,231,46,112,408,329,149,430),(24,244,47,97,409,314,150,443),(25,229,48,110,410,327,151,428),(26,242,49,95,411,312,152,441),(27,227,50,108,412,325,153,426),(28,240,51,93,413,310,154,439),(57,372,209,359,301,279,175,137),(58,385,210,344,302,264,176,122),(59,370,211,357,303,277,177,135),(60,383,212,342,304,262,178,120),(61,368,213,355,305,275,179,133),(62,381,214,340,306,260,180,118),(63,366,215,353,307,273,181,131),(64,379,216,338,308,258,182,116),(65,392,217,351,281,271,183,129),(66,377,218,364,282,256,184,114),(67,390,219,349,283,269,185,127),(68,375,220,362,284,254,186,140),(69,388,221,347,285,267,187,125),(70,373,222,360,286,280,188,138),(71,386,223,345,287,265,189,123),(72,371,224,358,288,278,190,136),(73,384,197,343,289,263,191,121),(74,369,198,356,290,276,192,134),(75,382,199,341,291,261,193,119),(76,367,200,354,292,274,194,132),(77,380,201,339,293,259,195,117),(78,365,202,352,294,272,196,130),(79,378,203,337,295,257,169,115),(80,391,204,350,296,270,170,128),(81,376,205,363,297,255,171,113),(82,389,206,348,298,268,172,126),(83,374,207,361,299,253,173,139),(84,387,208,346,300,266,174,124)], [(1,62,15,76),(2,61,16,75),(3,60,17,74),(4,59,18,73),(5,58,19,72),(6,57,20,71),(7,84,21,70),(8,83,22,69),(9,82,23,68),(10,81,24,67),(11,80,25,66),(12,79,26,65),(13,78,27,64),(14,77,28,63),(29,175,43,189),(30,174,44,188),(31,173,45,187),(32,172,46,186),(33,171,47,185),(34,170,48,184),(35,169,49,183),(36,196,50,182),(37,195,51,181),(38,194,52,180),(39,193,53,179),(40,192,54,178),(41,191,55,177),(42,190,56,176),(85,253,99,267),(86,280,100,266),(87,279,101,265),(88,278,102,264),(89,277,103,263),(90,276,104,262),(91,275,105,261),(92,274,106,260),(93,273,107,259),(94,272,108,258),(95,271,109,257),(96,270,110,256),(97,269,111,255),(98,268,112,254),(113,244,127,230),(114,243,128,229),(115,242,129,228),(116,241,130,227),(117,240,131,226),(118,239,132,225),(119,238,133,252),(120,237,134,251),(121,236,135,250),(122,235,136,249),(123,234,137,248),(124,233,138,247),(125,232,139,246),(126,231,140,245),(141,200,155,214),(142,199,156,213),(143,198,157,212),(144,197,158,211),(145,224,159,210),(146,223,160,209),(147,222,161,208),(148,221,162,207),(149,220,163,206),(150,219,164,205),(151,218,165,204),(152,217,166,203),(153,216,167,202),(154,215,168,201),(281,397,295,411),(282,396,296,410),(283,395,297,409),(284,394,298,408),(285,393,299,407),(286,420,300,406),(287,419,301,405),(288,418,302,404),(289,417,303,403),(290,416,304,402),(291,415,305,401),(292,414,306,400),(293,413,307,399),(294,412,308,398),(309,354,323,340),(310,353,324,339),(311,352,325,338),(312,351,326,337),(313,350,327,364),(314,349,328,363),(315,348,329,362),(316,347,330,361),(317,346,331,360),(318,345,332,359),(319,344,333,358),(320,343,334,357),(321,342,335,356),(322,341,336,355),(365,426,379,440),(366,425,380,439),(367,424,381,438),(368,423,382,437),(369,422,383,436),(370,421,384,435),(371,448,385,434),(372,447,386,433),(373,446,387,432),(374,445,388,431),(375,444,389,430),(376,443,390,429),(377,442,391,428),(378,441,392,427)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | D8 | Q16 | D14 | D14 | C7⋊D4 | D4⋊D7 | C7⋊Q16 | Q8×D7 |
| kernel | C28.17D8 | C4×C7⋊C8 | C28.Q8 | C28⋊2Q8 | C7×C4⋊Q8 | C7⋊C8 | C2×C28 | C4⋊Q8 | C28 | C28 | C42 | C4⋊C4 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 3 | 4 | 4 | 3 | 6 | 12 | 6 | 6 | 6 |
Matrix representation of C28.17D8 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 79 | 0 | 0 |
| 0 | 0 | 34 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 0 | 69 | 112 |
| 51 | 8 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 43 | 0 | 0 |
| 0 | 0 | 86 | 91 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 99 |
| 0 | 0 | 0 | 0 | 105 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 36 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 70 | 0 | 0 |
| 0 | 0 | 27 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 36 |
| 0 | 0 | 0 | 0 | 99 | 86 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,34,0,0,0,0,79,25,0,0,0,0,0,0,1,69,0,0,0,0,36,112],[51,14,0,0,0,0,8,0,0,0,0,0,0,0,22,86,0,0,0,0,43,91,0,0,0,0,0,0,62,105,0,0,0,0,99,0],[1,36,0,0,0,0,0,112,0,0,0,0,0,0,91,27,0,0,0,0,70,22,0,0,0,0,0,0,27,99,0,0,0,0,36,86] >;
C28.17D8 in GAP, Magma, Sage, TeX
C_{28}._{17}D_8 % in TeX
G:=Group("C28.17D8"); // GroupNames label
G:=SmallGroup(448,612);
// by ID
G=gap.SmallGroup(448,612);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,64,422,135,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations