metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.16SD16, C42.221D14, C7⋊C8⋊8Q8, C4⋊Q8.5D7, C7⋊3(C8⋊3Q8), C4.35(Q8×D7), C4⋊C4.80D14, C4.3(Q8⋊D7), C28.36(C2×Q8), (C2×C28).152D4, C14.31(C4⋊Q8), C4.4(D4.D7), C28⋊2Q8.19C2, C14.60(C2×SD16), (C4×C28).126C22, (C2×C28).397C23, C4.Dic14.16C2, C4⋊Dic7.157C22, C2.11(Dic7⋊Q8), (C4×C7⋊C8).12C2, (C7×C4⋊Q8).5C2, C2.11(C2×Q8⋊D7), C2.14(C2×D4.D7), (C2×C14).528(C2×D4), (C2×C7⋊C8).260C22, (C2×C4).134(C7⋊D4), (C7×C4⋊C4).127C22, (C2×C4).494(C22×D7), C22.200(C2×C7⋊D4), SmallGroup(448,613)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.SD16
G = < a,b,c | a28=b8=1, c2=a14, bab-1=a13, cac-1=a15, cbc-1=b3 >
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, C4.Q8, C4⋊Q8, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C8⋊3Q8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C4×C7⋊C8, C4.Dic14, C28⋊2Q8, C7×C4⋊Q8, C28.SD16
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, D14, C4⋊Q8, C2×SD16, C7⋊D4, C22×D7, C8⋊3Q8, D4.D7, Q8⋊D7, Q8×D7, C2×C7⋊D4, C2×D4.D7, C2×Q8⋊D7, Dic7⋊Q8, C28.SD16
►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)
(1 427 66 32 246 326 353 274)(2 440 67 45 247 311 354 259)(3 425 68 30 248 324 355 272)(4 438 69 43 249 309 356 257)(5 423 70 56 250 322 357 270)(6 436 71 41 251 335 358 255)(7 421 72 54 252 320 359 268)(8 434 73 39 225 333 360 253)(9 447 74 52 226 318 361 266)(10 432 75 37 227 331 362 279)(11 445 76 50 228 316 363 264)(12 430 77 35 229 329 364 277)(13 443 78 48 230 314 337 262)(14 428 79 33 231 327 338 275)(15 441 80 46 232 312 339 260)(16 426 81 31 233 325 340 273)(17 439 82 44 234 310 341 258)(18 424 83 29 235 323 342 271)(19 437 84 42 236 336 343 256)(20 422 57 55 237 321 344 269)(21 435 58 40 238 334 345 254)(22 448 59 53 239 319 346 267)(23 433 60 38 240 332 347 280)(24 446 61 51 241 317 348 265)(25 431 62 36 242 330 349 278)(26 444 63 49 243 315 350 263)(27 429 64 34 244 328 351 276)(28 442 65 47 245 313 352 261)(85 118 404 194 204 284 167 368)(86 131 405 179 205 297 168 381)(87 116 406 192 206 282 141 366)(88 129 407 177 207 295 142 379)(89 114 408 190 208 308 143 392)(90 127 409 175 209 293 144 377)(91 140 410 188 210 306 145 390)(92 125 411 173 211 291 146 375)(93 138 412 186 212 304 147 388)(94 123 413 171 213 289 148 373)(95 136 414 184 214 302 149 386)(96 121 415 169 215 287 150 371)(97 134 416 182 216 300 151 384)(98 119 417 195 217 285 152 369)(99 132 418 180 218 298 153 382)(100 117 419 193 219 283 154 367)(101 130 420 178 220 296 155 380)(102 115 393 191 221 281 156 365)(103 128 394 176 222 294 157 378)(104 113 395 189 223 307 158 391)(105 126 396 174 224 292 159 376)(106 139 397 187 197 305 160 389)(107 124 398 172 198 290 161 374)(108 137 399 185 199 303 162 387)(109 122 400 170 200 288 163 372)(110 135 401 183 201 301 164 385)(111 120 402 196 202 286 165 370)(112 133 403 181 203 299 166 383)
(1 188 15 174)(2 175 16 189)(3 190 17 176)(4 177 18 191)(5 192 19 178)(6 179 20 193)(7 194 21 180)(8 181 22 195)(9 196 23 182)(10 183 24 169)(11 170 25 184)(12 185 26 171)(13 172 27 186)(14 187 28 173)(29 221 43 207)(30 208 44 222)(31 223 45 209)(32 210 46 224)(33 197 47 211)(34 212 48 198)(35 199 49 213)(36 214 50 200)(37 201 51 215)(38 216 52 202)(39 203 53 217)(40 218 54 204)(41 205 55 219)(42 220 56 206)(57 117 71 131)(58 132 72 118)(59 119 73 133)(60 134 74 120)(61 121 75 135)(62 136 76 122)(63 123 77 137)(64 138 78 124)(65 125 79 139)(66 140 80 126)(67 127 81 113)(68 114 82 128)(69 129 83 115)(70 116 84 130)(85 254 99 268)(86 269 100 255)(87 256 101 270)(88 271 102 257)(89 258 103 272)(90 273 104 259)(91 260 105 274)(92 275 106 261)(93 262 107 276)(94 277 108 263)(95 264 109 278)(96 279 110 265)(97 266 111 280)(98 253 112 267)(141 437 155 423)(142 424 156 438)(143 439 157 425)(144 426 158 440)(145 441 159 427)(146 428 160 442)(147 443 161 429)(148 430 162 444)(149 445 163 431)(150 432 164 446)(151 447 165 433)(152 434 166 448)(153 421 167 435)(154 436 168 422)(225 383 239 369)(226 370 240 384)(227 385 241 371)(228 372 242 386)(229 387 243 373)(230 374 244 388)(231 389 245 375)(232 376 246 390)(233 391 247 377)(234 378 248 392)(235 365 249 379)(236 380 250 366)(237 367 251 381)(238 382 252 368)(281 356 295 342)(282 343 296 357)(283 358 297 344)(284 345 298 359)(285 360 299 346)(286 347 300 361)(287 362 301 348)(288 349 302 363)(289 364 303 350)(290 351 304 337)(291 338 305 352)(292 353 306 339)(293 340 307 354)(294 355 308 341)(309 407 323 393)(310 394 324 408)(311 409 325 395)(312 396 326 410)(313 411 327 397)(314 398 328 412)(315 413 329 399)(316 400 330 414)(317 415 331 401)(318 402 332 416)(319 417 333 403)(320 404 334 418)(321 419 335 405)(322 406 336 420)
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), (1,427,66,32,246,326,353,274)(2,440,67,45,247,311,354,259)(3,425,68,30,248,324,355,272)(4,438,69,43,249,309,356,257)(5,423,70,56,250,322,357,270)(6,436,71,41,251,335,358,255)(7,421,72,54,252,320,359,268)(8,434,73,39,225,333,360,253)(9,447,74,52,226,318,361,266)(10,432,75,37,227,331,362,279)(11,445,76,50,228,316,363,264)(12,430,77,35,229,329,364,277)(13,443,78,48,230,314,337,262)(14,428,79,33,231,327,338,275)(15,441,80,46,232,312,339,260)(16,426,81,31,233,325,340,273)(17,439,82,44,234,310,341,258)(18,424,83,29,235,323,342,271)(19,437,84,42,236,336,343,256)(20,422,57,55,237,321,344,269)(21,435,58,40,238,334,345,254)(22,448,59,53,239,319,346,267)(23,433,60,38,240,332,347,280)(24,446,61,51,241,317,348,265)(25,431,62,36,242,330,349,278)(26,444,63,49,243,315,350,263)(27,429,64,34,244,328,351,276)(28,442,65,47,245,313,352,261)(85,118,404,194,204,284,167,368)(86,131,405,179,205,297,168,381)(87,116,406,192,206,282,141,366)(88,129,407,177,207,295,142,379)(89,114,408,190,208,308,143,392)(90,127,409,175,209,293,144,377)(91,140,410,188,210,306,145,390)(92,125,411,173,211,291,146,375)(93,138,412,186,212,304,147,388)(94,123,413,171,213,289,148,373)(95,136,414,184,214,302,149,386)(96,121,415,169,215,287,150,371)(97,134,416,182,216,300,151,384)(98,119,417,195,217,285,152,369)(99,132,418,180,218,298,153,382)(100,117,419,193,219,283,154,367)(101,130,420,178,220,296,155,380)(102,115,393,191,221,281,156,365)(103,128,394,176,222,294,157,378)(104,113,395,189,223,307,158,391)(105,126,396,174,224,292,159,376)(106,139,397,187,197,305,160,389)(107,124,398,172,198,290,161,374)(108,137,399,185,199,303,162,387)(109,122,400,170,200,288,163,372)(110,135,401,183,201,301,164,385)(111,120,402,196,202,286,165,370)(112,133,403,181,203,299,166,383), (1,188,15,174)(2,175,16,189)(3,190,17,176)(4,177,18,191)(5,192,19,178)(6,179,20,193)(7,194,21,180)(8,181,22,195)(9,196,23,182)(10,183,24,169)(11,170,25,184)(12,185,26,171)(13,172,27,186)(14,187,28,173)(29,221,43,207)(30,208,44,222)(31,223,45,209)(32,210,46,224)(33,197,47,211)(34,212,48,198)(35,199,49,213)(36,214,50,200)(37,201,51,215)(38,216,52,202)(39,203,53,217)(40,218,54,204)(41,205,55,219)(42,220,56,206)(57,117,71,131)(58,132,72,118)(59,119,73,133)(60,134,74,120)(61,121,75,135)(62,136,76,122)(63,123,77,137)(64,138,78,124)(65,125,79,139)(66,140,80,126)(67,127,81,113)(68,114,82,128)(69,129,83,115)(70,116,84,130)(85,254,99,268)(86,269,100,255)(87,256,101,270)(88,271,102,257)(89,258,103,272)(90,273,104,259)(91,260,105,274)(92,275,106,261)(93,262,107,276)(94,277,108,263)(95,264,109,278)(96,279,110,265)(97,266,111,280)(98,253,112,267)(141,437,155,423)(142,424,156,438)(143,439,157,425)(144,426,158,440)(145,441,159,427)(146,428,160,442)(147,443,161,429)(148,430,162,444)(149,445,163,431)(150,432,164,446)(151,447,165,433)(152,434,166,448)(153,421,167,435)(154,436,168,422)(225,383,239,369)(226,370,240,384)(227,385,241,371)(228,372,242,386)(229,387,243,373)(230,374,244,388)(231,389,245,375)(232,376,246,390)(233,391,247,377)(234,378,248,392)(235,365,249,379)(236,380,250,366)(237,367,251,381)(238,382,252,368)(281,356,295,342)(282,343,296,357)(283,358,297,344)(284,345,298,359)(285,360,299,346)(286,347,300,361)(287,362,301,348)(288,349,302,363)(289,364,303,350)(290,351,304,337)(291,338,305,352)(292,353,306,339)(293,340,307,354)(294,355,308,341)(309,407,323,393)(310,394,324,408)(311,409,325,395)(312,396,326,410)(313,411,327,397)(314,398,328,412)(315,413,329,399)(316,400,330,414)(317,415,331,401)(318,402,332,416)(319,417,333,403)(320,404,334,418)(321,419,335,405)(322,406,336,420)>;
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), (1,427,66,32,246,326,353,274)(2,440,67,45,247,311,354,259)(3,425,68,30,248,324,355,272)(4,438,69,43,249,309,356,257)(5,423,70,56,250,322,357,270)(6,436,71,41,251,335,358,255)(7,421,72,54,252,320,359,268)(8,434,73,39,225,333,360,253)(9,447,74,52,226,318,361,266)(10,432,75,37,227,331,362,279)(11,445,76,50,228,316,363,264)(12,430,77,35,229,329,364,277)(13,443,78,48,230,314,337,262)(14,428,79,33,231,327,338,275)(15,441,80,46,232,312,339,260)(16,426,81,31,233,325,340,273)(17,439,82,44,234,310,341,258)(18,424,83,29,235,323,342,271)(19,437,84,42,236,336,343,256)(20,422,57,55,237,321,344,269)(21,435,58,40,238,334,345,254)(22,448,59,53,239,319,346,267)(23,433,60,38,240,332,347,280)(24,446,61,51,241,317,348,265)(25,431,62,36,242,330,349,278)(26,444,63,49,243,315,350,263)(27,429,64,34,244,328,351,276)(28,442,65,47,245,313,352,261)(85,118,404,194,204,284,167,368)(86,131,405,179,205,297,168,381)(87,116,406,192,206,282,141,366)(88,129,407,177,207,295,142,379)(89,114,408,190,208,308,143,392)(90,127,409,175,209,293,144,377)(91,140,410,188,210,306,145,390)(92,125,411,173,211,291,146,375)(93,138,412,186,212,304,147,388)(94,123,413,171,213,289,148,373)(95,136,414,184,214,302,149,386)(96,121,415,169,215,287,150,371)(97,134,416,182,216,300,151,384)(98,119,417,195,217,285,152,369)(99,132,418,180,218,298,153,382)(100,117,419,193,219,283,154,367)(101,130,420,178,220,296,155,380)(102,115,393,191,221,281,156,365)(103,128,394,176,222,294,157,378)(104,113,395,189,223,307,158,391)(105,126,396,174,224,292,159,376)(106,139,397,187,197,305,160,389)(107,124,398,172,198,290,161,374)(108,137,399,185,199,303,162,387)(109,122,400,170,200,288,163,372)(110,135,401,183,201,301,164,385)(111,120,402,196,202,286,165,370)(112,133,403,181,203,299,166,383), (1,188,15,174)(2,175,16,189)(3,190,17,176)(4,177,18,191)(5,192,19,178)(6,179,20,193)(7,194,21,180)(8,181,22,195)(9,196,23,182)(10,183,24,169)(11,170,25,184)(12,185,26,171)(13,172,27,186)(14,187,28,173)(29,221,43,207)(30,208,44,222)(31,223,45,209)(32,210,46,224)(33,197,47,211)(34,212,48,198)(35,199,49,213)(36,214,50,200)(37,201,51,215)(38,216,52,202)(39,203,53,217)(40,218,54,204)(41,205,55,219)(42,220,56,206)(57,117,71,131)(58,132,72,118)(59,119,73,133)(60,134,74,120)(61,121,75,135)(62,136,76,122)(63,123,77,137)(64,138,78,124)(65,125,79,139)(66,140,80,126)(67,127,81,113)(68,114,82,128)(69,129,83,115)(70,116,84,130)(85,254,99,268)(86,269,100,255)(87,256,101,270)(88,271,102,257)(89,258,103,272)(90,273,104,259)(91,260,105,274)(92,275,106,261)(93,262,107,276)(94,277,108,263)(95,264,109,278)(96,279,110,265)(97,266,111,280)(98,253,112,267)(141,437,155,423)(142,424,156,438)(143,439,157,425)(144,426,158,440)(145,441,159,427)(146,428,160,442)(147,443,161,429)(148,430,162,444)(149,445,163,431)(150,432,164,446)(151,447,165,433)(152,434,166,448)(153,421,167,435)(154,436,168,422)(225,383,239,369)(226,370,240,384)(227,385,241,371)(228,372,242,386)(229,387,243,373)(230,374,244,388)(231,389,245,375)(232,376,246,390)(233,391,247,377)(234,378,248,392)(235,365,249,379)(236,380,250,366)(237,367,251,381)(238,382,252,368)(281,356,295,342)(282,343,296,357)(283,358,297,344)(284,345,298,359)(285,360,299,346)(286,347,300,361)(287,362,301,348)(288,349,302,363)(289,364,303,350)(290,351,304,337)(291,338,305,352)(292,353,306,339)(293,340,307,354)(294,355,308,341)(309,407,323,393)(310,394,324,408)(311,409,325,395)(312,396,326,410)(313,411,327,397)(314,398,328,412)(315,413,329,399)(316,400,330,414)(317,415,331,401)(318,402,332,416)(319,417,333,403)(320,404,334,418)(321,419,335,405)(322,406,336,420) );
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)], [(1,427,66,32,246,326,353,274),(2,440,67,45,247,311,354,259),(3,425,68,30,248,324,355,272),(4,438,69,43,249,309,356,257),(5,423,70,56,250,322,357,270),(6,436,71,41,251,335,358,255),(7,421,72,54,252,320,359,268),(8,434,73,39,225,333,360,253),(9,447,74,52,226,318,361,266),(10,432,75,37,227,331,362,279),(11,445,76,50,228,316,363,264),(12,430,77,35,229,329,364,277),(13,443,78,48,230,314,337,262),(14,428,79,33,231,327,338,275),(15,441,80,46,232,312,339,260),(16,426,81,31,233,325,340,273),(17,439,82,44,234,310,341,258),(18,424,83,29,235,323,342,271),(19,437,84,42,236,336,343,256),(20,422,57,55,237,321,344,269),(21,435,58,40,238,334,345,254),(22,448,59,53,239,319,346,267),(23,433,60,38,240,332,347,280),(24,446,61,51,241,317,348,265),(25,431,62,36,242,330,349,278),(26,444,63,49,243,315,350,263),(27,429,64,34,244,328,351,276),(28,442,65,47,245,313,352,261),(85,118,404,194,204,284,167,368),(86,131,405,179,205,297,168,381),(87,116,406,192,206,282,141,366),(88,129,407,177,207,295,142,379),(89,114,408,190,208,308,143,392),(90,127,409,175,209,293,144,377),(91,140,410,188,210,306,145,390),(92,125,411,173,211,291,146,375),(93,138,412,186,212,304,147,388),(94,123,413,171,213,289,148,373),(95,136,414,184,214,302,149,386),(96,121,415,169,215,287,150,371),(97,134,416,182,216,300,151,384),(98,119,417,195,217,285,152,369),(99,132,418,180,218,298,153,382),(100,117,419,193,219,283,154,367),(101,130,420,178,220,296,155,380),(102,115,393,191,221,281,156,365),(103,128,394,176,222,294,157,378),(104,113,395,189,223,307,158,391),(105,126,396,174,224,292,159,376),(106,139,397,187,197,305,160,389),(107,124,398,172,198,290,161,374),(108,137,399,185,199,303,162,387),(109,122,400,170,200,288,163,372),(110,135,401,183,201,301,164,385),(111,120,402,196,202,286,165,370),(112,133,403,181,203,299,166,383)], [(1,188,15,174),(2,175,16,189),(3,190,17,176),(4,177,18,191),(5,192,19,178),(6,179,20,193),(7,194,21,180),(8,181,22,195),(9,196,23,182),(10,183,24,169),(11,170,25,184),(12,185,26,171),(13,172,27,186),(14,187,28,173),(29,221,43,207),(30,208,44,222),(31,223,45,209),(32,210,46,224),(33,197,47,211),(34,212,48,198),(35,199,49,213),(36,214,50,200),(37,201,51,215),(38,216,52,202),(39,203,53,217),(40,218,54,204),(41,205,55,219),(42,220,56,206),(57,117,71,131),(58,132,72,118),(59,119,73,133),(60,134,74,120),(61,121,75,135),(62,136,76,122),(63,123,77,137),(64,138,78,124),(65,125,79,139),(66,140,80,126),(67,127,81,113),(68,114,82,128),(69,129,83,115),(70,116,84,130),(85,254,99,268),(86,269,100,255),(87,256,101,270),(88,271,102,257),(89,258,103,272),(90,273,104,259),(91,260,105,274),(92,275,106,261),(93,262,107,276),(94,277,108,263),(95,264,109,278),(96,279,110,265),(97,266,111,280),(98,253,112,267),(141,437,155,423),(142,424,156,438),(143,439,157,425),(144,426,158,440),(145,441,159,427),(146,428,160,442),(147,443,161,429),(148,430,162,444),(149,445,163,431),(150,432,164,446),(151,447,165,433),(152,434,166,448),(153,421,167,435),(154,436,168,422),(225,383,239,369),(226,370,240,384),(227,385,241,371),(228,372,242,386),(229,387,243,373),(230,374,244,388),(231,389,245,375),(232,376,246,390),(233,391,247,377),(234,378,248,392),(235,365,249,379),(236,380,250,366),(237,367,251,381),(238,382,252,368),(281,356,295,342),(282,343,296,357),(283,358,297,344),(284,345,298,359),(285,360,299,346),(286,347,300,361),(287,362,301,348),(288,349,302,363),(289,364,303,350),(290,351,304,337),(291,338,305,352),(292,353,306,339),(293,340,307,354),(294,355,308,341),(309,407,323,393),(310,394,324,408),(311,409,325,395),(312,396,326,410),(313,411,327,397),(314,398,328,412),(315,413,329,399),(316,400,330,414),(317,415,331,401),(318,402,332,416),(319,417,333,403),(320,404,334,418),(321,419,335,405),(322,406,336,420)]])
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 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | SD16 | D14 | D14 | C7⋊D4 | D4.D7 | Q8⋊D7 | Q8×D7 |
| kernel | C28.SD16 | C4×C7⋊C8 | C4.Dic14 | C28⋊2Q8 | C7×C4⋊Q8 | C7⋊C8 | C2×C28 | C4⋊Q8 | C28 | C42 | C4⋊C4 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 3 | 8 | 3 | 6 | 12 | 6 | 6 | 6 |
Matrix representation of C28.SD16 ►in GL6(𝔽113)
| 1 | 22 | 0 | 0 | 0 | 0 |
| 41 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 104 | 0 | 0 |
| 0 | 0 | 9 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 87 | 53 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 60 | 0 | 0 |
| 0 | 0 | 31 | 66 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 106 | 87 |
| 83 | 13 | 0 | 0 | 0 | 0 |
| 35 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 77 | 85 |
| 0 | 0 | 0 | 0 | 18 | 36 |
G:=sub<GL(6,GF(113))| [1,41,0,0,0,0,22,112,0,0,0,0,0,0,1,9,0,0,0,0,104,33,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[87,32,0,0,0,0,53,0,0,0,0,0,0,0,47,31,0,0,0,0,60,66,0,0,0,0,0,0,0,106,0,0,0,0,16,87],[83,35,0,0,0,0,13,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,77,18,0,0,0,0,85,36] >;
C28.SD16 in GAP, Magma, Sage, TeX
C_{28}.{\rm SD}_{16} % in TeX
G:=Group("C28.SD16"); // GroupNames label
G:=SmallGroup(448,613);
// by ID
G=gap.SmallGroup(448,613);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,64,1094,135,58,438,102,18822]);
// 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^15,c*b*c^-1=b^3>;
// generators/relations