metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊3Dic14, C28.48SD16, C42.53D14, (C7×Q8)⋊3Q8, (C4×Q8).2D7, C7⋊4(Q8⋊Q8), (C2×C28).63D4, (Q8×C28).2C2, C28.28(C2×Q8), C4⋊C4.248D14, C4.13(Q8⋊D7), C28⋊C8.14C2, C4.62(C4○D28), C28.55(C4○D4), (C4×C28).91C22, (C2×Q8).155D14, C28⋊2Q8.13C2, C14.68(C2×SD16), C4.12(C2×Dic14), Q8⋊Dic7.7C2, (C2×C28).342C23, C14.64(C22⋊Q8), C4.Dic14.10C2, C2.9(D4.9D14), C4⋊Dic7.139C22, (Q8×C14).190C22, C2.15(C28.48D4), C14.110(C8.C22), C2.6(C2×Q8⋊D7), (C2×C7⋊C8).97C22, (C2×C14).473(C2×D4), (C2×C4).247(C7⋊D4), (C7×C4⋊C4).279C22, (C2×C4).442(C22×D7), C22.152(C2×C7⋊D4), SmallGroup(448,554)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.48SD16
G = < a,b,c | a28=b8=1, c2=a14, bab-1=cac-1=a-1, cbc-1=a14b3 >
Subgroups: 388 in 96 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, Dic7, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, Q8⋊Q8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C28⋊C8, C4.Dic14, Q8⋊Dic7, C28⋊2Q8, Q8×C28, C28.48SD16
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×SD16, C8.C22, Dic14, C7⋊D4, C22×D7, Q8⋊Q8, Q8⋊D7, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4, C2×Q8⋊D7, D4.9D14, C28.48SD16
►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)
(1 364 157 233 101 313 400 189)(2 363 158 232 102 312 401 188)(3 362 159 231 103 311 402 187)(4 361 160 230 104 310 403 186)(5 360 161 229 105 309 404 185)(6 359 162 228 106 336 405 184)(7 358 163 227 107 335 406 183)(8 357 164 226 108 334 407 182)(9 356 165 225 109 333 408 181)(10 355 166 252 110 332 409 180)(11 354 167 251 111 331 410 179)(12 353 168 250 112 330 411 178)(13 352 141 249 85 329 412 177)(14 351 142 248 86 328 413 176)(15 350 143 247 87 327 414 175)(16 349 144 246 88 326 415 174)(17 348 145 245 89 325 416 173)(18 347 146 244 90 324 417 172)(19 346 147 243 91 323 418 171)(20 345 148 242 92 322 419 170)(21 344 149 241 93 321 420 169)(22 343 150 240 94 320 393 196)(23 342 151 239 95 319 394 195)(24 341 152 238 96 318 395 194)(25 340 153 237 97 317 396 193)(26 339 154 236 98 316 397 192)(27 338 155 235 99 315 398 191)(28 337 156 234 100 314 399 190)(29 307 385 429 61 219 123 264)(30 306 386 428 62 218 124 263)(31 305 387 427 63 217 125 262)(32 304 388 426 64 216 126 261)(33 303 389 425 65 215 127 260)(34 302 390 424 66 214 128 259)(35 301 391 423 67 213 129 258)(36 300 392 422 68 212 130 257)(37 299 365 421 69 211 131 256)(38 298 366 448 70 210 132 255)(39 297 367 447 71 209 133 254)(40 296 368 446 72 208 134 253)(41 295 369 445 73 207 135 280)(42 294 370 444 74 206 136 279)(43 293 371 443 75 205 137 278)(44 292 372 442 76 204 138 277)(45 291 373 441 77 203 139 276)(46 290 374 440 78 202 140 275)(47 289 375 439 79 201 113 274)(48 288 376 438 80 200 114 273)(49 287 377 437 81 199 115 272)(50 286 378 436 82 198 116 271)(51 285 379 435 83 197 117 270)(52 284 380 434 84 224 118 269)(53 283 381 433 57 223 119 268)(54 282 382 432 58 222 120 267)(55 281 383 431 59 221 121 266)(56 308 384 430 60 220 122 265)
(1 440 15 426)(2 439 16 425)(3 438 17 424)(4 437 18 423)(5 436 19 422)(6 435 20 421)(7 434 21 448)(8 433 22 447)(9 432 23 446)(10 431 24 445)(11 430 25 444)(12 429 26 443)(13 428 27 442)(14 427 28 441)(29 178 43 192)(30 177 44 191)(31 176 45 190)(32 175 46 189)(33 174 47 188)(34 173 48 187)(35 172 49 186)(36 171 50 185)(37 170 51 184)(38 169 52 183)(39 196 53 182)(40 195 54 181)(41 194 55 180)(42 193 56 179)(57 226 71 240)(58 225 72 239)(59 252 73 238)(60 251 74 237)(61 250 75 236)(62 249 76 235)(63 248 77 234)(64 247 78 233)(65 246 79 232)(66 245 80 231)(67 244 81 230)(68 243 82 229)(69 242 83 228)(70 241 84 227)(85 263 99 277)(86 262 100 276)(87 261 101 275)(88 260 102 274)(89 259 103 273)(90 258 104 272)(91 257 105 271)(92 256 106 270)(93 255 107 269)(94 254 108 268)(95 253 109 267)(96 280 110 266)(97 279 111 265)(98 278 112 264)(113 363 127 349)(114 362 128 348)(115 361 129 347)(116 360 130 346)(117 359 131 345)(118 358 132 344)(119 357 133 343)(120 356 134 342)(121 355 135 341)(122 354 136 340)(123 353 137 339)(124 352 138 338)(125 351 139 337)(126 350 140 364)(141 306 155 292)(142 305 156 291)(143 304 157 290)(144 303 158 289)(145 302 159 288)(146 301 160 287)(147 300 161 286)(148 299 162 285)(149 298 163 284)(150 297 164 283)(151 296 165 282)(152 295 166 281)(153 294 167 308)(154 293 168 307)(197 419 211 405)(198 418 212 404)(199 417 213 403)(200 416 214 402)(201 415 215 401)(202 414 216 400)(203 413 217 399)(204 412 218 398)(205 411 219 397)(206 410 220 396)(207 409 221 395)(208 408 222 394)(209 407 223 393)(210 406 224 420)(309 392 323 378)(310 391 324 377)(311 390 325 376)(312 389 326 375)(313 388 327 374)(314 387 328 373)(315 386 329 372)(316 385 330 371)(317 384 331 370)(318 383 332 369)(319 382 333 368)(320 381 334 367)(321 380 335 366)(322 379 336 365)
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), (1,364,157,233,101,313,400,189)(2,363,158,232,102,312,401,188)(3,362,159,231,103,311,402,187)(4,361,160,230,104,310,403,186)(5,360,161,229,105,309,404,185)(6,359,162,228,106,336,405,184)(7,358,163,227,107,335,406,183)(8,357,164,226,108,334,407,182)(9,356,165,225,109,333,408,181)(10,355,166,252,110,332,409,180)(11,354,167,251,111,331,410,179)(12,353,168,250,112,330,411,178)(13,352,141,249,85,329,412,177)(14,351,142,248,86,328,413,176)(15,350,143,247,87,327,414,175)(16,349,144,246,88,326,415,174)(17,348,145,245,89,325,416,173)(18,347,146,244,90,324,417,172)(19,346,147,243,91,323,418,171)(20,345,148,242,92,322,419,170)(21,344,149,241,93,321,420,169)(22,343,150,240,94,320,393,196)(23,342,151,239,95,319,394,195)(24,341,152,238,96,318,395,194)(25,340,153,237,97,317,396,193)(26,339,154,236,98,316,397,192)(27,338,155,235,99,315,398,191)(28,337,156,234,100,314,399,190)(29,307,385,429,61,219,123,264)(30,306,386,428,62,218,124,263)(31,305,387,427,63,217,125,262)(32,304,388,426,64,216,126,261)(33,303,389,425,65,215,127,260)(34,302,390,424,66,214,128,259)(35,301,391,423,67,213,129,258)(36,300,392,422,68,212,130,257)(37,299,365,421,69,211,131,256)(38,298,366,448,70,210,132,255)(39,297,367,447,71,209,133,254)(40,296,368,446,72,208,134,253)(41,295,369,445,73,207,135,280)(42,294,370,444,74,206,136,279)(43,293,371,443,75,205,137,278)(44,292,372,442,76,204,138,277)(45,291,373,441,77,203,139,276)(46,290,374,440,78,202,140,275)(47,289,375,439,79,201,113,274)(48,288,376,438,80,200,114,273)(49,287,377,437,81,199,115,272)(50,286,378,436,82,198,116,271)(51,285,379,435,83,197,117,270)(52,284,380,434,84,224,118,269)(53,283,381,433,57,223,119,268)(54,282,382,432,58,222,120,267)(55,281,383,431,59,221,121,266)(56,308,384,430,60,220,122,265), (1,440,15,426)(2,439,16,425)(3,438,17,424)(4,437,18,423)(5,436,19,422)(6,435,20,421)(7,434,21,448)(8,433,22,447)(9,432,23,446)(10,431,24,445)(11,430,25,444)(12,429,26,443)(13,428,27,442)(14,427,28,441)(29,178,43,192)(30,177,44,191)(31,176,45,190)(32,175,46,189)(33,174,47,188)(34,173,48,187)(35,172,49,186)(36,171,50,185)(37,170,51,184)(38,169,52,183)(39,196,53,182)(40,195,54,181)(41,194,55,180)(42,193,56,179)(57,226,71,240)(58,225,72,239)(59,252,73,238)(60,251,74,237)(61,250,75,236)(62,249,76,235)(63,248,77,234)(64,247,78,233)(65,246,79,232)(66,245,80,231)(67,244,81,230)(68,243,82,229)(69,242,83,228)(70,241,84,227)(85,263,99,277)(86,262,100,276)(87,261,101,275)(88,260,102,274)(89,259,103,273)(90,258,104,272)(91,257,105,271)(92,256,106,270)(93,255,107,269)(94,254,108,268)(95,253,109,267)(96,280,110,266)(97,279,111,265)(98,278,112,264)(113,363,127,349)(114,362,128,348)(115,361,129,347)(116,360,130,346)(117,359,131,345)(118,358,132,344)(119,357,133,343)(120,356,134,342)(121,355,135,341)(122,354,136,340)(123,353,137,339)(124,352,138,338)(125,351,139,337)(126,350,140,364)(141,306,155,292)(142,305,156,291)(143,304,157,290)(144,303,158,289)(145,302,159,288)(146,301,160,287)(147,300,161,286)(148,299,162,285)(149,298,163,284)(150,297,164,283)(151,296,165,282)(152,295,166,281)(153,294,167,308)(154,293,168,307)(197,419,211,405)(198,418,212,404)(199,417,213,403)(200,416,214,402)(201,415,215,401)(202,414,216,400)(203,413,217,399)(204,412,218,398)(205,411,219,397)(206,410,220,396)(207,409,221,395)(208,408,222,394)(209,407,223,393)(210,406,224,420)(309,392,323,378)(310,391,324,377)(311,390,325,376)(312,389,326,375)(313,388,327,374)(314,387,328,373)(315,386,329,372)(316,385,330,371)(317,384,331,370)(318,383,332,369)(319,382,333,368)(320,381,334,367)(321,380,335,366)(322,379,336,365)>;
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), (1,364,157,233,101,313,400,189)(2,363,158,232,102,312,401,188)(3,362,159,231,103,311,402,187)(4,361,160,230,104,310,403,186)(5,360,161,229,105,309,404,185)(6,359,162,228,106,336,405,184)(7,358,163,227,107,335,406,183)(8,357,164,226,108,334,407,182)(9,356,165,225,109,333,408,181)(10,355,166,252,110,332,409,180)(11,354,167,251,111,331,410,179)(12,353,168,250,112,330,411,178)(13,352,141,249,85,329,412,177)(14,351,142,248,86,328,413,176)(15,350,143,247,87,327,414,175)(16,349,144,246,88,326,415,174)(17,348,145,245,89,325,416,173)(18,347,146,244,90,324,417,172)(19,346,147,243,91,323,418,171)(20,345,148,242,92,322,419,170)(21,344,149,241,93,321,420,169)(22,343,150,240,94,320,393,196)(23,342,151,239,95,319,394,195)(24,341,152,238,96,318,395,194)(25,340,153,237,97,317,396,193)(26,339,154,236,98,316,397,192)(27,338,155,235,99,315,398,191)(28,337,156,234,100,314,399,190)(29,307,385,429,61,219,123,264)(30,306,386,428,62,218,124,263)(31,305,387,427,63,217,125,262)(32,304,388,426,64,216,126,261)(33,303,389,425,65,215,127,260)(34,302,390,424,66,214,128,259)(35,301,391,423,67,213,129,258)(36,300,392,422,68,212,130,257)(37,299,365,421,69,211,131,256)(38,298,366,448,70,210,132,255)(39,297,367,447,71,209,133,254)(40,296,368,446,72,208,134,253)(41,295,369,445,73,207,135,280)(42,294,370,444,74,206,136,279)(43,293,371,443,75,205,137,278)(44,292,372,442,76,204,138,277)(45,291,373,441,77,203,139,276)(46,290,374,440,78,202,140,275)(47,289,375,439,79,201,113,274)(48,288,376,438,80,200,114,273)(49,287,377,437,81,199,115,272)(50,286,378,436,82,198,116,271)(51,285,379,435,83,197,117,270)(52,284,380,434,84,224,118,269)(53,283,381,433,57,223,119,268)(54,282,382,432,58,222,120,267)(55,281,383,431,59,221,121,266)(56,308,384,430,60,220,122,265), (1,440,15,426)(2,439,16,425)(3,438,17,424)(4,437,18,423)(5,436,19,422)(6,435,20,421)(7,434,21,448)(8,433,22,447)(9,432,23,446)(10,431,24,445)(11,430,25,444)(12,429,26,443)(13,428,27,442)(14,427,28,441)(29,178,43,192)(30,177,44,191)(31,176,45,190)(32,175,46,189)(33,174,47,188)(34,173,48,187)(35,172,49,186)(36,171,50,185)(37,170,51,184)(38,169,52,183)(39,196,53,182)(40,195,54,181)(41,194,55,180)(42,193,56,179)(57,226,71,240)(58,225,72,239)(59,252,73,238)(60,251,74,237)(61,250,75,236)(62,249,76,235)(63,248,77,234)(64,247,78,233)(65,246,79,232)(66,245,80,231)(67,244,81,230)(68,243,82,229)(69,242,83,228)(70,241,84,227)(85,263,99,277)(86,262,100,276)(87,261,101,275)(88,260,102,274)(89,259,103,273)(90,258,104,272)(91,257,105,271)(92,256,106,270)(93,255,107,269)(94,254,108,268)(95,253,109,267)(96,280,110,266)(97,279,111,265)(98,278,112,264)(113,363,127,349)(114,362,128,348)(115,361,129,347)(116,360,130,346)(117,359,131,345)(118,358,132,344)(119,357,133,343)(120,356,134,342)(121,355,135,341)(122,354,136,340)(123,353,137,339)(124,352,138,338)(125,351,139,337)(126,350,140,364)(141,306,155,292)(142,305,156,291)(143,304,157,290)(144,303,158,289)(145,302,159,288)(146,301,160,287)(147,300,161,286)(148,299,162,285)(149,298,163,284)(150,297,164,283)(151,296,165,282)(152,295,166,281)(153,294,167,308)(154,293,168,307)(197,419,211,405)(198,418,212,404)(199,417,213,403)(200,416,214,402)(201,415,215,401)(202,414,216,400)(203,413,217,399)(204,412,218,398)(205,411,219,397)(206,410,220,396)(207,409,221,395)(208,408,222,394)(209,407,223,393)(210,406,224,420)(309,392,323,378)(310,391,324,377)(311,390,325,376)(312,389,326,375)(313,388,327,374)(314,387,328,373)(315,386,329,372)(316,385,330,371)(317,384,331,370)(318,383,332,369)(319,382,333,368)(320,381,334,367)(321,380,335,366)(322,379,336,365) );
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)], [(1,364,157,233,101,313,400,189),(2,363,158,232,102,312,401,188),(3,362,159,231,103,311,402,187),(4,361,160,230,104,310,403,186),(5,360,161,229,105,309,404,185),(6,359,162,228,106,336,405,184),(7,358,163,227,107,335,406,183),(8,357,164,226,108,334,407,182),(9,356,165,225,109,333,408,181),(10,355,166,252,110,332,409,180),(11,354,167,251,111,331,410,179),(12,353,168,250,112,330,411,178),(13,352,141,249,85,329,412,177),(14,351,142,248,86,328,413,176),(15,350,143,247,87,327,414,175),(16,349,144,246,88,326,415,174),(17,348,145,245,89,325,416,173),(18,347,146,244,90,324,417,172),(19,346,147,243,91,323,418,171),(20,345,148,242,92,322,419,170),(21,344,149,241,93,321,420,169),(22,343,150,240,94,320,393,196),(23,342,151,239,95,319,394,195),(24,341,152,238,96,318,395,194),(25,340,153,237,97,317,396,193),(26,339,154,236,98,316,397,192),(27,338,155,235,99,315,398,191),(28,337,156,234,100,314,399,190),(29,307,385,429,61,219,123,264),(30,306,386,428,62,218,124,263),(31,305,387,427,63,217,125,262),(32,304,388,426,64,216,126,261),(33,303,389,425,65,215,127,260),(34,302,390,424,66,214,128,259),(35,301,391,423,67,213,129,258),(36,300,392,422,68,212,130,257),(37,299,365,421,69,211,131,256),(38,298,366,448,70,210,132,255),(39,297,367,447,71,209,133,254),(40,296,368,446,72,208,134,253),(41,295,369,445,73,207,135,280),(42,294,370,444,74,206,136,279),(43,293,371,443,75,205,137,278),(44,292,372,442,76,204,138,277),(45,291,373,441,77,203,139,276),(46,290,374,440,78,202,140,275),(47,289,375,439,79,201,113,274),(48,288,376,438,80,200,114,273),(49,287,377,437,81,199,115,272),(50,286,378,436,82,198,116,271),(51,285,379,435,83,197,117,270),(52,284,380,434,84,224,118,269),(53,283,381,433,57,223,119,268),(54,282,382,432,58,222,120,267),(55,281,383,431,59,221,121,266),(56,308,384,430,60,220,122,265)], [(1,440,15,426),(2,439,16,425),(3,438,17,424),(4,437,18,423),(5,436,19,422),(6,435,20,421),(7,434,21,448),(8,433,22,447),(9,432,23,446),(10,431,24,445),(11,430,25,444),(12,429,26,443),(13,428,27,442),(14,427,28,441),(29,178,43,192),(30,177,44,191),(31,176,45,190),(32,175,46,189),(33,174,47,188),(34,173,48,187),(35,172,49,186),(36,171,50,185),(37,170,51,184),(38,169,52,183),(39,196,53,182),(40,195,54,181),(41,194,55,180),(42,193,56,179),(57,226,71,240),(58,225,72,239),(59,252,73,238),(60,251,74,237),(61,250,75,236),(62,249,76,235),(63,248,77,234),(64,247,78,233),(65,246,79,232),(66,245,80,231),(67,244,81,230),(68,243,82,229),(69,242,83,228),(70,241,84,227),(85,263,99,277),(86,262,100,276),(87,261,101,275),(88,260,102,274),(89,259,103,273),(90,258,104,272),(91,257,105,271),(92,256,106,270),(93,255,107,269),(94,254,108,268),(95,253,109,267),(96,280,110,266),(97,279,111,265),(98,278,112,264),(113,363,127,349),(114,362,128,348),(115,361,129,347),(116,360,130,346),(117,359,131,345),(118,358,132,344),(119,357,133,343),(120,356,134,342),(121,355,135,341),(122,354,136,340),(123,353,137,339),(124,352,138,338),(125,351,139,337),(126,350,140,364),(141,306,155,292),(142,305,156,291),(143,304,157,290),(144,303,158,289),(145,302,159,288),(146,301,160,287),(147,300,161,286),(148,299,162,285),(149,298,163,284),(150,297,164,283),(151,296,165,282),(152,295,166,281),(153,294,167,308),(154,293,168,307),(197,419,211,405),(198,418,212,404),(199,417,213,403),(200,416,214,402),(201,415,215,401),(202,414,216,400),(203,413,217,399),(204,412,218,398),(205,411,219,397),(206,410,220,396),(207,409,221,395),(208,408,222,394),(209,407,223,393),(210,406,224,420),(309,392,323,378),(310,391,324,377),(311,390,325,376),(312,389,326,375),(313,388,327,374),(314,387,328,373),(315,386,329,372),(316,385,330,371),(317,384,331,370),(318,383,332,369),(319,382,333,368),(320,381,334,367),(321,380,335,366),(322,379,336,365)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | SD16 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | Dic14 | C4○D28 | C8.C22 | Q8⋊D7 | D4.9D14 |
| kernel | C28.48SD16 | C28⋊C8 | C4.Dic14 | Q8⋊Dic7 | C28⋊2Q8 | Q8×C28 | C2×C28 | C7×Q8 | C4×Q8 | C28 | C28 | C42 | C4⋊C4 | C2×Q8 | C2×C4 | Q8 | C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of C28.48SD16 ►in GL4(𝔽113) generated by
| 36 | 13 | 0 | 0 |
| 56 | 105 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 29 | 76 | 0 | 0 |
| 96 | 84 | 0 | 0 |
| 0 | 0 | 0 | 97 |
| 0 | 0 | 7 | 87 |
| 72 | 48 | 0 | 0 |
| 45 | 41 | 0 | 0 |
| 0 | 0 | 4 | 94 |
| 0 | 0 | 90 | 109 |
G:=sub<GL(4,GF(113))| [36,56,0,0,13,105,0,0,0,0,112,0,0,0,0,112],[29,96,0,0,76,84,0,0,0,0,0,7,0,0,97,87],[72,45,0,0,48,41,0,0,0,0,4,90,0,0,94,109] >;
C28.48SD16 in GAP, Magma, Sage, TeX
C_{28}._{48}{\rm SD}_{16} % in TeX
G:=Group("C28.48SD16"); // GroupNames label
G:=SmallGroup(448,554);
// by ID
G=gap.SmallGroup(448,554);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,254,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^14*b^3>;
// generators/relations