metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.5Q16, C28.9SD16, C42.10D14, C4⋊Q8.1D7, C4.8(Q8⋊D7), (Q8×C14).2C4, (C2×C28).107D4, (C2×Q8).2Dic7, C28⋊C8.12C2, C4.6(C7⋊Q16), C7⋊2(C4.6Q16), (C4×C28).48C22, C2.4(Q8⋊Dic7), C2.5(C28.D4), C14.12(Q8⋊C4), C14.10(C4.D4), C22.42(C23.D7), (C7×C4⋊Q8).1C2, (C2×C28).172(C2×C4), (C2×C4).12(C2×Dic7), (C2×C4).177(C7⋊D4), (C2×C14).104(C22⋊C4), SmallGroup(448,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.5Q16
G = < a,b,c | a28=b8=1, c2=a14b4, bab-1=a-1, cac-1=a15, cbc-1=a7b-1 >
Subgroups: 204 in 64 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C4⋊C8, C4⋊Q8, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×Q8, C4.6Q16, C2×C7⋊C8, C4×C28, C7×C4⋊C4, Q8×C14, C28⋊C8, C7×C4⋊Q8, C28.5Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, SD16, Q16, Dic7, D14, C4.D4, Q8⋊C4, C2×Dic7, C7⋊D4, C4.6Q16, Q8⋊D7, C7⋊Q16, C23.D7, C28.D4, Q8⋊Dic7, C28.5Q16
►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)
(1 136 299 337 315 429 103 178)(2 135 300 364 316 428 104 177)(3 134 301 363 317 427 105 176)(4 133 302 362 318 426 106 175)(5 132 303 361 319 425 107 174)(6 131 304 360 320 424 108 173)(7 130 305 359 321 423 109 172)(8 129 306 358 322 422 110 171)(9 128 307 357 323 421 111 170)(10 127 308 356 324 448 112 169)(11 126 281 355 325 447 85 196)(12 125 282 354 326 446 86 195)(13 124 283 353 327 445 87 194)(14 123 284 352 328 444 88 193)(15 122 285 351 329 443 89 192)(16 121 286 350 330 442 90 191)(17 120 287 349 331 441 91 190)(18 119 288 348 332 440 92 189)(19 118 289 347 333 439 93 188)(20 117 290 346 334 438 94 187)(21 116 291 345 335 437 95 186)(22 115 292 344 336 436 96 185)(23 114 293 343 309 435 97 184)(24 113 294 342 310 434 98 183)(25 140 295 341 311 433 99 182)(26 139 296 340 312 432 100 181)(27 138 297 339 313 431 101 180)(28 137 298 338 314 430 102 179)(29 158 205 84 232 376 407 253)(30 157 206 83 233 375 408 280)(31 156 207 82 234 374 409 279)(32 155 208 81 235 373 410 278)(33 154 209 80 236 372 411 277)(34 153 210 79 237 371 412 276)(35 152 211 78 238 370 413 275)(36 151 212 77 239 369 414 274)(37 150 213 76 240 368 415 273)(38 149 214 75 241 367 416 272)(39 148 215 74 242 366 417 271)(40 147 216 73 243 365 418 270)(41 146 217 72 244 392 419 269)(42 145 218 71 245 391 420 268)(43 144 219 70 246 390 393 267)(44 143 220 69 247 389 394 266)(45 142 221 68 248 388 395 265)(46 141 222 67 249 387 396 264)(47 168 223 66 250 386 397 263)(48 167 224 65 251 385 398 262)(49 166 197 64 252 384 399 261)(50 165 198 63 225 383 400 260)(51 164 199 62 226 382 401 259)(52 163 200 61 227 381 402 258)(53 162 201 60 228 380 403 257)(54 161 202 59 229 379 404 256)(55 160 203 58 230 378 405 255)(56 159 204 57 231 377 406 254)
(1 64 329 275)(2 79 330 262)(3 66 331 277)(4 81 332 264)(5 68 333 279)(6 83 334 266)(7 70 335 253)(8 57 336 268)(9 72 309 255)(10 59 310 270)(11 74 311 257)(12 61 312 272)(13 76 313 259)(14 63 314 274)(15 78 315 261)(16 65 316 276)(17 80 317 263)(18 67 318 278)(19 82 319 265)(20 69 320 280)(21 84 321 267)(22 71 322 254)(23 58 323 269)(24 73 324 256)(25 60 325 271)(26 75 326 258)(27 62 327 273)(28 77 328 260)(29 193 246 338)(30 180 247 353)(31 195 248 340)(32 182 249 355)(33 169 250 342)(34 184 251 357)(35 171 252 344)(36 186 225 359)(37 173 226 346)(38 188 227 361)(39 175 228 348)(40 190 229 363)(41 177 230 350)(42 192 231 337)(43 179 232 352)(44 194 233 339)(45 181 234 354)(46 196 235 341)(47 183 236 356)(48 170 237 343)(49 185 238 358)(50 172 239 345)(51 187 240 360)(52 174 241 347)(53 189 242 362)(54 176 243 349)(55 191 244 364)(56 178 245 351)(85 366 295 162)(86 381 296 149)(87 368 297 164)(88 383 298 151)(89 370 299 166)(90 385 300 153)(91 372 301 168)(92 387 302 155)(93 374 303 142)(94 389 304 157)(95 376 305 144)(96 391 306 159)(97 378 307 146)(98 365 308 161)(99 380 281 148)(100 367 282 163)(101 382 283 150)(102 369 284 165)(103 384 285 152)(104 371 286 167)(105 386 287 154)(106 373 288 141)(107 388 289 156)(108 375 290 143)(109 390 291 158)(110 377 292 145)(111 392 293 160)(112 379 294 147)(113 209 448 397)(114 224 421 412)(115 211 422 399)(116 198 423 414)(117 213 424 401)(118 200 425 416)(119 215 426 403)(120 202 427 418)(121 217 428 405)(122 204 429 420)(123 219 430 407)(124 206 431 394)(125 221 432 409)(126 208 433 396)(127 223 434 411)(128 210 435 398)(129 197 436 413)(130 212 437 400)(131 199 438 415)(132 214 439 402)(133 201 440 417)(134 216 441 404)(135 203 442 419)(136 218 443 406)(137 205 444 393)(138 220 445 408)(139 207 446 395)(140 222 447 410)
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), (1,136,299,337,315,429,103,178)(2,135,300,364,316,428,104,177)(3,134,301,363,317,427,105,176)(4,133,302,362,318,426,106,175)(5,132,303,361,319,425,107,174)(6,131,304,360,320,424,108,173)(7,130,305,359,321,423,109,172)(8,129,306,358,322,422,110,171)(9,128,307,357,323,421,111,170)(10,127,308,356,324,448,112,169)(11,126,281,355,325,447,85,196)(12,125,282,354,326,446,86,195)(13,124,283,353,327,445,87,194)(14,123,284,352,328,444,88,193)(15,122,285,351,329,443,89,192)(16,121,286,350,330,442,90,191)(17,120,287,349,331,441,91,190)(18,119,288,348,332,440,92,189)(19,118,289,347,333,439,93,188)(20,117,290,346,334,438,94,187)(21,116,291,345,335,437,95,186)(22,115,292,344,336,436,96,185)(23,114,293,343,309,435,97,184)(24,113,294,342,310,434,98,183)(25,140,295,341,311,433,99,182)(26,139,296,340,312,432,100,181)(27,138,297,339,313,431,101,180)(28,137,298,338,314,430,102,179)(29,158,205,84,232,376,407,253)(30,157,206,83,233,375,408,280)(31,156,207,82,234,374,409,279)(32,155,208,81,235,373,410,278)(33,154,209,80,236,372,411,277)(34,153,210,79,237,371,412,276)(35,152,211,78,238,370,413,275)(36,151,212,77,239,369,414,274)(37,150,213,76,240,368,415,273)(38,149,214,75,241,367,416,272)(39,148,215,74,242,366,417,271)(40,147,216,73,243,365,418,270)(41,146,217,72,244,392,419,269)(42,145,218,71,245,391,420,268)(43,144,219,70,246,390,393,267)(44,143,220,69,247,389,394,266)(45,142,221,68,248,388,395,265)(46,141,222,67,249,387,396,264)(47,168,223,66,250,386,397,263)(48,167,224,65,251,385,398,262)(49,166,197,64,252,384,399,261)(50,165,198,63,225,383,400,260)(51,164,199,62,226,382,401,259)(52,163,200,61,227,381,402,258)(53,162,201,60,228,380,403,257)(54,161,202,59,229,379,404,256)(55,160,203,58,230,378,405,255)(56,159,204,57,231,377,406,254), (1,64,329,275)(2,79,330,262)(3,66,331,277)(4,81,332,264)(5,68,333,279)(6,83,334,266)(7,70,335,253)(8,57,336,268)(9,72,309,255)(10,59,310,270)(11,74,311,257)(12,61,312,272)(13,76,313,259)(14,63,314,274)(15,78,315,261)(16,65,316,276)(17,80,317,263)(18,67,318,278)(19,82,319,265)(20,69,320,280)(21,84,321,267)(22,71,322,254)(23,58,323,269)(24,73,324,256)(25,60,325,271)(26,75,326,258)(27,62,327,273)(28,77,328,260)(29,193,246,338)(30,180,247,353)(31,195,248,340)(32,182,249,355)(33,169,250,342)(34,184,251,357)(35,171,252,344)(36,186,225,359)(37,173,226,346)(38,188,227,361)(39,175,228,348)(40,190,229,363)(41,177,230,350)(42,192,231,337)(43,179,232,352)(44,194,233,339)(45,181,234,354)(46,196,235,341)(47,183,236,356)(48,170,237,343)(49,185,238,358)(50,172,239,345)(51,187,240,360)(52,174,241,347)(53,189,242,362)(54,176,243,349)(55,191,244,364)(56,178,245,351)(85,366,295,162)(86,381,296,149)(87,368,297,164)(88,383,298,151)(89,370,299,166)(90,385,300,153)(91,372,301,168)(92,387,302,155)(93,374,303,142)(94,389,304,157)(95,376,305,144)(96,391,306,159)(97,378,307,146)(98,365,308,161)(99,380,281,148)(100,367,282,163)(101,382,283,150)(102,369,284,165)(103,384,285,152)(104,371,286,167)(105,386,287,154)(106,373,288,141)(107,388,289,156)(108,375,290,143)(109,390,291,158)(110,377,292,145)(111,392,293,160)(112,379,294,147)(113,209,448,397)(114,224,421,412)(115,211,422,399)(116,198,423,414)(117,213,424,401)(118,200,425,416)(119,215,426,403)(120,202,427,418)(121,217,428,405)(122,204,429,420)(123,219,430,407)(124,206,431,394)(125,221,432,409)(126,208,433,396)(127,223,434,411)(128,210,435,398)(129,197,436,413)(130,212,437,400)(131,199,438,415)(132,214,439,402)(133,201,440,417)(134,216,441,404)(135,203,442,419)(136,218,443,406)(137,205,444,393)(138,220,445,408)(139,207,446,395)(140,222,447,410)>;
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), (1,136,299,337,315,429,103,178)(2,135,300,364,316,428,104,177)(3,134,301,363,317,427,105,176)(4,133,302,362,318,426,106,175)(5,132,303,361,319,425,107,174)(6,131,304,360,320,424,108,173)(7,130,305,359,321,423,109,172)(8,129,306,358,322,422,110,171)(9,128,307,357,323,421,111,170)(10,127,308,356,324,448,112,169)(11,126,281,355,325,447,85,196)(12,125,282,354,326,446,86,195)(13,124,283,353,327,445,87,194)(14,123,284,352,328,444,88,193)(15,122,285,351,329,443,89,192)(16,121,286,350,330,442,90,191)(17,120,287,349,331,441,91,190)(18,119,288,348,332,440,92,189)(19,118,289,347,333,439,93,188)(20,117,290,346,334,438,94,187)(21,116,291,345,335,437,95,186)(22,115,292,344,336,436,96,185)(23,114,293,343,309,435,97,184)(24,113,294,342,310,434,98,183)(25,140,295,341,311,433,99,182)(26,139,296,340,312,432,100,181)(27,138,297,339,313,431,101,180)(28,137,298,338,314,430,102,179)(29,158,205,84,232,376,407,253)(30,157,206,83,233,375,408,280)(31,156,207,82,234,374,409,279)(32,155,208,81,235,373,410,278)(33,154,209,80,236,372,411,277)(34,153,210,79,237,371,412,276)(35,152,211,78,238,370,413,275)(36,151,212,77,239,369,414,274)(37,150,213,76,240,368,415,273)(38,149,214,75,241,367,416,272)(39,148,215,74,242,366,417,271)(40,147,216,73,243,365,418,270)(41,146,217,72,244,392,419,269)(42,145,218,71,245,391,420,268)(43,144,219,70,246,390,393,267)(44,143,220,69,247,389,394,266)(45,142,221,68,248,388,395,265)(46,141,222,67,249,387,396,264)(47,168,223,66,250,386,397,263)(48,167,224,65,251,385,398,262)(49,166,197,64,252,384,399,261)(50,165,198,63,225,383,400,260)(51,164,199,62,226,382,401,259)(52,163,200,61,227,381,402,258)(53,162,201,60,228,380,403,257)(54,161,202,59,229,379,404,256)(55,160,203,58,230,378,405,255)(56,159,204,57,231,377,406,254), (1,64,329,275)(2,79,330,262)(3,66,331,277)(4,81,332,264)(5,68,333,279)(6,83,334,266)(7,70,335,253)(8,57,336,268)(9,72,309,255)(10,59,310,270)(11,74,311,257)(12,61,312,272)(13,76,313,259)(14,63,314,274)(15,78,315,261)(16,65,316,276)(17,80,317,263)(18,67,318,278)(19,82,319,265)(20,69,320,280)(21,84,321,267)(22,71,322,254)(23,58,323,269)(24,73,324,256)(25,60,325,271)(26,75,326,258)(27,62,327,273)(28,77,328,260)(29,193,246,338)(30,180,247,353)(31,195,248,340)(32,182,249,355)(33,169,250,342)(34,184,251,357)(35,171,252,344)(36,186,225,359)(37,173,226,346)(38,188,227,361)(39,175,228,348)(40,190,229,363)(41,177,230,350)(42,192,231,337)(43,179,232,352)(44,194,233,339)(45,181,234,354)(46,196,235,341)(47,183,236,356)(48,170,237,343)(49,185,238,358)(50,172,239,345)(51,187,240,360)(52,174,241,347)(53,189,242,362)(54,176,243,349)(55,191,244,364)(56,178,245,351)(85,366,295,162)(86,381,296,149)(87,368,297,164)(88,383,298,151)(89,370,299,166)(90,385,300,153)(91,372,301,168)(92,387,302,155)(93,374,303,142)(94,389,304,157)(95,376,305,144)(96,391,306,159)(97,378,307,146)(98,365,308,161)(99,380,281,148)(100,367,282,163)(101,382,283,150)(102,369,284,165)(103,384,285,152)(104,371,286,167)(105,386,287,154)(106,373,288,141)(107,388,289,156)(108,375,290,143)(109,390,291,158)(110,377,292,145)(111,392,293,160)(112,379,294,147)(113,209,448,397)(114,224,421,412)(115,211,422,399)(116,198,423,414)(117,213,424,401)(118,200,425,416)(119,215,426,403)(120,202,427,418)(121,217,428,405)(122,204,429,420)(123,219,430,407)(124,206,431,394)(125,221,432,409)(126,208,433,396)(127,223,434,411)(128,210,435,398)(129,197,436,413)(130,212,437,400)(131,199,438,415)(132,214,439,402)(133,201,440,417)(134,216,441,404)(135,203,442,419)(136,218,443,406)(137,205,444,393)(138,220,445,408)(139,207,446,395)(140,222,447,410) );
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)], [(1,136,299,337,315,429,103,178),(2,135,300,364,316,428,104,177),(3,134,301,363,317,427,105,176),(4,133,302,362,318,426,106,175),(5,132,303,361,319,425,107,174),(6,131,304,360,320,424,108,173),(7,130,305,359,321,423,109,172),(8,129,306,358,322,422,110,171),(9,128,307,357,323,421,111,170),(10,127,308,356,324,448,112,169),(11,126,281,355,325,447,85,196),(12,125,282,354,326,446,86,195),(13,124,283,353,327,445,87,194),(14,123,284,352,328,444,88,193),(15,122,285,351,329,443,89,192),(16,121,286,350,330,442,90,191),(17,120,287,349,331,441,91,190),(18,119,288,348,332,440,92,189),(19,118,289,347,333,439,93,188),(20,117,290,346,334,438,94,187),(21,116,291,345,335,437,95,186),(22,115,292,344,336,436,96,185),(23,114,293,343,309,435,97,184),(24,113,294,342,310,434,98,183),(25,140,295,341,311,433,99,182),(26,139,296,340,312,432,100,181),(27,138,297,339,313,431,101,180),(28,137,298,338,314,430,102,179),(29,158,205,84,232,376,407,253),(30,157,206,83,233,375,408,280),(31,156,207,82,234,374,409,279),(32,155,208,81,235,373,410,278),(33,154,209,80,236,372,411,277),(34,153,210,79,237,371,412,276),(35,152,211,78,238,370,413,275),(36,151,212,77,239,369,414,274),(37,150,213,76,240,368,415,273),(38,149,214,75,241,367,416,272),(39,148,215,74,242,366,417,271),(40,147,216,73,243,365,418,270),(41,146,217,72,244,392,419,269),(42,145,218,71,245,391,420,268),(43,144,219,70,246,390,393,267),(44,143,220,69,247,389,394,266),(45,142,221,68,248,388,395,265),(46,141,222,67,249,387,396,264),(47,168,223,66,250,386,397,263),(48,167,224,65,251,385,398,262),(49,166,197,64,252,384,399,261),(50,165,198,63,225,383,400,260),(51,164,199,62,226,382,401,259),(52,163,200,61,227,381,402,258),(53,162,201,60,228,380,403,257),(54,161,202,59,229,379,404,256),(55,160,203,58,230,378,405,255),(56,159,204,57,231,377,406,254)], [(1,64,329,275),(2,79,330,262),(3,66,331,277),(4,81,332,264),(5,68,333,279),(6,83,334,266),(7,70,335,253),(8,57,336,268),(9,72,309,255),(10,59,310,270),(11,74,311,257),(12,61,312,272),(13,76,313,259),(14,63,314,274),(15,78,315,261),(16,65,316,276),(17,80,317,263),(18,67,318,278),(19,82,319,265),(20,69,320,280),(21,84,321,267),(22,71,322,254),(23,58,323,269),(24,73,324,256),(25,60,325,271),(26,75,326,258),(27,62,327,273),(28,77,328,260),(29,193,246,338),(30,180,247,353),(31,195,248,340),(32,182,249,355),(33,169,250,342),(34,184,251,357),(35,171,252,344),(36,186,225,359),(37,173,226,346),(38,188,227,361),(39,175,228,348),(40,190,229,363),(41,177,230,350),(42,192,231,337),(43,179,232,352),(44,194,233,339),(45,181,234,354),(46,196,235,341),(47,183,236,356),(48,170,237,343),(49,185,238,358),(50,172,239,345),(51,187,240,360),(52,174,241,347),(53,189,242,362),(54,176,243,349),(55,191,244,364),(56,178,245,351),(85,366,295,162),(86,381,296,149),(87,368,297,164),(88,383,298,151),(89,370,299,166),(90,385,300,153),(91,372,301,168),(92,387,302,155),(93,374,303,142),(94,389,304,157),(95,376,305,144),(96,391,306,159),(97,378,307,146),(98,365,308,161),(99,380,281,148),(100,367,282,163),(101,382,283,150),(102,369,284,165),(103,384,285,152),(104,371,286,167),(105,386,287,154),(106,373,288,141),(107,388,289,156),(108,375,290,143),(109,390,291,158),(110,377,292,145),(111,392,293,160),(112,379,294,147),(113,209,448,397),(114,224,421,412),(115,211,422,399),(116,198,423,414),(117,213,424,401),(118,200,425,416),(119,215,426,403),(120,202,427,418),(121,217,428,405),(122,204,429,420),(123,219,430,407),(124,206,431,394),(125,221,432,409),(126,208,433,396),(127,223,434,411),(128,210,435,398),(129,197,436,413),(130,212,437,400),(131,199,438,415),(132,214,439,402),(133,201,440,417),(134,216,441,404),(135,203,442,419),(136,218,443,406),(137,205,444,393),(138,220,445,408),(139,207,446,395),(140,222,447,410)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | D4 | D7 | SD16 | Q16 | D14 | Dic7 | C7⋊D4 | C4.D4 | Q8⋊D7 | C7⋊Q16 | C28.D4 |
| kernel | C28.5Q16 | C28⋊C8 | C7×C4⋊Q8 | Q8×C14 | C2×C28 | C4⋊Q8 | C28 | C28 | C42 | C2×Q8 | C2×C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 3 | 4 | 4 | 3 | 6 | 12 | 1 | 6 | 6 | 6 |
Matrix representation of C28.5Q16 ►in GL6(𝔽113)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 104 | 0 | 0 |
| 0 | 0 | 9 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 87 | 0 | 0 | 0 | 0 |
| 87 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 28 | 0 | 0 |
| 0 | 0 | 108 | 79 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 100 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 67 | 97 | 0 | 0 | 0 | 0 |
| 97 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 |
G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,0,0,0,0,0,0,0,1,9,0,0,0,0,104,33,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,87,0,0,0,0,87,112,0,0,0,0,0,0,34,108,0,0,0,0,28,79,0,0,0,0,0,0,100,13,0,0,0,0,100,100],[67,97,0,0,0,0,97,46,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,0,0,0,0,0,98] >;
C28.5Q16 in GAP, Magma, Sage, TeX
C_{28}._5Q_{16} % in TeX
G:=Group("C28.5Q16"); // GroupNames label
G:=SmallGroup(448,103);
// by ID
G=gap.SmallGroup(448,103);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,219,100,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14*b^4,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^7*b^-1>;
// generators/relations