metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.14Q16, C4.3Dic28, C28.25SD16, C42.253D14, (C4×C8).5D7, (C4×C56).5C2, (C2×C4).77D28, C14.1(C2×Q16), (C2×C28).374D4, (C2×C8).281D14, C4.4(C56⋊C2), C28⋊2Q8.2C2, C14.2(C2×SD16), C2.4(C2×Dic28), C7⋊1(C4.SD16), C4.99(C4○D28), C22.86(C2×D28), C4⋊Dic7.2C22, C28.215(C4○D4), (C2×C28).717C23, (C2×C56).340C22, (C4×C28).303C22, C28.44D4.1C2, C14.4(C4.4D4), C2.9(C4.D28), (C2×Dic14).1C22, C2.6(C2×C56⋊C2), (C2×C14).100(C2×D4), (C2×C4).660(C22×D7), SmallGroup(448,215)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.14Q16
G = < a,b,c | a28=b8=1, c2=a14b4, ab=ba, cac-1=a-1, cbc-1=a14b-1 >
Subgroups: 516 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C2×C14, C4×C8, Q8⋊C4, C4⋊Q8, C56, Dic14, C2×Dic7, C2×C28, C4.SD16, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C2×Dic14, C28.44D4, C4×C56, C28⋊2Q8, C28.14Q16
Quotients: C1, C2, C22, D4, C23, D7, SD16, Q16, C2×D4, C4○D4, D14, C4.4D4, C2×SD16, C2×Q16, D28, C22×D7, C4.SD16, C56⋊C2, Dic28, C2×D28, C4○D28, C4.D28, C2×C56⋊C2, C2×Dic28, C28.14Q16
►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)
(1 121 442 348 420 76 142 34)(2 122 443 349 393 77 143 35)(3 123 444 350 394 78 144 36)(4 124 445 351 395 79 145 37)(5 125 446 352 396 80 146 38)(6 126 447 353 397 81 147 39)(7 127 448 354 398 82 148 40)(8 128 421 355 399 83 149 41)(9 129 422 356 400 84 150 42)(10 130 423 357 401 57 151 43)(11 131 424 358 402 58 152 44)(12 132 425 359 403 59 153 45)(13 133 426 360 404 60 154 46)(14 134 427 361 405 61 155 47)(15 135 428 362 406 62 156 48)(16 136 429 363 407 63 157 49)(17 137 430 364 408 64 158 50)(18 138 431 337 409 65 159 51)(19 139 432 338 410 66 160 52)(20 140 433 339 411 67 161 53)(21 113 434 340 412 68 162 54)(22 114 435 341 413 69 163 55)(23 115 436 342 414 70 164 56)(24 116 437 343 415 71 165 29)(25 117 438 344 416 72 166 30)(26 118 439 345 417 73 167 31)(27 119 440 346 418 74 168 32)(28 120 441 347 419 75 141 33)(85 299 321 224 192 233 260 371)(86 300 322 197 193 234 261 372)(87 301 323 198 194 235 262 373)(88 302 324 199 195 236 263 374)(89 303 325 200 196 237 264 375)(90 304 326 201 169 238 265 376)(91 305 327 202 170 239 266 377)(92 306 328 203 171 240 267 378)(93 307 329 204 172 241 268 379)(94 308 330 205 173 242 269 380)(95 281 331 206 174 243 270 381)(96 282 332 207 175 244 271 382)(97 283 333 208 176 245 272 383)(98 284 334 209 177 246 273 384)(99 285 335 210 178 247 274 385)(100 286 336 211 179 248 275 386)(101 287 309 212 180 249 276 387)(102 288 310 213 181 250 277 388)(103 289 311 214 182 251 278 389)(104 290 312 215 183 252 279 390)(105 291 313 216 184 225 280 391)(106 292 314 217 185 226 253 392)(107 293 315 218 186 227 254 365)(108 294 316 219 187 228 255 366)(109 295 317 220 188 229 256 367)(110 296 318 221 189 230 257 368)(111 297 319 222 190 231 258 369)(112 298 320 223 191 232 259 370)
(1 324 406 277)(2 323 407 276)(3 322 408 275)(4 321 409 274)(5 320 410 273)(6 319 411 272)(7 318 412 271)(8 317 413 270)(9 316 414 269)(10 315 415 268)(11 314 416 267)(12 313 417 266)(13 312 418 265)(14 311 419 264)(15 310 420 263)(16 309 393 262)(17 336 394 261)(18 335 395 260)(19 334 396 259)(20 333 397 258)(21 332 398 257)(22 331 399 256)(23 330 400 255)(24 329 401 254)(25 328 402 253)(26 327 403 280)(27 326 404 279)(28 325 405 278)(29 218 357 379)(30 217 358 378)(31 216 359 377)(32 215 360 376)(33 214 361 375)(34 213 362 374)(35 212 363 373)(36 211 364 372)(37 210 337 371)(38 209 338 370)(39 208 339 369)(40 207 340 368)(41 206 341 367)(42 205 342 366)(43 204 343 365)(44 203 344 392)(45 202 345 391)(46 201 346 390)(47 200 347 389)(48 199 348 388)(49 198 349 387)(50 197 350 386)(51 224 351 385)(52 223 352 384)(53 222 353 383)(54 221 354 382)(55 220 355 381)(56 219 356 380)(57 241 116 293)(58 240 117 292)(59 239 118 291)(60 238 119 290)(61 237 120 289)(62 236 121 288)(63 235 122 287)(64 234 123 286)(65 233 124 285)(66 232 125 284)(67 231 126 283)(68 230 127 282)(69 229 128 281)(70 228 129 308)(71 227 130 307)(72 226 131 306)(73 225 132 305)(74 252 133 304)(75 251 134 303)(76 250 135 302)(77 249 136 301)(78 248 137 300)(79 247 138 299)(80 246 139 298)(81 245 140 297)(82 244 113 296)(83 243 114 295)(84 242 115 294)(85 159 178 445)(86 158 179 444)(87 157 180 443)(88 156 181 442)(89 155 182 441)(90 154 183 440)(91 153 184 439)(92 152 185 438)(93 151 186 437)(94 150 187 436)(95 149 188 435)(96 148 189 434)(97 147 190 433)(98 146 191 432)(99 145 192 431)(100 144 193 430)(101 143 194 429)(102 142 195 428)(103 141 196 427)(104 168 169 426)(105 167 170 425)(106 166 171 424)(107 165 172 423)(108 164 173 422)(109 163 174 421)(110 162 175 448)(111 161 176 447)(112 160 177 446)
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), (1,121,442,348,420,76,142,34)(2,122,443,349,393,77,143,35)(3,123,444,350,394,78,144,36)(4,124,445,351,395,79,145,37)(5,125,446,352,396,80,146,38)(6,126,447,353,397,81,147,39)(7,127,448,354,398,82,148,40)(8,128,421,355,399,83,149,41)(9,129,422,356,400,84,150,42)(10,130,423,357,401,57,151,43)(11,131,424,358,402,58,152,44)(12,132,425,359,403,59,153,45)(13,133,426,360,404,60,154,46)(14,134,427,361,405,61,155,47)(15,135,428,362,406,62,156,48)(16,136,429,363,407,63,157,49)(17,137,430,364,408,64,158,50)(18,138,431,337,409,65,159,51)(19,139,432,338,410,66,160,52)(20,140,433,339,411,67,161,53)(21,113,434,340,412,68,162,54)(22,114,435,341,413,69,163,55)(23,115,436,342,414,70,164,56)(24,116,437,343,415,71,165,29)(25,117,438,344,416,72,166,30)(26,118,439,345,417,73,167,31)(27,119,440,346,418,74,168,32)(28,120,441,347,419,75,141,33)(85,299,321,224,192,233,260,371)(86,300,322,197,193,234,261,372)(87,301,323,198,194,235,262,373)(88,302,324,199,195,236,263,374)(89,303,325,200,196,237,264,375)(90,304,326,201,169,238,265,376)(91,305,327,202,170,239,266,377)(92,306,328,203,171,240,267,378)(93,307,329,204,172,241,268,379)(94,308,330,205,173,242,269,380)(95,281,331,206,174,243,270,381)(96,282,332,207,175,244,271,382)(97,283,333,208,176,245,272,383)(98,284,334,209,177,246,273,384)(99,285,335,210,178,247,274,385)(100,286,336,211,179,248,275,386)(101,287,309,212,180,249,276,387)(102,288,310,213,181,250,277,388)(103,289,311,214,182,251,278,389)(104,290,312,215,183,252,279,390)(105,291,313,216,184,225,280,391)(106,292,314,217,185,226,253,392)(107,293,315,218,186,227,254,365)(108,294,316,219,187,228,255,366)(109,295,317,220,188,229,256,367)(110,296,318,221,189,230,257,368)(111,297,319,222,190,231,258,369)(112,298,320,223,191,232,259,370), (1,324,406,277)(2,323,407,276)(3,322,408,275)(4,321,409,274)(5,320,410,273)(6,319,411,272)(7,318,412,271)(8,317,413,270)(9,316,414,269)(10,315,415,268)(11,314,416,267)(12,313,417,266)(13,312,418,265)(14,311,419,264)(15,310,420,263)(16,309,393,262)(17,336,394,261)(18,335,395,260)(19,334,396,259)(20,333,397,258)(21,332,398,257)(22,331,399,256)(23,330,400,255)(24,329,401,254)(25,328,402,253)(26,327,403,280)(27,326,404,279)(28,325,405,278)(29,218,357,379)(30,217,358,378)(31,216,359,377)(32,215,360,376)(33,214,361,375)(34,213,362,374)(35,212,363,373)(36,211,364,372)(37,210,337,371)(38,209,338,370)(39,208,339,369)(40,207,340,368)(41,206,341,367)(42,205,342,366)(43,204,343,365)(44,203,344,392)(45,202,345,391)(46,201,346,390)(47,200,347,389)(48,199,348,388)(49,198,349,387)(50,197,350,386)(51,224,351,385)(52,223,352,384)(53,222,353,383)(54,221,354,382)(55,220,355,381)(56,219,356,380)(57,241,116,293)(58,240,117,292)(59,239,118,291)(60,238,119,290)(61,237,120,289)(62,236,121,288)(63,235,122,287)(64,234,123,286)(65,233,124,285)(66,232,125,284)(67,231,126,283)(68,230,127,282)(69,229,128,281)(70,228,129,308)(71,227,130,307)(72,226,131,306)(73,225,132,305)(74,252,133,304)(75,251,134,303)(76,250,135,302)(77,249,136,301)(78,248,137,300)(79,247,138,299)(80,246,139,298)(81,245,140,297)(82,244,113,296)(83,243,114,295)(84,242,115,294)(85,159,178,445)(86,158,179,444)(87,157,180,443)(88,156,181,442)(89,155,182,441)(90,154,183,440)(91,153,184,439)(92,152,185,438)(93,151,186,437)(94,150,187,436)(95,149,188,435)(96,148,189,434)(97,147,190,433)(98,146,191,432)(99,145,192,431)(100,144,193,430)(101,143,194,429)(102,142,195,428)(103,141,196,427)(104,168,169,426)(105,167,170,425)(106,166,171,424)(107,165,172,423)(108,164,173,422)(109,163,174,421)(110,162,175,448)(111,161,176,447)(112,160,177,446)>;
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), (1,121,442,348,420,76,142,34)(2,122,443,349,393,77,143,35)(3,123,444,350,394,78,144,36)(4,124,445,351,395,79,145,37)(5,125,446,352,396,80,146,38)(6,126,447,353,397,81,147,39)(7,127,448,354,398,82,148,40)(8,128,421,355,399,83,149,41)(9,129,422,356,400,84,150,42)(10,130,423,357,401,57,151,43)(11,131,424,358,402,58,152,44)(12,132,425,359,403,59,153,45)(13,133,426,360,404,60,154,46)(14,134,427,361,405,61,155,47)(15,135,428,362,406,62,156,48)(16,136,429,363,407,63,157,49)(17,137,430,364,408,64,158,50)(18,138,431,337,409,65,159,51)(19,139,432,338,410,66,160,52)(20,140,433,339,411,67,161,53)(21,113,434,340,412,68,162,54)(22,114,435,341,413,69,163,55)(23,115,436,342,414,70,164,56)(24,116,437,343,415,71,165,29)(25,117,438,344,416,72,166,30)(26,118,439,345,417,73,167,31)(27,119,440,346,418,74,168,32)(28,120,441,347,419,75,141,33)(85,299,321,224,192,233,260,371)(86,300,322,197,193,234,261,372)(87,301,323,198,194,235,262,373)(88,302,324,199,195,236,263,374)(89,303,325,200,196,237,264,375)(90,304,326,201,169,238,265,376)(91,305,327,202,170,239,266,377)(92,306,328,203,171,240,267,378)(93,307,329,204,172,241,268,379)(94,308,330,205,173,242,269,380)(95,281,331,206,174,243,270,381)(96,282,332,207,175,244,271,382)(97,283,333,208,176,245,272,383)(98,284,334,209,177,246,273,384)(99,285,335,210,178,247,274,385)(100,286,336,211,179,248,275,386)(101,287,309,212,180,249,276,387)(102,288,310,213,181,250,277,388)(103,289,311,214,182,251,278,389)(104,290,312,215,183,252,279,390)(105,291,313,216,184,225,280,391)(106,292,314,217,185,226,253,392)(107,293,315,218,186,227,254,365)(108,294,316,219,187,228,255,366)(109,295,317,220,188,229,256,367)(110,296,318,221,189,230,257,368)(111,297,319,222,190,231,258,369)(112,298,320,223,191,232,259,370), (1,324,406,277)(2,323,407,276)(3,322,408,275)(4,321,409,274)(5,320,410,273)(6,319,411,272)(7,318,412,271)(8,317,413,270)(9,316,414,269)(10,315,415,268)(11,314,416,267)(12,313,417,266)(13,312,418,265)(14,311,419,264)(15,310,420,263)(16,309,393,262)(17,336,394,261)(18,335,395,260)(19,334,396,259)(20,333,397,258)(21,332,398,257)(22,331,399,256)(23,330,400,255)(24,329,401,254)(25,328,402,253)(26,327,403,280)(27,326,404,279)(28,325,405,278)(29,218,357,379)(30,217,358,378)(31,216,359,377)(32,215,360,376)(33,214,361,375)(34,213,362,374)(35,212,363,373)(36,211,364,372)(37,210,337,371)(38,209,338,370)(39,208,339,369)(40,207,340,368)(41,206,341,367)(42,205,342,366)(43,204,343,365)(44,203,344,392)(45,202,345,391)(46,201,346,390)(47,200,347,389)(48,199,348,388)(49,198,349,387)(50,197,350,386)(51,224,351,385)(52,223,352,384)(53,222,353,383)(54,221,354,382)(55,220,355,381)(56,219,356,380)(57,241,116,293)(58,240,117,292)(59,239,118,291)(60,238,119,290)(61,237,120,289)(62,236,121,288)(63,235,122,287)(64,234,123,286)(65,233,124,285)(66,232,125,284)(67,231,126,283)(68,230,127,282)(69,229,128,281)(70,228,129,308)(71,227,130,307)(72,226,131,306)(73,225,132,305)(74,252,133,304)(75,251,134,303)(76,250,135,302)(77,249,136,301)(78,248,137,300)(79,247,138,299)(80,246,139,298)(81,245,140,297)(82,244,113,296)(83,243,114,295)(84,242,115,294)(85,159,178,445)(86,158,179,444)(87,157,180,443)(88,156,181,442)(89,155,182,441)(90,154,183,440)(91,153,184,439)(92,152,185,438)(93,151,186,437)(94,150,187,436)(95,149,188,435)(96,148,189,434)(97,147,190,433)(98,146,191,432)(99,145,192,431)(100,144,193,430)(101,143,194,429)(102,142,195,428)(103,141,196,427)(104,168,169,426)(105,167,170,425)(106,166,171,424)(107,165,172,423)(108,164,173,422)(109,163,174,421)(110,162,175,448)(111,161,176,447)(112,160,177,446) );
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)], [(1,121,442,348,420,76,142,34),(2,122,443,349,393,77,143,35),(3,123,444,350,394,78,144,36),(4,124,445,351,395,79,145,37),(5,125,446,352,396,80,146,38),(6,126,447,353,397,81,147,39),(7,127,448,354,398,82,148,40),(8,128,421,355,399,83,149,41),(9,129,422,356,400,84,150,42),(10,130,423,357,401,57,151,43),(11,131,424,358,402,58,152,44),(12,132,425,359,403,59,153,45),(13,133,426,360,404,60,154,46),(14,134,427,361,405,61,155,47),(15,135,428,362,406,62,156,48),(16,136,429,363,407,63,157,49),(17,137,430,364,408,64,158,50),(18,138,431,337,409,65,159,51),(19,139,432,338,410,66,160,52),(20,140,433,339,411,67,161,53),(21,113,434,340,412,68,162,54),(22,114,435,341,413,69,163,55),(23,115,436,342,414,70,164,56),(24,116,437,343,415,71,165,29),(25,117,438,344,416,72,166,30),(26,118,439,345,417,73,167,31),(27,119,440,346,418,74,168,32),(28,120,441,347,419,75,141,33),(85,299,321,224,192,233,260,371),(86,300,322,197,193,234,261,372),(87,301,323,198,194,235,262,373),(88,302,324,199,195,236,263,374),(89,303,325,200,196,237,264,375),(90,304,326,201,169,238,265,376),(91,305,327,202,170,239,266,377),(92,306,328,203,171,240,267,378),(93,307,329,204,172,241,268,379),(94,308,330,205,173,242,269,380),(95,281,331,206,174,243,270,381),(96,282,332,207,175,244,271,382),(97,283,333,208,176,245,272,383),(98,284,334,209,177,246,273,384),(99,285,335,210,178,247,274,385),(100,286,336,211,179,248,275,386),(101,287,309,212,180,249,276,387),(102,288,310,213,181,250,277,388),(103,289,311,214,182,251,278,389),(104,290,312,215,183,252,279,390),(105,291,313,216,184,225,280,391),(106,292,314,217,185,226,253,392),(107,293,315,218,186,227,254,365),(108,294,316,219,187,228,255,366),(109,295,317,220,188,229,256,367),(110,296,318,221,189,230,257,368),(111,297,319,222,190,231,258,369),(112,298,320,223,191,232,259,370)], [(1,324,406,277),(2,323,407,276),(3,322,408,275),(4,321,409,274),(5,320,410,273),(6,319,411,272),(7,318,412,271),(8,317,413,270),(9,316,414,269),(10,315,415,268),(11,314,416,267),(12,313,417,266),(13,312,418,265),(14,311,419,264),(15,310,420,263),(16,309,393,262),(17,336,394,261),(18,335,395,260),(19,334,396,259),(20,333,397,258),(21,332,398,257),(22,331,399,256),(23,330,400,255),(24,329,401,254),(25,328,402,253),(26,327,403,280),(27,326,404,279),(28,325,405,278),(29,218,357,379),(30,217,358,378),(31,216,359,377),(32,215,360,376),(33,214,361,375),(34,213,362,374),(35,212,363,373),(36,211,364,372),(37,210,337,371),(38,209,338,370),(39,208,339,369),(40,207,340,368),(41,206,341,367),(42,205,342,366),(43,204,343,365),(44,203,344,392),(45,202,345,391),(46,201,346,390),(47,200,347,389),(48,199,348,388),(49,198,349,387),(50,197,350,386),(51,224,351,385),(52,223,352,384),(53,222,353,383),(54,221,354,382),(55,220,355,381),(56,219,356,380),(57,241,116,293),(58,240,117,292),(59,239,118,291),(60,238,119,290),(61,237,120,289),(62,236,121,288),(63,235,122,287),(64,234,123,286),(65,233,124,285),(66,232,125,284),(67,231,126,283),(68,230,127,282),(69,229,128,281),(70,228,129,308),(71,227,130,307),(72,226,131,306),(73,225,132,305),(74,252,133,304),(75,251,134,303),(76,250,135,302),(77,249,136,301),(78,248,137,300),(79,247,138,299),(80,246,139,298),(81,245,140,297),(82,244,113,296),(83,243,114,295),(84,242,115,294),(85,159,178,445),(86,158,179,444),(87,157,180,443),(88,156,181,442),(89,155,182,441),(90,154,183,440),(91,153,184,439),(92,152,185,438),(93,151,186,437),(94,150,187,436),(95,149,188,435),(96,148,189,434),(97,147,190,433),(98,146,191,432),(99,145,192,431),(100,144,193,430),(101,143,194,429),(102,142,195,428),(103,141,196,427),(104,168,169,426),(105,167,170,425),(106,166,171,424),(107,165,172,423),(108,164,173,422),(109,163,174,421),(110,162,175,448),(111,161,176,447),(112,160,177,446)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | D4 | D7 | SD16 | Q16 | C4○D4 | D14 | D14 | D28 | C56⋊C2 | Dic28 | C4○D28 |
| kernel | C28.14Q16 | C28.44D4 | C4×C56 | C28⋊2Q8 | C2×C28 | C4×C8 | C28 | C28 | C28 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 3 | 6 | 12 | 24 | 24 | 24 |
Matrix representation of C28.14Q16 ►in GL4(𝔽113) generated by
| 112 | 24 | 0 | 0 |
| 89 | 10 | 0 | 0 |
| 0 | 0 | 96 | 77 |
| 0 | 0 | 36 | 23 |
| 106 | 22 | 0 | 0 |
| 91 | 69 | 0 | 0 |
| 0 | 0 | 105 | 9 |
| 0 | 0 | 104 | 95 |
| 107 | 6 | 0 | 0 |
| 88 | 6 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 24 | 103 |
G:=sub<GL(4,GF(113))| [112,89,0,0,24,10,0,0,0,0,96,36,0,0,77,23],[106,91,0,0,22,69,0,0,0,0,105,104,0,0,9,95],[107,88,0,0,6,6,0,0,0,0,10,24,0,0,10,103] >;
C28.14Q16 in GAP, Magma, Sage, TeX
C_{28}._{14}Q_{16} % in TeX
G:=Group("C28.14Q16"); // GroupNames label
G:=SmallGroup(448,215);
// by ID
G=gap.SmallGroup(448,215);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,254,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14*b^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^14*b^-1>;
// generators/relations