metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊4Dic14, C28.23Q16, C42.54D14, (C7×Q8)⋊4Q8, (C4×Q8).3D7, C7⋊4(C4.Q16), (C2×C28).64D4, (Q8×C28).3C2, C28.29(C2×Q8), C4⋊C4.249D14, C14.34(C2×Q16), C28⋊C8.15C2, C4.63(C4○D28), C28.56(C4○D4), (C4×C28).92C22, (C2×Q8).156D14, C28⋊2Q8.14C2, C4.13(C2×Dic14), C4.11(C7⋊Q16), Q8⋊Dic7.8C2, C2.9(D4⋊D14), (C2×C28).343C23, C28.Q8.10C2, C14.65(C22⋊Q8), C14.110(C8⋊C22), C4⋊Dic7.140C22, (Q8×C14).191C22, C2.16(C28.48D4), C2.6(C2×C7⋊Q16), (C2×C7⋊C8).98C22, (C2×C14).474(C2×D4), (C2×C4).248(C7⋊D4), (C7×C4⋊C4).280C22, (C2×C4).443(C22×D7), C22.153(C2×C7⋊D4), SmallGroup(448,555)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.23Q16
G = < a,b,c | a28=b8=1, c2=a14b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
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, C2.D8, C4×Q8, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C4.Q16, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C28⋊C8, C28.Q8, Q8⋊Dic7, C28⋊2Q8, Q8×C28, C28.23Q16
Quotients: C1, C2, C22, D4, Q8, C23, D7, Q16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×Q16, C8⋊C22, Dic14, C7⋊D4, C22×D7, C4.Q16, C7⋊Q16, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4, C2×C7⋊Q16, D4⋊D14, C28.23Q16
►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)
(1 281 142 81 260 175 383 39)(2 308 143 80 261 174 384 38)(3 307 144 79 262 173 385 37)(4 306 145 78 263 172 386 36)(5 305 146 77 264 171 387 35)(6 304 147 76 265 170 388 34)(7 303 148 75 266 169 389 33)(8 302 149 74 267 196 390 32)(9 301 150 73 268 195 391 31)(10 300 151 72 269 194 392 30)(11 299 152 71 270 193 365 29)(12 298 153 70 271 192 366 56)(13 297 154 69 272 191 367 55)(14 296 155 68 273 190 368 54)(15 295 156 67 274 189 369 53)(16 294 157 66 275 188 370 52)(17 293 158 65 276 187 371 51)(18 292 159 64 277 186 372 50)(19 291 160 63 278 185 373 49)(20 290 161 62 279 184 374 48)(21 289 162 61 280 183 375 47)(22 288 163 60 253 182 376 46)(23 287 164 59 254 181 377 45)(24 286 165 58 255 180 378 44)(25 285 166 57 256 179 379 43)(26 284 167 84 257 178 380 42)(27 283 168 83 258 177 381 41)(28 282 141 82 259 176 382 40)(85 244 134 362 318 413 221 433)(86 243 135 361 319 412 222 432)(87 242 136 360 320 411 223 431)(88 241 137 359 321 410 224 430)(89 240 138 358 322 409 197 429)(90 239 139 357 323 408 198 428)(91 238 140 356 324 407 199 427)(92 237 113 355 325 406 200 426)(93 236 114 354 326 405 201 425)(94 235 115 353 327 404 202 424)(95 234 116 352 328 403 203 423)(96 233 117 351 329 402 204 422)(97 232 118 350 330 401 205 421)(98 231 119 349 331 400 206 448)(99 230 120 348 332 399 207 447)(100 229 121 347 333 398 208 446)(101 228 122 346 334 397 209 445)(102 227 123 345 335 396 210 444)(103 226 124 344 336 395 211 443)(104 225 125 343 309 394 212 442)(105 252 126 342 310 393 213 441)(106 251 127 341 311 420 214 440)(107 250 128 340 312 419 215 439)(108 249 129 339 313 418 216 438)(109 248 130 338 314 417 217 437)(110 247 131 337 315 416 218 436)(111 246 132 364 316 415 219 435)(112 245 133 363 317 414 220 434)
(1 339 274 424)(2 340 275 425)(3 341 276 426)(4 342 277 427)(5 343 278 428)(6 344 279 429)(7 345 280 430)(8 346 253 431)(9 347 254 432)(10 348 255 433)(11 349 256 434)(12 350 257 435)(13 351 258 436)(14 352 259 437)(15 353 260 438)(16 354 261 439)(17 355 262 440)(18 356 263 441)(19 357 264 442)(20 358 265 443)(21 359 266 444)(22 360 267 445)(23 361 268 446)(24 362 269 447)(25 363 270 448)(26 364 271 421)(27 337 272 422)(28 338 273 423)(29 331 57 112)(30 332 58 85)(31 333 59 86)(32 334 60 87)(33 335 61 88)(34 336 62 89)(35 309 63 90)(36 310 64 91)(37 311 65 92)(38 312 66 93)(39 313 67 94)(40 314 68 95)(41 315 69 96)(42 316 70 97)(43 317 71 98)(44 318 72 99)(45 319 73 100)(46 320 74 101)(47 321 75 102)(48 322 76 103)(49 323 77 104)(50 324 78 105)(51 325 79 106)(52 326 80 107)(53 327 81 108)(54 328 82 109)(55 329 83 110)(56 330 84 111)(113 173 214 293)(114 174 215 294)(115 175 216 295)(116 176 217 296)(117 177 218 297)(118 178 219 298)(119 179 220 299)(120 180 221 300)(121 181 222 301)(122 182 223 302)(123 183 224 303)(124 184 197 304)(125 185 198 305)(126 186 199 306)(127 187 200 307)(128 188 201 308)(129 189 202 281)(130 190 203 282)(131 191 204 283)(132 192 205 284)(133 193 206 285)(134 194 207 286)(135 195 208 287)(136 196 209 288)(137 169 210 289)(138 170 211 290)(139 171 212 291)(140 172 213 292)(141 248 368 403)(142 249 369 404)(143 250 370 405)(144 251 371 406)(145 252 372 407)(146 225 373 408)(147 226 374 409)(148 227 375 410)(149 228 376 411)(150 229 377 412)(151 230 378 413)(152 231 379 414)(153 232 380 415)(154 233 381 416)(155 234 382 417)(156 235 383 418)(157 236 384 419)(158 237 385 420)(159 238 386 393)(160 239 387 394)(161 240 388 395)(162 241 389 396)(163 242 390 397)(164 243 391 398)(165 244 392 399)(166 245 365 400)(167 246 366 401)(168 247 367 402)
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), (1,281,142,81,260,175,383,39)(2,308,143,80,261,174,384,38)(3,307,144,79,262,173,385,37)(4,306,145,78,263,172,386,36)(5,305,146,77,264,171,387,35)(6,304,147,76,265,170,388,34)(7,303,148,75,266,169,389,33)(8,302,149,74,267,196,390,32)(9,301,150,73,268,195,391,31)(10,300,151,72,269,194,392,30)(11,299,152,71,270,193,365,29)(12,298,153,70,271,192,366,56)(13,297,154,69,272,191,367,55)(14,296,155,68,273,190,368,54)(15,295,156,67,274,189,369,53)(16,294,157,66,275,188,370,52)(17,293,158,65,276,187,371,51)(18,292,159,64,277,186,372,50)(19,291,160,63,278,185,373,49)(20,290,161,62,279,184,374,48)(21,289,162,61,280,183,375,47)(22,288,163,60,253,182,376,46)(23,287,164,59,254,181,377,45)(24,286,165,58,255,180,378,44)(25,285,166,57,256,179,379,43)(26,284,167,84,257,178,380,42)(27,283,168,83,258,177,381,41)(28,282,141,82,259,176,382,40)(85,244,134,362,318,413,221,433)(86,243,135,361,319,412,222,432)(87,242,136,360,320,411,223,431)(88,241,137,359,321,410,224,430)(89,240,138,358,322,409,197,429)(90,239,139,357,323,408,198,428)(91,238,140,356,324,407,199,427)(92,237,113,355,325,406,200,426)(93,236,114,354,326,405,201,425)(94,235,115,353,327,404,202,424)(95,234,116,352,328,403,203,423)(96,233,117,351,329,402,204,422)(97,232,118,350,330,401,205,421)(98,231,119,349,331,400,206,448)(99,230,120,348,332,399,207,447)(100,229,121,347,333,398,208,446)(101,228,122,346,334,397,209,445)(102,227,123,345,335,396,210,444)(103,226,124,344,336,395,211,443)(104,225,125,343,309,394,212,442)(105,252,126,342,310,393,213,441)(106,251,127,341,311,420,214,440)(107,250,128,340,312,419,215,439)(108,249,129,339,313,418,216,438)(109,248,130,338,314,417,217,437)(110,247,131,337,315,416,218,436)(111,246,132,364,316,415,219,435)(112,245,133,363,317,414,220,434), (1,339,274,424)(2,340,275,425)(3,341,276,426)(4,342,277,427)(5,343,278,428)(6,344,279,429)(7,345,280,430)(8,346,253,431)(9,347,254,432)(10,348,255,433)(11,349,256,434)(12,350,257,435)(13,351,258,436)(14,352,259,437)(15,353,260,438)(16,354,261,439)(17,355,262,440)(18,356,263,441)(19,357,264,442)(20,358,265,443)(21,359,266,444)(22,360,267,445)(23,361,268,446)(24,362,269,447)(25,363,270,448)(26,364,271,421)(27,337,272,422)(28,338,273,423)(29,331,57,112)(30,332,58,85)(31,333,59,86)(32,334,60,87)(33,335,61,88)(34,336,62,89)(35,309,63,90)(36,310,64,91)(37,311,65,92)(38,312,66,93)(39,313,67,94)(40,314,68,95)(41,315,69,96)(42,316,70,97)(43,317,71,98)(44,318,72,99)(45,319,73,100)(46,320,74,101)(47,321,75,102)(48,322,76,103)(49,323,77,104)(50,324,78,105)(51,325,79,106)(52,326,80,107)(53,327,81,108)(54,328,82,109)(55,329,83,110)(56,330,84,111)(113,173,214,293)(114,174,215,294)(115,175,216,295)(116,176,217,296)(117,177,218,297)(118,178,219,298)(119,179,220,299)(120,180,221,300)(121,181,222,301)(122,182,223,302)(123,183,224,303)(124,184,197,304)(125,185,198,305)(126,186,199,306)(127,187,200,307)(128,188,201,308)(129,189,202,281)(130,190,203,282)(131,191,204,283)(132,192,205,284)(133,193,206,285)(134,194,207,286)(135,195,208,287)(136,196,209,288)(137,169,210,289)(138,170,211,290)(139,171,212,291)(140,172,213,292)(141,248,368,403)(142,249,369,404)(143,250,370,405)(144,251,371,406)(145,252,372,407)(146,225,373,408)(147,226,374,409)(148,227,375,410)(149,228,376,411)(150,229,377,412)(151,230,378,413)(152,231,379,414)(153,232,380,415)(154,233,381,416)(155,234,382,417)(156,235,383,418)(157,236,384,419)(158,237,385,420)(159,238,386,393)(160,239,387,394)(161,240,388,395)(162,241,389,396)(163,242,390,397)(164,243,391,398)(165,244,392,399)(166,245,365,400)(167,246,366,401)(168,247,367,402)>;
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), (1,281,142,81,260,175,383,39)(2,308,143,80,261,174,384,38)(3,307,144,79,262,173,385,37)(4,306,145,78,263,172,386,36)(5,305,146,77,264,171,387,35)(6,304,147,76,265,170,388,34)(7,303,148,75,266,169,389,33)(8,302,149,74,267,196,390,32)(9,301,150,73,268,195,391,31)(10,300,151,72,269,194,392,30)(11,299,152,71,270,193,365,29)(12,298,153,70,271,192,366,56)(13,297,154,69,272,191,367,55)(14,296,155,68,273,190,368,54)(15,295,156,67,274,189,369,53)(16,294,157,66,275,188,370,52)(17,293,158,65,276,187,371,51)(18,292,159,64,277,186,372,50)(19,291,160,63,278,185,373,49)(20,290,161,62,279,184,374,48)(21,289,162,61,280,183,375,47)(22,288,163,60,253,182,376,46)(23,287,164,59,254,181,377,45)(24,286,165,58,255,180,378,44)(25,285,166,57,256,179,379,43)(26,284,167,84,257,178,380,42)(27,283,168,83,258,177,381,41)(28,282,141,82,259,176,382,40)(85,244,134,362,318,413,221,433)(86,243,135,361,319,412,222,432)(87,242,136,360,320,411,223,431)(88,241,137,359,321,410,224,430)(89,240,138,358,322,409,197,429)(90,239,139,357,323,408,198,428)(91,238,140,356,324,407,199,427)(92,237,113,355,325,406,200,426)(93,236,114,354,326,405,201,425)(94,235,115,353,327,404,202,424)(95,234,116,352,328,403,203,423)(96,233,117,351,329,402,204,422)(97,232,118,350,330,401,205,421)(98,231,119,349,331,400,206,448)(99,230,120,348,332,399,207,447)(100,229,121,347,333,398,208,446)(101,228,122,346,334,397,209,445)(102,227,123,345,335,396,210,444)(103,226,124,344,336,395,211,443)(104,225,125,343,309,394,212,442)(105,252,126,342,310,393,213,441)(106,251,127,341,311,420,214,440)(107,250,128,340,312,419,215,439)(108,249,129,339,313,418,216,438)(109,248,130,338,314,417,217,437)(110,247,131,337,315,416,218,436)(111,246,132,364,316,415,219,435)(112,245,133,363,317,414,220,434), (1,339,274,424)(2,340,275,425)(3,341,276,426)(4,342,277,427)(5,343,278,428)(6,344,279,429)(7,345,280,430)(8,346,253,431)(9,347,254,432)(10,348,255,433)(11,349,256,434)(12,350,257,435)(13,351,258,436)(14,352,259,437)(15,353,260,438)(16,354,261,439)(17,355,262,440)(18,356,263,441)(19,357,264,442)(20,358,265,443)(21,359,266,444)(22,360,267,445)(23,361,268,446)(24,362,269,447)(25,363,270,448)(26,364,271,421)(27,337,272,422)(28,338,273,423)(29,331,57,112)(30,332,58,85)(31,333,59,86)(32,334,60,87)(33,335,61,88)(34,336,62,89)(35,309,63,90)(36,310,64,91)(37,311,65,92)(38,312,66,93)(39,313,67,94)(40,314,68,95)(41,315,69,96)(42,316,70,97)(43,317,71,98)(44,318,72,99)(45,319,73,100)(46,320,74,101)(47,321,75,102)(48,322,76,103)(49,323,77,104)(50,324,78,105)(51,325,79,106)(52,326,80,107)(53,327,81,108)(54,328,82,109)(55,329,83,110)(56,330,84,111)(113,173,214,293)(114,174,215,294)(115,175,216,295)(116,176,217,296)(117,177,218,297)(118,178,219,298)(119,179,220,299)(120,180,221,300)(121,181,222,301)(122,182,223,302)(123,183,224,303)(124,184,197,304)(125,185,198,305)(126,186,199,306)(127,187,200,307)(128,188,201,308)(129,189,202,281)(130,190,203,282)(131,191,204,283)(132,192,205,284)(133,193,206,285)(134,194,207,286)(135,195,208,287)(136,196,209,288)(137,169,210,289)(138,170,211,290)(139,171,212,291)(140,172,213,292)(141,248,368,403)(142,249,369,404)(143,250,370,405)(144,251,371,406)(145,252,372,407)(146,225,373,408)(147,226,374,409)(148,227,375,410)(149,228,376,411)(150,229,377,412)(151,230,378,413)(152,231,379,414)(153,232,380,415)(154,233,381,416)(155,234,382,417)(156,235,383,418)(157,236,384,419)(158,237,385,420)(159,238,386,393)(160,239,387,394)(161,240,388,395)(162,241,389,396)(163,242,390,397)(164,243,391,398)(165,244,392,399)(166,245,365,400)(167,246,366,401)(168,247,367,402) );
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)], [(1,281,142,81,260,175,383,39),(2,308,143,80,261,174,384,38),(3,307,144,79,262,173,385,37),(4,306,145,78,263,172,386,36),(5,305,146,77,264,171,387,35),(6,304,147,76,265,170,388,34),(7,303,148,75,266,169,389,33),(8,302,149,74,267,196,390,32),(9,301,150,73,268,195,391,31),(10,300,151,72,269,194,392,30),(11,299,152,71,270,193,365,29),(12,298,153,70,271,192,366,56),(13,297,154,69,272,191,367,55),(14,296,155,68,273,190,368,54),(15,295,156,67,274,189,369,53),(16,294,157,66,275,188,370,52),(17,293,158,65,276,187,371,51),(18,292,159,64,277,186,372,50),(19,291,160,63,278,185,373,49),(20,290,161,62,279,184,374,48),(21,289,162,61,280,183,375,47),(22,288,163,60,253,182,376,46),(23,287,164,59,254,181,377,45),(24,286,165,58,255,180,378,44),(25,285,166,57,256,179,379,43),(26,284,167,84,257,178,380,42),(27,283,168,83,258,177,381,41),(28,282,141,82,259,176,382,40),(85,244,134,362,318,413,221,433),(86,243,135,361,319,412,222,432),(87,242,136,360,320,411,223,431),(88,241,137,359,321,410,224,430),(89,240,138,358,322,409,197,429),(90,239,139,357,323,408,198,428),(91,238,140,356,324,407,199,427),(92,237,113,355,325,406,200,426),(93,236,114,354,326,405,201,425),(94,235,115,353,327,404,202,424),(95,234,116,352,328,403,203,423),(96,233,117,351,329,402,204,422),(97,232,118,350,330,401,205,421),(98,231,119,349,331,400,206,448),(99,230,120,348,332,399,207,447),(100,229,121,347,333,398,208,446),(101,228,122,346,334,397,209,445),(102,227,123,345,335,396,210,444),(103,226,124,344,336,395,211,443),(104,225,125,343,309,394,212,442),(105,252,126,342,310,393,213,441),(106,251,127,341,311,420,214,440),(107,250,128,340,312,419,215,439),(108,249,129,339,313,418,216,438),(109,248,130,338,314,417,217,437),(110,247,131,337,315,416,218,436),(111,246,132,364,316,415,219,435),(112,245,133,363,317,414,220,434)], [(1,339,274,424),(2,340,275,425),(3,341,276,426),(4,342,277,427),(5,343,278,428),(6,344,279,429),(7,345,280,430),(8,346,253,431),(9,347,254,432),(10,348,255,433),(11,349,256,434),(12,350,257,435),(13,351,258,436),(14,352,259,437),(15,353,260,438),(16,354,261,439),(17,355,262,440),(18,356,263,441),(19,357,264,442),(20,358,265,443),(21,359,266,444),(22,360,267,445),(23,361,268,446),(24,362,269,447),(25,363,270,448),(26,364,271,421),(27,337,272,422),(28,338,273,423),(29,331,57,112),(30,332,58,85),(31,333,59,86),(32,334,60,87),(33,335,61,88),(34,336,62,89),(35,309,63,90),(36,310,64,91),(37,311,65,92),(38,312,66,93),(39,313,67,94),(40,314,68,95),(41,315,69,96),(42,316,70,97),(43,317,71,98),(44,318,72,99),(45,319,73,100),(46,320,74,101),(47,321,75,102),(48,322,76,103),(49,323,77,104),(50,324,78,105),(51,325,79,106),(52,326,80,107),(53,327,81,108),(54,328,82,109),(55,329,83,110),(56,330,84,111),(113,173,214,293),(114,174,215,294),(115,175,216,295),(116,176,217,296),(117,177,218,297),(118,178,219,298),(119,179,220,299),(120,180,221,300),(121,181,222,301),(122,182,223,302),(123,183,224,303),(124,184,197,304),(125,185,198,305),(126,186,199,306),(127,187,200,307),(128,188,201,308),(129,189,202,281),(130,190,203,282),(131,191,204,283),(132,192,205,284),(133,193,206,285),(134,194,207,286),(135,195,208,287),(136,196,209,288),(137,169,210,289),(138,170,211,290),(139,171,212,291),(140,172,213,292),(141,248,368,403),(142,249,369,404),(143,250,370,405),(144,251,371,406),(145,252,372,407),(146,225,373,408),(147,226,374,409),(148,227,375,410),(149,228,376,411),(150,229,377,412),(151,230,378,413),(152,231,379,414),(153,232,380,415),(154,233,381,416),(155,234,382,417),(156,235,383,418),(157,236,384,419),(158,237,385,420),(159,238,386,393),(160,239,387,394),(161,240,388,395),(162,241,389,396),(163,242,390,397),(164,243,391,398),(165,244,392,399),(166,245,365,400),(167,246,366,401),(168,247,367,402)]])
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 | Q16 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | Dic14 | C4○D28 | C8⋊C22 | C7⋊Q16 | D4⋊D14 |
| kernel | C28.23Q16 | C28⋊C8 | C28.Q8 | 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 C28.23Q16 ►in GL4(𝔽113) generated by
| 78 | 32 | 0 | 0 |
| 49 | 94 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 105 | 109 | 0 | 0 |
| 101 | 8 | 0 | 0 |
| 0 | 0 | 0 | 62 |
| 0 | 0 | 82 | 62 |
| 66 | 75 | 0 | 0 |
| 76 | 47 | 0 | 0 |
| 0 | 0 | 89 | 34 |
| 0 | 0 | 106 | 24 |
G:=sub<GL(4,GF(113))| [78,49,0,0,32,94,0,0,0,0,112,0,0,0,0,112],[105,101,0,0,109,8,0,0,0,0,0,82,0,0,62,62],[66,76,0,0,75,47,0,0,0,0,89,106,0,0,34,24] >;
C28.23Q16 in GAP, Magma, Sage, TeX
C_{28}._{23}Q_{16} % in TeX
G:=Group("C28.23Q16"); // GroupNames label
G:=SmallGroup(448,555);
// by ID
G=gap.SmallGroup(448,555);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,336,253,120,254,268,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations