metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊4Q16, C8.9D28, C4⋊1Dic28, C56.59D4, C42.266D14, (C4×C8).9D7, (C4×C56).11C2, C7⋊1(C4⋊Q16), (C2×C4).82D28, C4.33(C2×D28), C14.4(C2×Q16), (C2×C8).302D14, C28.276(C2×D4), (C2×C28).379D4, C28⋊2Q8.4C2, C2.6(C2×Dic28), C14.6(C4⋊1D4), C2.8(C28⋊4D4), (C2×Dic28).2C2, C22.94(C2×D28), (C2×C56).375C22, (C4×C28).311C22, (C2×C28).728C23, (C2×Dic14).5C22, (C2×C14).111(C2×D4), (C2×C4).671(C22×D7), SmallGroup(448,233)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊4Q16
G = < a,b,c | a28=b8=1, c2=b4, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 644 in 122 conjugacy classes, 55 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, Q16, C2×Q8, Dic7, C28, C2×C14, C4×C8, C4⋊Q8, C2×Q16, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C4⋊Q16, Dic28, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C4×C56, C28⋊2Q8, C2×Dic28, C28⋊4Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C4⋊1D4, C2×Q16, D28, C22×D7, C4⋊Q16, Dic28, C2×D28, C28⋊4D4, C2×Dic28, C28⋊4Q16
►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)
(1 374 222 331 432 251 186 132)(2 375 223 332 433 252 187 133)(3 376 224 333 434 225 188 134)(4 377 197 334 435 226 189 135)(5 378 198 335 436 227 190 136)(6 379 199 336 437 228 191 137)(7 380 200 309 438 229 192 138)(8 381 201 310 439 230 193 139)(9 382 202 311 440 231 194 140)(10 383 203 312 441 232 195 113)(11 384 204 313 442 233 196 114)(12 385 205 314 443 234 169 115)(13 386 206 315 444 235 170 116)(14 387 207 316 445 236 171 117)(15 388 208 317 446 237 172 118)(16 389 209 318 447 238 173 119)(17 390 210 319 448 239 174 120)(18 391 211 320 421 240 175 121)(19 392 212 321 422 241 176 122)(20 365 213 322 423 242 177 123)(21 366 214 323 424 243 178 124)(22 367 215 324 425 244 179 125)(23 368 216 325 426 245 180 126)(24 369 217 326 427 246 181 127)(25 370 218 327 428 247 182 128)(26 371 219 328 429 248 183 129)(27 372 220 329 430 249 184 130)(28 373 221 330 431 250 185 131)(29 253 163 308 102 399 350 58)(30 254 164 281 103 400 351 59)(31 255 165 282 104 401 352 60)(32 256 166 283 105 402 353 61)(33 257 167 284 106 403 354 62)(34 258 168 285 107 404 355 63)(35 259 141 286 108 405 356 64)(36 260 142 287 109 406 357 65)(37 261 143 288 110 407 358 66)(38 262 144 289 111 408 359 67)(39 263 145 290 112 409 360 68)(40 264 146 291 85 410 361 69)(41 265 147 292 86 411 362 70)(42 266 148 293 87 412 363 71)(43 267 149 294 88 413 364 72)(44 268 150 295 89 414 337 73)(45 269 151 296 90 415 338 74)(46 270 152 297 91 416 339 75)(47 271 153 298 92 417 340 76)(48 272 154 299 93 418 341 77)(49 273 155 300 94 419 342 78)(50 274 156 301 95 420 343 79)(51 275 157 302 96 393 344 80)(52 276 158 303 97 394 345 81)(53 277 159 304 98 395 346 82)(54 278 160 305 99 396 347 83)(55 279 161 306 100 397 348 84)(56 280 162 307 101 398 349 57)
(1 352 432 165)(2 351 433 164)(3 350 434 163)(4 349 435 162)(5 348 436 161)(6 347 437 160)(7 346 438 159)(8 345 439 158)(9 344 440 157)(10 343 441 156)(11 342 442 155)(12 341 443 154)(13 340 444 153)(14 339 445 152)(15 338 446 151)(16 337 447 150)(17 364 448 149)(18 363 421 148)(19 362 422 147)(20 361 423 146)(21 360 424 145)(22 359 425 144)(23 358 426 143)(24 357 427 142)(25 356 428 141)(26 355 429 168)(27 354 430 167)(28 353 431 166)(29 224 102 188)(30 223 103 187)(31 222 104 186)(32 221 105 185)(33 220 106 184)(34 219 107 183)(35 218 108 182)(36 217 109 181)(37 216 110 180)(38 215 111 179)(39 214 112 178)(40 213 85 177)(41 212 86 176)(42 211 87 175)(43 210 88 174)(44 209 89 173)(45 208 90 172)(46 207 91 171)(47 206 92 170)(48 205 93 169)(49 204 94 196)(50 203 95 195)(51 202 96 194)(52 201 97 193)(53 200 98 192)(54 199 99 191)(55 198 100 190)(56 197 101 189)(57 334 307 135)(58 333 308 134)(59 332 281 133)(60 331 282 132)(61 330 283 131)(62 329 284 130)(63 328 285 129)(64 327 286 128)(65 326 287 127)(66 325 288 126)(67 324 289 125)(68 323 290 124)(69 322 291 123)(70 321 292 122)(71 320 293 121)(72 319 294 120)(73 318 295 119)(74 317 296 118)(75 316 297 117)(76 315 298 116)(77 314 299 115)(78 313 300 114)(79 312 301 113)(80 311 302 140)(81 310 303 139)(82 309 304 138)(83 336 305 137)(84 335 306 136)(225 253 376 399)(226 280 377 398)(227 279 378 397)(228 278 379 396)(229 277 380 395)(230 276 381 394)(231 275 382 393)(232 274 383 420)(233 273 384 419)(234 272 385 418)(235 271 386 417)(236 270 387 416)(237 269 388 415)(238 268 389 414)(239 267 390 413)(240 266 391 412)(241 265 392 411)(242 264 365 410)(243 263 366 409)(244 262 367 408)(245 261 368 407)(246 260 369 406)(247 259 370 405)(248 258 371 404)(249 257 372 403)(250 256 373 402)(251 255 374 401)(252 254 375 400)
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), (1,374,222,331,432,251,186,132)(2,375,223,332,433,252,187,133)(3,376,224,333,434,225,188,134)(4,377,197,334,435,226,189,135)(5,378,198,335,436,227,190,136)(6,379,199,336,437,228,191,137)(7,380,200,309,438,229,192,138)(8,381,201,310,439,230,193,139)(9,382,202,311,440,231,194,140)(10,383,203,312,441,232,195,113)(11,384,204,313,442,233,196,114)(12,385,205,314,443,234,169,115)(13,386,206,315,444,235,170,116)(14,387,207,316,445,236,171,117)(15,388,208,317,446,237,172,118)(16,389,209,318,447,238,173,119)(17,390,210,319,448,239,174,120)(18,391,211,320,421,240,175,121)(19,392,212,321,422,241,176,122)(20,365,213,322,423,242,177,123)(21,366,214,323,424,243,178,124)(22,367,215,324,425,244,179,125)(23,368,216,325,426,245,180,126)(24,369,217,326,427,246,181,127)(25,370,218,327,428,247,182,128)(26,371,219,328,429,248,183,129)(27,372,220,329,430,249,184,130)(28,373,221,330,431,250,185,131)(29,253,163,308,102,399,350,58)(30,254,164,281,103,400,351,59)(31,255,165,282,104,401,352,60)(32,256,166,283,105,402,353,61)(33,257,167,284,106,403,354,62)(34,258,168,285,107,404,355,63)(35,259,141,286,108,405,356,64)(36,260,142,287,109,406,357,65)(37,261,143,288,110,407,358,66)(38,262,144,289,111,408,359,67)(39,263,145,290,112,409,360,68)(40,264,146,291,85,410,361,69)(41,265,147,292,86,411,362,70)(42,266,148,293,87,412,363,71)(43,267,149,294,88,413,364,72)(44,268,150,295,89,414,337,73)(45,269,151,296,90,415,338,74)(46,270,152,297,91,416,339,75)(47,271,153,298,92,417,340,76)(48,272,154,299,93,418,341,77)(49,273,155,300,94,419,342,78)(50,274,156,301,95,420,343,79)(51,275,157,302,96,393,344,80)(52,276,158,303,97,394,345,81)(53,277,159,304,98,395,346,82)(54,278,160,305,99,396,347,83)(55,279,161,306,100,397,348,84)(56,280,162,307,101,398,349,57), (1,352,432,165)(2,351,433,164)(3,350,434,163)(4,349,435,162)(5,348,436,161)(6,347,437,160)(7,346,438,159)(8,345,439,158)(9,344,440,157)(10,343,441,156)(11,342,442,155)(12,341,443,154)(13,340,444,153)(14,339,445,152)(15,338,446,151)(16,337,447,150)(17,364,448,149)(18,363,421,148)(19,362,422,147)(20,361,423,146)(21,360,424,145)(22,359,425,144)(23,358,426,143)(24,357,427,142)(25,356,428,141)(26,355,429,168)(27,354,430,167)(28,353,431,166)(29,224,102,188)(30,223,103,187)(31,222,104,186)(32,221,105,185)(33,220,106,184)(34,219,107,183)(35,218,108,182)(36,217,109,181)(37,216,110,180)(38,215,111,179)(39,214,112,178)(40,213,85,177)(41,212,86,176)(42,211,87,175)(43,210,88,174)(44,209,89,173)(45,208,90,172)(46,207,91,171)(47,206,92,170)(48,205,93,169)(49,204,94,196)(50,203,95,195)(51,202,96,194)(52,201,97,193)(53,200,98,192)(54,199,99,191)(55,198,100,190)(56,197,101,189)(57,334,307,135)(58,333,308,134)(59,332,281,133)(60,331,282,132)(61,330,283,131)(62,329,284,130)(63,328,285,129)(64,327,286,128)(65,326,287,127)(66,325,288,126)(67,324,289,125)(68,323,290,124)(69,322,291,123)(70,321,292,122)(71,320,293,121)(72,319,294,120)(73,318,295,119)(74,317,296,118)(75,316,297,117)(76,315,298,116)(77,314,299,115)(78,313,300,114)(79,312,301,113)(80,311,302,140)(81,310,303,139)(82,309,304,138)(83,336,305,137)(84,335,306,136)(225,253,376,399)(226,280,377,398)(227,279,378,397)(228,278,379,396)(229,277,380,395)(230,276,381,394)(231,275,382,393)(232,274,383,420)(233,273,384,419)(234,272,385,418)(235,271,386,417)(236,270,387,416)(237,269,388,415)(238,268,389,414)(239,267,390,413)(240,266,391,412)(241,265,392,411)(242,264,365,410)(243,263,366,409)(244,262,367,408)(245,261,368,407)(246,260,369,406)(247,259,370,405)(248,258,371,404)(249,257,372,403)(250,256,373,402)(251,255,374,401)(252,254,375,400)>;
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), (1,374,222,331,432,251,186,132)(2,375,223,332,433,252,187,133)(3,376,224,333,434,225,188,134)(4,377,197,334,435,226,189,135)(5,378,198,335,436,227,190,136)(6,379,199,336,437,228,191,137)(7,380,200,309,438,229,192,138)(8,381,201,310,439,230,193,139)(9,382,202,311,440,231,194,140)(10,383,203,312,441,232,195,113)(11,384,204,313,442,233,196,114)(12,385,205,314,443,234,169,115)(13,386,206,315,444,235,170,116)(14,387,207,316,445,236,171,117)(15,388,208,317,446,237,172,118)(16,389,209,318,447,238,173,119)(17,390,210,319,448,239,174,120)(18,391,211,320,421,240,175,121)(19,392,212,321,422,241,176,122)(20,365,213,322,423,242,177,123)(21,366,214,323,424,243,178,124)(22,367,215,324,425,244,179,125)(23,368,216,325,426,245,180,126)(24,369,217,326,427,246,181,127)(25,370,218,327,428,247,182,128)(26,371,219,328,429,248,183,129)(27,372,220,329,430,249,184,130)(28,373,221,330,431,250,185,131)(29,253,163,308,102,399,350,58)(30,254,164,281,103,400,351,59)(31,255,165,282,104,401,352,60)(32,256,166,283,105,402,353,61)(33,257,167,284,106,403,354,62)(34,258,168,285,107,404,355,63)(35,259,141,286,108,405,356,64)(36,260,142,287,109,406,357,65)(37,261,143,288,110,407,358,66)(38,262,144,289,111,408,359,67)(39,263,145,290,112,409,360,68)(40,264,146,291,85,410,361,69)(41,265,147,292,86,411,362,70)(42,266,148,293,87,412,363,71)(43,267,149,294,88,413,364,72)(44,268,150,295,89,414,337,73)(45,269,151,296,90,415,338,74)(46,270,152,297,91,416,339,75)(47,271,153,298,92,417,340,76)(48,272,154,299,93,418,341,77)(49,273,155,300,94,419,342,78)(50,274,156,301,95,420,343,79)(51,275,157,302,96,393,344,80)(52,276,158,303,97,394,345,81)(53,277,159,304,98,395,346,82)(54,278,160,305,99,396,347,83)(55,279,161,306,100,397,348,84)(56,280,162,307,101,398,349,57), (1,352,432,165)(2,351,433,164)(3,350,434,163)(4,349,435,162)(5,348,436,161)(6,347,437,160)(7,346,438,159)(8,345,439,158)(9,344,440,157)(10,343,441,156)(11,342,442,155)(12,341,443,154)(13,340,444,153)(14,339,445,152)(15,338,446,151)(16,337,447,150)(17,364,448,149)(18,363,421,148)(19,362,422,147)(20,361,423,146)(21,360,424,145)(22,359,425,144)(23,358,426,143)(24,357,427,142)(25,356,428,141)(26,355,429,168)(27,354,430,167)(28,353,431,166)(29,224,102,188)(30,223,103,187)(31,222,104,186)(32,221,105,185)(33,220,106,184)(34,219,107,183)(35,218,108,182)(36,217,109,181)(37,216,110,180)(38,215,111,179)(39,214,112,178)(40,213,85,177)(41,212,86,176)(42,211,87,175)(43,210,88,174)(44,209,89,173)(45,208,90,172)(46,207,91,171)(47,206,92,170)(48,205,93,169)(49,204,94,196)(50,203,95,195)(51,202,96,194)(52,201,97,193)(53,200,98,192)(54,199,99,191)(55,198,100,190)(56,197,101,189)(57,334,307,135)(58,333,308,134)(59,332,281,133)(60,331,282,132)(61,330,283,131)(62,329,284,130)(63,328,285,129)(64,327,286,128)(65,326,287,127)(66,325,288,126)(67,324,289,125)(68,323,290,124)(69,322,291,123)(70,321,292,122)(71,320,293,121)(72,319,294,120)(73,318,295,119)(74,317,296,118)(75,316,297,117)(76,315,298,116)(77,314,299,115)(78,313,300,114)(79,312,301,113)(80,311,302,140)(81,310,303,139)(82,309,304,138)(83,336,305,137)(84,335,306,136)(225,253,376,399)(226,280,377,398)(227,279,378,397)(228,278,379,396)(229,277,380,395)(230,276,381,394)(231,275,382,393)(232,274,383,420)(233,273,384,419)(234,272,385,418)(235,271,386,417)(236,270,387,416)(237,269,388,415)(238,268,389,414)(239,267,390,413)(240,266,391,412)(241,265,392,411)(242,264,365,410)(243,263,366,409)(244,262,367,408)(245,261,368,407)(246,260,369,406)(247,259,370,405)(248,258,371,404)(249,257,372,403)(250,256,373,402)(251,255,374,401)(252,254,375,400) );
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)], [(1,374,222,331,432,251,186,132),(2,375,223,332,433,252,187,133),(3,376,224,333,434,225,188,134),(4,377,197,334,435,226,189,135),(5,378,198,335,436,227,190,136),(6,379,199,336,437,228,191,137),(7,380,200,309,438,229,192,138),(8,381,201,310,439,230,193,139),(9,382,202,311,440,231,194,140),(10,383,203,312,441,232,195,113),(11,384,204,313,442,233,196,114),(12,385,205,314,443,234,169,115),(13,386,206,315,444,235,170,116),(14,387,207,316,445,236,171,117),(15,388,208,317,446,237,172,118),(16,389,209,318,447,238,173,119),(17,390,210,319,448,239,174,120),(18,391,211,320,421,240,175,121),(19,392,212,321,422,241,176,122),(20,365,213,322,423,242,177,123),(21,366,214,323,424,243,178,124),(22,367,215,324,425,244,179,125),(23,368,216,325,426,245,180,126),(24,369,217,326,427,246,181,127),(25,370,218,327,428,247,182,128),(26,371,219,328,429,248,183,129),(27,372,220,329,430,249,184,130),(28,373,221,330,431,250,185,131),(29,253,163,308,102,399,350,58),(30,254,164,281,103,400,351,59),(31,255,165,282,104,401,352,60),(32,256,166,283,105,402,353,61),(33,257,167,284,106,403,354,62),(34,258,168,285,107,404,355,63),(35,259,141,286,108,405,356,64),(36,260,142,287,109,406,357,65),(37,261,143,288,110,407,358,66),(38,262,144,289,111,408,359,67),(39,263,145,290,112,409,360,68),(40,264,146,291,85,410,361,69),(41,265,147,292,86,411,362,70),(42,266,148,293,87,412,363,71),(43,267,149,294,88,413,364,72),(44,268,150,295,89,414,337,73),(45,269,151,296,90,415,338,74),(46,270,152,297,91,416,339,75),(47,271,153,298,92,417,340,76),(48,272,154,299,93,418,341,77),(49,273,155,300,94,419,342,78),(50,274,156,301,95,420,343,79),(51,275,157,302,96,393,344,80),(52,276,158,303,97,394,345,81),(53,277,159,304,98,395,346,82),(54,278,160,305,99,396,347,83),(55,279,161,306,100,397,348,84),(56,280,162,307,101,398,349,57)], [(1,352,432,165),(2,351,433,164),(3,350,434,163),(4,349,435,162),(5,348,436,161),(6,347,437,160),(7,346,438,159),(8,345,439,158),(9,344,440,157),(10,343,441,156),(11,342,442,155),(12,341,443,154),(13,340,444,153),(14,339,445,152),(15,338,446,151),(16,337,447,150),(17,364,448,149),(18,363,421,148),(19,362,422,147),(20,361,423,146),(21,360,424,145),(22,359,425,144),(23,358,426,143),(24,357,427,142),(25,356,428,141),(26,355,429,168),(27,354,430,167),(28,353,431,166),(29,224,102,188),(30,223,103,187),(31,222,104,186),(32,221,105,185),(33,220,106,184),(34,219,107,183),(35,218,108,182),(36,217,109,181),(37,216,110,180),(38,215,111,179),(39,214,112,178),(40,213,85,177),(41,212,86,176),(42,211,87,175),(43,210,88,174),(44,209,89,173),(45,208,90,172),(46,207,91,171),(47,206,92,170),(48,205,93,169),(49,204,94,196),(50,203,95,195),(51,202,96,194),(52,201,97,193),(53,200,98,192),(54,199,99,191),(55,198,100,190),(56,197,101,189),(57,334,307,135),(58,333,308,134),(59,332,281,133),(60,331,282,132),(61,330,283,131),(62,329,284,130),(63,328,285,129),(64,327,286,128),(65,326,287,127),(66,325,288,126),(67,324,289,125),(68,323,290,124),(69,322,291,123),(70,321,292,122),(71,320,293,121),(72,319,294,120),(73,318,295,119),(74,317,296,118),(75,316,297,117),(76,315,298,116),(77,314,299,115),(78,313,300,114),(79,312,301,113),(80,311,302,140),(81,310,303,139),(82,309,304,138),(83,336,305,137),(84,335,306,136),(225,253,376,399),(226,280,377,398),(227,279,378,397),(228,278,379,396),(229,277,380,395),(230,276,381,394),(231,275,382,393),(232,274,383,420),(233,273,384,419),(234,272,385,418),(235,271,386,417),(236,270,387,416),(237,269,388,415),(238,268,389,414),(239,267,390,413),(240,266,391,412),(241,265,392,411),(242,264,365,410),(243,263,366,409),(244,262,367,408),(245,261,368,407),(246,260,369,406),(247,259,370,405),(248,258,371,404),(249,257,372,403),(250,256,373,402),(251,255,374,401),(252,254,375,400)]])
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 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D7 | Q16 | D14 | D14 | D28 | D28 | Dic28 |
| kernel | C28⋊4Q16 | C4×C56 | C28⋊2Q8 | C2×Dic28 | C56 | C2×C28 | C4×C8 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 4 | 2 | 3 | 8 | 3 | 6 | 24 | 12 | 48 |
Matrix representation of C28⋊4Q16 ►in GL4(𝔽113) generated by
| 1 | 104 | 0 | 0 |
| 9 | 33 | 0 | 0 |
| 0 | 0 | 108 | 100 |
| 0 | 0 | 26 | 90 |
| 21 | 65 | 0 | 0 |
| 48 | 41 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 85 | 79 | 0 | 0 |
| 53 | 28 | 0 | 0 |
| 0 | 0 | 29 | 46 |
| 0 | 0 | 80 | 84 |
G:=sub<GL(4,GF(113))| [1,9,0,0,104,33,0,0,0,0,108,26,0,0,100,90],[21,48,0,0,65,41,0,0,0,0,112,0,0,0,0,112],[85,53,0,0,79,28,0,0,0,0,29,80,0,0,46,84] >;
C28⋊4Q16 in GAP, Magma, Sage, TeX
C_{28}\rtimes_4Q_{16} % in TeX
G:=Group("C28:4Q16"); // GroupNames label
G:=SmallGroup(448,233);
// by ID
G=gap.SmallGroup(448,233);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,226,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=b^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations