metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊7Q16, Q8.2D28, C42.60D14, (C4×Q8).8D7, C4.18(C2×D28), (C2×C28).68D4, C4⋊3(C7⋊Q16), C7⋊3(C4⋊2Q16), C28.22(C2×D4), (Q8×C28).9C2, (C7×Q8).19D4, C4⋊C4.257D14, C14.36(C2×Q16), C28⋊C8.19C2, C28.64(C4○D4), C4.14(C4○D28), (C2×Q8).164D14, C28⋊2Q8.15C2, C2.17(C28⋊7D4), C14.69(C4⋊D4), (C4×C28).102C22, (C2×C28).351C23, C14.Q16.11C2, (Q8×C14).199C22, C2.10(D4.9D14), C14.112(C8.C22), (C2×Dic14).102C22, C2.7(C2×C7⋊Q16), (C2×C7⋊Q16).5C2, (C2×C14).482(C2×D4), (C2×C7⋊C8).104C22, (C2×C4).250(C7⋊D4), (C7×C4⋊C4).288C22, (C2×C4).451(C22×D7), C22.157(C2×C7⋊D4), SmallGroup(448,565)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊7Q16
G = < a,b,c | a28=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Subgroups: 452 in 108 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, Q16, C2×Q8, C2×Q8, Dic7, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C4⋊2Q16, C2×C7⋊C8, C4⋊Dic7, C7⋊Q16, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C28⋊C8, C14.Q16, C28⋊2Q8, C2×C7⋊Q16, Q8×C28, C28⋊7Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C4○D4, D14, C4⋊D4, C2×Q16, C8.C22, D28, C7⋊D4, C22×D7, C4⋊2Q16, C7⋊Q16, C2×D28, C4○D28, C2×C7⋊D4, C28⋊7D4, C2×C7⋊Q16, D4.9D14, C28⋊7Q16
►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)
(1 343 193 207 34 335 145 371)(2 342 194 206 35 334 146 370)(3 341 195 205 36 333 147 369)(4 340 196 204 37 332 148 368)(5 339 169 203 38 331 149 367)(6 338 170 202 39 330 150 366)(7 337 171 201 40 329 151 365)(8 364 172 200 41 328 152 392)(9 363 173 199 42 327 153 391)(10 362 174 198 43 326 154 390)(11 361 175 197 44 325 155 389)(12 360 176 224 45 324 156 388)(13 359 177 223 46 323 157 387)(14 358 178 222 47 322 158 386)(15 357 179 221 48 321 159 385)(16 356 180 220 49 320 160 384)(17 355 181 219 50 319 161 383)(18 354 182 218 51 318 162 382)(19 353 183 217 52 317 163 381)(20 352 184 216 53 316 164 380)(21 351 185 215 54 315 165 379)(22 350 186 214 55 314 166 378)(23 349 187 213 56 313 167 377)(24 348 188 212 29 312 168 376)(25 347 189 211 30 311 141 375)(26 346 190 210 31 310 142 374)(27 345 191 209 32 309 143 373)(28 344 192 208 33 336 144 372)(57 259 238 412 135 95 286 426)(58 258 239 411 136 94 287 425)(59 257 240 410 137 93 288 424)(60 256 241 409 138 92 289 423)(61 255 242 408 139 91 290 422)(62 254 243 407 140 90 291 421)(63 253 244 406 113 89 292 448)(64 280 245 405 114 88 293 447)(65 279 246 404 115 87 294 446)(66 278 247 403 116 86 295 445)(67 277 248 402 117 85 296 444)(68 276 249 401 118 112 297 443)(69 275 250 400 119 111 298 442)(70 274 251 399 120 110 299 441)(71 273 252 398 121 109 300 440)(72 272 225 397 122 108 301 439)(73 271 226 396 123 107 302 438)(74 270 227 395 124 106 303 437)(75 269 228 394 125 105 304 436)(76 268 229 393 126 104 305 435)(77 267 230 420 127 103 306 434)(78 266 231 419 128 102 307 433)(79 265 232 418 129 101 308 432)(80 264 233 417 130 100 281 431)(81 263 234 416 131 99 282 430)(82 262 235 415 132 98 283 429)(83 261 236 414 133 97 284 428)(84 260 237 413 134 96 285 427)
(1 131 34 81)(2 132 35 82)(3 133 36 83)(4 134 37 84)(5 135 38 57)(6 136 39 58)(7 137 40 59)(8 138 41 60)(9 139 42 61)(10 140 43 62)(11 113 44 63)(12 114 45 64)(13 115 46 65)(14 116 47 66)(15 117 48 67)(16 118 49 68)(17 119 50 69)(18 120 51 70)(19 121 52 71)(20 122 53 72)(21 123 54 73)(22 124 55 74)(23 125 56 75)(24 126 29 76)(25 127 30 77)(26 128 31 78)(27 129 32 79)(28 130 33 80)(85 221 277 385)(86 222 278 386)(87 223 279 387)(88 224 280 388)(89 197 253 389)(90 198 254 390)(91 199 255 391)(92 200 256 392)(93 201 257 365)(94 202 258 366)(95 203 259 367)(96 204 260 368)(97 205 261 369)(98 206 262 370)(99 207 263 371)(100 208 264 372)(101 209 265 373)(102 210 266 374)(103 211 267 375)(104 212 268 376)(105 213 269 377)(106 214 270 378)(107 215 271 379)(108 216 272 380)(109 217 273 381)(110 218 274 382)(111 219 275 383)(112 220 276 384)(141 306 189 230)(142 307 190 231)(143 308 191 232)(144 281 192 233)(145 282 193 234)(146 283 194 235)(147 284 195 236)(148 285 196 237)(149 286 169 238)(150 287 170 239)(151 288 171 240)(152 289 172 241)(153 290 173 242)(154 291 174 243)(155 292 175 244)(156 293 176 245)(157 294 177 246)(158 295 178 247)(159 296 179 248)(160 297 180 249)(161 298 181 250)(162 299 182 251)(163 300 183 252)(164 301 184 225)(165 302 185 226)(166 303 186 227)(167 304 187 228)(168 305 188 229)(309 432 345 418)(310 433 346 419)(311 434 347 420)(312 435 348 393)(313 436 349 394)(314 437 350 395)(315 438 351 396)(316 439 352 397)(317 440 353 398)(318 441 354 399)(319 442 355 400)(320 443 356 401)(321 444 357 402)(322 445 358 403)(323 446 359 404)(324 447 360 405)(325 448 361 406)(326 421 362 407)(327 422 363 408)(328 423 364 409)(329 424 337 410)(330 425 338 411)(331 426 339 412)(332 427 340 413)(333 428 341 414)(334 429 342 415)(335 430 343 416)(336 431 344 417)
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), (1,343,193,207,34,335,145,371)(2,342,194,206,35,334,146,370)(3,341,195,205,36,333,147,369)(4,340,196,204,37,332,148,368)(5,339,169,203,38,331,149,367)(6,338,170,202,39,330,150,366)(7,337,171,201,40,329,151,365)(8,364,172,200,41,328,152,392)(9,363,173,199,42,327,153,391)(10,362,174,198,43,326,154,390)(11,361,175,197,44,325,155,389)(12,360,176,224,45,324,156,388)(13,359,177,223,46,323,157,387)(14,358,178,222,47,322,158,386)(15,357,179,221,48,321,159,385)(16,356,180,220,49,320,160,384)(17,355,181,219,50,319,161,383)(18,354,182,218,51,318,162,382)(19,353,183,217,52,317,163,381)(20,352,184,216,53,316,164,380)(21,351,185,215,54,315,165,379)(22,350,186,214,55,314,166,378)(23,349,187,213,56,313,167,377)(24,348,188,212,29,312,168,376)(25,347,189,211,30,311,141,375)(26,346,190,210,31,310,142,374)(27,345,191,209,32,309,143,373)(28,344,192,208,33,336,144,372)(57,259,238,412,135,95,286,426)(58,258,239,411,136,94,287,425)(59,257,240,410,137,93,288,424)(60,256,241,409,138,92,289,423)(61,255,242,408,139,91,290,422)(62,254,243,407,140,90,291,421)(63,253,244,406,113,89,292,448)(64,280,245,405,114,88,293,447)(65,279,246,404,115,87,294,446)(66,278,247,403,116,86,295,445)(67,277,248,402,117,85,296,444)(68,276,249,401,118,112,297,443)(69,275,250,400,119,111,298,442)(70,274,251,399,120,110,299,441)(71,273,252,398,121,109,300,440)(72,272,225,397,122,108,301,439)(73,271,226,396,123,107,302,438)(74,270,227,395,124,106,303,437)(75,269,228,394,125,105,304,436)(76,268,229,393,126,104,305,435)(77,267,230,420,127,103,306,434)(78,266,231,419,128,102,307,433)(79,265,232,418,129,101,308,432)(80,264,233,417,130,100,281,431)(81,263,234,416,131,99,282,430)(82,262,235,415,132,98,283,429)(83,261,236,414,133,97,284,428)(84,260,237,413,134,96,285,427), (1,131,34,81)(2,132,35,82)(3,133,36,83)(4,134,37,84)(5,135,38,57)(6,136,39,58)(7,137,40,59)(8,138,41,60)(9,139,42,61)(10,140,43,62)(11,113,44,63)(12,114,45,64)(13,115,46,65)(14,116,47,66)(15,117,48,67)(16,118,49,68)(17,119,50,69)(18,120,51,70)(19,121,52,71)(20,122,53,72)(21,123,54,73)(22,124,55,74)(23,125,56,75)(24,126,29,76)(25,127,30,77)(26,128,31,78)(27,129,32,79)(28,130,33,80)(85,221,277,385)(86,222,278,386)(87,223,279,387)(88,224,280,388)(89,197,253,389)(90,198,254,390)(91,199,255,391)(92,200,256,392)(93,201,257,365)(94,202,258,366)(95,203,259,367)(96,204,260,368)(97,205,261,369)(98,206,262,370)(99,207,263,371)(100,208,264,372)(101,209,265,373)(102,210,266,374)(103,211,267,375)(104,212,268,376)(105,213,269,377)(106,214,270,378)(107,215,271,379)(108,216,272,380)(109,217,273,381)(110,218,274,382)(111,219,275,383)(112,220,276,384)(141,306,189,230)(142,307,190,231)(143,308,191,232)(144,281,192,233)(145,282,193,234)(146,283,194,235)(147,284,195,236)(148,285,196,237)(149,286,169,238)(150,287,170,239)(151,288,171,240)(152,289,172,241)(153,290,173,242)(154,291,174,243)(155,292,175,244)(156,293,176,245)(157,294,177,246)(158,295,178,247)(159,296,179,248)(160,297,180,249)(161,298,181,250)(162,299,182,251)(163,300,183,252)(164,301,184,225)(165,302,185,226)(166,303,186,227)(167,304,187,228)(168,305,188,229)(309,432,345,418)(310,433,346,419)(311,434,347,420)(312,435,348,393)(313,436,349,394)(314,437,350,395)(315,438,351,396)(316,439,352,397)(317,440,353,398)(318,441,354,399)(319,442,355,400)(320,443,356,401)(321,444,357,402)(322,445,358,403)(323,446,359,404)(324,447,360,405)(325,448,361,406)(326,421,362,407)(327,422,363,408)(328,423,364,409)(329,424,337,410)(330,425,338,411)(331,426,339,412)(332,427,340,413)(333,428,341,414)(334,429,342,415)(335,430,343,416)(336,431,344,417)>;
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), (1,343,193,207,34,335,145,371)(2,342,194,206,35,334,146,370)(3,341,195,205,36,333,147,369)(4,340,196,204,37,332,148,368)(5,339,169,203,38,331,149,367)(6,338,170,202,39,330,150,366)(7,337,171,201,40,329,151,365)(8,364,172,200,41,328,152,392)(9,363,173,199,42,327,153,391)(10,362,174,198,43,326,154,390)(11,361,175,197,44,325,155,389)(12,360,176,224,45,324,156,388)(13,359,177,223,46,323,157,387)(14,358,178,222,47,322,158,386)(15,357,179,221,48,321,159,385)(16,356,180,220,49,320,160,384)(17,355,181,219,50,319,161,383)(18,354,182,218,51,318,162,382)(19,353,183,217,52,317,163,381)(20,352,184,216,53,316,164,380)(21,351,185,215,54,315,165,379)(22,350,186,214,55,314,166,378)(23,349,187,213,56,313,167,377)(24,348,188,212,29,312,168,376)(25,347,189,211,30,311,141,375)(26,346,190,210,31,310,142,374)(27,345,191,209,32,309,143,373)(28,344,192,208,33,336,144,372)(57,259,238,412,135,95,286,426)(58,258,239,411,136,94,287,425)(59,257,240,410,137,93,288,424)(60,256,241,409,138,92,289,423)(61,255,242,408,139,91,290,422)(62,254,243,407,140,90,291,421)(63,253,244,406,113,89,292,448)(64,280,245,405,114,88,293,447)(65,279,246,404,115,87,294,446)(66,278,247,403,116,86,295,445)(67,277,248,402,117,85,296,444)(68,276,249,401,118,112,297,443)(69,275,250,400,119,111,298,442)(70,274,251,399,120,110,299,441)(71,273,252,398,121,109,300,440)(72,272,225,397,122,108,301,439)(73,271,226,396,123,107,302,438)(74,270,227,395,124,106,303,437)(75,269,228,394,125,105,304,436)(76,268,229,393,126,104,305,435)(77,267,230,420,127,103,306,434)(78,266,231,419,128,102,307,433)(79,265,232,418,129,101,308,432)(80,264,233,417,130,100,281,431)(81,263,234,416,131,99,282,430)(82,262,235,415,132,98,283,429)(83,261,236,414,133,97,284,428)(84,260,237,413,134,96,285,427), (1,131,34,81)(2,132,35,82)(3,133,36,83)(4,134,37,84)(5,135,38,57)(6,136,39,58)(7,137,40,59)(8,138,41,60)(9,139,42,61)(10,140,43,62)(11,113,44,63)(12,114,45,64)(13,115,46,65)(14,116,47,66)(15,117,48,67)(16,118,49,68)(17,119,50,69)(18,120,51,70)(19,121,52,71)(20,122,53,72)(21,123,54,73)(22,124,55,74)(23,125,56,75)(24,126,29,76)(25,127,30,77)(26,128,31,78)(27,129,32,79)(28,130,33,80)(85,221,277,385)(86,222,278,386)(87,223,279,387)(88,224,280,388)(89,197,253,389)(90,198,254,390)(91,199,255,391)(92,200,256,392)(93,201,257,365)(94,202,258,366)(95,203,259,367)(96,204,260,368)(97,205,261,369)(98,206,262,370)(99,207,263,371)(100,208,264,372)(101,209,265,373)(102,210,266,374)(103,211,267,375)(104,212,268,376)(105,213,269,377)(106,214,270,378)(107,215,271,379)(108,216,272,380)(109,217,273,381)(110,218,274,382)(111,219,275,383)(112,220,276,384)(141,306,189,230)(142,307,190,231)(143,308,191,232)(144,281,192,233)(145,282,193,234)(146,283,194,235)(147,284,195,236)(148,285,196,237)(149,286,169,238)(150,287,170,239)(151,288,171,240)(152,289,172,241)(153,290,173,242)(154,291,174,243)(155,292,175,244)(156,293,176,245)(157,294,177,246)(158,295,178,247)(159,296,179,248)(160,297,180,249)(161,298,181,250)(162,299,182,251)(163,300,183,252)(164,301,184,225)(165,302,185,226)(166,303,186,227)(167,304,187,228)(168,305,188,229)(309,432,345,418)(310,433,346,419)(311,434,347,420)(312,435,348,393)(313,436,349,394)(314,437,350,395)(315,438,351,396)(316,439,352,397)(317,440,353,398)(318,441,354,399)(319,442,355,400)(320,443,356,401)(321,444,357,402)(322,445,358,403)(323,446,359,404)(324,447,360,405)(325,448,361,406)(326,421,362,407)(327,422,363,408)(328,423,364,409)(329,424,337,410)(330,425,338,411)(331,426,339,412)(332,427,340,413)(333,428,341,414)(334,429,342,415)(335,430,343,416)(336,431,344,417) );
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)], [(1,343,193,207,34,335,145,371),(2,342,194,206,35,334,146,370),(3,341,195,205,36,333,147,369),(4,340,196,204,37,332,148,368),(5,339,169,203,38,331,149,367),(6,338,170,202,39,330,150,366),(7,337,171,201,40,329,151,365),(8,364,172,200,41,328,152,392),(9,363,173,199,42,327,153,391),(10,362,174,198,43,326,154,390),(11,361,175,197,44,325,155,389),(12,360,176,224,45,324,156,388),(13,359,177,223,46,323,157,387),(14,358,178,222,47,322,158,386),(15,357,179,221,48,321,159,385),(16,356,180,220,49,320,160,384),(17,355,181,219,50,319,161,383),(18,354,182,218,51,318,162,382),(19,353,183,217,52,317,163,381),(20,352,184,216,53,316,164,380),(21,351,185,215,54,315,165,379),(22,350,186,214,55,314,166,378),(23,349,187,213,56,313,167,377),(24,348,188,212,29,312,168,376),(25,347,189,211,30,311,141,375),(26,346,190,210,31,310,142,374),(27,345,191,209,32,309,143,373),(28,344,192,208,33,336,144,372),(57,259,238,412,135,95,286,426),(58,258,239,411,136,94,287,425),(59,257,240,410,137,93,288,424),(60,256,241,409,138,92,289,423),(61,255,242,408,139,91,290,422),(62,254,243,407,140,90,291,421),(63,253,244,406,113,89,292,448),(64,280,245,405,114,88,293,447),(65,279,246,404,115,87,294,446),(66,278,247,403,116,86,295,445),(67,277,248,402,117,85,296,444),(68,276,249,401,118,112,297,443),(69,275,250,400,119,111,298,442),(70,274,251,399,120,110,299,441),(71,273,252,398,121,109,300,440),(72,272,225,397,122,108,301,439),(73,271,226,396,123,107,302,438),(74,270,227,395,124,106,303,437),(75,269,228,394,125,105,304,436),(76,268,229,393,126,104,305,435),(77,267,230,420,127,103,306,434),(78,266,231,419,128,102,307,433),(79,265,232,418,129,101,308,432),(80,264,233,417,130,100,281,431),(81,263,234,416,131,99,282,430),(82,262,235,415,132,98,283,429),(83,261,236,414,133,97,284,428),(84,260,237,413,134,96,285,427)], [(1,131,34,81),(2,132,35,82),(3,133,36,83),(4,134,37,84),(5,135,38,57),(6,136,39,58),(7,137,40,59),(8,138,41,60),(9,139,42,61),(10,140,43,62),(11,113,44,63),(12,114,45,64),(13,115,46,65),(14,116,47,66),(15,117,48,67),(16,118,49,68),(17,119,50,69),(18,120,51,70),(19,121,52,71),(20,122,53,72),(21,123,54,73),(22,124,55,74),(23,125,56,75),(24,126,29,76),(25,127,30,77),(26,128,31,78),(27,129,32,79),(28,130,33,80),(85,221,277,385),(86,222,278,386),(87,223,279,387),(88,224,280,388),(89,197,253,389),(90,198,254,390),(91,199,255,391),(92,200,256,392),(93,201,257,365),(94,202,258,366),(95,203,259,367),(96,204,260,368),(97,205,261,369),(98,206,262,370),(99,207,263,371),(100,208,264,372),(101,209,265,373),(102,210,266,374),(103,211,267,375),(104,212,268,376),(105,213,269,377),(106,214,270,378),(107,215,271,379),(108,216,272,380),(109,217,273,381),(110,218,274,382),(111,219,275,383),(112,220,276,384),(141,306,189,230),(142,307,190,231),(143,308,191,232),(144,281,192,233),(145,282,193,234),(146,283,194,235),(147,284,195,236),(148,285,196,237),(149,286,169,238),(150,287,170,239),(151,288,171,240),(152,289,172,241),(153,290,173,242),(154,291,174,243),(155,292,175,244),(156,293,176,245),(157,294,177,246),(158,295,178,247),(159,296,179,248),(160,297,180,249),(161,298,181,250),(162,299,182,251),(163,300,183,252),(164,301,184,225),(165,302,185,226),(166,303,186,227),(167,304,187,228),(168,305,188,229),(309,432,345,418),(310,433,346,419),(311,434,347,420),(312,435,348,393),(313,436,349,394),(314,437,350,395),(315,438,351,396),(316,439,352,397),(317,440,353,398),(318,441,354,399),(319,442,355,400),(320,443,356,401),(321,444,357,402),(322,445,358,403),(323,446,359,404),(324,447,360,405),(325,448,361,406),(326,421,362,407),(327,422,363,408),(328,423,364,409),(329,424,337,410),(330,425,338,411),(331,426,339,412),(332,427,340,413),(333,428,341,414),(334,429,342,415),(335,430,343,416),(336,431,344,417)]])
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 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | D28 | C4○D28 | C8.C22 | C7⋊Q16 | D4.9D14 |
| kernel | C28⋊7Q16 | C28⋊C8 | C14.Q16 | C28⋊2Q8 | C2×C7⋊Q16 | 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 | 1 | 2 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of C28⋊7Q16 ►in GL4(𝔽113) generated by
| 96 | 77 | 0 | 0 |
| 36 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 89 | 1 | 0 | 0 |
| 103 | 24 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 31 | 51 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 9 |
| 0 | 0 | 81 | 93 |
G:=sub<GL(4,GF(113))| [96,36,0,0,77,23,0,0,0,0,1,0,0,0,0,1],[89,103,0,0,1,24,0,0,0,0,0,31,0,0,51,51],[1,0,0,0,0,1,0,0,0,0,20,81,0,0,9,93] >;
C28⋊7Q16 in GAP, Magma, Sage, TeX
C_{28}\rtimes_7Q_{16} % in TeX
G:=Group("C28:7Q16"); // GroupNames label
G:=SmallGroup(448,565);
// by ID
G=gap.SmallGroup(448,565);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations