metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.7Q16, Dic14⋊3Q8, C4.7Dic28, C42.41D14, C4⋊C8.8D7, C4.46(Q8×D7), C7⋊2(C4.Q16), (C2×C8).24D14, C14.8(C2×Q16), C56⋊1C4.9C2, (C2×C28).126D4, (C2×C4).137D28, C28.105(C2×Q8), (C4×C28).76C22, (C2×C56).28C22, C28⋊2Q8.11C2, C2.10(C2×Dic28), C28.289(C4○D4), C2.21(C8⋊D14), C14.18(C8⋊C22), (C2×C28).760C23, (C4×Dic14).12C2, C28.44D4.4C2, C22.123(C2×D28), C14.33(C22⋊Q8), C4⋊Dic7.21C22, C4.113(D4⋊2D7), C2.14(D14⋊2Q8), (C2×Dic14).216C22, (C7×C4⋊C8).13C2, (C2×C14).143(C2×D4), (C2×C4).705(C22×D7), SmallGroup(448,384)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.7Q16
G = < a,b,c | a28=b8=1, c2=b4, bab-1=a15, cac-1=a13, cbc-1=a14b-1 >
Subgroups: 484 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C56, Dic14, Dic14, C2×Dic7, C2×C28, C4.Q16, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C2×Dic14, C28.44D4, C56⋊1C4, C7×C4⋊C8, C4×Dic14, C28⋊2Q8, C28.7Q16
Quotients: C1, C2, C22, D4, Q8, C23, D7, Q16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×Q16, C8⋊C22, D28, C22×D7, C4.Q16, Dic28, C2×D28, D4⋊2D7, Q8×D7, D14⋊2Q8, C2×Dic28, C8⋊D14, 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)
(1 235 186 76 158 263 388 136)(2 250 187 63 159 278 389 123)(3 237 188 78 160 265 390 138)(4 252 189 65 161 280 391 125)(5 239 190 80 162 267 392 140)(6 226 191 67 163 254 365 127)(7 241 192 82 164 269 366 114)(8 228 193 69 165 256 367 129)(9 243 194 84 166 271 368 116)(10 230 195 71 167 258 369 131)(11 245 196 58 168 273 370 118)(12 232 169 73 141 260 371 133)(13 247 170 60 142 275 372 120)(14 234 171 75 143 262 373 135)(15 249 172 62 144 277 374 122)(16 236 173 77 145 264 375 137)(17 251 174 64 146 279 376 124)(18 238 175 79 147 266 377 139)(19 225 176 66 148 253 378 126)(20 240 177 81 149 268 379 113)(21 227 178 68 150 255 380 128)(22 242 179 83 151 270 381 115)(23 229 180 70 152 257 382 130)(24 244 181 57 153 272 383 117)(25 231 182 72 154 259 384 132)(26 246 183 59 155 274 385 119)(27 233 184 74 156 261 386 134)(28 248 185 61 157 276 387 121)(29 209 351 414 305 312 434 89)(30 224 352 401 306 327 435 104)(31 211 353 416 307 314 436 91)(32 198 354 403 308 329 437 106)(33 213 355 418 281 316 438 93)(34 200 356 405 282 331 439 108)(35 215 357 420 283 318 440 95)(36 202 358 407 284 333 441 110)(37 217 359 394 285 320 442 97)(38 204 360 409 286 335 443 112)(39 219 361 396 287 322 444 99)(40 206 362 411 288 309 445 86)(41 221 363 398 289 324 446 101)(42 208 364 413 290 311 447 88)(43 223 337 400 291 326 448 103)(44 210 338 415 292 313 421 90)(45 197 339 402 293 328 422 105)(46 212 340 417 294 315 423 92)(47 199 341 404 295 330 424 107)(48 214 342 419 296 317 425 94)(49 201 343 406 297 332 426 109)(50 216 344 393 298 319 427 96)(51 203 345 408 299 334 428 111)(52 218 346 395 300 321 429 98)(53 205 347 410 301 336 430 85)(54 220 348 397 302 323 431 100)(55 207 349 412 303 310 432 87)(56 222 350 399 304 325 433 102)
(1 32 158 308)(2 45 159 293)(3 30 160 306)(4 43 161 291)(5 56 162 304)(6 41 163 289)(7 54 164 302)(8 39 165 287)(9 52 166 300)(10 37 167 285)(11 50 168 298)(12 35 141 283)(13 48 142 296)(14 33 143 281)(15 46 144 294)(16 31 145 307)(17 44 146 292)(18 29 147 305)(19 42 148 290)(20 55 149 303)(21 40 150 288)(22 53 151 301)(23 38 152 286)(24 51 153 299)(25 36 154 284)(26 49 155 297)(27 34 156 282)(28 47 157 295)(57 320 117 217)(58 333 118 202)(59 318 119 215)(60 331 120 200)(61 316 121 213)(62 329 122 198)(63 314 123 211)(64 327 124 224)(65 312 125 209)(66 325 126 222)(67 310 127 207)(68 323 128 220)(69 336 129 205)(70 321 130 218)(71 334 131 203)(72 319 132 216)(73 332 133 201)(74 317 134 214)(75 330 135 199)(76 315 136 212)(77 328 137 197)(78 313 138 210)(79 326 139 223)(80 311 140 208)(81 324 113 221)(82 309 114 206)(83 322 115 219)(84 335 116 204)(85 256 410 228)(86 269 411 241)(87 254 412 226)(88 267 413 239)(89 280 414 252)(90 265 415 237)(91 278 416 250)(92 263 417 235)(93 276 418 248)(94 261 419 233)(95 274 420 246)(96 259 393 231)(97 272 394 244)(98 257 395 229)(99 270 396 242)(100 255 397 227)(101 268 398 240)(102 253 399 225)(103 266 400 238)(104 279 401 251)(105 264 402 236)(106 277 403 249)(107 262 404 234)(108 275 405 247)(109 260 406 232)(110 273 407 245)(111 258 408 230)(112 271 409 243)(169 440 371 357)(170 425 372 342)(171 438 373 355)(172 423 374 340)(173 436 375 353)(174 421 376 338)(175 434 377 351)(176 447 378 364)(177 432 379 349)(178 445 380 362)(179 430 381 347)(180 443 382 360)(181 428 383 345)(182 441 384 358)(183 426 385 343)(184 439 386 356)(185 424 387 341)(186 437 388 354)(187 422 389 339)(188 435 390 352)(189 448 391 337)(190 433 392 350)(191 446 365 363)(192 431 366 348)(193 444 367 361)(194 429 368 346)(195 442 369 359)(196 427 370 344)
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), (1,235,186,76,158,263,388,136)(2,250,187,63,159,278,389,123)(3,237,188,78,160,265,390,138)(4,252,189,65,161,280,391,125)(5,239,190,80,162,267,392,140)(6,226,191,67,163,254,365,127)(7,241,192,82,164,269,366,114)(8,228,193,69,165,256,367,129)(9,243,194,84,166,271,368,116)(10,230,195,71,167,258,369,131)(11,245,196,58,168,273,370,118)(12,232,169,73,141,260,371,133)(13,247,170,60,142,275,372,120)(14,234,171,75,143,262,373,135)(15,249,172,62,144,277,374,122)(16,236,173,77,145,264,375,137)(17,251,174,64,146,279,376,124)(18,238,175,79,147,266,377,139)(19,225,176,66,148,253,378,126)(20,240,177,81,149,268,379,113)(21,227,178,68,150,255,380,128)(22,242,179,83,151,270,381,115)(23,229,180,70,152,257,382,130)(24,244,181,57,153,272,383,117)(25,231,182,72,154,259,384,132)(26,246,183,59,155,274,385,119)(27,233,184,74,156,261,386,134)(28,248,185,61,157,276,387,121)(29,209,351,414,305,312,434,89)(30,224,352,401,306,327,435,104)(31,211,353,416,307,314,436,91)(32,198,354,403,308,329,437,106)(33,213,355,418,281,316,438,93)(34,200,356,405,282,331,439,108)(35,215,357,420,283,318,440,95)(36,202,358,407,284,333,441,110)(37,217,359,394,285,320,442,97)(38,204,360,409,286,335,443,112)(39,219,361,396,287,322,444,99)(40,206,362,411,288,309,445,86)(41,221,363,398,289,324,446,101)(42,208,364,413,290,311,447,88)(43,223,337,400,291,326,448,103)(44,210,338,415,292,313,421,90)(45,197,339,402,293,328,422,105)(46,212,340,417,294,315,423,92)(47,199,341,404,295,330,424,107)(48,214,342,419,296,317,425,94)(49,201,343,406,297,332,426,109)(50,216,344,393,298,319,427,96)(51,203,345,408,299,334,428,111)(52,218,346,395,300,321,429,98)(53,205,347,410,301,336,430,85)(54,220,348,397,302,323,431,100)(55,207,349,412,303,310,432,87)(56,222,350,399,304,325,433,102), (1,32,158,308)(2,45,159,293)(3,30,160,306)(4,43,161,291)(5,56,162,304)(6,41,163,289)(7,54,164,302)(8,39,165,287)(9,52,166,300)(10,37,167,285)(11,50,168,298)(12,35,141,283)(13,48,142,296)(14,33,143,281)(15,46,144,294)(16,31,145,307)(17,44,146,292)(18,29,147,305)(19,42,148,290)(20,55,149,303)(21,40,150,288)(22,53,151,301)(23,38,152,286)(24,51,153,299)(25,36,154,284)(26,49,155,297)(27,34,156,282)(28,47,157,295)(57,320,117,217)(58,333,118,202)(59,318,119,215)(60,331,120,200)(61,316,121,213)(62,329,122,198)(63,314,123,211)(64,327,124,224)(65,312,125,209)(66,325,126,222)(67,310,127,207)(68,323,128,220)(69,336,129,205)(70,321,130,218)(71,334,131,203)(72,319,132,216)(73,332,133,201)(74,317,134,214)(75,330,135,199)(76,315,136,212)(77,328,137,197)(78,313,138,210)(79,326,139,223)(80,311,140,208)(81,324,113,221)(82,309,114,206)(83,322,115,219)(84,335,116,204)(85,256,410,228)(86,269,411,241)(87,254,412,226)(88,267,413,239)(89,280,414,252)(90,265,415,237)(91,278,416,250)(92,263,417,235)(93,276,418,248)(94,261,419,233)(95,274,420,246)(96,259,393,231)(97,272,394,244)(98,257,395,229)(99,270,396,242)(100,255,397,227)(101,268,398,240)(102,253,399,225)(103,266,400,238)(104,279,401,251)(105,264,402,236)(106,277,403,249)(107,262,404,234)(108,275,405,247)(109,260,406,232)(110,273,407,245)(111,258,408,230)(112,271,409,243)(169,440,371,357)(170,425,372,342)(171,438,373,355)(172,423,374,340)(173,436,375,353)(174,421,376,338)(175,434,377,351)(176,447,378,364)(177,432,379,349)(178,445,380,362)(179,430,381,347)(180,443,382,360)(181,428,383,345)(182,441,384,358)(183,426,385,343)(184,439,386,356)(185,424,387,341)(186,437,388,354)(187,422,389,339)(188,435,390,352)(189,448,391,337)(190,433,392,350)(191,446,365,363)(192,431,366,348)(193,444,367,361)(194,429,368,346)(195,442,369,359)(196,427,370,344)>;
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), (1,235,186,76,158,263,388,136)(2,250,187,63,159,278,389,123)(3,237,188,78,160,265,390,138)(4,252,189,65,161,280,391,125)(5,239,190,80,162,267,392,140)(6,226,191,67,163,254,365,127)(7,241,192,82,164,269,366,114)(8,228,193,69,165,256,367,129)(9,243,194,84,166,271,368,116)(10,230,195,71,167,258,369,131)(11,245,196,58,168,273,370,118)(12,232,169,73,141,260,371,133)(13,247,170,60,142,275,372,120)(14,234,171,75,143,262,373,135)(15,249,172,62,144,277,374,122)(16,236,173,77,145,264,375,137)(17,251,174,64,146,279,376,124)(18,238,175,79,147,266,377,139)(19,225,176,66,148,253,378,126)(20,240,177,81,149,268,379,113)(21,227,178,68,150,255,380,128)(22,242,179,83,151,270,381,115)(23,229,180,70,152,257,382,130)(24,244,181,57,153,272,383,117)(25,231,182,72,154,259,384,132)(26,246,183,59,155,274,385,119)(27,233,184,74,156,261,386,134)(28,248,185,61,157,276,387,121)(29,209,351,414,305,312,434,89)(30,224,352,401,306,327,435,104)(31,211,353,416,307,314,436,91)(32,198,354,403,308,329,437,106)(33,213,355,418,281,316,438,93)(34,200,356,405,282,331,439,108)(35,215,357,420,283,318,440,95)(36,202,358,407,284,333,441,110)(37,217,359,394,285,320,442,97)(38,204,360,409,286,335,443,112)(39,219,361,396,287,322,444,99)(40,206,362,411,288,309,445,86)(41,221,363,398,289,324,446,101)(42,208,364,413,290,311,447,88)(43,223,337,400,291,326,448,103)(44,210,338,415,292,313,421,90)(45,197,339,402,293,328,422,105)(46,212,340,417,294,315,423,92)(47,199,341,404,295,330,424,107)(48,214,342,419,296,317,425,94)(49,201,343,406,297,332,426,109)(50,216,344,393,298,319,427,96)(51,203,345,408,299,334,428,111)(52,218,346,395,300,321,429,98)(53,205,347,410,301,336,430,85)(54,220,348,397,302,323,431,100)(55,207,349,412,303,310,432,87)(56,222,350,399,304,325,433,102), (1,32,158,308)(2,45,159,293)(3,30,160,306)(4,43,161,291)(5,56,162,304)(6,41,163,289)(7,54,164,302)(8,39,165,287)(9,52,166,300)(10,37,167,285)(11,50,168,298)(12,35,141,283)(13,48,142,296)(14,33,143,281)(15,46,144,294)(16,31,145,307)(17,44,146,292)(18,29,147,305)(19,42,148,290)(20,55,149,303)(21,40,150,288)(22,53,151,301)(23,38,152,286)(24,51,153,299)(25,36,154,284)(26,49,155,297)(27,34,156,282)(28,47,157,295)(57,320,117,217)(58,333,118,202)(59,318,119,215)(60,331,120,200)(61,316,121,213)(62,329,122,198)(63,314,123,211)(64,327,124,224)(65,312,125,209)(66,325,126,222)(67,310,127,207)(68,323,128,220)(69,336,129,205)(70,321,130,218)(71,334,131,203)(72,319,132,216)(73,332,133,201)(74,317,134,214)(75,330,135,199)(76,315,136,212)(77,328,137,197)(78,313,138,210)(79,326,139,223)(80,311,140,208)(81,324,113,221)(82,309,114,206)(83,322,115,219)(84,335,116,204)(85,256,410,228)(86,269,411,241)(87,254,412,226)(88,267,413,239)(89,280,414,252)(90,265,415,237)(91,278,416,250)(92,263,417,235)(93,276,418,248)(94,261,419,233)(95,274,420,246)(96,259,393,231)(97,272,394,244)(98,257,395,229)(99,270,396,242)(100,255,397,227)(101,268,398,240)(102,253,399,225)(103,266,400,238)(104,279,401,251)(105,264,402,236)(106,277,403,249)(107,262,404,234)(108,275,405,247)(109,260,406,232)(110,273,407,245)(111,258,408,230)(112,271,409,243)(169,440,371,357)(170,425,372,342)(171,438,373,355)(172,423,374,340)(173,436,375,353)(174,421,376,338)(175,434,377,351)(176,447,378,364)(177,432,379,349)(178,445,380,362)(179,430,381,347)(180,443,382,360)(181,428,383,345)(182,441,384,358)(183,426,385,343)(184,439,386,356)(185,424,387,341)(186,437,388,354)(187,422,389,339)(188,435,390,352)(189,448,391,337)(190,433,392,350)(191,446,365,363)(192,431,366,348)(193,444,367,361)(194,429,368,346)(195,442,369,359)(196,427,370,344) );
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)], [(1,235,186,76,158,263,388,136),(2,250,187,63,159,278,389,123),(3,237,188,78,160,265,390,138),(4,252,189,65,161,280,391,125),(5,239,190,80,162,267,392,140),(6,226,191,67,163,254,365,127),(7,241,192,82,164,269,366,114),(8,228,193,69,165,256,367,129),(9,243,194,84,166,271,368,116),(10,230,195,71,167,258,369,131),(11,245,196,58,168,273,370,118),(12,232,169,73,141,260,371,133),(13,247,170,60,142,275,372,120),(14,234,171,75,143,262,373,135),(15,249,172,62,144,277,374,122),(16,236,173,77,145,264,375,137),(17,251,174,64,146,279,376,124),(18,238,175,79,147,266,377,139),(19,225,176,66,148,253,378,126),(20,240,177,81,149,268,379,113),(21,227,178,68,150,255,380,128),(22,242,179,83,151,270,381,115),(23,229,180,70,152,257,382,130),(24,244,181,57,153,272,383,117),(25,231,182,72,154,259,384,132),(26,246,183,59,155,274,385,119),(27,233,184,74,156,261,386,134),(28,248,185,61,157,276,387,121),(29,209,351,414,305,312,434,89),(30,224,352,401,306,327,435,104),(31,211,353,416,307,314,436,91),(32,198,354,403,308,329,437,106),(33,213,355,418,281,316,438,93),(34,200,356,405,282,331,439,108),(35,215,357,420,283,318,440,95),(36,202,358,407,284,333,441,110),(37,217,359,394,285,320,442,97),(38,204,360,409,286,335,443,112),(39,219,361,396,287,322,444,99),(40,206,362,411,288,309,445,86),(41,221,363,398,289,324,446,101),(42,208,364,413,290,311,447,88),(43,223,337,400,291,326,448,103),(44,210,338,415,292,313,421,90),(45,197,339,402,293,328,422,105),(46,212,340,417,294,315,423,92),(47,199,341,404,295,330,424,107),(48,214,342,419,296,317,425,94),(49,201,343,406,297,332,426,109),(50,216,344,393,298,319,427,96),(51,203,345,408,299,334,428,111),(52,218,346,395,300,321,429,98),(53,205,347,410,301,336,430,85),(54,220,348,397,302,323,431,100),(55,207,349,412,303,310,432,87),(56,222,350,399,304,325,433,102)], [(1,32,158,308),(2,45,159,293),(3,30,160,306),(4,43,161,291),(5,56,162,304),(6,41,163,289),(7,54,164,302),(8,39,165,287),(9,52,166,300),(10,37,167,285),(11,50,168,298),(12,35,141,283),(13,48,142,296),(14,33,143,281),(15,46,144,294),(16,31,145,307),(17,44,146,292),(18,29,147,305),(19,42,148,290),(20,55,149,303),(21,40,150,288),(22,53,151,301),(23,38,152,286),(24,51,153,299),(25,36,154,284),(26,49,155,297),(27,34,156,282),(28,47,157,295),(57,320,117,217),(58,333,118,202),(59,318,119,215),(60,331,120,200),(61,316,121,213),(62,329,122,198),(63,314,123,211),(64,327,124,224),(65,312,125,209),(66,325,126,222),(67,310,127,207),(68,323,128,220),(69,336,129,205),(70,321,130,218),(71,334,131,203),(72,319,132,216),(73,332,133,201),(74,317,134,214),(75,330,135,199),(76,315,136,212),(77,328,137,197),(78,313,138,210),(79,326,139,223),(80,311,140,208),(81,324,113,221),(82,309,114,206),(83,322,115,219),(84,335,116,204),(85,256,410,228),(86,269,411,241),(87,254,412,226),(88,267,413,239),(89,280,414,252),(90,265,415,237),(91,278,416,250),(92,263,417,235),(93,276,418,248),(94,261,419,233),(95,274,420,246),(96,259,393,231),(97,272,394,244),(98,257,395,229),(99,270,396,242),(100,255,397,227),(101,268,398,240),(102,253,399,225),(103,266,400,238),(104,279,401,251),(105,264,402,236),(106,277,403,249),(107,262,404,234),(108,275,405,247),(109,260,406,232),(110,273,407,245),(111,258,408,230),(112,271,409,243),(169,440,371,357),(170,425,372,342),(171,438,373,355),(172,423,374,340),(173,436,375,353),(174,421,376,338),(175,434,377,351),(176,447,378,364),(177,432,379,349),(178,445,380,362),(179,430,381,347),(180,443,382,360),(181,428,383,345),(182,441,384,358),(183,426,385,343),(184,439,386,356),(185,424,387,341),(186,437,388,354),(187,422,389,339),(188,435,390,352),(189,448,391,337),(190,433,392,350),(191,446,365,363),(192,431,366,348),(193,444,367,361),(194,429,368,346),(195,442,369,359),(196,427,370,344)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | + | + | + | - | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | D28 | Dic28 | C8⋊C22 | D4⋊2D7 | Q8×D7 | C8⋊D14 |
| kernel | C28.7Q16 | C28.44D4 | C56⋊1C4 | C7×C4⋊C8 | C4×Dic14 | C28⋊2Q8 | Dic14 | C2×C28 | C4⋊C8 | C28 | C28 | C42 | C2×C8 | C2×C4 | C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 24 | 1 | 3 | 3 | 6 |
Matrix representation of C28.7Q16 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 89 | 0 | 0 |
| 0 | 0 | 24 | 103 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 26 |
| 0 | 0 | 0 | 0 | 26 | 112 |
| 0 | 62 | 0 | 0 | 0 | 0 |
| 82 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 95 | 55 | 0 | 0 | 0 | 0 |
| 66 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 91 | 0 | 0 |
| 0 | 0 | 33 | 106 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,89,103,0,0,0,0,0,0,1,26,0,0,0,0,26,112],[0,82,0,0,0,0,62,62,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[95,66,0,0,0,0,55,18,0,0,0,0,0,0,7,33,0,0,0,0,91,106,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;
C28.7Q16 in GAP, Magma, Sage, TeX
C_{28}._7Q_{16} % in TeX
G:=Group("C28.7Q16"); // GroupNames label
G:=SmallGroup(448,384);
// by ID
G=gap.SmallGroup(448,384);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,254,219,310,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=b^4,b*a*b^-1=a^15,c*a*c^-1=a^13,c*b*c^-1=a^14*b^-1>;
// generators/relations