metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.9C42, C22.11D56, C23.54D28, C22.4Dic28, (C2×C56)⋊6C4, C4⋊Dic7⋊6C4, (C2×C8)⋊4Dic7, (C2×C14).17D8, C28.30(C4⋊C4), (C2×C28).51Q8, (C2×C14).8Q16, C4.9(C4×Dic7), (C22×C8).3D7, (C2×C28).467D4, (C22×C56).2C2, C2.2(C8⋊Dic7), C2.2(C56⋊1C4), C14.7(C4.Q8), C7⋊2(C22.4Q16), (C2×C14).19SD16, (C2×C4).43Dic14, C14.10(C2.D8), C2.2(C2.D56), C28.85(C22⋊C4), C4.23(Dic7⋊C4), C14.9(Q8⋊C4), (C22×C4).413D14, (C22×C14).133D4, C22.9(C56⋊C2), C4.15(C23.D7), C14.20(D4⋊C4), C22.40(D14⋊C4), C2.2(C28.44D4), C22.18(C4⋊Dic7), (C22×C28).512C22, C14.11(C2.C42), C2.11(C14.C42), (C2×C4).99(C4×D7), (C2×C4⋊Dic7).1C2, (C2×C14).38(C4⋊C4), (C2×C28).293(C2×C4), (C2×C4).73(C2×Dic7), (C2×C4).233(C7⋊D4), (C2×C14).52(C22⋊C4), SmallGroup(448,108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.9C42
G = < a,b,c | a28=b4=1, c4=a14, bab-1=a-1, ac=ca, cbc-1=a7b >
Subgroups: 516 in 114 conjugacy classes, 67 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C4⋊C4, C22×C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C22.4Q16, C4⋊Dic7, C4⋊Dic7, C2×C56, C2×C56, C22×Dic7, C22×C28, C2×C4⋊Dic7, C22×C56, C28.9C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, Dic7, D14, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C22.4Q16, C56⋊C2, D56, Dic28, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C28.44D4, C8⋊Dic7, C56⋊1C4, C2.D56, C14.C42, C28.9C42
►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)
(1 189 83 354)(2 188 84 353)(3 187 57 352)(4 186 58 351)(5 185 59 350)(6 184 60 349)(7 183 61 348)(8 182 62 347)(9 181 63 346)(10 180 64 345)(11 179 65 344)(12 178 66 343)(13 177 67 342)(14 176 68 341)(15 175 69 340)(16 174 70 339)(17 173 71 338)(18 172 72 337)(19 171 73 364)(20 170 74 363)(21 169 75 362)(22 196 76 361)(23 195 77 360)(24 194 78 359)(25 193 79 358)(26 192 80 357)(27 191 81 356)(28 190 82 355)(29 370 210 247)(30 369 211 246)(31 368 212 245)(32 367 213 244)(33 366 214 243)(34 365 215 242)(35 392 216 241)(36 391 217 240)(37 390 218 239)(38 389 219 238)(39 388 220 237)(40 387 221 236)(41 386 222 235)(42 385 223 234)(43 384 224 233)(44 383 197 232)(45 382 198 231)(46 381 199 230)(47 380 200 229)(48 379 201 228)(49 378 202 227)(50 377 203 226)(51 376 204 225)(52 375 205 252)(53 374 206 251)(54 373 207 250)(55 372 208 249)(56 371 209 248)(85 412 160 265)(86 411 161 264)(87 410 162 263)(88 409 163 262)(89 408 164 261)(90 407 165 260)(91 406 166 259)(92 405 167 258)(93 404 168 257)(94 403 141 256)(95 402 142 255)(96 401 143 254)(97 400 144 253)(98 399 145 280)(99 398 146 279)(100 397 147 278)(101 396 148 277)(102 395 149 276)(103 394 150 275)(104 393 151 274)(105 420 152 273)(106 419 153 272)(107 418 154 271)(108 417 155 270)(109 416 156 269)(110 415 157 268)(111 414 158 267)(112 413 159 266)(113 307 443 327)(114 306 444 326)(115 305 445 325)(116 304 446 324)(117 303 447 323)(118 302 448 322)(119 301 421 321)(120 300 422 320)(121 299 423 319)(122 298 424 318)(123 297 425 317)(124 296 426 316)(125 295 427 315)(126 294 428 314)(127 293 429 313)(128 292 430 312)(129 291 431 311)(130 290 432 310)(131 289 433 309)(132 288 434 336)(133 287 435 335)(134 286 436 334)(135 285 437 333)(136 284 438 332)(137 283 439 331)(138 282 440 330)(139 281 441 329)(140 308 442 328)
(1 133 416 223 15 119 402 209)(2 134 417 224 16 120 403 210)(3 135 418 197 17 121 404 211)(4 136 419 198 18 122 405 212)(5 137 420 199 19 123 406 213)(6 138 393 200 20 124 407 214)(7 139 394 201 21 125 408 215)(8 140 395 202 22 126 409 216)(9 113 396 203 23 127 410 217)(10 114 397 204 24 128 411 218)(11 115 398 205 25 129 412 219)(12 116 399 206 26 130 413 220)(13 117 400 207 27 131 414 221)(14 118 401 208 28 132 415 222)(29 84 436 270 43 70 422 256)(30 57 437 271 44 71 423 257)(31 58 438 272 45 72 424 258)(32 59 439 273 46 73 425 259)(33 60 440 274 47 74 426 260)(34 61 441 275 48 75 427 261)(35 62 442 276 49 76 428 262)(36 63 443 277 50 77 429 263)(37 64 444 278 51 78 430 264)(38 65 445 279 52 79 431 265)(39 66 446 280 53 80 432 266)(40 67 447 253 54 81 433 267)(41 68 448 254 55 82 434 268)(42 69 421 255 56 83 435 269)(85 368 358 318 99 382 344 332)(86 369 359 319 100 383 345 333)(87 370 360 320 101 384 346 334)(88 371 361 321 102 385 347 335)(89 372 362 322 103 386 348 336)(90 373 363 323 104 387 349 309)(91 374 364 324 105 388 350 310)(92 375 337 325 106 389 351 311)(93 376 338 326 107 390 352 312)(94 377 339 327 108 391 353 313)(95 378 340 328 109 392 354 314)(96 379 341 329 110 365 355 315)(97 380 342 330 111 366 356 316)(98 381 343 331 112 367 357 317)(141 226 174 307 155 240 188 293)(142 227 175 308 156 241 189 294)(143 228 176 281 157 242 190 295)(144 229 177 282 158 243 191 296)(145 230 178 283 159 244 192 297)(146 231 179 284 160 245 193 298)(147 232 180 285 161 246 194 299)(148 233 181 286 162 247 195 300)(149 234 182 287 163 248 196 301)(150 235 183 288 164 249 169 302)(151 236 184 289 165 250 170 303)(152 237 185 290 166 251 171 304)(153 238 186 291 167 252 172 305)(154 239 187 292 168 225 173 306)
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), (1,189,83,354)(2,188,84,353)(3,187,57,352)(4,186,58,351)(5,185,59,350)(6,184,60,349)(7,183,61,348)(8,182,62,347)(9,181,63,346)(10,180,64,345)(11,179,65,344)(12,178,66,343)(13,177,67,342)(14,176,68,341)(15,175,69,340)(16,174,70,339)(17,173,71,338)(18,172,72,337)(19,171,73,364)(20,170,74,363)(21,169,75,362)(22,196,76,361)(23,195,77,360)(24,194,78,359)(25,193,79,358)(26,192,80,357)(27,191,81,356)(28,190,82,355)(29,370,210,247)(30,369,211,246)(31,368,212,245)(32,367,213,244)(33,366,214,243)(34,365,215,242)(35,392,216,241)(36,391,217,240)(37,390,218,239)(38,389,219,238)(39,388,220,237)(40,387,221,236)(41,386,222,235)(42,385,223,234)(43,384,224,233)(44,383,197,232)(45,382,198,231)(46,381,199,230)(47,380,200,229)(48,379,201,228)(49,378,202,227)(50,377,203,226)(51,376,204,225)(52,375,205,252)(53,374,206,251)(54,373,207,250)(55,372,208,249)(56,371,209,248)(85,412,160,265)(86,411,161,264)(87,410,162,263)(88,409,163,262)(89,408,164,261)(90,407,165,260)(91,406,166,259)(92,405,167,258)(93,404,168,257)(94,403,141,256)(95,402,142,255)(96,401,143,254)(97,400,144,253)(98,399,145,280)(99,398,146,279)(100,397,147,278)(101,396,148,277)(102,395,149,276)(103,394,150,275)(104,393,151,274)(105,420,152,273)(106,419,153,272)(107,418,154,271)(108,417,155,270)(109,416,156,269)(110,415,157,268)(111,414,158,267)(112,413,159,266)(113,307,443,327)(114,306,444,326)(115,305,445,325)(116,304,446,324)(117,303,447,323)(118,302,448,322)(119,301,421,321)(120,300,422,320)(121,299,423,319)(122,298,424,318)(123,297,425,317)(124,296,426,316)(125,295,427,315)(126,294,428,314)(127,293,429,313)(128,292,430,312)(129,291,431,311)(130,290,432,310)(131,289,433,309)(132,288,434,336)(133,287,435,335)(134,286,436,334)(135,285,437,333)(136,284,438,332)(137,283,439,331)(138,282,440,330)(139,281,441,329)(140,308,442,328), (1,133,416,223,15,119,402,209)(2,134,417,224,16,120,403,210)(3,135,418,197,17,121,404,211)(4,136,419,198,18,122,405,212)(5,137,420,199,19,123,406,213)(6,138,393,200,20,124,407,214)(7,139,394,201,21,125,408,215)(8,140,395,202,22,126,409,216)(9,113,396,203,23,127,410,217)(10,114,397,204,24,128,411,218)(11,115,398,205,25,129,412,219)(12,116,399,206,26,130,413,220)(13,117,400,207,27,131,414,221)(14,118,401,208,28,132,415,222)(29,84,436,270,43,70,422,256)(30,57,437,271,44,71,423,257)(31,58,438,272,45,72,424,258)(32,59,439,273,46,73,425,259)(33,60,440,274,47,74,426,260)(34,61,441,275,48,75,427,261)(35,62,442,276,49,76,428,262)(36,63,443,277,50,77,429,263)(37,64,444,278,51,78,430,264)(38,65,445,279,52,79,431,265)(39,66,446,280,53,80,432,266)(40,67,447,253,54,81,433,267)(41,68,448,254,55,82,434,268)(42,69,421,255,56,83,435,269)(85,368,358,318,99,382,344,332)(86,369,359,319,100,383,345,333)(87,370,360,320,101,384,346,334)(88,371,361,321,102,385,347,335)(89,372,362,322,103,386,348,336)(90,373,363,323,104,387,349,309)(91,374,364,324,105,388,350,310)(92,375,337,325,106,389,351,311)(93,376,338,326,107,390,352,312)(94,377,339,327,108,391,353,313)(95,378,340,328,109,392,354,314)(96,379,341,329,110,365,355,315)(97,380,342,330,111,366,356,316)(98,381,343,331,112,367,357,317)(141,226,174,307,155,240,188,293)(142,227,175,308,156,241,189,294)(143,228,176,281,157,242,190,295)(144,229,177,282,158,243,191,296)(145,230,178,283,159,244,192,297)(146,231,179,284,160,245,193,298)(147,232,180,285,161,246,194,299)(148,233,181,286,162,247,195,300)(149,234,182,287,163,248,196,301)(150,235,183,288,164,249,169,302)(151,236,184,289,165,250,170,303)(152,237,185,290,166,251,171,304)(153,238,186,291,167,252,172,305)(154,239,187,292,168,225,173,306)>;
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), (1,189,83,354)(2,188,84,353)(3,187,57,352)(4,186,58,351)(5,185,59,350)(6,184,60,349)(7,183,61,348)(8,182,62,347)(9,181,63,346)(10,180,64,345)(11,179,65,344)(12,178,66,343)(13,177,67,342)(14,176,68,341)(15,175,69,340)(16,174,70,339)(17,173,71,338)(18,172,72,337)(19,171,73,364)(20,170,74,363)(21,169,75,362)(22,196,76,361)(23,195,77,360)(24,194,78,359)(25,193,79,358)(26,192,80,357)(27,191,81,356)(28,190,82,355)(29,370,210,247)(30,369,211,246)(31,368,212,245)(32,367,213,244)(33,366,214,243)(34,365,215,242)(35,392,216,241)(36,391,217,240)(37,390,218,239)(38,389,219,238)(39,388,220,237)(40,387,221,236)(41,386,222,235)(42,385,223,234)(43,384,224,233)(44,383,197,232)(45,382,198,231)(46,381,199,230)(47,380,200,229)(48,379,201,228)(49,378,202,227)(50,377,203,226)(51,376,204,225)(52,375,205,252)(53,374,206,251)(54,373,207,250)(55,372,208,249)(56,371,209,248)(85,412,160,265)(86,411,161,264)(87,410,162,263)(88,409,163,262)(89,408,164,261)(90,407,165,260)(91,406,166,259)(92,405,167,258)(93,404,168,257)(94,403,141,256)(95,402,142,255)(96,401,143,254)(97,400,144,253)(98,399,145,280)(99,398,146,279)(100,397,147,278)(101,396,148,277)(102,395,149,276)(103,394,150,275)(104,393,151,274)(105,420,152,273)(106,419,153,272)(107,418,154,271)(108,417,155,270)(109,416,156,269)(110,415,157,268)(111,414,158,267)(112,413,159,266)(113,307,443,327)(114,306,444,326)(115,305,445,325)(116,304,446,324)(117,303,447,323)(118,302,448,322)(119,301,421,321)(120,300,422,320)(121,299,423,319)(122,298,424,318)(123,297,425,317)(124,296,426,316)(125,295,427,315)(126,294,428,314)(127,293,429,313)(128,292,430,312)(129,291,431,311)(130,290,432,310)(131,289,433,309)(132,288,434,336)(133,287,435,335)(134,286,436,334)(135,285,437,333)(136,284,438,332)(137,283,439,331)(138,282,440,330)(139,281,441,329)(140,308,442,328), (1,133,416,223,15,119,402,209)(2,134,417,224,16,120,403,210)(3,135,418,197,17,121,404,211)(4,136,419,198,18,122,405,212)(5,137,420,199,19,123,406,213)(6,138,393,200,20,124,407,214)(7,139,394,201,21,125,408,215)(8,140,395,202,22,126,409,216)(9,113,396,203,23,127,410,217)(10,114,397,204,24,128,411,218)(11,115,398,205,25,129,412,219)(12,116,399,206,26,130,413,220)(13,117,400,207,27,131,414,221)(14,118,401,208,28,132,415,222)(29,84,436,270,43,70,422,256)(30,57,437,271,44,71,423,257)(31,58,438,272,45,72,424,258)(32,59,439,273,46,73,425,259)(33,60,440,274,47,74,426,260)(34,61,441,275,48,75,427,261)(35,62,442,276,49,76,428,262)(36,63,443,277,50,77,429,263)(37,64,444,278,51,78,430,264)(38,65,445,279,52,79,431,265)(39,66,446,280,53,80,432,266)(40,67,447,253,54,81,433,267)(41,68,448,254,55,82,434,268)(42,69,421,255,56,83,435,269)(85,368,358,318,99,382,344,332)(86,369,359,319,100,383,345,333)(87,370,360,320,101,384,346,334)(88,371,361,321,102,385,347,335)(89,372,362,322,103,386,348,336)(90,373,363,323,104,387,349,309)(91,374,364,324,105,388,350,310)(92,375,337,325,106,389,351,311)(93,376,338,326,107,390,352,312)(94,377,339,327,108,391,353,313)(95,378,340,328,109,392,354,314)(96,379,341,329,110,365,355,315)(97,380,342,330,111,366,356,316)(98,381,343,331,112,367,357,317)(141,226,174,307,155,240,188,293)(142,227,175,308,156,241,189,294)(143,228,176,281,157,242,190,295)(144,229,177,282,158,243,191,296)(145,230,178,283,159,244,192,297)(146,231,179,284,160,245,193,298)(147,232,180,285,161,246,194,299)(148,233,181,286,162,247,195,300)(149,234,182,287,163,248,196,301)(150,235,183,288,164,249,169,302)(151,236,184,289,165,250,170,303)(152,237,185,290,166,251,171,304)(153,238,186,291,167,252,172,305)(154,239,187,292,168,225,173,306) );
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)], [(1,189,83,354),(2,188,84,353),(3,187,57,352),(4,186,58,351),(5,185,59,350),(6,184,60,349),(7,183,61,348),(8,182,62,347),(9,181,63,346),(10,180,64,345),(11,179,65,344),(12,178,66,343),(13,177,67,342),(14,176,68,341),(15,175,69,340),(16,174,70,339),(17,173,71,338),(18,172,72,337),(19,171,73,364),(20,170,74,363),(21,169,75,362),(22,196,76,361),(23,195,77,360),(24,194,78,359),(25,193,79,358),(26,192,80,357),(27,191,81,356),(28,190,82,355),(29,370,210,247),(30,369,211,246),(31,368,212,245),(32,367,213,244),(33,366,214,243),(34,365,215,242),(35,392,216,241),(36,391,217,240),(37,390,218,239),(38,389,219,238),(39,388,220,237),(40,387,221,236),(41,386,222,235),(42,385,223,234),(43,384,224,233),(44,383,197,232),(45,382,198,231),(46,381,199,230),(47,380,200,229),(48,379,201,228),(49,378,202,227),(50,377,203,226),(51,376,204,225),(52,375,205,252),(53,374,206,251),(54,373,207,250),(55,372,208,249),(56,371,209,248),(85,412,160,265),(86,411,161,264),(87,410,162,263),(88,409,163,262),(89,408,164,261),(90,407,165,260),(91,406,166,259),(92,405,167,258),(93,404,168,257),(94,403,141,256),(95,402,142,255),(96,401,143,254),(97,400,144,253),(98,399,145,280),(99,398,146,279),(100,397,147,278),(101,396,148,277),(102,395,149,276),(103,394,150,275),(104,393,151,274),(105,420,152,273),(106,419,153,272),(107,418,154,271),(108,417,155,270),(109,416,156,269),(110,415,157,268),(111,414,158,267),(112,413,159,266),(113,307,443,327),(114,306,444,326),(115,305,445,325),(116,304,446,324),(117,303,447,323),(118,302,448,322),(119,301,421,321),(120,300,422,320),(121,299,423,319),(122,298,424,318),(123,297,425,317),(124,296,426,316),(125,295,427,315),(126,294,428,314),(127,293,429,313),(128,292,430,312),(129,291,431,311),(130,290,432,310),(131,289,433,309),(132,288,434,336),(133,287,435,335),(134,286,436,334),(135,285,437,333),(136,284,438,332),(137,283,439,331),(138,282,440,330),(139,281,441,329),(140,308,442,328)], [(1,133,416,223,15,119,402,209),(2,134,417,224,16,120,403,210),(3,135,418,197,17,121,404,211),(4,136,419,198,18,122,405,212),(5,137,420,199,19,123,406,213),(6,138,393,200,20,124,407,214),(7,139,394,201,21,125,408,215),(8,140,395,202,22,126,409,216),(9,113,396,203,23,127,410,217),(10,114,397,204,24,128,411,218),(11,115,398,205,25,129,412,219),(12,116,399,206,26,130,413,220),(13,117,400,207,27,131,414,221),(14,118,401,208,28,132,415,222),(29,84,436,270,43,70,422,256),(30,57,437,271,44,71,423,257),(31,58,438,272,45,72,424,258),(32,59,439,273,46,73,425,259),(33,60,440,274,47,74,426,260),(34,61,441,275,48,75,427,261),(35,62,442,276,49,76,428,262),(36,63,443,277,50,77,429,263),(37,64,444,278,51,78,430,264),(38,65,445,279,52,79,431,265),(39,66,446,280,53,80,432,266),(40,67,447,253,54,81,433,267),(41,68,448,254,55,82,434,268),(42,69,421,255,56,83,435,269),(85,368,358,318,99,382,344,332),(86,369,359,319,100,383,345,333),(87,370,360,320,101,384,346,334),(88,371,361,321,102,385,347,335),(89,372,362,322,103,386,348,336),(90,373,363,323,104,387,349,309),(91,374,364,324,105,388,350,310),(92,375,337,325,106,389,351,311),(93,376,338,326,107,390,352,312),(94,377,339,327,108,391,353,313),(95,378,340,328,109,392,354,314),(96,379,341,329,110,365,355,315),(97,380,342,330,111,366,356,316),(98,381,343,331,112,367,357,317),(141,226,174,307,155,240,188,293),(142,227,175,308,156,241,189,294),(143,228,176,281,157,242,190,295),(144,229,177,282,158,243,191,296),(145,230,178,283,159,244,192,297),(146,231,179,284,160,245,193,298),(147,232,180,285,161,246,194,299),(148,233,181,286,162,247,195,300),(149,234,182,287,163,248,196,301),(150,235,183,288,164,249,169,302),(151,236,184,289,165,250,170,303),(152,237,185,290,166,251,171,304),(153,238,186,291,167,252,172,305),(154,239,187,292,168,225,173,306)]])
124 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - | - | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | D7 | D8 | SD16 | Q16 | Dic7 | D14 | Dic14 | C4×D7 | C7⋊D4 | D28 | C56⋊C2 | D56 | Dic28 |
| kernel | C28.9C42 | C2×C4⋊Dic7 | C22×C56 | C4⋊Dic7 | C2×C56 | C2×C28 | C2×C28 | C22×C14 | C22×C8 | C2×C14 | C2×C14 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 3 | 2 | 4 | 2 | 6 | 3 | 6 | 12 | 12 | 6 | 24 | 12 | 12 |
Matrix representation of C28.9C42 ►in GL5(𝔽113)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 90 | 81 |
| 0 | 0 | 0 | 13 | 77 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 68 | 102 | 0 | 0 |
| 0 | 102 | 45 | 0 | 0 |
| 0 | 0 | 0 | 51 | 49 |
| 0 | 0 | 0 | 83 | 62 |
| 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 105 | 71 |
| 0 | 0 | 0 | 10 | 95 |
G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,90,13,0,0,0,81,77],[1,0,0,0,0,0,68,102,0,0,0,102,45,0,0,0,0,0,51,83,0,0,0,49,62],[15,0,0,0,0,0,0,112,0,0,0,1,0,0,0,0,0,0,105,10,0,0,0,71,95] >;
C28.9C42 in GAP, Magma, Sage, TeX
C_{28}._9C_4^2 % in TeX
G:=Group("C28.9C4^2"); // GroupNames label
G:=SmallGroup(448,108);
// by ID
G=gap.SmallGroup(448,108);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,176,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=1,c^4=a^14,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations