metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.6D4, C8.16D28, C14.4Q32, Dic28⋊5C4, C28.5SD16, C14.4SD32, C8.13(C4×D7), C56.10(C2×C4), C2.D8.3D7, (C2×C28).92D4, (C2×C14).34D8, C4.2(Q8⋊D7), C4.3(D14⋊C4), C7⋊1(C2.Q32), (C2×C8).222D14, C2.2(D8.D7), C2.2(C7⋊Q32), C28.3(C22⋊C4), (C2×C56).74C22, (C2×Dic28).8C2, C14.6(D4⋊C4), C2.8(C14.D8), C22.15(D4⋊D7), (C2×C7⋊C16).4C2, (C7×C2.D8).3C2, (C2×C4).116(C7⋊D4), SmallGroup(448,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.6D4
G = < a,b,c | a56=b4=1, c2=a21, bab-1=a15, cac-1=a41, cbc-1=a49b-1 >
Subgroups: 300 in 58 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C16, C4⋊C4, C2×C8, Q16, C2×Q8, Dic7, C28, C28, C2×C14, C2.D8, C2×C16, C2×Q16, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C2.Q32, C7⋊C16, Dic28, Dic28, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C7⋊C16, C7×C2.D8, C2×Dic28, C56.6D4
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, D4⋊C4, SD32, Q32, C4×D7, D28, C7⋊D4, C2.Q32, D14⋊C4, D4⋊D7, Q8⋊D7, C14.D8, D8.D7, C7⋊Q32, C56.6D4
►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)
(1 142 322 96)(2 157 323 111)(3 116 324 70)(4 131 325 85)(5 146 326 100)(6 161 327 59)(7 120 328 74)(8 135 329 89)(9 150 330 104)(10 165 331 63)(11 124 332 78)(12 139 333 93)(13 154 334 108)(14 113 335 67)(15 128 336 82)(16 143 281 97)(17 158 282 112)(18 117 283 71)(19 132 284 86)(20 147 285 101)(21 162 286 60)(22 121 287 75)(23 136 288 90)(24 151 289 105)(25 166 290 64)(26 125 291 79)(27 140 292 94)(28 155 293 109)(29 114 294 68)(30 129 295 83)(31 144 296 98)(32 159 297 57)(33 118 298 72)(34 133 299 87)(35 148 300 102)(36 163 301 61)(37 122 302 76)(38 137 303 91)(39 152 304 106)(40 167 305 65)(41 126 306 80)(42 141 307 95)(43 156 308 110)(44 115 309 69)(45 130 310 84)(46 145 311 99)(47 160 312 58)(48 119 313 73)(49 134 314 88)(50 149 315 103)(51 164 316 62)(52 123 317 77)(53 138 318 92)(54 153 319 107)(55 168 320 66)(56 127 321 81)(169 337 236 417)(170 352 237 432)(171 367 238 447)(172 382 239 406)(173 341 240 421)(174 356 241 436)(175 371 242 395)(176 386 243 410)(177 345 244 425)(178 360 245 440)(179 375 246 399)(180 390 247 414)(181 349 248 429)(182 364 249 444)(183 379 250 403)(184 338 251 418)(185 353 252 433)(186 368 253 448)(187 383 254 407)(188 342 255 422)(189 357 256 437)(190 372 257 396)(191 387 258 411)(192 346 259 426)(193 361 260 441)(194 376 261 400)(195 391 262 415)(196 350 263 430)(197 365 264 445)(198 380 265 404)(199 339 266 419)(200 354 267 434)(201 369 268 393)(202 384 269 408)(203 343 270 423)(204 358 271 438)(205 373 272 397)(206 388 273 412)(207 347 274 427)(208 362 275 442)(209 377 276 401)(210 392 277 416)(211 351 278 431)(212 366 279 446)(213 381 280 405)(214 340 225 420)(215 355 226 435)(216 370 227 394)(217 385 228 409)(218 344 229 424)(219 359 230 439)(220 374 231 398)(221 389 232 413)(222 348 233 428)(223 363 234 443)(224 378 235 402)
(1 276 22 241 43 262 8 227 29 248 50 269 15 234 36 255)(2 261 23 226 44 247 9 268 30 233 51 254 16 275 37 240)(3 246 24 267 45 232 10 253 31 274 52 239 17 260 38 225)(4 231 25 252 46 273 11 238 32 259 53 280 18 245 39 266)(5 272 26 237 47 258 12 279 33 244 54 265 19 230 40 251)(6 257 27 278 48 243 13 264 34 229 55 250 20 271 41 236)(7 242 28 263 49 228 14 249 35 270 56 235 21 256 42 277)(57 419 78 440 99 405 64 426 85 447 106 412 71 433 92 398)(58 404 79 425 100 446 65 411 86 432 107 397 72 418 93 439)(59 445 80 410 101 431 66 396 87 417 108 438 73 403 94 424)(60 430 81 395 102 416 67 437 88 402 109 423 74 444 95 409)(61 415 82 436 103 401 68 422 89 443 110 408 75 429 96 394)(62 400 83 421 104 442 69 407 90 428 111 393 76 414 97 435)(63 441 84 406 105 427 70 448 91 413 112 434 77 399 98 420)(113 357 134 378 155 343 120 364 141 385 162 350 127 371 148 392)(114 342 135 363 156 384 121 349 142 370 163 391 128 356 149 377)(115 383 136 348 157 369 122 390 143 355 164 376 129 341 150 362)(116 368 137 389 158 354 123 375 144 340 165 361 130 382 151 347)(117 353 138 374 159 339 124 360 145 381 166 346 131 367 152 388)(118 338 139 359 160 380 125 345 146 366 167 387 132 352 153 373)(119 379 140 344 161 365 126 386 147 351 168 372 133 337 154 358)(169 327 190 292 211 313 176 334 197 299 218 320 183 285 204 306)(170 312 191 333 212 298 177 319 198 284 219 305 184 326 205 291)(171 297 192 318 213 283 178 304 199 325 220 290 185 311 206 332)(172 282 193 303 214 324 179 289 200 310 221 331 186 296 207 317)(173 323 194 288 215 309 180 330 201 295 222 316 187 281 208 302)(174 308 195 329 216 294 181 315 202 336 223 301 188 322 209 287)(175 293 196 314 217 335 182 300 203 321 224 286 189 307 210 328)
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), (1,142,322,96)(2,157,323,111)(3,116,324,70)(4,131,325,85)(5,146,326,100)(6,161,327,59)(7,120,328,74)(8,135,329,89)(9,150,330,104)(10,165,331,63)(11,124,332,78)(12,139,333,93)(13,154,334,108)(14,113,335,67)(15,128,336,82)(16,143,281,97)(17,158,282,112)(18,117,283,71)(19,132,284,86)(20,147,285,101)(21,162,286,60)(22,121,287,75)(23,136,288,90)(24,151,289,105)(25,166,290,64)(26,125,291,79)(27,140,292,94)(28,155,293,109)(29,114,294,68)(30,129,295,83)(31,144,296,98)(32,159,297,57)(33,118,298,72)(34,133,299,87)(35,148,300,102)(36,163,301,61)(37,122,302,76)(38,137,303,91)(39,152,304,106)(40,167,305,65)(41,126,306,80)(42,141,307,95)(43,156,308,110)(44,115,309,69)(45,130,310,84)(46,145,311,99)(47,160,312,58)(48,119,313,73)(49,134,314,88)(50,149,315,103)(51,164,316,62)(52,123,317,77)(53,138,318,92)(54,153,319,107)(55,168,320,66)(56,127,321,81)(169,337,236,417)(170,352,237,432)(171,367,238,447)(172,382,239,406)(173,341,240,421)(174,356,241,436)(175,371,242,395)(176,386,243,410)(177,345,244,425)(178,360,245,440)(179,375,246,399)(180,390,247,414)(181,349,248,429)(182,364,249,444)(183,379,250,403)(184,338,251,418)(185,353,252,433)(186,368,253,448)(187,383,254,407)(188,342,255,422)(189,357,256,437)(190,372,257,396)(191,387,258,411)(192,346,259,426)(193,361,260,441)(194,376,261,400)(195,391,262,415)(196,350,263,430)(197,365,264,445)(198,380,265,404)(199,339,266,419)(200,354,267,434)(201,369,268,393)(202,384,269,408)(203,343,270,423)(204,358,271,438)(205,373,272,397)(206,388,273,412)(207,347,274,427)(208,362,275,442)(209,377,276,401)(210,392,277,416)(211,351,278,431)(212,366,279,446)(213,381,280,405)(214,340,225,420)(215,355,226,435)(216,370,227,394)(217,385,228,409)(218,344,229,424)(219,359,230,439)(220,374,231,398)(221,389,232,413)(222,348,233,428)(223,363,234,443)(224,378,235,402), (1,276,22,241,43,262,8,227,29,248,50,269,15,234,36,255)(2,261,23,226,44,247,9,268,30,233,51,254,16,275,37,240)(3,246,24,267,45,232,10,253,31,274,52,239,17,260,38,225)(4,231,25,252,46,273,11,238,32,259,53,280,18,245,39,266)(5,272,26,237,47,258,12,279,33,244,54,265,19,230,40,251)(6,257,27,278,48,243,13,264,34,229,55,250,20,271,41,236)(7,242,28,263,49,228,14,249,35,270,56,235,21,256,42,277)(57,419,78,440,99,405,64,426,85,447,106,412,71,433,92,398)(58,404,79,425,100,446,65,411,86,432,107,397,72,418,93,439)(59,445,80,410,101,431,66,396,87,417,108,438,73,403,94,424)(60,430,81,395,102,416,67,437,88,402,109,423,74,444,95,409)(61,415,82,436,103,401,68,422,89,443,110,408,75,429,96,394)(62,400,83,421,104,442,69,407,90,428,111,393,76,414,97,435)(63,441,84,406,105,427,70,448,91,413,112,434,77,399,98,420)(113,357,134,378,155,343,120,364,141,385,162,350,127,371,148,392)(114,342,135,363,156,384,121,349,142,370,163,391,128,356,149,377)(115,383,136,348,157,369,122,390,143,355,164,376,129,341,150,362)(116,368,137,389,158,354,123,375,144,340,165,361,130,382,151,347)(117,353,138,374,159,339,124,360,145,381,166,346,131,367,152,388)(118,338,139,359,160,380,125,345,146,366,167,387,132,352,153,373)(119,379,140,344,161,365,126,386,147,351,168,372,133,337,154,358)(169,327,190,292,211,313,176,334,197,299,218,320,183,285,204,306)(170,312,191,333,212,298,177,319,198,284,219,305,184,326,205,291)(171,297,192,318,213,283,178,304,199,325,220,290,185,311,206,332)(172,282,193,303,214,324,179,289,200,310,221,331,186,296,207,317)(173,323,194,288,215,309,180,330,201,295,222,316,187,281,208,302)(174,308,195,329,216,294,181,315,202,336,223,301,188,322,209,287)(175,293,196,314,217,335,182,300,203,321,224,286,189,307,210,328)>;
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), (1,142,322,96)(2,157,323,111)(3,116,324,70)(4,131,325,85)(5,146,326,100)(6,161,327,59)(7,120,328,74)(8,135,329,89)(9,150,330,104)(10,165,331,63)(11,124,332,78)(12,139,333,93)(13,154,334,108)(14,113,335,67)(15,128,336,82)(16,143,281,97)(17,158,282,112)(18,117,283,71)(19,132,284,86)(20,147,285,101)(21,162,286,60)(22,121,287,75)(23,136,288,90)(24,151,289,105)(25,166,290,64)(26,125,291,79)(27,140,292,94)(28,155,293,109)(29,114,294,68)(30,129,295,83)(31,144,296,98)(32,159,297,57)(33,118,298,72)(34,133,299,87)(35,148,300,102)(36,163,301,61)(37,122,302,76)(38,137,303,91)(39,152,304,106)(40,167,305,65)(41,126,306,80)(42,141,307,95)(43,156,308,110)(44,115,309,69)(45,130,310,84)(46,145,311,99)(47,160,312,58)(48,119,313,73)(49,134,314,88)(50,149,315,103)(51,164,316,62)(52,123,317,77)(53,138,318,92)(54,153,319,107)(55,168,320,66)(56,127,321,81)(169,337,236,417)(170,352,237,432)(171,367,238,447)(172,382,239,406)(173,341,240,421)(174,356,241,436)(175,371,242,395)(176,386,243,410)(177,345,244,425)(178,360,245,440)(179,375,246,399)(180,390,247,414)(181,349,248,429)(182,364,249,444)(183,379,250,403)(184,338,251,418)(185,353,252,433)(186,368,253,448)(187,383,254,407)(188,342,255,422)(189,357,256,437)(190,372,257,396)(191,387,258,411)(192,346,259,426)(193,361,260,441)(194,376,261,400)(195,391,262,415)(196,350,263,430)(197,365,264,445)(198,380,265,404)(199,339,266,419)(200,354,267,434)(201,369,268,393)(202,384,269,408)(203,343,270,423)(204,358,271,438)(205,373,272,397)(206,388,273,412)(207,347,274,427)(208,362,275,442)(209,377,276,401)(210,392,277,416)(211,351,278,431)(212,366,279,446)(213,381,280,405)(214,340,225,420)(215,355,226,435)(216,370,227,394)(217,385,228,409)(218,344,229,424)(219,359,230,439)(220,374,231,398)(221,389,232,413)(222,348,233,428)(223,363,234,443)(224,378,235,402), (1,276,22,241,43,262,8,227,29,248,50,269,15,234,36,255)(2,261,23,226,44,247,9,268,30,233,51,254,16,275,37,240)(3,246,24,267,45,232,10,253,31,274,52,239,17,260,38,225)(4,231,25,252,46,273,11,238,32,259,53,280,18,245,39,266)(5,272,26,237,47,258,12,279,33,244,54,265,19,230,40,251)(6,257,27,278,48,243,13,264,34,229,55,250,20,271,41,236)(7,242,28,263,49,228,14,249,35,270,56,235,21,256,42,277)(57,419,78,440,99,405,64,426,85,447,106,412,71,433,92,398)(58,404,79,425,100,446,65,411,86,432,107,397,72,418,93,439)(59,445,80,410,101,431,66,396,87,417,108,438,73,403,94,424)(60,430,81,395,102,416,67,437,88,402,109,423,74,444,95,409)(61,415,82,436,103,401,68,422,89,443,110,408,75,429,96,394)(62,400,83,421,104,442,69,407,90,428,111,393,76,414,97,435)(63,441,84,406,105,427,70,448,91,413,112,434,77,399,98,420)(113,357,134,378,155,343,120,364,141,385,162,350,127,371,148,392)(114,342,135,363,156,384,121,349,142,370,163,391,128,356,149,377)(115,383,136,348,157,369,122,390,143,355,164,376,129,341,150,362)(116,368,137,389,158,354,123,375,144,340,165,361,130,382,151,347)(117,353,138,374,159,339,124,360,145,381,166,346,131,367,152,388)(118,338,139,359,160,380,125,345,146,366,167,387,132,352,153,373)(119,379,140,344,161,365,126,386,147,351,168,372,133,337,154,358)(169,327,190,292,211,313,176,334,197,299,218,320,183,285,204,306)(170,312,191,333,212,298,177,319,198,284,219,305,184,326,205,291)(171,297,192,318,213,283,178,304,199,325,220,290,185,311,206,332)(172,282,193,303,214,324,179,289,200,310,221,331,186,296,207,317)(173,323,194,288,215,309,180,330,201,295,222,316,187,281,208,302)(174,308,195,329,216,294,181,315,202,336,223,301,188,322,209,287)(175,293,196,314,217,335,182,300,203,321,224,286,189,307,210,328) );
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)], [(1,142,322,96),(2,157,323,111),(3,116,324,70),(4,131,325,85),(5,146,326,100),(6,161,327,59),(7,120,328,74),(8,135,329,89),(9,150,330,104),(10,165,331,63),(11,124,332,78),(12,139,333,93),(13,154,334,108),(14,113,335,67),(15,128,336,82),(16,143,281,97),(17,158,282,112),(18,117,283,71),(19,132,284,86),(20,147,285,101),(21,162,286,60),(22,121,287,75),(23,136,288,90),(24,151,289,105),(25,166,290,64),(26,125,291,79),(27,140,292,94),(28,155,293,109),(29,114,294,68),(30,129,295,83),(31,144,296,98),(32,159,297,57),(33,118,298,72),(34,133,299,87),(35,148,300,102),(36,163,301,61),(37,122,302,76),(38,137,303,91),(39,152,304,106),(40,167,305,65),(41,126,306,80),(42,141,307,95),(43,156,308,110),(44,115,309,69),(45,130,310,84),(46,145,311,99),(47,160,312,58),(48,119,313,73),(49,134,314,88),(50,149,315,103),(51,164,316,62),(52,123,317,77),(53,138,318,92),(54,153,319,107),(55,168,320,66),(56,127,321,81),(169,337,236,417),(170,352,237,432),(171,367,238,447),(172,382,239,406),(173,341,240,421),(174,356,241,436),(175,371,242,395),(176,386,243,410),(177,345,244,425),(178,360,245,440),(179,375,246,399),(180,390,247,414),(181,349,248,429),(182,364,249,444),(183,379,250,403),(184,338,251,418),(185,353,252,433),(186,368,253,448),(187,383,254,407),(188,342,255,422),(189,357,256,437),(190,372,257,396),(191,387,258,411),(192,346,259,426),(193,361,260,441),(194,376,261,400),(195,391,262,415),(196,350,263,430),(197,365,264,445),(198,380,265,404),(199,339,266,419),(200,354,267,434),(201,369,268,393),(202,384,269,408),(203,343,270,423),(204,358,271,438),(205,373,272,397),(206,388,273,412),(207,347,274,427),(208,362,275,442),(209,377,276,401),(210,392,277,416),(211,351,278,431),(212,366,279,446),(213,381,280,405),(214,340,225,420),(215,355,226,435),(216,370,227,394),(217,385,228,409),(218,344,229,424),(219,359,230,439),(220,374,231,398),(221,389,232,413),(222,348,233,428),(223,363,234,443),(224,378,235,402)], [(1,276,22,241,43,262,8,227,29,248,50,269,15,234,36,255),(2,261,23,226,44,247,9,268,30,233,51,254,16,275,37,240),(3,246,24,267,45,232,10,253,31,274,52,239,17,260,38,225),(4,231,25,252,46,273,11,238,32,259,53,280,18,245,39,266),(5,272,26,237,47,258,12,279,33,244,54,265,19,230,40,251),(6,257,27,278,48,243,13,264,34,229,55,250,20,271,41,236),(7,242,28,263,49,228,14,249,35,270,56,235,21,256,42,277),(57,419,78,440,99,405,64,426,85,447,106,412,71,433,92,398),(58,404,79,425,100,446,65,411,86,432,107,397,72,418,93,439),(59,445,80,410,101,431,66,396,87,417,108,438,73,403,94,424),(60,430,81,395,102,416,67,437,88,402,109,423,74,444,95,409),(61,415,82,436,103,401,68,422,89,443,110,408,75,429,96,394),(62,400,83,421,104,442,69,407,90,428,111,393,76,414,97,435),(63,441,84,406,105,427,70,448,91,413,112,434,77,399,98,420),(113,357,134,378,155,343,120,364,141,385,162,350,127,371,148,392),(114,342,135,363,156,384,121,349,142,370,163,391,128,356,149,377),(115,383,136,348,157,369,122,390,143,355,164,376,129,341,150,362),(116,368,137,389,158,354,123,375,144,340,165,361,130,382,151,347),(117,353,138,374,159,339,124,360,145,381,166,346,131,367,152,388),(118,338,139,359,160,380,125,345,146,366,167,387,132,352,153,373),(119,379,140,344,161,365,126,386,147,351,168,372,133,337,154,358),(169,327,190,292,211,313,176,334,197,299,218,320,183,285,204,306),(170,312,191,333,212,298,177,319,198,284,219,305,184,326,205,291),(171,297,192,318,213,283,178,304,199,325,220,290,185,311,206,332),(172,282,193,303,214,324,179,289,200,310,221,331,186,296,207,317),(173,323,194,288,215,309,180,330,201,295,222,316,187,281,208,302),(174,308,195,329,216,294,181,315,202,336,223,301,188,322,209,287),(175,293,196,314,217,335,182,300,203,321,224,286,189,307,210,328)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | SD16 | D8 | D14 | SD32 | Q32 | C4×D7 | D28 | C7⋊D4 | Q8⋊D7 | D4⋊D7 | D8.D7 | C7⋊Q32 |
| kernel | C56.6D4 | C2×C7⋊C16 | C7×C2.D8 | C2×Dic28 | Dic28 | C56 | C2×C28 | C2.D8 | C28 | C2×C14 | C2×C8 | C14 | C14 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 3 | 3 | 6 | 6 |
Matrix representation of C56.6D4 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 82 | 31 | 0 | 0 |
| 0 | 82 | 82 | 0 | 0 |
| 0 | 0 | 0 | 1 | 112 |
| 0 | 0 | 0 | 11 | 103 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 77 | 20 | 0 | 0 |
| 0 | 20 | 36 | 0 | 0 |
| 0 | 0 | 0 | 29 | 5 |
| 0 | 0 | 0 | 58 | 84 |
| 15 | 0 | 0 | 0 | 0 |
| 0 | 44 | 60 | 0 | 0 |
| 0 | 53 | 44 | 0 | 0 |
| 0 | 0 | 0 | 11 | 73 |
| 0 | 0 | 0 | 20 | 102 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,82,82,0,0,0,31,82,0,0,0,0,0,1,11,0,0,0,112,103],[98,0,0,0,0,0,77,20,0,0,0,20,36,0,0,0,0,0,29,58,0,0,0,5,84],[15,0,0,0,0,0,44,53,0,0,0,60,44,0,0,0,0,0,11,20,0,0,0,73,102] >;
C56.6D4 in GAP, Magma, Sage, TeX
C_{56}._6D_4 % in TeX
G:=Group("C56.6D4"); // GroupNames label
G:=SmallGroup(448,49);
// by ID
G=gap.SmallGroup(448,49);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,675,346,192,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^21,b*a*b^-1=a^15,c*a*c^-1=a^41,c*b*c^-1=a^49*b^-1>;
// generators/relations