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## G = C56.78D4order 448 = 26·7

### 1st non-split extension by C56 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C56.78D4
 Chief series C1 — C7 — C14 — C28 — C56 — C2×C56 — C56⋊1C4 — C56.78D4
 Lower central C7 — C14 — C28 — C56 — C56.78D4
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C56.78D4
G = < a,b,c | a56=b4=1, c2=a28, bab-1=cac-1=a-1, cbc-1=a49b-1 >

Subgroups: 372 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, C2×C14, C2.D8, C2×C16, C2×Q16, C56, Dic14, C2×Dic7, C2×C28, C2.Q32, C112, Dic28, Dic28, C4⋊Dic7, C2×C56, C2×Dic14, C561C4, C2×C112, C2×Dic28, C56.78D4
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, C56⋊C2, D56, D14⋊C4, C112⋊C2, Dic56, C2.D56, C56.78D4

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

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

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

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

118 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 ··· 28L 56A ··· 56X 112A ··· 112AV order 1 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 1 1 2 2 56 56 56 56 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

118 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + - + + - image C1 C2 C2 C2 C4 D4 D4 D7 SD16 D8 D14 SD32 Q32 C4×D7 C7⋊D4 D28 C56⋊C2 D56 C112⋊C2 Dic56 kernel C56.78D4 C56⋊1C4 C2×C112 C2×Dic28 Dic28 C56 C2×C28 C2×C16 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 12 12 24 24

Matrix representation of C56.78D4 in GL4(𝔽113) generated by

 0 1 0 0 112 9 0 0 0 0 48 41 0 0 72 78
,
 79 108 0 0 28 34 0 0 0 0 55 67 0 0 110 58
,
 1 0 0 0 9 112 0 0 0 0 32 89 0 0 38 81
`G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,48,72,0,0,41,78],[79,28,0,0,108,34,0,0,0,0,55,110,0,0,67,58],[1,9,0,0,0,112,0,0,0,0,32,38,0,0,89,81] >;`

C56.78D4 in GAP, Magma, Sage, TeX

`C_{56}._{78}D_4`
`% in TeX`

`G:=Group("C56.78D4");`
`// GroupNames label`

`G:=SmallGroup(448,60);`
`// by ID`

`G=gap.SmallGroup(448,60);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,280,85,204,422,268,1684,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^49*b^-1>;`
`// generators/relations`

׿
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