metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.20D14, C56⋊C4.7C2, (C2×C8).174D14, Q8⋊C4.8D7, (C2×Q8).12D14, C4.30(C4○D28), C28.16(C4○D4), (C2×Dic7).27D4, C14.Q16.3C2, Q8⋊Dic7.5C2, C22.191(D4×D7), C28.3Q8.2C2, C4.56(D4⋊2D7), (C2×C28).237C23, (C2×C56).194C22, Dic7⋊Q8.4C2, C28.44D4.9C2, C4⋊Dic7.86C22, (Q8×C14).20C22, C2.10(Q16⋊D7), C14.28(C4.4D4), C2.16(SD16⋊D7), C14.55(C8.C22), (C4×Dic7).21C22, C7⋊2(C42.30C22), (C2×Dic14).66C22, C2.18(Dic7.D4), (C2×C7⋊C8).32C22, (C2×C14).250(C2×D4), (C7×C4⋊C4).38C22, (C7×Q8⋊C4).10C2, (C2×C4).344(C22×D7), SmallGroup(448,331)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊C4.C2
G = < a,b,c | a56=b4=1, c2=a28, bab-1=a13, cac-1=a43b2, cbc-1=b-1 >
Subgroups: 404 in 90 conjugacy classes, 37 normal (all characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C42.30C22, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, Q8×C14, C14.Q16, C56⋊C4, C28.44D4, Q8⋊Dic7, C7×Q8⋊C4, C28.3Q8, Dic7⋊Q8, C56⋊C4.C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C8.C22, C22×D7, C42.30C22, C4○D28, D4×D7, D4⋊2D7, Dic7.D4, SD16⋊D7, Q16⋊D7, C56⋊C4.C2
►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)
(1 367 414 207)(2 380 415 220)(3 337 416 177)(4 350 417 190)(5 363 418 203)(6 376 419 216)(7 389 420 173)(8 346 421 186)(9 359 422 199)(10 372 423 212)(11 385 424 169)(12 342 425 182)(13 355 426 195)(14 368 427 208)(15 381 428 221)(16 338 429 178)(17 351 430 191)(18 364 431 204)(19 377 432 217)(20 390 433 174)(21 347 434 187)(22 360 435 200)(23 373 436 213)(24 386 437 170)(25 343 438 183)(26 356 439 196)(27 369 440 209)(28 382 441 222)(29 339 442 179)(30 352 443 192)(31 365 444 205)(32 378 445 218)(33 391 446 175)(34 348 447 188)(35 361 448 201)(36 374 393 214)(37 387 394 171)(38 344 395 184)(39 357 396 197)(40 370 397 210)(41 383 398 223)(42 340 399 180)(43 353 400 193)(44 366 401 206)(45 379 402 219)(46 392 403 176)(47 349 404 189)(48 362 405 202)(49 375 406 215)(50 388 407 172)(51 345 408 185)(52 358 409 198)(53 371 410 211)(54 384 411 224)(55 341 412 181)(56 354 413 194)(57 250 163 312)(58 263 164 325)(59 276 165 282)(60 233 166 295)(61 246 167 308)(62 259 168 321)(63 272 113 334)(64 229 114 291)(65 242 115 304)(66 255 116 317)(67 268 117 330)(68 225 118 287)(69 238 119 300)(70 251 120 313)(71 264 121 326)(72 277 122 283)(73 234 123 296)(74 247 124 309)(75 260 125 322)(76 273 126 335)(77 230 127 292)(78 243 128 305)(79 256 129 318)(80 269 130 331)(81 226 131 288)(82 239 132 301)(83 252 133 314)(84 265 134 327)(85 278 135 284)(86 235 136 297)(87 248 137 310)(88 261 138 323)(89 274 139 336)(90 231 140 293)(91 244 141 306)(92 257 142 319)(93 270 143 332)(94 227 144 289)(95 240 145 302)(96 253 146 315)(97 266 147 328)(98 279 148 285)(99 236 149 298)(100 249 150 311)(101 262 151 324)(102 275 152 281)(103 232 153 294)(104 245 154 307)(105 258 155 320)(106 271 156 333)(107 228 157 290)(108 241 158 303)(109 254 159 316)(110 267 160 329)(111 280 161 286)(112 237 162 299)
(1 302 29 330)(2 227 30 255)(3 332 31 304)(4 257 32 229)(5 306 33 334)(6 231 34 259)(7 336 35 308)(8 261 36 233)(9 310 37 282)(10 235 38 263)(11 284 39 312)(12 265 40 237)(13 314 41 286)(14 239 42 267)(15 288 43 316)(16 269 44 241)(17 318 45 290)(18 243 46 271)(19 292 47 320)(20 273 48 245)(21 322 49 294)(22 247 50 275)(23 296 51 324)(24 277 52 249)(25 326 53 298)(26 251 54 279)(27 300 55 328)(28 225 56 253)(57 169 85 197)(58 372 86 344)(59 199 87 171)(60 346 88 374)(61 173 89 201)(62 376 90 348)(63 203 91 175)(64 350 92 378)(65 177 93 205)(66 380 94 352)(67 207 95 179)(68 354 96 382)(69 181 97 209)(70 384 98 356)(71 211 99 183)(72 358 100 386)(73 185 101 213)(74 388 102 360)(75 215 103 187)(76 362 104 390)(77 189 105 217)(78 392 106 364)(79 219 107 191)(80 366 108 338)(81 193 109 221)(82 340 110 368)(83 223 111 195)(84 370 112 342)(113 363 141 391)(114 190 142 218)(115 337 143 365)(116 220 144 192)(117 367 145 339)(118 194 146 222)(119 341 147 369)(120 224 148 196)(121 371 149 343)(122 198 150 170)(123 345 151 373)(124 172 152 200)(125 375 153 347)(126 202 154 174)(127 349 155 377)(128 176 156 204)(129 379 157 351)(130 206 158 178)(131 353 159 381)(132 180 160 208)(133 383 161 355)(134 210 162 182)(135 357 163 385)(136 184 164 212)(137 387 165 359)(138 214 166 186)(139 361 167 389)(140 188 168 216)(226 400 254 428)(228 430 256 402)(230 404 258 432)(232 434 260 406)(234 408 262 436)(236 438 264 410)(238 412 266 440)(240 442 268 414)(242 416 270 444)(244 446 272 418)(246 420 274 448)(248 394 276 422)(250 424 278 396)(252 398 280 426)(281 435 309 407)(283 409 311 437)(285 439 313 411)(287 413 315 441)(289 443 317 415)(291 417 319 445)(293 447 321 419)(295 421 323 393)(297 395 325 423)(299 425 327 397)(301 399 329 427)(303 429 331 401)(305 403 333 431)(307 433 335 405)
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), (1,367,414,207)(2,380,415,220)(3,337,416,177)(4,350,417,190)(5,363,418,203)(6,376,419,216)(7,389,420,173)(8,346,421,186)(9,359,422,199)(10,372,423,212)(11,385,424,169)(12,342,425,182)(13,355,426,195)(14,368,427,208)(15,381,428,221)(16,338,429,178)(17,351,430,191)(18,364,431,204)(19,377,432,217)(20,390,433,174)(21,347,434,187)(22,360,435,200)(23,373,436,213)(24,386,437,170)(25,343,438,183)(26,356,439,196)(27,369,440,209)(28,382,441,222)(29,339,442,179)(30,352,443,192)(31,365,444,205)(32,378,445,218)(33,391,446,175)(34,348,447,188)(35,361,448,201)(36,374,393,214)(37,387,394,171)(38,344,395,184)(39,357,396,197)(40,370,397,210)(41,383,398,223)(42,340,399,180)(43,353,400,193)(44,366,401,206)(45,379,402,219)(46,392,403,176)(47,349,404,189)(48,362,405,202)(49,375,406,215)(50,388,407,172)(51,345,408,185)(52,358,409,198)(53,371,410,211)(54,384,411,224)(55,341,412,181)(56,354,413,194)(57,250,163,312)(58,263,164,325)(59,276,165,282)(60,233,166,295)(61,246,167,308)(62,259,168,321)(63,272,113,334)(64,229,114,291)(65,242,115,304)(66,255,116,317)(67,268,117,330)(68,225,118,287)(69,238,119,300)(70,251,120,313)(71,264,121,326)(72,277,122,283)(73,234,123,296)(74,247,124,309)(75,260,125,322)(76,273,126,335)(77,230,127,292)(78,243,128,305)(79,256,129,318)(80,269,130,331)(81,226,131,288)(82,239,132,301)(83,252,133,314)(84,265,134,327)(85,278,135,284)(86,235,136,297)(87,248,137,310)(88,261,138,323)(89,274,139,336)(90,231,140,293)(91,244,141,306)(92,257,142,319)(93,270,143,332)(94,227,144,289)(95,240,145,302)(96,253,146,315)(97,266,147,328)(98,279,148,285)(99,236,149,298)(100,249,150,311)(101,262,151,324)(102,275,152,281)(103,232,153,294)(104,245,154,307)(105,258,155,320)(106,271,156,333)(107,228,157,290)(108,241,158,303)(109,254,159,316)(110,267,160,329)(111,280,161,286)(112,237,162,299), (1,302,29,330)(2,227,30,255)(3,332,31,304)(4,257,32,229)(5,306,33,334)(6,231,34,259)(7,336,35,308)(8,261,36,233)(9,310,37,282)(10,235,38,263)(11,284,39,312)(12,265,40,237)(13,314,41,286)(14,239,42,267)(15,288,43,316)(16,269,44,241)(17,318,45,290)(18,243,46,271)(19,292,47,320)(20,273,48,245)(21,322,49,294)(22,247,50,275)(23,296,51,324)(24,277,52,249)(25,326,53,298)(26,251,54,279)(27,300,55,328)(28,225,56,253)(57,169,85,197)(58,372,86,344)(59,199,87,171)(60,346,88,374)(61,173,89,201)(62,376,90,348)(63,203,91,175)(64,350,92,378)(65,177,93,205)(66,380,94,352)(67,207,95,179)(68,354,96,382)(69,181,97,209)(70,384,98,356)(71,211,99,183)(72,358,100,386)(73,185,101,213)(74,388,102,360)(75,215,103,187)(76,362,104,390)(77,189,105,217)(78,392,106,364)(79,219,107,191)(80,366,108,338)(81,193,109,221)(82,340,110,368)(83,223,111,195)(84,370,112,342)(113,363,141,391)(114,190,142,218)(115,337,143,365)(116,220,144,192)(117,367,145,339)(118,194,146,222)(119,341,147,369)(120,224,148,196)(121,371,149,343)(122,198,150,170)(123,345,151,373)(124,172,152,200)(125,375,153,347)(126,202,154,174)(127,349,155,377)(128,176,156,204)(129,379,157,351)(130,206,158,178)(131,353,159,381)(132,180,160,208)(133,383,161,355)(134,210,162,182)(135,357,163,385)(136,184,164,212)(137,387,165,359)(138,214,166,186)(139,361,167,389)(140,188,168,216)(226,400,254,428)(228,430,256,402)(230,404,258,432)(232,434,260,406)(234,408,262,436)(236,438,264,410)(238,412,266,440)(240,442,268,414)(242,416,270,444)(244,446,272,418)(246,420,274,448)(248,394,276,422)(250,424,278,396)(252,398,280,426)(281,435,309,407)(283,409,311,437)(285,439,313,411)(287,413,315,441)(289,443,317,415)(291,417,319,445)(293,447,321,419)(295,421,323,393)(297,395,325,423)(299,425,327,397)(301,399,329,427)(303,429,331,401)(305,403,333,431)(307,433,335,405)>;
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), (1,367,414,207)(2,380,415,220)(3,337,416,177)(4,350,417,190)(5,363,418,203)(6,376,419,216)(7,389,420,173)(8,346,421,186)(9,359,422,199)(10,372,423,212)(11,385,424,169)(12,342,425,182)(13,355,426,195)(14,368,427,208)(15,381,428,221)(16,338,429,178)(17,351,430,191)(18,364,431,204)(19,377,432,217)(20,390,433,174)(21,347,434,187)(22,360,435,200)(23,373,436,213)(24,386,437,170)(25,343,438,183)(26,356,439,196)(27,369,440,209)(28,382,441,222)(29,339,442,179)(30,352,443,192)(31,365,444,205)(32,378,445,218)(33,391,446,175)(34,348,447,188)(35,361,448,201)(36,374,393,214)(37,387,394,171)(38,344,395,184)(39,357,396,197)(40,370,397,210)(41,383,398,223)(42,340,399,180)(43,353,400,193)(44,366,401,206)(45,379,402,219)(46,392,403,176)(47,349,404,189)(48,362,405,202)(49,375,406,215)(50,388,407,172)(51,345,408,185)(52,358,409,198)(53,371,410,211)(54,384,411,224)(55,341,412,181)(56,354,413,194)(57,250,163,312)(58,263,164,325)(59,276,165,282)(60,233,166,295)(61,246,167,308)(62,259,168,321)(63,272,113,334)(64,229,114,291)(65,242,115,304)(66,255,116,317)(67,268,117,330)(68,225,118,287)(69,238,119,300)(70,251,120,313)(71,264,121,326)(72,277,122,283)(73,234,123,296)(74,247,124,309)(75,260,125,322)(76,273,126,335)(77,230,127,292)(78,243,128,305)(79,256,129,318)(80,269,130,331)(81,226,131,288)(82,239,132,301)(83,252,133,314)(84,265,134,327)(85,278,135,284)(86,235,136,297)(87,248,137,310)(88,261,138,323)(89,274,139,336)(90,231,140,293)(91,244,141,306)(92,257,142,319)(93,270,143,332)(94,227,144,289)(95,240,145,302)(96,253,146,315)(97,266,147,328)(98,279,148,285)(99,236,149,298)(100,249,150,311)(101,262,151,324)(102,275,152,281)(103,232,153,294)(104,245,154,307)(105,258,155,320)(106,271,156,333)(107,228,157,290)(108,241,158,303)(109,254,159,316)(110,267,160,329)(111,280,161,286)(112,237,162,299), (1,302,29,330)(2,227,30,255)(3,332,31,304)(4,257,32,229)(5,306,33,334)(6,231,34,259)(7,336,35,308)(8,261,36,233)(9,310,37,282)(10,235,38,263)(11,284,39,312)(12,265,40,237)(13,314,41,286)(14,239,42,267)(15,288,43,316)(16,269,44,241)(17,318,45,290)(18,243,46,271)(19,292,47,320)(20,273,48,245)(21,322,49,294)(22,247,50,275)(23,296,51,324)(24,277,52,249)(25,326,53,298)(26,251,54,279)(27,300,55,328)(28,225,56,253)(57,169,85,197)(58,372,86,344)(59,199,87,171)(60,346,88,374)(61,173,89,201)(62,376,90,348)(63,203,91,175)(64,350,92,378)(65,177,93,205)(66,380,94,352)(67,207,95,179)(68,354,96,382)(69,181,97,209)(70,384,98,356)(71,211,99,183)(72,358,100,386)(73,185,101,213)(74,388,102,360)(75,215,103,187)(76,362,104,390)(77,189,105,217)(78,392,106,364)(79,219,107,191)(80,366,108,338)(81,193,109,221)(82,340,110,368)(83,223,111,195)(84,370,112,342)(113,363,141,391)(114,190,142,218)(115,337,143,365)(116,220,144,192)(117,367,145,339)(118,194,146,222)(119,341,147,369)(120,224,148,196)(121,371,149,343)(122,198,150,170)(123,345,151,373)(124,172,152,200)(125,375,153,347)(126,202,154,174)(127,349,155,377)(128,176,156,204)(129,379,157,351)(130,206,158,178)(131,353,159,381)(132,180,160,208)(133,383,161,355)(134,210,162,182)(135,357,163,385)(136,184,164,212)(137,387,165,359)(138,214,166,186)(139,361,167,389)(140,188,168,216)(226,400,254,428)(228,430,256,402)(230,404,258,432)(232,434,260,406)(234,408,262,436)(236,438,264,410)(238,412,266,440)(240,442,268,414)(242,416,270,444)(244,446,272,418)(246,420,274,448)(248,394,276,422)(250,424,278,396)(252,398,280,426)(281,435,309,407)(283,409,311,437)(285,439,313,411)(287,413,315,441)(289,443,317,415)(291,417,319,445)(293,447,321,419)(295,421,323,393)(297,395,325,423)(299,425,327,397)(301,399,329,427)(303,429,331,401)(305,403,333,431)(307,433,335,405) );
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)], [(1,367,414,207),(2,380,415,220),(3,337,416,177),(4,350,417,190),(5,363,418,203),(6,376,419,216),(7,389,420,173),(8,346,421,186),(9,359,422,199),(10,372,423,212),(11,385,424,169),(12,342,425,182),(13,355,426,195),(14,368,427,208),(15,381,428,221),(16,338,429,178),(17,351,430,191),(18,364,431,204),(19,377,432,217),(20,390,433,174),(21,347,434,187),(22,360,435,200),(23,373,436,213),(24,386,437,170),(25,343,438,183),(26,356,439,196),(27,369,440,209),(28,382,441,222),(29,339,442,179),(30,352,443,192),(31,365,444,205),(32,378,445,218),(33,391,446,175),(34,348,447,188),(35,361,448,201),(36,374,393,214),(37,387,394,171),(38,344,395,184),(39,357,396,197),(40,370,397,210),(41,383,398,223),(42,340,399,180),(43,353,400,193),(44,366,401,206),(45,379,402,219),(46,392,403,176),(47,349,404,189),(48,362,405,202),(49,375,406,215),(50,388,407,172),(51,345,408,185),(52,358,409,198),(53,371,410,211),(54,384,411,224),(55,341,412,181),(56,354,413,194),(57,250,163,312),(58,263,164,325),(59,276,165,282),(60,233,166,295),(61,246,167,308),(62,259,168,321),(63,272,113,334),(64,229,114,291),(65,242,115,304),(66,255,116,317),(67,268,117,330),(68,225,118,287),(69,238,119,300),(70,251,120,313),(71,264,121,326),(72,277,122,283),(73,234,123,296),(74,247,124,309),(75,260,125,322),(76,273,126,335),(77,230,127,292),(78,243,128,305),(79,256,129,318),(80,269,130,331),(81,226,131,288),(82,239,132,301),(83,252,133,314),(84,265,134,327),(85,278,135,284),(86,235,136,297),(87,248,137,310),(88,261,138,323),(89,274,139,336),(90,231,140,293),(91,244,141,306),(92,257,142,319),(93,270,143,332),(94,227,144,289),(95,240,145,302),(96,253,146,315),(97,266,147,328),(98,279,148,285),(99,236,149,298),(100,249,150,311),(101,262,151,324),(102,275,152,281),(103,232,153,294),(104,245,154,307),(105,258,155,320),(106,271,156,333),(107,228,157,290),(108,241,158,303),(109,254,159,316),(110,267,160,329),(111,280,161,286),(112,237,162,299)], [(1,302,29,330),(2,227,30,255),(3,332,31,304),(4,257,32,229),(5,306,33,334),(6,231,34,259),(7,336,35,308),(8,261,36,233),(9,310,37,282),(10,235,38,263),(11,284,39,312),(12,265,40,237),(13,314,41,286),(14,239,42,267),(15,288,43,316),(16,269,44,241),(17,318,45,290),(18,243,46,271),(19,292,47,320),(20,273,48,245),(21,322,49,294),(22,247,50,275),(23,296,51,324),(24,277,52,249),(25,326,53,298),(26,251,54,279),(27,300,55,328),(28,225,56,253),(57,169,85,197),(58,372,86,344),(59,199,87,171),(60,346,88,374),(61,173,89,201),(62,376,90,348),(63,203,91,175),(64,350,92,378),(65,177,93,205),(66,380,94,352),(67,207,95,179),(68,354,96,382),(69,181,97,209),(70,384,98,356),(71,211,99,183),(72,358,100,386),(73,185,101,213),(74,388,102,360),(75,215,103,187),(76,362,104,390),(77,189,105,217),(78,392,106,364),(79,219,107,191),(80,366,108,338),(81,193,109,221),(82,340,110,368),(83,223,111,195),(84,370,112,342),(113,363,141,391),(114,190,142,218),(115,337,143,365),(116,220,144,192),(117,367,145,339),(118,194,146,222),(119,341,147,369),(120,224,148,196),(121,371,149,343),(122,198,150,170),(123,345,151,373),(124,172,152,200),(125,375,153,347),(126,202,154,174),(127,349,155,377),(128,176,156,204),(129,379,157,351),(130,206,158,178),(131,353,159,381),(132,180,160,208),(133,383,161,355),(134,210,162,182),(135,357,163,385),(136,184,164,212),(137,387,165,359),(138,214,166,186),(139,361,167,389),(140,188,168,216),(226,400,254,428),(228,430,256,402),(230,404,258,432),(232,434,260,406),(234,408,262,436),(236,438,264,410),(238,412,266,440),(240,442,268,414),(242,416,270,444),(244,446,272,418),(246,420,274,448),(248,394,276,422),(250,424,278,396),(252,398,280,426),(281,435,309,407),(283,409,311,437),(285,439,313,411),(287,413,315,441),(289,443,317,415),(291,417,319,445),(293,447,321,419),(295,421,323,393),(297,395,325,423),(299,425,327,397),(301,399,329,427),(303,429,331,401),(305,403,333,431),(307,433,335,405)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | C8.C22 | D4⋊2D7 | D4×D7 | SD16⋊D7 | Q16⋊D7 |
| kernel | C56⋊C4.C2 | C14.Q16 | C56⋊C4 | C28.44D4 | Q8⋊Dic7 | C7×Q8⋊C4 | C28.3Q8 | Dic7⋊Q8 | C2×Dic7 | Q8⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 3 | 3 | 12 | 2 | 3 | 3 | 6 | 6 |
Matrix representation of C56⋊C4.C2 ►in GL8(𝔽113)
| 10 | 63 | 0 | 0 | 0 | 0 | 0 | 0 |
| 63 | 103 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 52 | 44 | 82 |
| 0 | 0 | 0 | 0 | 87 | 61 | 110 | 69 |
| 0 | 0 | 0 | 0 | 75 | 6 | 52 | 52 |
| 0 | 0 | 0 | 0 | 41 | 38 | 87 | 0 |
| 37 | 41 | 0 | 0 | 0 | 0 | 0 | 0 |
| 41 | 76 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 85 | 108 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 76 | 0 | 75 | 82 |
| 0 | 0 | 0 | 0 | 0 | 76 | 79 | 38 |
| 0 | 0 | 0 | 0 | 38 | 31 | 37 | 0 |
| 0 | 0 | 0 | 0 | 34 | 75 | 0 | 37 |
| 50 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 63 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 75 | 6 | 52 | 52 |
| 0 | 0 | 0 | 0 | 41 | 38 | 87 | 0 |
| 0 | 0 | 0 | 0 | 0 | 52 | 44 | 82 |
| 0 | 0 | 0 | 0 | 87 | 61 | 110 | 69 |
G:=sub<GL(8,GF(113))| [10,63,0,0,0,0,0,0,63,103,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,9,0,0,0,0,0,0,0,0,0,87,75,41,0,0,0,0,52,61,6,38,0,0,0,0,44,110,52,87,0,0,0,0,82,69,52,0],[37,41,0,0,0,0,0,0,41,76,0,0,0,0,0,0,0,0,85,21,0,0,0,0,0,0,108,28,0,0,0,0,0,0,0,0,76,0,38,34,0,0,0,0,0,76,31,75,0,0,0,0,75,79,37,0,0,0,0,0,82,38,0,37],[50,10,0,0,0,0,0,0,10,63,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,75,41,0,87,0,0,0,0,6,38,52,61,0,0,0,0,52,87,44,110,0,0,0,0,52,0,82,69] >;
C56⋊C4.C2 in GAP, Magma, Sage, TeX
C_{56}\rtimes C_4.C_2 % in TeX
G:=Group("C56:C4.C2"); // GroupNames label
G:=SmallGroup(448,331);
// by ID
G=gap.SmallGroup(448,331);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,232,1094,135,100,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=a^13,c*a*c^-1=a^43*b^2,c*b*c^-1=b^-1>;
// generators/relations