metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊3Q8, C8⋊2Dic14, C7⋊C8⋊1Q8, C7⋊2(C8⋊Q8), C28⋊Q8.6C2, C4.21(Q8×D7), C4⋊C4.33D14, (C2×C8).59D14, C4.Q8.3D7, C2.9(C28⋊Q8), C28.57(C2×Q8), C56⋊C4.2C2, C14.14(C4⋊Q8), C56⋊1C4.17C2, (C2×Dic7).38D4, C4.21(C2×Dic14), C28.Q8.4C2, C22.211(D4×D7), C4.Dic14.6C2, C28.3Q8.4C2, C2.20(D56⋊C2), C14.67(C8⋊C22), (C2×C28).272C23, (C2×C56).108C22, C2.21(SD16⋊D7), C14.39(C8.C22), C4⋊Dic7.104C22, (C4×Dic7).29C22, (C7×C4.Q8).3C2, (C2×C7⋊C8).53C22, (C2×C14).277(C2×D4), (C7×C4⋊C4).65C22, (C2×C4).375(C22×D7), SmallGroup(448,390)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊3Q8
G = < a,b,c | a56=b4=1, c2=b2, bab-1=a43, cac-1=a13, cbc-1=b-1 >
Subgroups: 428 in 90 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, C4.Q8, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C8⋊Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C28.Q8, C4.Dic14, C56⋊C4, C56⋊1C4, C7×C4.Q8, C28⋊Q8, C28.3Q8, C56⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, C8⋊C22, C8.C22, Dic14, C22×D7, C8⋊Q8, C2×Dic14, D4×D7, Q8×D7, C28⋊Q8, D56⋊C2, SD16⋊D7, C56⋊3Q8
►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)
(1 177 139 283)(2 220 140 326)(3 207 141 313)(4 194 142 300)(5 181 143 287)(6 224 144 330)(7 211 145 317)(8 198 146 304)(9 185 147 291)(10 172 148 334)(11 215 149 321)(12 202 150 308)(13 189 151 295)(14 176 152 282)(15 219 153 325)(16 206 154 312)(17 193 155 299)(18 180 156 286)(19 223 157 329)(20 210 158 316)(21 197 159 303)(22 184 160 290)(23 171 161 333)(24 214 162 320)(25 201 163 307)(26 188 164 294)(27 175 165 281)(28 218 166 324)(29 205 167 311)(30 192 168 298)(31 179 113 285)(32 222 114 328)(33 209 115 315)(34 196 116 302)(35 183 117 289)(36 170 118 332)(37 213 119 319)(38 200 120 306)(39 187 121 293)(40 174 122 336)(41 217 123 323)(42 204 124 310)(43 191 125 297)(44 178 126 284)(45 221 127 327)(46 208 128 314)(47 195 129 301)(48 182 130 288)(49 169 131 331)(50 212 132 318)(51 199 133 305)(52 186 134 292)(53 173 135 335)(54 216 136 322)(55 203 137 309)(56 190 138 296)(57 256 439 390)(58 243 440 377)(59 230 441 364)(60 273 442 351)(61 260 443 338)(62 247 444 381)(63 234 445 368)(64 277 446 355)(65 264 447 342)(66 251 448 385)(67 238 393 372)(68 225 394 359)(69 268 395 346)(70 255 396 389)(71 242 397 376)(72 229 398 363)(73 272 399 350)(74 259 400 337)(75 246 401 380)(76 233 402 367)(77 276 403 354)(78 263 404 341)(79 250 405 384)(80 237 406 371)(81 280 407 358)(82 267 408 345)(83 254 409 388)(84 241 410 375)(85 228 411 362)(86 271 412 349)(87 258 413 392)(88 245 414 379)(89 232 415 366)(90 275 416 353)(91 262 417 340)(92 249 418 383)(93 236 419 370)(94 279 420 357)(95 266 421 344)(96 253 422 387)(97 240 423 374)(98 227 424 361)(99 270 425 348)(100 257 426 391)(101 244 427 378)(102 231 428 365)(103 274 429 352)(104 261 430 339)(105 248 431 382)(106 235 432 369)(107 278 433 356)(108 265 434 343)(109 252 435 386)(110 239 436 373)(111 226 437 360)(112 269 438 347)
(1 337 139 259)(2 350 140 272)(3 363 141 229)(4 376 142 242)(5 389 143 255)(6 346 144 268)(7 359 145 225)(8 372 146 238)(9 385 147 251)(10 342 148 264)(11 355 149 277)(12 368 150 234)(13 381 151 247)(14 338 152 260)(15 351 153 273)(16 364 154 230)(17 377 155 243)(18 390 156 256)(19 347 157 269)(20 360 158 226)(21 373 159 239)(22 386 160 252)(23 343 161 265)(24 356 162 278)(25 369 163 235)(26 382 164 248)(27 339 165 261)(28 352 166 274)(29 365 167 231)(30 378 168 244)(31 391 113 257)(32 348 114 270)(33 361 115 227)(34 374 116 240)(35 387 117 253)(36 344 118 266)(37 357 119 279)(38 370 120 236)(39 383 121 249)(40 340 122 262)(41 353 123 275)(42 366 124 232)(43 379 125 245)(44 392 126 258)(45 349 127 271)(46 362 128 228)(47 375 129 241)(48 388 130 254)(49 345 131 267)(50 358 132 280)(51 371 133 237)(52 384 134 250)(53 341 135 263)(54 354 136 276)(55 367 137 233)(56 380 138 246)(57 180 439 286)(58 193 440 299)(59 206 441 312)(60 219 442 325)(61 176 443 282)(62 189 444 295)(63 202 445 308)(64 215 446 321)(65 172 447 334)(66 185 448 291)(67 198 393 304)(68 211 394 317)(69 224 395 330)(70 181 396 287)(71 194 397 300)(72 207 398 313)(73 220 399 326)(74 177 400 283)(75 190 401 296)(76 203 402 309)(77 216 403 322)(78 173 404 335)(79 186 405 292)(80 199 406 305)(81 212 407 318)(82 169 408 331)(83 182 409 288)(84 195 410 301)(85 208 411 314)(86 221 412 327)(87 178 413 284)(88 191 414 297)(89 204 415 310)(90 217 416 323)(91 174 417 336)(92 187 418 293)(93 200 419 306)(94 213 420 319)(95 170 421 332)(96 183 422 289)(97 196 423 302)(98 209 424 315)(99 222 425 328)(100 179 426 285)(101 192 427 298)(102 205 428 311)(103 218 429 324)(104 175 430 281)(105 188 431 294)(106 201 432 307)(107 214 433 320)(108 171 434 333)(109 184 435 290)(110 197 436 303)(111 210 437 316)(112 223 438 329)
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), (1,177,139,283)(2,220,140,326)(3,207,141,313)(4,194,142,300)(5,181,143,287)(6,224,144,330)(7,211,145,317)(8,198,146,304)(9,185,147,291)(10,172,148,334)(11,215,149,321)(12,202,150,308)(13,189,151,295)(14,176,152,282)(15,219,153,325)(16,206,154,312)(17,193,155,299)(18,180,156,286)(19,223,157,329)(20,210,158,316)(21,197,159,303)(22,184,160,290)(23,171,161,333)(24,214,162,320)(25,201,163,307)(26,188,164,294)(27,175,165,281)(28,218,166,324)(29,205,167,311)(30,192,168,298)(31,179,113,285)(32,222,114,328)(33,209,115,315)(34,196,116,302)(35,183,117,289)(36,170,118,332)(37,213,119,319)(38,200,120,306)(39,187,121,293)(40,174,122,336)(41,217,123,323)(42,204,124,310)(43,191,125,297)(44,178,126,284)(45,221,127,327)(46,208,128,314)(47,195,129,301)(48,182,130,288)(49,169,131,331)(50,212,132,318)(51,199,133,305)(52,186,134,292)(53,173,135,335)(54,216,136,322)(55,203,137,309)(56,190,138,296)(57,256,439,390)(58,243,440,377)(59,230,441,364)(60,273,442,351)(61,260,443,338)(62,247,444,381)(63,234,445,368)(64,277,446,355)(65,264,447,342)(66,251,448,385)(67,238,393,372)(68,225,394,359)(69,268,395,346)(70,255,396,389)(71,242,397,376)(72,229,398,363)(73,272,399,350)(74,259,400,337)(75,246,401,380)(76,233,402,367)(77,276,403,354)(78,263,404,341)(79,250,405,384)(80,237,406,371)(81,280,407,358)(82,267,408,345)(83,254,409,388)(84,241,410,375)(85,228,411,362)(86,271,412,349)(87,258,413,392)(88,245,414,379)(89,232,415,366)(90,275,416,353)(91,262,417,340)(92,249,418,383)(93,236,419,370)(94,279,420,357)(95,266,421,344)(96,253,422,387)(97,240,423,374)(98,227,424,361)(99,270,425,348)(100,257,426,391)(101,244,427,378)(102,231,428,365)(103,274,429,352)(104,261,430,339)(105,248,431,382)(106,235,432,369)(107,278,433,356)(108,265,434,343)(109,252,435,386)(110,239,436,373)(111,226,437,360)(112,269,438,347), (1,337,139,259)(2,350,140,272)(3,363,141,229)(4,376,142,242)(5,389,143,255)(6,346,144,268)(7,359,145,225)(8,372,146,238)(9,385,147,251)(10,342,148,264)(11,355,149,277)(12,368,150,234)(13,381,151,247)(14,338,152,260)(15,351,153,273)(16,364,154,230)(17,377,155,243)(18,390,156,256)(19,347,157,269)(20,360,158,226)(21,373,159,239)(22,386,160,252)(23,343,161,265)(24,356,162,278)(25,369,163,235)(26,382,164,248)(27,339,165,261)(28,352,166,274)(29,365,167,231)(30,378,168,244)(31,391,113,257)(32,348,114,270)(33,361,115,227)(34,374,116,240)(35,387,117,253)(36,344,118,266)(37,357,119,279)(38,370,120,236)(39,383,121,249)(40,340,122,262)(41,353,123,275)(42,366,124,232)(43,379,125,245)(44,392,126,258)(45,349,127,271)(46,362,128,228)(47,375,129,241)(48,388,130,254)(49,345,131,267)(50,358,132,280)(51,371,133,237)(52,384,134,250)(53,341,135,263)(54,354,136,276)(55,367,137,233)(56,380,138,246)(57,180,439,286)(58,193,440,299)(59,206,441,312)(60,219,442,325)(61,176,443,282)(62,189,444,295)(63,202,445,308)(64,215,446,321)(65,172,447,334)(66,185,448,291)(67,198,393,304)(68,211,394,317)(69,224,395,330)(70,181,396,287)(71,194,397,300)(72,207,398,313)(73,220,399,326)(74,177,400,283)(75,190,401,296)(76,203,402,309)(77,216,403,322)(78,173,404,335)(79,186,405,292)(80,199,406,305)(81,212,407,318)(82,169,408,331)(83,182,409,288)(84,195,410,301)(85,208,411,314)(86,221,412,327)(87,178,413,284)(88,191,414,297)(89,204,415,310)(90,217,416,323)(91,174,417,336)(92,187,418,293)(93,200,419,306)(94,213,420,319)(95,170,421,332)(96,183,422,289)(97,196,423,302)(98,209,424,315)(99,222,425,328)(100,179,426,285)(101,192,427,298)(102,205,428,311)(103,218,429,324)(104,175,430,281)(105,188,431,294)(106,201,432,307)(107,214,433,320)(108,171,434,333)(109,184,435,290)(110,197,436,303)(111,210,437,316)(112,223,438,329)>;
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), (1,177,139,283)(2,220,140,326)(3,207,141,313)(4,194,142,300)(5,181,143,287)(6,224,144,330)(7,211,145,317)(8,198,146,304)(9,185,147,291)(10,172,148,334)(11,215,149,321)(12,202,150,308)(13,189,151,295)(14,176,152,282)(15,219,153,325)(16,206,154,312)(17,193,155,299)(18,180,156,286)(19,223,157,329)(20,210,158,316)(21,197,159,303)(22,184,160,290)(23,171,161,333)(24,214,162,320)(25,201,163,307)(26,188,164,294)(27,175,165,281)(28,218,166,324)(29,205,167,311)(30,192,168,298)(31,179,113,285)(32,222,114,328)(33,209,115,315)(34,196,116,302)(35,183,117,289)(36,170,118,332)(37,213,119,319)(38,200,120,306)(39,187,121,293)(40,174,122,336)(41,217,123,323)(42,204,124,310)(43,191,125,297)(44,178,126,284)(45,221,127,327)(46,208,128,314)(47,195,129,301)(48,182,130,288)(49,169,131,331)(50,212,132,318)(51,199,133,305)(52,186,134,292)(53,173,135,335)(54,216,136,322)(55,203,137,309)(56,190,138,296)(57,256,439,390)(58,243,440,377)(59,230,441,364)(60,273,442,351)(61,260,443,338)(62,247,444,381)(63,234,445,368)(64,277,446,355)(65,264,447,342)(66,251,448,385)(67,238,393,372)(68,225,394,359)(69,268,395,346)(70,255,396,389)(71,242,397,376)(72,229,398,363)(73,272,399,350)(74,259,400,337)(75,246,401,380)(76,233,402,367)(77,276,403,354)(78,263,404,341)(79,250,405,384)(80,237,406,371)(81,280,407,358)(82,267,408,345)(83,254,409,388)(84,241,410,375)(85,228,411,362)(86,271,412,349)(87,258,413,392)(88,245,414,379)(89,232,415,366)(90,275,416,353)(91,262,417,340)(92,249,418,383)(93,236,419,370)(94,279,420,357)(95,266,421,344)(96,253,422,387)(97,240,423,374)(98,227,424,361)(99,270,425,348)(100,257,426,391)(101,244,427,378)(102,231,428,365)(103,274,429,352)(104,261,430,339)(105,248,431,382)(106,235,432,369)(107,278,433,356)(108,265,434,343)(109,252,435,386)(110,239,436,373)(111,226,437,360)(112,269,438,347), (1,337,139,259)(2,350,140,272)(3,363,141,229)(4,376,142,242)(5,389,143,255)(6,346,144,268)(7,359,145,225)(8,372,146,238)(9,385,147,251)(10,342,148,264)(11,355,149,277)(12,368,150,234)(13,381,151,247)(14,338,152,260)(15,351,153,273)(16,364,154,230)(17,377,155,243)(18,390,156,256)(19,347,157,269)(20,360,158,226)(21,373,159,239)(22,386,160,252)(23,343,161,265)(24,356,162,278)(25,369,163,235)(26,382,164,248)(27,339,165,261)(28,352,166,274)(29,365,167,231)(30,378,168,244)(31,391,113,257)(32,348,114,270)(33,361,115,227)(34,374,116,240)(35,387,117,253)(36,344,118,266)(37,357,119,279)(38,370,120,236)(39,383,121,249)(40,340,122,262)(41,353,123,275)(42,366,124,232)(43,379,125,245)(44,392,126,258)(45,349,127,271)(46,362,128,228)(47,375,129,241)(48,388,130,254)(49,345,131,267)(50,358,132,280)(51,371,133,237)(52,384,134,250)(53,341,135,263)(54,354,136,276)(55,367,137,233)(56,380,138,246)(57,180,439,286)(58,193,440,299)(59,206,441,312)(60,219,442,325)(61,176,443,282)(62,189,444,295)(63,202,445,308)(64,215,446,321)(65,172,447,334)(66,185,448,291)(67,198,393,304)(68,211,394,317)(69,224,395,330)(70,181,396,287)(71,194,397,300)(72,207,398,313)(73,220,399,326)(74,177,400,283)(75,190,401,296)(76,203,402,309)(77,216,403,322)(78,173,404,335)(79,186,405,292)(80,199,406,305)(81,212,407,318)(82,169,408,331)(83,182,409,288)(84,195,410,301)(85,208,411,314)(86,221,412,327)(87,178,413,284)(88,191,414,297)(89,204,415,310)(90,217,416,323)(91,174,417,336)(92,187,418,293)(93,200,419,306)(94,213,420,319)(95,170,421,332)(96,183,422,289)(97,196,423,302)(98,209,424,315)(99,222,425,328)(100,179,426,285)(101,192,427,298)(102,205,428,311)(103,218,429,324)(104,175,430,281)(105,188,431,294)(106,201,432,307)(107,214,433,320)(108,171,434,333)(109,184,435,290)(110,197,436,303)(111,210,437,316)(112,223,438,329) );
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)], [(1,177,139,283),(2,220,140,326),(3,207,141,313),(4,194,142,300),(5,181,143,287),(6,224,144,330),(7,211,145,317),(8,198,146,304),(9,185,147,291),(10,172,148,334),(11,215,149,321),(12,202,150,308),(13,189,151,295),(14,176,152,282),(15,219,153,325),(16,206,154,312),(17,193,155,299),(18,180,156,286),(19,223,157,329),(20,210,158,316),(21,197,159,303),(22,184,160,290),(23,171,161,333),(24,214,162,320),(25,201,163,307),(26,188,164,294),(27,175,165,281),(28,218,166,324),(29,205,167,311),(30,192,168,298),(31,179,113,285),(32,222,114,328),(33,209,115,315),(34,196,116,302),(35,183,117,289),(36,170,118,332),(37,213,119,319),(38,200,120,306),(39,187,121,293),(40,174,122,336),(41,217,123,323),(42,204,124,310),(43,191,125,297),(44,178,126,284),(45,221,127,327),(46,208,128,314),(47,195,129,301),(48,182,130,288),(49,169,131,331),(50,212,132,318),(51,199,133,305),(52,186,134,292),(53,173,135,335),(54,216,136,322),(55,203,137,309),(56,190,138,296),(57,256,439,390),(58,243,440,377),(59,230,441,364),(60,273,442,351),(61,260,443,338),(62,247,444,381),(63,234,445,368),(64,277,446,355),(65,264,447,342),(66,251,448,385),(67,238,393,372),(68,225,394,359),(69,268,395,346),(70,255,396,389),(71,242,397,376),(72,229,398,363),(73,272,399,350),(74,259,400,337),(75,246,401,380),(76,233,402,367),(77,276,403,354),(78,263,404,341),(79,250,405,384),(80,237,406,371),(81,280,407,358),(82,267,408,345),(83,254,409,388),(84,241,410,375),(85,228,411,362),(86,271,412,349),(87,258,413,392),(88,245,414,379),(89,232,415,366),(90,275,416,353),(91,262,417,340),(92,249,418,383),(93,236,419,370),(94,279,420,357),(95,266,421,344),(96,253,422,387),(97,240,423,374),(98,227,424,361),(99,270,425,348),(100,257,426,391),(101,244,427,378),(102,231,428,365),(103,274,429,352),(104,261,430,339),(105,248,431,382),(106,235,432,369),(107,278,433,356),(108,265,434,343),(109,252,435,386),(110,239,436,373),(111,226,437,360),(112,269,438,347)], [(1,337,139,259),(2,350,140,272),(3,363,141,229),(4,376,142,242),(5,389,143,255),(6,346,144,268),(7,359,145,225),(8,372,146,238),(9,385,147,251),(10,342,148,264),(11,355,149,277),(12,368,150,234),(13,381,151,247),(14,338,152,260),(15,351,153,273),(16,364,154,230),(17,377,155,243),(18,390,156,256),(19,347,157,269),(20,360,158,226),(21,373,159,239),(22,386,160,252),(23,343,161,265),(24,356,162,278),(25,369,163,235),(26,382,164,248),(27,339,165,261),(28,352,166,274),(29,365,167,231),(30,378,168,244),(31,391,113,257),(32,348,114,270),(33,361,115,227),(34,374,116,240),(35,387,117,253),(36,344,118,266),(37,357,119,279),(38,370,120,236),(39,383,121,249),(40,340,122,262),(41,353,123,275),(42,366,124,232),(43,379,125,245),(44,392,126,258),(45,349,127,271),(46,362,128,228),(47,375,129,241),(48,388,130,254),(49,345,131,267),(50,358,132,280),(51,371,133,237),(52,384,134,250),(53,341,135,263),(54,354,136,276),(55,367,137,233),(56,380,138,246),(57,180,439,286),(58,193,440,299),(59,206,441,312),(60,219,442,325),(61,176,443,282),(62,189,444,295),(63,202,445,308),(64,215,446,321),(65,172,447,334),(66,185,448,291),(67,198,393,304),(68,211,394,317),(69,224,395,330),(70,181,396,287),(71,194,397,300),(72,207,398,313),(73,220,399,326),(74,177,400,283),(75,190,401,296),(76,203,402,309),(77,216,403,322),(78,173,404,335),(79,186,405,292),(80,199,406,305),(81,212,407,318),(82,169,408,331),(83,182,409,288),(84,195,410,301),(85,208,411,314),(86,221,412,327),(87,178,413,284),(88,191,414,297),(89,204,415,310),(90,217,416,323),(91,174,417,336),(92,187,418,293),(93,200,419,306),(94,213,420,319),(95,170,421,332),(96,183,422,289),(97,196,423,302),(98,209,424,315),(99,222,425,328),(100,179,426,285),(101,192,427,298),(102,205,428,311),(103,218,429,324),(104,175,430,281),(105,188,431,294),(106,201,432,307),(107,214,433,320),(108,171,434,333),(109,184,435,290),(110,197,436,303),(111,210,437,316),(112,223,438,329)]])
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 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | - | + | - | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | Q8 | D4 | D7 | D14 | D14 | Dic14 | C8⋊C22 | C8.C22 | Q8×D7 | D4×D7 | D56⋊C2 | SD16⋊D7 |
| kernel | C56⋊3Q8 | C28.Q8 | C4.Dic14 | C56⋊C4 | C56⋊1C4 | C7×C4.Q8 | C28⋊Q8 | C28.3Q8 | C7⋊C8 | C56 | C2×Dic7 | C4.Q8 | C4⋊C4 | C2×C8 | C8 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 6 | 3 | 12 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C56⋊3Q8 ►in GL8(𝔽113)
| 100 | 41 | 83 | 42 | 0 | 0 | 0 | 0 |
| 72 | 67 | 71 | 74 | 0 | 0 | 0 | 0 |
| 85 | 62 | 13 | 72 | 0 | 0 | 0 | 0 |
| 51 | 104 | 41 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 21 | 29 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 74 |
| 0 | 0 | 0 | 0 | 76 | 36 | 68 | 20 |
| 0 | 0 | 0 | 0 | 93 | 21 | 73 | 78 |
| 1 | 0 | 111 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 111 | 0 | 0 | 0 | 0 |
| 1 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 28 | 43 | 40 |
| 0 | 0 | 0 | 0 | 85 | 79 | 18 | 7 |
| 0 | 0 | 0 | 0 | 28 | 103 | 54 | 85 |
| 0 | 0 | 0 | 0 | 4 | 37 | 56 | 40 |
| 78 | 85 | 85 | 68 | 0 | 0 | 0 | 0 |
| 36 | 35 | 74 | 28 | 0 | 0 | 0 | 0 |
| 64 | 6 | 35 | 28 | 0 | 0 | 0 | 0 |
| 73 | 49 | 77 | 78 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 70 | 27 | 12 | 93 |
| 0 | 0 | 0 | 0 | 92 | 43 | 17 | 16 |
| 0 | 0 | 0 | 0 | 105 | 103 | 112 | 7 |
| 0 | 0 | 0 | 0 | 65 | 107 | 70 | 1 |
G:=sub<GL(8,GF(113))| [100,72,85,51,0,0,0,0,41,67,62,104,0,0,0,0,83,71,13,41,0,0,0,0,42,74,72,46,0,0,0,0,0,0,0,0,0,0,76,93,0,0,0,0,0,0,36,21,0,0,0,0,21,3,68,73,0,0,0,0,29,74,20,78],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,111,0,112,0,0,0,0,0,0,111,0,112,0,0,0,0,0,0,0,0,53,85,28,4,0,0,0,0,28,79,103,37,0,0,0,0,43,18,54,56,0,0,0,0,40,7,85,40],[78,36,64,73,0,0,0,0,85,35,6,49,0,0,0,0,85,74,35,77,0,0,0,0,68,28,28,78,0,0,0,0,0,0,0,0,70,92,105,65,0,0,0,0,27,43,103,107,0,0,0,0,12,17,112,70,0,0,0,0,93,16,7,1] >;
C56⋊3Q8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_3Q_8 % in TeX
G:=Group("C56:3Q8"); // GroupNames label
G:=SmallGroup(448,390);
// by ID
G=gap.SmallGroup(448,390);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,120,254,555,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=b^2,b*a*b^-1=a^43,c*a*c^-1=a^13,c*b*c^-1=b^-1>;
// generators/relations