metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊8Q8, C4.4D56, C8⋊7Dic14, C28.29D8, C28.15Q16, C4.4Dic28, C42.254D14, (C4×C8).7D7, (C4×C56).9C2, C14.1(C2×D8), C7⋊1(C8⋊2Q8), C2.4(C2×D56), (C2×C4).78D28, C14.3(C4⋊Q8), C14.2(C2×Q16), C28.70(C2×Q8), C56⋊1C4.4C2, (C2×C28).375D4, (C2×C8).300D14, C28⋊2Q8.3C2, C2.5(C2×Dic28), C2.7(C28⋊2Q8), C4.36(C2×Dic14), C22.87(C2×D28), C4⋊Dic7.3C22, (C2×C28).718C23, (C4×C28).304C22, (C2×C56).373C22, (C2×C14).101(C2×D4), (C2×C4).661(C22×D7), SmallGroup(448,216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊8Q8
G = < a,b,c | a56=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 516 in 98 conjugacy classes, 55 normal (21 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C2×C14, C4×C8, C2.D8, C4⋊Q8, C56, Dic14, C2×Dic7, C2×C28, C8⋊2Q8, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C56⋊1C4, C4×C56, C28⋊2Q8, C56⋊8Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, D8, Q16, C2×D4, C2×Q8, D14, C4⋊Q8, C2×D8, C2×Q16, Dic14, D28, C22×D7, C8⋊2Q8, D56, Dic28, C2×Dic14, C2×D28, C28⋊2Q8, C2×D56, C2×Dic28, C56⋊8Q8
►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)
(1 354 411 325)(2 355 412 326)(3 356 413 327)(4 357 414 328)(5 358 415 329)(6 359 416 330)(7 360 417 331)(8 361 418 332)(9 362 419 333)(10 363 420 334)(11 364 421 335)(12 365 422 336)(13 366 423 281)(14 367 424 282)(15 368 425 283)(16 369 426 284)(17 370 427 285)(18 371 428 286)(19 372 429 287)(20 373 430 288)(21 374 431 289)(22 375 432 290)(23 376 433 291)(24 377 434 292)(25 378 435 293)(26 379 436 294)(27 380 437 295)(28 381 438 296)(29 382 439 297)(30 383 440 298)(31 384 441 299)(32 385 442 300)(33 386 443 301)(34 387 444 302)(35 388 445 303)(36 389 446 304)(37 390 447 305)(38 391 448 306)(39 392 393 307)(40 337 394 308)(41 338 395 309)(42 339 396 310)(43 340 397 311)(44 341 398 312)(45 342 399 313)(46 343 400 314)(47 344 401 315)(48 345 402 316)(49 346 403 317)(50 347 404 318)(51 348 405 319)(52 349 406 320)(53 350 407 321)(54 351 408 322)(55 352 409 323)(56 353 410 324)(57 254 224 130)(58 255 169 131)(59 256 170 132)(60 257 171 133)(61 258 172 134)(62 259 173 135)(63 260 174 136)(64 261 175 137)(65 262 176 138)(66 263 177 139)(67 264 178 140)(68 265 179 141)(69 266 180 142)(70 267 181 143)(71 268 182 144)(72 269 183 145)(73 270 184 146)(74 271 185 147)(75 272 186 148)(76 273 187 149)(77 274 188 150)(78 275 189 151)(79 276 190 152)(80 277 191 153)(81 278 192 154)(82 279 193 155)(83 280 194 156)(84 225 195 157)(85 226 196 158)(86 227 197 159)(87 228 198 160)(88 229 199 161)(89 230 200 162)(90 231 201 163)(91 232 202 164)(92 233 203 165)(93 234 204 166)(94 235 205 167)(95 236 206 168)(96 237 207 113)(97 238 208 114)(98 239 209 115)(99 240 210 116)(100 241 211 117)(101 242 212 118)(102 243 213 119)(103 244 214 120)(104 245 215 121)(105 246 216 122)(106 247 217 123)(107 248 218 124)(108 249 219 125)(109 250 220 126)(110 251 221 127)(111 252 222 128)(112 253 223 129)
(1 153 411 277)(2 152 412 276)(3 151 413 275)(4 150 414 274)(5 149 415 273)(6 148 416 272)(7 147 417 271)(8 146 418 270)(9 145 419 269)(10 144 420 268)(11 143 421 267)(12 142 422 266)(13 141 423 265)(14 140 424 264)(15 139 425 263)(16 138 426 262)(17 137 427 261)(18 136 428 260)(19 135 429 259)(20 134 430 258)(21 133 431 257)(22 132 432 256)(23 131 433 255)(24 130 434 254)(25 129 435 253)(26 128 436 252)(27 127 437 251)(28 126 438 250)(29 125 439 249)(30 124 440 248)(31 123 441 247)(32 122 442 246)(33 121 443 245)(34 120 444 244)(35 119 445 243)(36 118 446 242)(37 117 447 241)(38 116 448 240)(39 115 393 239)(40 114 394 238)(41 113 395 237)(42 168 396 236)(43 167 397 235)(44 166 398 234)(45 165 399 233)(46 164 400 232)(47 163 401 231)(48 162 402 230)(49 161 403 229)(50 160 404 228)(51 159 405 227)(52 158 406 226)(53 157 407 225)(54 156 408 280)(55 155 409 279)(56 154 410 278)(57 377 224 292)(58 376 169 291)(59 375 170 290)(60 374 171 289)(61 373 172 288)(62 372 173 287)(63 371 174 286)(64 370 175 285)(65 369 176 284)(66 368 177 283)(67 367 178 282)(68 366 179 281)(69 365 180 336)(70 364 181 335)(71 363 182 334)(72 362 183 333)(73 361 184 332)(74 360 185 331)(75 359 186 330)(76 358 187 329)(77 357 188 328)(78 356 189 327)(79 355 190 326)(80 354 191 325)(81 353 192 324)(82 352 193 323)(83 351 194 322)(84 350 195 321)(85 349 196 320)(86 348 197 319)(87 347 198 318)(88 346 199 317)(89 345 200 316)(90 344 201 315)(91 343 202 314)(92 342 203 313)(93 341 204 312)(94 340 205 311)(95 339 206 310)(96 338 207 309)(97 337 208 308)(98 392 209 307)(99 391 210 306)(100 390 211 305)(101 389 212 304)(102 388 213 303)(103 387 214 302)(104 386 215 301)(105 385 216 300)(106 384 217 299)(107 383 218 298)(108 382 219 297)(109 381 220 296)(110 380 221 295)(111 379 222 294)(112 378 223 293)
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), (1,354,411,325)(2,355,412,326)(3,356,413,327)(4,357,414,328)(5,358,415,329)(6,359,416,330)(7,360,417,331)(8,361,418,332)(9,362,419,333)(10,363,420,334)(11,364,421,335)(12,365,422,336)(13,366,423,281)(14,367,424,282)(15,368,425,283)(16,369,426,284)(17,370,427,285)(18,371,428,286)(19,372,429,287)(20,373,430,288)(21,374,431,289)(22,375,432,290)(23,376,433,291)(24,377,434,292)(25,378,435,293)(26,379,436,294)(27,380,437,295)(28,381,438,296)(29,382,439,297)(30,383,440,298)(31,384,441,299)(32,385,442,300)(33,386,443,301)(34,387,444,302)(35,388,445,303)(36,389,446,304)(37,390,447,305)(38,391,448,306)(39,392,393,307)(40,337,394,308)(41,338,395,309)(42,339,396,310)(43,340,397,311)(44,341,398,312)(45,342,399,313)(46,343,400,314)(47,344,401,315)(48,345,402,316)(49,346,403,317)(50,347,404,318)(51,348,405,319)(52,349,406,320)(53,350,407,321)(54,351,408,322)(55,352,409,323)(56,353,410,324)(57,254,224,130)(58,255,169,131)(59,256,170,132)(60,257,171,133)(61,258,172,134)(62,259,173,135)(63,260,174,136)(64,261,175,137)(65,262,176,138)(66,263,177,139)(67,264,178,140)(68,265,179,141)(69,266,180,142)(70,267,181,143)(71,268,182,144)(72,269,183,145)(73,270,184,146)(74,271,185,147)(75,272,186,148)(76,273,187,149)(77,274,188,150)(78,275,189,151)(79,276,190,152)(80,277,191,153)(81,278,192,154)(82,279,193,155)(83,280,194,156)(84,225,195,157)(85,226,196,158)(86,227,197,159)(87,228,198,160)(88,229,199,161)(89,230,200,162)(90,231,201,163)(91,232,202,164)(92,233,203,165)(93,234,204,166)(94,235,205,167)(95,236,206,168)(96,237,207,113)(97,238,208,114)(98,239,209,115)(99,240,210,116)(100,241,211,117)(101,242,212,118)(102,243,213,119)(103,244,214,120)(104,245,215,121)(105,246,216,122)(106,247,217,123)(107,248,218,124)(108,249,219,125)(109,250,220,126)(110,251,221,127)(111,252,222,128)(112,253,223,129), (1,153,411,277)(2,152,412,276)(3,151,413,275)(4,150,414,274)(5,149,415,273)(6,148,416,272)(7,147,417,271)(8,146,418,270)(9,145,419,269)(10,144,420,268)(11,143,421,267)(12,142,422,266)(13,141,423,265)(14,140,424,264)(15,139,425,263)(16,138,426,262)(17,137,427,261)(18,136,428,260)(19,135,429,259)(20,134,430,258)(21,133,431,257)(22,132,432,256)(23,131,433,255)(24,130,434,254)(25,129,435,253)(26,128,436,252)(27,127,437,251)(28,126,438,250)(29,125,439,249)(30,124,440,248)(31,123,441,247)(32,122,442,246)(33,121,443,245)(34,120,444,244)(35,119,445,243)(36,118,446,242)(37,117,447,241)(38,116,448,240)(39,115,393,239)(40,114,394,238)(41,113,395,237)(42,168,396,236)(43,167,397,235)(44,166,398,234)(45,165,399,233)(46,164,400,232)(47,163,401,231)(48,162,402,230)(49,161,403,229)(50,160,404,228)(51,159,405,227)(52,158,406,226)(53,157,407,225)(54,156,408,280)(55,155,409,279)(56,154,410,278)(57,377,224,292)(58,376,169,291)(59,375,170,290)(60,374,171,289)(61,373,172,288)(62,372,173,287)(63,371,174,286)(64,370,175,285)(65,369,176,284)(66,368,177,283)(67,367,178,282)(68,366,179,281)(69,365,180,336)(70,364,181,335)(71,363,182,334)(72,362,183,333)(73,361,184,332)(74,360,185,331)(75,359,186,330)(76,358,187,329)(77,357,188,328)(78,356,189,327)(79,355,190,326)(80,354,191,325)(81,353,192,324)(82,352,193,323)(83,351,194,322)(84,350,195,321)(85,349,196,320)(86,348,197,319)(87,347,198,318)(88,346,199,317)(89,345,200,316)(90,344,201,315)(91,343,202,314)(92,342,203,313)(93,341,204,312)(94,340,205,311)(95,339,206,310)(96,338,207,309)(97,337,208,308)(98,392,209,307)(99,391,210,306)(100,390,211,305)(101,389,212,304)(102,388,213,303)(103,387,214,302)(104,386,215,301)(105,385,216,300)(106,384,217,299)(107,383,218,298)(108,382,219,297)(109,381,220,296)(110,380,221,295)(111,379,222,294)(112,378,223,293)>;
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), (1,354,411,325)(2,355,412,326)(3,356,413,327)(4,357,414,328)(5,358,415,329)(6,359,416,330)(7,360,417,331)(8,361,418,332)(9,362,419,333)(10,363,420,334)(11,364,421,335)(12,365,422,336)(13,366,423,281)(14,367,424,282)(15,368,425,283)(16,369,426,284)(17,370,427,285)(18,371,428,286)(19,372,429,287)(20,373,430,288)(21,374,431,289)(22,375,432,290)(23,376,433,291)(24,377,434,292)(25,378,435,293)(26,379,436,294)(27,380,437,295)(28,381,438,296)(29,382,439,297)(30,383,440,298)(31,384,441,299)(32,385,442,300)(33,386,443,301)(34,387,444,302)(35,388,445,303)(36,389,446,304)(37,390,447,305)(38,391,448,306)(39,392,393,307)(40,337,394,308)(41,338,395,309)(42,339,396,310)(43,340,397,311)(44,341,398,312)(45,342,399,313)(46,343,400,314)(47,344,401,315)(48,345,402,316)(49,346,403,317)(50,347,404,318)(51,348,405,319)(52,349,406,320)(53,350,407,321)(54,351,408,322)(55,352,409,323)(56,353,410,324)(57,254,224,130)(58,255,169,131)(59,256,170,132)(60,257,171,133)(61,258,172,134)(62,259,173,135)(63,260,174,136)(64,261,175,137)(65,262,176,138)(66,263,177,139)(67,264,178,140)(68,265,179,141)(69,266,180,142)(70,267,181,143)(71,268,182,144)(72,269,183,145)(73,270,184,146)(74,271,185,147)(75,272,186,148)(76,273,187,149)(77,274,188,150)(78,275,189,151)(79,276,190,152)(80,277,191,153)(81,278,192,154)(82,279,193,155)(83,280,194,156)(84,225,195,157)(85,226,196,158)(86,227,197,159)(87,228,198,160)(88,229,199,161)(89,230,200,162)(90,231,201,163)(91,232,202,164)(92,233,203,165)(93,234,204,166)(94,235,205,167)(95,236,206,168)(96,237,207,113)(97,238,208,114)(98,239,209,115)(99,240,210,116)(100,241,211,117)(101,242,212,118)(102,243,213,119)(103,244,214,120)(104,245,215,121)(105,246,216,122)(106,247,217,123)(107,248,218,124)(108,249,219,125)(109,250,220,126)(110,251,221,127)(111,252,222,128)(112,253,223,129), (1,153,411,277)(2,152,412,276)(3,151,413,275)(4,150,414,274)(5,149,415,273)(6,148,416,272)(7,147,417,271)(8,146,418,270)(9,145,419,269)(10,144,420,268)(11,143,421,267)(12,142,422,266)(13,141,423,265)(14,140,424,264)(15,139,425,263)(16,138,426,262)(17,137,427,261)(18,136,428,260)(19,135,429,259)(20,134,430,258)(21,133,431,257)(22,132,432,256)(23,131,433,255)(24,130,434,254)(25,129,435,253)(26,128,436,252)(27,127,437,251)(28,126,438,250)(29,125,439,249)(30,124,440,248)(31,123,441,247)(32,122,442,246)(33,121,443,245)(34,120,444,244)(35,119,445,243)(36,118,446,242)(37,117,447,241)(38,116,448,240)(39,115,393,239)(40,114,394,238)(41,113,395,237)(42,168,396,236)(43,167,397,235)(44,166,398,234)(45,165,399,233)(46,164,400,232)(47,163,401,231)(48,162,402,230)(49,161,403,229)(50,160,404,228)(51,159,405,227)(52,158,406,226)(53,157,407,225)(54,156,408,280)(55,155,409,279)(56,154,410,278)(57,377,224,292)(58,376,169,291)(59,375,170,290)(60,374,171,289)(61,373,172,288)(62,372,173,287)(63,371,174,286)(64,370,175,285)(65,369,176,284)(66,368,177,283)(67,367,178,282)(68,366,179,281)(69,365,180,336)(70,364,181,335)(71,363,182,334)(72,362,183,333)(73,361,184,332)(74,360,185,331)(75,359,186,330)(76,358,187,329)(77,357,188,328)(78,356,189,327)(79,355,190,326)(80,354,191,325)(81,353,192,324)(82,352,193,323)(83,351,194,322)(84,350,195,321)(85,349,196,320)(86,348,197,319)(87,347,198,318)(88,346,199,317)(89,345,200,316)(90,344,201,315)(91,343,202,314)(92,342,203,313)(93,341,204,312)(94,340,205,311)(95,339,206,310)(96,338,207,309)(97,337,208,308)(98,392,209,307)(99,391,210,306)(100,390,211,305)(101,389,212,304)(102,388,213,303)(103,387,214,302)(104,386,215,301)(105,385,216,300)(106,384,217,299)(107,383,218,298)(108,382,219,297)(109,381,220,296)(110,380,221,295)(111,379,222,294)(112,378,223,293) );
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)], [(1,354,411,325),(2,355,412,326),(3,356,413,327),(4,357,414,328),(5,358,415,329),(6,359,416,330),(7,360,417,331),(8,361,418,332),(9,362,419,333),(10,363,420,334),(11,364,421,335),(12,365,422,336),(13,366,423,281),(14,367,424,282),(15,368,425,283),(16,369,426,284),(17,370,427,285),(18,371,428,286),(19,372,429,287),(20,373,430,288),(21,374,431,289),(22,375,432,290),(23,376,433,291),(24,377,434,292),(25,378,435,293),(26,379,436,294),(27,380,437,295),(28,381,438,296),(29,382,439,297),(30,383,440,298),(31,384,441,299),(32,385,442,300),(33,386,443,301),(34,387,444,302),(35,388,445,303),(36,389,446,304),(37,390,447,305),(38,391,448,306),(39,392,393,307),(40,337,394,308),(41,338,395,309),(42,339,396,310),(43,340,397,311),(44,341,398,312),(45,342,399,313),(46,343,400,314),(47,344,401,315),(48,345,402,316),(49,346,403,317),(50,347,404,318),(51,348,405,319),(52,349,406,320),(53,350,407,321),(54,351,408,322),(55,352,409,323),(56,353,410,324),(57,254,224,130),(58,255,169,131),(59,256,170,132),(60,257,171,133),(61,258,172,134),(62,259,173,135),(63,260,174,136),(64,261,175,137),(65,262,176,138),(66,263,177,139),(67,264,178,140),(68,265,179,141),(69,266,180,142),(70,267,181,143),(71,268,182,144),(72,269,183,145),(73,270,184,146),(74,271,185,147),(75,272,186,148),(76,273,187,149),(77,274,188,150),(78,275,189,151),(79,276,190,152),(80,277,191,153),(81,278,192,154),(82,279,193,155),(83,280,194,156),(84,225,195,157),(85,226,196,158),(86,227,197,159),(87,228,198,160),(88,229,199,161),(89,230,200,162),(90,231,201,163),(91,232,202,164),(92,233,203,165),(93,234,204,166),(94,235,205,167),(95,236,206,168),(96,237,207,113),(97,238,208,114),(98,239,209,115),(99,240,210,116),(100,241,211,117),(101,242,212,118),(102,243,213,119),(103,244,214,120),(104,245,215,121),(105,246,216,122),(106,247,217,123),(107,248,218,124),(108,249,219,125),(109,250,220,126),(110,251,221,127),(111,252,222,128),(112,253,223,129)], [(1,153,411,277),(2,152,412,276),(3,151,413,275),(4,150,414,274),(5,149,415,273),(6,148,416,272),(7,147,417,271),(8,146,418,270),(9,145,419,269),(10,144,420,268),(11,143,421,267),(12,142,422,266),(13,141,423,265),(14,140,424,264),(15,139,425,263),(16,138,426,262),(17,137,427,261),(18,136,428,260),(19,135,429,259),(20,134,430,258),(21,133,431,257),(22,132,432,256),(23,131,433,255),(24,130,434,254),(25,129,435,253),(26,128,436,252),(27,127,437,251),(28,126,438,250),(29,125,439,249),(30,124,440,248),(31,123,441,247),(32,122,442,246),(33,121,443,245),(34,120,444,244),(35,119,445,243),(36,118,446,242),(37,117,447,241),(38,116,448,240),(39,115,393,239),(40,114,394,238),(41,113,395,237),(42,168,396,236),(43,167,397,235),(44,166,398,234),(45,165,399,233),(46,164,400,232),(47,163,401,231),(48,162,402,230),(49,161,403,229),(50,160,404,228),(51,159,405,227),(52,158,406,226),(53,157,407,225),(54,156,408,280),(55,155,409,279),(56,154,410,278),(57,377,224,292),(58,376,169,291),(59,375,170,290),(60,374,171,289),(61,373,172,288),(62,372,173,287),(63,371,174,286),(64,370,175,285),(65,369,176,284),(66,368,177,283),(67,367,178,282),(68,366,179,281),(69,365,180,336),(70,364,181,335),(71,363,182,334),(72,362,183,333),(73,361,184,332),(74,360,185,331),(75,359,186,330),(76,358,187,329),(77,357,188,328),(78,356,189,327),(79,355,190,326),(80,354,191,325),(81,353,192,324),(82,352,193,323),(83,351,194,322),(84,350,195,321),(85,349,196,320),(86,348,197,319),(87,347,198,318),(88,346,199,317),(89,345,200,316),(90,344,201,315),(91,343,202,314),(92,342,203,313),(93,341,204,312),(94,340,205,311),(95,339,206,310),(96,338,207,309),(97,337,208,308),(98,392,209,307),(99,391,210,306),(100,390,211,305),(101,389,212,304),(102,388,213,303),(103,387,214,302),(104,386,215,301),(105,385,216,300),(106,384,217,299),(107,383,218,298),(108,382,219,297),(109,381,220,296),(110,380,221,295),(111,379,222,294),(112,378,223,293)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D4 | D7 | D8 | Q16 | D14 | D14 | Dic14 | D28 | D56 | Dic28 |
| kernel | C56⋊8Q8 | C56⋊1C4 | C4×C56 | C28⋊2Q8 | C56 | C2×C28 | C4×C8 | C28 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 3 | 4 | 4 | 3 | 6 | 24 | 12 | 24 | 24 |
Matrix representation of C56⋊8Q8 ►in GL4(𝔽113) generated by
| 80 | 22 | 0 | 0 |
| 60 | 49 | 0 | 0 |
| 0 | 0 | 61 | 1 |
| 0 | 0 | 7 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 112 |
| 0 | 0 | 106 | 61 |
| 49 | 91 | 0 | 0 |
| 68 | 64 | 0 | 0 |
| 0 | 0 | 108 | 97 |
| 0 | 0 | 44 | 5 |
G:=sub<GL(4,GF(113))| [80,60,0,0,22,49,0,0,0,0,61,7,0,0,1,52],[1,0,0,0,0,1,0,0,0,0,52,106,0,0,112,61],[49,68,0,0,91,64,0,0,0,0,108,44,0,0,97,5] >;
C56⋊8Q8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_8Q_8 % in TeX
G:=Group("C56:8Q8"); // GroupNames label
G:=SmallGroup(448,216);
// by ID
G=gap.SmallGroup(448,216);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,254,226,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations