metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.8Q8, C8.8Dic14, C7⋊C8.1Q8, C4.22(Q8×D7), C4⋊C4.35D14, C4.Q8.7D7, C28.58(C2×Q8), C7⋊2(C8.5Q8), (C2×C8).257D14, C14.15(C4⋊Q8), C2.10(C28⋊Q8), (C8×Dic7).7C2, C8⋊Dic7.14C2, C14.53(C4○D8), (C2×Dic7).97D4, C4.22(C2×Dic14), C28.Q8.6C2, C22.213(D4×D7), C28.3Q8.6C2, (C2×C28).274C23, (C2×C56).158C22, C4⋊Dic7.106C22, C2.21(SD16⋊3D7), (C4×Dic7).230C22, (C7×C4.Q8).5C2, (C2×C14).279(C2×D4), (C7×C4⋊C4).67C22, (C2×C7⋊C8).225C22, (C2×C4).377(C22×D7), SmallGroup(448,392)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.8Q8
G = < a,b,c | a56=b4=1, c2=b2, bab-1=a43, cac-1=a41, cbc-1=a28b-1 >
Subgroups: 364 in 86 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Dic7, C28, C28, C2×C14, C4×C8, C4.Q8, C4.Q8, C2.D8, C42.C2, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C8.5Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C2×C56, C28.Q8, C8×Dic7, C8⋊Dic7, C7×C4.Q8, C28.3Q8, C56.8Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, C4○D8, Dic14, C22×D7, C8.5Q8, C2×Dic14, D4×D7, Q8×D7, C28⋊Q8, SD16⋊3D7, 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 62 229 376)(2 105 230 363)(3 92 231 350)(4 79 232 337)(5 66 233 380)(6 109 234 367)(7 96 235 354)(8 83 236 341)(9 70 237 384)(10 57 238 371)(11 100 239 358)(12 87 240 345)(13 74 241 388)(14 61 242 375)(15 104 243 362)(16 91 244 349)(17 78 245 392)(18 65 246 379)(19 108 247 366)(20 95 248 353)(21 82 249 340)(22 69 250 383)(23 112 251 370)(24 99 252 357)(25 86 253 344)(26 73 254 387)(27 60 255 374)(28 103 256 361)(29 90 257 348)(30 77 258 391)(31 64 259 378)(32 107 260 365)(33 94 261 352)(34 81 262 339)(35 68 263 382)(36 111 264 369)(37 98 265 356)(38 85 266 343)(39 72 267 386)(40 59 268 373)(41 102 269 360)(42 89 270 347)(43 76 271 390)(44 63 272 377)(45 106 273 364)(46 93 274 351)(47 80 275 338)(48 67 276 381)(49 110 277 368)(50 97 278 355)(51 84 279 342)(52 71 280 385)(53 58 225 372)(54 101 226 359)(55 88 227 346)(56 75 228 389)(113 211 439 294)(114 198 440 281)(115 185 441 324)(116 172 442 311)(117 215 443 298)(118 202 444 285)(119 189 445 328)(120 176 446 315)(121 219 447 302)(122 206 448 289)(123 193 393 332)(124 180 394 319)(125 223 395 306)(126 210 396 293)(127 197 397 336)(128 184 398 323)(129 171 399 310)(130 214 400 297)(131 201 401 284)(132 188 402 327)(133 175 403 314)(134 218 404 301)(135 205 405 288)(136 192 406 331)(137 179 407 318)(138 222 408 305)(139 209 409 292)(140 196 410 335)(141 183 411 322)(142 170 412 309)(143 213 413 296)(144 200 414 283)(145 187 415 326)(146 174 416 313)(147 217 417 300)(148 204 418 287)(149 191 419 330)(150 178 420 317)(151 221 421 304)(152 208 422 291)(153 195 423 334)(154 182 424 321)(155 169 425 308)(156 212 426 295)(157 199 427 282)(158 186 428 325)(159 173 429 312)(160 216 430 299)(161 203 431 286)(162 190 432 329)(163 177 433 316)(164 220 434 303)(165 207 435 290)(166 194 436 333)(167 181 437 320)(168 224 438 307)
(1 201 229 284)(2 186 230 325)(3 171 231 310)(4 212 232 295)(5 197 233 336)(6 182 234 321)(7 223 235 306)(8 208 236 291)(9 193 237 332)(10 178 238 317)(11 219 239 302)(12 204 240 287)(13 189 241 328)(14 174 242 313)(15 215 243 298)(16 200 244 283)(17 185 245 324)(18 170 246 309)(19 211 247 294)(20 196 248 335)(21 181 249 320)(22 222 250 305)(23 207 251 290)(24 192 252 331)(25 177 253 316)(26 218 254 301)(27 203 255 286)(28 188 256 327)(29 173 257 312)(30 214 258 297)(31 199 259 282)(32 184 260 323)(33 169 261 308)(34 210 262 293)(35 195 263 334)(36 180 264 319)(37 221 265 304)(38 206 266 289)(39 191 267 330)(40 176 268 315)(41 217 269 300)(42 202 270 285)(43 187 271 326)(44 172 272 311)(45 213 273 296)(46 198 274 281)(47 183 275 322)(48 224 276 307)(49 209 277 292)(50 194 278 333)(51 179 279 318)(52 220 280 303)(53 205 225 288)(54 190 226 329)(55 175 227 314)(56 216 228 299)(57 122 371 448)(58 163 372 433)(59 148 373 418)(60 133 374 403)(61 118 375 444)(62 159 376 429)(63 144 377 414)(64 129 378 399)(65 114 379 440)(66 155 380 425)(67 140 381 410)(68 125 382 395)(69 166 383 436)(70 151 384 421)(71 136 385 406)(72 121 386 447)(73 162 387 432)(74 147 388 417)(75 132 389 402)(76 117 390 443)(77 158 391 428)(78 143 392 413)(79 128 337 398)(80 113 338 439)(81 154 339 424)(82 139 340 409)(83 124 341 394)(84 165 342 435)(85 150 343 420)(86 135 344 405)(87 120 345 446)(88 161 346 431)(89 146 347 416)(90 131 348 401)(91 116 349 442)(92 157 350 427)(93 142 351 412)(94 127 352 397)(95 168 353 438)(96 153 354 423)(97 138 355 408)(98 123 356 393)(99 164 357 434)(100 149 358 419)(101 134 359 404)(102 119 360 445)(103 160 361 430)(104 145 362 415)(105 130 363 400)(106 115 364 441)(107 156 365 426)(108 141 366 411)(109 126 367 396)(110 167 368 437)(111 152 369 422)(112 137 370 407)
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), (1,62,229,376)(2,105,230,363)(3,92,231,350)(4,79,232,337)(5,66,233,380)(6,109,234,367)(7,96,235,354)(8,83,236,341)(9,70,237,384)(10,57,238,371)(11,100,239,358)(12,87,240,345)(13,74,241,388)(14,61,242,375)(15,104,243,362)(16,91,244,349)(17,78,245,392)(18,65,246,379)(19,108,247,366)(20,95,248,353)(21,82,249,340)(22,69,250,383)(23,112,251,370)(24,99,252,357)(25,86,253,344)(26,73,254,387)(27,60,255,374)(28,103,256,361)(29,90,257,348)(30,77,258,391)(31,64,259,378)(32,107,260,365)(33,94,261,352)(34,81,262,339)(35,68,263,382)(36,111,264,369)(37,98,265,356)(38,85,266,343)(39,72,267,386)(40,59,268,373)(41,102,269,360)(42,89,270,347)(43,76,271,390)(44,63,272,377)(45,106,273,364)(46,93,274,351)(47,80,275,338)(48,67,276,381)(49,110,277,368)(50,97,278,355)(51,84,279,342)(52,71,280,385)(53,58,225,372)(54,101,226,359)(55,88,227,346)(56,75,228,389)(113,211,439,294)(114,198,440,281)(115,185,441,324)(116,172,442,311)(117,215,443,298)(118,202,444,285)(119,189,445,328)(120,176,446,315)(121,219,447,302)(122,206,448,289)(123,193,393,332)(124,180,394,319)(125,223,395,306)(126,210,396,293)(127,197,397,336)(128,184,398,323)(129,171,399,310)(130,214,400,297)(131,201,401,284)(132,188,402,327)(133,175,403,314)(134,218,404,301)(135,205,405,288)(136,192,406,331)(137,179,407,318)(138,222,408,305)(139,209,409,292)(140,196,410,335)(141,183,411,322)(142,170,412,309)(143,213,413,296)(144,200,414,283)(145,187,415,326)(146,174,416,313)(147,217,417,300)(148,204,418,287)(149,191,419,330)(150,178,420,317)(151,221,421,304)(152,208,422,291)(153,195,423,334)(154,182,424,321)(155,169,425,308)(156,212,426,295)(157,199,427,282)(158,186,428,325)(159,173,429,312)(160,216,430,299)(161,203,431,286)(162,190,432,329)(163,177,433,316)(164,220,434,303)(165,207,435,290)(166,194,436,333)(167,181,437,320)(168,224,438,307), (1,201,229,284)(2,186,230,325)(3,171,231,310)(4,212,232,295)(5,197,233,336)(6,182,234,321)(7,223,235,306)(8,208,236,291)(9,193,237,332)(10,178,238,317)(11,219,239,302)(12,204,240,287)(13,189,241,328)(14,174,242,313)(15,215,243,298)(16,200,244,283)(17,185,245,324)(18,170,246,309)(19,211,247,294)(20,196,248,335)(21,181,249,320)(22,222,250,305)(23,207,251,290)(24,192,252,331)(25,177,253,316)(26,218,254,301)(27,203,255,286)(28,188,256,327)(29,173,257,312)(30,214,258,297)(31,199,259,282)(32,184,260,323)(33,169,261,308)(34,210,262,293)(35,195,263,334)(36,180,264,319)(37,221,265,304)(38,206,266,289)(39,191,267,330)(40,176,268,315)(41,217,269,300)(42,202,270,285)(43,187,271,326)(44,172,272,311)(45,213,273,296)(46,198,274,281)(47,183,275,322)(48,224,276,307)(49,209,277,292)(50,194,278,333)(51,179,279,318)(52,220,280,303)(53,205,225,288)(54,190,226,329)(55,175,227,314)(56,216,228,299)(57,122,371,448)(58,163,372,433)(59,148,373,418)(60,133,374,403)(61,118,375,444)(62,159,376,429)(63,144,377,414)(64,129,378,399)(65,114,379,440)(66,155,380,425)(67,140,381,410)(68,125,382,395)(69,166,383,436)(70,151,384,421)(71,136,385,406)(72,121,386,447)(73,162,387,432)(74,147,388,417)(75,132,389,402)(76,117,390,443)(77,158,391,428)(78,143,392,413)(79,128,337,398)(80,113,338,439)(81,154,339,424)(82,139,340,409)(83,124,341,394)(84,165,342,435)(85,150,343,420)(86,135,344,405)(87,120,345,446)(88,161,346,431)(89,146,347,416)(90,131,348,401)(91,116,349,442)(92,157,350,427)(93,142,351,412)(94,127,352,397)(95,168,353,438)(96,153,354,423)(97,138,355,408)(98,123,356,393)(99,164,357,434)(100,149,358,419)(101,134,359,404)(102,119,360,445)(103,160,361,430)(104,145,362,415)(105,130,363,400)(106,115,364,441)(107,156,365,426)(108,141,366,411)(109,126,367,396)(110,167,368,437)(111,152,369,422)(112,137,370,407)>;
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), (1,62,229,376)(2,105,230,363)(3,92,231,350)(4,79,232,337)(5,66,233,380)(6,109,234,367)(7,96,235,354)(8,83,236,341)(9,70,237,384)(10,57,238,371)(11,100,239,358)(12,87,240,345)(13,74,241,388)(14,61,242,375)(15,104,243,362)(16,91,244,349)(17,78,245,392)(18,65,246,379)(19,108,247,366)(20,95,248,353)(21,82,249,340)(22,69,250,383)(23,112,251,370)(24,99,252,357)(25,86,253,344)(26,73,254,387)(27,60,255,374)(28,103,256,361)(29,90,257,348)(30,77,258,391)(31,64,259,378)(32,107,260,365)(33,94,261,352)(34,81,262,339)(35,68,263,382)(36,111,264,369)(37,98,265,356)(38,85,266,343)(39,72,267,386)(40,59,268,373)(41,102,269,360)(42,89,270,347)(43,76,271,390)(44,63,272,377)(45,106,273,364)(46,93,274,351)(47,80,275,338)(48,67,276,381)(49,110,277,368)(50,97,278,355)(51,84,279,342)(52,71,280,385)(53,58,225,372)(54,101,226,359)(55,88,227,346)(56,75,228,389)(113,211,439,294)(114,198,440,281)(115,185,441,324)(116,172,442,311)(117,215,443,298)(118,202,444,285)(119,189,445,328)(120,176,446,315)(121,219,447,302)(122,206,448,289)(123,193,393,332)(124,180,394,319)(125,223,395,306)(126,210,396,293)(127,197,397,336)(128,184,398,323)(129,171,399,310)(130,214,400,297)(131,201,401,284)(132,188,402,327)(133,175,403,314)(134,218,404,301)(135,205,405,288)(136,192,406,331)(137,179,407,318)(138,222,408,305)(139,209,409,292)(140,196,410,335)(141,183,411,322)(142,170,412,309)(143,213,413,296)(144,200,414,283)(145,187,415,326)(146,174,416,313)(147,217,417,300)(148,204,418,287)(149,191,419,330)(150,178,420,317)(151,221,421,304)(152,208,422,291)(153,195,423,334)(154,182,424,321)(155,169,425,308)(156,212,426,295)(157,199,427,282)(158,186,428,325)(159,173,429,312)(160,216,430,299)(161,203,431,286)(162,190,432,329)(163,177,433,316)(164,220,434,303)(165,207,435,290)(166,194,436,333)(167,181,437,320)(168,224,438,307), (1,201,229,284)(2,186,230,325)(3,171,231,310)(4,212,232,295)(5,197,233,336)(6,182,234,321)(7,223,235,306)(8,208,236,291)(9,193,237,332)(10,178,238,317)(11,219,239,302)(12,204,240,287)(13,189,241,328)(14,174,242,313)(15,215,243,298)(16,200,244,283)(17,185,245,324)(18,170,246,309)(19,211,247,294)(20,196,248,335)(21,181,249,320)(22,222,250,305)(23,207,251,290)(24,192,252,331)(25,177,253,316)(26,218,254,301)(27,203,255,286)(28,188,256,327)(29,173,257,312)(30,214,258,297)(31,199,259,282)(32,184,260,323)(33,169,261,308)(34,210,262,293)(35,195,263,334)(36,180,264,319)(37,221,265,304)(38,206,266,289)(39,191,267,330)(40,176,268,315)(41,217,269,300)(42,202,270,285)(43,187,271,326)(44,172,272,311)(45,213,273,296)(46,198,274,281)(47,183,275,322)(48,224,276,307)(49,209,277,292)(50,194,278,333)(51,179,279,318)(52,220,280,303)(53,205,225,288)(54,190,226,329)(55,175,227,314)(56,216,228,299)(57,122,371,448)(58,163,372,433)(59,148,373,418)(60,133,374,403)(61,118,375,444)(62,159,376,429)(63,144,377,414)(64,129,378,399)(65,114,379,440)(66,155,380,425)(67,140,381,410)(68,125,382,395)(69,166,383,436)(70,151,384,421)(71,136,385,406)(72,121,386,447)(73,162,387,432)(74,147,388,417)(75,132,389,402)(76,117,390,443)(77,158,391,428)(78,143,392,413)(79,128,337,398)(80,113,338,439)(81,154,339,424)(82,139,340,409)(83,124,341,394)(84,165,342,435)(85,150,343,420)(86,135,344,405)(87,120,345,446)(88,161,346,431)(89,146,347,416)(90,131,348,401)(91,116,349,442)(92,157,350,427)(93,142,351,412)(94,127,352,397)(95,168,353,438)(96,153,354,423)(97,138,355,408)(98,123,356,393)(99,164,357,434)(100,149,358,419)(101,134,359,404)(102,119,360,445)(103,160,361,430)(104,145,362,415)(105,130,363,400)(106,115,364,441)(107,156,365,426)(108,141,366,411)(109,126,367,396)(110,167,368,437)(111,152,369,422)(112,137,370,407) );
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)], [(1,62,229,376),(2,105,230,363),(3,92,231,350),(4,79,232,337),(5,66,233,380),(6,109,234,367),(7,96,235,354),(8,83,236,341),(9,70,237,384),(10,57,238,371),(11,100,239,358),(12,87,240,345),(13,74,241,388),(14,61,242,375),(15,104,243,362),(16,91,244,349),(17,78,245,392),(18,65,246,379),(19,108,247,366),(20,95,248,353),(21,82,249,340),(22,69,250,383),(23,112,251,370),(24,99,252,357),(25,86,253,344),(26,73,254,387),(27,60,255,374),(28,103,256,361),(29,90,257,348),(30,77,258,391),(31,64,259,378),(32,107,260,365),(33,94,261,352),(34,81,262,339),(35,68,263,382),(36,111,264,369),(37,98,265,356),(38,85,266,343),(39,72,267,386),(40,59,268,373),(41,102,269,360),(42,89,270,347),(43,76,271,390),(44,63,272,377),(45,106,273,364),(46,93,274,351),(47,80,275,338),(48,67,276,381),(49,110,277,368),(50,97,278,355),(51,84,279,342),(52,71,280,385),(53,58,225,372),(54,101,226,359),(55,88,227,346),(56,75,228,389),(113,211,439,294),(114,198,440,281),(115,185,441,324),(116,172,442,311),(117,215,443,298),(118,202,444,285),(119,189,445,328),(120,176,446,315),(121,219,447,302),(122,206,448,289),(123,193,393,332),(124,180,394,319),(125,223,395,306),(126,210,396,293),(127,197,397,336),(128,184,398,323),(129,171,399,310),(130,214,400,297),(131,201,401,284),(132,188,402,327),(133,175,403,314),(134,218,404,301),(135,205,405,288),(136,192,406,331),(137,179,407,318),(138,222,408,305),(139,209,409,292),(140,196,410,335),(141,183,411,322),(142,170,412,309),(143,213,413,296),(144,200,414,283),(145,187,415,326),(146,174,416,313),(147,217,417,300),(148,204,418,287),(149,191,419,330),(150,178,420,317),(151,221,421,304),(152,208,422,291),(153,195,423,334),(154,182,424,321),(155,169,425,308),(156,212,426,295),(157,199,427,282),(158,186,428,325),(159,173,429,312),(160,216,430,299),(161,203,431,286),(162,190,432,329),(163,177,433,316),(164,220,434,303),(165,207,435,290),(166,194,436,333),(167,181,437,320),(168,224,438,307)], [(1,201,229,284),(2,186,230,325),(3,171,231,310),(4,212,232,295),(5,197,233,336),(6,182,234,321),(7,223,235,306),(8,208,236,291),(9,193,237,332),(10,178,238,317),(11,219,239,302),(12,204,240,287),(13,189,241,328),(14,174,242,313),(15,215,243,298),(16,200,244,283),(17,185,245,324),(18,170,246,309),(19,211,247,294),(20,196,248,335),(21,181,249,320),(22,222,250,305),(23,207,251,290),(24,192,252,331),(25,177,253,316),(26,218,254,301),(27,203,255,286),(28,188,256,327),(29,173,257,312),(30,214,258,297),(31,199,259,282),(32,184,260,323),(33,169,261,308),(34,210,262,293),(35,195,263,334),(36,180,264,319),(37,221,265,304),(38,206,266,289),(39,191,267,330),(40,176,268,315),(41,217,269,300),(42,202,270,285),(43,187,271,326),(44,172,272,311),(45,213,273,296),(46,198,274,281),(47,183,275,322),(48,224,276,307),(49,209,277,292),(50,194,278,333),(51,179,279,318),(52,220,280,303),(53,205,225,288),(54,190,226,329),(55,175,227,314),(56,216,228,299),(57,122,371,448),(58,163,372,433),(59,148,373,418),(60,133,374,403),(61,118,375,444),(62,159,376,429),(63,144,377,414),(64,129,378,399),(65,114,379,440),(66,155,380,425),(67,140,381,410),(68,125,382,395),(69,166,383,436),(70,151,384,421),(71,136,385,406),(72,121,386,447),(73,162,387,432),(74,147,388,417),(75,132,389,402),(76,117,390,443),(77,158,391,428),(78,143,392,413),(79,128,337,398),(80,113,338,439),(81,154,339,424),(82,139,340,409),(83,124,341,394),(84,165,342,435),(85,150,343,420),(86,135,344,405),(87,120,345,446),(88,161,346,431),(89,146,347,416),(90,131,348,401),(91,116,349,442),(92,157,350,427),(93,142,351,412),(94,127,352,397),(95,168,353,438),(96,153,354,423),(97,138,355,408),(98,123,356,393),(99,164,357,434),(100,149,358,419),(101,134,359,404),(102,119,360,445),(103,160,361,430),(104,145,362,415),(105,130,363,400),(106,115,364,441),(107,156,365,426),(108,141,366,411),(109,126,367,396),(110,167,368,437),(111,152,369,422),(112,137,370,407)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 14 | 14 | 14 | 14 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | Q8 | D4 | D7 | D14 | D14 | C4○D8 | Dic14 | Q8×D7 | D4×D7 | SD16⋊3D7 |
| kernel | C56.8Q8 | C28.Q8 | C8×Dic7 | C8⋊Dic7 | C7×C4.Q8 | C28.3Q8 | C7⋊C8 | C56 | C2×Dic7 | C4.Q8 | C4⋊C4 | C2×C8 | C14 | C8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 6 | 3 | 8 | 12 | 3 | 3 | 12 |
Matrix representation of C56.8Q8 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 24 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 87 | 26 |
| 96 | 105 | 0 | 0 |
| 8 | 17 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 83 | 15 |
| 41 | 61 | 0 | 0 |
| 28 | 72 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,24,0,0,0,0,0,87,0,0,13,26],[96,8,0,0,105,17,0,0,0,0,98,83,0,0,0,15],[41,28,0,0,61,72,0,0,0,0,15,0,0,0,0,15] >;
C56.8Q8 in GAP, Magma, Sage, TeX
C_{56}._8Q_8 % in TeX
G:=Group("C56.8Q8"); // GroupNames label
G:=SmallGroup(448,392);
// by ID
G=gap.SmallGroup(448,392);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,926,555,58,438,102,18822]);
// 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^41,c*b*c^-1=a^28*b^-1>;
// generators/relations