metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊9Q8, C8⋊8Dic14, C28.24SD16, C42.252D14, C7⋊1(C8⋊3Q8), (C4×C8).13D7, (C4×C56).15C2, (C2×C4).76D28, C14.2(C4⋊Q8), C28.69(C2×Q8), C8⋊Dic7.4C2, (C2×C8).315D14, (C2×C28).373D4, C4.3(C56⋊C2), C28⋊2Q8.1C2, C14.1(C2×SD16), C2.6(C28⋊2Q8), C4.35(C2×Dic14), C22.85(C2×D28), C4⋊Dic7.1C22, (C2×C28).716C23, (C2×C56).387C22, (C4×C28).302C22, C2.5(C2×C56⋊C2), (C2×C14).99(C2×D4), (C2×C4).659(C22×D7), SmallGroup(448,214)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊9Q8
G = < a,b,c | a56=b4=1, c2=b2, ab=ba, cac-1=a27, cbc-1=b-1 >
Subgroups: 516 in 98 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C2×C14, C4×C8, C4.Q8, C4⋊Q8, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C8⋊3Q8, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C8⋊Dic7, C4×C56, C28⋊2Q8, C56⋊9Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, D14, C4⋊Q8, C2×SD16, Dic14, D28, C22×D7, C8⋊3Q8, C56⋊C2, C2×Dic14, C2×D28, C28⋊2Q8, C2×C56⋊C2, C56⋊9Q8
►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)
(1 124 69 362)(2 125 70 363)(3 126 71 364)(4 127 72 365)(5 128 73 366)(6 129 74 367)(7 130 75 368)(8 131 76 369)(9 132 77 370)(10 133 78 371)(11 134 79 372)(12 135 80 373)(13 136 81 374)(14 137 82 375)(15 138 83 376)(16 139 84 377)(17 140 85 378)(18 141 86 379)(19 142 87 380)(20 143 88 381)(21 144 89 382)(22 145 90 383)(23 146 91 384)(24 147 92 385)(25 148 93 386)(26 149 94 387)(27 150 95 388)(28 151 96 389)(29 152 97 390)(30 153 98 391)(31 154 99 392)(32 155 100 337)(33 156 101 338)(34 157 102 339)(35 158 103 340)(36 159 104 341)(37 160 105 342)(38 161 106 343)(39 162 107 344)(40 163 108 345)(41 164 109 346)(42 165 110 347)(43 166 111 348)(44 167 112 349)(45 168 57 350)(46 113 58 351)(47 114 59 352)(48 115 60 353)(49 116 61 354)(50 117 62 355)(51 118 63 356)(52 119 64 357)(53 120 65 358)(54 121 66 359)(55 122 67 360)(56 123 68 361)(169 325 407 239)(170 326 408 240)(171 327 409 241)(172 328 410 242)(173 329 411 243)(174 330 412 244)(175 331 413 245)(176 332 414 246)(177 333 415 247)(178 334 416 248)(179 335 417 249)(180 336 418 250)(181 281 419 251)(182 282 420 252)(183 283 421 253)(184 284 422 254)(185 285 423 255)(186 286 424 256)(187 287 425 257)(188 288 426 258)(189 289 427 259)(190 290 428 260)(191 291 429 261)(192 292 430 262)(193 293 431 263)(194 294 432 264)(195 295 433 265)(196 296 434 266)(197 297 435 267)(198 298 436 268)(199 299 437 269)(200 300 438 270)(201 301 439 271)(202 302 440 272)(203 303 441 273)(204 304 442 274)(205 305 443 275)(206 306 444 276)(207 307 445 277)(208 308 446 278)(209 309 447 279)(210 310 448 280)(211 311 393 225)(212 312 394 226)(213 313 395 227)(214 314 396 228)(215 315 397 229)(216 316 398 230)(217 317 399 231)(218 318 400 232)(219 319 401 233)(220 320 402 234)(221 321 403 235)(222 322 404 236)(223 323 405 237)(224 324 406 238)
(1 311 69 225)(2 282 70 252)(3 309 71 279)(4 336 72 250)(5 307 73 277)(6 334 74 248)(7 305 75 275)(8 332 76 246)(9 303 77 273)(10 330 78 244)(11 301 79 271)(12 328 80 242)(13 299 81 269)(14 326 82 240)(15 297 83 267)(16 324 84 238)(17 295 85 265)(18 322 86 236)(19 293 87 263)(20 320 88 234)(21 291 89 261)(22 318 90 232)(23 289 91 259)(24 316 92 230)(25 287 93 257)(26 314 94 228)(27 285 95 255)(28 312 96 226)(29 283 97 253)(30 310 98 280)(31 281 99 251)(32 308 100 278)(33 335 101 249)(34 306 102 276)(35 333 103 247)(36 304 104 274)(37 331 105 245)(38 302 106 272)(39 329 107 243)(40 300 108 270)(41 327 109 241)(42 298 110 268)(43 325 111 239)(44 296 112 266)(45 323 57 237)(46 294 58 264)(47 321 59 235)(48 292 60 262)(49 319 61 233)(50 290 62 260)(51 317 63 231)(52 288 64 258)(53 315 65 229)(54 286 66 256)(55 313 67 227)(56 284 68 254)(113 194 351 432)(114 221 352 403)(115 192 353 430)(116 219 354 401)(117 190 355 428)(118 217 356 399)(119 188 357 426)(120 215 358 397)(121 186 359 424)(122 213 360 395)(123 184 361 422)(124 211 362 393)(125 182 363 420)(126 209 364 447)(127 180 365 418)(128 207 366 445)(129 178 367 416)(130 205 368 443)(131 176 369 414)(132 203 370 441)(133 174 371 412)(134 201 372 439)(135 172 373 410)(136 199 374 437)(137 170 375 408)(138 197 376 435)(139 224 377 406)(140 195 378 433)(141 222 379 404)(142 193 380 431)(143 220 381 402)(144 191 382 429)(145 218 383 400)(146 189 384 427)(147 216 385 398)(148 187 386 425)(149 214 387 396)(150 185 388 423)(151 212 389 394)(152 183 390 421)(153 210 391 448)(154 181 392 419)(155 208 337 446)(156 179 338 417)(157 206 339 444)(158 177 340 415)(159 204 341 442)(160 175 342 413)(161 202 343 440)(162 173 344 411)(163 200 345 438)(164 171 346 409)(165 198 347 436)(166 169 348 407)(167 196 349 434)(168 223 350 405)
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), (1,124,69,362)(2,125,70,363)(3,126,71,364)(4,127,72,365)(5,128,73,366)(6,129,74,367)(7,130,75,368)(8,131,76,369)(9,132,77,370)(10,133,78,371)(11,134,79,372)(12,135,80,373)(13,136,81,374)(14,137,82,375)(15,138,83,376)(16,139,84,377)(17,140,85,378)(18,141,86,379)(19,142,87,380)(20,143,88,381)(21,144,89,382)(22,145,90,383)(23,146,91,384)(24,147,92,385)(25,148,93,386)(26,149,94,387)(27,150,95,388)(28,151,96,389)(29,152,97,390)(30,153,98,391)(31,154,99,392)(32,155,100,337)(33,156,101,338)(34,157,102,339)(35,158,103,340)(36,159,104,341)(37,160,105,342)(38,161,106,343)(39,162,107,344)(40,163,108,345)(41,164,109,346)(42,165,110,347)(43,166,111,348)(44,167,112,349)(45,168,57,350)(46,113,58,351)(47,114,59,352)(48,115,60,353)(49,116,61,354)(50,117,62,355)(51,118,63,356)(52,119,64,357)(53,120,65,358)(54,121,66,359)(55,122,67,360)(56,123,68,361)(169,325,407,239)(170,326,408,240)(171,327,409,241)(172,328,410,242)(173,329,411,243)(174,330,412,244)(175,331,413,245)(176,332,414,246)(177,333,415,247)(178,334,416,248)(179,335,417,249)(180,336,418,250)(181,281,419,251)(182,282,420,252)(183,283,421,253)(184,284,422,254)(185,285,423,255)(186,286,424,256)(187,287,425,257)(188,288,426,258)(189,289,427,259)(190,290,428,260)(191,291,429,261)(192,292,430,262)(193,293,431,263)(194,294,432,264)(195,295,433,265)(196,296,434,266)(197,297,435,267)(198,298,436,268)(199,299,437,269)(200,300,438,270)(201,301,439,271)(202,302,440,272)(203,303,441,273)(204,304,442,274)(205,305,443,275)(206,306,444,276)(207,307,445,277)(208,308,446,278)(209,309,447,279)(210,310,448,280)(211,311,393,225)(212,312,394,226)(213,313,395,227)(214,314,396,228)(215,315,397,229)(216,316,398,230)(217,317,399,231)(218,318,400,232)(219,319,401,233)(220,320,402,234)(221,321,403,235)(222,322,404,236)(223,323,405,237)(224,324,406,238), (1,311,69,225)(2,282,70,252)(3,309,71,279)(4,336,72,250)(5,307,73,277)(6,334,74,248)(7,305,75,275)(8,332,76,246)(9,303,77,273)(10,330,78,244)(11,301,79,271)(12,328,80,242)(13,299,81,269)(14,326,82,240)(15,297,83,267)(16,324,84,238)(17,295,85,265)(18,322,86,236)(19,293,87,263)(20,320,88,234)(21,291,89,261)(22,318,90,232)(23,289,91,259)(24,316,92,230)(25,287,93,257)(26,314,94,228)(27,285,95,255)(28,312,96,226)(29,283,97,253)(30,310,98,280)(31,281,99,251)(32,308,100,278)(33,335,101,249)(34,306,102,276)(35,333,103,247)(36,304,104,274)(37,331,105,245)(38,302,106,272)(39,329,107,243)(40,300,108,270)(41,327,109,241)(42,298,110,268)(43,325,111,239)(44,296,112,266)(45,323,57,237)(46,294,58,264)(47,321,59,235)(48,292,60,262)(49,319,61,233)(50,290,62,260)(51,317,63,231)(52,288,64,258)(53,315,65,229)(54,286,66,256)(55,313,67,227)(56,284,68,254)(113,194,351,432)(114,221,352,403)(115,192,353,430)(116,219,354,401)(117,190,355,428)(118,217,356,399)(119,188,357,426)(120,215,358,397)(121,186,359,424)(122,213,360,395)(123,184,361,422)(124,211,362,393)(125,182,363,420)(126,209,364,447)(127,180,365,418)(128,207,366,445)(129,178,367,416)(130,205,368,443)(131,176,369,414)(132,203,370,441)(133,174,371,412)(134,201,372,439)(135,172,373,410)(136,199,374,437)(137,170,375,408)(138,197,376,435)(139,224,377,406)(140,195,378,433)(141,222,379,404)(142,193,380,431)(143,220,381,402)(144,191,382,429)(145,218,383,400)(146,189,384,427)(147,216,385,398)(148,187,386,425)(149,214,387,396)(150,185,388,423)(151,212,389,394)(152,183,390,421)(153,210,391,448)(154,181,392,419)(155,208,337,446)(156,179,338,417)(157,206,339,444)(158,177,340,415)(159,204,341,442)(160,175,342,413)(161,202,343,440)(162,173,344,411)(163,200,345,438)(164,171,346,409)(165,198,347,436)(166,169,348,407)(167,196,349,434)(168,223,350,405)>;
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), (1,124,69,362)(2,125,70,363)(3,126,71,364)(4,127,72,365)(5,128,73,366)(6,129,74,367)(7,130,75,368)(8,131,76,369)(9,132,77,370)(10,133,78,371)(11,134,79,372)(12,135,80,373)(13,136,81,374)(14,137,82,375)(15,138,83,376)(16,139,84,377)(17,140,85,378)(18,141,86,379)(19,142,87,380)(20,143,88,381)(21,144,89,382)(22,145,90,383)(23,146,91,384)(24,147,92,385)(25,148,93,386)(26,149,94,387)(27,150,95,388)(28,151,96,389)(29,152,97,390)(30,153,98,391)(31,154,99,392)(32,155,100,337)(33,156,101,338)(34,157,102,339)(35,158,103,340)(36,159,104,341)(37,160,105,342)(38,161,106,343)(39,162,107,344)(40,163,108,345)(41,164,109,346)(42,165,110,347)(43,166,111,348)(44,167,112,349)(45,168,57,350)(46,113,58,351)(47,114,59,352)(48,115,60,353)(49,116,61,354)(50,117,62,355)(51,118,63,356)(52,119,64,357)(53,120,65,358)(54,121,66,359)(55,122,67,360)(56,123,68,361)(169,325,407,239)(170,326,408,240)(171,327,409,241)(172,328,410,242)(173,329,411,243)(174,330,412,244)(175,331,413,245)(176,332,414,246)(177,333,415,247)(178,334,416,248)(179,335,417,249)(180,336,418,250)(181,281,419,251)(182,282,420,252)(183,283,421,253)(184,284,422,254)(185,285,423,255)(186,286,424,256)(187,287,425,257)(188,288,426,258)(189,289,427,259)(190,290,428,260)(191,291,429,261)(192,292,430,262)(193,293,431,263)(194,294,432,264)(195,295,433,265)(196,296,434,266)(197,297,435,267)(198,298,436,268)(199,299,437,269)(200,300,438,270)(201,301,439,271)(202,302,440,272)(203,303,441,273)(204,304,442,274)(205,305,443,275)(206,306,444,276)(207,307,445,277)(208,308,446,278)(209,309,447,279)(210,310,448,280)(211,311,393,225)(212,312,394,226)(213,313,395,227)(214,314,396,228)(215,315,397,229)(216,316,398,230)(217,317,399,231)(218,318,400,232)(219,319,401,233)(220,320,402,234)(221,321,403,235)(222,322,404,236)(223,323,405,237)(224,324,406,238), (1,311,69,225)(2,282,70,252)(3,309,71,279)(4,336,72,250)(5,307,73,277)(6,334,74,248)(7,305,75,275)(8,332,76,246)(9,303,77,273)(10,330,78,244)(11,301,79,271)(12,328,80,242)(13,299,81,269)(14,326,82,240)(15,297,83,267)(16,324,84,238)(17,295,85,265)(18,322,86,236)(19,293,87,263)(20,320,88,234)(21,291,89,261)(22,318,90,232)(23,289,91,259)(24,316,92,230)(25,287,93,257)(26,314,94,228)(27,285,95,255)(28,312,96,226)(29,283,97,253)(30,310,98,280)(31,281,99,251)(32,308,100,278)(33,335,101,249)(34,306,102,276)(35,333,103,247)(36,304,104,274)(37,331,105,245)(38,302,106,272)(39,329,107,243)(40,300,108,270)(41,327,109,241)(42,298,110,268)(43,325,111,239)(44,296,112,266)(45,323,57,237)(46,294,58,264)(47,321,59,235)(48,292,60,262)(49,319,61,233)(50,290,62,260)(51,317,63,231)(52,288,64,258)(53,315,65,229)(54,286,66,256)(55,313,67,227)(56,284,68,254)(113,194,351,432)(114,221,352,403)(115,192,353,430)(116,219,354,401)(117,190,355,428)(118,217,356,399)(119,188,357,426)(120,215,358,397)(121,186,359,424)(122,213,360,395)(123,184,361,422)(124,211,362,393)(125,182,363,420)(126,209,364,447)(127,180,365,418)(128,207,366,445)(129,178,367,416)(130,205,368,443)(131,176,369,414)(132,203,370,441)(133,174,371,412)(134,201,372,439)(135,172,373,410)(136,199,374,437)(137,170,375,408)(138,197,376,435)(139,224,377,406)(140,195,378,433)(141,222,379,404)(142,193,380,431)(143,220,381,402)(144,191,382,429)(145,218,383,400)(146,189,384,427)(147,216,385,398)(148,187,386,425)(149,214,387,396)(150,185,388,423)(151,212,389,394)(152,183,390,421)(153,210,391,448)(154,181,392,419)(155,208,337,446)(156,179,338,417)(157,206,339,444)(158,177,340,415)(159,204,341,442)(160,175,342,413)(161,202,343,440)(162,173,344,411)(163,200,345,438)(164,171,346,409)(165,198,347,436)(166,169,348,407)(167,196,349,434)(168,223,350,405) );
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)], [(1,124,69,362),(2,125,70,363),(3,126,71,364),(4,127,72,365),(5,128,73,366),(6,129,74,367),(7,130,75,368),(8,131,76,369),(9,132,77,370),(10,133,78,371),(11,134,79,372),(12,135,80,373),(13,136,81,374),(14,137,82,375),(15,138,83,376),(16,139,84,377),(17,140,85,378),(18,141,86,379),(19,142,87,380),(20,143,88,381),(21,144,89,382),(22,145,90,383),(23,146,91,384),(24,147,92,385),(25,148,93,386),(26,149,94,387),(27,150,95,388),(28,151,96,389),(29,152,97,390),(30,153,98,391),(31,154,99,392),(32,155,100,337),(33,156,101,338),(34,157,102,339),(35,158,103,340),(36,159,104,341),(37,160,105,342),(38,161,106,343),(39,162,107,344),(40,163,108,345),(41,164,109,346),(42,165,110,347),(43,166,111,348),(44,167,112,349),(45,168,57,350),(46,113,58,351),(47,114,59,352),(48,115,60,353),(49,116,61,354),(50,117,62,355),(51,118,63,356),(52,119,64,357),(53,120,65,358),(54,121,66,359),(55,122,67,360),(56,123,68,361),(169,325,407,239),(170,326,408,240),(171,327,409,241),(172,328,410,242),(173,329,411,243),(174,330,412,244),(175,331,413,245),(176,332,414,246),(177,333,415,247),(178,334,416,248),(179,335,417,249),(180,336,418,250),(181,281,419,251),(182,282,420,252),(183,283,421,253),(184,284,422,254),(185,285,423,255),(186,286,424,256),(187,287,425,257),(188,288,426,258),(189,289,427,259),(190,290,428,260),(191,291,429,261),(192,292,430,262),(193,293,431,263),(194,294,432,264),(195,295,433,265),(196,296,434,266),(197,297,435,267),(198,298,436,268),(199,299,437,269),(200,300,438,270),(201,301,439,271),(202,302,440,272),(203,303,441,273),(204,304,442,274),(205,305,443,275),(206,306,444,276),(207,307,445,277),(208,308,446,278),(209,309,447,279),(210,310,448,280),(211,311,393,225),(212,312,394,226),(213,313,395,227),(214,314,396,228),(215,315,397,229),(216,316,398,230),(217,317,399,231),(218,318,400,232),(219,319,401,233),(220,320,402,234),(221,321,403,235),(222,322,404,236),(223,323,405,237),(224,324,406,238)], [(1,311,69,225),(2,282,70,252),(3,309,71,279),(4,336,72,250),(5,307,73,277),(6,334,74,248),(7,305,75,275),(8,332,76,246),(9,303,77,273),(10,330,78,244),(11,301,79,271),(12,328,80,242),(13,299,81,269),(14,326,82,240),(15,297,83,267),(16,324,84,238),(17,295,85,265),(18,322,86,236),(19,293,87,263),(20,320,88,234),(21,291,89,261),(22,318,90,232),(23,289,91,259),(24,316,92,230),(25,287,93,257),(26,314,94,228),(27,285,95,255),(28,312,96,226),(29,283,97,253),(30,310,98,280),(31,281,99,251),(32,308,100,278),(33,335,101,249),(34,306,102,276),(35,333,103,247),(36,304,104,274),(37,331,105,245),(38,302,106,272),(39,329,107,243),(40,300,108,270),(41,327,109,241),(42,298,110,268),(43,325,111,239),(44,296,112,266),(45,323,57,237),(46,294,58,264),(47,321,59,235),(48,292,60,262),(49,319,61,233),(50,290,62,260),(51,317,63,231),(52,288,64,258),(53,315,65,229),(54,286,66,256),(55,313,67,227),(56,284,68,254),(113,194,351,432),(114,221,352,403),(115,192,353,430),(116,219,354,401),(117,190,355,428),(118,217,356,399),(119,188,357,426),(120,215,358,397),(121,186,359,424),(122,213,360,395),(123,184,361,422),(124,211,362,393),(125,182,363,420),(126,209,364,447),(127,180,365,418),(128,207,366,445),(129,178,367,416),(130,205,368,443),(131,176,369,414),(132,203,370,441),(133,174,371,412),(134,201,372,439),(135,172,373,410),(136,199,374,437),(137,170,375,408),(138,197,376,435),(139,224,377,406),(140,195,378,433),(141,222,379,404),(142,193,380,431),(143,220,381,402),(144,191,382,429),(145,218,383,400),(146,189,384,427),(147,216,385,398),(148,187,386,425),(149,214,387,396),(150,185,388,423),(151,212,389,394),(152,183,390,421),(153,210,391,448),(154,181,392,419),(155,208,337,446),(156,179,338,417),(157,206,339,444),(158,177,340,415),(159,204,341,442),(160,175,342,413),(161,202,343,440),(162,173,344,411),(163,200,345,438),(164,171,346,409),(165,198,347,436),(166,169,348,407),(167,196,349,434),(168,223,350,405)]])
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 |
| type | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | Q8 | D4 | D7 | SD16 | D14 | D14 | Dic14 | D28 | C56⋊C2 |
| kernel | C56⋊9Q8 | C8⋊Dic7 | C4×C56 | C28⋊2Q8 | C56 | C2×C28 | C4×C8 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 3 | 8 | 3 | 6 | 24 | 12 | 48 |
Matrix representation of C56⋊9Q8 ►in GL4(𝔽113) generated by
| 25 | 112 | 0 | 0 |
| 60 | 79 | 0 | 0 |
| 0 | 0 | 42 | 24 |
| 0 | 0 | 89 | 32 |
| 9 | 38 | 0 | 0 |
| 93 | 104 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 38 | 56 | 0 | 0 |
| 65 | 75 | 0 | 0 |
| 0 | 0 | 109 | 32 |
| 0 | 0 | 109 | 4 |
G:=sub<GL(4,GF(113))| [25,60,0,0,112,79,0,0,0,0,42,89,0,0,24,32],[9,93,0,0,38,104,0,0,0,0,112,0,0,0,0,112],[38,65,0,0,56,75,0,0,0,0,109,109,0,0,32,4] >;
C56⋊9Q8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_9Q_8 % in TeX
G:=Group("C56:9Q8"); // GroupNames label
G:=SmallGroup(448,214);
// by ID
G=gap.SmallGroup(448,214);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,254,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^27,c*b*c^-1=b^-1>;
// generators/relations