metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊5Q8, C8⋊5Dic14, Dic7.3SD16, C7⋊C8⋊5Q8, C7⋊2(C8⋊3Q8), C28⋊Q8.5C2, C4.20(Q8×D7), C4⋊C4.32D14, C4.Q8.5D7, C2.8(C28⋊Q8), C28.10(C2×Q8), (C2×C8).256D14, C14.13(C4⋊Q8), (C8×Dic7).6C2, C8⋊Dic7.13C2, C2.21(D7×SD16), (C2×Dic7).96D4, C14.36(C2×SD16), C4.20(C2×Dic14), C22.210(D4×D7), C4.Dic14.5C2, (C2×C56).157C22, (C2×C28).271C23, C4⋊Dic7.103C22, (C4×Dic7).229C22, (C7×C4.Q8).4C2, (C2×C14).276(C2×D4), (C7×C4⋊C4).64C22, (C2×C7⋊C8).224C22, (C2×C4).374(C22×D7), SmallGroup(448,389)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊5Q8
G = < a,b,c | a56=b4=1, c2=b2, bab-1=a43, cac-1=a41, cbc-1=b-1 >
Subgroups: 492 in 98 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×C8, C4.Q8, C4.Q8, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C8⋊3Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C4.Dic14, C8×Dic7, C8⋊Dic7, C7×C4.Q8, C28⋊Q8, C56⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, D14, C4⋊Q8, C2×SD16, Dic14, C22×D7, C8⋊3Q8, C2×Dic14, D4×D7, Q8×D7, C28⋊Q8, D7×SD16, C56⋊5Q8
►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)
(1 313 107 151)(2 300 108 138)(3 287 109 125)(4 330 110 168)(5 317 111 155)(6 304 112 142)(7 291 57 129)(8 334 58 116)(9 321 59 159)(10 308 60 146)(11 295 61 133)(12 282 62 120)(13 325 63 163)(14 312 64 150)(15 299 65 137)(16 286 66 124)(17 329 67 167)(18 316 68 154)(19 303 69 141)(20 290 70 128)(21 333 71 115)(22 320 72 158)(23 307 73 145)(24 294 74 132)(25 281 75 119)(26 324 76 162)(27 311 77 149)(28 298 78 136)(29 285 79 123)(30 328 80 166)(31 315 81 153)(32 302 82 140)(33 289 83 127)(34 332 84 114)(35 319 85 157)(36 306 86 144)(37 293 87 131)(38 336 88 118)(39 323 89 161)(40 310 90 148)(41 297 91 135)(42 284 92 122)(43 327 93 165)(44 314 94 152)(45 301 95 139)(46 288 96 126)(47 331 97 113)(48 318 98 156)(49 305 99 143)(50 292 100 130)(51 335 101 117)(52 322 102 160)(53 309 103 147)(54 296 104 134)(55 283 105 121)(56 326 106 164)(169 443 232 361)(170 430 233 348)(171 417 234 391)(172 404 235 378)(173 447 236 365)(174 434 237 352)(175 421 238 339)(176 408 239 382)(177 395 240 369)(178 438 241 356)(179 425 242 343)(180 412 243 386)(181 399 244 373)(182 442 245 360)(183 429 246 347)(184 416 247 390)(185 403 248 377)(186 446 249 364)(187 433 250 351)(188 420 251 338)(189 407 252 381)(190 394 253 368)(191 437 254 355)(192 424 255 342)(193 411 256 385)(194 398 257 372)(195 441 258 359)(196 428 259 346)(197 415 260 389)(198 402 261 376)(199 445 262 363)(200 432 263 350)(201 419 264 337)(202 406 265 380)(203 393 266 367)(204 436 267 354)(205 423 268 341)(206 410 269 384)(207 397 270 371)(208 440 271 358)(209 427 272 345)(210 414 273 388)(211 401 274 375)(212 444 275 362)(213 431 276 349)(214 418 277 392)(215 405 278 379)(216 448 279 366)(217 435 280 353)(218 422 225 340)(219 409 226 383)(220 396 227 370)(221 439 228 357)(222 426 229 344)(223 413 230 387)(224 400 231 374)
(1 272 107 209)(2 257 108 194)(3 242 109 179)(4 227 110 220)(5 268 111 205)(6 253 112 190)(7 238 57 175)(8 279 58 216)(9 264 59 201)(10 249 60 186)(11 234 61 171)(12 275 62 212)(13 260 63 197)(14 245 64 182)(15 230 65 223)(16 271 66 208)(17 256 67 193)(18 241 68 178)(19 226 69 219)(20 267 70 204)(21 252 71 189)(22 237 72 174)(23 278 73 215)(24 263 74 200)(25 248 75 185)(26 233 76 170)(27 274 77 211)(28 259 78 196)(29 244 79 181)(30 229 80 222)(31 270 81 207)(32 255 82 192)(33 240 83 177)(34 225 84 218)(35 266 85 203)(36 251 86 188)(37 236 87 173)(38 277 88 214)(39 262 89 199)(40 247 90 184)(41 232 91 169)(42 273 92 210)(43 258 93 195)(44 243 94 180)(45 228 95 221)(46 269 96 206)(47 254 97 191)(48 239 98 176)(49 280 99 217)(50 265 100 202)(51 250 101 187)(52 235 102 172)(53 276 103 213)(54 261 104 198)(55 246 105 183)(56 231 106 224)(113 355 331 437)(114 340 332 422)(115 381 333 407)(116 366 334 448)(117 351 335 433)(118 392 336 418)(119 377 281 403)(120 362 282 444)(121 347 283 429)(122 388 284 414)(123 373 285 399)(124 358 286 440)(125 343 287 425)(126 384 288 410)(127 369 289 395)(128 354 290 436)(129 339 291 421)(130 380 292 406)(131 365 293 447)(132 350 294 432)(133 391 295 417)(134 376 296 402)(135 361 297 443)(136 346 298 428)(137 387 299 413)(138 372 300 398)(139 357 301 439)(140 342 302 424)(141 383 303 409)(142 368 304 394)(143 353 305 435)(144 338 306 420)(145 379 307 405)(146 364 308 446)(147 349 309 431)(148 390 310 416)(149 375 311 401)(150 360 312 442)(151 345 313 427)(152 386 314 412)(153 371 315 397)(154 356 316 438)(155 341 317 423)(156 382 318 408)(157 367 319 393)(158 352 320 434)(159 337 321 419)(160 378 322 404)(161 363 323 445)(162 348 324 430)(163 389 325 415)(164 374 326 400)(165 359 327 441)(166 344 328 426)(167 385 329 411)(168 370 330 396)
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), (1,313,107,151)(2,300,108,138)(3,287,109,125)(4,330,110,168)(5,317,111,155)(6,304,112,142)(7,291,57,129)(8,334,58,116)(9,321,59,159)(10,308,60,146)(11,295,61,133)(12,282,62,120)(13,325,63,163)(14,312,64,150)(15,299,65,137)(16,286,66,124)(17,329,67,167)(18,316,68,154)(19,303,69,141)(20,290,70,128)(21,333,71,115)(22,320,72,158)(23,307,73,145)(24,294,74,132)(25,281,75,119)(26,324,76,162)(27,311,77,149)(28,298,78,136)(29,285,79,123)(30,328,80,166)(31,315,81,153)(32,302,82,140)(33,289,83,127)(34,332,84,114)(35,319,85,157)(36,306,86,144)(37,293,87,131)(38,336,88,118)(39,323,89,161)(40,310,90,148)(41,297,91,135)(42,284,92,122)(43,327,93,165)(44,314,94,152)(45,301,95,139)(46,288,96,126)(47,331,97,113)(48,318,98,156)(49,305,99,143)(50,292,100,130)(51,335,101,117)(52,322,102,160)(53,309,103,147)(54,296,104,134)(55,283,105,121)(56,326,106,164)(169,443,232,361)(170,430,233,348)(171,417,234,391)(172,404,235,378)(173,447,236,365)(174,434,237,352)(175,421,238,339)(176,408,239,382)(177,395,240,369)(178,438,241,356)(179,425,242,343)(180,412,243,386)(181,399,244,373)(182,442,245,360)(183,429,246,347)(184,416,247,390)(185,403,248,377)(186,446,249,364)(187,433,250,351)(188,420,251,338)(189,407,252,381)(190,394,253,368)(191,437,254,355)(192,424,255,342)(193,411,256,385)(194,398,257,372)(195,441,258,359)(196,428,259,346)(197,415,260,389)(198,402,261,376)(199,445,262,363)(200,432,263,350)(201,419,264,337)(202,406,265,380)(203,393,266,367)(204,436,267,354)(205,423,268,341)(206,410,269,384)(207,397,270,371)(208,440,271,358)(209,427,272,345)(210,414,273,388)(211,401,274,375)(212,444,275,362)(213,431,276,349)(214,418,277,392)(215,405,278,379)(216,448,279,366)(217,435,280,353)(218,422,225,340)(219,409,226,383)(220,396,227,370)(221,439,228,357)(222,426,229,344)(223,413,230,387)(224,400,231,374), (1,272,107,209)(2,257,108,194)(3,242,109,179)(4,227,110,220)(5,268,111,205)(6,253,112,190)(7,238,57,175)(8,279,58,216)(9,264,59,201)(10,249,60,186)(11,234,61,171)(12,275,62,212)(13,260,63,197)(14,245,64,182)(15,230,65,223)(16,271,66,208)(17,256,67,193)(18,241,68,178)(19,226,69,219)(20,267,70,204)(21,252,71,189)(22,237,72,174)(23,278,73,215)(24,263,74,200)(25,248,75,185)(26,233,76,170)(27,274,77,211)(28,259,78,196)(29,244,79,181)(30,229,80,222)(31,270,81,207)(32,255,82,192)(33,240,83,177)(34,225,84,218)(35,266,85,203)(36,251,86,188)(37,236,87,173)(38,277,88,214)(39,262,89,199)(40,247,90,184)(41,232,91,169)(42,273,92,210)(43,258,93,195)(44,243,94,180)(45,228,95,221)(46,269,96,206)(47,254,97,191)(48,239,98,176)(49,280,99,217)(50,265,100,202)(51,250,101,187)(52,235,102,172)(53,276,103,213)(54,261,104,198)(55,246,105,183)(56,231,106,224)(113,355,331,437)(114,340,332,422)(115,381,333,407)(116,366,334,448)(117,351,335,433)(118,392,336,418)(119,377,281,403)(120,362,282,444)(121,347,283,429)(122,388,284,414)(123,373,285,399)(124,358,286,440)(125,343,287,425)(126,384,288,410)(127,369,289,395)(128,354,290,436)(129,339,291,421)(130,380,292,406)(131,365,293,447)(132,350,294,432)(133,391,295,417)(134,376,296,402)(135,361,297,443)(136,346,298,428)(137,387,299,413)(138,372,300,398)(139,357,301,439)(140,342,302,424)(141,383,303,409)(142,368,304,394)(143,353,305,435)(144,338,306,420)(145,379,307,405)(146,364,308,446)(147,349,309,431)(148,390,310,416)(149,375,311,401)(150,360,312,442)(151,345,313,427)(152,386,314,412)(153,371,315,397)(154,356,316,438)(155,341,317,423)(156,382,318,408)(157,367,319,393)(158,352,320,434)(159,337,321,419)(160,378,322,404)(161,363,323,445)(162,348,324,430)(163,389,325,415)(164,374,326,400)(165,359,327,441)(166,344,328,426)(167,385,329,411)(168,370,330,396)>;
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), (1,313,107,151)(2,300,108,138)(3,287,109,125)(4,330,110,168)(5,317,111,155)(6,304,112,142)(7,291,57,129)(8,334,58,116)(9,321,59,159)(10,308,60,146)(11,295,61,133)(12,282,62,120)(13,325,63,163)(14,312,64,150)(15,299,65,137)(16,286,66,124)(17,329,67,167)(18,316,68,154)(19,303,69,141)(20,290,70,128)(21,333,71,115)(22,320,72,158)(23,307,73,145)(24,294,74,132)(25,281,75,119)(26,324,76,162)(27,311,77,149)(28,298,78,136)(29,285,79,123)(30,328,80,166)(31,315,81,153)(32,302,82,140)(33,289,83,127)(34,332,84,114)(35,319,85,157)(36,306,86,144)(37,293,87,131)(38,336,88,118)(39,323,89,161)(40,310,90,148)(41,297,91,135)(42,284,92,122)(43,327,93,165)(44,314,94,152)(45,301,95,139)(46,288,96,126)(47,331,97,113)(48,318,98,156)(49,305,99,143)(50,292,100,130)(51,335,101,117)(52,322,102,160)(53,309,103,147)(54,296,104,134)(55,283,105,121)(56,326,106,164)(169,443,232,361)(170,430,233,348)(171,417,234,391)(172,404,235,378)(173,447,236,365)(174,434,237,352)(175,421,238,339)(176,408,239,382)(177,395,240,369)(178,438,241,356)(179,425,242,343)(180,412,243,386)(181,399,244,373)(182,442,245,360)(183,429,246,347)(184,416,247,390)(185,403,248,377)(186,446,249,364)(187,433,250,351)(188,420,251,338)(189,407,252,381)(190,394,253,368)(191,437,254,355)(192,424,255,342)(193,411,256,385)(194,398,257,372)(195,441,258,359)(196,428,259,346)(197,415,260,389)(198,402,261,376)(199,445,262,363)(200,432,263,350)(201,419,264,337)(202,406,265,380)(203,393,266,367)(204,436,267,354)(205,423,268,341)(206,410,269,384)(207,397,270,371)(208,440,271,358)(209,427,272,345)(210,414,273,388)(211,401,274,375)(212,444,275,362)(213,431,276,349)(214,418,277,392)(215,405,278,379)(216,448,279,366)(217,435,280,353)(218,422,225,340)(219,409,226,383)(220,396,227,370)(221,439,228,357)(222,426,229,344)(223,413,230,387)(224,400,231,374), (1,272,107,209)(2,257,108,194)(3,242,109,179)(4,227,110,220)(5,268,111,205)(6,253,112,190)(7,238,57,175)(8,279,58,216)(9,264,59,201)(10,249,60,186)(11,234,61,171)(12,275,62,212)(13,260,63,197)(14,245,64,182)(15,230,65,223)(16,271,66,208)(17,256,67,193)(18,241,68,178)(19,226,69,219)(20,267,70,204)(21,252,71,189)(22,237,72,174)(23,278,73,215)(24,263,74,200)(25,248,75,185)(26,233,76,170)(27,274,77,211)(28,259,78,196)(29,244,79,181)(30,229,80,222)(31,270,81,207)(32,255,82,192)(33,240,83,177)(34,225,84,218)(35,266,85,203)(36,251,86,188)(37,236,87,173)(38,277,88,214)(39,262,89,199)(40,247,90,184)(41,232,91,169)(42,273,92,210)(43,258,93,195)(44,243,94,180)(45,228,95,221)(46,269,96,206)(47,254,97,191)(48,239,98,176)(49,280,99,217)(50,265,100,202)(51,250,101,187)(52,235,102,172)(53,276,103,213)(54,261,104,198)(55,246,105,183)(56,231,106,224)(113,355,331,437)(114,340,332,422)(115,381,333,407)(116,366,334,448)(117,351,335,433)(118,392,336,418)(119,377,281,403)(120,362,282,444)(121,347,283,429)(122,388,284,414)(123,373,285,399)(124,358,286,440)(125,343,287,425)(126,384,288,410)(127,369,289,395)(128,354,290,436)(129,339,291,421)(130,380,292,406)(131,365,293,447)(132,350,294,432)(133,391,295,417)(134,376,296,402)(135,361,297,443)(136,346,298,428)(137,387,299,413)(138,372,300,398)(139,357,301,439)(140,342,302,424)(141,383,303,409)(142,368,304,394)(143,353,305,435)(144,338,306,420)(145,379,307,405)(146,364,308,446)(147,349,309,431)(148,390,310,416)(149,375,311,401)(150,360,312,442)(151,345,313,427)(152,386,314,412)(153,371,315,397)(154,356,316,438)(155,341,317,423)(156,382,318,408)(157,367,319,393)(158,352,320,434)(159,337,321,419)(160,378,322,404)(161,363,323,445)(162,348,324,430)(163,389,325,415)(164,374,326,400)(165,359,327,441)(166,344,328,426)(167,385,329,411)(168,370,330,396) );
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)], [(1,313,107,151),(2,300,108,138),(3,287,109,125),(4,330,110,168),(5,317,111,155),(6,304,112,142),(7,291,57,129),(8,334,58,116),(9,321,59,159),(10,308,60,146),(11,295,61,133),(12,282,62,120),(13,325,63,163),(14,312,64,150),(15,299,65,137),(16,286,66,124),(17,329,67,167),(18,316,68,154),(19,303,69,141),(20,290,70,128),(21,333,71,115),(22,320,72,158),(23,307,73,145),(24,294,74,132),(25,281,75,119),(26,324,76,162),(27,311,77,149),(28,298,78,136),(29,285,79,123),(30,328,80,166),(31,315,81,153),(32,302,82,140),(33,289,83,127),(34,332,84,114),(35,319,85,157),(36,306,86,144),(37,293,87,131),(38,336,88,118),(39,323,89,161),(40,310,90,148),(41,297,91,135),(42,284,92,122),(43,327,93,165),(44,314,94,152),(45,301,95,139),(46,288,96,126),(47,331,97,113),(48,318,98,156),(49,305,99,143),(50,292,100,130),(51,335,101,117),(52,322,102,160),(53,309,103,147),(54,296,104,134),(55,283,105,121),(56,326,106,164),(169,443,232,361),(170,430,233,348),(171,417,234,391),(172,404,235,378),(173,447,236,365),(174,434,237,352),(175,421,238,339),(176,408,239,382),(177,395,240,369),(178,438,241,356),(179,425,242,343),(180,412,243,386),(181,399,244,373),(182,442,245,360),(183,429,246,347),(184,416,247,390),(185,403,248,377),(186,446,249,364),(187,433,250,351),(188,420,251,338),(189,407,252,381),(190,394,253,368),(191,437,254,355),(192,424,255,342),(193,411,256,385),(194,398,257,372),(195,441,258,359),(196,428,259,346),(197,415,260,389),(198,402,261,376),(199,445,262,363),(200,432,263,350),(201,419,264,337),(202,406,265,380),(203,393,266,367),(204,436,267,354),(205,423,268,341),(206,410,269,384),(207,397,270,371),(208,440,271,358),(209,427,272,345),(210,414,273,388),(211,401,274,375),(212,444,275,362),(213,431,276,349),(214,418,277,392),(215,405,278,379),(216,448,279,366),(217,435,280,353),(218,422,225,340),(219,409,226,383),(220,396,227,370),(221,439,228,357),(222,426,229,344),(223,413,230,387),(224,400,231,374)], [(1,272,107,209),(2,257,108,194),(3,242,109,179),(4,227,110,220),(5,268,111,205),(6,253,112,190),(7,238,57,175),(8,279,58,216),(9,264,59,201),(10,249,60,186),(11,234,61,171),(12,275,62,212),(13,260,63,197),(14,245,64,182),(15,230,65,223),(16,271,66,208),(17,256,67,193),(18,241,68,178),(19,226,69,219),(20,267,70,204),(21,252,71,189),(22,237,72,174),(23,278,73,215),(24,263,74,200),(25,248,75,185),(26,233,76,170),(27,274,77,211),(28,259,78,196),(29,244,79,181),(30,229,80,222),(31,270,81,207),(32,255,82,192),(33,240,83,177),(34,225,84,218),(35,266,85,203),(36,251,86,188),(37,236,87,173),(38,277,88,214),(39,262,89,199),(40,247,90,184),(41,232,91,169),(42,273,92,210),(43,258,93,195),(44,243,94,180),(45,228,95,221),(46,269,96,206),(47,254,97,191),(48,239,98,176),(49,280,99,217),(50,265,100,202),(51,250,101,187),(52,235,102,172),(53,276,103,213),(54,261,104,198),(55,246,105,183),(56,231,106,224),(113,355,331,437),(114,340,332,422),(115,381,333,407),(116,366,334,448),(117,351,335,433),(118,392,336,418),(119,377,281,403),(120,362,282,444),(121,347,283,429),(122,388,284,414),(123,373,285,399),(124,358,286,440),(125,343,287,425),(126,384,288,410),(127,369,289,395),(128,354,290,436),(129,339,291,421),(130,380,292,406),(131,365,293,447),(132,350,294,432),(133,391,295,417),(134,376,296,402),(135,361,297,443),(136,346,298,428),(137,387,299,413),(138,372,300,398),(139,357,301,439),(140,342,302,424),(141,383,303,409),(142,368,304,394),(143,353,305,435),(144,338,306,420),(145,379,307,405),(146,364,308,446),(147,349,309,431),(148,390,310,416),(149,375,311,401),(150,360,312,442),(151,345,313,427),(152,386,314,412),(153,371,315,397),(154,356,316,438),(155,341,317,423),(156,382,318,408),(157,367,319,393),(158,352,320,434),(159,337,321,419),(160,378,322,404),(161,363,323,445),(162,348,324,430),(163,389,325,415),(164,374,326,400),(165,359,327,441),(166,344,328,426),(167,385,329,411),(168,370,330,396)]])
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 | SD16 | D14 | D14 | Dic14 | Q8×D7 | D4×D7 | D7×SD16 |
| kernel | C56⋊5Q8 | C4.Dic14 | C8×Dic7 | C8⋊Dic7 | C7×C4.Q8 | C28⋊Q8 | C7⋊C8 | C56 | C2×Dic7 | C4.Q8 | Dic7 | C4⋊C4 | C2×C8 | C8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 8 | 6 | 3 | 12 | 3 | 3 | 12 |
Matrix representation of C56⋊5Q8 ►in GL6(𝔽113)
| 86 | 53 | 0 | 0 | 0 | 0 |
| 31 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 112 | 0 | 0 |
| 0 | 0 | 81 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 13 |
| 0 | 0 | 0 | 0 | 100 | 100 |
| 88 | 22 | 0 | 0 | 0 | 0 |
| 64 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 41 |
| 0 | 0 | 0 | 0 | 41 | 76 |
| 86 | 53 | 0 | 0 | 0 | 0 |
| 31 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 53 | 0 | 0 |
| 0 | 0 | 54 | 90 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(113))| [86,31,0,0,0,0,53,27,0,0,0,0,0,0,1,81,0,0,0,0,112,33,0,0,0,0,0,0,100,100,0,0,0,0,13,100],[88,64,0,0,0,0,22,25,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,37,41,0,0,0,0,41,76],[86,31,0,0,0,0,53,27,0,0,0,0,0,0,23,54,0,0,0,0,53,90,0,0,0,0,0,0,0,1,0,0,0,0,112,0] >;
C56⋊5Q8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_5Q_8 % in TeX
G:=Group("C56:5Q8"); // GroupNames label
G:=SmallGroup(448,389);
// by ID
G=gap.SmallGroup(448,389);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,254,555,58,438,102,18822]);
// 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^41,c*b*c^-1=b^-1>;
// generators/relations