metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊11Q8, C8⋊9Dic14, C42.251D14, C28.14M4(2), C7⋊1(C8⋊4Q8), (C4×C8).16D7, C14.6(C4×Q8), (C4×C56).22C2, C4⋊Dic7.6C4, C28.80(C2×Q8), C56⋊C4.5C2, C28⋊C8.4C2, C14.2(C8○D4), (C2×C8).280D14, C4.6(C8⋊D7), Dic7⋊C8.1C2, Dic7⋊C4.1C4, C2.8(C4×Dic14), (C4×Dic14).2C2, (C2×Dic14).6C4, C4.45(C2×Dic14), C14.1(C2×M4(2)), C28.239(C4○D4), C4.123(C4○D28), (C2×C28).801C23, (C2×C56).339C22, (C4×C28).322C22, C2.6(D28.2C4), (C4×Dic7).178C22, C2.6(C2×C8⋊D7), C22.94(C2×C4×D7), (C2×C4).102(C4×D7), (C2×C28).219(C2×C4), (C2×C7⋊C8).186C22, (C2×C14).56(C22×C4), (C2×Dic7).10(C2×C4), (C2×C4).743(C22×D7), SmallGroup(448,213)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊11Q8
G = < a,b,c | a56=b4=1, c2=b2, ab=ba, cac-1=a13, cbc-1=b-1 >
Subgroups: 324 in 94 conjugacy classes, 55 normal (33 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, C28, C28, C28, C2×C14, C4×C8, C8⋊C4, C4⋊C8, C4×Q8, C7⋊C8, C56, C56, Dic14, C2×Dic7, C2×C28, C8⋊4Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28⋊C8, Dic7⋊C8, C56⋊C4, C4×C56, C4×Dic14, C56⋊11Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, M4(2), C22×C4, C2×Q8, C4○D4, D14, C4×Q8, C2×M4(2), C8○D4, Dic14, C4×D7, C22×D7, C8⋊4Q8, C8⋊D7, C2×Dic14, C2×C4×D7, C4○D28, C4×Dic14, C2×C8⋊D7, D28.2C4, C56⋊11Q8
►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)
(1 233 443 370)(2 234 444 371)(3 235 445 372)(4 236 446 373)(5 237 447 374)(6 238 448 375)(7 239 393 376)(8 240 394 377)(9 241 395 378)(10 242 396 379)(11 243 397 380)(12 244 398 381)(13 245 399 382)(14 246 400 383)(15 247 401 384)(16 248 402 385)(17 249 403 386)(18 250 404 387)(19 251 405 388)(20 252 406 389)(21 253 407 390)(22 254 408 391)(23 255 409 392)(24 256 410 337)(25 257 411 338)(26 258 412 339)(27 259 413 340)(28 260 414 341)(29 261 415 342)(30 262 416 343)(31 263 417 344)(32 264 418 345)(33 265 419 346)(34 266 420 347)(35 267 421 348)(36 268 422 349)(37 269 423 350)(38 270 424 351)(39 271 425 352)(40 272 426 353)(41 273 427 354)(42 274 428 355)(43 275 429 356)(44 276 430 357)(45 277 431 358)(46 278 432 359)(47 279 433 360)(48 280 434 361)(49 225 435 362)(50 226 436 363)(51 227 437 364)(52 228 438 365)(53 229 439 366)(54 230 440 367)(55 231 441 368)(56 232 442 369)(57 205 125 314)(58 206 126 315)(59 207 127 316)(60 208 128 317)(61 209 129 318)(62 210 130 319)(63 211 131 320)(64 212 132 321)(65 213 133 322)(66 214 134 323)(67 215 135 324)(68 216 136 325)(69 217 137 326)(70 218 138 327)(71 219 139 328)(72 220 140 329)(73 221 141 330)(74 222 142 331)(75 223 143 332)(76 224 144 333)(77 169 145 334)(78 170 146 335)(79 171 147 336)(80 172 148 281)(81 173 149 282)(82 174 150 283)(83 175 151 284)(84 176 152 285)(85 177 153 286)(86 178 154 287)(87 179 155 288)(88 180 156 289)(89 181 157 290)(90 182 158 291)(91 183 159 292)(92 184 160 293)(93 185 161 294)(94 186 162 295)(95 187 163 296)(96 188 164 297)(97 189 165 298)(98 190 166 299)(99 191 167 300)(100 192 168 301)(101 193 113 302)(102 194 114 303)(103 195 115 304)(104 196 116 305)(105 197 117 306)(106 198 118 307)(107 199 119 308)(108 200 120 309)(109 201 121 310)(110 202 122 311)(111 203 123 312)(112 204 124 313)
(1 91 443 159)(2 104 444 116)(3 61 445 129)(4 74 446 142)(5 87 447 155)(6 100 448 168)(7 57 393 125)(8 70 394 138)(9 83 395 151)(10 96 396 164)(11 109 397 121)(12 66 398 134)(13 79 399 147)(14 92 400 160)(15 105 401 117)(16 62 402 130)(17 75 403 143)(18 88 404 156)(19 101 405 113)(20 58 406 126)(21 71 407 139)(22 84 408 152)(23 97 409 165)(24 110 410 122)(25 67 411 135)(26 80 412 148)(27 93 413 161)(28 106 414 118)(29 63 415 131)(30 76 416 144)(31 89 417 157)(32 102 418 114)(33 59 419 127)(34 72 420 140)(35 85 421 153)(36 98 422 166)(37 111 423 123)(38 68 424 136)(39 81 425 149)(40 94 426 162)(41 107 427 119)(42 64 428 132)(43 77 429 145)(44 90 430 158)(45 103 431 115)(46 60 432 128)(47 73 433 141)(48 86 434 154)(49 99 435 167)(50 112 436 124)(51 69 437 137)(52 82 438 150)(53 95 439 163)(54 108 440 120)(55 65 441 133)(56 78 442 146)(169 275 334 356)(170 232 335 369)(171 245 336 382)(172 258 281 339)(173 271 282 352)(174 228 283 365)(175 241 284 378)(176 254 285 391)(177 267 286 348)(178 280 287 361)(179 237 288 374)(180 250 289 387)(181 263 290 344)(182 276 291 357)(183 233 292 370)(184 246 293 383)(185 259 294 340)(186 272 295 353)(187 229 296 366)(188 242 297 379)(189 255 298 392)(190 268 299 349)(191 225 300 362)(192 238 301 375)(193 251 302 388)(194 264 303 345)(195 277 304 358)(196 234 305 371)(197 247 306 384)(198 260 307 341)(199 273 308 354)(200 230 309 367)(201 243 310 380)(202 256 311 337)(203 269 312 350)(204 226 313 363)(205 239 314 376)(206 252 315 389)(207 265 316 346)(208 278 317 359)(209 235 318 372)(210 248 319 385)(211 261 320 342)(212 274 321 355)(213 231 322 368)(214 244 323 381)(215 257 324 338)(216 270 325 351)(217 227 326 364)(218 240 327 377)(219 253 328 390)(220 266 329 347)(221 279 330 360)(222 236 331 373)(223 249 332 386)(224 262 333 343)
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), (1,233,443,370)(2,234,444,371)(3,235,445,372)(4,236,446,373)(5,237,447,374)(6,238,448,375)(7,239,393,376)(8,240,394,377)(9,241,395,378)(10,242,396,379)(11,243,397,380)(12,244,398,381)(13,245,399,382)(14,246,400,383)(15,247,401,384)(16,248,402,385)(17,249,403,386)(18,250,404,387)(19,251,405,388)(20,252,406,389)(21,253,407,390)(22,254,408,391)(23,255,409,392)(24,256,410,337)(25,257,411,338)(26,258,412,339)(27,259,413,340)(28,260,414,341)(29,261,415,342)(30,262,416,343)(31,263,417,344)(32,264,418,345)(33,265,419,346)(34,266,420,347)(35,267,421,348)(36,268,422,349)(37,269,423,350)(38,270,424,351)(39,271,425,352)(40,272,426,353)(41,273,427,354)(42,274,428,355)(43,275,429,356)(44,276,430,357)(45,277,431,358)(46,278,432,359)(47,279,433,360)(48,280,434,361)(49,225,435,362)(50,226,436,363)(51,227,437,364)(52,228,438,365)(53,229,439,366)(54,230,440,367)(55,231,441,368)(56,232,442,369)(57,205,125,314)(58,206,126,315)(59,207,127,316)(60,208,128,317)(61,209,129,318)(62,210,130,319)(63,211,131,320)(64,212,132,321)(65,213,133,322)(66,214,134,323)(67,215,135,324)(68,216,136,325)(69,217,137,326)(70,218,138,327)(71,219,139,328)(72,220,140,329)(73,221,141,330)(74,222,142,331)(75,223,143,332)(76,224,144,333)(77,169,145,334)(78,170,146,335)(79,171,147,336)(80,172,148,281)(81,173,149,282)(82,174,150,283)(83,175,151,284)(84,176,152,285)(85,177,153,286)(86,178,154,287)(87,179,155,288)(88,180,156,289)(89,181,157,290)(90,182,158,291)(91,183,159,292)(92,184,160,293)(93,185,161,294)(94,186,162,295)(95,187,163,296)(96,188,164,297)(97,189,165,298)(98,190,166,299)(99,191,167,300)(100,192,168,301)(101,193,113,302)(102,194,114,303)(103,195,115,304)(104,196,116,305)(105,197,117,306)(106,198,118,307)(107,199,119,308)(108,200,120,309)(109,201,121,310)(110,202,122,311)(111,203,123,312)(112,204,124,313), (1,91,443,159)(2,104,444,116)(3,61,445,129)(4,74,446,142)(5,87,447,155)(6,100,448,168)(7,57,393,125)(8,70,394,138)(9,83,395,151)(10,96,396,164)(11,109,397,121)(12,66,398,134)(13,79,399,147)(14,92,400,160)(15,105,401,117)(16,62,402,130)(17,75,403,143)(18,88,404,156)(19,101,405,113)(20,58,406,126)(21,71,407,139)(22,84,408,152)(23,97,409,165)(24,110,410,122)(25,67,411,135)(26,80,412,148)(27,93,413,161)(28,106,414,118)(29,63,415,131)(30,76,416,144)(31,89,417,157)(32,102,418,114)(33,59,419,127)(34,72,420,140)(35,85,421,153)(36,98,422,166)(37,111,423,123)(38,68,424,136)(39,81,425,149)(40,94,426,162)(41,107,427,119)(42,64,428,132)(43,77,429,145)(44,90,430,158)(45,103,431,115)(46,60,432,128)(47,73,433,141)(48,86,434,154)(49,99,435,167)(50,112,436,124)(51,69,437,137)(52,82,438,150)(53,95,439,163)(54,108,440,120)(55,65,441,133)(56,78,442,146)(169,275,334,356)(170,232,335,369)(171,245,336,382)(172,258,281,339)(173,271,282,352)(174,228,283,365)(175,241,284,378)(176,254,285,391)(177,267,286,348)(178,280,287,361)(179,237,288,374)(180,250,289,387)(181,263,290,344)(182,276,291,357)(183,233,292,370)(184,246,293,383)(185,259,294,340)(186,272,295,353)(187,229,296,366)(188,242,297,379)(189,255,298,392)(190,268,299,349)(191,225,300,362)(192,238,301,375)(193,251,302,388)(194,264,303,345)(195,277,304,358)(196,234,305,371)(197,247,306,384)(198,260,307,341)(199,273,308,354)(200,230,309,367)(201,243,310,380)(202,256,311,337)(203,269,312,350)(204,226,313,363)(205,239,314,376)(206,252,315,389)(207,265,316,346)(208,278,317,359)(209,235,318,372)(210,248,319,385)(211,261,320,342)(212,274,321,355)(213,231,322,368)(214,244,323,381)(215,257,324,338)(216,270,325,351)(217,227,326,364)(218,240,327,377)(219,253,328,390)(220,266,329,347)(221,279,330,360)(222,236,331,373)(223,249,332,386)(224,262,333,343)>;
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), (1,233,443,370)(2,234,444,371)(3,235,445,372)(4,236,446,373)(5,237,447,374)(6,238,448,375)(7,239,393,376)(8,240,394,377)(9,241,395,378)(10,242,396,379)(11,243,397,380)(12,244,398,381)(13,245,399,382)(14,246,400,383)(15,247,401,384)(16,248,402,385)(17,249,403,386)(18,250,404,387)(19,251,405,388)(20,252,406,389)(21,253,407,390)(22,254,408,391)(23,255,409,392)(24,256,410,337)(25,257,411,338)(26,258,412,339)(27,259,413,340)(28,260,414,341)(29,261,415,342)(30,262,416,343)(31,263,417,344)(32,264,418,345)(33,265,419,346)(34,266,420,347)(35,267,421,348)(36,268,422,349)(37,269,423,350)(38,270,424,351)(39,271,425,352)(40,272,426,353)(41,273,427,354)(42,274,428,355)(43,275,429,356)(44,276,430,357)(45,277,431,358)(46,278,432,359)(47,279,433,360)(48,280,434,361)(49,225,435,362)(50,226,436,363)(51,227,437,364)(52,228,438,365)(53,229,439,366)(54,230,440,367)(55,231,441,368)(56,232,442,369)(57,205,125,314)(58,206,126,315)(59,207,127,316)(60,208,128,317)(61,209,129,318)(62,210,130,319)(63,211,131,320)(64,212,132,321)(65,213,133,322)(66,214,134,323)(67,215,135,324)(68,216,136,325)(69,217,137,326)(70,218,138,327)(71,219,139,328)(72,220,140,329)(73,221,141,330)(74,222,142,331)(75,223,143,332)(76,224,144,333)(77,169,145,334)(78,170,146,335)(79,171,147,336)(80,172,148,281)(81,173,149,282)(82,174,150,283)(83,175,151,284)(84,176,152,285)(85,177,153,286)(86,178,154,287)(87,179,155,288)(88,180,156,289)(89,181,157,290)(90,182,158,291)(91,183,159,292)(92,184,160,293)(93,185,161,294)(94,186,162,295)(95,187,163,296)(96,188,164,297)(97,189,165,298)(98,190,166,299)(99,191,167,300)(100,192,168,301)(101,193,113,302)(102,194,114,303)(103,195,115,304)(104,196,116,305)(105,197,117,306)(106,198,118,307)(107,199,119,308)(108,200,120,309)(109,201,121,310)(110,202,122,311)(111,203,123,312)(112,204,124,313), (1,91,443,159)(2,104,444,116)(3,61,445,129)(4,74,446,142)(5,87,447,155)(6,100,448,168)(7,57,393,125)(8,70,394,138)(9,83,395,151)(10,96,396,164)(11,109,397,121)(12,66,398,134)(13,79,399,147)(14,92,400,160)(15,105,401,117)(16,62,402,130)(17,75,403,143)(18,88,404,156)(19,101,405,113)(20,58,406,126)(21,71,407,139)(22,84,408,152)(23,97,409,165)(24,110,410,122)(25,67,411,135)(26,80,412,148)(27,93,413,161)(28,106,414,118)(29,63,415,131)(30,76,416,144)(31,89,417,157)(32,102,418,114)(33,59,419,127)(34,72,420,140)(35,85,421,153)(36,98,422,166)(37,111,423,123)(38,68,424,136)(39,81,425,149)(40,94,426,162)(41,107,427,119)(42,64,428,132)(43,77,429,145)(44,90,430,158)(45,103,431,115)(46,60,432,128)(47,73,433,141)(48,86,434,154)(49,99,435,167)(50,112,436,124)(51,69,437,137)(52,82,438,150)(53,95,439,163)(54,108,440,120)(55,65,441,133)(56,78,442,146)(169,275,334,356)(170,232,335,369)(171,245,336,382)(172,258,281,339)(173,271,282,352)(174,228,283,365)(175,241,284,378)(176,254,285,391)(177,267,286,348)(178,280,287,361)(179,237,288,374)(180,250,289,387)(181,263,290,344)(182,276,291,357)(183,233,292,370)(184,246,293,383)(185,259,294,340)(186,272,295,353)(187,229,296,366)(188,242,297,379)(189,255,298,392)(190,268,299,349)(191,225,300,362)(192,238,301,375)(193,251,302,388)(194,264,303,345)(195,277,304,358)(196,234,305,371)(197,247,306,384)(198,260,307,341)(199,273,308,354)(200,230,309,367)(201,243,310,380)(202,256,311,337)(203,269,312,350)(204,226,313,363)(205,239,314,376)(206,252,315,389)(207,265,316,346)(208,278,317,359)(209,235,318,372)(210,248,319,385)(211,261,320,342)(212,274,321,355)(213,231,322,368)(214,244,323,381)(215,257,324,338)(216,270,325,351)(217,227,326,364)(218,240,327,377)(219,253,328,390)(220,266,329,347)(221,279,330,360)(222,236,331,373)(223,249,332,386)(224,262,333,343) );
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)], [(1,233,443,370),(2,234,444,371),(3,235,445,372),(4,236,446,373),(5,237,447,374),(6,238,448,375),(7,239,393,376),(8,240,394,377),(9,241,395,378),(10,242,396,379),(11,243,397,380),(12,244,398,381),(13,245,399,382),(14,246,400,383),(15,247,401,384),(16,248,402,385),(17,249,403,386),(18,250,404,387),(19,251,405,388),(20,252,406,389),(21,253,407,390),(22,254,408,391),(23,255,409,392),(24,256,410,337),(25,257,411,338),(26,258,412,339),(27,259,413,340),(28,260,414,341),(29,261,415,342),(30,262,416,343),(31,263,417,344),(32,264,418,345),(33,265,419,346),(34,266,420,347),(35,267,421,348),(36,268,422,349),(37,269,423,350),(38,270,424,351),(39,271,425,352),(40,272,426,353),(41,273,427,354),(42,274,428,355),(43,275,429,356),(44,276,430,357),(45,277,431,358),(46,278,432,359),(47,279,433,360),(48,280,434,361),(49,225,435,362),(50,226,436,363),(51,227,437,364),(52,228,438,365),(53,229,439,366),(54,230,440,367),(55,231,441,368),(56,232,442,369),(57,205,125,314),(58,206,126,315),(59,207,127,316),(60,208,128,317),(61,209,129,318),(62,210,130,319),(63,211,131,320),(64,212,132,321),(65,213,133,322),(66,214,134,323),(67,215,135,324),(68,216,136,325),(69,217,137,326),(70,218,138,327),(71,219,139,328),(72,220,140,329),(73,221,141,330),(74,222,142,331),(75,223,143,332),(76,224,144,333),(77,169,145,334),(78,170,146,335),(79,171,147,336),(80,172,148,281),(81,173,149,282),(82,174,150,283),(83,175,151,284),(84,176,152,285),(85,177,153,286),(86,178,154,287),(87,179,155,288),(88,180,156,289),(89,181,157,290),(90,182,158,291),(91,183,159,292),(92,184,160,293),(93,185,161,294),(94,186,162,295),(95,187,163,296),(96,188,164,297),(97,189,165,298),(98,190,166,299),(99,191,167,300),(100,192,168,301),(101,193,113,302),(102,194,114,303),(103,195,115,304),(104,196,116,305),(105,197,117,306),(106,198,118,307),(107,199,119,308),(108,200,120,309),(109,201,121,310),(110,202,122,311),(111,203,123,312),(112,204,124,313)], [(1,91,443,159),(2,104,444,116),(3,61,445,129),(4,74,446,142),(5,87,447,155),(6,100,448,168),(7,57,393,125),(8,70,394,138),(9,83,395,151),(10,96,396,164),(11,109,397,121),(12,66,398,134),(13,79,399,147),(14,92,400,160),(15,105,401,117),(16,62,402,130),(17,75,403,143),(18,88,404,156),(19,101,405,113),(20,58,406,126),(21,71,407,139),(22,84,408,152),(23,97,409,165),(24,110,410,122),(25,67,411,135),(26,80,412,148),(27,93,413,161),(28,106,414,118),(29,63,415,131),(30,76,416,144),(31,89,417,157),(32,102,418,114),(33,59,419,127),(34,72,420,140),(35,85,421,153),(36,98,422,166),(37,111,423,123),(38,68,424,136),(39,81,425,149),(40,94,426,162),(41,107,427,119),(42,64,428,132),(43,77,429,145),(44,90,430,158),(45,103,431,115),(46,60,432,128),(47,73,433,141),(48,86,434,154),(49,99,435,167),(50,112,436,124),(51,69,437,137),(52,82,438,150),(53,95,439,163),(54,108,440,120),(55,65,441,133),(56,78,442,146),(169,275,334,356),(170,232,335,369),(171,245,336,382),(172,258,281,339),(173,271,282,352),(174,228,283,365),(175,241,284,378),(176,254,285,391),(177,267,286,348),(178,280,287,361),(179,237,288,374),(180,250,289,387),(181,263,290,344),(182,276,291,357),(183,233,292,370),(184,246,293,383),(185,259,294,340),(186,272,295,353),(187,229,296,366),(188,242,297,379),(189,255,298,392),(190,268,299,349),(191,225,300,362),(192,238,301,375),(193,251,302,388),(194,264,303,345),(195,277,304,358),(196,234,305,371),(197,247,306,384),(198,260,307,341),(199,273,308,354),(200,230,309,367),(201,243,310,380),(202,256,311,337),(203,269,312,350),(204,226,313,363),(205,239,314,376),(206,252,315,389),(207,265,316,346),(208,278,317,359),(209,235,318,372),(210,248,319,385),(211,261,320,342),(212,274,321,355),(213,231,322,368),(214,244,323,381),(215,257,324,338),(216,270,325,351),(217,227,326,364),(218,240,327,377),(219,253,328,390),(220,266,329,347),(221,279,330,360),(222,236,331,373),(223,249,332,386),(224,262,333,343)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | D7 | M4(2) | C4○D4 | D14 | D14 | C8○D4 | Dic14 | C4×D7 | C8⋊D7 | C4○D28 | D28.2C4 |
| kernel | C56⋊11Q8 | C28⋊C8 | Dic7⋊C8 | C56⋊C4 | C4×C56 | C4×Dic14 | Dic7⋊C4 | C4⋊Dic7 | C2×Dic14 | C56 | C4×C8 | C28 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 4 | 12 | 12 | 24 | 12 | 24 |
Matrix representation of C56⋊11Q8 ►in GL4(𝔽113) generated by
| 0 | 15 | 0 | 0 |
| 98 | 55 | 0 | 0 |
| 0 | 0 | 83 | 33 |
| 0 | 0 | 80 | 82 |
| 104 | 46 | 0 | 0 |
| 67 | 9 | 0 | 0 |
| 0 | 0 | 96 | 8 |
| 0 | 0 | 105 | 17 |
| 15 | 58 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 92 | 76 |
| 0 | 0 | 15 | 21 |
G:=sub<GL(4,GF(113))| [0,98,0,0,15,55,0,0,0,0,83,80,0,0,33,82],[104,67,0,0,46,9,0,0,0,0,96,105,0,0,8,17],[15,0,0,0,58,98,0,0,0,0,92,15,0,0,76,21] >;
C56⋊11Q8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{11}Q_8 % in TeX
G:=Group("C56:11Q8"); // GroupNames label
G:=SmallGroup(448,213);
// by ID
G=gap.SmallGroup(448,213);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,758,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^13,c*b*c^-1=b^-1>;
// generators/relations