metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.4Q8, C8.6Dic14, C7⋊C8.2Q8, C4.28(Q8×D7), C4⋊C4.46D14, C2.D8.7D7, C28.60(C2×Q8), C7⋊3(C8.5Q8), (C2×C8).228D14, C2.13(C28⋊Q8), C14.18(C4⋊Q8), (C8×Dic7).3C2, C56⋊1C4.15C2, C14.74(C4○D8), (C2×C56).80C22, C4.25(C2×Dic14), C22.227(D4×D7), C28.3Q8.9C2, C4.Dic14.9C2, C2.13(D8⋊3D7), (C2×C28).294C23, (C2×Dic7).101D4, C2.12(Q8.D14), C4⋊Dic7.120C22, (C4×Dic7).234C22, (C7×C2.D8).6C2, (C2×C14).299(C2×D4), (C7×C4⋊C4).87C22, (C2×C7⋊C8).232C22, (C2×C4).397(C22×D7), SmallGroup(448,412)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.4Q8
G = < a,b,c | a56=b4=1, c2=b2, bab-1=a15, cac-1=a41, cbc-1=a28b-1 >
Subgroups: 364 in 86 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Dic7, C28, C28, C2×C14, C4×C8, C4.Q8, C2.D8, 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, C4.Dic14, C8×Dic7, C56⋊1C4, C7×C2.D8, C28.3Q8, C56.4Q8
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, D8⋊3D7, Q8.D14, C56.4Q8
►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)
(1 107 163 302)(2 66 164 317)(3 81 165 332)(4 96 166 291)(5 111 167 306)(6 70 168 321)(7 85 113 336)(8 100 114 295)(9 59 115 310)(10 74 116 325)(11 89 117 284)(12 104 118 299)(13 63 119 314)(14 78 120 329)(15 93 121 288)(16 108 122 303)(17 67 123 318)(18 82 124 333)(19 97 125 292)(20 112 126 307)(21 71 127 322)(22 86 128 281)(23 101 129 296)(24 60 130 311)(25 75 131 326)(26 90 132 285)(27 105 133 300)(28 64 134 315)(29 79 135 330)(30 94 136 289)(31 109 137 304)(32 68 138 319)(33 83 139 334)(34 98 140 293)(35 57 141 308)(36 72 142 323)(37 87 143 282)(38 102 144 297)(39 61 145 312)(40 76 146 327)(41 91 147 286)(42 106 148 301)(43 65 149 316)(44 80 150 331)(45 95 151 290)(46 110 152 305)(47 69 153 320)(48 84 154 335)(49 99 155 294)(50 58 156 309)(51 73 157 324)(52 88 158 283)(53 103 159 298)(54 62 160 313)(55 77 161 328)(56 92 162 287)(169 420 340 248)(170 435 341 263)(171 394 342 278)(172 409 343 237)(173 424 344 252)(174 439 345 267)(175 398 346 226)(176 413 347 241)(177 428 348 256)(178 443 349 271)(179 402 350 230)(180 417 351 245)(181 432 352 260)(182 447 353 275)(183 406 354 234)(184 421 355 249)(185 436 356 264)(186 395 357 279)(187 410 358 238)(188 425 359 253)(189 440 360 268)(190 399 361 227)(191 414 362 242)(192 429 363 257)(193 444 364 272)(194 403 365 231)(195 418 366 246)(196 433 367 261)(197 448 368 276)(198 407 369 235)(199 422 370 250)(200 437 371 265)(201 396 372 280)(202 411 373 239)(203 426 374 254)(204 441 375 269)(205 400 376 228)(206 415 377 243)(207 430 378 258)(208 445 379 273)(209 404 380 232)(210 419 381 247)(211 434 382 262)(212 393 383 277)(213 408 384 236)(214 423 385 251)(215 438 386 266)(216 397 387 225)(217 412 388 240)(218 427 389 255)(219 442 390 270)(220 401 391 229)(221 416 392 244)(222 431 337 259)(223 446 338 274)(224 405 339 233)
(1 170 163 341)(2 211 164 382)(3 196 165 367)(4 181 166 352)(5 222 167 337)(6 207 168 378)(7 192 113 363)(8 177 114 348)(9 218 115 389)(10 203 116 374)(11 188 117 359)(12 173 118 344)(13 214 119 385)(14 199 120 370)(15 184 121 355)(16 169 122 340)(17 210 123 381)(18 195 124 366)(19 180 125 351)(20 221 126 392)(21 206 127 377)(22 191 128 362)(23 176 129 347)(24 217 130 388)(25 202 131 373)(26 187 132 358)(27 172 133 343)(28 213 134 384)(29 198 135 369)(30 183 136 354)(31 224 137 339)(32 209 138 380)(33 194 139 365)(34 179 140 350)(35 220 141 391)(36 205 142 376)(37 190 143 361)(38 175 144 346)(39 216 145 387)(40 201 146 372)(41 186 147 357)(42 171 148 342)(43 212 149 383)(44 197 150 368)(45 182 151 353)(46 223 152 338)(47 208 153 379)(48 193 154 364)(49 178 155 349)(50 219 156 390)(51 204 157 375)(52 189 158 360)(53 174 159 345)(54 215 160 386)(55 200 161 371)(56 185 162 356)(57 257 308 429)(58 242 309 414)(59 227 310 399)(60 268 311 440)(61 253 312 425)(62 238 313 410)(63 279 314 395)(64 264 315 436)(65 249 316 421)(66 234 317 406)(67 275 318 447)(68 260 319 432)(69 245 320 417)(70 230 321 402)(71 271 322 443)(72 256 323 428)(73 241 324 413)(74 226 325 398)(75 267 326 439)(76 252 327 424)(77 237 328 409)(78 278 329 394)(79 263 330 435)(80 248 331 420)(81 233 332 405)(82 274 333 446)(83 259 334 431)(84 244 335 416)(85 229 336 401)(86 270 281 442)(87 255 282 427)(88 240 283 412)(89 225 284 397)(90 266 285 438)(91 251 286 423)(92 236 287 408)(93 277 288 393)(94 262 289 434)(95 247 290 419)(96 232 291 404)(97 273 292 445)(98 258 293 430)(99 243 294 415)(100 228 295 400)(101 269 296 441)(102 254 297 426)(103 239 298 411)(104 280 299 396)(105 265 300 437)(106 250 301 422)(107 235 302 407)(108 276 303 448)(109 261 304 433)(110 246 305 418)(111 231 306 403)(112 272 307 444)
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), (1,107,163,302)(2,66,164,317)(3,81,165,332)(4,96,166,291)(5,111,167,306)(6,70,168,321)(7,85,113,336)(8,100,114,295)(9,59,115,310)(10,74,116,325)(11,89,117,284)(12,104,118,299)(13,63,119,314)(14,78,120,329)(15,93,121,288)(16,108,122,303)(17,67,123,318)(18,82,124,333)(19,97,125,292)(20,112,126,307)(21,71,127,322)(22,86,128,281)(23,101,129,296)(24,60,130,311)(25,75,131,326)(26,90,132,285)(27,105,133,300)(28,64,134,315)(29,79,135,330)(30,94,136,289)(31,109,137,304)(32,68,138,319)(33,83,139,334)(34,98,140,293)(35,57,141,308)(36,72,142,323)(37,87,143,282)(38,102,144,297)(39,61,145,312)(40,76,146,327)(41,91,147,286)(42,106,148,301)(43,65,149,316)(44,80,150,331)(45,95,151,290)(46,110,152,305)(47,69,153,320)(48,84,154,335)(49,99,155,294)(50,58,156,309)(51,73,157,324)(52,88,158,283)(53,103,159,298)(54,62,160,313)(55,77,161,328)(56,92,162,287)(169,420,340,248)(170,435,341,263)(171,394,342,278)(172,409,343,237)(173,424,344,252)(174,439,345,267)(175,398,346,226)(176,413,347,241)(177,428,348,256)(178,443,349,271)(179,402,350,230)(180,417,351,245)(181,432,352,260)(182,447,353,275)(183,406,354,234)(184,421,355,249)(185,436,356,264)(186,395,357,279)(187,410,358,238)(188,425,359,253)(189,440,360,268)(190,399,361,227)(191,414,362,242)(192,429,363,257)(193,444,364,272)(194,403,365,231)(195,418,366,246)(196,433,367,261)(197,448,368,276)(198,407,369,235)(199,422,370,250)(200,437,371,265)(201,396,372,280)(202,411,373,239)(203,426,374,254)(204,441,375,269)(205,400,376,228)(206,415,377,243)(207,430,378,258)(208,445,379,273)(209,404,380,232)(210,419,381,247)(211,434,382,262)(212,393,383,277)(213,408,384,236)(214,423,385,251)(215,438,386,266)(216,397,387,225)(217,412,388,240)(218,427,389,255)(219,442,390,270)(220,401,391,229)(221,416,392,244)(222,431,337,259)(223,446,338,274)(224,405,339,233), (1,170,163,341)(2,211,164,382)(3,196,165,367)(4,181,166,352)(5,222,167,337)(6,207,168,378)(7,192,113,363)(8,177,114,348)(9,218,115,389)(10,203,116,374)(11,188,117,359)(12,173,118,344)(13,214,119,385)(14,199,120,370)(15,184,121,355)(16,169,122,340)(17,210,123,381)(18,195,124,366)(19,180,125,351)(20,221,126,392)(21,206,127,377)(22,191,128,362)(23,176,129,347)(24,217,130,388)(25,202,131,373)(26,187,132,358)(27,172,133,343)(28,213,134,384)(29,198,135,369)(30,183,136,354)(31,224,137,339)(32,209,138,380)(33,194,139,365)(34,179,140,350)(35,220,141,391)(36,205,142,376)(37,190,143,361)(38,175,144,346)(39,216,145,387)(40,201,146,372)(41,186,147,357)(42,171,148,342)(43,212,149,383)(44,197,150,368)(45,182,151,353)(46,223,152,338)(47,208,153,379)(48,193,154,364)(49,178,155,349)(50,219,156,390)(51,204,157,375)(52,189,158,360)(53,174,159,345)(54,215,160,386)(55,200,161,371)(56,185,162,356)(57,257,308,429)(58,242,309,414)(59,227,310,399)(60,268,311,440)(61,253,312,425)(62,238,313,410)(63,279,314,395)(64,264,315,436)(65,249,316,421)(66,234,317,406)(67,275,318,447)(68,260,319,432)(69,245,320,417)(70,230,321,402)(71,271,322,443)(72,256,323,428)(73,241,324,413)(74,226,325,398)(75,267,326,439)(76,252,327,424)(77,237,328,409)(78,278,329,394)(79,263,330,435)(80,248,331,420)(81,233,332,405)(82,274,333,446)(83,259,334,431)(84,244,335,416)(85,229,336,401)(86,270,281,442)(87,255,282,427)(88,240,283,412)(89,225,284,397)(90,266,285,438)(91,251,286,423)(92,236,287,408)(93,277,288,393)(94,262,289,434)(95,247,290,419)(96,232,291,404)(97,273,292,445)(98,258,293,430)(99,243,294,415)(100,228,295,400)(101,269,296,441)(102,254,297,426)(103,239,298,411)(104,280,299,396)(105,265,300,437)(106,250,301,422)(107,235,302,407)(108,276,303,448)(109,261,304,433)(110,246,305,418)(111,231,306,403)(112,272,307,444)>;
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), (1,107,163,302)(2,66,164,317)(3,81,165,332)(4,96,166,291)(5,111,167,306)(6,70,168,321)(7,85,113,336)(8,100,114,295)(9,59,115,310)(10,74,116,325)(11,89,117,284)(12,104,118,299)(13,63,119,314)(14,78,120,329)(15,93,121,288)(16,108,122,303)(17,67,123,318)(18,82,124,333)(19,97,125,292)(20,112,126,307)(21,71,127,322)(22,86,128,281)(23,101,129,296)(24,60,130,311)(25,75,131,326)(26,90,132,285)(27,105,133,300)(28,64,134,315)(29,79,135,330)(30,94,136,289)(31,109,137,304)(32,68,138,319)(33,83,139,334)(34,98,140,293)(35,57,141,308)(36,72,142,323)(37,87,143,282)(38,102,144,297)(39,61,145,312)(40,76,146,327)(41,91,147,286)(42,106,148,301)(43,65,149,316)(44,80,150,331)(45,95,151,290)(46,110,152,305)(47,69,153,320)(48,84,154,335)(49,99,155,294)(50,58,156,309)(51,73,157,324)(52,88,158,283)(53,103,159,298)(54,62,160,313)(55,77,161,328)(56,92,162,287)(169,420,340,248)(170,435,341,263)(171,394,342,278)(172,409,343,237)(173,424,344,252)(174,439,345,267)(175,398,346,226)(176,413,347,241)(177,428,348,256)(178,443,349,271)(179,402,350,230)(180,417,351,245)(181,432,352,260)(182,447,353,275)(183,406,354,234)(184,421,355,249)(185,436,356,264)(186,395,357,279)(187,410,358,238)(188,425,359,253)(189,440,360,268)(190,399,361,227)(191,414,362,242)(192,429,363,257)(193,444,364,272)(194,403,365,231)(195,418,366,246)(196,433,367,261)(197,448,368,276)(198,407,369,235)(199,422,370,250)(200,437,371,265)(201,396,372,280)(202,411,373,239)(203,426,374,254)(204,441,375,269)(205,400,376,228)(206,415,377,243)(207,430,378,258)(208,445,379,273)(209,404,380,232)(210,419,381,247)(211,434,382,262)(212,393,383,277)(213,408,384,236)(214,423,385,251)(215,438,386,266)(216,397,387,225)(217,412,388,240)(218,427,389,255)(219,442,390,270)(220,401,391,229)(221,416,392,244)(222,431,337,259)(223,446,338,274)(224,405,339,233), (1,170,163,341)(2,211,164,382)(3,196,165,367)(4,181,166,352)(5,222,167,337)(6,207,168,378)(7,192,113,363)(8,177,114,348)(9,218,115,389)(10,203,116,374)(11,188,117,359)(12,173,118,344)(13,214,119,385)(14,199,120,370)(15,184,121,355)(16,169,122,340)(17,210,123,381)(18,195,124,366)(19,180,125,351)(20,221,126,392)(21,206,127,377)(22,191,128,362)(23,176,129,347)(24,217,130,388)(25,202,131,373)(26,187,132,358)(27,172,133,343)(28,213,134,384)(29,198,135,369)(30,183,136,354)(31,224,137,339)(32,209,138,380)(33,194,139,365)(34,179,140,350)(35,220,141,391)(36,205,142,376)(37,190,143,361)(38,175,144,346)(39,216,145,387)(40,201,146,372)(41,186,147,357)(42,171,148,342)(43,212,149,383)(44,197,150,368)(45,182,151,353)(46,223,152,338)(47,208,153,379)(48,193,154,364)(49,178,155,349)(50,219,156,390)(51,204,157,375)(52,189,158,360)(53,174,159,345)(54,215,160,386)(55,200,161,371)(56,185,162,356)(57,257,308,429)(58,242,309,414)(59,227,310,399)(60,268,311,440)(61,253,312,425)(62,238,313,410)(63,279,314,395)(64,264,315,436)(65,249,316,421)(66,234,317,406)(67,275,318,447)(68,260,319,432)(69,245,320,417)(70,230,321,402)(71,271,322,443)(72,256,323,428)(73,241,324,413)(74,226,325,398)(75,267,326,439)(76,252,327,424)(77,237,328,409)(78,278,329,394)(79,263,330,435)(80,248,331,420)(81,233,332,405)(82,274,333,446)(83,259,334,431)(84,244,335,416)(85,229,336,401)(86,270,281,442)(87,255,282,427)(88,240,283,412)(89,225,284,397)(90,266,285,438)(91,251,286,423)(92,236,287,408)(93,277,288,393)(94,262,289,434)(95,247,290,419)(96,232,291,404)(97,273,292,445)(98,258,293,430)(99,243,294,415)(100,228,295,400)(101,269,296,441)(102,254,297,426)(103,239,298,411)(104,280,299,396)(105,265,300,437)(106,250,301,422)(107,235,302,407)(108,276,303,448)(109,261,304,433)(110,246,305,418)(111,231,306,403)(112,272,307,444) );
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)], [(1,107,163,302),(2,66,164,317),(3,81,165,332),(4,96,166,291),(5,111,167,306),(6,70,168,321),(7,85,113,336),(8,100,114,295),(9,59,115,310),(10,74,116,325),(11,89,117,284),(12,104,118,299),(13,63,119,314),(14,78,120,329),(15,93,121,288),(16,108,122,303),(17,67,123,318),(18,82,124,333),(19,97,125,292),(20,112,126,307),(21,71,127,322),(22,86,128,281),(23,101,129,296),(24,60,130,311),(25,75,131,326),(26,90,132,285),(27,105,133,300),(28,64,134,315),(29,79,135,330),(30,94,136,289),(31,109,137,304),(32,68,138,319),(33,83,139,334),(34,98,140,293),(35,57,141,308),(36,72,142,323),(37,87,143,282),(38,102,144,297),(39,61,145,312),(40,76,146,327),(41,91,147,286),(42,106,148,301),(43,65,149,316),(44,80,150,331),(45,95,151,290),(46,110,152,305),(47,69,153,320),(48,84,154,335),(49,99,155,294),(50,58,156,309),(51,73,157,324),(52,88,158,283),(53,103,159,298),(54,62,160,313),(55,77,161,328),(56,92,162,287),(169,420,340,248),(170,435,341,263),(171,394,342,278),(172,409,343,237),(173,424,344,252),(174,439,345,267),(175,398,346,226),(176,413,347,241),(177,428,348,256),(178,443,349,271),(179,402,350,230),(180,417,351,245),(181,432,352,260),(182,447,353,275),(183,406,354,234),(184,421,355,249),(185,436,356,264),(186,395,357,279),(187,410,358,238),(188,425,359,253),(189,440,360,268),(190,399,361,227),(191,414,362,242),(192,429,363,257),(193,444,364,272),(194,403,365,231),(195,418,366,246),(196,433,367,261),(197,448,368,276),(198,407,369,235),(199,422,370,250),(200,437,371,265),(201,396,372,280),(202,411,373,239),(203,426,374,254),(204,441,375,269),(205,400,376,228),(206,415,377,243),(207,430,378,258),(208,445,379,273),(209,404,380,232),(210,419,381,247),(211,434,382,262),(212,393,383,277),(213,408,384,236),(214,423,385,251),(215,438,386,266),(216,397,387,225),(217,412,388,240),(218,427,389,255),(219,442,390,270),(220,401,391,229),(221,416,392,244),(222,431,337,259),(223,446,338,274),(224,405,339,233)], [(1,170,163,341),(2,211,164,382),(3,196,165,367),(4,181,166,352),(5,222,167,337),(6,207,168,378),(7,192,113,363),(8,177,114,348),(9,218,115,389),(10,203,116,374),(11,188,117,359),(12,173,118,344),(13,214,119,385),(14,199,120,370),(15,184,121,355),(16,169,122,340),(17,210,123,381),(18,195,124,366),(19,180,125,351),(20,221,126,392),(21,206,127,377),(22,191,128,362),(23,176,129,347),(24,217,130,388),(25,202,131,373),(26,187,132,358),(27,172,133,343),(28,213,134,384),(29,198,135,369),(30,183,136,354),(31,224,137,339),(32,209,138,380),(33,194,139,365),(34,179,140,350),(35,220,141,391),(36,205,142,376),(37,190,143,361),(38,175,144,346),(39,216,145,387),(40,201,146,372),(41,186,147,357),(42,171,148,342),(43,212,149,383),(44,197,150,368),(45,182,151,353),(46,223,152,338),(47,208,153,379),(48,193,154,364),(49,178,155,349),(50,219,156,390),(51,204,157,375),(52,189,158,360),(53,174,159,345),(54,215,160,386),(55,200,161,371),(56,185,162,356),(57,257,308,429),(58,242,309,414),(59,227,310,399),(60,268,311,440),(61,253,312,425),(62,238,313,410),(63,279,314,395),(64,264,315,436),(65,249,316,421),(66,234,317,406),(67,275,318,447),(68,260,319,432),(69,245,320,417),(70,230,321,402),(71,271,322,443),(72,256,323,428),(73,241,324,413),(74,226,325,398),(75,267,326,439),(76,252,327,424),(77,237,328,409),(78,278,329,394),(79,263,330,435),(80,248,331,420),(81,233,332,405),(82,274,333,446),(83,259,334,431),(84,244,335,416),(85,229,336,401),(86,270,281,442),(87,255,282,427),(88,240,283,412),(89,225,284,397),(90,266,285,438),(91,251,286,423),(92,236,287,408),(93,277,288,393),(94,262,289,434),(95,247,290,419),(96,232,291,404),(97,273,292,445),(98,258,293,430),(99,243,294,415),(100,228,295,400),(101,269,296,441),(102,254,297,426),(103,239,298,411),(104,280,299,396),(105,265,300,437),(106,250,301,422),(107,235,302,407),(108,276,303,448),(109,261,304,433),(110,246,305,418),(111,231,306,403),(112,272,307,444)]])
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 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | + | + | - | - | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | Q8 | D4 | D7 | D14 | D14 | C4○D8 | Dic14 | Q8×D7 | D4×D7 | D8⋊3D7 | Q8.D14 |
| kernel | C56.4Q8 | C4.Dic14 | C8×Dic7 | C56⋊1C4 | C7×C2.D8 | C28.3Q8 | C7⋊C8 | C56 | C2×Dic7 | C2.D8 | C4⋊C4 | C2×C8 | C14 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 6 | 3 | 8 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of C56.4Q8 ►in GL4(𝔽113) generated by
| 34 | 1 | 0 | 0 |
| 111 | 103 | 0 | 0 |
| 0 | 0 | 62 | 82 |
| 0 | 0 | 62 | 0 |
| 50 | 105 | 0 | 0 |
| 16 | 63 | 0 | 0 |
| 0 | 0 | 10 | 86 |
| 0 | 0 | 79 | 103 |
| 32 | 87 | 0 | 0 |
| 9 | 81 | 0 | 0 |
| 0 | 0 | 98 | 15 |
| 0 | 0 | 83 | 15 |
G:=sub<GL(4,GF(113))| [34,111,0,0,1,103,0,0,0,0,62,62,0,0,82,0],[50,16,0,0,105,63,0,0,0,0,10,79,0,0,86,103],[32,9,0,0,87,81,0,0,0,0,98,83,0,0,15,15] >;
C56.4Q8 in GAP, Magma, Sage, TeX
C_{56}._4Q_8 % in TeX
G:=Group("C56.4Q8"); // GroupNames label
G:=SmallGroup(448,412);
// by ID
G=gap.SmallGroup(448,412);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,926,219,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=b^2,b*a*b^-1=a^15,c*a*c^-1=a^41,c*b*c^-1=a^28*b^-1>;
// generators/relations