metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊4Q8, C8⋊3Dic14, C7⋊C8⋊2Q8, C7⋊3(C8⋊Q8), C28⋊Q8.9C2, C4.27(Q8×D7), C4⋊C4.44D14, (C2×C8).64D14, C2.D8.8D7, C28.59(C2×Q8), C56⋊C4.3C2, C2.12(C28⋊Q8), C14.17(C4⋊Q8), C8⋊Dic7.10C2, (C2×Dic7).46D4, C4.24(C2×Dic14), C28.Q8.9C2, C22.225(D4×D7), C4.Dic14.7C2, C28.3Q8.7C2, C2.21(D8⋊D7), C14.39(C8⋊C22), (C2×C28).292C23, (C2×C56).142C22, C2.20(Q16⋊D7), C14.67(C8.C22), C4⋊Dic7.118C22, (C4×Dic7).35C22, (C7×C2.D8).7C2, (C2×C7⋊C8).66C22, (C2×C14).297(C2×D4), (C7×C4⋊C4).85C22, (C2×C4).395(C22×D7), SmallGroup(448,410)
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=a13, cbc-1=b-1 >
Subgroups: 428 in 90 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, C4.Q8, C2.D8, C2.D8, C42.C2, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C8⋊Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C28.Q8, C4.Dic14, C56⋊C4, C8⋊Dic7, C7×C2.D8, C28⋊Q8, C28.3Q8, C56⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, C8⋊C22, C8.C22, Dic14, C22×D7, C8⋊Q8, C2×Dic14, D4×D7, Q8×D7, C28⋊Q8, D8⋊D7, Q16⋊D7, 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 93 168 245)(2 108 113 260)(3 67 114 275)(4 82 115 234)(5 97 116 249)(6 112 117 264)(7 71 118 279)(8 86 119 238)(9 101 120 253)(10 60 121 268)(11 75 122 227)(12 90 123 242)(13 105 124 257)(14 64 125 272)(15 79 126 231)(16 94 127 246)(17 109 128 261)(18 68 129 276)(19 83 130 235)(20 98 131 250)(21 57 132 265)(22 72 133 280)(23 87 134 239)(24 102 135 254)(25 61 136 269)(26 76 137 228)(27 91 138 243)(28 106 139 258)(29 65 140 273)(30 80 141 232)(31 95 142 247)(32 110 143 262)(33 69 144 277)(34 84 145 236)(35 99 146 251)(36 58 147 266)(37 73 148 225)(38 88 149 240)(39 103 150 255)(40 62 151 270)(41 77 152 229)(42 92 153 244)(43 107 154 259)(44 66 155 274)(45 81 156 233)(46 96 157 248)(47 111 158 263)(48 70 159 278)(49 85 160 237)(50 100 161 252)(51 59 162 267)(52 74 163 226)(53 89 164 241)(54 104 165 256)(55 63 166 271)(56 78 167 230)(169 430 282 339)(170 445 283 354)(171 404 284 369)(172 419 285 384)(173 434 286 343)(174 393 287 358)(175 408 288 373)(176 423 289 388)(177 438 290 347)(178 397 291 362)(179 412 292 377)(180 427 293 392)(181 442 294 351)(182 401 295 366)(183 416 296 381)(184 431 297 340)(185 446 298 355)(186 405 299 370)(187 420 300 385)(188 435 301 344)(189 394 302 359)(190 409 303 374)(191 424 304 389)(192 439 305 348)(193 398 306 363)(194 413 307 378)(195 428 308 337)(196 443 309 352)(197 402 310 367)(198 417 311 382)(199 432 312 341)(200 447 313 356)(201 406 314 371)(202 421 315 386)(203 436 316 345)(204 395 317 360)(205 410 318 375)(206 425 319 390)(207 440 320 349)(208 399 321 364)(209 414 322 379)(210 429 323 338)(211 444 324 353)(212 403 325 368)(213 418 326 383)(214 433 327 342)(215 448 328 357)(216 407 329 372)(217 422 330 387)(218 437 331 346)(219 396 332 361)(220 411 333 376)(221 426 334 391)(222 441 335 350)(223 400 336 365)(224 415 281 380)
(1 213 168 326)(2 170 113 283)(3 183 114 296)(4 196 115 309)(5 209 116 322)(6 222 117 335)(7 179 118 292)(8 192 119 305)(9 205 120 318)(10 218 121 331)(11 175 122 288)(12 188 123 301)(13 201 124 314)(14 214 125 327)(15 171 126 284)(16 184 127 297)(17 197 128 310)(18 210 129 323)(19 223 130 336)(20 180 131 293)(21 193 132 306)(22 206 133 319)(23 219 134 332)(24 176 135 289)(25 189 136 302)(26 202 137 315)(27 215 138 328)(28 172 139 285)(29 185 140 298)(30 198 141 311)(31 211 142 324)(32 224 143 281)(33 181 144 294)(34 194 145 307)(35 207 146 320)(36 220 147 333)(37 177 148 290)(38 190 149 303)(39 203 150 316)(40 216 151 329)(41 173 152 286)(42 186 153 299)(43 199 154 312)(44 212 155 325)(45 169 156 282)(46 182 157 295)(47 195 158 308)(48 208 159 321)(49 221 160 334)(50 178 161 291)(51 191 162 304)(52 204 163 317)(53 217 164 330)(54 174 165 287)(55 187 166 300)(56 200 167 313)(57 363 265 398)(58 376 266 411)(59 389 267 424)(60 346 268 437)(61 359 269 394)(62 372 270 407)(63 385 271 420)(64 342 272 433)(65 355 273 446)(66 368 274 403)(67 381 275 416)(68 338 276 429)(69 351 277 442)(70 364 278 399)(71 377 279 412)(72 390 280 425)(73 347 225 438)(74 360 226 395)(75 373 227 408)(76 386 228 421)(77 343 229 434)(78 356 230 447)(79 369 231 404)(80 382 232 417)(81 339 233 430)(82 352 234 443)(83 365 235 400)(84 378 236 413)(85 391 237 426)(86 348 238 439)(87 361 239 396)(88 374 240 409)(89 387 241 422)(90 344 242 435)(91 357 243 448)(92 370 244 405)(93 383 245 418)(94 340 246 431)(95 353 247 444)(96 366 248 401)(97 379 249 414)(98 392 250 427)(99 349 251 440)(100 362 252 397)(101 375 253 410)(102 388 254 423)(103 345 255 436)(104 358 256 393)(105 371 257 406)(106 384 258 419)(107 341 259 432)(108 354 260 445)(109 367 261 402)(110 380 262 415)(111 337 263 428)(112 350 264 441)
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), (1,93,168,245)(2,108,113,260)(3,67,114,275)(4,82,115,234)(5,97,116,249)(6,112,117,264)(7,71,118,279)(8,86,119,238)(9,101,120,253)(10,60,121,268)(11,75,122,227)(12,90,123,242)(13,105,124,257)(14,64,125,272)(15,79,126,231)(16,94,127,246)(17,109,128,261)(18,68,129,276)(19,83,130,235)(20,98,131,250)(21,57,132,265)(22,72,133,280)(23,87,134,239)(24,102,135,254)(25,61,136,269)(26,76,137,228)(27,91,138,243)(28,106,139,258)(29,65,140,273)(30,80,141,232)(31,95,142,247)(32,110,143,262)(33,69,144,277)(34,84,145,236)(35,99,146,251)(36,58,147,266)(37,73,148,225)(38,88,149,240)(39,103,150,255)(40,62,151,270)(41,77,152,229)(42,92,153,244)(43,107,154,259)(44,66,155,274)(45,81,156,233)(46,96,157,248)(47,111,158,263)(48,70,159,278)(49,85,160,237)(50,100,161,252)(51,59,162,267)(52,74,163,226)(53,89,164,241)(54,104,165,256)(55,63,166,271)(56,78,167,230)(169,430,282,339)(170,445,283,354)(171,404,284,369)(172,419,285,384)(173,434,286,343)(174,393,287,358)(175,408,288,373)(176,423,289,388)(177,438,290,347)(178,397,291,362)(179,412,292,377)(180,427,293,392)(181,442,294,351)(182,401,295,366)(183,416,296,381)(184,431,297,340)(185,446,298,355)(186,405,299,370)(187,420,300,385)(188,435,301,344)(189,394,302,359)(190,409,303,374)(191,424,304,389)(192,439,305,348)(193,398,306,363)(194,413,307,378)(195,428,308,337)(196,443,309,352)(197,402,310,367)(198,417,311,382)(199,432,312,341)(200,447,313,356)(201,406,314,371)(202,421,315,386)(203,436,316,345)(204,395,317,360)(205,410,318,375)(206,425,319,390)(207,440,320,349)(208,399,321,364)(209,414,322,379)(210,429,323,338)(211,444,324,353)(212,403,325,368)(213,418,326,383)(214,433,327,342)(215,448,328,357)(216,407,329,372)(217,422,330,387)(218,437,331,346)(219,396,332,361)(220,411,333,376)(221,426,334,391)(222,441,335,350)(223,400,336,365)(224,415,281,380), (1,213,168,326)(2,170,113,283)(3,183,114,296)(4,196,115,309)(5,209,116,322)(6,222,117,335)(7,179,118,292)(8,192,119,305)(9,205,120,318)(10,218,121,331)(11,175,122,288)(12,188,123,301)(13,201,124,314)(14,214,125,327)(15,171,126,284)(16,184,127,297)(17,197,128,310)(18,210,129,323)(19,223,130,336)(20,180,131,293)(21,193,132,306)(22,206,133,319)(23,219,134,332)(24,176,135,289)(25,189,136,302)(26,202,137,315)(27,215,138,328)(28,172,139,285)(29,185,140,298)(30,198,141,311)(31,211,142,324)(32,224,143,281)(33,181,144,294)(34,194,145,307)(35,207,146,320)(36,220,147,333)(37,177,148,290)(38,190,149,303)(39,203,150,316)(40,216,151,329)(41,173,152,286)(42,186,153,299)(43,199,154,312)(44,212,155,325)(45,169,156,282)(46,182,157,295)(47,195,158,308)(48,208,159,321)(49,221,160,334)(50,178,161,291)(51,191,162,304)(52,204,163,317)(53,217,164,330)(54,174,165,287)(55,187,166,300)(56,200,167,313)(57,363,265,398)(58,376,266,411)(59,389,267,424)(60,346,268,437)(61,359,269,394)(62,372,270,407)(63,385,271,420)(64,342,272,433)(65,355,273,446)(66,368,274,403)(67,381,275,416)(68,338,276,429)(69,351,277,442)(70,364,278,399)(71,377,279,412)(72,390,280,425)(73,347,225,438)(74,360,226,395)(75,373,227,408)(76,386,228,421)(77,343,229,434)(78,356,230,447)(79,369,231,404)(80,382,232,417)(81,339,233,430)(82,352,234,443)(83,365,235,400)(84,378,236,413)(85,391,237,426)(86,348,238,439)(87,361,239,396)(88,374,240,409)(89,387,241,422)(90,344,242,435)(91,357,243,448)(92,370,244,405)(93,383,245,418)(94,340,246,431)(95,353,247,444)(96,366,248,401)(97,379,249,414)(98,392,250,427)(99,349,251,440)(100,362,252,397)(101,375,253,410)(102,388,254,423)(103,345,255,436)(104,358,256,393)(105,371,257,406)(106,384,258,419)(107,341,259,432)(108,354,260,445)(109,367,261,402)(110,380,262,415)(111,337,263,428)(112,350,264,441)>;
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), (1,93,168,245)(2,108,113,260)(3,67,114,275)(4,82,115,234)(5,97,116,249)(6,112,117,264)(7,71,118,279)(8,86,119,238)(9,101,120,253)(10,60,121,268)(11,75,122,227)(12,90,123,242)(13,105,124,257)(14,64,125,272)(15,79,126,231)(16,94,127,246)(17,109,128,261)(18,68,129,276)(19,83,130,235)(20,98,131,250)(21,57,132,265)(22,72,133,280)(23,87,134,239)(24,102,135,254)(25,61,136,269)(26,76,137,228)(27,91,138,243)(28,106,139,258)(29,65,140,273)(30,80,141,232)(31,95,142,247)(32,110,143,262)(33,69,144,277)(34,84,145,236)(35,99,146,251)(36,58,147,266)(37,73,148,225)(38,88,149,240)(39,103,150,255)(40,62,151,270)(41,77,152,229)(42,92,153,244)(43,107,154,259)(44,66,155,274)(45,81,156,233)(46,96,157,248)(47,111,158,263)(48,70,159,278)(49,85,160,237)(50,100,161,252)(51,59,162,267)(52,74,163,226)(53,89,164,241)(54,104,165,256)(55,63,166,271)(56,78,167,230)(169,430,282,339)(170,445,283,354)(171,404,284,369)(172,419,285,384)(173,434,286,343)(174,393,287,358)(175,408,288,373)(176,423,289,388)(177,438,290,347)(178,397,291,362)(179,412,292,377)(180,427,293,392)(181,442,294,351)(182,401,295,366)(183,416,296,381)(184,431,297,340)(185,446,298,355)(186,405,299,370)(187,420,300,385)(188,435,301,344)(189,394,302,359)(190,409,303,374)(191,424,304,389)(192,439,305,348)(193,398,306,363)(194,413,307,378)(195,428,308,337)(196,443,309,352)(197,402,310,367)(198,417,311,382)(199,432,312,341)(200,447,313,356)(201,406,314,371)(202,421,315,386)(203,436,316,345)(204,395,317,360)(205,410,318,375)(206,425,319,390)(207,440,320,349)(208,399,321,364)(209,414,322,379)(210,429,323,338)(211,444,324,353)(212,403,325,368)(213,418,326,383)(214,433,327,342)(215,448,328,357)(216,407,329,372)(217,422,330,387)(218,437,331,346)(219,396,332,361)(220,411,333,376)(221,426,334,391)(222,441,335,350)(223,400,336,365)(224,415,281,380), (1,213,168,326)(2,170,113,283)(3,183,114,296)(4,196,115,309)(5,209,116,322)(6,222,117,335)(7,179,118,292)(8,192,119,305)(9,205,120,318)(10,218,121,331)(11,175,122,288)(12,188,123,301)(13,201,124,314)(14,214,125,327)(15,171,126,284)(16,184,127,297)(17,197,128,310)(18,210,129,323)(19,223,130,336)(20,180,131,293)(21,193,132,306)(22,206,133,319)(23,219,134,332)(24,176,135,289)(25,189,136,302)(26,202,137,315)(27,215,138,328)(28,172,139,285)(29,185,140,298)(30,198,141,311)(31,211,142,324)(32,224,143,281)(33,181,144,294)(34,194,145,307)(35,207,146,320)(36,220,147,333)(37,177,148,290)(38,190,149,303)(39,203,150,316)(40,216,151,329)(41,173,152,286)(42,186,153,299)(43,199,154,312)(44,212,155,325)(45,169,156,282)(46,182,157,295)(47,195,158,308)(48,208,159,321)(49,221,160,334)(50,178,161,291)(51,191,162,304)(52,204,163,317)(53,217,164,330)(54,174,165,287)(55,187,166,300)(56,200,167,313)(57,363,265,398)(58,376,266,411)(59,389,267,424)(60,346,268,437)(61,359,269,394)(62,372,270,407)(63,385,271,420)(64,342,272,433)(65,355,273,446)(66,368,274,403)(67,381,275,416)(68,338,276,429)(69,351,277,442)(70,364,278,399)(71,377,279,412)(72,390,280,425)(73,347,225,438)(74,360,226,395)(75,373,227,408)(76,386,228,421)(77,343,229,434)(78,356,230,447)(79,369,231,404)(80,382,232,417)(81,339,233,430)(82,352,234,443)(83,365,235,400)(84,378,236,413)(85,391,237,426)(86,348,238,439)(87,361,239,396)(88,374,240,409)(89,387,241,422)(90,344,242,435)(91,357,243,448)(92,370,244,405)(93,383,245,418)(94,340,246,431)(95,353,247,444)(96,366,248,401)(97,379,249,414)(98,392,250,427)(99,349,251,440)(100,362,252,397)(101,375,253,410)(102,388,254,423)(103,345,255,436)(104,358,256,393)(105,371,257,406)(106,384,258,419)(107,341,259,432)(108,354,260,445)(109,367,261,402)(110,380,262,415)(111,337,263,428)(112,350,264,441) );
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)], [(1,93,168,245),(2,108,113,260),(3,67,114,275),(4,82,115,234),(5,97,116,249),(6,112,117,264),(7,71,118,279),(8,86,119,238),(9,101,120,253),(10,60,121,268),(11,75,122,227),(12,90,123,242),(13,105,124,257),(14,64,125,272),(15,79,126,231),(16,94,127,246),(17,109,128,261),(18,68,129,276),(19,83,130,235),(20,98,131,250),(21,57,132,265),(22,72,133,280),(23,87,134,239),(24,102,135,254),(25,61,136,269),(26,76,137,228),(27,91,138,243),(28,106,139,258),(29,65,140,273),(30,80,141,232),(31,95,142,247),(32,110,143,262),(33,69,144,277),(34,84,145,236),(35,99,146,251),(36,58,147,266),(37,73,148,225),(38,88,149,240),(39,103,150,255),(40,62,151,270),(41,77,152,229),(42,92,153,244),(43,107,154,259),(44,66,155,274),(45,81,156,233),(46,96,157,248),(47,111,158,263),(48,70,159,278),(49,85,160,237),(50,100,161,252),(51,59,162,267),(52,74,163,226),(53,89,164,241),(54,104,165,256),(55,63,166,271),(56,78,167,230),(169,430,282,339),(170,445,283,354),(171,404,284,369),(172,419,285,384),(173,434,286,343),(174,393,287,358),(175,408,288,373),(176,423,289,388),(177,438,290,347),(178,397,291,362),(179,412,292,377),(180,427,293,392),(181,442,294,351),(182,401,295,366),(183,416,296,381),(184,431,297,340),(185,446,298,355),(186,405,299,370),(187,420,300,385),(188,435,301,344),(189,394,302,359),(190,409,303,374),(191,424,304,389),(192,439,305,348),(193,398,306,363),(194,413,307,378),(195,428,308,337),(196,443,309,352),(197,402,310,367),(198,417,311,382),(199,432,312,341),(200,447,313,356),(201,406,314,371),(202,421,315,386),(203,436,316,345),(204,395,317,360),(205,410,318,375),(206,425,319,390),(207,440,320,349),(208,399,321,364),(209,414,322,379),(210,429,323,338),(211,444,324,353),(212,403,325,368),(213,418,326,383),(214,433,327,342),(215,448,328,357),(216,407,329,372),(217,422,330,387),(218,437,331,346),(219,396,332,361),(220,411,333,376),(221,426,334,391),(222,441,335,350),(223,400,336,365),(224,415,281,380)], [(1,213,168,326),(2,170,113,283),(3,183,114,296),(4,196,115,309),(5,209,116,322),(6,222,117,335),(7,179,118,292),(8,192,119,305),(9,205,120,318),(10,218,121,331),(11,175,122,288),(12,188,123,301),(13,201,124,314),(14,214,125,327),(15,171,126,284),(16,184,127,297),(17,197,128,310),(18,210,129,323),(19,223,130,336),(20,180,131,293),(21,193,132,306),(22,206,133,319),(23,219,134,332),(24,176,135,289),(25,189,136,302),(26,202,137,315),(27,215,138,328),(28,172,139,285),(29,185,140,298),(30,198,141,311),(31,211,142,324),(32,224,143,281),(33,181,144,294),(34,194,145,307),(35,207,146,320),(36,220,147,333),(37,177,148,290),(38,190,149,303),(39,203,150,316),(40,216,151,329),(41,173,152,286),(42,186,153,299),(43,199,154,312),(44,212,155,325),(45,169,156,282),(46,182,157,295),(47,195,158,308),(48,208,159,321),(49,221,160,334),(50,178,161,291),(51,191,162,304),(52,204,163,317),(53,217,164,330),(54,174,165,287),(55,187,166,300),(56,200,167,313),(57,363,265,398),(58,376,266,411),(59,389,267,424),(60,346,268,437),(61,359,269,394),(62,372,270,407),(63,385,271,420),(64,342,272,433),(65,355,273,446),(66,368,274,403),(67,381,275,416),(68,338,276,429),(69,351,277,442),(70,364,278,399),(71,377,279,412),(72,390,280,425),(73,347,225,438),(74,360,226,395),(75,373,227,408),(76,386,228,421),(77,343,229,434),(78,356,230,447),(79,369,231,404),(80,382,232,417),(81,339,233,430),(82,352,234,443),(83,365,235,400),(84,378,236,413),(85,391,237,426),(86,348,238,439),(87,361,239,396),(88,374,240,409),(89,387,241,422),(90,344,242,435),(91,357,243,448),(92,370,244,405),(93,383,245,418),(94,340,246,431),(95,353,247,444),(96,366,248,401),(97,379,249,414),(98,392,250,427),(99,349,251,440),(100,362,252,397),(101,375,253,410),(102,388,254,423),(103,345,255,436),(104,358,256,393),(105,371,257,406),(106,384,258,419),(107,341,259,432),(108,354,260,445),(109,367,261,402),(110,380,262,415),(111,337,263,428),(112,350,264,441)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | - | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | Q8 | D4 | D7 | D14 | D14 | Dic14 | C8⋊C22 | C8.C22 | Q8×D7 | D4×D7 | D8⋊D7 | Q16⋊D7 |
| kernel | C56⋊4Q8 | C28.Q8 | C4.Dic14 | C56⋊C4 | C8⋊Dic7 | C7×C2.D8 | C28⋊Q8 | C28.3Q8 | C7⋊C8 | C56 | C2×Dic7 | C2.D8 | C4⋊C4 | C2×C8 | C8 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 6 | 3 | 12 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C56⋊4Q8 ►in GL6(𝔽113)
| 40 | 6 | 0 | 0 | 0 | 0 |
| 91 | 73 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 37 | 42 | 37 |
| 0 | 0 | 76 | 36 | 76 | 36 |
| 0 | 0 | 71 | 76 | 42 | 37 |
| 0 | 0 | 37 | 77 | 76 | 36 |
| 95 | 7 | 0 | 0 | 0 | 0 |
| 2 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 81 | 94 | 1 | 95 |
| 0 | 0 | 19 | 23 | 18 | 65 |
| 0 | 0 | 1 | 95 | 32 | 19 |
| 0 | 0 | 18 | 65 | 94 | 90 |
| 73 | 107 | 0 | 0 | 0 | 0 |
| 22 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 90 | 54 | 85 | 89 |
| 0 | 0 | 73 | 23 | 63 | 28 |
| 0 | 0 | 28 | 24 | 90 | 54 |
| 0 | 0 | 50 | 85 | 73 | 23 |
G:=sub<GL(6,GF(113))| [40,91,0,0,0,0,6,73,0,0,0,0,0,0,42,76,71,37,0,0,37,36,76,77,0,0,42,76,42,76,0,0,37,36,37,36],[95,2,0,0,0,0,7,18,0,0,0,0,0,0,81,19,1,18,0,0,94,23,95,65,0,0,1,18,32,94,0,0,95,65,19,90],[73,22,0,0,0,0,107,40,0,0,0,0,0,0,90,73,28,50,0,0,54,23,24,85,0,0,85,63,90,73,0,0,89,28,54,23] >;
C56⋊4Q8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_4Q_8 % in TeX
G:=Group("C56:4Q8"); // GroupNames label
G:=SmallGroup(448,410);
// by ID
G=gap.SmallGroup(448,410);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,120,254,219,58,438,102,18822]);
// 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^13,c*b*c^-1=b^-1>;
// generators/relations