metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊6Q8, C8⋊6Dic14, C42.12D14, Dic7.1M4(2), C7⋊2(C8⋊4Q8), C14.7(C4×Q8), C8⋊C4.5D7, C4⋊Dic7.8C4, C28.81(C2×Q8), C28⋊C8.5C2, (C2×C8).154D14, Dic7⋊C4.4C4, C56⋊C4.11C2, C2.9(C4×Dic14), C2.8(D7×M4(2)), C14.20(C8○D4), Dic7⋊C8.15C2, C2.6(D28.C4), (C4×C28).12C22, (C2×Dic14).7C4, (C4×Dic14).4C2, (C8×Dic7).16C2, C4.46(C2×Dic14), C4.128(C4○D28), C28.244(C4○D4), (C2×C56).224C22, (C2×C28).809C23, C14.14(C2×M4(2)), (C4×Dic7).180C22, (C2×C4).28(C4×D7), (C7×C8⋊C4).4C2, C22.97(C2×C4×D7), (C2×C28).36(C2×C4), (C2×C7⋊C8).293C22, (C2×C14).64(C22×C4), (C2×Dic7).13(C2×C4), (C2×C4).751(C22×D7), SmallGroup(448,235)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊Q8
G = < a,b,c | a56=b4=1, c2=b2, bab-1=a29, cac-1=a41, cbc-1=b-1 >
Subgroups: 324 in 94 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×C8, C8⋊C4, 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, C8×Dic7, Dic7⋊C8, C56⋊C4, C7×C8⋊C4, C4×Dic14, C56⋊Q8
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, C2×Dic14, C2×C4×D7, C4○D28, C4×Dic14, D7×M4(2), D28.C4, C56⋊Q8
►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)
(1 287 243 157)(2 316 244 130)(3 289 245 159)(4 318 246 132)(5 291 247 161)(6 320 248 134)(7 293 249 163)(8 322 250 136)(9 295 251 165)(10 324 252 138)(11 297 253 167)(12 326 254 140)(13 299 255 113)(14 328 256 142)(15 301 257 115)(16 330 258 144)(17 303 259 117)(18 332 260 146)(19 305 261 119)(20 334 262 148)(21 307 263 121)(22 336 264 150)(23 309 265 123)(24 282 266 152)(25 311 267 125)(26 284 268 154)(27 313 269 127)(28 286 270 156)(29 315 271 129)(30 288 272 158)(31 317 273 131)(32 290 274 160)(33 319 275 133)(34 292 276 162)(35 321 277 135)(36 294 278 164)(37 323 279 137)(38 296 280 166)(39 325 225 139)(40 298 226 168)(41 327 227 141)(42 300 228 114)(43 329 229 143)(44 302 230 116)(45 331 231 145)(46 304 232 118)(47 333 233 147)(48 306 234 120)(49 335 235 149)(50 308 236 122)(51 281 237 151)(52 310 238 124)(53 283 239 153)(54 312 240 126)(55 285 241 155)(56 314 242 128)(57 395 207 388)(58 424 208 361)(59 397 209 390)(60 426 210 363)(61 399 211 392)(62 428 212 365)(63 401 213 338)(64 430 214 367)(65 403 215 340)(66 432 216 369)(67 405 217 342)(68 434 218 371)(69 407 219 344)(70 436 220 373)(71 409 221 346)(72 438 222 375)(73 411 223 348)(74 440 224 377)(75 413 169 350)(76 442 170 379)(77 415 171 352)(78 444 172 381)(79 417 173 354)(80 446 174 383)(81 419 175 356)(82 448 176 385)(83 421 177 358)(84 394 178 387)(85 423 179 360)(86 396 180 389)(87 425 181 362)(88 398 182 391)(89 427 183 364)(90 400 184 337)(91 429 185 366)(92 402 186 339)(93 431 187 368)(94 404 188 341)(95 433 189 370)(96 406 190 343)(97 435 191 372)(98 408 192 345)(99 437 193 374)(100 410 194 347)(101 439 195 376)(102 412 196 349)(103 441 197 378)(104 414 198 351)(105 443 199 380)(106 416 200 353)(107 445 201 382)(108 418 202 355)(109 447 203 384)(110 420 204 357)(111 393 205 386)(112 422 206 359)
(1 219 243 69)(2 204 244 110)(3 189 245 95)(4 174 246 80)(5 215 247 65)(6 200 248 106)(7 185 249 91)(8 170 250 76)(9 211 251 61)(10 196 252 102)(11 181 253 87)(12 222 254 72)(13 207 255 57)(14 192 256 98)(15 177 257 83)(16 218 258 68)(17 203 259 109)(18 188 260 94)(19 173 261 79)(20 214 262 64)(21 199 263 105)(22 184 264 90)(23 169 265 75)(24 210 266 60)(25 195 267 101)(26 180 268 86)(27 221 269 71)(28 206 270 112)(29 191 271 97)(30 176 272 82)(31 217 273 67)(32 202 274 108)(33 187 275 93)(34 172 276 78)(35 213 277 63)(36 198 278 104)(37 183 279 89)(38 224 280 74)(39 209 225 59)(40 194 226 100)(41 179 227 85)(42 220 228 70)(43 205 229 111)(44 190 230 96)(45 175 231 81)(46 216 232 66)(47 201 233 107)(48 186 234 92)(49 171 235 77)(50 212 236 62)(51 197 237 103)(52 182 238 88)(53 223 239 73)(54 208 240 58)(55 193 241 99)(56 178 242 84)(113 388 299 395)(114 373 300 436)(115 358 301 421)(116 343 302 406)(117 384 303 447)(118 369 304 432)(119 354 305 417)(120 339 306 402)(121 380 307 443)(122 365 308 428)(123 350 309 413)(124 391 310 398)(125 376 311 439)(126 361 312 424)(127 346 313 409)(128 387 314 394)(129 372 315 435)(130 357 316 420)(131 342 317 405)(132 383 318 446)(133 368 319 431)(134 353 320 416)(135 338 321 401)(136 379 322 442)(137 364 323 427)(138 349 324 412)(139 390 325 397)(140 375 326 438)(141 360 327 423)(142 345 328 408)(143 386 329 393)(144 371 330 434)(145 356 331 419)(146 341 332 404)(147 382 333 445)(148 367 334 430)(149 352 335 415)(150 337 336 400)(151 378 281 441)(152 363 282 426)(153 348 283 411)(154 389 284 396)(155 374 285 437)(156 359 286 422)(157 344 287 407)(158 385 288 448)(159 370 289 433)(160 355 290 418)(161 340 291 403)(162 381 292 444)(163 366 293 429)(164 351 294 414)(165 392 295 399)(166 377 296 440)(167 362 297 425)(168 347 298 410)
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), (1,287,243,157)(2,316,244,130)(3,289,245,159)(4,318,246,132)(5,291,247,161)(6,320,248,134)(7,293,249,163)(8,322,250,136)(9,295,251,165)(10,324,252,138)(11,297,253,167)(12,326,254,140)(13,299,255,113)(14,328,256,142)(15,301,257,115)(16,330,258,144)(17,303,259,117)(18,332,260,146)(19,305,261,119)(20,334,262,148)(21,307,263,121)(22,336,264,150)(23,309,265,123)(24,282,266,152)(25,311,267,125)(26,284,268,154)(27,313,269,127)(28,286,270,156)(29,315,271,129)(30,288,272,158)(31,317,273,131)(32,290,274,160)(33,319,275,133)(34,292,276,162)(35,321,277,135)(36,294,278,164)(37,323,279,137)(38,296,280,166)(39,325,225,139)(40,298,226,168)(41,327,227,141)(42,300,228,114)(43,329,229,143)(44,302,230,116)(45,331,231,145)(46,304,232,118)(47,333,233,147)(48,306,234,120)(49,335,235,149)(50,308,236,122)(51,281,237,151)(52,310,238,124)(53,283,239,153)(54,312,240,126)(55,285,241,155)(56,314,242,128)(57,395,207,388)(58,424,208,361)(59,397,209,390)(60,426,210,363)(61,399,211,392)(62,428,212,365)(63,401,213,338)(64,430,214,367)(65,403,215,340)(66,432,216,369)(67,405,217,342)(68,434,218,371)(69,407,219,344)(70,436,220,373)(71,409,221,346)(72,438,222,375)(73,411,223,348)(74,440,224,377)(75,413,169,350)(76,442,170,379)(77,415,171,352)(78,444,172,381)(79,417,173,354)(80,446,174,383)(81,419,175,356)(82,448,176,385)(83,421,177,358)(84,394,178,387)(85,423,179,360)(86,396,180,389)(87,425,181,362)(88,398,182,391)(89,427,183,364)(90,400,184,337)(91,429,185,366)(92,402,186,339)(93,431,187,368)(94,404,188,341)(95,433,189,370)(96,406,190,343)(97,435,191,372)(98,408,192,345)(99,437,193,374)(100,410,194,347)(101,439,195,376)(102,412,196,349)(103,441,197,378)(104,414,198,351)(105,443,199,380)(106,416,200,353)(107,445,201,382)(108,418,202,355)(109,447,203,384)(110,420,204,357)(111,393,205,386)(112,422,206,359), (1,219,243,69)(2,204,244,110)(3,189,245,95)(4,174,246,80)(5,215,247,65)(6,200,248,106)(7,185,249,91)(8,170,250,76)(9,211,251,61)(10,196,252,102)(11,181,253,87)(12,222,254,72)(13,207,255,57)(14,192,256,98)(15,177,257,83)(16,218,258,68)(17,203,259,109)(18,188,260,94)(19,173,261,79)(20,214,262,64)(21,199,263,105)(22,184,264,90)(23,169,265,75)(24,210,266,60)(25,195,267,101)(26,180,268,86)(27,221,269,71)(28,206,270,112)(29,191,271,97)(30,176,272,82)(31,217,273,67)(32,202,274,108)(33,187,275,93)(34,172,276,78)(35,213,277,63)(36,198,278,104)(37,183,279,89)(38,224,280,74)(39,209,225,59)(40,194,226,100)(41,179,227,85)(42,220,228,70)(43,205,229,111)(44,190,230,96)(45,175,231,81)(46,216,232,66)(47,201,233,107)(48,186,234,92)(49,171,235,77)(50,212,236,62)(51,197,237,103)(52,182,238,88)(53,223,239,73)(54,208,240,58)(55,193,241,99)(56,178,242,84)(113,388,299,395)(114,373,300,436)(115,358,301,421)(116,343,302,406)(117,384,303,447)(118,369,304,432)(119,354,305,417)(120,339,306,402)(121,380,307,443)(122,365,308,428)(123,350,309,413)(124,391,310,398)(125,376,311,439)(126,361,312,424)(127,346,313,409)(128,387,314,394)(129,372,315,435)(130,357,316,420)(131,342,317,405)(132,383,318,446)(133,368,319,431)(134,353,320,416)(135,338,321,401)(136,379,322,442)(137,364,323,427)(138,349,324,412)(139,390,325,397)(140,375,326,438)(141,360,327,423)(142,345,328,408)(143,386,329,393)(144,371,330,434)(145,356,331,419)(146,341,332,404)(147,382,333,445)(148,367,334,430)(149,352,335,415)(150,337,336,400)(151,378,281,441)(152,363,282,426)(153,348,283,411)(154,389,284,396)(155,374,285,437)(156,359,286,422)(157,344,287,407)(158,385,288,448)(159,370,289,433)(160,355,290,418)(161,340,291,403)(162,381,292,444)(163,366,293,429)(164,351,294,414)(165,392,295,399)(166,377,296,440)(167,362,297,425)(168,347,298,410)>;
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), (1,287,243,157)(2,316,244,130)(3,289,245,159)(4,318,246,132)(5,291,247,161)(6,320,248,134)(7,293,249,163)(8,322,250,136)(9,295,251,165)(10,324,252,138)(11,297,253,167)(12,326,254,140)(13,299,255,113)(14,328,256,142)(15,301,257,115)(16,330,258,144)(17,303,259,117)(18,332,260,146)(19,305,261,119)(20,334,262,148)(21,307,263,121)(22,336,264,150)(23,309,265,123)(24,282,266,152)(25,311,267,125)(26,284,268,154)(27,313,269,127)(28,286,270,156)(29,315,271,129)(30,288,272,158)(31,317,273,131)(32,290,274,160)(33,319,275,133)(34,292,276,162)(35,321,277,135)(36,294,278,164)(37,323,279,137)(38,296,280,166)(39,325,225,139)(40,298,226,168)(41,327,227,141)(42,300,228,114)(43,329,229,143)(44,302,230,116)(45,331,231,145)(46,304,232,118)(47,333,233,147)(48,306,234,120)(49,335,235,149)(50,308,236,122)(51,281,237,151)(52,310,238,124)(53,283,239,153)(54,312,240,126)(55,285,241,155)(56,314,242,128)(57,395,207,388)(58,424,208,361)(59,397,209,390)(60,426,210,363)(61,399,211,392)(62,428,212,365)(63,401,213,338)(64,430,214,367)(65,403,215,340)(66,432,216,369)(67,405,217,342)(68,434,218,371)(69,407,219,344)(70,436,220,373)(71,409,221,346)(72,438,222,375)(73,411,223,348)(74,440,224,377)(75,413,169,350)(76,442,170,379)(77,415,171,352)(78,444,172,381)(79,417,173,354)(80,446,174,383)(81,419,175,356)(82,448,176,385)(83,421,177,358)(84,394,178,387)(85,423,179,360)(86,396,180,389)(87,425,181,362)(88,398,182,391)(89,427,183,364)(90,400,184,337)(91,429,185,366)(92,402,186,339)(93,431,187,368)(94,404,188,341)(95,433,189,370)(96,406,190,343)(97,435,191,372)(98,408,192,345)(99,437,193,374)(100,410,194,347)(101,439,195,376)(102,412,196,349)(103,441,197,378)(104,414,198,351)(105,443,199,380)(106,416,200,353)(107,445,201,382)(108,418,202,355)(109,447,203,384)(110,420,204,357)(111,393,205,386)(112,422,206,359), (1,219,243,69)(2,204,244,110)(3,189,245,95)(4,174,246,80)(5,215,247,65)(6,200,248,106)(7,185,249,91)(8,170,250,76)(9,211,251,61)(10,196,252,102)(11,181,253,87)(12,222,254,72)(13,207,255,57)(14,192,256,98)(15,177,257,83)(16,218,258,68)(17,203,259,109)(18,188,260,94)(19,173,261,79)(20,214,262,64)(21,199,263,105)(22,184,264,90)(23,169,265,75)(24,210,266,60)(25,195,267,101)(26,180,268,86)(27,221,269,71)(28,206,270,112)(29,191,271,97)(30,176,272,82)(31,217,273,67)(32,202,274,108)(33,187,275,93)(34,172,276,78)(35,213,277,63)(36,198,278,104)(37,183,279,89)(38,224,280,74)(39,209,225,59)(40,194,226,100)(41,179,227,85)(42,220,228,70)(43,205,229,111)(44,190,230,96)(45,175,231,81)(46,216,232,66)(47,201,233,107)(48,186,234,92)(49,171,235,77)(50,212,236,62)(51,197,237,103)(52,182,238,88)(53,223,239,73)(54,208,240,58)(55,193,241,99)(56,178,242,84)(113,388,299,395)(114,373,300,436)(115,358,301,421)(116,343,302,406)(117,384,303,447)(118,369,304,432)(119,354,305,417)(120,339,306,402)(121,380,307,443)(122,365,308,428)(123,350,309,413)(124,391,310,398)(125,376,311,439)(126,361,312,424)(127,346,313,409)(128,387,314,394)(129,372,315,435)(130,357,316,420)(131,342,317,405)(132,383,318,446)(133,368,319,431)(134,353,320,416)(135,338,321,401)(136,379,322,442)(137,364,323,427)(138,349,324,412)(139,390,325,397)(140,375,326,438)(141,360,327,423)(142,345,328,408)(143,386,329,393)(144,371,330,434)(145,356,331,419)(146,341,332,404)(147,382,333,445)(148,367,334,430)(149,352,335,415)(150,337,336,400)(151,378,281,441)(152,363,282,426)(153,348,283,411)(154,389,284,396)(155,374,285,437)(156,359,286,422)(157,344,287,407)(158,385,288,448)(159,370,289,433)(160,355,290,418)(161,340,291,403)(162,381,292,444)(163,366,293,429)(164,351,294,414)(165,392,295,399)(166,377,296,440)(167,362,297,425)(168,347,298,410) );
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)], [(1,287,243,157),(2,316,244,130),(3,289,245,159),(4,318,246,132),(5,291,247,161),(6,320,248,134),(7,293,249,163),(8,322,250,136),(9,295,251,165),(10,324,252,138),(11,297,253,167),(12,326,254,140),(13,299,255,113),(14,328,256,142),(15,301,257,115),(16,330,258,144),(17,303,259,117),(18,332,260,146),(19,305,261,119),(20,334,262,148),(21,307,263,121),(22,336,264,150),(23,309,265,123),(24,282,266,152),(25,311,267,125),(26,284,268,154),(27,313,269,127),(28,286,270,156),(29,315,271,129),(30,288,272,158),(31,317,273,131),(32,290,274,160),(33,319,275,133),(34,292,276,162),(35,321,277,135),(36,294,278,164),(37,323,279,137),(38,296,280,166),(39,325,225,139),(40,298,226,168),(41,327,227,141),(42,300,228,114),(43,329,229,143),(44,302,230,116),(45,331,231,145),(46,304,232,118),(47,333,233,147),(48,306,234,120),(49,335,235,149),(50,308,236,122),(51,281,237,151),(52,310,238,124),(53,283,239,153),(54,312,240,126),(55,285,241,155),(56,314,242,128),(57,395,207,388),(58,424,208,361),(59,397,209,390),(60,426,210,363),(61,399,211,392),(62,428,212,365),(63,401,213,338),(64,430,214,367),(65,403,215,340),(66,432,216,369),(67,405,217,342),(68,434,218,371),(69,407,219,344),(70,436,220,373),(71,409,221,346),(72,438,222,375),(73,411,223,348),(74,440,224,377),(75,413,169,350),(76,442,170,379),(77,415,171,352),(78,444,172,381),(79,417,173,354),(80,446,174,383),(81,419,175,356),(82,448,176,385),(83,421,177,358),(84,394,178,387),(85,423,179,360),(86,396,180,389),(87,425,181,362),(88,398,182,391),(89,427,183,364),(90,400,184,337),(91,429,185,366),(92,402,186,339),(93,431,187,368),(94,404,188,341),(95,433,189,370),(96,406,190,343),(97,435,191,372),(98,408,192,345),(99,437,193,374),(100,410,194,347),(101,439,195,376),(102,412,196,349),(103,441,197,378),(104,414,198,351),(105,443,199,380),(106,416,200,353),(107,445,201,382),(108,418,202,355),(109,447,203,384),(110,420,204,357),(111,393,205,386),(112,422,206,359)], [(1,219,243,69),(2,204,244,110),(3,189,245,95),(4,174,246,80),(5,215,247,65),(6,200,248,106),(7,185,249,91),(8,170,250,76),(9,211,251,61),(10,196,252,102),(11,181,253,87),(12,222,254,72),(13,207,255,57),(14,192,256,98),(15,177,257,83),(16,218,258,68),(17,203,259,109),(18,188,260,94),(19,173,261,79),(20,214,262,64),(21,199,263,105),(22,184,264,90),(23,169,265,75),(24,210,266,60),(25,195,267,101),(26,180,268,86),(27,221,269,71),(28,206,270,112),(29,191,271,97),(30,176,272,82),(31,217,273,67),(32,202,274,108),(33,187,275,93),(34,172,276,78),(35,213,277,63),(36,198,278,104),(37,183,279,89),(38,224,280,74),(39,209,225,59),(40,194,226,100),(41,179,227,85),(42,220,228,70),(43,205,229,111),(44,190,230,96),(45,175,231,81),(46,216,232,66),(47,201,233,107),(48,186,234,92),(49,171,235,77),(50,212,236,62),(51,197,237,103),(52,182,238,88),(53,223,239,73),(54,208,240,58),(55,193,241,99),(56,178,242,84),(113,388,299,395),(114,373,300,436),(115,358,301,421),(116,343,302,406),(117,384,303,447),(118,369,304,432),(119,354,305,417),(120,339,306,402),(121,380,307,443),(122,365,308,428),(123,350,309,413),(124,391,310,398),(125,376,311,439),(126,361,312,424),(127,346,313,409),(128,387,314,394),(129,372,315,435),(130,357,316,420),(131,342,317,405),(132,383,318,446),(133,368,319,431),(134,353,320,416),(135,338,321,401),(136,379,322,442),(137,364,323,427),(138,349,324,412),(139,390,325,397),(140,375,326,438),(141,360,327,423),(142,345,328,408),(143,386,329,393),(144,371,330,434),(145,356,331,419),(146,341,332,404),(147,382,333,445),(148,367,334,430),(149,352,335,415),(150,337,336,400),(151,378,281,441),(152,363,282,426),(153,348,283,411),(154,389,284,396),(155,374,285,437),(156,359,286,422),(157,344,287,407),(158,385,288,448),(159,370,289,433),(160,355,290,418),(161,340,291,403),(162,381,292,444),(163,366,293,429),(164,351,294,414),(165,392,295,399),(166,377,296,440),(167,362,297,425),(168,347,298,410)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| 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 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | D7 | M4(2) | C4○D4 | D14 | D14 | C8○D4 | Dic14 | C4×D7 | C4○D28 | D7×M4(2) | D28.C4 |
| kernel | C56⋊Q8 | C28⋊C8 | C8×Dic7 | Dic7⋊C8 | C56⋊C4 | C7×C8⋊C4 | C4×Dic14 | Dic7⋊C4 | C4⋊Dic7 | C2×Dic14 | C56 | C8⋊C4 | Dic7 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 4 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of C56⋊Q8 ►in GL4(𝔽113) generated by
| 92 | 15 | 0 | 0 |
| 82 | 76 | 0 | 0 |
| 0 | 0 | 95 | 0 |
| 0 | 0 | 109 | 18 |
| 17 | 46 | 0 | 0 |
| 33 | 96 | 0 | 0 |
| 0 | 0 | 88 | 112 |
| 0 | 0 | 59 | 25 |
| 26 | 44 | 0 | 0 |
| 18 | 87 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [92,82,0,0,15,76,0,0,0,0,95,109,0,0,0,18],[17,33,0,0,46,96,0,0,0,0,88,59,0,0,112,25],[26,18,0,0,44,87,0,0,0,0,112,0,0,0,0,112] >;
C56⋊Q8 in GAP, Magma, Sage, TeX
C_{56}\rtimes Q_8 % in TeX
G:=Group("C56:Q8"); // GroupNames label
G:=SmallGroup(448,235);
// by ID
G=gap.SmallGroup(448,235);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,387,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=b^2,b*a*b^-1=a^29,c*a*c^-1=a^41,c*b*c^-1=b^-1>;
// generators/relations