direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C56, (C4×C56).6C2, C4.4(C2×C56), (C4×C8).3C14, C4⋊C4.12C28, C4⋊C8.11C14, C2.2(Q8×C28), C28.33(C2×C8), (C2×Q8).9C28, C4.24(Q8×C14), C14.33(C4×Q8), C2.5(C22×C56), (Q8×C14).19C4, (C4×Q8).11C14, (Q8×C28).24C2, C28.130(C2×Q8), C14.51(C8○D4), C42.74(C2×C14), C14.34(C22×C8), C28.356(C4○D4), (C2×C28).993C23, (C4×C28).359C22, (C2×C56).362C22, C22.23(C22×C28), C2.3(C7×C8○D4), (C7×C4⋊C8).24C2, (C7×C4⋊C4).24C4, C4.54(C7×C4○D4), (C2×C4).37(C2×C28), (C2×C8).108(C2×C14), (C2×C28).214(C2×C4), (C2×C14).243(C22×C4), (C2×C4).161(C22×C14), SmallGroup(448,853)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C56
G = < a,b,c | a56=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 114 in 102 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, C28, C28, C28, C2×C14, C4×C8, C4⋊C8, C4×Q8, C56, C56, C2×C28, C2×C28, C7×Q8, C8×Q8, C4×C28, C7×C4⋊C4, C2×C56, C2×C56, Q8×C14, C4×C56, C7×C4⋊C8, Q8×C28, Q8×C56
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, Q8, C23, C14, C2×C8, C22×C4, C2×Q8, C4○D4, C28, C2×C14, C4×Q8, C22×C8, C8○D4, C56, C2×C28, C7×Q8, C22×C14, C8×Q8, C2×C56, C22×C28, Q8×C14, C7×C4○D4, Q8×C28, C22×C56, C7×C8○D4, Q8×C56
►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)
(1 380 93 221)(2 381 94 222)(3 382 95 223)(4 383 96 224)(5 384 97 169)(6 385 98 170)(7 386 99 171)(8 387 100 172)(9 388 101 173)(10 389 102 174)(11 390 103 175)(12 391 104 176)(13 392 105 177)(14 337 106 178)(15 338 107 179)(16 339 108 180)(17 340 109 181)(18 341 110 182)(19 342 111 183)(20 343 112 184)(21 344 57 185)(22 345 58 186)(23 346 59 187)(24 347 60 188)(25 348 61 189)(26 349 62 190)(27 350 63 191)(28 351 64 192)(29 352 65 193)(30 353 66 194)(31 354 67 195)(32 355 68 196)(33 356 69 197)(34 357 70 198)(35 358 71 199)(36 359 72 200)(37 360 73 201)(38 361 74 202)(39 362 75 203)(40 363 76 204)(41 364 77 205)(42 365 78 206)(43 366 79 207)(44 367 80 208)(45 368 81 209)(46 369 82 210)(47 370 83 211)(48 371 84 212)(49 372 85 213)(50 373 86 214)(51 374 87 215)(52 375 88 216)(53 376 89 217)(54 377 90 218)(55 378 91 219)(56 379 92 220)(113 272 301 393)(114 273 302 394)(115 274 303 395)(116 275 304 396)(117 276 305 397)(118 277 306 398)(119 278 307 399)(120 279 308 400)(121 280 309 401)(122 225 310 402)(123 226 311 403)(124 227 312 404)(125 228 313 405)(126 229 314 406)(127 230 315 407)(128 231 316 408)(129 232 317 409)(130 233 318 410)(131 234 319 411)(132 235 320 412)(133 236 321 413)(134 237 322 414)(135 238 323 415)(136 239 324 416)(137 240 325 417)(138 241 326 418)(139 242 327 419)(140 243 328 420)(141 244 329 421)(142 245 330 422)(143 246 331 423)(144 247 332 424)(145 248 333 425)(146 249 334 426)(147 250 335 427)(148 251 336 428)(149 252 281 429)(150 253 282 430)(151 254 283 431)(152 255 284 432)(153 256 285 433)(154 257 286 434)(155 258 287 435)(156 259 288 436)(157 260 289 437)(158 261 290 438)(159 262 291 439)(160 263 292 440)(161 264 293 441)(162 265 294 442)(163 266 295 443)(164 267 296 444)(165 268 297 445)(166 269 298 446)(167 270 299 447)(168 271 300 448)
(1 117 93 305)(2 118 94 306)(3 119 95 307)(4 120 96 308)(5 121 97 309)(6 122 98 310)(7 123 99 311)(8 124 100 312)(9 125 101 313)(10 126 102 314)(11 127 103 315)(12 128 104 316)(13 129 105 317)(14 130 106 318)(15 131 107 319)(16 132 108 320)(17 133 109 321)(18 134 110 322)(19 135 111 323)(20 136 112 324)(21 137 57 325)(22 138 58 326)(23 139 59 327)(24 140 60 328)(25 141 61 329)(26 142 62 330)(27 143 63 331)(28 144 64 332)(29 145 65 333)(30 146 66 334)(31 147 67 335)(32 148 68 336)(33 149 69 281)(34 150 70 282)(35 151 71 283)(36 152 72 284)(37 153 73 285)(38 154 74 286)(39 155 75 287)(40 156 76 288)(41 157 77 289)(42 158 78 290)(43 159 79 291)(44 160 80 292)(45 161 81 293)(46 162 82 294)(47 163 83 295)(48 164 84 296)(49 165 85 297)(50 166 86 298)(51 167 87 299)(52 168 88 300)(53 113 89 301)(54 114 90 302)(55 115 91 303)(56 116 92 304)(169 280 384 401)(170 225 385 402)(171 226 386 403)(172 227 387 404)(173 228 388 405)(174 229 389 406)(175 230 390 407)(176 231 391 408)(177 232 392 409)(178 233 337 410)(179 234 338 411)(180 235 339 412)(181 236 340 413)(182 237 341 414)(183 238 342 415)(184 239 343 416)(185 240 344 417)(186 241 345 418)(187 242 346 419)(188 243 347 420)(189 244 348 421)(190 245 349 422)(191 246 350 423)(192 247 351 424)(193 248 352 425)(194 249 353 426)(195 250 354 427)(196 251 355 428)(197 252 356 429)(198 253 357 430)(199 254 358 431)(200 255 359 432)(201 256 360 433)(202 257 361 434)(203 258 362 435)(204 259 363 436)(205 260 364 437)(206 261 365 438)(207 262 366 439)(208 263 367 440)(209 264 368 441)(210 265 369 442)(211 266 370 443)(212 267 371 444)(213 268 372 445)(214 269 373 446)(215 270 374 447)(216 271 375 448)(217 272 376 393)(218 273 377 394)(219 274 378 395)(220 275 379 396)(221 276 380 397)(222 277 381 398)(223 278 382 399)(224 279 383 400)
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), (1,380,93,221)(2,381,94,222)(3,382,95,223)(4,383,96,224)(5,384,97,169)(6,385,98,170)(7,386,99,171)(8,387,100,172)(9,388,101,173)(10,389,102,174)(11,390,103,175)(12,391,104,176)(13,392,105,177)(14,337,106,178)(15,338,107,179)(16,339,108,180)(17,340,109,181)(18,341,110,182)(19,342,111,183)(20,343,112,184)(21,344,57,185)(22,345,58,186)(23,346,59,187)(24,347,60,188)(25,348,61,189)(26,349,62,190)(27,350,63,191)(28,351,64,192)(29,352,65,193)(30,353,66,194)(31,354,67,195)(32,355,68,196)(33,356,69,197)(34,357,70,198)(35,358,71,199)(36,359,72,200)(37,360,73,201)(38,361,74,202)(39,362,75,203)(40,363,76,204)(41,364,77,205)(42,365,78,206)(43,366,79,207)(44,367,80,208)(45,368,81,209)(46,369,82,210)(47,370,83,211)(48,371,84,212)(49,372,85,213)(50,373,86,214)(51,374,87,215)(52,375,88,216)(53,376,89,217)(54,377,90,218)(55,378,91,219)(56,379,92,220)(113,272,301,393)(114,273,302,394)(115,274,303,395)(116,275,304,396)(117,276,305,397)(118,277,306,398)(119,278,307,399)(120,279,308,400)(121,280,309,401)(122,225,310,402)(123,226,311,403)(124,227,312,404)(125,228,313,405)(126,229,314,406)(127,230,315,407)(128,231,316,408)(129,232,317,409)(130,233,318,410)(131,234,319,411)(132,235,320,412)(133,236,321,413)(134,237,322,414)(135,238,323,415)(136,239,324,416)(137,240,325,417)(138,241,326,418)(139,242,327,419)(140,243,328,420)(141,244,329,421)(142,245,330,422)(143,246,331,423)(144,247,332,424)(145,248,333,425)(146,249,334,426)(147,250,335,427)(148,251,336,428)(149,252,281,429)(150,253,282,430)(151,254,283,431)(152,255,284,432)(153,256,285,433)(154,257,286,434)(155,258,287,435)(156,259,288,436)(157,260,289,437)(158,261,290,438)(159,262,291,439)(160,263,292,440)(161,264,293,441)(162,265,294,442)(163,266,295,443)(164,267,296,444)(165,268,297,445)(166,269,298,446)(167,270,299,447)(168,271,300,448), (1,117,93,305)(2,118,94,306)(3,119,95,307)(4,120,96,308)(5,121,97,309)(6,122,98,310)(7,123,99,311)(8,124,100,312)(9,125,101,313)(10,126,102,314)(11,127,103,315)(12,128,104,316)(13,129,105,317)(14,130,106,318)(15,131,107,319)(16,132,108,320)(17,133,109,321)(18,134,110,322)(19,135,111,323)(20,136,112,324)(21,137,57,325)(22,138,58,326)(23,139,59,327)(24,140,60,328)(25,141,61,329)(26,142,62,330)(27,143,63,331)(28,144,64,332)(29,145,65,333)(30,146,66,334)(31,147,67,335)(32,148,68,336)(33,149,69,281)(34,150,70,282)(35,151,71,283)(36,152,72,284)(37,153,73,285)(38,154,74,286)(39,155,75,287)(40,156,76,288)(41,157,77,289)(42,158,78,290)(43,159,79,291)(44,160,80,292)(45,161,81,293)(46,162,82,294)(47,163,83,295)(48,164,84,296)(49,165,85,297)(50,166,86,298)(51,167,87,299)(52,168,88,300)(53,113,89,301)(54,114,90,302)(55,115,91,303)(56,116,92,304)(169,280,384,401)(170,225,385,402)(171,226,386,403)(172,227,387,404)(173,228,388,405)(174,229,389,406)(175,230,390,407)(176,231,391,408)(177,232,392,409)(178,233,337,410)(179,234,338,411)(180,235,339,412)(181,236,340,413)(182,237,341,414)(183,238,342,415)(184,239,343,416)(185,240,344,417)(186,241,345,418)(187,242,346,419)(188,243,347,420)(189,244,348,421)(190,245,349,422)(191,246,350,423)(192,247,351,424)(193,248,352,425)(194,249,353,426)(195,250,354,427)(196,251,355,428)(197,252,356,429)(198,253,357,430)(199,254,358,431)(200,255,359,432)(201,256,360,433)(202,257,361,434)(203,258,362,435)(204,259,363,436)(205,260,364,437)(206,261,365,438)(207,262,366,439)(208,263,367,440)(209,264,368,441)(210,265,369,442)(211,266,370,443)(212,267,371,444)(213,268,372,445)(214,269,373,446)(215,270,374,447)(216,271,375,448)(217,272,376,393)(218,273,377,394)(219,274,378,395)(220,275,379,396)(221,276,380,397)(222,277,381,398)(223,278,382,399)(224,279,383,400)>;
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), (1,380,93,221)(2,381,94,222)(3,382,95,223)(4,383,96,224)(5,384,97,169)(6,385,98,170)(7,386,99,171)(8,387,100,172)(9,388,101,173)(10,389,102,174)(11,390,103,175)(12,391,104,176)(13,392,105,177)(14,337,106,178)(15,338,107,179)(16,339,108,180)(17,340,109,181)(18,341,110,182)(19,342,111,183)(20,343,112,184)(21,344,57,185)(22,345,58,186)(23,346,59,187)(24,347,60,188)(25,348,61,189)(26,349,62,190)(27,350,63,191)(28,351,64,192)(29,352,65,193)(30,353,66,194)(31,354,67,195)(32,355,68,196)(33,356,69,197)(34,357,70,198)(35,358,71,199)(36,359,72,200)(37,360,73,201)(38,361,74,202)(39,362,75,203)(40,363,76,204)(41,364,77,205)(42,365,78,206)(43,366,79,207)(44,367,80,208)(45,368,81,209)(46,369,82,210)(47,370,83,211)(48,371,84,212)(49,372,85,213)(50,373,86,214)(51,374,87,215)(52,375,88,216)(53,376,89,217)(54,377,90,218)(55,378,91,219)(56,379,92,220)(113,272,301,393)(114,273,302,394)(115,274,303,395)(116,275,304,396)(117,276,305,397)(118,277,306,398)(119,278,307,399)(120,279,308,400)(121,280,309,401)(122,225,310,402)(123,226,311,403)(124,227,312,404)(125,228,313,405)(126,229,314,406)(127,230,315,407)(128,231,316,408)(129,232,317,409)(130,233,318,410)(131,234,319,411)(132,235,320,412)(133,236,321,413)(134,237,322,414)(135,238,323,415)(136,239,324,416)(137,240,325,417)(138,241,326,418)(139,242,327,419)(140,243,328,420)(141,244,329,421)(142,245,330,422)(143,246,331,423)(144,247,332,424)(145,248,333,425)(146,249,334,426)(147,250,335,427)(148,251,336,428)(149,252,281,429)(150,253,282,430)(151,254,283,431)(152,255,284,432)(153,256,285,433)(154,257,286,434)(155,258,287,435)(156,259,288,436)(157,260,289,437)(158,261,290,438)(159,262,291,439)(160,263,292,440)(161,264,293,441)(162,265,294,442)(163,266,295,443)(164,267,296,444)(165,268,297,445)(166,269,298,446)(167,270,299,447)(168,271,300,448), (1,117,93,305)(2,118,94,306)(3,119,95,307)(4,120,96,308)(5,121,97,309)(6,122,98,310)(7,123,99,311)(8,124,100,312)(9,125,101,313)(10,126,102,314)(11,127,103,315)(12,128,104,316)(13,129,105,317)(14,130,106,318)(15,131,107,319)(16,132,108,320)(17,133,109,321)(18,134,110,322)(19,135,111,323)(20,136,112,324)(21,137,57,325)(22,138,58,326)(23,139,59,327)(24,140,60,328)(25,141,61,329)(26,142,62,330)(27,143,63,331)(28,144,64,332)(29,145,65,333)(30,146,66,334)(31,147,67,335)(32,148,68,336)(33,149,69,281)(34,150,70,282)(35,151,71,283)(36,152,72,284)(37,153,73,285)(38,154,74,286)(39,155,75,287)(40,156,76,288)(41,157,77,289)(42,158,78,290)(43,159,79,291)(44,160,80,292)(45,161,81,293)(46,162,82,294)(47,163,83,295)(48,164,84,296)(49,165,85,297)(50,166,86,298)(51,167,87,299)(52,168,88,300)(53,113,89,301)(54,114,90,302)(55,115,91,303)(56,116,92,304)(169,280,384,401)(170,225,385,402)(171,226,386,403)(172,227,387,404)(173,228,388,405)(174,229,389,406)(175,230,390,407)(176,231,391,408)(177,232,392,409)(178,233,337,410)(179,234,338,411)(180,235,339,412)(181,236,340,413)(182,237,341,414)(183,238,342,415)(184,239,343,416)(185,240,344,417)(186,241,345,418)(187,242,346,419)(188,243,347,420)(189,244,348,421)(190,245,349,422)(191,246,350,423)(192,247,351,424)(193,248,352,425)(194,249,353,426)(195,250,354,427)(196,251,355,428)(197,252,356,429)(198,253,357,430)(199,254,358,431)(200,255,359,432)(201,256,360,433)(202,257,361,434)(203,258,362,435)(204,259,363,436)(205,260,364,437)(206,261,365,438)(207,262,366,439)(208,263,367,440)(209,264,368,441)(210,265,369,442)(211,266,370,443)(212,267,371,444)(213,268,372,445)(214,269,373,446)(215,270,374,447)(216,271,375,448)(217,272,376,393)(218,273,377,394)(219,274,378,395)(220,275,379,396)(221,276,380,397)(222,277,381,398)(223,278,382,399)(224,279,383,400) );
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)], [(1,380,93,221),(2,381,94,222),(3,382,95,223),(4,383,96,224),(5,384,97,169),(6,385,98,170),(7,386,99,171),(8,387,100,172),(9,388,101,173),(10,389,102,174),(11,390,103,175),(12,391,104,176),(13,392,105,177),(14,337,106,178),(15,338,107,179),(16,339,108,180),(17,340,109,181),(18,341,110,182),(19,342,111,183),(20,343,112,184),(21,344,57,185),(22,345,58,186),(23,346,59,187),(24,347,60,188),(25,348,61,189),(26,349,62,190),(27,350,63,191),(28,351,64,192),(29,352,65,193),(30,353,66,194),(31,354,67,195),(32,355,68,196),(33,356,69,197),(34,357,70,198),(35,358,71,199),(36,359,72,200),(37,360,73,201),(38,361,74,202),(39,362,75,203),(40,363,76,204),(41,364,77,205),(42,365,78,206),(43,366,79,207),(44,367,80,208),(45,368,81,209),(46,369,82,210),(47,370,83,211),(48,371,84,212),(49,372,85,213),(50,373,86,214),(51,374,87,215),(52,375,88,216),(53,376,89,217),(54,377,90,218),(55,378,91,219),(56,379,92,220),(113,272,301,393),(114,273,302,394),(115,274,303,395),(116,275,304,396),(117,276,305,397),(118,277,306,398),(119,278,307,399),(120,279,308,400),(121,280,309,401),(122,225,310,402),(123,226,311,403),(124,227,312,404),(125,228,313,405),(126,229,314,406),(127,230,315,407),(128,231,316,408),(129,232,317,409),(130,233,318,410),(131,234,319,411),(132,235,320,412),(133,236,321,413),(134,237,322,414),(135,238,323,415),(136,239,324,416),(137,240,325,417),(138,241,326,418),(139,242,327,419),(140,243,328,420),(141,244,329,421),(142,245,330,422),(143,246,331,423),(144,247,332,424),(145,248,333,425),(146,249,334,426),(147,250,335,427),(148,251,336,428),(149,252,281,429),(150,253,282,430),(151,254,283,431),(152,255,284,432),(153,256,285,433),(154,257,286,434),(155,258,287,435),(156,259,288,436),(157,260,289,437),(158,261,290,438),(159,262,291,439),(160,263,292,440),(161,264,293,441),(162,265,294,442),(163,266,295,443),(164,267,296,444),(165,268,297,445),(166,269,298,446),(167,270,299,447),(168,271,300,448)], [(1,117,93,305),(2,118,94,306),(3,119,95,307),(4,120,96,308),(5,121,97,309),(6,122,98,310),(7,123,99,311),(8,124,100,312),(9,125,101,313),(10,126,102,314),(11,127,103,315),(12,128,104,316),(13,129,105,317),(14,130,106,318),(15,131,107,319),(16,132,108,320),(17,133,109,321),(18,134,110,322),(19,135,111,323),(20,136,112,324),(21,137,57,325),(22,138,58,326),(23,139,59,327),(24,140,60,328),(25,141,61,329),(26,142,62,330),(27,143,63,331),(28,144,64,332),(29,145,65,333),(30,146,66,334),(31,147,67,335),(32,148,68,336),(33,149,69,281),(34,150,70,282),(35,151,71,283),(36,152,72,284),(37,153,73,285),(38,154,74,286),(39,155,75,287),(40,156,76,288),(41,157,77,289),(42,158,78,290),(43,159,79,291),(44,160,80,292),(45,161,81,293),(46,162,82,294),(47,163,83,295),(48,164,84,296),(49,165,85,297),(50,166,86,298),(51,167,87,299),(52,168,88,300),(53,113,89,301),(54,114,90,302),(55,115,91,303),(56,116,92,304),(169,280,384,401),(170,225,385,402),(171,226,386,403),(172,227,387,404),(173,228,388,405),(174,229,389,406),(175,230,390,407),(176,231,391,408),(177,232,392,409),(178,233,337,410),(179,234,338,411),(180,235,339,412),(181,236,340,413),(182,237,341,414),(183,238,342,415),(184,239,343,416),(185,240,344,417),(186,241,345,418),(187,242,346,419),(188,243,347,420),(189,244,348,421),(190,245,349,422),(191,246,350,423),(192,247,351,424),(193,248,352,425),(194,249,353,426),(195,250,354,427),(196,251,355,428),(197,252,356,429),(198,253,357,430),(199,254,358,431),(200,255,359,432),(201,256,360,433),(202,257,361,434),(203,258,362,435),(204,259,363,436),(205,260,364,437),(206,261,365,438),(207,262,366,439),(208,263,367,440),(209,264,368,441),(210,265,369,442),(211,266,370,443),(212,267,371,444),(213,268,372,445),(214,269,373,446),(215,270,374,447),(216,271,375,448),(217,272,376,393),(218,273,377,394),(219,274,378,395),(220,275,379,396),(221,276,380,397),(222,277,381,398),(223,278,382,399),(224,279,383,400)]])
280 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 7A | ··· | 7F | 8A | ··· | 8H | 8I | ··· | 8T | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28CR | 56A | ··· | 56AV | 56AW | ··· | 56DP |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C14 | C28 | C28 | C56 | Q8 | C4○D4 | C8○D4 | C7×Q8 | C7×C4○D4 | C7×C8○D4 |
| kernel | Q8×C56 | C4×C56 | C7×C4⋊C8 | Q8×C28 | C7×C4⋊C4 | Q8×C14 | C8×Q8 | C7×Q8 | C4×C8 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | Q8 | C56 | C28 | C14 | C8 | C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 6 | 16 | 18 | 18 | 6 | 36 | 12 | 96 | 2 | 2 | 4 | 12 | 12 | 24 |
Matrix representation of Q8×C56 ►in GL3(𝔽113) generated by
| 95 | 0 | 0 |
| 0 | 83 | 0 |
| 0 | 0 | 83 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 112 | 0 |
| 1 | 0 | 0 |
| 0 | 81 | 52 |
| 0 | 52 | 32 |
G:=sub<GL(3,GF(113))| [95,0,0,0,83,0,0,0,83],[1,0,0,0,0,112,0,1,0],[1,0,0,0,81,52,0,52,32] >;
Q8×C56 in GAP, Magma, Sage, TeX
Q_8\times C_{56} % in TeX
G:=Group("Q8xC56"); // GroupNames label
G:=SmallGroup(448,853);
// by ID
G=gap.SmallGroup(448,853);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,898,124]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations