metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.13Q8, C8.12Dic14, C42.255D14, (C4×C8).10D7, (C4×C56).12C2, (C2×C4).59D28, C14.4(C4⋊Q8), C28.71(C2×Q8), C8⋊Dic7.5C2, C56⋊1C4.5C2, C7⋊1(C8.5Q8), C14.1(C4○D8), (C2×C8).316D14, (C2×C28).349D4, C2.8(C28⋊2Q8), C4.37(C2×Dic14), C22.88(C2×D28), C4⋊Dic7.4C22, C28.6Q8.1C2, C2.6(D56⋊7C2), (C2×C56).388C22, (C4×C28).305C22, (C2×C28).719C23, (C2×C14).102(C2×D4), (C2×C4).662(C22×D7), SmallGroup(448,217)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.13Q8
G = < a,b,c | a56=b4=1, c2=a28b2, ab=ba, cac-1=a-1, cbc-1=a28b-1 >
Subgroups: 388 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, C14, C14, C42, C4⋊C4, C2×C8, Dic7, C28, C28, C2×C14, C4×C8, C4.Q8, C2.D8, C42.C2, C56, C2×Dic7, C2×C28, C2×C28, C8.5Q8, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C8⋊Dic7, C56⋊1C4, C4×C56, C28.6Q8, C56.13Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, C4○D8, Dic14, D28, C22×D7, C8.5Q8, C2×Dic14, C2×D28, C28⋊2Q8, D56⋊7C2, C56.13Q8
►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)
(1 300 345 430)(2 301 346 431)(3 302 347 432)(4 303 348 433)(5 304 349 434)(6 305 350 435)(7 306 351 436)(8 307 352 437)(9 308 353 438)(10 309 354 439)(11 310 355 440)(12 311 356 441)(13 312 357 442)(14 313 358 443)(15 314 359 444)(16 315 360 445)(17 316 361 446)(18 317 362 447)(19 318 363 448)(20 319 364 393)(21 320 365 394)(22 321 366 395)(23 322 367 396)(24 323 368 397)(25 324 369 398)(26 325 370 399)(27 326 371 400)(28 327 372 401)(29 328 373 402)(30 329 374 403)(31 330 375 404)(32 331 376 405)(33 332 377 406)(34 333 378 407)(35 334 379 408)(36 335 380 409)(37 336 381 410)(38 281 382 411)(39 282 383 412)(40 283 384 413)(41 284 385 414)(42 285 386 415)(43 286 387 416)(44 287 388 417)(45 288 389 418)(46 289 390 419)(47 290 391 420)(48 291 392 421)(49 292 337 422)(50 293 338 423)(51 294 339 424)(52 295 340 425)(53 296 341 426)(54 297 342 427)(55 298 343 428)(56 299 344 429)(57 115 257 180)(58 116 258 181)(59 117 259 182)(60 118 260 183)(61 119 261 184)(62 120 262 185)(63 121 263 186)(64 122 264 187)(65 123 265 188)(66 124 266 189)(67 125 267 190)(68 126 268 191)(69 127 269 192)(70 128 270 193)(71 129 271 194)(72 130 272 195)(73 131 273 196)(74 132 274 197)(75 133 275 198)(76 134 276 199)(77 135 277 200)(78 136 278 201)(79 137 279 202)(80 138 280 203)(81 139 225 204)(82 140 226 205)(83 141 227 206)(84 142 228 207)(85 143 229 208)(86 144 230 209)(87 145 231 210)(88 146 232 211)(89 147 233 212)(90 148 234 213)(91 149 235 214)(92 150 236 215)(93 151 237 216)(94 152 238 217)(95 153 239 218)(96 154 240 219)(97 155 241 220)(98 156 242 221)(99 157 243 222)(100 158 244 223)(101 159 245 224)(102 160 246 169)(103 161 247 170)(104 162 248 171)(105 163 249 172)(106 164 250 173)(107 165 251 174)(108 166 252 175)(109 167 253 176)(110 168 254 177)(111 113 255 178)(112 114 256 179)
(1 240 373 68)(2 239 374 67)(3 238 375 66)(4 237 376 65)(5 236 377 64)(6 235 378 63)(7 234 379 62)(8 233 380 61)(9 232 381 60)(10 231 382 59)(11 230 383 58)(12 229 384 57)(13 228 385 112)(14 227 386 111)(15 226 387 110)(16 225 388 109)(17 280 389 108)(18 279 390 107)(19 278 391 106)(20 277 392 105)(21 276 337 104)(22 275 338 103)(23 274 339 102)(24 273 340 101)(25 272 341 100)(26 271 342 99)(27 270 343 98)(28 269 344 97)(29 268 345 96)(30 267 346 95)(31 266 347 94)(32 265 348 93)(33 264 349 92)(34 263 350 91)(35 262 351 90)(36 261 352 89)(37 260 353 88)(38 259 354 87)(39 258 355 86)(40 257 356 85)(41 256 357 84)(42 255 358 83)(43 254 359 82)(44 253 360 81)(45 252 361 80)(46 251 362 79)(47 250 363 78)(48 249 364 77)(49 248 365 76)(50 247 366 75)(51 246 367 74)(52 245 368 73)(53 244 369 72)(54 243 370 71)(55 242 371 70)(56 241 372 69)(113 415 206 313)(114 414 207 312)(115 413 208 311)(116 412 209 310)(117 411 210 309)(118 410 211 308)(119 409 212 307)(120 408 213 306)(121 407 214 305)(122 406 215 304)(123 405 216 303)(124 404 217 302)(125 403 218 301)(126 402 219 300)(127 401 220 299)(128 400 221 298)(129 399 222 297)(130 398 223 296)(131 397 224 295)(132 396 169 294)(133 395 170 293)(134 394 171 292)(135 393 172 291)(136 448 173 290)(137 447 174 289)(138 446 175 288)(139 445 176 287)(140 444 177 286)(141 443 178 285)(142 442 179 284)(143 441 180 283)(144 440 181 282)(145 439 182 281)(146 438 183 336)(147 437 184 335)(148 436 185 334)(149 435 186 333)(150 434 187 332)(151 433 188 331)(152 432 189 330)(153 431 190 329)(154 430 191 328)(155 429 192 327)(156 428 193 326)(157 427 194 325)(158 426 195 324)(159 425 196 323)(160 424 197 322)(161 423 198 321)(162 422 199 320)(163 421 200 319)(164 420 201 318)(165 419 202 317)(166 418 203 316)(167 417 204 315)(168 416 205 314)
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), (1,300,345,430)(2,301,346,431)(3,302,347,432)(4,303,348,433)(5,304,349,434)(6,305,350,435)(7,306,351,436)(8,307,352,437)(9,308,353,438)(10,309,354,439)(11,310,355,440)(12,311,356,441)(13,312,357,442)(14,313,358,443)(15,314,359,444)(16,315,360,445)(17,316,361,446)(18,317,362,447)(19,318,363,448)(20,319,364,393)(21,320,365,394)(22,321,366,395)(23,322,367,396)(24,323,368,397)(25,324,369,398)(26,325,370,399)(27,326,371,400)(28,327,372,401)(29,328,373,402)(30,329,374,403)(31,330,375,404)(32,331,376,405)(33,332,377,406)(34,333,378,407)(35,334,379,408)(36,335,380,409)(37,336,381,410)(38,281,382,411)(39,282,383,412)(40,283,384,413)(41,284,385,414)(42,285,386,415)(43,286,387,416)(44,287,388,417)(45,288,389,418)(46,289,390,419)(47,290,391,420)(48,291,392,421)(49,292,337,422)(50,293,338,423)(51,294,339,424)(52,295,340,425)(53,296,341,426)(54,297,342,427)(55,298,343,428)(56,299,344,429)(57,115,257,180)(58,116,258,181)(59,117,259,182)(60,118,260,183)(61,119,261,184)(62,120,262,185)(63,121,263,186)(64,122,264,187)(65,123,265,188)(66,124,266,189)(67,125,267,190)(68,126,268,191)(69,127,269,192)(70,128,270,193)(71,129,271,194)(72,130,272,195)(73,131,273,196)(74,132,274,197)(75,133,275,198)(76,134,276,199)(77,135,277,200)(78,136,278,201)(79,137,279,202)(80,138,280,203)(81,139,225,204)(82,140,226,205)(83,141,227,206)(84,142,228,207)(85,143,229,208)(86,144,230,209)(87,145,231,210)(88,146,232,211)(89,147,233,212)(90,148,234,213)(91,149,235,214)(92,150,236,215)(93,151,237,216)(94,152,238,217)(95,153,239,218)(96,154,240,219)(97,155,241,220)(98,156,242,221)(99,157,243,222)(100,158,244,223)(101,159,245,224)(102,160,246,169)(103,161,247,170)(104,162,248,171)(105,163,249,172)(106,164,250,173)(107,165,251,174)(108,166,252,175)(109,167,253,176)(110,168,254,177)(111,113,255,178)(112,114,256,179), (1,240,373,68)(2,239,374,67)(3,238,375,66)(4,237,376,65)(5,236,377,64)(6,235,378,63)(7,234,379,62)(8,233,380,61)(9,232,381,60)(10,231,382,59)(11,230,383,58)(12,229,384,57)(13,228,385,112)(14,227,386,111)(15,226,387,110)(16,225,388,109)(17,280,389,108)(18,279,390,107)(19,278,391,106)(20,277,392,105)(21,276,337,104)(22,275,338,103)(23,274,339,102)(24,273,340,101)(25,272,341,100)(26,271,342,99)(27,270,343,98)(28,269,344,97)(29,268,345,96)(30,267,346,95)(31,266,347,94)(32,265,348,93)(33,264,349,92)(34,263,350,91)(35,262,351,90)(36,261,352,89)(37,260,353,88)(38,259,354,87)(39,258,355,86)(40,257,356,85)(41,256,357,84)(42,255,358,83)(43,254,359,82)(44,253,360,81)(45,252,361,80)(46,251,362,79)(47,250,363,78)(48,249,364,77)(49,248,365,76)(50,247,366,75)(51,246,367,74)(52,245,368,73)(53,244,369,72)(54,243,370,71)(55,242,371,70)(56,241,372,69)(113,415,206,313)(114,414,207,312)(115,413,208,311)(116,412,209,310)(117,411,210,309)(118,410,211,308)(119,409,212,307)(120,408,213,306)(121,407,214,305)(122,406,215,304)(123,405,216,303)(124,404,217,302)(125,403,218,301)(126,402,219,300)(127,401,220,299)(128,400,221,298)(129,399,222,297)(130,398,223,296)(131,397,224,295)(132,396,169,294)(133,395,170,293)(134,394,171,292)(135,393,172,291)(136,448,173,290)(137,447,174,289)(138,446,175,288)(139,445,176,287)(140,444,177,286)(141,443,178,285)(142,442,179,284)(143,441,180,283)(144,440,181,282)(145,439,182,281)(146,438,183,336)(147,437,184,335)(148,436,185,334)(149,435,186,333)(150,434,187,332)(151,433,188,331)(152,432,189,330)(153,431,190,329)(154,430,191,328)(155,429,192,327)(156,428,193,326)(157,427,194,325)(158,426,195,324)(159,425,196,323)(160,424,197,322)(161,423,198,321)(162,422,199,320)(163,421,200,319)(164,420,201,318)(165,419,202,317)(166,418,203,316)(167,417,204,315)(168,416,205,314)>;
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), (1,300,345,430)(2,301,346,431)(3,302,347,432)(4,303,348,433)(5,304,349,434)(6,305,350,435)(7,306,351,436)(8,307,352,437)(9,308,353,438)(10,309,354,439)(11,310,355,440)(12,311,356,441)(13,312,357,442)(14,313,358,443)(15,314,359,444)(16,315,360,445)(17,316,361,446)(18,317,362,447)(19,318,363,448)(20,319,364,393)(21,320,365,394)(22,321,366,395)(23,322,367,396)(24,323,368,397)(25,324,369,398)(26,325,370,399)(27,326,371,400)(28,327,372,401)(29,328,373,402)(30,329,374,403)(31,330,375,404)(32,331,376,405)(33,332,377,406)(34,333,378,407)(35,334,379,408)(36,335,380,409)(37,336,381,410)(38,281,382,411)(39,282,383,412)(40,283,384,413)(41,284,385,414)(42,285,386,415)(43,286,387,416)(44,287,388,417)(45,288,389,418)(46,289,390,419)(47,290,391,420)(48,291,392,421)(49,292,337,422)(50,293,338,423)(51,294,339,424)(52,295,340,425)(53,296,341,426)(54,297,342,427)(55,298,343,428)(56,299,344,429)(57,115,257,180)(58,116,258,181)(59,117,259,182)(60,118,260,183)(61,119,261,184)(62,120,262,185)(63,121,263,186)(64,122,264,187)(65,123,265,188)(66,124,266,189)(67,125,267,190)(68,126,268,191)(69,127,269,192)(70,128,270,193)(71,129,271,194)(72,130,272,195)(73,131,273,196)(74,132,274,197)(75,133,275,198)(76,134,276,199)(77,135,277,200)(78,136,278,201)(79,137,279,202)(80,138,280,203)(81,139,225,204)(82,140,226,205)(83,141,227,206)(84,142,228,207)(85,143,229,208)(86,144,230,209)(87,145,231,210)(88,146,232,211)(89,147,233,212)(90,148,234,213)(91,149,235,214)(92,150,236,215)(93,151,237,216)(94,152,238,217)(95,153,239,218)(96,154,240,219)(97,155,241,220)(98,156,242,221)(99,157,243,222)(100,158,244,223)(101,159,245,224)(102,160,246,169)(103,161,247,170)(104,162,248,171)(105,163,249,172)(106,164,250,173)(107,165,251,174)(108,166,252,175)(109,167,253,176)(110,168,254,177)(111,113,255,178)(112,114,256,179), (1,240,373,68)(2,239,374,67)(3,238,375,66)(4,237,376,65)(5,236,377,64)(6,235,378,63)(7,234,379,62)(8,233,380,61)(9,232,381,60)(10,231,382,59)(11,230,383,58)(12,229,384,57)(13,228,385,112)(14,227,386,111)(15,226,387,110)(16,225,388,109)(17,280,389,108)(18,279,390,107)(19,278,391,106)(20,277,392,105)(21,276,337,104)(22,275,338,103)(23,274,339,102)(24,273,340,101)(25,272,341,100)(26,271,342,99)(27,270,343,98)(28,269,344,97)(29,268,345,96)(30,267,346,95)(31,266,347,94)(32,265,348,93)(33,264,349,92)(34,263,350,91)(35,262,351,90)(36,261,352,89)(37,260,353,88)(38,259,354,87)(39,258,355,86)(40,257,356,85)(41,256,357,84)(42,255,358,83)(43,254,359,82)(44,253,360,81)(45,252,361,80)(46,251,362,79)(47,250,363,78)(48,249,364,77)(49,248,365,76)(50,247,366,75)(51,246,367,74)(52,245,368,73)(53,244,369,72)(54,243,370,71)(55,242,371,70)(56,241,372,69)(113,415,206,313)(114,414,207,312)(115,413,208,311)(116,412,209,310)(117,411,210,309)(118,410,211,308)(119,409,212,307)(120,408,213,306)(121,407,214,305)(122,406,215,304)(123,405,216,303)(124,404,217,302)(125,403,218,301)(126,402,219,300)(127,401,220,299)(128,400,221,298)(129,399,222,297)(130,398,223,296)(131,397,224,295)(132,396,169,294)(133,395,170,293)(134,394,171,292)(135,393,172,291)(136,448,173,290)(137,447,174,289)(138,446,175,288)(139,445,176,287)(140,444,177,286)(141,443,178,285)(142,442,179,284)(143,441,180,283)(144,440,181,282)(145,439,182,281)(146,438,183,336)(147,437,184,335)(148,436,185,334)(149,435,186,333)(150,434,187,332)(151,433,188,331)(152,432,189,330)(153,431,190,329)(154,430,191,328)(155,429,192,327)(156,428,193,326)(157,427,194,325)(158,426,195,324)(159,425,196,323)(160,424,197,322)(161,423,198,321)(162,422,199,320)(163,421,200,319)(164,420,201,318)(165,419,202,317)(166,418,203,316)(167,417,204,315)(168,416,205,314) );
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)], [(1,300,345,430),(2,301,346,431),(3,302,347,432),(4,303,348,433),(5,304,349,434),(6,305,350,435),(7,306,351,436),(8,307,352,437),(9,308,353,438),(10,309,354,439),(11,310,355,440),(12,311,356,441),(13,312,357,442),(14,313,358,443),(15,314,359,444),(16,315,360,445),(17,316,361,446),(18,317,362,447),(19,318,363,448),(20,319,364,393),(21,320,365,394),(22,321,366,395),(23,322,367,396),(24,323,368,397),(25,324,369,398),(26,325,370,399),(27,326,371,400),(28,327,372,401),(29,328,373,402),(30,329,374,403),(31,330,375,404),(32,331,376,405),(33,332,377,406),(34,333,378,407),(35,334,379,408),(36,335,380,409),(37,336,381,410),(38,281,382,411),(39,282,383,412),(40,283,384,413),(41,284,385,414),(42,285,386,415),(43,286,387,416),(44,287,388,417),(45,288,389,418),(46,289,390,419),(47,290,391,420),(48,291,392,421),(49,292,337,422),(50,293,338,423),(51,294,339,424),(52,295,340,425),(53,296,341,426),(54,297,342,427),(55,298,343,428),(56,299,344,429),(57,115,257,180),(58,116,258,181),(59,117,259,182),(60,118,260,183),(61,119,261,184),(62,120,262,185),(63,121,263,186),(64,122,264,187),(65,123,265,188),(66,124,266,189),(67,125,267,190),(68,126,268,191),(69,127,269,192),(70,128,270,193),(71,129,271,194),(72,130,272,195),(73,131,273,196),(74,132,274,197),(75,133,275,198),(76,134,276,199),(77,135,277,200),(78,136,278,201),(79,137,279,202),(80,138,280,203),(81,139,225,204),(82,140,226,205),(83,141,227,206),(84,142,228,207),(85,143,229,208),(86,144,230,209),(87,145,231,210),(88,146,232,211),(89,147,233,212),(90,148,234,213),(91,149,235,214),(92,150,236,215),(93,151,237,216),(94,152,238,217),(95,153,239,218),(96,154,240,219),(97,155,241,220),(98,156,242,221),(99,157,243,222),(100,158,244,223),(101,159,245,224),(102,160,246,169),(103,161,247,170),(104,162,248,171),(105,163,249,172),(106,164,250,173),(107,165,251,174),(108,166,252,175),(109,167,253,176),(110,168,254,177),(111,113,255,178),(112,114,256,179)], [(1,240,373,68),(2,239,374,67),(3,238,375,66),(4,237,376,65),(5,236,377,64),(6,235,378,63),(7,234,379,62),(8,233,380,61),(9,232,381,60),(10,231,382,59),(11,230,383,58),(12,229,384,57),(13,228,385,112),(14,227,386,111),(15,226,387,110),(16,225,388,109),(17,280,389,108),(18,279,390,107),(19,278,391,106),(20,277,392,105),(21,276,337,104),(22,275,338,103),(23,274,339,102),(24,273,340,101),(25,272,341,100),(26,271,342,99),(27,270,343,98),(28,269,344,97),(29,268,345,96),(30,267,346,95),(31,266,347,94),(32,265,348,93),(33,264,349,92),(34,263,350,91),(35,262,351,90),(36,261,352,89),(37,260,353,88),(38,259,354,87),(39,258,355,86),(40,257,356,85),(41,256,357,84),(42,255,358,83),(43,254,359,82),(44,253,360,81),(45,252,361,80),(46,251,362,79),(47,250,363,78),(48,249,364,77),(49,248,365,76),(50,247,366,75),(51,246,367,74),(52,245,368,73),(53,244,369,72),(54,243,370,71),(55,242,371,70),(56,241,372,69),(113,415,206,313),(114,414,207,312),(115,413,208,311),(116,412,209,310),(117,411,210,309),(118,410,211,308),(119,409,212,307),(120,408,213,306),(121,407,214,305),(122,406,215,304),(123,405,216,303),(124,404,217,302),(125,403,218,301),(126,402,219,300),(127,401,220,299),(128,400,221,298),(129,399,222,297),(130,398,223,296),(131,397,224,295),(132,396,169,294),(133,395,170,293),(134,394,171,292),(135,393,172,291),(136,448,173,290),(137,447,174,289),(138,446,175,288),(139,445,176,287),(140,444,177,286),(141,443,178,285),(142,442,179,284),(143,441,180,283),(144,440,181,282),(145,439,182,281),(146,438,183,336),(147,437,184,335),(148,436,185,334),(149,435,186,333),(150,434,187,332),(151,433,188,331),(152,432,189,330),(153,431,190,329),(154,430,191,328),(155,429,192,327),(156,428,193,326),(157,427,194,325),(158,426,195,324),(159,425,196,323),(160,424,197,322),(161,423,198,321),(162,422,199,320),(163,421,200,319),(164,420,201,318),(165,419,202,317),(166,418,203,316),(167,417,204,315),(168,416,205,314)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | D14 | D14 | C4○D8 | Dic14 | D28 | D56⋊7C2 |
| kernel | C56.13Q8 | C8⋊Dic7 | C56⋊1C4 | C4×C56 | C28.6Q8 | C56 | C2×C28 | C4×C8 | C42 | C2×C8 | C14 | C8 | C2×C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 3 | 3 | 6 | 8 | 24 | 12 | 48 |
Matrix representation of C56.13Q8 ►in GL4(𝔽113) generated by
| 13 | 21 | 0 | 0 |
| 71 | 80 | 0 | 0 |
| 0 | 0 | 96 | 84 |
| 0 | 0 | 89 | 72 |
| 86 | 108 | 0 | 0 |
| 10 | 27 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 96 | 86 | 0 | 0 |
| 107 | 17 | 0 | 0 |
| 0 | 0 | 74 | 39 |
| 0 | 0 | 103 | 39 |
G:=sub<GL(4,GF(113))| [13,71,0,0,21,80,0,0,0,0,96,89,0,0,84,72],[86,10,0,0,108,27,0,0,0,0,98,0,0,0,0,98],[96,107,0,0,86,17,0,0,0,0,74,103,0,0,39,39] >;
C56.13Q8 in GAP, Magma, Sage, TeX
C_{56}._{13}Q_8 % in TeX
G:=Group("C56.13Q8"); // GroupNames label
G:=SmallGroup(448,217);
// by ID
G=gap.SmallGroup(448,217);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,344,254,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^28*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^28*b^-1>;
// generators/relations