metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊2Q8, C8⋊4Dic14, Dic7.2D8, Dic7.2Q16, C7⋊C8⋊6Q8, C2.12(D7×D8), C7⋊2(C8⋊2Q8), C28⋊Q8.7C2, C4.26(Q8×D7), C4⋊C4.43D14, C14.27(C2×D8), C2.D8.4D7, C28.17(C2×Q8), C2.12(D7×Q16), (C2×C8).227D14, C14.16(C4⋊Q8), C2.11(C28⋊Q8), C14.21(C2×Q16), (C8×Dic7).2C2, C56⋊1C4.14C2, (C2×C56).79C22, C4.23(C2×Dic14), C28.Q8.7C2, C22.223(D4×D7), (C2×C28).290C23, (C2×Dic7).100D4, C4⋊Dic7.116C22, (C4×Dic7).233C22, (C7×C2.D8).5C2, (C2×C14).295(C2×D4), (C7×C4⋊C4).83C22, (C2×C7⋊C8).231C22, (C2×C4).393(C22×D7), SmallGroup(448,408)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊2Q8
G = < a,b,c | a56=b4=1, c2=b2, bab-1=a15, cac-1=a41, cbc-1=b-1 >
Subgroups: 492 in 98 conjugacy classes, 47 normal (27 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, Dic7, C28, C28, C2×C14, C4×C8, C2.D8, C2.D8, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C8⋊2Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C28.Q8, C8×Dic7, C56⋊1C4, C7×C2.D8, C28⋊Q8, C56⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, D8, Q16, C2×D4, C2×Q8, D14, C4⋊Q8, C2×D8, C2×Q16, Dic14, C22×D7, C8⋊2Q8, C2×Dic14, D4×D7, Q8×D7, C28⋊Q8, D7×D8, D7×Q16, C56⋊2Q8
►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)
(1 413 245 334)(2 428 246 293)(3 443 247 308)(4 402 248 323)(5 417 249 282)(6 432 250 297)(7 447 251 312)(8 406 252 327)(9 421 253 286)(10 436 254 301)(11 395 255 316)(12 410 256 331)(13 425 257 290)(14 440 258 305)(15 399 259 320)(16 414 260 335)(17 429 261 294)(18 444 262 309)(19 403 263 324)(20 418 264 283)(21 433 265 298)(22 448 266 313)(23 407 267 328)(24 422 268 287)(25 437 269 302)(26 396 270 317)(27 411 271 332)(28 426 272 291)(29 441 273 306)(30 400 274 321)(31 415 275 336)(32 430 276 295)(33 445 277 310)(34 404 278 325)(35 419 279 284)(36 434 280 299)(37 393 225 314)(38 408 226 329)(39 423 227 288)(40 438 228 303)(41 397 229 318)(42 412 230 333)(43 427 231 292)(44 442 232 307)(45 401 233 322)(46 416 234 281)(47 431 235 296)(48 446 236 311)(49 405 237 326)(50 420 238 285)(51 435 239 300)(52 394 240 315)(53 409 241 330)(54 424 242 289)(55 439 243 304)(56 398 244 319)(57 132 210 389)(58 147 211 348)(59 162 212 363)(60 121 213 378)(61 136 214 337)(62 151 215 352)(63 166 216 367)(64 125 217 382)(65 140 218 341)(66 155 219 356)(67 114 220 371)(68 129 221 386)(69 144 222 345)(70 159 223 360)(71 118 224 375)(72 133 169 390)(73 148 170 349)(74 163 171 364)(75 122 172 379)(76 137 173 338)(77 152 174 353)(78 167 175 368)(79 126 176 383)(80 141 177 342)(81 156 178 357)(82 115 179 372)(83 130 180 387)(84 145 181 346)(85 160 182 361)(86 119 183 376)(87 134 184 391)(88 149 185 350)(89 164 186 365)(90 123 187 380)(91 138 188 339)(92 153 189 354)(93 168 190 369)(94 127 191 384)(95 142 192 343)(96 157 193 358)(97 116 194 373)(98 131 195 388)(99 146 196 347)(100 161 197 362)(101 120 198 377)(102 135 199 392)(103 150 200 351)(104 165 201 366)(105 124 202 381)(106 139 203 340)(107 154 204 355)(108 113 205 370)(109 128 206 385)(110 143 207 344)(111 158 208 359)(112 117 209 374)
(1 189 245 92)(2 174 246 77)(3 215 247 62)(4 200 248 103)(5 185 249 88)(6 170 250 73)(7 211 251 58)(8 196 252 99)(9 181 253 84)(10 222 254 69)(11 207 255 110)(12 192 256 95)(13 177 257 80)(14 218 258 65)(15 203 259 106)(16 188 260 91)(17 173 261 76)(18 214 262 61)(19 199 263 102)(20 184 264 87)(21 169 265 72)(22 210 266 57)(23 195 267 98)(24 180 268 83)(25 221 269 68)(26 206 270 109)(27 191 271 94)(28 176 272 79)(29 217 273 64)(30 202 274 105)(31 187 275 90)(32 172 276 75)(33 213 277 60)(34 198 278 101)(35 183 279 86)(36 224 280 71)(37 209 225 112)(38 194 226 97)(39 179 227 82)(40 220 228 67)(41 205 229 108)(42 190 230 93)(43 175 231 78)(44 216 232 63)(45 201 233 104)(46 186 234 89)(47 171 235 74)(48 212 236 59)(49 197 237 100)(50 182 238 85)(51 223 239 70)(52 208 240 111)(53 193 241 96)(54 178 242 81)(55 219 243 66)(56 204 244 107)(113 318 370 397)(114 303 371 438)(115 288 372 423)(116 329 373 408)(117 314 374 393)(118 299 375 434)(119 284 376 419)(120 325 377 404)(121 310 378 445)(122 295 379 430)(123 336 380 415)(124 321 381 400)(125 306 382 441)(126 291 383 426)(127 332 384 411)(128 317 385 396)(129 302 386 437)(130 287 387 422)(131 328 388 407)(132 313 389 448)(133 298 390 433)(134 283 391 418)(135 324 392 403)(136 309 337 444)(137 294 338 429)(138 335 339 414)(139 320 340 399)(140 305 341 440)(141 290 342 425)(142 331 343 410)(143 316 344 395)(144 301 345 436)(145 286 346 421)(146 327 347 406)(147 312 348 447)(148 297 349 432)(149 282 350 417)(150 323 351 402)(151 308 352 443)(152 293 353 428)(153 334 354 413)(154 319 355 398)(155 304 356 439)(156 289 357 424)(157 330 358 409)(158 315 359 394)(159 300 360 435)(160 285 361 420)(161 326 362 405)(162 311 363 446)(163 296 364 431)(164 281 365 416)(165 322 366 401)(166 307 367 442)(167 292 368 427)(168 333 369 412)
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), (1,413,245,334)(2,428,246,293)(3,443,247,308)(4,402,248,323)(5,417,249,282)(6,432,250,297)(7,447,251,312)(8,406,252,327)(9,421,253,286)(10,436,254,301)(11,395,255,316)(12,410,256,331)(13,425,257,290)(14,440,258,305)(15,399,259,320)(16,414,260,335)(17,429,261,294)(18,444,262,309)(19,403,263,324)(20,418,264,283)(21,433,265,298)(22,448,266,313)(23,407,267,328)(24,422,268,287)(25,437,269,302)(26,396,270,317)(27,411,271,332)(28,426,272,291)(29,441,273,306)(30,400,274,321)(31,415,275,336)(32,430,276,295)(33,445,277,310)(34,404,278,325)(35,419,279,284)(36,434,280,299)(37,393,225,314)(38,408,226,329)(39,423,227,288)(40,438,228,303)(41,397,229,318)(42,412,230,333)(43,427,231,292)(44,442,232,307)(45,401,233,322)(46,416,234,281)(47,431,235,296)(48,446,236,311)(49,405,237,326)(50,420,238,285)(51,435,239,300)(52,394,240,315)(53,409,241,330)(54,424,242,289)(55,439,243,304)(56,398,244,319)(57,132,210,389)(58,147,211,348)(59,162,212,363)(60,121,213,378)(61,136,214,337)(62,151,215,352)(63,166,216,367)(64,125,217,382)(65,140,218,341)(66,155,219,356)(67,114,220,371)(68,129,221,386)(69,144,222,345)(70,159,223,360)(71,118,224,375)(72,133,169,390)(73,148,170,349)(74,163,171,364)(75,122,172,379)(76,137,173,338)(77,152,174,353)(78,167,175,368)(79,126,176,383)(80,141,177,342)(81,156,178,357)(82,115,179,372)(83,130,180,387)(84,145,181,346)(85,160,182,361)(86,119,183,376)(87,134,184,391)(88,149,185,350)(89,164,186,365)(90,123,187,380)(91,138,188,339)(92,153,189,354)(93,168,190,369)(94,127,191,384)(95,142,192,343)(96,157,193,358)(97,116,194,373)(98,131,195,388)(99,146,196,347)(100,161,197,362)(101,120,198,377)(102,135,199,392)(103,150,200,351)(104,165,201,366)(105,124,202,381)(106,139,203,340)(107,154,204,355)(108,113,205,370)(109,128,206,385)(110,143,207,344)(111,158,208,359)(112,117,209,374), (1,189,245,92)(2,174,246,77)(3,215,247,62)(4,200,248,103)(5,185,249,88)(6,170,250,73)(7,211,251,58)(8,196,252,99)(9,181,253,84)(10,222,254,69)(11,207,255,110)(12,192,256,95)(13,177,257,80)(14,218,258,65)(15,203,259,106)(16,188,260,91)(17,173,261,76)(18,214,262,61)(19,199,263,102)(20,184,264,87)(21,169,265,72)(22,210,266,57)(23,195,267,98)(24,180,268,83)(25,221,269,68)(26,206,270,109)(27,191,271,94)(28,176,272,79)(29,217,273,64)(30,202,274,105)(31,187,275,90)(32,172,276,75)(33,213,277,60)(34,198,278,101)(35,183,279,86)(36,224,280,71)(37,209,225,112)(38,194,226,97)(39,179,227,82)(40,220,228,67)(41,205,229,108)(42,190,230,93)(43,175,231,78)(44,216,232,63)(45,201,233,104)(46,186,234,89)(47,171,235,74)(48,212,236,59)(49,197,237,100)(50,182,238,85)(51,223,239,70)(52,208,240,111)(53,193,241,96)(54,178,242,81)(55,219,243,66)(56,204,244,107)(113,318,370,397)(114,303,371,438)(115,288,372,423)(116,329,373,408)(117,314,374,393)(118,299,375,434)(119,284,376,419)(120,325,377,404)(121,310,378,445)(122,295,379,430)(123,336,380,415)(124,321,381,400)(125,306,382,441)(126,291,383,426)(127,332,384,411)(128,317,385,396)(129,302,386,437)(130,287,387,422)(131,328,388,407)(132,313,389,448)(133,298,390,433)(134,283,391,418)(135,324,392,403)(136,309,337,444)(137,294,338,429)(138,335,339,414)(139,320,340,399)(140,305,341,440)(141,290,342,425)(142,331,343,410)(143,316,344,395)(144,301,345,436)(145,286,346,421)(146,327,347,406)(147,312,348,447)(148,297,349,432)(149,282,350,417)(150,323,351,402)(151,308,352,443)(152,293,353,428)(153,334,354,413)(154,319,355,398)(155,304,356,439)(156,289,357,424)(157,330,358,409)(158,315,359,394)(159,300,360,435)(160,285,361,420)(161,326,362,405)(162,311,363,446)(163,296,364,431)(164,281,365,416)(165,322,366,401)(166,307,367,442)(167,292,368,427)(168,333,369,412)>;
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), (1,413,245,334)(2,428,246,293)(3,443,247,308)(4,402,248,323)(5,417,249,282)(6,432,250,297)(7,447,251,312)(8,406,252,327)(9,421,253,286)(10,436,254,301)(11,395,255,316)(12,410,256,331)(13,425,257,290)(14,440,258,305)(15,399,259,320)(16,414,260,335)(17,429,261,294)(18,444,262,309)(19,403,263,324)(20,418,264,283)(21,433,265,298)(22,448,266,313)(23,407,267,328)(24,422,268,287)(25,437,269,302)(26,396,270,317)(27,411,271,332)(28,426,272,291)(29,441,273,306)(30,400,274,321)(31,415,275,336)(32,430,276,295)(33,445,277,310)(34,404,278,325)(35,419,279,284)(36,434,280,299)(37,393,225,314)(38,408,226,329)(39,423,227,288)(40,438,228,303)(41,397,229,318)(42,412,230,333)(43,427,231,292)(44,442,232,307)(45,401,233,322)(46,416,234,281)(47,431,235,296)(48,446,236,311)(49,405,237,326)(50,420,238,285)(51,435,239,300)(52,394,240,315)(53,409,241,330)(54,424,242,289)(55,439,243,304)(56,398,244,319)(57,132,210,389)(58,147,211,348)(59,162,212,363)(60,121,213,378)(61,136,214,337)(62,151,215,352)(63,166,216,367)(64,125,217,382)(65,140,218,341)(66,155,219,356)(67,114,220,371)(68,129,221,386)(69,144,222,345)(70,159,223,360)(71,118,224,375)(72,133,169,390)(73,148,170,349)(74,163,171,364)(75,122,172,379)(76,137,173,338)(77,152,174,353)(78,167,175,368)(79,126,176,383)(80,141,177,342)(81,156,178,357)(82,115,179,372)(83,130,180,387)(84,145,181,346)(85,160,182,361)(86,119,183,376)(87,134,184,391)(88,149,185,350)(89,164,186,365)(90,123,187,380)(91,138,188,339)(92,153,189,354)(93,168,190,369)(94,127,191,384)(95,142,192,343)(96,157,193,358)(97,116,194,373)(98,131,195,388)(99,146,196,347)(100,161,197,362)(101,120,198,377)(102,135,199,392)(103,150,200,351)(104,165,201,366)(105,124,202,381)(106,139,203,340)(107,154,204,355)(108,113,205,370)(109,128,206,385)(110,143,207,344)(111,158,208,359)(112,117,209,374), (1,189,245,92)(2,174,246,77)(3,215,247,62)(4,200,248,103)(5,185,249,88)(6,170,250,73)(7,211,251,58)(8,196,252,99)(9,181,253,84)(10,222,254,69)(11,207,255,110)(12,192,256,95)(13,177,257,80)(14,218,258,65)(15,203,259,106)(16,188,260,91)(17,173,261,76)(18,214,262,61)(19,199,263,102)(20,184,264,87)(21,169,265,72)(22,210,266,57)(23,195,267,98)(24,180,268,83)(25,221,269,68)(26,206,270,109)(27,191,271,94)(28,176,272,79)(29,217,273,64)(30,202,274,105)(31,187,275,90)(32,172,276,75)(33,213,277,60)(34,198,278,101)(35,183,279,86)(36,224,280,71)(37,209,225,112)(38,194,226,97)(39,179,227,82)(40,220,228,67)(41,205,229,108)(42,190,230,93)(43,175,231,78)(44,216,232,63)(45,201,233,104)(46,186,234,89)(47,171,235,74)(48,212,236,59)(49,197,237,100)(50,182,238,85)(51,223,239,70)(52,208,240,111)(53,193,241,96)(54,178,242,81)(55,219,243,66)(56,204,244,107)(113,318,370,397)(114,303,371,438)(115,288,372,423)(116,329,373,408)(117,314,374,393)(118,299,375,434)(119,284,376,419)(120,325,377,404)(121,310,378,445)(122,295,379,430)(123,336,380,415)(124,321,381,400)(125,306,382,441)(126,291,383,426)(127,332,384,411)(128,317,385,396)(129,302,386,437)(130,287,387,422)(131,328,388,407)(132,313,389,448)(133,298,390,433)(134,283,391,418)(135,324,392,403)(136,309,337,444)(137,294,338,429)(138,335,339,414)(139,320,340,399)(140,305,341,440)(141,290,342,425)(142,331,343,410)(143,316,344,395)(144,301,345,436)(145,286,346,421)(146,327,347,406)(147,312,348,447)(148,297,349,432)(149,282,350,417)(150,323,351,402)(151,308,352,443)(152,293,353,428)(153,334,354,413)(154,319,355,398)(155,304,356,439)(156,289,357,424)(157,330,358,409)(158,315,359,394)(159,300,360,435)(160,285,361,420)(161,326,362,405)(162,311,363,446)(163,296,364,431)(164,281,365,416)(165,322,366,401)(166,307,367,442)(167,292,368,427)(168,333,369,412) );
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)], [(1,413,245,334),(2,428,246,293),(3,443,247,308),(4,402,248,323),(5,417,249,282),(6,432,250,297),(7,447,251,312),(8,406,252,327),(9,421,253,286),(10,436,254,301),(11,395,255,316),(12,410,256,331),(13,425,257,290),(14,440,258,305),(15,399,259,320),(16,414,260,335),(17,429,261,294),(18,444,262,309),(19,403,263,324),(20,418,264,283),(21,433,265,298),(22,448,266,313),(23,407,267,328),(24,422,268,287),(25,437,269,302),(26,396,270,317),(27,411,271,332),(28,426,272,291),(29,441,273,306),(30,400,274,321),(31,415,275,336),(32,430,276,295),(33,445,277,310),(34,404,278,325),(35,419,279,284),(36,434,280,299),(37,393,225,314),(38,408,226,329),(39,423,227,288),(40,438,228,303),(41,397,229,318),(42,412,230,333),(43,427,231,292),(44,442,232,307),(45,401,233,322),(46,416,234,281),(47,431,235,296),(48,446,236,311),(49,405,237,326),(50,420,238,285),(51,435,239,300),(52,394,240,315),(53,409,241,330),(54,424,242,289),(55,439,243,304),(56,398,244,319),(57,132,210,389),(58,147,211,348),(59,162,212,363),(60,121,213,378),(61,136,214,337),(62,151,215,352),(63,166,216,367),(64,125,217,382),(65,140,218,341),(66,155,219,356),(67,114,220,371),(68,129,221,386),(69,144,222,345),(70,159,223,360),(71,118,224,375),(72,133,169,390),(73,148,170,349),(74,163,171,364),(75,122,172,379),(76,137,173,338),(77,152,174,353),(78,167,175,368),(79,126,176,383),(80,141,177,342),(81,156,178,357),(82,115,179,372),(83,130,180,387),(84,145,181,346),(85,160,182,361),(86,119,183,376),(87,134,184,391),(88,149,185,350),(89,164,186,365),(90,123,187,380),(91,138,188,339),(92,153,189,354),(93,168,190,369),(94,127,191,384),(95,142,192,343),(96,157,193,358),(97,116,194,373),(98,131,195,388),(99,146,196,347),(100,161,197,362),(101,120,198,377),(102,135,199,392),(103,150,200,351),(104,165,201,366),(105,124,202,381),(106,139,203,340),(107,154,204,355),(108,113,205,370),(109,128,206,385),(110,143,207,344),(111,158,208,359),(112,117,209,374)], [(1,189,245,92),(2,174,246,77),(3,215,247,62),(4,200,248,103),(5,185,249,88),(6,170,250,73),(7,211,251,58),(8,196,252,99),(9,181,253,84),(10,222,254,69),(11,207,255,110),(12,192,256,95),(13,177,257,80),(14,218,258,65),(15,203,259,106),(16,188,260,91),(17,173,261,76),(18,214,262,61),(19,199,263,102),(20,184,264,87),(21,169,265,72),(22,210,266,57),(23,195,267,98),(24,180,268,83),(25,221,269,68),(26,206,270,109),(27,191,271,94),(28,176,272,79),(29,217,273,64),(30,202,274,105),(31,187,275,90),(32,172,276,75),(33,213,277,60),(34,198,278,101),(35,183,279,86),(36,224,280,71),(37,209,225,112),(38,194,226,97),(39,179,227,82),(40,220,228,67),(41,205,229,108),(42,190,230,93),(43,175,231,78),(44,216,232,63),(45,201,233,104),(46,186,234,89),(47,171,235,74),(48,212,236,59),(49,197,237,100),(50,182,238,85),(51,223,239,70),(52,208,240,111),(53,193,241,96),(54,178,242,81),(55,219,243,66),(56,204,244,107),(113,318,370,397),(114,303,371,438),(115,288,372,423),(116,329,373,408),(117,314,374,393),(118,299,375,434),(119,284,376,419),(120,325,377,404),(121,310,378,445),(122,295,379,430),(123,336,380,415),(124,321,381,400),(125,306,382,441),(126,291,383,426),(127,332,384,411),(128,317,385,396),(129,302,386,437),(130,287,387,422),(131,328,388,407),(132,313,389,448),(133,298,390,433),(134,283,391,418),(135,324,392,403),(136,309,337,444),(137,294,338,429),(138,335,339,414),(139,320,340,399),(140,305,341,440),(141,290,342,425),(142,331,343,410),(143,316,344,395),(144,301,345,436),(145,286,346,421),(146,327,347,406),(147,312,348,447),(148,297,349,432),(149,282,350,417),(150,323,351,402),(151,308,352,443),(152,293,353,428),(153,334,354,413),(154,319,355,398),(155,304,356,439),(156,289,357,424),(157,330,358,409),(158,315,359,394),(159,300,360,435),(160,285,361,420),(161,326,362,405),(162,311,363,446),(163,296,364,431),(164,281,365,416),(165,322,366,401),(166,307,367,442),(167,292,368,427),(168,333,369,412)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 14 | 14 | 14 | 14 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | + | - | + | + | - | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | Q8 | D4 | D7 | D8 | Q16 | D14 | D14 | Dic14 | Q8×D7 | D4×D7 | D7×D8 | D7×Q16 |
| kernel | C56⋊2Q8 | C28.Q8 | C8×Dic7 | C56⋊1C4 | C7×C2.D8 | C28⋊Q8 | C7⋊C8 | C56 | C2×Dic7 | C2.D8 | Dic7 | Dic7 | C4⋊C4 | C2×C8 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 3 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of C56⋊2Q8 ►in GL6(𝔽113)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 85 |
| 0 | 0 | 0 | 0 | 109 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 91 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 51 |
| 0 | 0 | 0 | 0 | 57 | 91 |
| 35 | 111 | 0 | 0 | 0 | 0 |
| 47 | 78 | 0 | 0 | 0 | 0 |
| 0 | 0 | 93 | 26 | 0 | 0 |
| 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,109,0,0,0,0,85,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,72,0,0,0,0,91,1,0,0,0,0,0,0,22,57,0,0,0,0,51,91],[35,47,0,0,0,0,111,78,0,0,0,0,0,0,93,15,0,0,0,0,26,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C56⋊2Q8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_2Q_8 % in TeX
G:=Group("C56:2Q8"); // GroupNames label
G:=SmallGroup(448,408);
// by ID
G=gap.SmallGroup(448,408);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,254,219,58,438,102,18822]);
// 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^41,c*b*c^-1=b^-1>;
// generators/relations