metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C56)⋊5C4, C14.9(C4×C8), (C2×C8)⋊3Dic7, (C2×Dic7)⋊2C8, C28.43(C4⋊C4), C14.10(C4⋊C8), (C2×C28).64Q8, C2.5(C8×Dic7), (C22×C8).2D7, C2.2(D14⋊C8), (C2×C28).491D4, (C2×C4).164D28, (C22×C56).1C2, C2.3(C56⋊C4), C14.6(C8⋊C4), C22.12(C8×D7), C23.59(C4×D7), C4.48(D14⋊C4), C2.2(Dic7⋊C8), (C2×C14).19C42, C4.21(C4⋊Dic7), (C2×C4).54Dic14, C28.62(C22⋊C4), C14.11(C22⋊C8), C4.31(Dic7⋊C4), (C22×C4).452D14, C22.8(C8⋊D7), (C2×C14).10M4(2), C4.25(C23.D7), (C22×Dic7).6C4, C22.17(C4×Dic7), C22.39(D14⋊C4), C7⋊2(C22.7C42), C2.2(C14.C42), C22.21(Dic7⋊C4), (C22×C28).549C22, C14.10(C2.C42), (C2×C7⋊C8)⋊11C4, (C2×C14).13(C2×C8), (C2×C4).171(C4×D7), (C22×C7⋊C8).16C2, (C2×C4×Dic7).18C2, (C2×C14).37(C4⋊C4), (C2×C28).313(C2×C4), (C2×C4).94(C2×Dic7), (C2×C4).269(C7⋊D4), (C22×C14).87(C2×C4), (C2×C14).51(C22⋊C4), SmallGroup(448,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C56)⋊5C4
G = < a,b,c | a2=b56=c4=1, ab=ba, ac=ca, cbc-1=ab41 >
Subgroups: 388 in 118 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2×C42, C22×C8, C22×C8, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C22×C14, C22.7C42, C2×C7⋊C8, C2×C7⋊C8, C4×Dic7, C2×C56, C2×C56, C22×Dic7, C22×C28, C22×C7⋊C8, C2×C4×Dic7, C22×C56, (C2×C56)⋊5C4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), Dic7, D14, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C22.7C42, C8×D7, C8⋊D7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C8×Dic7, Dic7⋊C8, C56⋊C4, D14⋊C8, C14.C42, (C2×C56)⋊5C4
►Regular action on 448 points
Generators in S448
(1 205)(2 206)(3 207)(4 208)(5 209)(6 210)(7 211)(8 212)(9 213)(10 214)(11 215)(12 216)(13 217)(14 218)(15 219)(16 220)(17 221)(18 222)(19 223)(20 224)(21 169)(22 170)(23 171)(24 172)(25 173)(26 174)(27 175)(28 176)(29 177)(30 178)(31 179)(32 180)(33 181)(34 182)(35 183)(36 184)(37 185)(38 186)(39 187)(40 188)(41 189)(42 190)(43 191)(44 192)(45 193)(46 194)(47 195)(48 196)(49 197)(50 198)(51 199)(52 200)(53 201)(54 202)(55 203)(56 204)(57 266)(58 267)(59 268)(60 269)(61 270)(62 271)(63 272)(64 273)(65 274)(66 275)(67 276)(68 277)(69 278)(70 279)(71 280)(72 225)(73 226)(74 227)(75 228)(76 229)(77 230)(78 231)(79 232)(80 233)(81 234)(82 235)(83 236)(84 237)(85 238)(86 239)(87 240)(88 241)(89 242)(90 243)(91 244)(92 245)(93 246)(94 247)(95 248)(96 249)(97 250)(98 251)(99 252)(100 253)(101 254)(102 255)(103 256)(104 257)(105 258)(106 259)(107 260)(108 261)(109 262)(110 263)(111 264)(112 265)(113 416)(114 417)(115 418)(116 419)(117 420)(118 421)(119 422)(120 423)(121 424)(122 425)(123 426)(124 427)(125 428)(126 429)(127 430)(128 431)(129 432)(130 433)(131 434)(132 435)(133 436)(134 437)(135 438)(136 439)(137 440)(138 441)(139 442)(140 443)(141 444)(142 445)(143 446)(144 447)(145 448)(146 393)(147 394)(148 395)(149 396)(150 397)(151 398)(152 399)(153 400)(154 401)(155 402)(156 403)(157 404)(158 405)(159 406)(160 407)(161 408)(162 409)(163 410)(164 411)(165 412)(166 413)(167 414)(168 415)(281 380)(282 381)(283 382)(284 383)(285 384)(286 385)(287 386)(288 387)(289 388)(290 389)(291 390)(292 391)(293 392)(294 337)(295 338)(296 339)(297 340)(298 341)(299 342)(300 343)(301 344)(302 345)(303 346)(304 347)(305 348)(306 349)(307 350)(308 351)(309 352)(310 353)(311 354)(312 355)(313 356)(314 357)(315 358)(316 359)(317 360)(318 361)(319 362)(320 363)(321 364)(322 365)(323 366)(324 367)(325 368)(326 369)(327 370)(328 371)(329 372)(330 373)(331 374)(332 375)(333 376)(334 377)(335 378)(336 379)
(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)
(1 288 425 272)(2 372 426 104)(3 314 427 242)(4 342 428 74)(5 284 429 268)(6 368 430 100)(7 310 431 238)(8 338 432 70)(9 336 433 264)(10 364 434 96)(11 306 435 234)(12 390 436 66)(13 332 437 260)(14 360 438 92)(15 302 439 230)(16 386 440 62)(17 328 441 256)(18 356 442 88)(19 298 443 226)(20 382 444 58)(21 324 445 252)(22 352 446 84)(23 294 447 278)(24 378 448 110)(25 320 393 248)(26 348 394 80)(27 290 395 274)(28 374 396 106)(29 316 397 244)(30 344 398 76)(31 286 399 270)(32 370 400 102)(33 312 401 240)(34 340 402 72)(35 282 403 266)(36 366 404 98)(37 308 405 236)(38 392 406 68)(39 334 407 262)(40 362 408 94)(41 304 409 232)(42 388 410 64)(43 330 411 258)(44 358 412 90)(45 300 413 228)(46 384 414 60)(47 326 415 254)(48 354 416 86)(49 296 417 280)(50 380 418 112)(51 322 419 250)(52 350 420 82)(53 292 421 276)(54 376 422 108)(55 318 423 246)(56 346 424 78)(57 183 381 156)(59 209 383 126)(61 179 385 152)(63 205 387 122)(65 175 389 148)(67 201 391 118)(69 171 337 144)(71 197 339 114)(73 223 341 140)(75 193 343 166)(77 219 345 136)(79 189 347 162)(81 215 349 132)(83 185 351 158)(85 211 353 128)(87 181 355 154)(89 207 357 124)(91 177 359 150)(93 203 361 120)(95 173 363 146)(97 199 365 116)(99 169 367 142)(101 195 369 168)(103 221 371 138)(105 191 373 164)(107 217 375 134)(109 187 377 160)(111 213 379 130)(113 239 196 311)(115 265 198 281)(117 235 200 307)(119 261 202 333)(121 231 204 303)(123 257 206 329)(125 227 208 299)(127 253 210 325)(129 279 212 295)(131 249 214 321)(133 275 216 291)(135 245 218 317)(137 271 220 287)(139 241 222 313)(141 267 224 283)(143 237 170 309)(145 263 172 335)(147 233 174 305)(149 259 176 331)(151 229 178 301)(153 255 180 327)(155 225 182 297)(157 251 184 323)(159 277 186 293)(161 247 188 319)(163 273 190 289)(165 243 192 315)(167 269 194 285)
G:=sub<Sym(448)| (1,205)(2,206)(3,207)(4,208)(5,209)(6,210)(7,211)(8,212)(9,213)(10,214)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,221)(18,222)(19,223)(20,224)(21,169)(22,170)(23,171)(24,172)(25,173)(26,174)(27,175)(28,176)(29,177)(30,178)(31,179)(32,180)(33,181)(34,182)(35,183)(36,184)(37,185)(38,186)(39,187)(40,188)(41,189)(42,190)(43,191)(44,192)(45,193)(46,194)(47,195)(48,196)(49,197)(50,198)(51,199)(52,200)(53,201)(54,202)(55,203)(56,204)(57,266)(58,267)(59,268)(60,269)(61,270)(62,271)(63,272)(64,273)(65,274)(66,275)(67,276)(68,277)(69,278)(70,279)(71,280)(72,225)(73,226)(74,227)(75,228)(76,229)(77,230)(78,231)(79,232)(80,233)(81,234)(82,235)(83,236)(84,237)(85,238)(86,239)(87,240)(88,241)(89,242)(90,243)(91,244)(92,245)(93,246)(94,247)(95,248)(96,249)(97,250)(98,251)(99,252)(100,253)(101,254)(102,255)(103,256)(104,257)(105,258)(106,259)(107,260)(108,261)(109,262)(110,263)(111,264)(112,265)(113,416)(114,417)(115,418)(116,419)(117,420)(118,421)(119,422)(120,423)(121,424)(122,425)(123,426)(124,427)(125,428)(126,429)(127,430)(128,431)(129,432)(130,433)(131,434)(132,435)(133,436)(134,437)(135,438)(136,439)(137,440)(138,441)(139,442)(140,443)(141,444)(142,445)(143,446)(144,447)(145,448)(146,393)(147,394)(148,395)(149,396)(150,397)(151,398)(152,399)(153,400)(154,401)(155,402)(156,403)(157,404)(158,405)(159,406)(160,407)(161,408)(162,409)(163,410)(164,411)(165,412)(166,413)(167,414)(168,415)(281,380)(282,381)(283,382)(284,383)(285,384)(286,385)(287,386)(288,387)(289,388)(290,389)(291,390)(292,391)(293,392)(294,337)(295,338)(296,339)(297,340)(298,341)(299,342)(300,343)(301,344)(302,345)(303,346)(304,347)(305,348)(306,349)(307,350)(308,351)(309,352)(310,353)(311,354)(312,355)(313,356)(314,357)(315,358)(316,359)(317,360)(318,361)(319,362)(320,363)(321,364)(322,365)(323,366)(324,367)(325,368)(326,369)(327,370)(328,371)(329,372)(330,373)(331,374)(332,375)(333,376)(334,377)(335,378)(336,379), (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), (1,288,425,272)(2,372,426,104)(3,314,427,242)(4,342,428,74)(5,284,429,268)(6,368,430,100)(7,310,431,238)(8,338,432,70)(9,336,433,264)(10,364,434,96)(11,306,435,234)(12,390,436,66)(13,332,437,260)(14,360,438,92)(15,302,439,230)(16,386,440,62)(17,328,441,256)(18,356,442,88)(19,298,443,226)(20,382,444,58)(21,324,445,252)(22,352,446,84)(23,294,447,278)(24,378,448,110)(25,320,393,248)(26,348,394,80)(27,290,395,274)(28,374,396,106)(29,316,397,244)(30,344,398,76)(31,286,399,270)(32,370,400,102)(33,312,401,240)(34,340,402,72)(35,282,403,266)(36,366,404,98)(37,308,405,236)(38,392,406,68)(39,334,407,262)(40,362,408,94)(41,304,409,232)(42,388,410,64)(43,330,411,258)(44,358,412,90)(45,300,413,228)(46,384,414,60)(47,326,415,254)(48,354,416,86)(49,296,417,280)(50,380,418,112)(51,322,419,250)(52,350,420,82)(53,292,421,276)(54,376,422,108)(55,318,423,246)(56,346,424,78)(57,183,381,156)(59,209,383,126)(61,179,385,152)(63,205,387,122)(65,175,389,148)(67,201,391,118)(69,171,337,144)(71,197,339,114)(73,223,341,140)(75,193,343,166)(77,219,345,136)(79,189,347,162)(81,215,349,132)(83,185,351,158)(85,211,353,128)(87,181,355,154)(89,207,357,124)(91,177,359,150)(93,203,361,120)(95,173,363,146)(97,199,365,116)(99,169,367,142)(101,195,369,168)(103,221,371,138)(105,191,373,164)(107,217,375,134)(109,187,377,160)(111,213,379,130)(113,239,196,311)(115,265,198,281)(117,235,200,307)(119,261,202,333)(121,231,204,303)(123,257,206,329)(125,227,208,299)(127,253,210,325)(129,279,212,295)(131,249,214,321)(133,275,216,291)(135,245,218,317)(137,271,220,287)(139,241,222,313)(141,267,224,283)(143,237,170,309)(145,263,172,335)(147,233,174,305)(149,259,176,331)(151,229,178,301)(153,255,180,327)(155,225,182,297)(157,251,184,323)(159,277,186,293)(161,247,188,319)(163,273,190,289)(165,243,192,315)(167,269,194,285)>;
G:=Group( (1,205)(2,206)(3,207)(4,208)(5,209)(6,210)(7,211)(8,212)(9,213)(10,214)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,221)(18,222)(19,223)(20,224)(21,169)(22,170)(23,171)(24,172)(25,173)(26,174)(27,175)(28,176)(29,177)(30,178)(31,179)(32,180)(33,181)(34,182)(35,183)(36,184)(37,185)(38,186)(39,187)(40,188)(41,189)(42,190)(43,191)(44,192)(45,193)(46,194)(47,195)(48,196)(49,197)(50,198)(51,199)(52,200)(53,201)(54,202)(55,203)(56,204)(57,266)(58,267)(59,268)(60,269)(61,270)(62,271)(63,272)(64,273)(65,274)(66,275)(67,276)(68,277)(69,278)(70,279)(71,280)(72,225)(73,226)(74,227)(75,228)(76,229)(77,230)(78,231)(79,232)(80,233)(81,234)(82,235)(83,236)(84,237)(85,238)(86,239)(87,240)(88,241)(89,242)(90,243)(91,244)(92,245)(93,246)(94,247)(95,248)(96,249)(97,250)(98,251)(99,252)(100,253)(101,254)(102,255)(103,256)(104,257)(105,258)(106,259)(107,260)(108,261)(109,262)(110,263)(111,264)(112,265)(113,416)(114,417)(115,418)(116,419)(117,420)(118,421)(119,422)(120,423)(121,424)(122,425)(123,426)(124,427)(125,428)(126,429)(127,430)(128,431)(129,432)(130,433)(131,434)(132,435)(133,436)(134,437)(135,438)(136,439)(137,440)(138,441)(139,442)(140,443)(141,444)(142,445)(143,446)(144,447)(145,448)(146,393)(147,394)(148,395)(149,396)(150,397)(151,398)(152,399)(153,400)(154,401)(155,402)(156,403)(157,404)(158,405)(159,406)(160,407)(161,408)(162,409)(163,410)(164,411)(165,412)(166,413)(167,414)(168,415)(281,380)(282,381)(283,382)(284,383)(285,384)(286,385)(287,386)(288,387)(289,388)(290,389)(291,390)(292,391)(293,392)(294,337)(295,338)(296,339)(297,340)(298,341)(299,342)(300,343)(301,344)(302,345)(303,346)(304,347)(305,348)(306,349)(307,350)(308,351)(309,352)(310,353)(311,354)(312,355)(313,356)(314,357)(315,358)(316,359)(317,360)(318,361)(319,362)(320,363)(321,364)(322,365)(323,366)(324,367)(325,368)(326,369)(327,370)(328,371)(329,372)(330,373)(331,374)(332,375)(333,376)(334,377)(335,378)(336,379), (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), (1,288,425,272)(2,372,426,104)(3,314,427,242)(4,342,428,74)(5,284,429,268)(6,368,430,100)(7,310,431,238)(8,338,432,70)(9,336,433,264)(10,364,434,96)(11,306,435,234)(12,390,436,66)(13,332,437,260)(14,360,438,92)(15,302,439,230)(16,386,440,62)(17,328,441,256)(18,356,442,88)(19,298,443,226)(20,382,444,58)(21,324,445,252)(22,352,446,84)(23,294,447,278)(24,378,448,110)(25,320,393,248)(26,348,394,80)(27,290,395,274)(28,374,396,106)(29,316,397,244)(30,344,398,76)(31,286,399,270)(32,370,400,102)(33,312,401,240)(34,340,402,72)(35,282,403,266)(36,366,404,98)(37,308,405,236)(38,392,406,68)(39,334,407,262)(40,362,408,94)(41,304,409,232)(42,388,410,64)(43,330,411,258)(44,358,412,90)(45,300,413,228)(46,384,414,60)(47,326,415,254)(48,354,416,86)(49,296,417,280)(50,380,418,112)(51,322,419,250)(52,350,420,82)(53,292,421,276)(54,376,422,108)(55,318,423,246)(56,346,424,78)(57,183,381,156)(59,209,383,126)(61,179,385,152)(63,205,387,122)(65,175,389,148)(67,201,391,118)(69,171,337,144)(71,197,339,114)(73,223,341,140)(75,193,343,166)(77,219,345,136)(79,189,347,162)(81,215,349,132)(83,185,351,158)(85,211,353,128)(87,181,355,154)(89,207,357,124)(91,177,359,150)(93,203,361,120)(95,173,363,146)(97,199,365,116)(99,169,367,142)(101,195,369,168)(103,221,371,138)(105,191,373,164)(107,217,375,134)(109,187,377,160)(111,213,379,130)(113,239,196,311)(115,265,198,281)(117,235,200,307)(119,261,202,333)(121,231,204,303)(123,257,206,329)(125,227,208,299)(127,253,210,325)(129,279,212,295)(131,249,214,321)(133,275,216,291)(135,245,218,317)(137,271,220,287)(139,241,222,313)(141,267,224,283)(143,237,170,309)(145,263,172,335)(147,233,174,305)(149,259,176,331)(151,229,178,301)(153,255,180,327)(155,225,182,297)(157,251,184,323)(159,277,186,293)(161,247,188,319)(163,273,190,289)(165,243,192,315)(167,269,194,285) );
G=PermutationGroup([[(1,205),(2,206),(3,207),(4,208),(5,209),(6,210),(7,211),(8,212),(9,213),(10,214),(11,215),(12,216),(13,217),(14,218),(15,219),(16,220),(17,221),(18,222),(19,223),(20,224),(21,169),(22,170),(23,171),(24,172),(25,173),(26,174),(27,175),(28,176),(29,177),(30,178),(31,179),(32,180),(33,181),(34,182),(35,183),(36,184),(37,185),(38,186),(39,187),(40,188),(41,189),(42,190),(43,191),(44,192),(45,193),(46,194),(47,195),(48,196),(49,197),(50,198),(51,199),(52,200),(53,201),(54,202),(55,203),(56,204),(57,266),(58,267),(59,268),(60,269),(61,270),(62,271),(63,272),(64,273),(65,274),(66,275),(67,276),(68,277),(69,278),(70,279),(71,280),(72,225),(73,226),(74,227),(75,228),(76,229),(77,230),(78,231),(79,232),(80,233),(81,234),(82,235),(83,236),(84,237),(85,238),(86,239),(87,240),(88,241),(89,242),(90,243),(91,244),(92,245),(93,246),(94,247),(95,248),(96,249),(97,250),(98,251),(99,252),(100,253),(101,254),(102,255),(103,256),(104,257),(105,258),(106,259),(107,260),(108,261),(109,262),(110,263),(111,264),(112,265),(113,416),(114,417),(115,418),(116,419),(117,420),(118,421),(119,422),(120,423),(121,424),(122,425),(123,426),(124,427),(125,428),(126,429),(127,430),(128,431),(129,432),(130,433),(131,434),(132,435),(133,436),(134,437),(135,438),(136,439),(137,440),(138,441),(139,442),(140,443),(141,444),(142,445),(143,446),(144,447),(145,448),(146,393),(147,394),(148,395),(149,396),(150,397),(151,398),(152,399),(153,400),(154,401),(155,402),(156,403),(157,404),(158,405),(159,406),(160,407),(161,408),(162,409),(163,410),(164,411),(165,412),(166,413),(167,414),(168,415),(281,380),(282,381),(283,382),(284,383),(285,384),(286,385),(287,386),(288,387),(289,388),(290,389),(291,390),(292,391),(293,392),(294,337),(295,338),(296,339),(297,340),(298,341),(299,342),(300,343),(301,344),(302,345),(303,346),(304,347),(305,348),(306,349),(307,350),(308,351),(309,352),(310,353),(311,354),(312,355),(313,356),(314,357),(315,358),(316,359),(317,360),(318,361),(319,362),(320,363),(321,364),(322,365),(323,366),(324,367),(325,368),(326,369),(327,370),(328,371),(329,372),(330,373),(331,374),(332,375),(333,376),(334,377),(335,378),(336,379)], [(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)], [(1,288,425,272),(2,372,426,104),(3,314,427,242),(4,342,428,74),(5,284,429,268),(6,368,430,100),(7,310,431,238),(8,338,432,70),(9,336,433,264),(10,364,434,96),(11,306,435,234),(12,390,436,66),(13,332,437,260),(14,360,438,92),(15,302,439,230),(16,386,440,62),(17,328,441,256),(18,356,442,88),(19,298,443,226),(20,382,444,58),(21,324,445,252),(22,352,446,84),(23,294,447,278),(24,378,448,110),(25,320,393,248),(26,348,394,80),(27,290,395,274),(28,374,396,106),(29,316,397,244),(30,344,398,76),(31,286,399,270),(32,370,400,102),(33,312,401,240),(34,340,402,72),(35,282,403,266),(36,366,404,98),(37,308,405,236),(38,392,406,68),(39,334,407,262),(40,362,408,94),(41,304,409,232),(42,388,410,64),(43,330,411,258),(44,358,412,90),(45,300,413,228),(46,384,414,60),(47,326,415,254),(48,354,416,86),(49,296,417,280),(50,380,418,112),(51,322,419,250),(52,350,420,82),(53,292,421,276),(54,376,422,108),(55,318,423,246),(56,346,424,78),(57,183,381,156),(59,209,383,126),(61,179,385,152),(63,205,387,122),(65,175,389,148),(67,201,391,118),(69,171,337,144),(71,197,339,114),(73,223,341,140),(75,193,343,166),(77,219,345,136),(79,189,347,162),(81,215,349,132),(83,185,351,158),(85,211,353,128),(87,181,355,154),(89,207,357,124),(91,177,359,150),(93,203,361,120),(95,173,363,146),(97,199,365,116),(99,169,367,142),(101,195,369,168),(103,221,371,138),(105,191,373,164),(107,217,375,134),(109,187,377,160),(111,213,379,130),(113,239,196,311),(115,265,198,281),(117,235,200,307),(119,261,202,333),(121,231,204,303),(123,257,206,329),(125,227,208,299),(127,253,210,325),(129,279,212,295),(131,249,214,321),(133,275,216,291),(135,245,218,317),(137,271,220,287),(139,241,222,313),(141,267,224,283),(143,237,170,309),(145,263,172,335),(147,233,174,305),(149,259,176,331),(151,229,178,301),(153,255,180,327),(155,225,182,297),(157,251,184,323),(159,277,186,293),(161,247,188,319),(163,273,190,289),(165,243,192,315),(167,269,194,285)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | Q8 | D7 | M4(2) | Dic7 | D14 | Dic14 | C4×D7 | D28 | C7⋊D4 | C4×D7 | C8×D7 | C8⋊D7 |
| kernel | (C2×C56)⋊5C4 | C22×C7⋊C8 | C2×C4×Dic7 | C22×C56 | C2×C7⋊C8 | C2×C56 | C22×Dic7 | C2×Dic7 | C2×C28 | C2×C28 | C22×C8 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 3 | 1 | 3 | 4 | 6 | 3 | 6 | 6 | 6 | 12 | 6 | 24 | 24 |
Matrix representation of (C2×C56)⋊5C4 ►in GL5(𝔽113)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 95 | 66 | 0 | 0 |
| 0 | 0 | 0 | 89 | 29 |
| 0 | 0 | 0 | 55 | 56 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 4 | 13 | 0 | 0 |
| 0 | 103 | 109 | 0 | 0 |
| 0 | 0 | 0 | 21 | 17 |
| 0 | 0 | 0 | 14 | 92 |
G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,0,95,0,0,0,18,66,0,0,0,0,0,89,55,0,0,0,29,56],[98,0,0,0,0,0,4,103,0,0,0,13,109,0,0,0,0,0,21,14,0,0,0,17,92] >;
(C2×C56)⋊5C4 in GAP, Magma, Sage, TeX
(C_2\times C_{56})\rtimes_5C_4 % in TeX
G:=Group("(C2xC56):5C4"); // GroupNames label
G:=SmallGroup(448,107);
// by ID
G=gap.SmallGroup(448,107);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^56=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^41>;
// generators/relations