metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.279D14, C28.27M4(2), C7⋊C8⋊3C8, (C4×C8).1D7, C7⋊1(C8⋊C8), (C4×C56).1C2, C14.6(C4×C8), (C2×C56).7C4, C4.19(C8×D7), C28.24(C2×C8), C2.3(C8×Dic7), (C2×C8).4Dic7, C14.1(C8⋊C4), C2.1(C56⋊C4), C4.12(C8⋊D7), (C2×C14).14C42, C4.9(C4.Dic7), (C4×C28).335C22, C22.14(C4×Dic7), C2.1(C42.D7), (C4×C7⋊C8).16C2, (C2×C7⋊C8).10C4, (C2×C4).164(C4×D7), (C2×C28).238(C2×C4), (C2×C4).88(C2×Dic7), SmallGroup(448,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.279D14
G = < a,b,c,d | a4=b4=1, c14=b-1, d2=a-1b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2b2c13 >
Subgroups: 164 in 66 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C4, C22, C7, C8, C2×C4, C14, C42, C2×C8, C2×C8, C28, C2×C14, C4×C8, C4×C8, C7⋊C8, C7⋊C8, C56, C2×C28, C8⋊C8, C2×C7⋊C8, C4×C28, C2×C56, C4×C7⋊C8, C4×C56, C42.279D14
Quotients: C1, C2, C4, C22, C8, C2×C4, D7, C42, C2×C8, M4(2), Dic7, D14, C4×C8, C8⋊C4, C4×D7, C2×Dic7, C8⋊C8, C8×D7, C8⋊D7, C4.Dic7, C4×Dic7, C42.D7, C8×Dic7, C56⋊C4, C42.279D14
►Regular action on 448 points
Generators in S448
(1 353 295 426)(2 354 296 427)(3 355 297 428)(4 356 298 429)(5 357 299 430)(6 358 300 431)(7 359 301 432)(8 360 302 433)(9 361 303 434)(10 362 304 435)(11 363 305 436)(12 364 306 437)(13 365 307 438)(14 366 308 439)(15 367 309 440)(16 368 310 441)(17 369 311 442)(18 370 312 443)(19 371 313 444)(20 372 314 445)(21 373 315 446)(22 374 316 447)(23 375 317 448)(24 376 318 393)(25 377 319 394)(26 378 320 395)(27 379 321 396)(28 380 322 397)(29 381 323 398)(30 382 324 399)(31 383 325 400)(32 384 326 401)(33 385 327 402)(34 386 328 403)(35 387 329 404)(36 388 330 405)(37 389 331 406)(38 390 332 407)(39 391 333 408)(40 392 334 409)(41 337 335 410)(42 338 336 411)(43 339 281 412)(44 340 282 413)(45 341 283 414)(46 342 284 415)(47 343 285 416)(48 344 286 417)(49 345 287 418)(50 346 288 419)(51 347 289 420)(52 348 290 421)(53 349 291 422)(54 350 292 423)(55 351 293 424)(56 352 294 425)(57 145 176 231)(58 146 177 232)(59 147 178 233)(60 148 179 234)(61 149 180 235)(62 150 181 236)(63 151 182 237)(64 152 183 238)(65 153 184 239)(66 154 185 240)(67 155 186 241)(68 156 187 242)(69 157 188 243)(70 158 189 244)(71 159 190 245)(72 160 191 246)(73 161 192 247)(74 162 193 248)(75 163 194 249)(76 164 195 250)(77 165 196 251)(78 166 197 252)(79 167 198 253)(80 168 199 254)(81 113 200 255)(82 114 201 256)(83 115 202 257)(84 116 203 258)(85 117 204 259)(86 118 205 260)(87 119 206 261)(88 120 207 262)(89 121 208 263)(90 122 209 264)(91 123 210 265)(92 124 211 266)(93 125 212 267)(94 126 213 268)(95 127 214 269)(96 128 215 270)(97 129 216 271)(98 130 217 272)(99 131 218 273)(100 132 219 274)(101 133 220 275)(102 134 221 276)(103 135 222 277)(104 136 223 278)(105 137 224 279)(106 138 169 280)(107 139 170 225)(108 140 171 226)(109 141 172 227)(110 142 173 228)(111 143 174 229)(112 144 175 230)
(1 43 29 15)(2 44 30 16)(3 45 31 17)(4 46 32 18)(5 47 33 19)(6 48 34 20)(7 49 35 21)(8 50 36 22)(9 51 37 23)(10 52 38 24)(11 53 39 25)(12 54 40 26)(13 55 41 27)(14 56 42 28)(57 99 85 71)(58 100 86 72)(59 101 87 73)(60 102 88 74)(61 103 89 75)(62 104 90 76)(63 105 91 77)(64 106 92 78)(65 107 93 79)(66 108 94 80)(67 109 95 81)(68 110 96 82)(69 111 97 83)(70 112 98 84)(113 155 141 127)(114 156 142 128)(115 157 143 129)(116 158 144 130)(117 159 145 131)(118 160 146 132)(119 161 147 133)(120 162 148 134)(121 163 149 135)(122 164 150 136)(123 165 151 137)(124 166 152 138)(125 167 153 139)(126 168 154 140)(169 211 197 183)(170 212 198 184)(171 213 199 185)(172 214 200 186)(173 215 201 187)(174 216 202 188)(175 217 203 189)(176 218 204 190)(177 219 205 191)(178 220 206 192)(179 221 207 193)(180 222 208 194)(181 223 209 195)(182 224 210 196)(225 267 253 239)(226 268 254 240)(227 269 255 241)(228 270 256 242)(229 271 257 243)(230 272 258 244)(231 273 259 245)(232 274 260 246)(233 275 261 247)(234 276 262 248)(235 277 263 249)(236 278 264 250)(237 279 265 251)(238 280 266 252)(281 323 309 295)(282 324 310 296)(283 325 311 297)(284 326 312 298)(285 327 313 299)(286 328 314 300)(287 329 315 301)(288 330 316 302)(289 331 317 303)(290 332 318 304)(291 333 319 305)(292 334 320 306)(293 335 321 307)(294 336 322 308)(337 379 365 351)(338 380 366 352)(339 381 367 353)(340 382 368 354)(341 383 369 355)(342 384 370 356)(343 385 371 357)(344 386 372 358)(345 387 373 359)(346 388 374 360)(347 389 375 361)(348 390 376 362)(349 391 377 363)(350 392 378 364)(393 435 421 407)(394 436 422 408)(395 437 423 409)(396 438 424 410)(397 439 425 411)(398 440 426 412)(399 441 427 413)(400 442 428 414)(401 443 429 415)(402 444 430 416)(403 445 431 417)(404 446 432 418)(405 447 433 419)(406 448 434 420)
(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 88 412 248 323 179 367 134)(2 192 413 147 324 101 368 261)(3 58 414 274 325 205 369 160)(4 218 415 117 326 71 370 231)(5 84 416 244 327 175 371 130)(6 188 417 143 328 97 372 257)(7 110 418 270 329 201 373 156)(8 214 419 113 330 67 374 227)(9 80 420 240 331 171 375 126)(10 184 421 139 332 93 376 253)(11 106 422 266 333 197 377 152)(12 210 423 165 334 63 378 279)(13 76 424 236 335 223 379 122)(14 180 425 135 336 89 380 249)(15 102 426 262 281 193 381 148)(16 206 427 161 282 59 382 275)(17 72 428 232 283 219 383 118)(18 176 429 131 284 85 384 245)(19 98 430 258 285 189 385 144)(20 202 431 157 286 111 386 271)(21 68 432 228 287 215 387 114)(22 172 433 127 288 81 388 241)(23 94 434 254 289 185 389 140)(24 198 435 153 290 107 390 267)(25 64 436 280 291 211 391 166)(26 224 437 123 292 77 392 237)(27 90 438 250 293 181 337 136)(28 194 439 149 294 103 338 263)(29 60 440 276 295 207 339 162)(30 220 441 119 296 73 340 233)(31 86 442 246 297 177 341 132)(32 190 443 145 298 99 342 259)(33 112 444 272 299 203 343 158)(34 216 445 115 300 69 344 229)(35 82 446 242 301 173 345 128)(36 186 447 141 302 95 346 255)(37 108 448 268 303 199 347 154)(38 212 393 167 304 65 348 225)(39 78 394 238 305 169 349 124)(40 182 395 137 306 91 350 251)(41 104 396 264 307 195 351 150)(42 208 397 163 308 61 352 277)(43 74 398 234 309 221 353 120)(44 178 399 133 310 87 354 247)(45 100 400 260 311 191 355 146)(46 204 401 159 312 57 356 273)(47 70 402 230 313 217 357 116)(48 174 403 129 314 83 358 243)(49 96 404 256 315 187 359 142)(50 200 405 155 316 109 360 269)(51 66 406 226 317 213 361 168)(52 170 407 125 318 79 362 239)(53 92 408 252 319 183 363 138)(54 196 409 151 320 105 364 265)(55 62 410 278 321 209 365 164)(56 222 411 121 322 75 366 235)
G:=sub<Sym(448)| (1,353,295,426)(2,354,296,427)(3,355,297,428)(4,356,298,429)(5,357,299,430)(6,358,300,431)(7,359,301,432)(8,360,302,433)(9,361,303,434)(10,362,304,435)(11,363,305,436)(12,364,306,437)(13,365,307,438)(14,366,308,439)(15,367,309,440)(16,368,310,441)(17,369,311,442)(18,370,312,443)(19,371,313,444)(20,372,314,445)(21,373,315,446)(22,374,316,447)(23,375,317,448)(24,376,318,393)(25,377,319,394)(26,378,320,395)(27,379,321,396)(28,380,322,397)(29,381,323,398)(30,382,324,399)(31,383,325,400)(32,384,326,401)(33,385,327,402)(34,386,328,403)(35,387,329,404)(36,388,330,405)(37,389,331,406)(38,390,332,407)(39,391,333,408)(40,392,334,409)(41,337,335,410)(42,338,336,411)(43,339,281,412)(44,340,282,413)(45,341,283,414)(46,342,284,415)(47,343,285,416)(48,344,286,417)(49,345,287,418)(50,346,288,419)(51,347,289,420)(52,348,290,421)(53,349,291,422)(54,350,292,423)(55,351,293,424)(56,352,294,425)(57,145,176,231)(58,146,177,232)(59,147,178,233)(60,148,179,234)(61,149,180,235)(62,150,181,236)(63,151,182,237)(64,152,183,238)(65,153,184,239)(66,154,185,240)(67,155,186,241)(68,156,187,242)(69,157,188,243)(70,158,189,244)(71,159,190,245)(72,160,191,246)(73,161,192,247)(74,162,193,248)(75,163,194,249)(76,164,195,250)(77,165,196,251)(78,166,197,252)(79,167,198,253)(80,168,199,254)(81,113,200,255)(82,114,201,256)(83,115,202,257)(84,116,203,258)(85,117,204,259)(86,118,205,260)(87,119,206,261)(88,120,207,262)(89,121,208,263)(90,122,209,264)(91,123,210,265)(92,124,211,266)(93,125,212,267)(94,126,213,268)(95,127,214,269)(96,128,215,270)(97,129,216,271)(98,130,217,272)(99,131,218,273)(100,132,219,274)(101,133,220,275)(102,134,221,276)(103,135,222,277)(104,136,223,278)(105,137,224,279)(106,138,169,280)(107,139,170,225)(108,140,171,226)(109,141,172,227)(110,142,173,228)(111,143,174,229)(112,144,175,230), (1,43,29,15)(2,44,30,16)(3,45,31,17)(4,46,32,18)(5,47,33,19)(6,48,34,20)(7,49,35,21)(8,50,36,22)(9,51,37,23)(10,52,38,24)(11,53,39,25)(12,54,40,26)(13,55,41,27)(14,56,42,28)(57,99,85,71)(58,100,86,72)(59,101,87,73)(60,102,88,74)(61,103,89,75)(62,104,90,76)(63,105,91,77)(64,106,92,78)(65,107,93,79)(66,108,94,80)(67,109,95,81)(68,110,96,82)(69,111,97,83)(70,112,98,84)(113,155,141,127)(114,156,142,128)(115,157,143,129)(116,158,144,130)(117,159,145,131)(118,160,146,132)(119,161,147,133)(120,162,148,134)(121,163,149,135)(122,164,150,136)(123,165,151,137)(124,166,152,138)(125,167,153,139)(126,168,154,140)(169,211,197,183)(170,212,198,184)(171,213,199,185)(172,214,200,186)(173,215,201,187)(174,216,202,188)(175,217,203,189)(176,218,204,190)(177,219,205,191)(178,220,206,192)(179,221,207,193)(180,222,208,194)(181,223,209,195)(182,224,210,196)(225,267,253,239)(226,268,254,240)(227,269,255,241)(228,270,256,242)(229,271,257,243)(230,272,258,244)(231,273,259,245)(232,274,260,246)(233,275,261,247)(234,276,262,248)(235,277,263,249)(236,278,264,250)(237,279,265,251)(238,280,266,252)(281,323,309,295)(282,324,310,296)(283,325,311,297)(284,326,312,298)(285,327,313,299)(286,328,314,300)(287,329,315,301)(288,330,316,302)(289,331,317,303)(290,332,318,304)(291,333,319,305)(292,334,320,306)(293,335,321,307)(294,336,322,308)(337,379,365,351)(338,380,366,352)(339,381,367,353)(340,382,368,354)(341,383,369,355)(342,384,370,356)(343,385,371,357)(344,386,372,358)(345,387,373,359)(346,388,374,360)(347,389,375,361)(348,390,376,362)(349,391,377,363)(350,392,378,364)(393,435,421,407)(394,436,422,408)(395,437,423,409)(396,438,424,410)(397,439,425,411)(398,440,426,412)(399,441,427,413)(400,442,428,414)(401,443,429,415)(402,444,430,416)(403,445,431,417)(404,446,432,418)(405,447,433,419)(406,448,434,420), (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), (1,88,412,248,323,179,367,134)(2,192,413,147,324,101,368,261)(3,58,414,274,325,205,369,160)(4,218,415,117,326,71,370,231)(5,84,416,244,327,175,371,130)(6,188,417,143,328,97,372,257)(7,110,418,270,329,201,373,156)(8,214,419,113,330,67,374,227)(9,80,420,240,331,171,375,126)(10,184,421,139,332,93,376,253)(11,106,422,266,333,197,377,152)(12,210,423,165,334,63,378,279)(13,76,424,236,335,223,379,122)(14,180,425,135,336,89,380,249)(15,102,426,262,281,193,381,148)(16,206,427,161,282,59,382,275)(17,72,428,232,283,219,383,118)(18,176,429,131,284,85,384,245)(19,98,430,258,285,189,385,144)(20,202,431,157,286,111,386,271)(21,68,432,228,287,215,387,114)(22,172,433,127,288,81,388,241)(23,94,434,254,289,185,389,140)(24,198,435,153,290,107,390,267)(25,64,436,280,291,211,391,166)(26,224,437,123,292,77,392,237)(27,90,438,250,293,181,337,136)(28,194,439,149,294,103,338,263)(29,60,440,276,295,207,339,162)(30,220,441,119,296,73,340,233)(31,86,442,246,297,177,341,132)(32,190,443,145,298,99,342,259)(33,112,444,272,299,203,343,158)(34,216,445,115,300,69,344,229)(35,82,446,242,301,173,345,128)(36,186,447,141,302,95,346,255)(37,108,448,268,303,199,347,154)(38,212,393,167,304,65,348,225)(39,78,394,238,305,169,349,124)(40,182,395,137,306,91,350,251)(41,104,396,264,307,195,351,150)(42,208,397,163,308,61,352,277)(43,74,398,234,309,221,353,120)(44,178,399,133,310,87,354,247)(45,100,400,260,311,191,355,146)(46,204,401,159,312,57,356,273)(47,70,402,230,313,217,357,116)(48,174,403,129,314,83,358,243)(49,96,404,256,315,187,359,142)(50,200,405,155,316,109,360,269)(51,66,406,226,317,213,361,168)(52,170,407,125,318,79,362,239)(53,92,408,252,319,183,363,138)(54,196,409,151,320,105,364,265)(55,62,410,278,321,209,365,164)(56,222,411,121,322,75,366,235)>;
G:=Group( (1,353,295,426)(2,354,296,427)(3,355,297,428)(4,356,298,429)(5,357,299,430)(6,358,300,431)(7,359,301,432)(8,360,302,433)(9,361,303,434)(10,362,304,435)(11,363,305,436)(12,364,306,437)(13,365,307,438)(14,366,308,439)(15,367,309,440)(16,368,310,441)(17,369,311,442)(18,370,312,443)(19,371,313,444)(20,372,314,445)(21,373,315,446)(22,374,316,447)(23,375,317,448)(24,376,318,393)(25,377,319,394)(26,378,320,395)(27,379,321,396)(28,380,322,397)(29,381,323,398)(30,382,324,399)(31,383,325,400)(32,384,326,401)(33,385,327,402)(34,386,328,403)(35,387,329,404)(36,388,330,405)(37,389,331,406)(38,390,332,407)(39,391,333,408)(40,392,334,409)(41,337,335,410)(42,338,336,411)(43,339,281,412)(44,340,282,413)(45,341,283,414)(46,342,284,415)(47,343,285,416)(48,344,286,417)(49,345,287,418)(50,346,288,419)(51,347,289,420)(52,348,290,421)(53,349,291,422)(54,350,292,423)(55,351,293,424)(56,352,294,425)(57,145,176,231)(58,146,177,232)(59,147,178,233)(60,148,179,234)(61,149,180,235)(62,150,181,236)(63,151,182,237)(64,152,183,238)(65,153,184,239)(66,154,185,240)(67,155,186,241)(68,156,187,242)(69,157,188,243)(70,158,189,244)(71,159,190,245)(72,160,191,246)(73,161,192,247)(74,162,193,248)(75,163,194,249)(76,164,195,250)(77,165,196,251)(78,166,197,252)(79,167,198,253)(80,168,199,254)(81,113,200,255)(82,114,201,256)(83,115,202,257)(84,116,203,258)(85,117,204,259)(86,118,205,260)(87,119,206,261)(88,120,207,262)(89,121,208,263)(90,122,209,264)(91,123,210,265)(92,124,211,266)(93,125,212,267)(94,126,213,268)(95,127,214,269)(96,128,215,270)(97,129,216,271)(98,130,217,272)(99,131,218,273)(100,132,219,274)(101,133,220,275)(102,134,221,276)(103,135,222,277)(104,136,223,278)(105,137,224,279)(106,138,169,280)(107,139,170,225)(108,140,171,226)(109,141,172,227)(110,142,173,228)(111,143,174,229)(112,144,175,230), (1,43,29,15)(2,44,30,16)(3,45,31,17)(4,46,32,18)(5,47,33,19)(6,48,34,20)(7,49,35,21)(8,50,36,22)(9,51,37,23)(10,52,38,24)(11,53,39,25)(12,54,40,26)(13,55,41,27)(14,56,42,28)(57,99,85,71)(58,100,86,72)(59,101,87,73)(60,102,88,74)(61,103,89,75)(62,104,90,76)(63,105,91,77)(64,106,92,78)(65,107,93,79)(66,108,94,80)(67,109,95,81)(68,110,96,82)(69,111,97,83)(70,112,98,84)(113,155,141,127)(114,156,142,128)(115,157,143,129)(116,158,144,130)(117,159,145,131)(118,160,146,132)(119,161,147,133)(120,162,148,134)(121,163,149,135)(122,164,150,136)(123,165,151,137)(124,166,152,138)(125,167,153,139)(126,168,154,140)(169,211,197,183)(170,212,198,184)(171,213,199,185)(172,214,200,186)(173,215,201,187)(174,216,202,188)(175,217,203,189)(176,218,204,190)(177,219,205,191)(178,220,206,192)(179,221,207,193)(180,222,208,194)(181,223,209,195)(182,224,210,196)(225,267,253,239)(226,268,254,240)(227,269,255,241)(228,270,256,242)(229,271,257,243)(230,272,258,244)(231,273,259,245)(232,274,260,246)(233,275,261,247)(234,276,262,248)(235,277,263,249)(236,278,264,250)(237,279,265,251)(238,280,266,252)(281,323,309,295)(282,324,310,296)(283,325,311,297)(284,326,312,298)(285,327,313,299)(286,328,314,300)(287,329,315,301)(288,330,316,302)(289,331,317,303)(290,332,318,304)(291,333,319,305)(292,334,320,306)(293,335,321,307)(294,336,322,308)(337,379,365,351)(338,380,366,352)(339,381,367,353)(340,382,368,354)(341,383,369,355)(342,384,370,356)(343,385,371,357)(344,386,372,358)(345,387,373,359)(346,388,374,360)(347,389,375,361)(348,390,376,362)(349,391,377,363)(350,392,378,364)(393,435,421,407)(394,436,422,408)(395,437,423,409)(396,438,424,410)(397,439,425,411)(398,440,426,412)(399,441,427,413)(400,442,428,414)(401,443,429,415)(402,444,430,416)(403,445,431,417)(404,446,432,418)(405,447,433,419)(406,448,434,420), (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), (1,88,412,248,323,179,367,134)(2,192,413,147,324,101,368,261)(3,58,414,274,325,205,369,160)(4,218,415,117,326,71,370,231)(5,84,416,244,327,175,371,130)(6,188,417,143,328,97,372,257)(7,110,418,270,329,201,373,156)(8,214,419,113,330,67,374,227)(9,80,420,240,331,171,375,126)(10,184,421,139,332,93,376,253)(11,106,422,266,333,197,377,152)(12,210,423,165,334,63,378,279)(13,76,424,236,335,223,379,122)(14,180,425,135,336,89,380,249)(15,102,426,262,281,193,381,148)(16,206,427,161,282,59,382,275)(17,72,428,232,283,219,383,118)(18,176,429,131,284,85,384,245)(19,98,430,258,285,189,385,144)(20,202,431,157,286,111,386,271)(21,68,432,228,287,215,387,114)(22,172,433,127,288,81,388,241)(23,94,434,254,289,185,389,140)(24,198,435,153,290,107,390,267)(25,64,436,280,291,211,391,166)(26,224,437,123,292,77,392,237)(27,90,438,250,293,181,337,136)(28,194,439,149,294,103,338,263)(29,60,440,276,295,207,339,162)(30,220,441,119,296,73,340,233)(31,86,442,246,297,177,341,132)(32,190,443,145,298,99,342,259)(33,112,444,272,299,203,343,158)(34,216,445,115,300,69,344,229)(35,82,446,242,301,173,345,128)(36,186,447,141,302,95,346,255)(37,108,448,268,303,199,347,154)(38,212,393,167,304,65,348,225)(39,78,394,238,305,169,349,124)(40,182,395,137,306,91,350,251)(41,104,396,264,307,195,351,150)(42,208,397,163,308,61,352,277)(43,74,398,234,309,221,353,120)(44,178,399,133,310,87,354,247)(45,100,400,260,311,191,355,146)(46,204,401,159,312,57,356,273)(47,70,402,230,313,217,357,116)(48,174,403,129,314,83,358,243)(49,96,404,256,315,187,359,142)(50,200,405,155,316,109,360,269)(51,66,406,226,317,213,361,168)(52,170,407,125,318,79,362,239)(53,92,408,252,319,183,363,138)(54,196,409,151,320,105,364,265)(55,62,410,278,321,209,365,164)(56,222,411,121,322,75,366,235) );
G=PermutationGroup([[(1,353,295,426),(2,354,296,427),(3,355,297,428),(4,356,298,429),(5,357,299,430),(6,358,300,431),(7,359,301,432),(8,360,302,433),(9,361,303,434),(10,362,304,435),(11,363,305,436),(12,364,306,437),(13,365,307,438),(14,366,308,439),(15,367,309,440),(16,368,310,441),(17,369,311,442),(18,370,312,443),(19,371,313,444),(20,372,314,445),(21,373,315,446),(22,374,316,447),(23,375,317,448),(24,376,318,393),(25,377,319,394),(26,378,320,395),(27,379,321,396),(28,380,322,397),(29,381,323,398),(30,382,324,399),(31,383,325,400),(32,384,326,401),(33,385,327,402),(34,386,328,403),(35,387,329,404),(36,388,330,405),(37,389,331,406),(38,390,332,407),(39,391,333,408),(40,392,334,409),(41,337,335,410),(42,338,336,411),(43,339,281,412),(44,340,282,413),(45,341,283,414),(46,342,284,415),(47,343,285,416),(48,344,286,417),(49,345,287,418),(50,346,288,419),(51,347,289,420),(52,348,290,421),(53,349,291,422),(54,350,292,423),(55,351,293,424),(56,352,294,425),(57,145,176,231),(58,146,177,232),(59,147,178,233),(60,148,179,234),(61,149,180,235),(62,150,181,236),(63,151,182,237),(64,152,183,238),(65,153,184,239),(66,154,185,240),(67,155,186,241),(68,156,187,242),(69,157,188,243),(70,158,189,244),(71,159,190,245),(72,160,191,246),(73,161,192,247),(74,162,193,248),(75,163,194,249),(76,164,195,250),(77,165,196,251),(78,166,197,252),(79,167,198,253),(80,168,199,254),(81,113,200,255),(82,114,201,256),(83,115,202,257),(84,116,203,258),(85,117,204,259),(86,118,205,260),(87,119,206,261),(88,120,207,262),(89,121,208,263),(90,122,209,264),(91,123,210,265),(92,124,211,266),(93,125,212,267),(94,126,213,268),(95,127,214,269),(96,128,215,270),(97,129,216,271),(98,130,217,272),(99,131,218,273),(100,132,219,274),(101,133,220,275),(102,134,221,276),(103,135,222,277),(104,136,223,278),(105,137,224,279),(106,138,169,280),(107,139,170,225),(108,140,171,226),(109,141,172,227),(110,142,173,228),(111,143,174,229),(112,144,175,230)], [(1,43,29,15),(2,44,30,16),(3,45,31,17),(4,46,32,18),(5,47,33,19),(6,48,34,20),(7,49,35,21),(8,50,36,22),(9,51,37,23),(10,52,38,24),(11,53,39,25),(12,54,40,26),(13,55,41,27),(14,56,42,28),(57,99,85,71),(58,100,86,72),(59,101,87,73),(60,102,88,74),(61,103,89,75),(62,104,90,76),(63,105,91,77),(64,106,92,78),(65,107,93,79),(66,108,94,80),(67,109,95,81),(68,110,96,82),(69,111,97,83),(70,112,98,84),(113,155,141,127),(114,156,142,128),(115,157,143,129),(116,158,144,130),(117,159,145,131),(118,160,146,132),(119,161,147,133),(120,162,148,134),(121,163,149,135),(122,164,150,136),(123,165,151,137),(124,166,152,138),(125,167,153,139),(126,168,154,140),(169,211,197,183),(170,212,198,184),(171,213,199,185),(172,214,200,186),(173,215,201,187),(174,216,202,188),(175,217,203,189),(176,218,204,190),(177,219,205,191),(178,220,206,192),(179,221,207,193),(180,222,208,194),(181,223,209,195),(182,224,210,196),(225,267,253,239),(226,268,254,240),(227,269,255,241),(228,270,256,242),(229,271,257,243),(230,272,258,244),(231,273,259,245),(232,274,260,246),(233,275,261,247),(234,276,262,248),(235,277,263,249),(236,278,264,250),(237,279,265,251),(238,280,266,252),(281,323,309,295),(282,324,310,296),(283,325,311,297),(284,326,312,298),(285,327,313,299),(286,328,314,300),(287,329,315,301),(288,330,316,302),(289,331,317,303),(290,332,318,304),(291,333,319,305),(292,334,320,306),(293,335,321,307),(294,336,322,308),(337,379,365,351),(338,380,366,352),(339,381,367,353),(340,382,368,354),(341,383,369,355),(342,384,370,356),(343,385,371,357),(344,386,372,358),(345,387,373,359),(346,388,374,360),(347,389,375,361),(348,390,376,362),(349,391,377,363),(350,392,378,364),(393,435,421,407),(394,436,422,408),(395,437,423,409),(396,438,424,410),(397,439,425,411),(398,440,426,412),(399,441,427,413),(400,442,428,414),(401,443,429,415),(402,444,430,416),(403,445,431,417),(404,446,432,418),(405,447,433,419),(406,448,434,420)], [(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)], [(1,88,412,248,323,179,367,134),(2,192,413,147,324,101,368,261),(3,58,414,274,325,205,369,160),(4,218,415,117,326,71,370,231),(5,84,416,244,327,175,371,130),(6,188,417,143,328,97,372,257),(7,110,418,270,329,201,373,156),(8,214,419,113,330,67,374,227),(9,80,420,240,331,171,375,126),(10,184,421,139,332,93,376,253),(11,106,422,266,333,197,377,152),(12,210,423,165,334,63,378,279),(13,76,424,236,335,223,379,122),(14,180,425,135,336,89,380,249),(15,102,426,262,281,193,381,148),(16,206,427,161,282,59,382,275),(17,72,428,232,283,219,383,118),(18,176,429,131,284,85,384,245),(19,98,430,258,285,189,385,144),(20,202,431,157,286,111,386,271),(21,68,432,228,287,215,387,114),(22,172,433,127,288,81,388,241),(23,94,434,254,289,185,389,140),(24,198,435,153,290,107,390,267),(25,64,436,280,291,211,391,166),(26,224,437,123,292,77,392,237),(27,90,438,250,293,181,337,136),(28,194,439,149,294,103,338,263),(29,60,440,276,295,207,339,162),(30,220,441,119,296,73,340,233),(31,86,442,246,297,177,341,132),(32,190,443,145,298,99,342,259),(33,112,444,272,299,203,343,158),(34,216,445,115,300,69,344,229),(35,82,446,242,301,173,345,128),(36,186,447,141,302,95,346,255),(37,108,448,268,303,199,347,154),(38,212,393,167,304,65,348,225),(39,78,394,238,305,169,349,124),(40,182,395,137,306,91,350,251),(41,104,396,264,307,195,351,150),(42,208,397,163,308,61,352,277),(43,74,398,234,309,221,353,120),(44,178,399,133,310,87,354,247),(45,100,400,260,311,191,355,146),(46,204,401,159,312,57,356,273),(47,70,402,230,313,217,357,116),(48,174,403,129,314,83,358,243),(49,96,404,256,315,187,359,142),(50,200,405,155,316,109,360,269),(51,66,406,226,317,213,361,168),(52,170,407,125,318,79,362,239),(53,92,408,252,319,183,363,138),(54,196,409,151,320,105,364,265),(55,62,410,278,321,209,365,164),(56,222,411,121,322,75,366,235)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8X | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D7 | M4(2) | D14 | Dic7 | C4×D7 | C8×D7 | C8⋊D7 | C4.Dic7 |
| kernel | C42.279D14 | C4×C7⋊C8 | C4×C56 | C2×C7⋊C8 | C2×C56 | C7⋊C8 | C4×C8 | C28 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 3 | 8 | 3 | 6 | 12 | 24 | 24 | 24 |
Matrix representation of C42.279D14 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 98 | 0 |
| 0 | 0 | 98 |
| 15 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 15 |
| 44 | 0 | 0 |
| 0 | 30 | 75 |
| 0 | 38 | 38 |
| 44 | 0 | 0 |
| 0 | 100 | 10 |
| 0 | 96 | 13 |
G:=sub<GL(3,GF(113))| [1,0,0,0,98,0,0,0,98],[15,0,0,0,15,0,0,0,15],[44,0,0,0,30,38,0,75,38],[44,0,0,0,100,96,0,10,13] >;
C42.279D14 in GAP, Magma, Sage, TeX
C_4^2._{279}D_{14} % in TeX
G:=Group("C4^2.279D14"); // GroupNames label
G:=SmallGroup(448,11);
// by ID
G=gap.SmallGroup(448,11);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,477,64,184,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^-1,d^2=a^-1*b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^13>;
// generators/relations