metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.14D14, (C2×C4).24D28, C8⋊C4.6D7, (C2×C28).35D4, (C2×C8).155D14, (C4×C28).2C22, C28⋊2Q8.6C2, C22.96(C2×D28), C4⋊Dic7.8C22, C28.6Q8.3C2, C28.221(C4○D4), C4.105(C4○D28), (C2×C56).310C22, (C2×C28).730C23, C14.7(C4.4D4), C2.7(C8.D14), C14.2(C8.C22), C28.44D4.15C2, C2.12(C4.D28), (C2×Dic14).6C22, C7⋊1(C42.30C22), (C7×C8⋊C4).10C2, (C2×C14).113(C2×D4), (C2×C4).674(C22×D7), SmallGroup(448,237)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.14D14
G = < a,b,c,d | a4=b4=1, c14=a2b, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b-1c13 >
Subgroups: 452 in 90 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C42.30C22, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28.44D4, C7×C8⋊C4, C28⋊2Q8, C28.6Q8, C42.14D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C8.C22, D28, C22×D7, C42.30C22, C2×D28, C4○D28, C4.D28, C8.D14, C42.14D14
►Regular action on 448 points
Generators in S448
(1 188 263 351)(2 217 264 380)(3 190 265 353)(4 219 266 382)(5 192 267 355)(6 221 268 384)(7 194 269 357)(8 223 270 386)(9 196 271 359)(10 169 272 388)(11 198 273 361)(12 171 274 390)(13 200 275 363)(14 173 276 392)(15 202 277 365)(16 175 278 338)(17 204 279 367)(18 177 280 340)(19 206 225 369)(20 179 226 342)(21 208 227 371)(22 181 228 344)(23 210 229 373)(24 183 230 346)(25 212 231 375)(26 185 232 348)(27 214 233 377)(28 187 234 350)(29 216 235 379)(30 189 236 352)(31 218 237 381)(32 191 238 354)(33 220 239 383)(34 193 240 356)(35 222 241 385)(36 195 242 358)(37 224 243 387)(38 197 244 360)(39 170 245 389)(40 199 246 362)(41 172 247 391)(42 201 248 364)(43 174 249 337)(44 203 250 366)(45 176 251 339)(46 205 252 368)(47 178 253 341)(48 207 254 370)(49 180 255 343)(50 209 256 372)(51 182 257 345)(52 211 258 374)(53 184 259 347)(54 213 260 376)(55 186 261 349)(56 215 262 378)(57 283 429 161)(58 312 430 134)(59 285 431 163)(60 314 432 136)(61 287 433 165)(62 316 434 138)(63 289 435 167)(64 318 436 140)(65 291 437 113)(66 320 438 142)(67 293 439 115)(68 322 440 144)(69 295 441 117)(70 324 442 146)(71 297 443 119)(72 326 444 148)(73 299 445 121)(74 328 446 150)(75 301 447 123)(76 330 448 152)(77 303 393 125)(78 332 394 154)(79 305 395 127)(80 334 396 156)(81 307 397 129)(82 336 398 158)(83 309 399 131)(84 282 400 160)(85 311 401 133)(86 284 402 162)(87 313 403 135)(88 286 404 164)(89 315 405 137)(90 288 406 166)(91 317 407 139)(92 290 408 168)(93 319 409 141)(94 292 410 114)(95 321 411 143)(96 294 412 116)(97 323 413 145)(98 296 414 118)(99 325 415 147)(100 298 416 120)(101 327 417 149)(102 300 418 122)(103 329 419 151)(104 302 420 124)(105 331 421 153)(106 304 422 126)(107 333 423 155)(108 306 424 128)(109 335 425 157)(110 308 426 130)(111 281 427 159)(112 310 428 132)
(1 277 29 249)(2 278 30 250)(3 279 31 251)(4 280 32 252)(5 225 33 253)(6 226 34 254)(7 227 35 255)(8 228 36 256)(9 229 37 257)(10 230 38 258)(11 231 39 259)(12 232 40 260)(13 233 41 261)(14 234 42 262)(15 235 43 263)(16 236 44 264)(17 237 45 265)(18 238 46 266)(19 239 47 267)(20 240 48 268)(21 241 49 269)(22 242 50 270)(23 243 51 271)(24 244 52 272)(25 245 53 273)(26 246 54 274)(27 247 55 275)(28 248 56 276)(57 443 85 415)(58 444 86 416)(59 445 87 417)(60 446 88 418)(61 447 89 419)(62 448 90 420)(63 393 91 421)(64 394 92 422)(65 395 93 423)(66 396 94 424)(67 397 95 425)(68 398 96 426)(69 399 97 427)(70 400 98 428)(71 401 99 429)(72 402 100 430)(73 403 101 431)(74 404 102 432)(75 405 103 433)(76 406 104 434)(77 407 105 435)(78 408 106 436)(79 409 107 437)(80 410 108 438)(81 411 109 439)(82 412 110 440)(83 413 111 441)(84 414 112 442)(113 305 141 333)(114 306 142 334)(115 307 143 335)(116 308 144 336)(117 309 145 281)(118 310 146 282)(119 311 147 283)(120 312 148 284)(121 313 149 285)(122 314 150 286)(123 315 151 287)(124 316 152 288)(125 317 153 289)(126 318 154 290)(127 319 155 291)(128 320 156 292)(129 321 157 293)(130 322 158 294)(131 323 159 295)(132 324 160 296)(133 325 161 297)(134 326 162 298)(135 327 163 299)(136 328 164 300)(137 329 165 301)(138 330 166 302)(139 331 167 303)(140 332 168 304)(169 346 197 374)(170 347 198 375)(171 348 199 376)(172 349 200 377)(173 350 201 378)(174 351 202 379)(175 352 203 380)(176 353 204 381)(177 354 205 382)(178 355 206 383)(179 356 207 384)(180 357 208 385)(181 358 209 386)(182 359 210 387)(183 360 211 388)(184 361 212 389)(185 362 213 390)(186 363 214 391)(187 364 215 392)(188 365 216 337)(189 366 217 338)(190 367 218 339)(191 368 219 340)(192 369 220 341)(193 370 221 342)(194 371 222 343)(195 372 223 344)(196 373 224 345)
(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)
(1 305 29 333)(2 126 30 154)(3 303 31 331)(4 124 32 152)(5 301 33 329)(6 122 34 150)(7 299 35 327)(8 120 36 148)(9 297 37 325)(10 118 38 146)(11 295 39 323)(12 116 40 144)(13 293 41 321)(14 114 42 142)(15 291 43 319)(16 168 44 140)(17 289 45 317)(18 166 46 138)(19 287 47 315)(20 164 48 136)(21 285 49 313)(22 162 50 134)(23 283 51 311)(24 160 52 132)(25 281 53 309)(26 158 54 130)(27 335 55 307)(28 156 56 128)(57 182 85 210)(58 344 86 372)(59 180 87 208)(60 342 88 370)(61 178 89 206)(62 340 90 368)(63 176 91 204)(64 338 92 366)(65 174 93 202)(66 392 94 364)(67 172 95 200)(68 390 96 362)(69 170 97 198)(70 388 98 360)(71 224 99 196)(72 386 100 358)(73 222 101 194)(74 384 102 356)(75 220 103 192)(76 382 104 354)(77 218 105 190)(78 380 106 352)(79 216 107 188)(80 378 108 350)(81 214 109 186)(82 376 110 348)(83 212 111 184)(84 374 112 346)(113 249 141 277)(115 247 143 275)(117 245 145 273)(119 243 147 271)(121 241 149 269)(123 239 151 267)(125 237 153 265)(127 235 155 263)(129 233 157 261)(131 231 159 259)(133 229 161 257)(135 227 163 255)(137 225 165 253)(139 279 167 251)(169 414 197 442)(171 412 199 440)(173 410 201 438)(175 408 203 436)(177 406 205 434)(179 404 207 432)(181 402 209 430)(183 400 211 428)(185 398 213 426)(187 396 215 424)(189 394 217 422)(191 448 219 420)(193 446 221 418)(195 444 223 416)(226 286 254 314)(228 284 256 312)(230 282 258 310)(232 336 260 308)(234 334 262 306)(236 332 264 304)(238 330 266 302)(240 328 268 300)(242 326 270 298)(244 324 272 296)(246 322 274 294)(248 320 276 292)(250 318 278 290)(252 316 280 288)(337 409 365 437)(339 407 367 435)(341 405 369 433)(343 403 371 431)(345 401 373 429)(347 399 375 427)(349 397 377 425)(351 395 379 423)(353 393 381 421)(355 447 383 419)(357 445 385 417)(359 443 387 415)(361 441 389 413)(363 439 391 411)
G:=sub<Sym(448)| (1,188,263,351)(2,217,264,380)(3,190,265,353)(4,219,266,382)(5,192,267,355)(6,221,268,384)(7,194,269,357)(8,223,270,386)(9,196,271,359)(10,169,272,388)(11,198,273,361)(12,171,274,390)(13,200,275,363)(14,173,276,392)(15,202,277,365)(16,175,278,338)(17,204,279,367)(18,177,280,340)(19,206,225,369)(20,179,226,342)(21,208,227,371)(22,181,228,344)(23,210,229,373)(24,183,230,346)(25,212,231,375)(26,185,232,348)(27,214,233,377)(28,187,234,350)(29,216,235,379)(30,189,236,352)(31,218,237,381)(32,191,238,354)(33,220,239,383)(34,193,240,356)(35,222,241,385)(36,195,242,358)(37,224,243,387)(38,197,244,360)(39,170,245,389)(40,199,246,362)(41,172,247,391)(42,201,248,364)(43,174,249,337)(44,203,250,366)(45,176,251,339)(46,205,252,368)(47,178,253,341)(48,207,254,370)(49,180,255,343)(50,209,256,372)(51,182,257,345)(52,211,258,374)(53,184,259,347)(54,213,260,376)(55,186,261,349)(56,215,262,378)(57,283,429,161)(58,312,430,134)(59,285,431,163)(60,314,432,136)(61,287,433,165)(62,316,434,138)(63,289,435,167)(64,318,436,140)(65,291,437,113)(66,320,438,142)(67,293,439,115)(68,322,440,144)(69,295,441,117)(70,324,442,146)(71,297,443,119)(72,326,444,148)(73,299,445,121)(74,328,446,150)(75,301,447,123)(76,330,448,152)(77,303,393,125)(78,332,394,154)(79,305,395,127)(80,334,396,156)(81,307,397,129)(82,336,398,158)(83,309,399,131)(84,282,400,160)(85,311,401,133)(86,284,402,162)(87,313,403,135)(88,286,404,164)(89,315,405,137)(90,288,406,166)(91,317,407,139)(92,290,408,168)(93,319,409,141)(94,292,410,114)(95,321,411,143)(96,294,412,116)(97,323,413,145)(98,296,414,118)(99,325,415,147)(100,298,416,120)(101,327,417,149)(102,300,418,122)(103,329,419,151)(104,302,420,124)(105,331,421,153)(106,304,422,126)(107,333,423,155)(108,306,424,128)(109,335,425,157)(110,308,426,130)(111,281,427,159)(112,310,428,132), (1,277,29,249)(2,278,30,250)(3,279,31,251)(4,280,32,252)(5,225,33,253)(6,226,34,254)(7,227,35,255)(8,228,36,256)(9,229,37,257)(10,230,38,258)(11,231,39,259)(12,232,40,260)(13,233,41,261)(14,234,42,262)(15,235,43,263)(16,236,44,264)(17,237,45,265)(18,238,46,266)(19,239,47,267)(20,240,48,268)(21,241,49,269)(22,242,50,270)(23,243,51,271)(24,244,52,272)(25,245,53,273)(26,246,54,274)(27,247,55,275)(28,248,56,276)(57,443,85,415)(58,444,86,416)(59,445,87,417)(60,446,88,418)(61,447,89,419)(62,448,90,420)(63,393,91,421)(64,394,92,422)(65,395,93,423)(66,396,94,424)(67,397,95,425)(68,398,96,426)(69,399,97,427)(70,400,98,428)(71,401,99,429)(72,402,100,430)(73,403,101,431)(74,404,102,432)(75,405,103,433)(76,406,104,434)(77,407,105,435)(78,408,106,436)(79,409,107,437)(80,410,108,438)(81,411,109,439)(82,412,110,440)(83,413,111,441)(84,414,112,442)(113,305,141,333)(114,306,142,334)(115,307,143,335)(116,308,144,336)(117,309,145,281)(118,310,146,282)(119,311,147,283)(120,312,148,284)(121,313,149,285)(122,314,150,286)(123,315,151,287)(124,316,152,288)(125,317,153,289)(126,318,154,290)(127,319,155,291)(128,320,156,292)(129,321,157,293)(130,322,158,294)(131,323,159,295)(132,324,160,296)(133,325,161,297)(134,326,162,298)(135,327,163,299)(136,328,164,300)(137,329,165,301)(138,330,166,302)(139,331,167,303)(140,332,168,304)(169,346,197,374)(170,347,198,375)(171,348,199,376)(172,349,200,377)(173,350,201,378)(174,351,202,379)(175,352,203,380)(176,353,204,381)(177,354,205,382)(178,355,206,383)(179,356,207,384)(180,357,208,385)(181,358,209,386)(182,359,210,387)(183,360,211,388)(184,361,212,389)(185,362,213,390)(186,363,214,391)(187,364,215,392)(188,365,216,337)(189,366,217,338)(190,367,218,339)(191,368,219,340)(192,369,220,341)(193,370,221,342)(194,371,222,343)(195,372,223,344)(196,373,224,345), (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), (1,305,29,333)(2,126,30,154)(3,303,31,331)(4,124,32,152)(5,301,33,329)(6,122,34,150)(7,299,35,327)(8,120,36,148)(9,297,37,325)(10,118,38,146)(11,295,39,323)(12,116,40,144)(13,293,41,321)(14,114,42,142)(15,291,43,319)(16,168,44,140)(17,289,45,317)(18,166,46,138)(19,287,47,315)(20,164,48,136)(21,285,49,313)(22,162,50,134)(23,283,51,311)(24,160,52,132)(25,281,53,309)(26,158,54,130)(27,335,55,307)(28,156,56,128)(57,182,85,210)(58,344,86,372)(59,180,87,208)(60,342,88,370)(61,178,89,206)(62,340,90,368)(63,176,91,204)(64,338,92,366)(65,174,93,202)(66,392,94,364)(67,172,95,200)(68,390,96,362)(69,170,97,198)(70,388,98,360)(71,224,99,196)(72,386,100,358)(73,222,101,194)(74,384,102,356)(75,220,103,192)(76,382,104,354)(77,218,105,190)(78,380,106,352)(79,216,107,188)(80,378,108,350)(81,214,109,186)(82,376,110,348)(83,212,111,184)(84,374,112,346)(113,249,141,277)(115,247,143,275)(117,245,145,273)(119,243,147,271)(121,241,149,269)(123,239,151,267)(125,237,153,265)(127,235,155,263)(129,233,157,261)(131,231,159,259)(133,229,161,257)(135,227,163,255)(137,225,165,253)(139,279,167,251)(169,414,197,442)(171,412,199,440)(173,410,201,438)(175,408,203,436)(177,406,205,434)(179,404,207,432)(181,402,209,430)(183,400,211,428)(185,398,213,426)(187,396,215,424)(189,394,217,422)(191,448,219,420)(193,446,221,418)(195,444,223,416)(226,286,254,314)(228,284,256,312)(230,282,258,310)(232,336,260,308)(234,334,262,306)(236,332,264,304)(238,330,266,302)(240,328,268,300)(242,326,270,298)(244,324,272,296)(246,322,274,294)(248,320,276,292)(250,318,278,290)(252,316,280,288)(337,409,365,437)(339,407,367,435)(341,405,369,433)(343,403,371,431)(345,401,373,429)(347,399,375,427)(349,397,377,425)(351,395,379,423)(353,393,381,421)(355,447,383,419)(357,445,385,417)(359,443,387,415)(361,441,389,413)(363,439,391,411)>;
G:=Group( (1,188,263,351)(2,217,264,380)(3,190,265,353)(4,219,266,382)(5,192,267,355)(6,221,268,384)(7,194,269,357)(8,223,270,386)(9,196,271,359)(10,169,272,388)(11,198,273,361)(12,171,274,390)(13,200,275,363)(14,173,276,392)(15,202,277,365)(16,175,278,338)(17,204,279,367)(18,177,280,340)(19,206,225,369)(20,179,226,342)(21,208,227,371)(22,181,228,344)(23,210,229,373)(24,183,230,346)(25,212,231,375)(26,185,232,348)(27,214,233,377)(28,187,234,350)(29,216,235,379)(30,189,236,352)(31,218,237,381)(32,191,238,354)(33,220,239,383)(34,193,240,356)(35,222,241,385)(36,195,242,358)(37,224,243,387)(38,197,244,360)(39,170,245,389)(40,199,246,362)(41,172,247,391)(42,201,248,364)(43,174,249,337)(44,203,250,366)(45,176,251,339)(46,205,252,368)(47,178,253,341)(48,207,254,370)(49,180,255,343)(50,209,256,372)(51,182,257,345)(52,211,258,374)(53,184,259,347)(54,213,260,376)(55,186,261,349)(56,215,262,378)(57,283,429,161)(58,312,430,134)(59,285,431,163)(60,314,432,136)(61,287,433,165)(62,316,434,138)(63,289,435,167)(64,318,436,140)(65,291,437,113)(66,320,438,142)(67,293,439,115)(68,322,440,144)(69,295,441,117)(70,324,442,146)(71,297,443,119)(72,326,444,148)(73,299,445,121)(74,328,446,150)(75,301,447,123)(76,330,448,152)(77,303,393,125)(78,332,394,154)(79,305,395,127)(80,334,396,156)(81,307,397,129)(82,336,398,158)(83,309,399,131)(84,282,400,160)(85,311,401,133)(86,284,402,162)(87,313,403,135)(88,286,404,164)(89,315,405,137)(90,288,406,166)(91,317,407,139)(92,290,408,168)(93,319,409,141)(94,292,410,114)(95,321,411,143)(96,294,412,116)(97,323,413,145)(98,296,414,118)(99,325,415,147)(100,298,416,120)(101,327,417,149)(102,300,418,122)(103,329,419,151)(104,302,420,124)(105,331,421,153)(106,304,422,126)(107,333,423,155)(108,306,424,128)(109,335,425,157)(110,308,426,130)(111,281,427,159)(112,310,428,132), (1,277,29,249)(2,278,30,250)(3,279,31,251)(4,280,32,252)(5,225,33,253)(6,226,34,254)(7,227,35,255)(8,228,36,256)(9,229,37,257)(10,230,38,258)(11,231,39,259)(12,232,40,260)(13,233,41,261)(14,234,42,262)(15,235,43,263)(16,236,44,264)(17,237,45,265)(18,238,46,266)(19,239,47,267)(20,240,48,268)(21,241,49,269)(22,242,50,270)(23,243,51,271)(24,244,52,272)(25,245,53,273)(26,246,54,274)(27,247,55,275)(28,248,56,276)(57,443,85,415)(58,444,86,416)(59,445,87,417)(60,446,88,418)(61,447,89,419)(62,448,90,420)(63,393,91,421)(64,394,92,422)(65,395,93,423)(66,396,94,424)(67,397,95,425)(68,398,96,426)(69,399,97,427)(70,400,98,428)(71,401,99,429)(72,402,100,430)(73,403,101,431)(74,404,102,432)(75,405,103,433)(76,406,104,434)(77,407,105,435)(78,408,106,436)(79,409,107,437)(80,410,108,438)(81,411,109,439)(82,412,110,440)(83,413,111,441)(84,414,112,442)(113,305,141,333)(114,306,142,334)(115,307,143,335)(116,308,144,336)(117,309,145,281)(118,310,146,282)(119,311,147,283)(120,312,148,284)(121,313,149,285)(122,314,150,286)(123,315,151,287)(124,316,152,288)(125,317,153,289)(126,318,154,290)(127,319,155,291)(128,320,156,292)(129,321,157,293)(130,322,158,294)(131,323,159,295)(132,324,160,296)(133,325,161,297)(134,326,162,298)(135,327,163,299)(136,328,164,300)(137,329,165,301)(138,330,166,302)(139,331,167,303)(140,332,168,304)(169,346,197,374)(170,347,198,375)(171,348,199,376)(172,349,200,377)(173,350,201,378)(174,351,202,379)(175,352,203,380)(176,353,204,381)(177,354,205,382)(178,355,206,383)(179,356,207,384)(180,357,208,385)(181,358,209,386)(182,359,210,387)(183,360,211,388)(184,361,212,389)(185,362,213,390)(186,363,214,391)(187,364,215,392)(188,365,216,337)(189,366,217,338)(190,367,218,339)(191,368,219,340)(192,369,220,341)(193,370,221,342)(194,371,222,343)(195,372,223,344)(196,373,224,345), (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), (1,305,29,333)(2,126,30,154)(3,303,31,331)(4,124,32,152)(5,301,33,329)(6,122,34,150)(7,299,35,327)(8,120,36,148)(9,297,37,325)(10,118,38,146)(11,295,39,323)(12,116,40,144)(13,293,41,321)(14,114,42,142)(15,291,43,319)(16,168,44,140)(17,289,45,317)(18,166,46,138)(19,287,47,315)(20,164,48,136)(21,285,49,313)(22,162,50,134)(23,283,51,311)(24,160,52,132)(25,281,53,309)(26,158,54,130)(27,335,55,307)(28,156,56,128)(57,182,85,210)(58,344,86,372)(59,180,87,208)(60,342,88,370)(61,178,89,206)(62,340,90,368)(63,176,91,204)(64,338,92,366)(65,174,93,202)(66,392,94,364)(67,172,95,200)(68,390,96,362)(69,170,97,198)(70,388,98,360)(71,224,99,196)(72,386,100,358)(73,222,101,194)(74,384,102,356)(75,220,103,192)(76,382,104,354)(77,218,105,190)(78,380,106,352)(79,216,107,188)(80,378,108,350)(81,214,109,186)(82,376,110,348)(83,212,111,184)(84,374,112,346)(113,249,141,277)(115,247,143,275)(117,245,145,273)(119,243,147,271)(121,241,149,269)(123,239,151,267)(125,237,153,265)(127,235,155,263)(129,233,157,261)(131,231,159,259)(133,229,161,257)(135,227,163,255)(137,225,165,253)(139,279,167,251)(169,414,197,442)(171,412,199,440)(173,410,201,438)(175,408,203,436)(177,406,205,434)(179,404,207,432)(181,402,209,430)(183,400,211,428)(185,398,213,426)(187,396,215,424)(189,394,217,422)(191,448,219,420)(193,446,221,418)(195,444,223,416)(226,286,254,314)(228,284,256,312)(230,282,258,310)(232,336,260,308)(234,334,262,306)(236,332,264,304)(238,330,266,302)(240,328,268,300)(242,326,270,298)(244,324,272,296)(246,322,274,294)(248,320,276,292)(250,318,278,290)(252,316,280,288)(337,409,365,437)(339,407,367,435)(341,405,369,433)(343,403,371,431)(345,401,373,429)(347,399,375,427)(349,397,377,425)(351,395,379,423)(353,393,381,421)(355,447,383,419)(357,445,385,417)(359,443,387,415)(361,441,389,413)(363,439,391,411) );
G=PermutationGroup([[(1,188,263,351),(2,217,264,380),(3,190,265,353),(4,219,266,382),(5,192,267,355),(6,221,268,384),(7,194,269,357),(8,223,270,386),(9,196,271,359),(10,169,272,388),(11,198,273,361),(12,171,274,390),(13,200,275,363),(14,173,276,392),(15,202,277,365),(16,175,278,338),(17,204,279,367),(18,177,280,340),(19,206,225,369),(20,179,226,342),(21,208,227,371),(22,181,228,344),(23,210,229,373),(24,183,230,346),(25,212,231,375),(26,185,232,348),(27,214,233,377),(28,187,234,350),(29,216,235,379),(30,189,236,352),(31,218,237,381),(32,191,238,354),(33,220,239,383),(34,193,240,356),(35,222,241,385),(36,195,242,358),(37,224,243,387),(38,197,244,360),(39,170,245,389),(40,199,246,362),(41,172,247,391),(42,201,248,364),(43,174,249,337),(44,203,250,366),(45,176,251,339),(46,205,252,368),(47,178,253,341),(48,207,254,370),(49,180,255,343),(50,209,256,372),(51,182,257,345),(52,211,258,374),(53,184,259,347),(54,213,260,376),(55,186,261,349),(56,215,262,378),(57,283,429,161),(58,312,430,134),(59,285,431,163),(60,314,432,136),(61,287,433,165),(62,316,434,138),(63,289,435,167),(64,318,436,140),(65,291,437,113),(66,320,438,142),(67,293,439,115),(68,322,440,144),(69,295,441,117),(70,324,442,146),(71,297,443,119),(72,326,444,148),(73,299,445,121),(74,328,446,150),(75,301,447,123),(76,330,448,152),(77,303,393,125),(78,332,394,154),(79,305,395,127),(80,334,396,156),(81,307,397,129),(82,336,398,158),(83,309,399,131),(84,282,400,160),(85,311,401,133),(86,284,402,162),(87,313,403,135),(88,286,404,164),(89,315,405,137),(90,288,406,166),(91,317,407,139),(92,290,408,168),(93,319,409,141),(94,292,410,114),(95,321,411,143),(96,294,412,116),(97,323,413,145),(98,296,414,118),(99,325,415,147),(100,298,416,120),(101,327,417,149),(102,300,418,122),(103,329,419,151),(104,302,420,124),(105,331,421,153),(106,304,422,126),(107,333,423,155),(108,306,424,128),(109,335,425,157),(110,308,426,130),(111,281,427,159),(112,310,428,132)], [(1,277,29,249),(2,278,30,250),(3,279,31,251),(4,280,32,252),(5,225,33,253),(6,226,34,254),(7,227,35,255),(8,228,36,256),(9,229,37,257),(10,230,38,258),(11,231,39,259),(12,232,40,260),(13,233,41,261),(14,234,42,262),(15,235,43,263),(16,236,44,264),(17,237,45,265),(18,238,46,266),(19,239,47,267),(20,240,48,268),(21,241,49,269),(22,242,50,270),(23,243,51,271),(24,244,52,272),(25,245,53,273),(26,246,54,274),(27,247,55,275),(28,248,56,276),(57,443,85,415),(58,444,86,416),(59,445,87,417),(60,446,88,418),(61,447,89,419),(62,448,90,420),(63,393,91,421),(64,394,92,422),(65,395,93,423),(66,396,94,424),(67,397,95,425),(68,398,96,426),(69,399,97,427),(70,400,98,428),(71,401,99,429),(72,402,100,430),(73,403,101,431),(74,404,102,432),(75,405,103,433),(76,406,104,434),(77,407,105,435),(78,408,106,436),(79,409,107,437),(80,410,108,438),(81,411,109,439),(82,412,110,440),(83,413,111,441),(84,414,112,442),(113,305,141,333),(114,306,142,334),(115,307,143,335),(116,308,144,336),(117,309,145,281),(118,310,146,282),(119,311,147,283),(120,312,148,284),(121,313,149,285),(122,314,150,286),(123,315,151,287),(124,316,152,288),(125,317,153,289),(126,318,154,290),(127,319,155,291),(128,320,156,292),(129,321,157,293),(130,322,158,294),(131,323,159,295),(132,324,160,296),(133,325,161,297),(134,326,162,298),(135,327,163,299),(136,328,164,300),(137,329,165,301),(138,330,166,302),(139,331,167,303),(140,332,168,304),(169,346,197,374),(170,347,198,375),(171,348,199,376),(172,349,200,377),(173,350,201,378),(174,351,202,379),(175,352,203,380),(176,353,204,381),(177,354,205,382),(178,355,206,383),(179,356,207,384),(180,357,208,385),(181,358,209,386),(182,359,210,387),(183,360,211,388),(184,361,212,389),(185,362,213,390),(186,363,214,391),(187,364,215,392),(188,365,216,337),(189,366,217,338),(190,367,218,339),(191,368,219,340),(192,369,220,341),(193,370,221,342),(194,371,222,343),(195,372,223,344),(196,373,224,345)], [(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)], [(1,305,29,333),(2,126,30,154),(3,303,31,331),(4,124,32,152),(5,301,33,329),(6,122,34,150),(7,299,35,327),(8,120,36,148),(9,297,37,325),(10,118,38,146),(11,295,39,323),(12,116,40,144),(13,293,41,321),(14,114,42,142),(15,291,43,319),(16,168,44,140),(17,289,45,317),(18,166,46,138),(19,287,47,315),(20,164,48,136),(21,285,49,313),(22,162,50,134),(23,283,51,311),(24,160,52,132),(25,281,53,309),(26,158,54,130),(27,335,55,307),(28,156,56,128),(57,182,85,210),(58,344,86,372),(59,180,87,208),(60,342,88,370),(61,178,89,206),(62,340,90,368),(63,176,91,204),(64,338,92,366),(65,174,93,202),(66,392,94,364),(67,172,95,200),(68,390,96,362),(69,170,97,198),(70,388,98,360),(71,224,99,196),(72,386,100,358),(73,222,101,194),(74,384,102,356),(75,220,103,192),(76,382,104,354),(77,218,105,190),(78,380,106,352),(79,216,107,188),(80,378,108,350),(81,214,109,186),(82,376,110,348),(83,212,111,184),(84,374,112,346),(113,249,141,277),(115,247,143,275),(117,245,145,273),(119,243,147,271),(121,241,149,269),(123,239,151,267),(125,237,153,265),(127,235,155,263),(129,233,157,261),(131,231,159,259),(133,229,161,257),(135,227,163,255),(137,225,165,253),(139,279,167,251),(169,414,197,442),(171,412,199,440),(173,410,201,438),(175,408,203,436),(177,406,205,434),(179,404,207,432),(181,402,209,430),(183,400,211,428),(185,398,213,426),(187,396,215,424),(189,394,217,422),(191,448,219,420),(193,446,221,418),(195,444,223,416),(226,286,254,314),(228,284,256,312),(230,282,258,310),(232,336,260,308),(234,334,262,306),(236,332,264,304),(238,330,266,302),(240,328,268,300),(242,326,270,298),(244,324,272,296),(246,322,274,294),(248,320,276,292),(250,318,278,290),(252,316,280,288),(337,409,365,437),(339,407,367,435),(341,405,369,433),(343,403,371,431),(345,401,373,429),(347,399,375,427),(349,397,377,425),(351,395,379,423),(353,393,381,421),(355,447,383,419),(357,445,385,417),(359,443,387,415),(361,441,389,413),(363,439,391,411)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D28 | C4○D28 | C8.C22 | C8.D14 |
| kernel | C42.14D14 | C28.44D4 | C7×C8⋊C4 | C28⋊2Q8 | C28.6Q8 | C2×C28 | C8⋊C4 | C28 | C42 | C2×C8 | C2×C4 | C4 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 12 | 24 | 2 | 12 |
Matrix representation of C42.14D14 ►in GL6(𝔽113)
| 1 | 8 | 0 | 0 | 0 | 0 |
| 28 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 59 | 52 | 36 | 51 |
| 0 | 0 | 61 | 65 | 22 | 91 |
| 0 | 0 | 31 | 103 | 93 | 61 |
| 0 | 0 | 5 | 18 | 1 | 9 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 67 | 0 | 0 |
| 0 | 0 | 80 | 17 | 0 | 0 |
| 0 | 0 | 29 | 0 | 9 | 67 |
| 0 | 0 | 49 | 84 | 46 | 104 |
| 15 | 7 | 0 | 0 | 0 | 0 |
| 81 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 73 | 19 | 62 |
| 0 | 0 | 107 | 62 | 55 | 77 |
| 0 | 0 | 84 | 14 | 112 | 40 |
| 0 | 0 | 9 | 57 | 112 | 39 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 62 | 1 | 7 |
| 0 | 0 | 11 | 48 | 110 | 54 |
| 0 | 0 | 87 | 38 | 109 | 11 |
| 0 | 0 | 56 | 87 | 42 | 60 |
G:=sub<GL(6,GF(113))| [1,28,0,0,0,0,8,112,0,0,0,0,0,0,59,61,31,5,0,0,52,65,103,18,0,0,36,22,93,1,0,0,51,91,61,9],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,96,80,29,49,0,0,67,17,0,84,0,0,0,0,9,46,0,0,0,0,67,104],[15,81,0,0,0,0,7,98,0,0,0,0,0,0,13,107,84,9,0,0,73,62,14,57,0,0,19,55,112,112,0,0,62,77,40,39],[0,36,0,0,0,0,22,0,0,0,0,0,0,0,9,11,87,56,0,0,62,48,38,87,0,0,1,110,109,42,0,0,7,54,11,60] >;
C42.14D14 in GAP, Magma, Sage, TeX
C_4^2._{14}D_{14} % in TeX
G:=Group("C4^2.14D14"); // GroupNames label
G:=SmallGroup(448,237);
// by ID
G=gap.SmallGroup(448,237);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,344,254,387,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2*b,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^13>;
// generators/relations