metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.71D14, C4⋊C4.78D14, (C2×C28).86D4, C42.C2.5D7, C28.73(C4○D4), C28⋊2Q8.17C2, (C4×C28).117C22, (C2×C28).387C23, C4.15(Q8⋊2D7), C14.Q16.13C2, C42.D7.6C2, C14.57(C4.4D4), C2.22(D4.9D14), C2.10(C28.23D4), C14.123(C8.C22), C7⋊3(C42.30C22), (C2×Dic14).110C22, (C2×C14).518(C2×D4), (C2×C4).68(C7⋊D4), (C2×C7⋊C8).128C22, (C7×C42.C2).4C2, (C7×C4⋊C4).125C22, (C2×C4).485(C22×D7), C22.191(C2×C7⋊D4), SmallGroup(448,603)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.71D14
G = < a,b,c,d | a4=b4=1, c14=a2, d2=a2b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c13 >
Subgroups: 380 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, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C2×C28, C42.30C22, C2×C7⋊C8, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C42.D7, C14.Q16, C28⋊2Q8, C7×C42.C2, C42.71D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C8.C22, C7⋊D4, C22×D7, C42.30C22, Q8⋊2D7, C2×C7⋊D4, C28.23D4, D4.9D14, C42.71D14
►Regular action on 448 points
Generators in S448
(1 78 15 64)(2 192 16 178)(3 80 17 66)(4 194 18 180)(5 82 19 68)(6 196 20 182)(7 84 21 70)(8 170 22 184)(9 58 23 72)(10 172 24 186)(11 60 25 74)(12 174 26 188)(13 62 27 76)(14 176 28 190)(29 87 43 101)(30 411 44 397)(31 89 45 103)(32 413 46 399)(33 91 47 105)(34 415 48 401)(35 93 49 107)(36 417 50 403)(37 95 51 109)(38 419 52 405)(39 97 53 111)(40 393 54 407)(41 99 55 85)(42 395 56 409)(57 296 71 282)(59 298 73 284)(61 300 75 286)(63 302 77 288)(65 304 79 290)(67 306 81 292)(69 308 83 294)(86 434 100 448)(88 436 102 422)(90 438 104 424)(92 440 106 426)(94 442 108 428)(96 444 110 430)(98 446 112 432)(113 380 127 366)(114 339 128 353)(115 382 129 368)(116 341 130 355)(117 384 131 370)(118 343 132 357)(119 386 133 372)(120 345 134 359)(121 388 135 374)(122 347 136 361)(123 390 137 376)(124 349 138 363)(125 392 139 378)(126 351 140 337)(141 227 155 241)(142 217 156 203)(143 229 157 243)(144 219 158 205)(145 231 159 245)(146 221 160 207)(147 233 161 247)(148 223 162 209)(149 235 163 249)(150 197 164 211)(151 237 165 251)(152 199 166 213)(153 239 167 225)(154 201 168 215)(169 295 183 281)(171 297 185 283)(173 299 187 285)(175 301 189 287)(177 303 191 289)(179 305 193 291)(181 307 195 293)(198 332 212 318)(200 334 214 320)(202 336 216 322)(204 310 218 324)(206 312 220 326)(208 314 222 328)(210 316 224 330)(226 321 240 335)(228 323 242 309)(230 325 244 311)(232 327 246 313)(234 329 248 315)(236 331 250 317)(238 333 252 319)(253 360 267 346)(254 375 268 389)(255 362 269 348)(256 377 270 391)(257 364 271 350)(258 379 272 365)(259 338 273 352)(260 381 274 367)(261 340 275 354)(262 383 276 369)(263 342 277 356)(264 385 278 371)(265 344 279 358)(266 387 280 373)(394 447 408 433)(396 421 410 435)(398 423 412 437)(400 425 414 439)(402 427 416 441)(404 429 418 443)(406 431 420 445)
(1 338 289 366)(2 367 290 339)(3 340 291 368)(4 369 292 341)(5 342 293 370)(6 371 294 343)(7 344 295 372)(8 373 296 345)(9 346 297 374)(10 375 298 347)(11 348 299 376)(12 377 300 349)(13 350 301 378)(14 379 302 351)(15 352 303 380)(16 381 304 353)(17 354 305 382)(18 383 306 355)(19 356 307 384)(20 385 308 357)(21 358 281 386)(22 387 282 359)(23 360 283 388)(24 389 284 361)(25 362 285 390)(26 391 286 363)(27 364 287 392)(28 365 288 337)(29 328 435 147)(30 148 436 329)(31 330 437 149)(32 150 438 331)(33 332 439 151)(34 152 440 333)(35 334 441 153)(36 154 442 335)(37 336 443 155)(38 156 444 309)(39 310 445 157)(40 158 446 311)(41 312 447 159)(42 160 448 313)(43 314 421 161)(44 162 422 315)(45 316 423 163)(46 164 424 317)(47 318 425 165)(48 166 426 319)(49 320 427 167)(50 168 428 321)(51 322 429 141)(52 142 430 323)(53 324 431 143)(54 144 432 325)(55 326 433 145)(56 146 434 327)(57 120 184 280)(58 253 185 121)(59 122 186 254)(60 255 187 123)(61 124 188 256)(62 257 189 125)(63 126 190 258)(64 259 191 127)(65 128 192 260)(66 261 193 129)(67 130 194 262)(68 263 195 131)(69 132 196 264)(70 265 169 133)(71 134 170 266)(72 267 171 135)(73 136 172 268)(74 269 173 137)(75 138 174 270)(76 271 175 139)(77 140 176 272)(78 273 177 113)(79 114 178 274)(80 275 179 115)(81 116 180 276)(82 277 181 117)(83 118 182 278)(84 279 183 119)(85 206 394 231)(86 232 395 207)(87 208 396 233)(88 234 397 209)(89 210 398 235)(90 236 399 211)(91 212 400 237)(92 238 401 213)(93 214 402 239)(94 240 403 215)(95 216 404 241)(96 242 405 217)(97 218 406 243)(98 244 407 219)(99 220 408 245)(100 246 409 221)(101 222 410 247)(102 248 411 223)(103 224 412 249)(104 250 413 197)(105 198 414 251)(106 252 415 199)(107 200 416 225)(108 226 417 201)(109 202 418 227)(110 228 419 203)(111 204 420 229)(112 230 393 205)
(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)
(1 209 352 102 289 234 380 411)(2 101 381 208 290 410 353 233)(3 207 354 100 291 232 382 409)(4 99 383 206 292 408 355 231)(5 205 356 98 293 230 384 407)(6 97 385 204 294 406 357 229)(7 203 358 96 295 228 386 405)(8 95 387 202 296 404 359 227)(9 201 360 94 297 226 388 403)(10 93 389 200 298 402 361 225)(11 199 362 92 299 252 390 401)(12 91 391 198 300 400 363 251)(13 197 364 90 301 250 392 399)(14 89 365 224 302 398 337 249)(15 223 338 88 303 248 366 397)(16 87 367 222 304 396 339 247)(17 221 340 86 305 246 368 395)(18 85 369 220 306 394 341 245)(19 219 342 112 307 244 370 393)(20 111 371 218 308 420 343 243)(21 217 344 110 281 242 372 419)(22 109 373 216 282 418 345 241)(23 215 346 108 283 240 374 417)(24 107 375 214 284 416 347 239)(25 213 348 106 285 238 376 415)(26 105 377 212 286 414 349 237)(27 211 350 104 287 236 378 413)(28 103 379 210 288 412 351 235)(29 114 314 192 435 274 161 65)(30 191 162 113 436 64 315 273)(31 140 316 190 437 272 163 63)(32 189 164 139 438 62 317 271)(33 138 318 188 439 270 165 61)(34 187 166 137 440 60 319 269)(35 136 320 186 441 268 167 59)(36 185 168 135 442 58 321 267)(37 134 322 184 443 266 141 57)(38 183 142 133 444 84 323 265)(39 132 324 182 445 264 143 83)(40 181 144 131 446 82 325 263)(41 130 326 180 447 262 145 81)(42 179 146 129 448 80 327 261)(43 128 328 178 421 260 147 79)(44 177 148 127 422 78 329 259)(45 126 330 176 423 258 149 77)(46 175 150 125 424 76 331 257)(47 124 332 174 425 256 151 75)(48 173 152 123 426 74 333 255)(49 122 334 172 427 254 153 73)(50 171 154 121 428 72 335 253)(51 120 336 170 429 280 155 71)(52 169 156 119 430 70 309 279)(53 118 310 196 431 278 157 69)(54 195 158 117 432 68 311 277)(55 116 312 194 433 276 159 67)(56 193 160 115 434 66 313 275)
G:=sub<Sym(448)| (1,78,15,64)(2,192,16,178)(3,80,17,66)(4,194,18,180)(5,82,19,68)(6,196,20,182)(7,84,21,70)(8,170,22,184)(9,58,23,72)(10,172,24,186)(11,60,25,74)(12,174,26,188)(13,62,27,76)(14,176,28,190)(29,87,43,101)(30,411,44,397)(31,89,45,103)(32,413,46,399)(33,91,47,105)(34,415,48,401)(35,93,49,107)(36,417,50,403)(37,95,51,109)(38,419,52,405)(39,97,53,111)(40,393,54,407)(41,99,55,85)(42,395,56,409)(57,296,71,282)(59,298,73,284)(61,300,75,286)(63,302,77,288)(65,304,79,290)(67,306,81,292)(69,308,83,294)(86,434,100,448)(88,436,102,422)(90,438,104,424)(92,440,106,426)(94,442,108,428)(96,444,110,430)(98,446,112,432)(113,380,127,366)(114,339,128,353)(115,382,129,368)(116,341,130,355)(117,384,131,370)(118,343,132,357)(119,386,133,372)(120,345,134,359)(121,388,135,374)(122,347,136,361)(123,390,137,376)(124,349,138,363)(125,392,139,378)(126,351,140,337)(141,227,155,241)(142,217,156,203)(143,229,157,243)(144,219,158,205)(145,231,159,245)(146,221,160,207)(147,233,161,247)(148,223,162,209)(149,235,163,249)(150,197,164,211)(151,237,165,251)(152,199,166,213)(153,239,167,225)(154,201,168,215)(169,295,183,281)(171,297,185,283)(173,299,187,285)(175,301,189,287)(177,303,191,289)(179,305,193,291)(181,307,195,293)(198,332,212,318)(200,334,214,320)(202,336,216,322)(204,310,218,324)(206,312,220,326)(208,314,222,328)(210,316,224,330)(226,321,240,335)(228,323,242,309)(230,325,244,311)(232,327,246,313)(234,329,248,315)(236,331,250,317)(238,333,252,319)(253,360,267,346)(254,375,268,389)(255,362,269,348)(256,377,270,391)(257,364,271,350)(258,379,272,365)(259,338,273,352)(260,381,274,367)(261,340,275,354)(262,383,276,369)(263,342,277,356)(264,385,278,371)(265,344,279,358)(266,387,280,373)(394,447,408,433)(396,421,410,435)(398,423,412,437)(400,425,414,439)(402,427,416,441)(404,429,418,443)(406,431,420,445), (1,338,289,366)(2,367,290,339)(3,340,291,368)(4,369,292,341)(5,342,293,370)(6,371,294,343)(7,344,295,372)(8,373,296,345)(9,346,297,374)(10,375,298,347)(11,348,299,376)(12,377,300,349)(13,350,301,378)(14,379,302,351)(15,352,303,380)(16,381,304,353)(17,354,305,382)(18,383,306,355)(19,356,307,384)(20,385,308,357)(21,358,281,386)(22,387,282,359)(23,360,283,388)(24,389,284,361)(25,362,285,390)(26,391,286,363)(27,364,287,392)(28,365,288,337)(29,328,435,147)(30,148,436,329)(31,330,437,149)(32,150,438,331)(33,332,439,151)(34,152,440,333)(35,334,441,153)(36,154,442,335)(37,336,443,155)(38,156,444,309)(39,310,445,157)(40,158,446,311)(41,312,447,159)(42,160,448,313)(43,314,421,161)(44,162,422,315)(45,316,423,163)(46,164,424,317)(47,318,425,165)(48,166,426,319)(49,320,427,167)(50,168,428,321)(51,322,429,141)(52,142,430,323)(53,324,431,143)(54,144,432,325)(55,326,433,145)(56,146,434,327)(57,120,184,280)(58,253,185,121)(59,122,186,254)(60,255,187,123)(61,124,188,256)(62,257,189,125)(63,126,190,258)(64,259,191,127)(65,128,192,260)(66,261,193,129)(67,130,194,262)(68,263,195,131)(69,132,196,264)(70,265,169,133)(71,134,170,266)(72,267,171,135)(73,136,172,268)(74,269,173,137)(75,138,174,270)(76,271,175,139)(77,140,176,272)(78,273,177,113)(79,114,178,274)(80,275,179,115)(81,116,180,276)(82,277,181,117)(83,118,182,278)(84,279,183,119)(85,206,394,231)(86,232,395,207)(87,208,396,233)(88,234,397,209)(89,210,398,235)(90,236,399,211)(91,212,400,237)(92,238,401,213)(93,214,402,239)(94,240,403,215)(95,216,404,241)(96,242,405,217)(97,218,406,243)(98,244,407,219)(99,220,408,245)(100,246,409,221)(101,222,410,247)(102,248,411,223)(103,224,412,249)(104,250,413,197)(105,198,414,251)(106,252,415,199)(107,200,416,225)(108,226,417,201)(109,202,418,227)(110,228,419,203)(111,204,420,229)(112,230,393,205), (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), (1,209,352,102,289,234,380,411)(2,101,381,208,290,410,353,233)(3,207,354,100,291,232,382,409)(4,99,383,206,292,408,355,231)(5,205,356,98,293,230,384,407)(6,97,385,204,294,406,357,229)(7,203,358,96,295,228,386,405)(8,95,387,202,296,404,359,227)(9,201,360,94,297,226,388,403)(10,93,389,200,298,402,361,225)(11,199,362,92,299,252,390,401)(12,91,391,198,300,400,363,251)(13,197,364,90,301,250,392,399)(14,89,365,224,302,398,337,249)(15,223,338,88,303,248,366,397)(16,87,367,222,304,396,339,247)(17,221,340,86,305,246,368,395)(18,85,369,220,306,394,341,245)(19,219,342,112,307,244,370,393)(20,111,371,218,308,420,343,243)(21,217,344,110,281,242,372,419)(22,109,373,216,282,418,345,241)(23,215,346,108,283,240,374,417)(24,107,375,214,284,416,347,239)(25,213,348,106,285,238,376,415)(26,105,377,212,286,414,349,237)(27,211,350,104,287,236,378,413)(28,103,379,210,288,412,351,235)(29,114,314,192,435,274,161,65)(30,191,162,113,436,64,315,273)(31,140,316,190,437,272,163,63)(32,189,164,139,438,62,317,271)(33,138,318,188,439,270,165,61)(34,187,166,137,440,60,319,269)(35,136,320,186,441,268,167,59)(36,185,168,135,442,58,321,267)(37,134,322,184,443,266,141,57)(38,183,142,133,444,84,323,265)(39,132,324,182,445,264,143,83)(40,181,144,131,446,82,325,263)(41,130,326,180,447,262,145,81)(42,179,146,129,448,80,327,261)(43,128,328,178,421,260,147,79)(44,177,148,127,422,78,329,259)(45,126,330,176,423,258,149,77)(46,175,150,125,424,76,331,257)(47,124,332,174,425,256,151,75)(48,173,152,123,426,74,333,255)(49,122,334,172,427,254,153,73)(50,171,154,121,428,72,335,253)(51,120,336,170,429,280,155,71)(52,169,156,119,430,70,309,279)(53,118,310,196,431,278,157,69)(54,195,158,117,432,68,311,277)(55,116,312,194,433,276,159,67)(56,193,160,115,434,66,313,275)>;
G:=Group( (1,78,15,64)(2,192,16,178)(3,80,17,66)(4,194,18,180)(5,82,19,68)(6,196,20,182)(7,84,21,70)(8,170,22,184)(9,58,23,72)(10,172,24,186)(11,60,25,74)(12,174,26,188)(13,62,27,76)(14,176,28,190)(29,87,43,101)(30,411,44,397)(31,89,45,103)(32,413,46,399)(33,91,47,105)(34,415,48,401)(35,93,49,107)(36,417,50,403)(37,95,51,109)(38,419,52,405)(39,97,53,111)(40,393,54,407)(41,99,55,85)(42,395,56,409)(57,296,71,282)(59,298,73,284)(61,300,75,286)(63,302,77,288)(65,304,79,290)(67,306,81,292)(69,308,83,294)(86,434,100,448)(88,436,102,422)(90,438,104,424)(92,440,106,426)(94,442,108,428)(96,444,110,430)(98,446,112,432)(113,380,127,366)(114,339,128,353)(115,382,129,368)(116,341,130,355)(117,384,131,370)(118,343,132,357)(119,386,133,372)(120,345,134,359)(121,388,135,374)(122,347,136,361)(123,390,137,376)(124,349,138,363)(125,392,139,378)(126,351,140,337)(141,227,155,241)(142,217,156,203)(143,229,157,243)(144,219,158,205)(145,231,159,245)(146,221,160,207)(147,233,161,247)(148,223,162,209)(149,235,163,249)(150,197,164,211)(151,237,165,251)(152,199,166,213)(153,239,167,225)(154,201,168,215)(169,295,183,281)(171,297,185,283)(173,299,187,285)(175,301,189,287)(177,303,191,289)(179,305,193,291)(181,307,195,293)(198,332,212,318)(200,334,214,320)(202,336,216,322)(204,310,218,324)(206,312,220,326)(208,314,222,328)(210,316,224,330)(226,321,240,335)(228,323,242,309)(230,325,244,311)(232,327,246,313)(234,329,248,315)(236,331,250,317)(238,333,252,319)(253,360,267,346)(254,375,268,389)(255,362,269,348)(256,377,270,391)(257,364,271,350)(258,379,272,365)(259,338,273,352)(260,381,274,367)(261,340,275,354)(262,383,276,369)(263,342,277,356)(264,385,278,371)(265,344,279,358)(266,387,280,373)(394,447,408,433)(396,421,410,435)(398,423,412,437)(400,425,414,439)(402,427,416,441)(404,429,418,443)(406,431,420,445), (1,338,289,366)(2,367,290,339)(3,340,291,368)(4,369,292,341)(5,342,293,370)(6,371,294,343)(7,344,295,372)(8,373,296,345)(9,346,297,374)(10,375,298,347)(11,348,299,376)(12,377,300,349)(13,350,301,378)(14,379,302,351)(15,352,303,380)(16,381,304,353)(17,354,305,382)(18,383,306,355)(19,356,307,384)(20,385,308,357)(21,358,281,386)(22,387,282,359)(23,360,283,388)(24,389,284,361)(25,362,285,390)(26,391,286,363)(27,364,287,392)(28,365,288,337)(29,328,435,147)(30,148,436,329)(31,330,437,149)(32,150,438,331)(33,332,439,151)(34,152,440,333)(35,334,441,153)(36,154,442,335)(37,336,443,155)(38,156,444,309)(39,310,445,157)(40,158,446,311)(41,312,447,159)(42,160,448,313)(43,314,421,161)(44,162,422,315)(45,316,423,163)(46,164,424,317)(47,318,425,165)(48,166,426,319)(49,320,427,167)(50,168,428,321)(51,322,429,141)(52,142,430,323)(53,324,431,143)(54,144,432,325)(55,326,433,145)(56,146,434,327)(57,120,184,280)(58,253,185,121)(59,122,186,254)(60,255,187,123)(61,124,188,256)(62,257,189,125)(63,126,190,258)(64,259,191,127)(65,128,192,260)(66,261,193,129)(67,130,194,262)(68,263,195,131)(69,132,196,264)(70,265,169,133)(71,134,170,266)(72,267,171,135)(73,136,172,268)(74,269,173,137)(75,138,174,270)(76,271,175,139)(77,140,176,272)(78,273,177,113)(79,114,178,274)(80,275,179,115)(81,116,180,276)(82,277,181,117)(83,118,182,278)(84,279,183,119)(85,206,394,231)(86,232,395,207)(87,208,396,233)(88,234,397,209)(89,210,398,235)(90,236,399,211)(91,212,400,237)(92,238,401,213)(93,214,402,239)(94,240,403,215)(95,216,404,241)(96,242,405,217)(97,218,406,243)(98,244,407,219)(99,220,408,245)(100,246,409,221)(101,222,410,247)(102,248,411,223)(103,224,412,249)(104,250,413,197)(105,198,414,251)(106,252,415,199)(107,200,416,225)(108,226,417,201)(109,202,418,227)(110,228,419,203)(111,204,420,229)(112,230,393,205), (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), (1,209,352,102,289,234,380,411)(2,101,381,208,290,410,353,233)(3,207,354,100,291,232,382,409)(4,99,383,206,292,408,355,231)(5,205,356,98,293,230,384,407)(6,97,385,204,294,406,357,229)(7,203,358,96,295,228,386,405)(8,95,387,202,296,404,359,227)(9,201,360,94,297,226,388,403)(10,93,389,200,298,402,361,225)(11,199,362,92,299,252,390,401)(12,91,391,198,300,400,363,251)(13,197,364,90,301,250,392,399)(14,89,365,224,302,398,337,249)(15,223,338,88,303,248,366,397)(16,87,367,222,304,396,339,247)(17,221,340,86,305,246,368,395)(18,85,369,220,306,394,341,245)(19,219,342,112,307,244,370,393)(20,111,371,218,308,420,343,243)(21,217,344,110,281,242,372,419)(22,109,373,216,282,418,345,241)(23,215,346,108,283,240,374,417)(24,107,375,214,284,416,347,239)(25,213,348,106,285,238,376,415)(26,105,377,212,286,414,349,237)(27,211,350,104,287,236,378,413)(28,103,379,210,288,412,351,235)(29,114,314,192,435,274,161,65)(30,191,162,113,436,64,315,273)(31,140,316,190,437,272,163,63)(32,189,164,139,438,62,317,271)(33,138,318,188,439,270,165,61)(34,187,166,137,440,60,319,269)(35,136,320,186,441,268,167,59)(36,185,168,135,442,58,321,267)(37,134,322,184,443,266,141,57)(38,183,142,133,444,84,323,265)(39,132,324,182,445,264,143,83)(40,181,144,131,446,82,325,263)(41,130,326,180,447,262,145,81)(42,179,146,129,448,80,327,261)(43,128,328,178,421,260,147,79)(44,177,148,127,422,78,329,259)(45,126,330,176,423,258,149,77)(46,175,150,125,424,76,331,257)(47,124,332,174,425,256,151,75)(48,173,152,123,426,74,333,255)(49,122,334,172,427,254,153,73)(50,171,154,121,428,72,335,253)(51,120,336,170,429,280,155,71)(52,169,156,119,430,70,309,279)(53,118,310,196,431,278,157,69)(54,195,158,117,432,68,311,277)(55,116,312,194,433,276,159,67)(56,193,160,115,434,66,313,275) );
G=PermutationGroup([[(1,78,15,64),(2,192,16,178),(3,80,17,66),(4,194,18,180),(5,82,19,68),(6,196,20,182),(7,84,21,70),(8,170,22,184),(9,58,23,72),(10,172,24,186),(11,60,25,74),(12,174,26,188),(13,62,27,76),(14,176,28,190),(29,87,43,101),(30,411,44,397),(31,89,45,103),(32,413,46,399),(33,91,47,105),(34,415,48,401),(35,93,49,107),(36,417,50,403),(37,95,51,109),(38,419,52,405),(39,97,53,111),(40,393,54,407),(41,99,55,85),(42,395,56,409),(57,296,71,282),(59,298,73,284),(61,300,75,286),(63,302,77,288),(65,304,79,290),(67,306,81,292),(69,308,83,294),(86,434,100,448),(88,436,102,422),(90,438,104,424),(92,440,106,426),(94,442,108,428),(96,444,110,430),(98,446,112,432),(113,380,127,366),(114,339,128,353),(115,382,129,368),(116,341,130,355),(117,384,131,370),(118,343,132,357),(119,386,133,372),(120,345,134,359),(121,388,135,374),(122,347,136,361),(123,390,137,376),(124,349,138,363),(125,392,139,378),(126,351,140,337),(141,227,155,241),(142,217,156,203),(143,229,157,243),(144,219,158,205),(145,231,159,245),(146,221,160,207),(147,233,161,247),(148,223,162,209),(149,235,163,249),(150,197,164,211),(151,237,165,251),(152,199,166,213),(153,239,167,225),(154,201,168,215),(169,295,183,281),(171,297,185,283),(173,299,187,285),(175,301,189,287),(177,303,191,289),(179,305,193,291),(181,307,195,293),(198,332,212,318),(200,334,214,320),(202,336,216,322),(204,310,218,324),(206,312,220,326),(208,314,222,328),(210,316,224,330),(226,321,240,335),(228,323,242,309),(230,325,244,311),(232,327,246,313),(234,329,248,315),(236,331,250,317),(238,333,252,319),(253,360,267,346),(254,375,268,389),(255,362,269,348),(256,377,270,391),(257,364,271,350),(258,379,272,365),(259,338,273,352),(260,381,274,367),(261,340,275,354),(262,383,276,369),(263,342,277,356),(264,385,278,371),(265,344,279,358),(266,387,280,373),(394,447,408,433),(396,421,410,435),(398,423,412,437),(400,425,414,439),(402,427,416,441),(404,429,418,443),(406,431,420,445)], [(1,338,289,366),(2,367,290,339),(3,340,291,368),(4,369,292,341),(5,342,293,370),(6,371,294,343),(7,344,295,372),(8,373,296,345),(9,346,297,374),(10,375,298,347),(11,348,299,376),(12,377,300,349),(13,350,301,378),(14,379,302,351),(15,352,303,380),(16,381,304,353),(17,354,305,382),(18,383,306,355),(19,356,307,384),(20,385,308,357),(21,358,281,386),(22,387,282,359),(23,360,283,388),(24,389,284,361),(25,362,285,390),(26,391,286,363),(27,364,287,392),(28,365,288,337),(29,328,435,147),(30,148,436,329),(31,330,437,149),(32,150,438,331),(33,332,439,151),(34,152,440,333),(35,334,441,153),(36,154,442,335),(37,336,443,155),(38,156,444,309),(39,310,445,157),(40,158,446,311),(41,312,447,159),(42,160,448,313),(43,314,421,161),(44,162,422,315),(45,316,423,163),(46,164,424,317),(47,318,425,165),(48,166,426,319),(49,320,427,167),(50,168,428,321),(51,322,429,141),(52,142,430,323),(53,324,431,143),(54,144,432,325),(55,326,433,145),(56,146,434,327),(57,120,184,280),(58,253,185,121),(59,122,186,254),(60,255,187,123),(61,124,188,256),(62,257,189,125),(63,126,190,258),(64,259,191,127),(65,128,192,260),(66,261,193,129),(67,130,194,262),(68,263,195,131),(69,132,196,264),(70,265,169,133),(71,134,170,266),(72,267,171,135),(73,136,172,268),(74,269,173,137),(75,138,174,270),(76,271,175,139),(77,140,176,272),(78,273,177,113),(79,114,178,274),(80,275,179,115),(81,116,180,276),(82,277,181,117),(83,118,182,278),(84,279,183,119),(85,206,394,231),(86,232,395,207),(87,208,396,233),(88,234,397,209),(89,210,398,235),(90,236,399,211),(91,212,400,237),(92,238,401,213),(93,214,402,239),(94,240,403,215),(95,216,404,241),(96,242,405,217),(97,218,406,243),(98,244,407,219),(99,220,408,245),(100,246,409,221),(101,222,410,247),(102,248,411,223),(103,224,412,249),(104,250,413,197),(105,198,414,251),(106,252,415,199),(107,200,416,225),(108,226,417,201),(109,202,418,227),(110,228,419,203),(111,204,420,229),(112,230,393,205)], [(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)], [(1,209,352,102,289,234,380,411),(2,101,381,208,290,410,353,233),(3,207,354,100,291,232,382,409),(4,99,383,206,292,408,355,231),(5,205,356,98,293,230,384,407),(6,97,385,204,294,406,357,229),(7,203,358,96,295,228,386,405),(8,95,387,202,296,404,359,227),(9,201,360,94,297,226,388,403),(10,93,389,200,298,402,361,225),(11,199,362,92,299,252,390,401),(12,91,391,198,300,400,363,251),(13,197,364,90,301,250,392,399),(14,89,365,224,302,398,337,249),(15,223,338,88,303,248,366,397),(16,87,367,222,304,396,339,247),(17,221,340,86,305,246,368,395),(18,85,369,220,306,394,341,245),(19,219,342,112,307,244,370,393),(20,111,371,218,308,420,343,243),(21,217,344,110,281,242,372,419),(22,109,373,216,282,418,345,241),(23,215,346,108,283,240,374,417),(24,107,375,214,284,416,347,239),(25,213,348,106,285,238,376,415),(26,105,377,212,286,414,349,237),(27,211,350,104,287,236,378,413),(28,103,379,210,288,412,351,235),(29,114,314,192,435,274,161,65),(30,191,162,113,436,64,315,273),(31,140,316,190,437,272,163,63),(32,189,164,139,438,62,317,271),(33,138,318,188,439,270,165,61),(34,187,166,137,440,60,319,269),(35,136,320,186,441,268,167,59),(36,185,168,135,442,58,321,267),(37,134,322,184,443,266,141,57),(38,183,142,133,444,84,323,265),(39,132,324,182,445,264,143,83),(40,181,144,131,446,82,325,263),(41,130,326,180,447,262,145,81),(42,179,146,129,448,80,327,261),(43,128,328,178,421,260,147,79),(44,177,148,127,422,78,329,259),(45,126,330,176,423,258,149,77),(46,175,150,125,424,76,331,257),(47,124,332,174,425,256,151,75),(48,173,152,123,426,74,333,255),(49,122,334,172,427,254,153,73),(50,171,154,121,428,72,335,253),(51,120,336,170,429,280,155,71),(52,169,156,119,430,70,309,279),(53,118,310,196,431,278,157,69),(54,195,158,117,432,68,311,277),(55,116,312,194,433,276,159,67),(56,193,160,115,434,66,313,275)]])
58 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 | ··· | 28R | 28S | ··· | 28AD |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C8.C22 | Q8⋊2D7 | D4.9D14 |
| kernel | C42.71D14 | C42.D7 | C14.Q16 | C28⋊2Q8 | C7×C42.C2 | C2×C28 | C42.C2 | C28 | C42 | C4⋊C4 | C2×C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 12 | 2 | 6 | 12 |
Matrix representation of C42.71D14 ►in GL6(𝔽113)
| 88 | 74 | 0 | 0 | 0 | 0 |
| 74 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 104 | 46 | 0 | 0 |
| 0 | 0 | 67 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 104 | 46 |
| 0 | 0 | 0 | 0 | 67 | 9 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 56 | 35 | 24 |
| 0 | 0 | 57 | 16 | 89 | 10 |
| 0 | 0 | 35 | 24 | 1 | 57 |
| 0 | 0 | 89 | 10 | 56 | 97 |
| 36 | 20 | 0 | 0 | 0 | 0 |
| 20 | 77 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 20 | 62 | 56 |
| 0 | 0 | 32 | 7 | 95 | 51 |
| 0 | 0 | 51 | 57 | 106 | 20 |
| 0 | 0 | 18 | 62 | 32 | 7 |
G:=sub<GL(6,GF(113))| [88,74,0,0,0,0,74,25,0,0,0,0,0,0,104,67,0,0,0,0,46,9,0,0,0,0,0,0,104,67,0,0,0,0,46,9],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,1,0,0,0,0,0,0,1,0,0],[0,112,0,0,0,0,1,0,0,0,0,0,0,0,112,57,35,89,0,0,56,16,24,10,0,0,35,89,1,56,0,0,24,10,57,97],[36,20,0,0,0,0,20,77,0,0,0,0,0,0,106,32,51,18,0,0,20,7,57,62,0,0,62,95,106,32,0,0,56,51,20,7] >;
C42.71D14 in GAP, Magma, Sage, TeX
C_4^2._{71}D_{14} % in TeX
G:=Group("C4^2.71D14"); // GroupNames label
G:=SmallGroup(448,603);
// by ID
G=gap.SmallGroup(448,603);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,555,100,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^13>;
// generators/relations