metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.27D14, Dic7.2M4(2), C7⋊C8⋊9Q8, C4⋊C8.11D7, C7⋊3(C8⋊4Q8), C4.51(Q8×D7), C14.16(C4×Q8), C56⋊C4.8C2, (C2×C8).212D14, Dic7⋊C8.7C2, C4⋊Dic7.11C4, C28.109(C2×Q8), C14.11(C8○D4), Dic7⋊C4.13C4, (C4×C28).54C22, (C8×Dic7).13C2, (C4×Dic14).7C2, (C2×Dic14).9C4, C2.15(D7×M4(2)), C28.301(C4○D4), (C2×C28).826C23, (C2×C56).205C22, C14.24(C2×M4(2)), C4.127(D4⋊2D7), C42.D7.2C2, C2.7(Dic7⋊3Q8), C2.13(D28.2C4), (C4×Dic7).273C22, (C7×C4⋊C8).16C2, (C2×C4).34(C4×D7), (C2×C28).42(C2×C4), C22.108(C2×C4×D7), (C2×C7⋊C8).193C22, (C2×C14).81(C22×C4), (C2×Dic7).20(C2×C4), (C2×C4).768(C22×D7), SmallGroup(448,362)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.27D14
G = < a,b,c,d | a4=b4=1, c14=b-1, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=b2c13 >
Subgroups: 324 in 94 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C7⋊C8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C8⋊4Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C42.D7, C8×Dic7, Dic7⋊C8, C56⋊C4, C7×C4⋊C8, C4×Dic14, C42.27D14
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, M4(2), C22×C4, C2×Q8, C4○D4, D14, C4×Q8, C2×M4(2), C8○D4, C4×D7, C22×D7, C8⋊4Q8, C2×C4×D7, D4⋊2D7, Q8×D7, Dic7⋊3Q8, D28.2C4, D7×M4(2), C42.27D14
►Regular action on 448 points
Generators in S448
(1 121 84 330)(2 331 85 122)(3 123 86 332)(4 333 87 124)(5 125 88 334)(6 335 89 126)(7 127 90 336)(8 281 91 128)(9 129 92 282)(10 283 93 130)(11 131 94 284)(12 285 95 132)(13 133 96 286)(14 287 97 134)(15 135 98 288)(16 289 99 136)(17 137 100 290)(18 291 101 138)(19 139 102 292)(20 293 103 140)(21 141 104 294)(22 295 105 142)(23 143 106 296)(24 297 107 144)(25 145 108 298)(26 299 109 146)(27 147 110 300)(28 301 111 148)(29 149 112 302)(30 303 57 150)(31 151 58 304)(32 305 59 152)(33 153 60 306)(34 307 61 154)(35 155 62 308)(36 309 63 156)(37 157 64 310)(38 311 65 158)(39 159 66 312)(40 313 67 160)(41 161 68 314)(42 315 69 162)(43 163 70 316)(44 317 71 164)(45 165 72 318)(46 319 73 166)(47 167 74 320)(48 321 75 168)(49 113 76 322)(50 323 77 114)(51 115 78 324)(52 325 79 116)(53 117 80 326)(54 327 81 118)(55 119 82 328)(56 329 83 120)(169 344 238 444)(170 445 239 345)(171 346 240 446)(172 447 241 347)(173 348 242 448)(174 393 243 349)(175 350 244 394)(176 395 245 351)(177 352 246 396)(178 397 247 353)(179 354 248 398)(180 399 249 355)(181 356 250 400)(182 401 251 357)(183 358 252 402)(184 403 253 359)(185 360 254 404)(186 405 255 361)(187 362 256 406)(188 407 257 363)(189 364 258 408)(190 409 259 365)(191 366 260 410)(192 411 261 367)(193 368 262 412)(194 413 263 369)(195 370 264 414)(196 415 265 371)(197 372 266 416)(198 417 267 373)(199 374 268 418)(200 419 269 375)(201 376 270 420)(202 421 271 377)(203 378 272 422)(204 423 273 379)(205 380 274 424)(206 425 275 381)(207 382 276 426)(208 427 277 383)(209 384 278 428)(210 429 279 385)(211 386 280 430)(212 431 225 387)(213 388 226 432)(214 433 227 389)(215 390 228 434)(216 435 229 391)(217 392 230 436)(218 437 231 337)(219 338 232 438)(220 439 233 339)(221 340 234 440)(222 441 235 341)(223 342 236 442)(224 443 237 343)
(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)
(1 229 70 202 29 257 98 174)(2 270 71 187 30 242 99 215)(3 255 72 172 31 227 100 200)(4 240 73 213 32 268 101 185)(5 225 74 198 33 253 102 170)(6 266 75 183 34 238 103 211)(7 251 76 224 35 279 104 196)(8 236 77 209 36 264 105 181)(9 277 78 194 37 249 106 222)(10 262 79 179 38 234 107 207)(11 247 80 220 39 275 108 192)(12 232 81 205 40 260 109 177)(13 273 82 190 41 245 110 218)(14 258 83 175 42 230 111 203)(15 243 84 216 43 271 112 188)(16 228 85 201 44 256 57 173)(17 269 86 186 45 241 58 214)(18 254 87 171 46 226 59 199)(19 239 88 212 47 267 60 184)(20 280 89 197 48 252 61 169)(21 265 90 182 49 237 62 210)(22 250 91 223 50 278 63 195)(23 235 92 208 51 263 64 180)(24 276 93 193 52 248 65 221)(25 261 94 178 53 233 66 206)(26 246 95 219 54 274 67 191)(27 231 96 204 55 259 68 176)(28 272 97 189 56 244 69 217)(113 371 308 401 141 343 336 429)(114 356 309 442 142 384 281 414)(115 341 310 427 143 369 282 399)(116 382 311 412 144 354 283 440)(117 367 312 397 145 339 284 425)(118 352 313 438 146 380 285 410)(119 337 314 423 147 365 286 395)(120 378 315 408 148 350 287 436)(121 363 316 393 149 391 288 421)(122 348 317 434 150 376 289 406)(123 389 318 419 151 361 290 447)(124 374 319 404 152 346 291 432)(125 359 320 445 153 387 292 417)(126 344 321 430 154 372 293 402)(127 385 322 415 155 357 294 443)(128 370 323 400 156 342 295 428)(129 355 324 441 157 383 296 413)(130 340 325 426 158 368 297 398)(131 381 326 411 159 353 298 439)(132 366 327 396 160 338 299 424)(133 351 328 437 161 379 300 409)(134 392 329 422 162 364 301 394)(135 377 330 407 163 349 302 435)(136 362 331 448 164 390 303 420)(137 347 332 433 165 375 304 405)(138 388 333 418 166 360 305 446)(139 373 334 403 167 345 306 431)(140 358 335 444 168 386 307 416)
G:=sub<Sym(448)| (1,121,84,330)(2,331,85,122)(3,123,86,332)(4,333,87,124)(5,125,88,334)(6,335,89,126)(7,127,90,336)(8,281,91,128)(9,129,92,282)(10,283,93,130)(11,131,94,284)(12,285,95,132)(13,133,96,286)(14,287,97,134)(15,135,98,288)(16,289,99,136)(17,137,100,290)(18,291,101,138)(19,139,102,292)(20,293,103,140)(21,141,104,294)(22,295,105,142)(23,143,106,296)(24,297,107,144)(25,145,108,298)(26,299,109,146)(27,147,110,300)(28,301,111,148)(29,149,112,302)(30,303,57,150)(31,151,58,304)(32,305,59,152)(33,153,60,306)(34,307,61,154)(35,155,62,308)(36,309,63,156)(37,157,64,310)(38,311,65,158)(39,159,66,312)(40,313,67,160)(41,161,68,314)(42,315,69,162)(43,163,70,316)(44,317,71,164)(45,165,72,318)(46,319,73,166)(47,167,74,320)(48,321,75,168)(49,113,76,322)(50,323,77,114)(51,115,78,324)(52,325,79,116)(53,117,80,326)(54,327,81,118)(55,119,82,328)(56,329,83,120)(169,344,238,444)(170,445,239,345)(171,346,240,446)(172,447,241,347)(173,348,242,448)(174,393,243,349)(175,350,244,394)(176,395,245,351)(177,352,246,396)(178,397,247,353)(179,354,248,398)(180,399,249,355)(181,356,250,400)(182,401,251,357)(183,358,252,402)(184,403,253,359)(185,360,254,404)(186,405,255,361)(187,362,256,406)(188,407,257,363)(189,364,258,408)(190,409,259,365)(191,366,260,410)(192,411,261,367)(193,368,262,412)(194,413,263,369)(195,370,264,414)(196,415,265,371)(197,372,266,416)(198,417,267,373)(199,374,268,418)(200,419,269,375)(201,376,270,420)(202,421,271,377)(203,378,272,422)(204,423,273,379)(205,380,274,424)(206,425,275,381)(207,382,276,426)(208,427,277,383)(209,384,278,428)(210,429,279,385)(211,386,280,430)(212,431,225,387)(213,388,226,432)(214,433,227,389)(215,390,228,434)(216,435,229,391)(217,392,230,436)(218,437,231,337)(219,338,232,438)(220,439,233,339)(221,340,234,440)(222,441,235,341)(223,342,236,442)(224,443,237,343), (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), (1,229,70,202,29,257,98,174)(2,270,71,187,30,242,99,215)(3,255,72,172,31,227,100,200)(4,240,73,213,32,268,101,185)(5,225,74,198,33,253,102,170)(6,266,75,183,34,238,103,211)(7,251,76,224,35,279,104,196)(8,236,77,209,36,264,105,181)(9,277,78,194,37,249,106,222)(10,262,79,179,38,234,107,207)(11,247,80,220,39,275,108,192)(12,232,81,205,40,260,109,177)(13,273,82,190,41,245,110,218)(14,258,83,175,42,230,111,203)(15,243,84,216,43,271,112,188)(16,228,85,201,44,256,57,173)(17,269,86,186,45,241,58,214)(18,254,87,171,46,226,59,199)(19,239,88,212,47,267,60,184)(20,280,89,197,48,252,61,169)(21,265,90,182,49,237,62,210)(22,250,91,223,50,278,63,195)(23,235,92,208,51,263,64,180)(24,276,93,193,52,248,65,221)(25,261,94,178,53,233,66,206)(26,246,95,219,54,274,67,191)(27,231,96,204,55,259,68,176)(28,272,97,189,56,244,69,217)(113,371,308,401,141,343,336,429)(114,356,309,442,142,384,281,414)(115,341,310,427,143,369,282,399)(116,382,311,412,144,354,283,440)(117,367,312,397,145,339,284,425)(118,352,313,438,146,380,285,410)(119,337,314,423,147,365,286,395)(120,378,315,408,148,350,287,436)(121,363,316,393,149,391,288,421)(122,348,317,434,150,376,289,406)(123,389,318,419,151,361,290,447)(124,374,319,404,152,346,291,432)(125,359,320,445,153,387,292,417)(126,344,321,430,154,372,293,402)(127,385,322,415,155,357,294,443)(128,370,323,400,156,342,295,428)(129,355,324,441,157,383,296,413)(130,340,325,426,158,368,297,398)(131,381,326,411,159,353,298,439)(132,366,327,396,160,338,299,424)(133,351,328,437,161,379,300,409)(134,392,329,422,162,364,301,394)(135,377,330,407,163,349,302,435)(136,362,331,448,164,390,303,420)(137,347,332,433,165,375,304,405)(138,388,333,418,166,360,305,446)(139,373,334,403,167,345,306,431)(140,358,335,444,168,386,307,416)>;
G:=Group( (1,121,84,330)(2,331,85,122)(3,123,86,332)(4,333,87,124)(5,125,88,334)(6,335,89,126)(7,127,90,336)(8,281,91,128)(9,129,92,282)(10,283,93,130)(11,131,94,284)(12,285,95,132)(13,133,96,286)(14,287,97,134)(15,135,98,288)(16,289,99,136)(17,137,100,290)(18,291,101,138)(19,139,102,292)(20,293,103,140)(21,141,104,294)(22,295,105,142)(23,143,106,296)(24,297,107,144)(25,145,108,298)(26,299,109,146)(27,147,110,300)(28,301,111,148)(29,149,112,302)(30,303,57,150)(31,151,58,304)(32,305,59,152)(33,153,60,306)(34,307,61,154)(35,155,62,308)(36,309,63,156)(37,157,64,310)(38,311,65,158)(39,159,66,312)(40,313,67,160)(41,161,68,314)(42,315,69,162)(43,163,70,316)(44,317,71,164)(45,165,72,318)(46,319,73,166)(47,167,74,320)(48,321,75,168)(49,113,76,322)(50,323,77,114)(51,115,78,324)(52,325,79,116)(53,117,80,326)(54,327,81,118)(55,119,82,328)(56,329,83,120)(169,344,238,444)(170,445,239,345)(171,346,240,446)(172,447,241,347)(173,348,242,448)(174,393,243,349)(175,350,244,394)(176,395,245,351)(177,352,246,396)(178,397,247,353)(179,354,248,398)(180,399,249,355)(181,356,250,400)(182,401,251,357)(183,358,252,402)(184,403,253,359)(185,360,254,404)(186,405,255,361)(187,362,256,406)(188,407,257,363)(189,364,258,408)(190,409,259,365)(191,366,260,410)(192,411,261,367)(193,368,262,412)(194,413,263,369)(195,370,264,414)(196,415,265,371)(197,372,266,416)(198,417,267,373)(199,374,268,418)(200,419,269,375)(201,376,270,420)(202,421,271,377)(203,378,272,422)(204,423,273,379)(205,380,274,424)(206,425,275,381)(207,382,276,426)(208,427,277,383)(209,384,278,428)(210,429,279,385)(211,386,280,430)(212,431,225,387)(213,388,226,432)(214,433,227,389)(215,390,228,434)(216,435,229,391)(217,392,230,436)(218,437,231,337)(219,338,232,438)(220,439,233,339)(221,340,234,440)(222,441,235,341)(223,342,236,442)(224,443,237,343), (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), (1,229,70,202,29,257,98,174)(2,270,71,187,30,242,99,215)(3,255,72,172,31,227,100,200)(4,240,73,213,32,268,101,185)(5,225,74,198,33,253,102,170)(6,266,75,183,34,238,103,211)(7,251,76,224,35,279,104,196)(8,236,77,209,36,264,105,181)(9,277,78,194,37,249,106,222)(10,262,79,179,38,234,107,207)(11,247,80,220,39,275,108,192)(12,232,81,205,40,260,109,177)(13,273,82,190,41,245,110,218)(14,258,83,175,42,230,111,203)(15,243,84,216,43,271,112,188)(16,228,85,201,44,256,57,173)(17,269,86,186,45,241,58,214)(18,254,87,171,46,226,59,199)(19,239,88,212,47,267,60,184)(20,280,89,197,48,252,61,169)(21,265,90,182,49,237,62,210)(22,250,91,223,50,278,63,195)(23,235,92,208,51,263,64,180)(24,276,93,193,52,248,65,221)(25,261,94,178,53,233,66,206)(26,246,95,219,54,274,67,191)(27,231,96,204,55,259,68,176)(28,272,97,189,56,244,69,217)(113,371,308,401,141,343,336,429)(114,356,309,442,142,384,281,414)(115,341,310,427,143,369,282,399)(116,382,311,412,144,354,283,440)(117,367,312,397,145,339,284,425)(118,352,313,438,146,380,285,410)(119,337,314,423,147,365,286,395)(120,378,315,408,148,350,287,436)(121,363,316,393,149,391,288,421)(122,348,317,434,150,376,289,406)(123,389,318,419,151,361,290,447)(124,374,319,404,152,346,291,432)(125,359,320,445,153,387,292,417)(126,344,321,430,154,372,293,402)(127,385,322,415,155,357,294,443)(128,370,323,400,156,342,295,428)(129,355,324,441,157,383,296,413)(130,340,325,426,158,368,297,398)(131,381,326,411,159,353,298,439)(132,366,327,396,160,338,299,424)(133,351,328,437,161,379,300,409)(134,392,329,422,162,364,301,394)(135,377,330,407,163,349,302,435)(136,362,331,448,164,390,303,420)(137,347,332,433,165,375,304,405)(138,388,333,418,166,360,305,446)(139,373,334,403,167,345,306,431)(140,358,335,444,168,386,307,416) );
G=PermutationGroup([[(1,121,84,330),(2,331,85,122),(3,123,86,332),(4,333,87,124),(5,125,88,334),(6,335,89,126),(7,127,90,336),(8,281,91,128),(9,129,92,282),(10,283,93,130),(11,131,94,284),(12,285,95,132),(13,133,96,286),(14,287,97,134),(15,135,98,288),(16,289,99,136),(17,137,100,290),(18,291,101,138),(19,139,102,292),(20,293,103,140),(21,141,104,294),(22,295,105,142),(23,143,106,296),(24,297,107,144),(25,145,108,298),(26,299,109,146),(27,147,110,300),(28,301,111,148),(29,149,112,302),(30,303,57,150),(31,151,58,304),(32,305,59,152),(33,153,60,306),(34,307,61,154),(35,155,62,308),(36,309,63,156),(37,157,64,310),(38,311,65,158),(39,159,66,312),(40,313,67,160),(41,161,68,314),(42,315,69,162),(43,163,70,316),(44,317,71,164),(45,165,72,318),(46,319,73,166),(47,167,74,320),(48,321,75,168),(49,113,76,322),(50,323,77,114),(51,115,78,324),(52,325,79,116),(53,117,80,326),(54,327,81,118),(55,119,82,328),(56,329,83,120),(169,344,238,444),(170,445,239,345),(171,346,240,446),(172,447,241,347),(173,348,242,448),(174,393,243,349),(175,350,244,394),(176,395,245,351),(177,352,246,396),(178,397,247,353),(179,354,248,398),(180,399,249,355),(181,356,250,400),(182,401,251,357),(183,358,252,402),(184,403,253,359),(185,360,254,404),(186,405,255,361),(187,362,256,406),(188,407,257,363),(189,364,258,408),(190,409,259,365),(191,366,260,410),(192,411,261,367),(193,368,262,412),(194,413,263,369),(195,370,264,414),(196,415,265,371),(197,372,266,416),(198,417,267,373),(199,374,268,418),(200,419,269,375),(201,376,270,420),(202,421,271,377),(203,378,272,422),(204,423,273,379),(205,380,274,424),(206,425,275,381),(207,382,276,426),(208,427,277,383),(209,384,278,428),(210,429,279,385),(211,386,280,430),(212,431,225,387),(213,388,226,432),(214,433,227,389),(215,390,228,434),(216,435,229,391),(217,392,230,436),(218,437,231,337),(219,338,232,438),(220,439,233,339),(221,340,234,440),(222,441,235,341),(223,342,236,442),(224,443,237,343)], [(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)], [(1,229,70,202,29,257,98,174),(2,270,71,187,30,242,99,215),(3,255,72,172,31,227,100,200),(4,240,73,213,32,268,101,185),(5,225,74,198,33,253,102,170),(6,266,75,183,34,238,103,211),(7,251,76,224,35,279,104,196),(8,236,77,209,36,264,105,181),(9,277,78,194,37,249,106,222),(10,262,79,179,38,234,107,207),(11,247,80,220,39,275,108,192),(12,232,81,205,40,260,109,177),(13,273,82,190,41,245,110,218),(14,258,83,175,42,230,111,203),(15,243,84,216,43,271,112,188),(16,228,85,201,44,256,57,173),(17,269,86,186,45,241,58,214),(18,254,87,171,46,226,59,199),(19,239,88,212,47,267,60,184),(20,280,89,197,48,252,61,169),(21,265,90,182,49,237,62,210),(22,250,91,223,50,278,63,195),(23,235,92,208,51,263,64,180),(24,276,93,193,52,248,65,221),(25,261,94,178,53,233,66,206),(26,246,95,219,54,274,67,191),(27,231,96,204,55,259,68,176),(28,272,97,189,56,244,69,217),(113,371,308,401,141,343,336,429),(114,356,309,442,142,384,281,414),(115,341,310,427,143,369,282,399),(116,382,311,412,144,354,283,440),(117,367,312,397,145,339,284,425),(118,352,313,438,146,380,285,410),(119,337,314,423,147,365,286,395),(120,378,315,408,148,350,287,436),(121,363,316,393,149,391,288,421),(122,348,317,434,150,376,289,406),(123,389,318,419,151,361,290,447),(124,374,319,404,152,346,291,432),(125,359,320,445,153,387,292,417),(126,344,321,430,154,372,293,402),(127,385,322,415,155,357,294,443),(128,370,323,400,156,342,295,428),(129,355,324,441,157,383,296,413),(130,340,325,426,158,368,297,398),(131,381,326,411,159,353,298,439),(132,366,327,396,160,338,299,424),(133,351,328,437,161,379,300,409),(134,392,329,422,162,364,301,394),(135,377,330,407,163,349,302,435),(136,362,331,448,164,390,303,420),(137,347,332,433,165,375,304,405),(138,388,333,418,166,360,305,446),(139,373,334,403,167,345,306,431),(140,358,335,444,168,386,307,416)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | D7 | M4(2) | C4○D4 | D14 | D14 | C8○D4 | C4×D7 | D28.2C4 | D4⋊2D7 | Q8×D7 | D7×M4(2) |
| kernel | C42.27D14 | C42.D7 | C8×Dic7 | Dic7⋊C8 | C56⋊C4 | C7×C4⋊C8 | C4×Dic14 | Dic7⋊C4 | C4⋊Dic7 | C2×Dic14 | C7⋊C8 | C4⋊C8 | Dic7 | C28 | C42 | C2×C8 | C14 | C2×C4 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 4 | 12 | 24 | 3 | 3 | 6 |
Matrix representation of C42.27D14 ►in GL4(𝔽113) generated by
| 96 | 106 | 0 | 0 |
| 106 | 17 | 0 | 0 |
| 0 | 0 | 34 | 101 |
| 0 | 0 | 68 | 79 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 56 | 57 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 29 |
| 0 | 0 | 74 | 109 |
G:=sub<GL(4,GF(113))| [96,106,0,0,106,17,0,0,0,0,34,68,0,0,101,79],[112,0,0,0,0,112,0,0,0,0,15,0,0,0,0,15],[0,112,0,0,1,0,0,0,0,0,0,56,0,0,30,57],[1,0,0,0,0,1,0,0,0,0,4,74,0,0,29,109] >;
C42.27D14 in GAP, Magma, Sage, TeX
C_4^2._{27}D_{14} % in TeX
G:=Group("C4^2.27D14"); // GroupNames label
G:=SmallGroup(448,362);
// by ID
G=gap.SmallGroup(448,362);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,120,219,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^-1,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^13>;
// generators/relations