metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.2D14, C14.6C4≀C2, C8⋊C4.3D7, C4⋊Dic7.1C4, (C2×C28).224D4, (C2×C4).106D28, (C4×C28).11C22, C28.6Q8.4C2, C2.7(Dic14⋊C4), C2.5(D28⋊4C4), C42.D7.1C2, C22.57(D14⋊C4), C14.1(C4.10D4), C2.3(C4.12D28), C7⋊1(C42.2C22), (C2×C4).11(C4×D7), (C7×C8⋊C4).7C2, (C2×C28).23(C2×C4), (C2×C4).207(C7⋊D4), (C2×C14).38(C22⋊C4), SmallGroup(448,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.2D14
G = < a,b,c,d | a4=b4=1, c14=a-1, d2=ba=ab, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=bc13 >
Subgroups: 260 in 60 conjugacy classes, 25 normal (all characteristic)
C1, C2, C4, C22, C7, C8, C2×C4, C2×C4, C14, C42, C4⋊C4, C2×C8, Dic7, C28, C2×C14, C8⋊C4, C8⋊C4, C42.C2, C7⋊C8, C56, C2×Dic7, C2×C28, C42.2C22, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C42.D7, C7×C8⋊C4, C28.6Q8, C42.2D14
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C4.10D4, C4≀C2, C4×D7, D28, C7⋊D4, C42.2C22, D14⋊C4, Dic14⋊C4, C4.12D28, D28⋊4C4, C42.2D14
►Regular action on 448 points
Generators in S448
(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)
(1 246 168 66)(2 275 113 95)(3 248 114 68)(4 277 115 97)(5 250 116 70)(6 279 117 99)(7 252 118 72)(8 225 119 101)(9 254 120 74)(10 227 121 103)(11 256 122 76)(12 229 123 105)(13 258 124 78)(14 231 125 107)(15 260 126 80)(16 233 127 109)(17 262 128 82)(18 235 129 111)(19 264 130 84)(20 237 131 57)(21 266 132 86)(22 239 133 59)(23 268 134 88)(24 241 135 61)(25 270 136 90)(26 243 137 63)(27 272 138 92)(28 245 139 65)(29 274 140 94)(30 247 141 67)(31 276 142 96)(32 249 143 69)(33 278 144 98)(34 251 145 71)(35 280 146 100)(36 253 147 73)(37 226 148 102)(38 255 149 75)(39 228 150 104)(40 257 151 77)(41 230 152 106)(42 259 153 79)(43 232 154 108)(44 261 155 81)(45 234 156 110)(46 263 157 83)(47 236 158 112)(48 265 159 85)(49 238 160 58)(50 267 161 87)(51 240 162 60)(52 269 163 89)(53 242 164 62)(54 271 165 91)(55 244 166 64)(56 273 167 93)(169 363 427 327)(170 392 428 300)(171 365 429 329)(172 338 430 302)(173 367 431 331)(174 340 432 304)(175 369 433 333)(176 342 434 306)(177 371 435 335)(178 344 436 308)(179 373 437 281)(180 346 438 310)(181 375 439 283)(182 348 440 312)(183 377 441 285)(184 350 442 314)(185 379 443 287)(186 352 444 316)(187 381 445 289)(188 354 446 318)(189 383 447 291)(190 356 448 320)(191 385 393 293)(192 358 394 322)(193 387 395 295)(194 360 396 324)(195 389 397 297)(196 362 398 326)(197 391 399 299)(198 364 400 328)(199 337 401 301)(200 366 402 330)(201 339 403 303)(202 368 404 332)(203 341 405 305)(204 370 406 334)(205 343 407 307)(206 372 408 336)(207 345 409 309)(208 374 410 282)(209 347 411 311)(210 376 412 284)(211 349 413 313)(212 378 414 286)(213 351 415 315)(214 380 416 288)(215 353 417 317)(216 382 418 290)(217 355 419 319)(218 384 420 292)(219 357 421 321)(220 386 422 294)(221 359 423 323)(222 388 424 296)(223 361 425 325)(224 390 426 298)
(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)
(1 434 232 292 140 204 80 356)(2 291 261 175 141 355 109 405)(3 174 234 382 142 404 82 318)(4 381 263 431 143 317 111 201)(5 430 236 288 144 200 84 352)(6 287 265 171 145 351 57 401)(7 170 238 378 146 400 86 314)(8 377 267 427 147 313 59 197)(9 426 240 284 148 196 88 348)(10 283 269 223 149 347 61 397)(11 222 242 374 150 396 90 310)(12 373 271 423 151 309 63 193)(13 422 244 336 152 192 92 344)(14 335 273 219 153 343 65 393)(15 218 246 370 154 448 94 306)(16 369 275 419 155 305 67 189)(17 418 248 332 156 188 96 340)(18 331 277 215 157 339 69 445)(19 214 250 366 158 444 98 302)(20 365 279 415 159 301 71 185)(21 414 252 328 160 184 100 392)(22 327 225 211 161 391 73 441)(23 210 254 362 162 440 102 298)(24 361 227 411 163 297 75 181)(25 410 256 324 164 180 104 388)(26 323 229 207 165 387 77 437)(27 206 258 358 166 436 106 294)(28 357 231 407 167 293 79 177)(29 406 260 320 168 176 108 384)(30 319 233 203 113 383 81 433)(31 202 262 354 114 432 110 290)(32 353 235 403 115 289 83 173)(33 402 264 316 116 172 112 380)(34 315 237 199 117 379 85 429)(35 198 266 350 118 428 58 286)(36 349 239 399 119 285 87 169)(37 398 268 312 120 224 60 376)(38 311 241 195 121 375 89 425)(39 194 270 346 122 424 62 282)(40 345 243 395 123 281 91 221)(41 394 272 308 124 220 64 372)(42 307 245 191 125 371 93 421)(43 190 274 342 126 420 66 334)(44 341 247 447 127 333 95 217)(45 446 276 304 128 216 68 368)(46 303 249 187 129 367 97 417)(47 186 278 338 130 416 70 330)(48 337 251 443 131 329 99 213)(49 442 280 300 132 212 72 364)(50 299 253 183 133 363 101 413)(51 182 226 390 134 412 74 326)(52 389 255 439 135 325 103 209)(53 438 228 296 136 208 76 360)(54 295 257 179 137 359 105 409)(55 178 230 386 138 408 78 322)(56 385 259 435 139 321 107 205)
G:=sub<Sym(448)| (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), (1,246,168,66)(2,275,113,95)(3,248,114,68)(4,277,115,97)(5,250,116,70)(6,279,117,99)(7,252,118,72)(8,225,119,101)(9,254,120,74)(10,227,121,103)(11,256,122,76)(12,229,123,105)(13,258,124,78)(14,231,125,107)(15,260,126,80)(16,233,127,109)(17,262,128,82)(18,235,129,111)(19,264,130,84)(20,237,131,57)(21,266,132,86)(22,239,133,59)(23,268,134,88)(24,241,135,61)(25,270,136,90)(26,243,137,63)(27,272,138,92)(28,245,139,65)(29,274,140,94)(30,247,141,67)(31,276,142,96)(32,249,143,69)(33,278,144,98)(34,251,145,71)(35,280,146,100)(36,253,147,73)(37,226,148,102)(38,255,149,75)(39,228,150,104)(40,257,151,77)(41,230,152,106)(42,259,153,79)(43,232,154,108)(44,261,155,81)(45,234,156,110)(46,263,157,83)(47,236,158,112)(48,265,159,85)(49,238,160,58)(50,267,161,87)(51,240,162,60)(52,269,163,89)(53,242,164,62)(54,271,165,91)(55,244,166,64)(56,273,167,93)(169,363,427,327)(170,392,428,300)(171,365,429,329)(172,338,430,302)(173,367,431,331)(174,340,432,304)(175,369,433,333)(176,342,434,306)(177,371,435,335)(178,344,436,308)(179,373,437,281)(180,346,438,310)(181,375,439,283)(182,348,440,312)(183,377,441,285)(184,350,442,314)(185,379,443,287)(186,352,444,316)(187,381,445,289)(188,354,446,318)(189,383,447,291)(190,356,448,320)(191,385,393,293)(192,358,394,322)(193,387,395,295)(194,360,396,324)(195,389,397,297)(196,362,398,326)(197,391,399,299)(198,364,400,328)(199,337,401,301)(200,366,402,330)(201,339,403,303)(202,368,404,332)(203,341,405,305)(204,370,406,334)(205,343,407,307)(206,372,408,336)(207,345,409,309)(208,374,410,282)(209,347,411,311)(210,376,412,284)(211,349,413,313)(212,378,414,286)(213,351,415,315)(214,380,416,288)(215,353,417,317)(216,382,418,290)(217,355,419,319)(218,384,420,292)(219,357,421,321)(220,386,422,294)(221,359,423,323)(222,388,424,296)(223,361,425,325)(224,390,426,298), (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), (1,434,232,292,140,204,80,356)(2,291,261,175,141,355,109,405)(3,174,234,382,142,404,82,318)(4,381,263,431,143,317,111,201)(5,430,236,288,144,200,84,352)(6,287,265,171,145,351,57,401)(7,170,238,378,146,400,86,314)(8,377,267,427,147,313,59,197)(9,426,240,284,148,196,88,348)(10,283,269,223,149,347,61,397)(11,222,242,374,150,396,90,310)(12,373,271,423,151,309,63,193)(13,422,244,336,152,192,92,344)(14,335,273,219,153,343,65,393)(15,218,246,370,154,448,94,306)(16,369,275,419,155,305,67,189)(17,418,248,332,156,188,96,340)(18,331,277,215,157,339,69,445)(19,214,250,366,158,444,98,302)(20,365,279,415,159,301,71,185)(21,414,252,328,160,184,100,392)(22,327,225,211,161,391,73,441)(23,210,254,362,162,440,102,298)(24,361,227,411,163,297,75,181)(25,410,256,324,164,180,104,388)(26,323,229,207,165,387,77,437)(27,206,258,358,166,436,106,294)(28,357,231,407,167,293,79,177)(29,406,260,320,168,176,108,384)(30,319,233,203,113,383,81,433)(31,202,262,354,114,432,110,290)(32,353,235,403,115,289,83,173)(33,402,264,316,116,172,112,380)(34,315,237,199,117,379,85,429)(35,198,266,350,118,428,58,286)(36,349,239,399,119,285,87,169)(37,398,268,312,120,224,60,376)(38,311,241,195,121,375,89,425)(39,194,270,346,122,424,62,282)(40,345,243,395,123,281,91,221)(41,394,272,308,124,220,64,372)(42,307,245,191,125,371,93,421)(43,190,274,342,126,420,66,334)(44,341,247,447,127,333,95,217)(45,446,276,304,128,216,68,368)(46,303,249,187,129,367,97,417)(47,186,278,338,130,416,70,330)(48,337,251,443,131,329,99,213)(49,442,280,300,132,212,72,364)(50,299,253,183,133,363,101,413)(51,182,226,390,134,412,74,326)(52,389,255,439,135,325,103,209)(53,438,228,296,136,208,76,360)(54,295,257,179,137,359,105,409)(55,178,230,386,138,408,78,322)(56,385,259,435,139,321,107,205)>;
G:=Group( (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), (1,246,168,66)(2,275,113,95)(3,248,114,68)(4,277,115,97)(5,250,116,70)(6,279,117,99)(7,252,118,72)(8,225,119,101)(9,254,120,74)(10,227,121,103)(11,256,122,76)(12,229,123,105)(13,258,124,78)(14,231,125,107)(15,260,126,80)(16,233,127,109)(17,262,128,82)(18,235,129,111)(19,264,130,84)(20,237,131,57)(21,266,132,86)(22,239,133,59)(23,268,134,88)(24,241,135,61)(25,270,136,90)(26,243,137,63)(27,272,138,92)(28,245,139,65)(29,274,140,94)(30,247,141,67)(31,276,142,96)(32,249,143,69)(33,278,144,98)(34,251,145,71)(35,280,146,100)(36,253,147,73)(37,226,148,102)(38,255,149,75)(39,228,150,104)(40,257,151,77)(41,230,152,106)(42,259,153,79)(43,232,154,108)(44,261,155,81)(45,234,156,110)(46,263,157,83)(47,236,158,112)(48,265,159,85)(49,238,160,58)(50,267,161,87)(51,240,162,60)(52,269,163,89)(53,242,164,62)(54,271,165,91)(55,244,166,64)(56,273,167,93)(169,363,427,327)(170,392,428,300)(171,365,429,329)(172,338,430,302)(173,367,431,331)(174,340,432,304)(175,369,433,333)(176,342,434,306)(177,371,435,335)(178,344,436,308)(179,373,437,281)(180,346,438,310)(181,375,439,283)(182,348,440,312)(183,377,441,285)(184,350,442,314)(185,379,443,287)(186,352,444,316)(187,381,445,289)(188,354,446,318)(189,383,447,291)(190,356,448,320)(191,385,393,293)(192,358,394,322)(193,387,395,295)(194,360,396,324)(195,389,397,297)(196,362,398,326)(197,391,399,299)(198,364,400,328)(199,337,401,301)(200,366,402,330)(201,339,403,303)(202,368,404,332)(203,341,405,305)(204,370,406,334)(205,343,407,307)(206,372,408,336)(207,345,409,309)(208,374,410,282)(209,347,411,311)(210,376,412,284)(211,349,413,313)(212,378,414,286)(213,351,415,315)(214,380,416,288)(215,353,417,317)(216,382,418,290)(217,355,419,319)(218,384,420,292)(219,357,421,321)(220,386,422,294)(221,359,423,323)(222,388,424,296)(223,361,425,325)(224,390,426,298), (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), (1,434,232,292,140,204,80,356)(2,291,261,175,141,355,109,405)(3,174,234,382,142,404,82,318)(4,381,263,431,143,317,111,201)(5,430,236,288,144,200,84,352)(6,287,265,171,145,351,57,401)(7,170,238,378,146,400,86,314)(8,377,267,427,147,313,59,197)(9,426,240,284,148,196,88,348)(10,283,269,223,149,347,61,397)(11,222,242,374,150,396,90,310)(12,373,271,423,151,309,63,193)(13,422,244,336,152,192,92,344)(14,335,273,219,153,343,65,393)(15,218,246,370,154,448,94,306)(16,369,275,419,155,305,67,189)(17,418,248,332,156,188,96,340)(18,331,277,215,157,339,69,445)(19,214,250,366,158,444,98,302)(20,365,279,415,159,301,71,185)(21,414,252,328,160,184,100,392)(22,327,225,211,161,391,73,441)(23,210,254,362,162,440,102,298)(24,361,227,411,163,297,75,181)(25,410,256,324,164,180,104,388)(26,323,229,207,165,387,77,437)(27,206,258,358,166,436,106,294)(28,357,231,407,167,293,79,177)(29,406,260,320,168,176,108,384)(30,319,233,203,113,383,81,433)(31,202,262,354,114,432,110,290)(32,353,235,403,115,289,83,173)(33,402,264,316,116,172,112,380)(34,315,237,199,117,379,85,429)(35,198,266,350,118,428,58,286)(36,349,239,399,119,285,87,169)(37,398,268,312,120,224,60,376)(38,311,241,195,121,375,89,425)(39,194,270,346,122,424,62,282)(40,345,243,395,123,281,91,221)(41,394,272,308,124,220,64,372)(42,307,245,191,125,371,93,421)(43,190,274,342,126,420,66,334)(44,341,247,447,127,333,95,217)(45,446,276,304,128,216,68,368)(46,303,249,187,129,367,97,417)(47,186,278,338,130,416,70,330)(48,337,251,443,131,329,99,213)(49,442,280,300,132,212,72,364)(50,299,253,183,133,363,101,413)(51,182,226,390,134,412,74,326)(52,389,255,439,135,325,103,209)(53,438,228,296,136,208,76,360)(54,295,257,179,137,359,105,409)(55,178,230,386,138,408,78,322)(56,385,259,435,139,321,107,205) );
G=PermutationGroup([[(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)], [(1,246,168,66),(2,275,113,95),(3,248,114,68),(4,277,115,97),(5,250,116,70),(6,279,117,99),(7,252,118,72),(8,225,119,101),(9,254,120,74),(10,227,121,103),(11,256,122,76),(12,229,123,105),(13,258,124,78),(14,231,125,107),(15,260,126,80),(16,233,127,109),(17,262,128,82),(18,235,129,111),(19,264,130,84),(20,237,131,57),(21,266,132,86),(22,239,133,59),(23,268,134,88),(24,241,135,61),(25,270,136,90),(26,243,137,63),(27,272,138,92),(28,245,139,65),(29,274,140,94),(30,247,141,67),(31,276,142,96),(32,249,143,69),(33,278,144,98),(34,251,145,71),(35,280,146,100),(36,253,147,73),(37,226,148,102),(38,255,149,75),(39,228,150,104),(40,257,151,77),(41,230,152,106),(42,259,153,79),(43,232,154,108),(44,261,155,81),(45,234,156,110),(46,263,157,83),(47,236,158,112),(48,265,159,85),(49,238,160,58),(50,267,161,87),(51,240,162,60),(52,269,163,89),(53,242,164,62),(54,271,165,91),(55,244,166,64),(56,273,167,93),(169,363,427,327),(170,392,428,300),(171,365,429,329),(172,338,430,302),(173,367,431,331),(174,340,432,304),(175,369,433,333),(176,342,434,306),(177,371,435,335),(178,344,436,308),(179,373,437,281),(180,346,438,310),(181,375,439,283),(182,348,440,312),(183,377,441,285),(184,350,442,314),(185,379,443,287),(186,352,444,316),(187,381,445,289),(188,354,446,318),(189,383,447,291),(190,356,448,320),(191,385,393,293),(192,358,394,322),(193,387,395,295),(194,360,396,324),(195,389,397,297),(196,362,398,326),(197,391,399,299),(198,364,400,328),(199,337,401,301),(200,366,402,330),(201,339,403,303),(202,368,404,332),(203,341,405,305),(204,370,406,334),(205,343,407,307),(206,372,408,336),(207,345,409,309),(208,374,410,282),(209,347,411,311),(210,376,412,284),(211,349,413,313),(212,378,414,286),(213,351,415,315),(214,380,416,288),(215,353,417,317),(216,382,418,290),(217,355,419,319),(218,384,420,292),(219,357,421,321),(220,386,422,294),(221,359,423,323),(222,388,424,296),(223,361,425,325),(224,390,426,298)], [(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)], [(1,434,232,292,140,204,80,356),(2,291,261,175,141,355,109,405),(3,174,234,382,142,404,82,318),(4,381,263,431,143,317,111,201),(5,430,236,288,144,200,84,352),(6,287,265,171,145,351,57,401),(7,170,238,378,146,400,86,314),(8,377,267,427,147,313,59,197),(9,426,240,284,148,196,88,348),(10,283,269,223,149,347,61,397),(11,222,242,374,150,396,90,310),(12,373,271,423,151,309,63,193),(13,422,244,336,152,192,92,344),(14,335,273,219,153,343,65,393),(15,218,246,370,154,448,94,306),(16,369,275,419,155,305,67,189),(17,418,248,332,156,188,96,340),(18,331,277,215,157,339,69,445),(19,214,250,366,158,444,98,302),(20,365,279,415,159,301,71,185),(21,414,252,328,160,184,100,392),(22,327,225,211,161,391,73,441),(23,210,254,362,162,440,102,298),(24,361,227,411,163,297,75,181),(25,410,256,324,164,180,104,388),(26,323,229,207,165,387,77,437),(27,206,258,358,166,436,106,294),(28,357,231,407,167,293,79,177),(29,406,260,320,168,176,108,384),(30,319,233,203,113,383,81,433),(31,202,262,354,114,432,110,290),(32,353,235,403,115,289,83,173),(33,402,264,316,116,172,112,380),(34,315,237,199,117,379,85,429),(35,198,266,350,118,428,58,286),(36,349,239,399,119,285,87,169),(37,398,268,312,120,224,60,376),(38,311,241,195,121,375,89,425),(39,194,270,346,122,424,62,282),(40,345,243,395,123,281,91,221),(41,394,272,308,124,220,64,372),(42,307,245,191,125,371,93,421),(43,190,274,342,126,420,66,334),(44,341,247,447,127,333,95,217),(45,446,276,304,128,216,68,368),(46,303,249,187,129,367,97,417),(47,186,278,338,130,416,70,330),(48,337,251,443,131,329,99,213),(49,442,280,300,132,212,72,364),(50,299,253,183,133,363,101,413),(51,182,226,390,134,412,74,326),(52,389,255,439,135,325,103,209),(53,438,228,296,136,208,76,360),(54,295,257,179,137,359,105,409),(55,178,230,386,138,408,78,322),(56,385,259,435,139,321,107,205)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D7 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | Dic14⋊C4 | C4.10D4 | C4.12D28 | D28⋊4C4 |
| kernel | C42.2D14 | C42.D7 | C7×C8⋊C4 | C28.6Q8 | C4⋊Dic7 | C2×C28 | C8⋊C4 | C42 | C14 | C2×C4 | C2×C4 | C2×C4 | C2 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 3 | 8 | 6 | 6 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of C42.2D14 ►in GL4(𝔽113) generated by
| 29 | 7 | 0 | 0 |
| 106 | 84 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 17 | 8 | 0 | 0 |
| 105 | 96 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 1 | 112 |
| 5 | 22 | 0 | 0 |
| 91 | 81 | 0 | 0 |
| 0 | 0 | 11 | 83 |
| 0 | 0 | 109 | 102 |
| 4 | 102 | 0 | 0 |
| 85 | 109 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 106 | 14 |
G:=sub<GL(4,GF(113))| [29,106,0,0,7,84,0,0,0,0,15,0,0,0,0,15],[17,105,0,0,8,96,0,0,0,0,1,1,0,0,111,112],[5,91,0,0,22,81,0,0,0,0,11,109,0,0,83,102],[4,85,0,0,102,109,0,0,0,0,0,106,0,0,14,14] >;
C42.2D14 in GAP, Magma, Sage, TeX
C_4^2._2D_{14} % in TeX
G:=Group("C4^2.2D14"); // GroupNames label
G:=SmallGroup(448,22);
// by ID
G=gap.SmallGroup(448,22);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,422,184,1571,570,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^-1,d^2=b*a=a*b,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=b*c^13>;
// generators/relations