metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.147D14, C14.272- (1+4), C4⋊C4.110D14, C42.C2.7D7, Dic7.Q8.3C2, (C2×C28).187C23, (C4×C28).223C22, (C2×C14).230C24, C7⋊(C22.58C24), C4.Dic14.13C2, C28.6Q8.12C2, Dic7⋊C4.85C22, C4⋊Dic7.237C22, C22.251(C23×D7), (C2×Dic7).120C23, (C4×Dic7).138C22, C2.56(D4.10D14), C2.28(Q8.10D14), (C7×C42.C2).6C2, (C7×C4⋊C4).185C22, (C2×C4).202(C22×D7), SmallGroup(448,1139)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 604 in 172 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2 [×2], C4 [×15], C22, C7, C2×C4, C2×C4 [×6], C2×C4 [×8], C14, C14 [×2], C42, C42 [×4], C4⋊C4 [×6], C4⋊C4 [×24], Dic7 [×8], C28 [×7], C2×C14, C42.C2, C42.C2 [×14], C2×Dic7 [×8], C2×C28, C2×C28 [×6], C22.58C24, C4×Dic7 [×4], Dic7⋊C4 [×16], C4⋊Dic7 [×8], C4×C28, C7×C4⋊C4 [×6], C28.6Q8 [×2], Dic7.Q8 [×8], C4.Dic14 [×4], C7×C42.C2, C42.147D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2- (1+4) [×3], C22×D7 [×7], C22.58C24, C23×D7, Q8.10D14, D4.10D14 [×2], C42.147D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=a2, d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c13 >
►Regular action on 448 points
Generators in S448
(1 97 15 111)(2 404 16 418)(3 99 17 85)(4 406 18 420)(5 101 19 87)(6 408 20 394)(7 103 21 89)(8 410 22 396)(9 105 23 91)(10 412 24 398)(11 107 25 93)(12 414 26 400)(13 109 27 95)(14 416 28 402)(29 220 43 206)(30 364 44 350)(31 222 45 208)(32 338 46 352)(33 224 47 210)(34 340 48 354)(35 198 49 212)(36 342 50 356)(37 200 51 214)(38 344 52 358)(39 202 53 216)(40 346 54 360)(41 204 55 218)(42 348 56 362)(57 116 71 130)(58 374 72 388)(59 118 73 132)(60 376 74 390)(61 120 75 134)(62 378 76 392)(63 122 77 136)(64 380 78 366)(65 124 79 138)(66 382 80 368)(67 126 81 140)(68 384 82 370)(69 128 83 114)(70 386 84 372)(86 437 100 423)(88 439 102 425)(90 441 104 427)(92 443 106 429)(94 445 108 431)(96 447 110 433)(98 421 112 435)(113 272 127 258)(115 274 129 260)(117 276 131 262)(119 278 133 264)(121 280 135 266)(123 254 137 268)(125 256 139 270)(141 232 155 246)(142 296 156 282)(143 234 157 248)(144 298 158 284)(145 236 159 250)(146 300 160 286)(147 238 161 252)(148 302 162 288)(149 240 163 226)(150 304 164 290)(151 242 165 228)(152 306 166 292)(153 244 167 230)(154 308 168 294)(169 243 183 229)(170 307 184 293)(171 245 185 231)(172 281 186 295)(173 247 187 233)(174 283 188 297)(175 249 189 235)(176 285 190 299)(177 251 191 237)(178 287 192 301)(179 225 193 239)(180 289 194 303)(181 227 195 241)(182 291 196 305)(197 323 211 309)(199 325 213 311)(201 327 215 313)(203 329 217 315)(205 331 219 317)(207 333 221 319)(209 335 223 321)(253 365 267 379)(255 367 269 381)(257 369 271 383)(259 371 273 385)(261 373 275 387)(263 375 277 389)(265 377 279 391)(310 341 324 355)(312 343 326 357)(314 345 328 359)(316 347 330 361)(318 349 332 363)(320 351 334 337)(322 353 336 339)(393 438 407 424)(395 440 409 426)(397 442 411 428)(399 444 413 430)(401 446 415 432)(403 448 417 434)(405 422 419 436)
(1 351 448 222)(2 223 421 352)(3 353 422 224)(4 197 423 354)(5 355 424 198)(6 199 425 356)(7 357 426 200)(8 201 427 358)(9 359 428 202)(10 203 429 360)(11 361 430 204)(12 205 431 362)(13 363 432 206)(14 207 433 364)(15 337 434 208)(16 209 435 338)(17 339 436 210)(18 211 437 340)(19 341 438 212)(20 213 439 342)(21 343 440 214)(22 215 441 344)(23 345 442 216)(24 217 443 346)(25 347 444 218)(26 219 445 348)(27 349 446 220)(28 221 447 350)(29 109 318 401)(30 402 319 110)(31 111 320 403)(32 404 321 112)(33 85 322 405)(34 406 323 86)(35 87 324 407)(36 408 325 88)(37 89 326 409)(38 410 327 90)(39 91 328 411)(40 412 329 92)(41 93 330 413)(42 414 331 94)(43 95 332 415)(44 416 333 96)(45 97 334 417)(46 418 335 98)(47 99 336 419)(48 420 309 100)(49 101 310 393)(50 394 311 102)(51 103 312 395)(52 396 313 104)(53 105 314 397)(54 398 315 106)(55 107 316 399)(56 400 317 108)(57 171 275 168)(58 141 276 172)(59 173 277 142)(60 143 278 174)(61 175 279 144)(62 145 280 176)(63 177 253 146)(64 147 254 178)(65 179 255 148)(66 149 256 180)(67 181 257 150)(68 151 258 182)(69 183 259 152)(70 153 260 184)(71 185 261 154)(72 155 262 186)(73 187 263 156)(74 157 264 188)(75 189 265 158)(76 159 266 190)(77 191 267 160)(78 161 268 192)(79 193 269 162)(80 163 270 194)(81 195 271 164)(82 165 272 196)(83 169 273 166)(84 167 274 170)(113 291 384 242)(114 243 385 292)(115 293 386 244)(116 245 387 294)(117 295 388 246)(118 247 389 296)(119 297 390 248)(120 249 391 298)(121 299 392 250)(122 251 365 300)(123 301 366 252)(124 225 367 302)(125 303 368 226)(126 227 369 304)(127 305 370 228)(128 229 371 306)(129 307 372 230)(130 231 373 308)(131 281 374 232)(132 233 375 282)(133 283 376 234)(134 235 377 284)(135 285 378 236)(136 237 379 286)(137 287 380 238)(138 239 381 288)(139 289 382 240)(140 241 383 290)
(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)
(1 114 434 371)(2 127 435 384)(3 140 436 369)(4 125 437 382)(5 138 438 367)(6 123 439 380)(7 136 440 365)(8 121 441 378)(9 134 442 391)(10 119 443 376)(11 132 444 389)(12 117 445 374)(13 130 446 387)(14 115 447 372)(15 128 448 385)(16 113 421 370)(17 126 422 383)(18 139 423 368)(19 124 424 381)(20 137 425 366)(21 122 426 379)(22 135 427 392)(23 120 428 377)(24 133 429 390)(25 118 430 375)(26 131 431 388)(27 116 432 373)(28 129 433 386)(29 171 332 154)(30 184 333 167)(31 169 334 152)(32 182 335 165)(33 195 336 150)(34 180 309 163)(35 193 310 148)(36 178 311 161)(37 191 312 146)(38 176 313 159)(39 189 314 144)(40 174 315 157)(41 187 316 142)(42 172 317 155)(43 185 318 168)(44 170 319 153)(45 183 320 166)(46 196 321 151)(47 181 322 164)(48 194 323 149)(49 179 324 162)(50 192 325 147)(51 177 326 160)(52 190 327 145)(53 175 328 158)(54 188 329 143)(55 173 330 156)(56 186 331 141)(57 401 261 95)(58 414 262 108)(59 399 263 93)(60 412 264 106)(61 397 265 91)(62 410 266 104)(63 395 267 89)(64 408 268 102)(65 393 269 87)(66 406 270 100)(67 419 271 85)(68 404 272 98)(69 417 273 111)(70 402 274 96)(71 415 275 109)(72 400 276 94)(73 413 277 107)(74 398 278 92)(75 411 279 105)(76 396 280 90)(77 409 253 103)(78 394 254 88)(79 407 255 101)(80 420 256 86)(81 405 257 99)(82 418 258 112)(83 403 259 97)(84 416 260 110)(197 240 340 303)(198 225 341 288)(199 238 342 301)(200 251 343 286)(201 236 344 299)(202 249 345 284)(203 234 346 297)(204 247 347 282)(205 232 348 295)(206 245 349 308)(207 230 350 293)(208 243 351 306)(209 228 352 291)(210 241 353 304)(211 226 354 289)(212 239 355 302)(213 252 356 287)(214 237 357 300)(215 250 358 285)(216 235 359 298)(217 248 360 283)(218 233 361 296)(219 246 362 281)(220 231 363 294)(221 244 364 307)(222 229 337 292)(223 242 338 305)(224 227 339 290)
G:=sub<Sym(448)| (1,97,15,111)(2,404,16,418)(3,99,17,85)(4,406,18,420)(5,101,19,87)(6,408,20,394)(7,103,21,89)(8,410,22,396)(9,105,23,91)(10,412,24,398)(11,107,25,93)(12,414,26,400)(13,109,27,95)(14,416,28,402)(29,220,43,206)(30,364,44,350)(31,222,45,208)(32,338,46,352)(33,224,47,210)(34,340,48,354)(35,198,49,212)(36,342,50,356)(37,200,51,214)(38,344,52,358)(39,202,53,216)(40,346,54,360)(41,204,55,218)(42,348,56,362)(57,116,71,130)(58,374,72,388)(59,118,73,132)(60,376,74,390)(61,120,75,134)(62,378,76,392)(63,122,77,136)(64,380,78,366)(65,124,79,138)(66,382,80,368)(67,126,81,140)(68,384,82,370)(69,128,83,114)(70,386,84,372)(86,437,100,423)(88,439,102,425)(90,441,104,427)(92,443,106,429)(94,445,108,431)(96,447,110,433)(98,421,112,435)(113,272,127,258)(115,274,129,260)(117,276,131,262)(119,278,133,264)(121,280,135,266)(123,254,137,268)(125,256,139,270)(141,232,155,246)(142,296,156,282)(143,234,157,248)(144,298,158,284)(145,236,159,250)(146,300,160,286)(147,238,161,252)(148,302,162,288)(149,240,163,226)(150,304,164,290)(151,242,165,228)(152,306,166,292)(153,244,167,230)(154,308,168,294)(169,243,183,229)(170,307,184,293)(171,245,185,231)(172,281,186,295)(173,247,187,233)(174,283,188,297)(175,249,189,235)(176,285,190,299)(177,251,191,237)(178,287,192,301)(179,225,193,239)(180,289,194,303)(181,227,195,241)(182,291,196,305)(197,323,211,309)(199,325,213,311)(201,327,215,313)(203,329,217,315)(205,331,219,317)(207,333,221,319)(209,335,223,321)(253,365,267,379)(255,367,269,381)(257,369,271,383)(259,371,273,385)(261,373,275,387)(263,375,277,389)(265,377,279,391)(310,341,324,355)(312,343,326,357)(314,345,328,359)(316,347,330,361)(318,349,332,363)(320,351,334,337)(322,353,336,339)(393,438,407,424)(395,440,409,426)(397,442,411,428)(399,444,413,430)(401,446,415,432)(403,448,417,434)(405,422,419,436), (1,351,448,222)(2,223,421,352)(3,353,422,224)(4,197,423,354)(5,355,424,198)(6,199,425,356)(7,357,426,200)(8,201,427,358)(9,359,428,202)(10,203,429,360)(11,361,430,204)(12,205,431,362)(13,363,432,206)(14,207,433,364)(15,337,434,208)(16,209,435,338)(17,339,436,210)(18,211,437,340)(19,341,438,212)(20,213,439,342)(21,343,440,214)(22,215,441,344)(23,345,442,216)(24,217,443,346)(25,347,444,218)(26,219,445,348)(27,349,446,220)(28,221,447,350)(29,109,318,401)(30,402,319,110)(31,111,320,403)(32,404,321,112)(33,85,322,405)(34,406,323,86)(35,87,324,407)(36,408,325,88)(37,89,326,409)(38,410,327,90)(39,91,328,411)(40,412,329,92)(41,93,330,413)(42,414,331,94)(43,95,332,415)(44,416,333,96)(45,97,334,417)(46,418,335,98)(47,99,336,419)(48,420,309,100)(49,101,310,393)(50,394,311,102)(51,103,312,395)(52,396,313,104)(53,105,314,397)(54,398,315,106)(55,107,316,399)(56,400,317,108)(57,171,275,168)(58,141,276,172)(59,173,277,142)(60,143,278,174)(61,175,279,144)(62,145,280,176)(63,177,253,146)(64,147,254,178)(65,179,255,148)(66,149,256,180)(67,181,257,150)(68,151,258,182)(69,183,259,152)(70,153,260,184)(71,185,261,154)(72,155,262,186)(73,187,263,156)(74,157,264,188)(75,189,265,158)(76,159,266,190)(77,191,267,160)(78,161,268,192)(79,193,269,162)(80,163,270,194)(81,195,271,164)(82,165,272,196)(83,169,273,166)(84,167,274,170)(113,291,384,242)(114,243,385,292)(115,293,386,244)(116,245,387,294)(117,295,388,246)(118,247,389,296)(119,297,390,248)(120,249,391,298)(121,299,392,250)(122,251,365,300)(123,301,366,252)(124,225,367,302)(125,303,368,226)(126,227,369,304)(127,305,370,228)(128,229,371,306)(129,307,372,230)(130,231,373,308)(131,281,374,232)(132,233,375,282)(133,283,376,234)(134,235,377,284)(135,285,378,236)(136,237,379,286)(137,287,380,238)(138,239,381,288)(139,289,382,240)(140,241,383,290), (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), (1,114,434,371)(2,127,435,384)(3,140,436,369)(4,125,437,382)(5,138,438,367)(6,123,439,380)(7,136,440,365)(8,121,441,378)(9,134,442,391)(10,119,443,376)(11,132,444,389)(12,117,445,374)(13,130,446,387)(14,115,447,372)(15,128,448,385)(16,113,421,370)(17,126,422,383)(18,139,423,368)(19,124,424,381)(20,137,425,366)(21,122,426,379)(22,135,427,392)(23,120,428,377)(24,133,429,390)(25,118,430,375)(26,131,431,388)(27,116,432,373)(28,129,433,386)(29,171,332,154)(30,184,333,167)(31,169,334,152)(32,182,335,165)(33,195,336,150)(34,180,309,163)(35,193,310,148)(36,178,311,161)(37,191,312,146)(38,176,313,159)(39,189,314,144)(40,174,315,157)(41,187,316,142)(42,172,317,155)(43,185,318,168)(44,170,319,153)(45,183,320,166)(46,196,321,151)(47,181,322,164)(48,194,323,149)(49,179,324,162)(50,192,325,147)(51,177,326,160)(52,190,327,145)(53,175,328,158)(54,188,329,143)(55,173,330,156)(56,186,331,141)(57,401,261,95)(58,414,262,108)(59,399,263,93)(60,412,264,106)(61,397,265,91)(62,410,266,104)(63,395,267,89)(64,408,268,102)(65,393,269,87)(66,406,270,100)(67,419,271,85)(68,404,272,98)(69,417,273,111)(70,402,274,96)(71,415,275,109)(72,400,276,94)(73,413,277,107)(74,398,278,92)(75,411,279,105)(76,396,280,90)(77,409,253,103)(78,394,254,88)(79,407,255,101)(80,420,256,86)(81,405,257,99)(82,418,258,112)(83,403,259,97)(84,416,260,110)(197,240,340,303)(198,225,341,288)(199,238,342,301)(200,251,343,286)(201,236,344,299)(202,249,345,284)(203,234,346,297)(204,247,347,282)(205,232,348,295)(206,245,349,308)(207,230,350,293)(208,243,351,306)(209,228,352,291)(210,241,353,304)(211,226,354,289)(212,239,355,302)(213,252,356,287)(214,237,357,300)(215,250,358,285)(216,235,359,298)(217,248,360,283)(218,233,361,296)(219,246,362,281)(220,231,363,294)(221,244,364,307)(222,229,337,292)(223,242,338,305)(224,227,339,290)>;
G:=Group( (1,97,15,111)(2,404,16,418)(3,99,17,85)(4,406,18,420)(5,101,19,87)(6,408,20,394)(7,103,21,89)(8,410,22,396)(9,105,23,91)(10,412,24,398)(11,107,25,93)(12,414,26,400)(13,109,27,95)(14,416,28,402)(29,220,43,206)(30,364,44,350)(31,222,45,208)(32,338,46,352)(33,224,47,210)(34,340,48,354)(35,198,49,212)(36,342,50,356)(37,200,51,214)(38,344,52,358)(39,202,53,216)(40,346,54,360)(41,204,55,218)(42,348,56,362)(57,116,71,130)(58,374,72,388)(59,118,73,132)(60,376,74,390)(61,120,75,134)(62,378,76,392)(63,122,77,136)(64,380,78,366)(65,124,79,138)(66,382,80,368)(67,126,81,140)(68,384,82,370)(69,128,83,114)(70,386,84,372)(86,437,100,423)(88,439,102,425)(90,441,104,427)(92,443,106,429)(94,445,108,431)(96,447,110,433)(98,421,112,435)(113,272,127,258)(115,274,129,260)(117,276,131,262)(119,278,133,264)(121,280,135,266)(123,254,137,268)(125,256,139,270)(141,232,155,246)(142,296,156,282)(143,234,157,248)(144,298,158,284)(145,236,159,250)(146,300,160,286)(147,238,161,252)(148,302,162,288)(149,240,163,226)(150,304,164,290)(151,242,165,228)(152,306,166,292)(153,244,167,230)(154,308,168,294)(169,243,183,229)(170,307,184,293)(171,245,185,231)(172,281,186,295)(173,247,187,233)(174,283,188,297)(175,249,189,235)(176,285,190,299)(177,251,191,237)(178,287,192,301)(179,225,193,239)(180,289,194,303)(181,227,195,241)(182,291,196,305)(197,323,211,309)(199,325,213,311)(201,327,215,313)(203,329,217,315)(205,331,219,317)(207,333,221,319)(209,335,223,321)(253,365,267,379)(255,367,269,381)(257,369,271,383)(259,371,273,385)(261,373,275,387)(263,375,277,389)(265,377,279,391)(310,341,324,355)(312,343,326,357)(314,345,328,359)(316,347,330,361)(318,349,332,363)(320,351,334,337)(322,353,336,339)(393,438,407,424)(395,440,409,426)(397,442,411,428)(399,444,413,430)(401,446,415,432)(403,448,417,434)(405,422,419,436), (1,351,448,222)(2,223,421,352)(3,353,422,224)(4,197,423,354)(5,355,424,198)(6,199,425,356)(7,357,426,200)(8,201,427,358)(9,359,428,202)(10,203,429,360)(11,361,430,204)(12,205,431,362)(13,363,432,206)(14,207,433,364)(15,337,434,208)(16,209,435,338)(17,339,436,210)(18,211,437,340)(19,341,438,212)(20,213,439,342)(21,343,440,214)(22,215,441,344)(23,345,442,216)(24,217,443,346)(25,347,444,218)(26,219,445,348)(27,349,446,220)(28,221,447,350)(29,109,318,401)(30,402,319,110)(31,111,320,403)(32,404,321,112)(33,85,322,405)(34,406,323,86)(35,87,324,407)(36,408,325,88)(37,89,326,409)(38,410,327,90)(39,91,328,411)(40,412,329,92)(41,93,330,413)(42,414,331,94)(43,95,332,415)(44,416,333,96)(45,97,334,417)(46,418,335,98)(47,99,336,419)(48,420,309,100)(49,101,310,393)(50,394,311,102)(51,103,312,395)(52,396,313,104)(53,105,314,397)(54,398,315,106)(55,107,316,399)(56,400,317,108)(57,171,275,168)(58,141,276,172)(59,173,277,142)(60,143,278,174)(61,175,279,144)(62,145,280,176)(63,177,253,146)(64,147,254,178)(65,179,255,148)(66,149,256,180)(67,181,257,150)(68,151,258,182)(69,183,259,152)(70,153,260,184)(71,185,261,154)(72,155,262,186)(73,187,263,156)(74,157,264,188)(75,189,265,158)(76,159,266,190)(77,191,267,160)(78,161,268,192)(79,193,269,162)(80,163,270,194)(81,195,271,164)(82,165,272,196)(83,169,273,166)(84,167,274,170)(113,291,384,242)(114,243,385,292)(115,293,386,244)(116,245,387,294)(117,295,388,246)(118,247,389,296)(119,297,390,248)(120,249,391,298)(121,299,392,250)(122,251,365,300)(123,301,366,252)(124,225,367,302)(125,303,368,226)(126,227,369,304)(127,305,370,228)(128,229,371,306)(129,307,372,230)(130,231,373,308)(131,281,374,232)(132,233,375,282)(133,283,376,234)(134,235,377,284)(135,285,378,236)(136,237,379,286)(137,287,380,238)(138,239,381,288)(139,289,382,240)(140,241,383,290), (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), (1,114,434,371)(2,127,435,384)(3,140,436,369)(4,125,437,382)(5,138,438,367)(6,123,439,380)(7,136,440,365)(8,121,441,378)(9,134,442,391)(10,119,443,376)(11,132,444,389)(12,117,445,374)(13,130,446,387)(14,115,447,372)(15,128,448,385)(16,113,421,370)(17,126,422,383)(18,139,423,368)(19,124,424,381)(20,137,425,366)(21,122,426,379)(22,135,427,392)(23,120,428,377)(24,133,429,390)(25,118,430,375)(26,131,431,388)(27,116,432,373)(28,129,433,386)(29,171,332,154)(30,184,333,167)(31,169,334,152)(32,182,335,165)(33,195,336,150)(34,180,309,163)(35,193,310,148)(36,178,311,161)(37,191,312,146)(38,176,313,159)(39,189,314,144)(40,174,315,157)(41,187,316,142)(42,172,317,155)(43,185,318,168)(44,170,319,153)(45,183,320,166)(46,196,321,151)(47,181,322,164)(48,194,323,149)(49,179,324,162)(50,192,325,147)(51,177,326,160)(52,190,327,145)(53,175,328,158)(54,188,329,143)(55,173,330,156)(56,186,331,141)(57,401,261,95)(58,414,262,108)(59,399,263,93)(60,412,264,106)(61,397,265,91)(62,410,266,104)(63,395,267,89)(64,408,268,102)(65,393,269,87)(66,406,270,100)(67,419,271,85)(68,404,272,98)(69,417,273,111)(70,402,274,96)(71,415,275,109)(72,400,276,94)(73,413,277,107)(74,398,278,92)(75,411,279,105)(76,396,280,90)(77,409,253,103)(78,394,254,88)(79,407,255,101)(80,420,256,86)(81,405,257,99)(82,418,258,112)(83,403,259,97)(84,416,260,110)(197,240,340,303)(198,225,341,288)(199,238,342,301)(200,251,343,286)(201,236,344,299)(202,249,345,284)(203,234,346,297)(204,247,347,282)(205,232,348,295)(206,245,349,308)(207,230,350,293)(208,243,351,306)(209,228,352,291)(210,241,353,304)(211,226,354,289)(212,239,355,302)(213,252,356,287)(214,237,357,300)(215,250,358,285)(216,235,359,298)(217,248,360,283)(218,233,361,296)(219,246,362,281)(220,231,363,294)(221,244,364,307)(222,229,337,292)(223,242,338,305)(224,227,339,290) );
G=PermutationGroup([(1,97,15,111),(2,404,16,418),(3,99,17,85),(4,406,18,420),(5,101,19,87),(6,408,20,394),(7,103,21,89),(8,410,22,396),(9,105,23,91),(10,412,24,398),(11,107,25,93),(12,414,26,400),(13,109,27,95),(14,416,28,402),(29,220,43,206),(30,364,44,350),(31,222,45,208),(32,338,46,352),(33,224,47,210),(34,340,48,354),(35,198,49,212),(36,342,50,356),(37,200,51,214),(38,344,52,358),(39,202,53,216),(40,346,54,360),(41,204,55,218),(42,348,56,362),(57,116,71,130),(58,374,72,388),(59,118,73,132),(60,376,74,390),(61,120,75,134),(62,378,76,392),(63,122,77,136),(64,380,78,366),(65,124,79,138),(66,382,80,368),(67,126,81,140),(68,384,82,370),(69,128,83,114),(70,386,84,372),(86,437,100,423),(88,439,102,425),(90,441,104,427),(92,443,106,429),(94,445,108,431),(96,447,110,433),(98,421,112,435),(113,272,127,258),(115,274,129,260),(117,276,131,262),(119,278,133,264),(121,280,135,266),(123,254,137,268),(125,256,139,270),(141,232,155,246),(142,296,156,282),(143,234,157,248),(144,298,158,284),(145,236,159,250),(146,300,160,286),(147,238,161,252),(148,302,162,288),(149,240,163,226),(150,304,164,290),(151,242,165,228),(152,306,166,292),(153,244,167,230),(154,308,168,294),(169,243,183,229),(170,307,184,293),(171,245,185,231),(172,281,186,295),(173,247,187,233),(174,283,188,297),(175,249,189,235),(176,285,190,299),(177,251,191,237),(178,287,192,301),(179,225,193,239),(180,289,194,303),(181,227,195,241),(182,291,196,305),(197,323,211,309),(199,325,213,311),(201,327,215,313),(203,329,217,315),(205,331,219,317),(207,333,221,319),(209,335,223,321),(253,365,267,379),(255,367,269,381),(257,369,271,383),(259,371,273,385),(261,373,275,387),(263,375,277,389),(265,377,279,391),(310,341,324,355),(312,343,326,357),(314,345,328,359),(316,347,330,361),(318,349,332,363),(320,351,334,337),(322,353,336,339),(393,438,407,424),(395,440,409,426),(397,442,411,428),(399,444,413,430),(401,446,415,432),(403,448,417,434),(405,422,419,436)], [(1,351,448,222),(2,223,421,352),(3,353,422,224),(4,197,423,354),(5,355,424,198),(6,199,425,356),(7,357,426,200),(8,201,427,358),(9,359,428,202),(10,203,429,360),(11,361,430,204),(12,205,431,362),(13,363,432,206),(14,207,433,364),(15,337,434,208),(16,209,435,338),(17,339,436,210),(18,211,437,340),(19,341,438,212),(20,213,439,342),(21,343,440,214),(22,215,441,344),(23,345,442,216),(24,217,443,346),(25,347,444,218),(26,219,445,348),(27,349,446,220),(28,221,447,350),(29,109,318,401),(30,402,319,110),(31,111,320,403),(32,404,321,112),(33,85,322,405),(34,406,323,86),(35,87,324,407),(36,408,325,88),(37,89,326,409),(38,410,327,90),(39,91,328,411),(40,412,329,92),(41,93,330,413),(42,414,331,94),(43,95,332,415),(44,416,333,96),(45,97,334,417),(46,418,335,98),(47,99,336,419),(48,420,309,100),(49,101,310,393),(50,394,311,102),(51,103,312,395),(52,396,313,104),(53,105,314,397),(54,398,315,106),(55,107,316,399),(56,400,317,108),(57,171,275,168),(58,141,276,172),(59,173,277,142),(60,143,278,174),(61,175,279,144),(62,145,280,176),(63,177,253,146),(64,147,254,178),(65,179,255,148),(66,149,256,180),(67,181,257,150),(68,151,258,182),(69,183,259,152),(70,153,260,184),(71,185,261,154),(72,155,262,186),(73,187,263,156),(74,157,264,188),(75,189,265,158),(76,159,266,190),(77,191,267,160),(78,161,268,192),(79,193,269,162),(80,163,270,194),(81,195,271,164),(82,165,272,196),(83,169,273,166),(84,167,274,170),(113,291,384,242),(114,243,385,292),(115,293,386,244),(116,245,387,294),(117,295,388,246),(118,247,389,296),(119,297,390,248),(120,249,391,298),(121,299,392,250),(122,251,365,300),(123,301,366,252),(124,225,367,302),(125,303,368,226),(126,227,369,304),(127,305,370,228),(128,229,371,306),(129,307,372,230),(130,231,373,308),(131,281,374,232),(132,233,375,282),(133,283,376,234),(134,235,377,284),(135,285,378,236),(136,237,379,286),(137,287,380,238),(138,239,381,288),(139,289,382,240),(140,241,383,290)], [(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)], [(1,114,434,371),(2,127,435,384),(3,140,436,369),(4,125,437,382),(5,138,438,367),(6,123,439,380),(7,136,440,365),(8,121,441,378),(9,134,442,391),(10,119,443,376),(11,132,444,389),(12,117,445,374),(13,130,446,387),(14,115,447,372),(15,128,448,385),(16,113,421,370),(17,126,422,383),(18,139,423,368),(19,124,424,381),(20,137,425,366),(21,122,426,379),(22,135,427,392),(23,120,428,377),(24,133,429,390),(25,118,430,375),(26,131,431,388),(27,116,432,373),(28,129,433,386),(29,171,332,154),(30,184,333,167),(31,169,334,152),(32,182,335,165),(33,195,336,150),(34,180,309,163),(35,193,310,148),(36,178,311,161),(37,191,312,146),(38,176,313,159),(39,189,314,144),(40,174,315,157),(41,187,316,142),(42,172,317,155),(43,185,318,168),(44,170,319,153),(45,183,320,166),(46,196,321,151),(47,181,322,164),(48,194,323,149),(49,179,324,162),(50,192,325,147),(51,177,326,160),(52,190,327,145),(53,175,328,158),(54,188,329,143),(55,173,330,156),(56,186,331,141),(57,401,261,95),(58,414,262,108),(59,399,263,93),(60,412,264,106),(61,397,265,91),(62,410,266,104),(63,395,267,89),(64,408,268,102),(65,393,269,87),(66,406,270,100),(67,419,271,85),(68,404,272,98),(69,417,273,111),(70,402,274,96),(71,415,275,109),(72,400,276,94),(73,413,277,107),(74,398,278,92),(75,411,279,105),(76,396,280,90),(77,409,253,103),(78,394,254,88),(79,407,255,101),(80,420,256,86),(81,405,257,99),(82,418,258,112),(83,403,259,97),(84,416,260,110),(197,240,340,303),(198,225,341,288),(199,238,342,301),(200,251,343,286),(201,236,344,299),(202,249,345,284),(203,234,346,297),(204,247,347,282),(205,232,348,295),(206,245,349,308),(207,230,350,293),(208,243,351,306),(209,228,352,291),(210,241,353,304),(211,226,354,289),(212,239,355,302),(213,252,356,287),(214,237,357,300),(215,250,358,285),(216,235,359,298),(217,248,360,283),(218,233,361,296),(219,246,362,281),(220,231,363,294),(221,244,364,307),(222,229,337,292),(223,242,338,305),(224,227,339,290)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 1 | 0 | 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 1 | 11 | 15 | 0 | 0 | 0 | 0 |
| 23 | 13 | 28 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 | 25 | 22 |
| 0 | 0 | 0 | 0 | 18 | 2 | 7 | 4 |
| 0 | 0 | 0 | 0 | 2 | 18 | 2 | 18 |
| 0 | 0 | 0 | 0 | 11 | 27 | 11 | 27 |
| 11 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 27 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 19 | 11 | 2 | 0 | 0 | 0 | 0 |
| 15 | 5 | 22 | 3 | 0 | 0 | 0 | 0 |
| 12 | 18 | 6 | 10 | 0 | 0 | 0 | 0 |
| 18 | 27 | 19 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 15 | 23 | 5 |
| 0 | 0 | 0 | 0 | 14 | 4 | 24 | 9 |
| 0 | 0 | 0 | 0 | 9 | 2 | 12 | 14 |
| 0 | 0 | 0 | 0 | 27 | 15 | 15 | 25 |
| 18 | 1 | 26 | 13 | 0 | 0 | 0 | 0 |
| 7 | 11 | 25 | 5 | 0 | 0 | 0 | 0 |
| 10 | 3 | 14 | 19 | 0 | 0 | 0 | 0 |
| 8 | 23 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 19 | 28 |
| 0 | 0 | 0 | 0 | 6 | 28 | 27 | 10 |
| 0 | 0 | 0 | 0 | 23 | 11 | 28 | 26 |
| 0 | 0 | 0 | 0 | 22 | 6 | 23 | 1 |
G:=sub<GL(8,GF(29))| [1,0,23,16,0,0,0,0,0,1,13,0,0,0,0,0,0,11,28,0,0,0,0,0,18,15,0,28,0,0,0,0,0,0,0,0,27,18,2,11,0,0,0,0,11,2,18,27,0,0,0,0,25,7,2,11,0,0,0,0,22,4,18,27],[11,21,0,0,0,0,0,0,15,18,0,0,0,0,0,0,0,0,20,14,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,28,0,1,0,0,0,0,0,0,28,0,1,0,0,0,0,27,0,1,0,0,0,0,0,0,27,0,1],[0,15,12,18,0,0,0,0,19,5,18,27,0,0,0,0,11,22,6,19,0,0,0,0,2,3,10,18,0,0,0,0,0,0,0,0,17,14,9,27,0,0,0,0,15,4,2,15,0,0,0,0,23,24,12,15,0,0,0,0,5,9,14,25],[18,7,10,8,0,0,0,0,1,11,3,23,0,0,0,0,26,25,14,1,0,0,0,0,13,5,19,15,0,0,0,0,0,0,0,0,1,6,23,22,0,0,0,0,3,28,11,6,0,0,0,0,19,27,28,23,0,0,0,0,28,10,26,1] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4G | 4H | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | 2- (1+4) | Q8.10D14 | D4.10D14 |
| kernel | C42.147D14 | C28.6Q8 | Dic7.Q8 | C4.Dic14 | C7×C42.C2 | C42.C2 | C42 | C4⋊C4 | C14 | C2 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 3 | 3 | 18 | 3 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{147}D_{14} % in TeX
G:=Group("C4^2.147D14"); // GroupNames label
G:=SmallGroup(448,1139);
// by ID
G=gap.SmallGroup(448,1139);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,758,555,100,675,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^13>;
// generators/relations
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