direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C2×C14, (C22×C4)⋊9C28, C4⋊2(C22×C28), C28⋊10(C22×C4), (C22×C28)⋊20C4, (C23×C4).8C14, C2.2(C23×C28), C23.59(C7×D4), C23.11(C7×Q8), C24.39(C2×C14), C14.54(C23×C4), (C23×C28).10C2, C23.39(C2×C28), C22.58(D4×C14), (C22×C14).30Q8, C14.55(C22×Q8), C22.16(Q8×C14), (C2×C28).958C23, (C2×C14).333C24, C14.178(C22×D4), (C22×C14).220D4, C22.6(C23×C14), C23.66(C22×C14), C22.24(C22×C28), (C22×C28).507C22, (C23×C14).119C22, (C22×C14).466C23, C2.2(D4×C2×C14), C2.1(Q8×C2×C14), (C2×C28)⋊39(C2×C4), (C2×C4)⋊10(C2×C28), (C2×C14).680(C2×D4), (C2×C14).114(C2×Q8), (C2×C4).53(C22×C14), (C2×C14).245(C22×C4), (C22×C14).148(C2×C4), (C22×C4).123(C2×C14), SmallGroup(448,1296)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 498 in 418 conjugacy classes, 338 normal (16 characteristic)
C1, C2 [×3], C2 [×12], C4 [×8], C4 [×8], C22, C22 [×34], C7, C2×C4 [×36], C2×C4 [×24], C23 [×15], C14 [×3], C14 [×12], C4⋊C4 [×16], C22×C4 [×26], C22×C4 [×8], C24, C28 [×8], C28 [×8], C2×C14, C2×C14 [×34], C2×C4⋊C4 [×12], C23×C4, C23×C4 [×2], C2×C28 [×36], C2×C28 [×24], C22×C14 [×15], C22×C4⋊C4, C7×C4⋊C4 [×16], C22×C28 [×26], C22×C28 [×8], C23×C14, C14×C4⋊C4 [×12], C23×C28, C23×C28 [×2], C4⋊C4×C2×C14
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C7, C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C14 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C28 [×8], C2×C14 [×35], C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C2×C28 [×28], C7×D4 [×4], C7×Q8 [×4], C22×C14 [×15], C22×C4⋊C4, C7×C4⋊C4 [×16], C22×C28 [×14], D4×C14 [×6], Q8×C14 [×6], C23×C14, C14×C4⋊C4 [×12], C23×C28, D4×C2×C14, Q8×C2×C14, C4⋊C4×C2×C14
Generators and relations
G = < a,b,c,d | a2=b14=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 448 points
Generators in S448
(1 52)(2 53)(3 54)(4 55)(5 56)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 163)(24 164)(25 165)(26 166)(27 167)(28 168)(29 398)(30 399)(31 400)(32 401)(33 402)(34 403)(35 404)(36 405)(37 406)(38 393)(39 394)(40 395)(41 396)(42 397)(57 262)(58 263)(59 264)(60 265)(61 266)(62 253)(63 254)(64 255)(65 256)(66 257)(67 258)(68 259)(69 260)(70 261)(71 223)(72 224)(73 211)(74 212)(75 213)(76 214)(77 215)(78 216)(79 217)(80 218)(81 219)(82 220)(83 221)(84 222)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 141)(96 142)(97 143)(98 144)(99 196)(100 183)(101 184)(102 185)(103 186)(104 187)(105 188)(106 189)(107 190)(108 191)(109 192)(110 193)(111 194)(112 195)(113 295)(114 296)(115 297)(116 298)(117 299)(118 300)(119 301)(120 302)(121 303)(122 304)(123 305)(124 306)(125 307)(126 308)(127 294)(128 281)(129 282)(130 283)(131 284)(132 285)(133 286)(134 287)(135 288)(136 289)(137 290)(138 291)(139 292)(140 293)(169 231)(170 232)(171 233)(172 234)(173 235)(174 236)(175 237)(176 238)(177 225)(178 226)(179 227)(180 228)(181 229)(182 230)(197 426)(198 427)(199 428)(200 429)(201 430)(202 431)(203 432)(204 433)(205 434)(206 421)(207 422)(208 423)(209 424)(210 425)(239 316)(240 317)(241 318)(242 319)(243 320)(244 321)(245 322)(246 309)(247 310)(248 311)(249 312)(250 313)(251 314)(252 315)(267 373)(268 374)(269 375)(270 376)(271 377)(272 378)(273 365)(274 366)(275 367)(276 368)(277 369)(278 370)(279 371)(280 372)(323 382)(324 383)(325 384)(326 385)(327 386)(328 387)(329 388)(330 389)(331 390)(332 391)(333 392)(334 379)(335 380)(336 381)(337 436)(338 437)(339 438)(340 439)(341 440)(342 441)(343 442)(344 443)(345 444)(346 445)(347 446)(348 447)(349 448)(350 435)(351 410)(352 411)(353 412)(354 413)(355 414)(356 415)(357 416)(358 417)(359 418)(360 419)(361 420)(362 407)(363 408)(364 409)
(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 59 302 95)(2 60 303 96)(3 61 304 97)(4 62 305 98)(5 63 306 85)(6 64 307 86)(7 65 308 87)(8 66 295 88)(9 67 296 89)(10 68 297 90)(11 69 298 91)(12 70 299 92)(13 57 300 93)(14 58 301 94)(15 419 198 342)(16 420 199 343)(17 407 200 344)(18 408 201 345)(19 409 202 346)(20 410 203 347)(21 411 204 348)(22 412 205 349)(23 413 206 350)(24 414 207 337)(25 415 208 338)(26 416 209 339)(27 417 210 340)(28 418 197 341)(29 328 106 370)(30 329 107 371)(31 330 108 372)(32 331 109 373)(33 332 110 374)(34 333 111 375)(35 334 112 376)(36 335 99 377)(37 336 100 378)(38 323 101 365)(39 324 102 366)(40 325 103 367)(41 326 104 368)(42 327 105 369)(43 255 125 146)(44 256 126 147)(45 257 113 148)(46 258 114 149)(47 259 115 150)(48 260 116 151)(49 261 117 152)(50 262 118 153)(51 263 119 154)(52 264 120 141)(53 265 121 142)(54 266 122 143)(55 253 123 144)(56 254 124 145)(71 234 314 135)(72 235 315 136)(73 236 316 137)(74 237 317 138)(75 238 318 139)(76 225 319 140)(77 226 320 127)(78 227 321 128)(79 228 322 129)(80 229 309 130)(81 230 310 131)(82 231 311 132)(83 232 312 133)(84 233 313 134)(155 360 427 441)(156 361 428 442)(157 362 429 443)(158 363 430 444)(159 364 431 445)(160 351 432 446)(161 352 433 447)(162 353 434 448)(163 354 421 435)(164 355 422 436)(165 356 423 437)(166 357 424 438)(167 358 425 439)(168 359 426 440)(169 248 285 220)(170 249 286 221)(171 250 287 222)(172 251 288 223)(173 252 289 224)(174 239 290 211)(175 240 291 212)(176 241 292 213)(177 242 293 214)(178 243 294 215)(179 244 281 216)(180 245 282 217)(181 246 283 218)(182 247 284 219)(183 272 406 381)(184 273 393 382)(185 274 394 383)(186 275 395 384)(187 276 396 385)(188 277 397 386)(189 278 398 387)(190 279 399 388)(191 280 400 389)(192 267 401 390)(193 268 402 391)(194 269 403 392)(195 270 404 379)(196 271 405 380)
(1 134 352 329)(2 135 353 330)(3 136 354 331)(4 137 355 332)(5 138 356 333)(6 139 357 334)(7 140 358 335)(8 127 359 336)(9 128 360 323)(10 129 361 324)(11 130 362 325)(12 131 363 326)(13 132 364 327)(14 133 351 328)(15 184 149 216)(16 185 150 217)(17 186 151 218)(18 187 152 219)(19 188 153 220)(20 189 154 221)(21 190 141 222)(22 191 142 223)(23 192 143 224)(24 193 144 211)(25 194 145 212)(26 195 146 213)(27 196 147 214)(28 183 148 215)(29 58 312 432)(30 59 313 433)(31 60 314 434)(32 61 315 421)(33 62 316 422)(34 63 317 423)(35 64 318 424)(36 65 319 425)(37 66 320 426)(38 67 321 427)(39 68 322 428)(40 69 309 429)(41 70 310 430)(42 57 311 431)(43 292 416 379)(44 293 417 380)(45 294 418 381)(46 281 419 382)(47 282 420 383)(48 283 407 384)(49 284 408 385)(50 285 409 386)(51 286 410 387)(52 287 411 388)(53 288 412 389)(54 289 413 390)(55 290 414 391)(56 291 415 392)(71 162 108 96)(72 163 109 97)(73 164 110 98)(74 165 111 85)(75 166 112 86)(76 167 99 87)(77 168 100 88)(78 155 101 89)(79 156 102 90)(80 157 103 91)(81 158 104 92)(82 159 105 93)(83 160 106 94)(84 161 107 95)(113 178 341 272)(114 179 342 273)(115 180 343 274)(116 181 344 275)(117 182 345 276)(118 169 346 277)(119 170 347 278)(120 171 348 279)(121 172 349 280)(122 173 350 267)(123 174 337 268)(124 175 338 269)(125 176 339 270)(126 177 340 271)(197 406 257 243)(198 393 258 244)(199 394 259 245)(200 395 260 246)(201 396 261 247)(202 397 262 248)(203 398 263 249)(204 399 264 250)(205 400 265 251)(206 401 266 252)(207 402 253 239)(208 403 254 240)(209 404 255 241)(210 405 256 242)(225 439 377 308)(226 440 378 295)(227 441 365 296)(228 442 366 297)(229 443 367 298)(230 444 368 299)(231 445 369 300)(232 446 370 301)(233 447 371 302)(234 448 372 303)(235 435 373 304)(236 436 374 305)(237 437 375 306)(238 438 376 307)
G:=sub<Sym(448)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,166)(27,167)(28,168)(29,398)(30,399)(31,400)(32,401)(33,402)(34,403)(35,404)(36,405)(37,406)(38,393)(39,394)(40,395)(41,396)(42,397)(57,262)(58,263)(59,264)(60,265)(61,266)(62,253)(63,254)(64,255)(65,256)(66,257)(67,258)(68,259)(69,260)(70,261)(71,223)(72,224)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,217)(80,218)(81,219)(82,220)(83,221)(84,222)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,141)(96,142)(97,143)(98,144)(99,196)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)(108,191)(109,192)(110,193)(111,194)(112,195)(113,295)(114,296)(115,297)(116,298)(117,299)(118,300)(119,301)(120,302)(121,303)(122,304)(123,305)(124,306)(125,307)(126,308)(127,294)(128,281)(129,282)(130,283)(131,284)(132,285)(133,286)(134,287)(135,288)(136,289)(137,290)(138,291)(139,292)(140,293)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,225)(178,226)(179,227)(180,228)(181,229)(182,230)(197,426)(198,427)(199,428)(200,429)(201,430)(202,431)(203,432)(204,433)(205,434)(206,421)(207,422)(208,423)(209,424)(210,425)(239,316)(240,317)(241,318)(242,319)(243,320)(244,321)(245,322)(246,309)(247,310)(248,311)(249,312)(250,313)(251,314)(252,315)(267,373)(268,374)(269,375)(270,376)(271,377)(272,378)(273,365)(274,366)(275,367)(276,368)(277,369)(278,370)(279,371)(280,372)(323,382)(324,383)(325,384)(326,385)(327,386)(328,387)(329,388)(330,389)(331,390)(332,391)(333,392)(334,379)(335,380)(336,381)(337,436)(338,437)(339,438)(340,439)(341,440)(342,441)(343,442)(344,443)(345,444)(346,445)(347,446)(348,447)(349,448)(350,435)(351,410)(352,411)(353,412)(354,413)(355,414)(356,415)(357,416)(358,417)(359,418)(360,419)(361,420)(362,407)(363,408)(364,409), (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), (1,59,302,95)(2,60,303,96)(3,61,304,97)(4,62,305,98)(5,63,306,85)(6,64,307,86)(7,65,308,87)(8,66,295,88)(9,67,296,89)(10,68,297,90)(11,69,298,91)(12,70,299,92)(13,57,300,93)(14,58,301,94)(15,419,198,342)(16,420,199,343)(17,407,200,344)(18,408,201,345)(19,409,202,346)(20,410,203,347)(21,411,204,348)(22,412,205,349)(23,413,206,350)(24,414,207,337)(25,415,208,338)(26,416,209,339)(27,417,210,340)(28,418,197,341)(29,328,106,370)(30,329,107,371)(31,330,108,372)(32,331,109,373)(33,332,110,374)(34,333,111,375)(35,334,112,376)(36,335,99,377)(37,336,100,378)(38,323,101,365)(39,324,102,366)(40,325,103,367)(41,326,104,368)(42,327,105,369)(43,255,125,146)(44,256,126,147)(45,257,113,148)(46,258,114,149)(47,259,115,150)(48,260,116,151)(49,261,117,152)(50,262,118,153)(51,263,119,154)(52,264,120,141)(53,265,121,142)(54,266,122,143)(55,253,123,144)(56,254,124,145)(71,234,314,135)(72,235,315,136)(73,236,316,137)(74,237,317,138)(75,238,318,139)(76,225,319,140)(77,226,320,127)(78,227,321,128)(79,228,322,129)(80,229,309,130)(81,230,310,131)(82,231,311,132)(83,232,312,133)(84,233,313,134)(155,360,427,441)(156,361,428,442)(157,362,429,443)(158,363,430,444)(159,364,431,445)(160,351,432,446)(161,352,433,447)(162,353,434,448)(163,354,421,435)(164,355,422,436)(165,356,423,437)(166,357,424,438)(167,358,425,439)(168,359,426,440)(169,248,285,220)(170,249,286,221)(171,250,287,222)(172,251,288,223)(173,252,289,224)(174,239,290,211)(175,240,291,212)(176,241,292,213)(177,242,293,214)(178,243,294,215)(179,244,281,216)(180,245,282,217)(181,246,283,218)(182,247,284,219)(183,272,406,381)(184,273,393,382)(185,274,394,383)(186,275,395,384)(187,276,396,385)(188,277,397,386)(189,278,398,387)(190,279,399,388)(191,280,400,389)(192,267,401,390)(193,268,402,391)(194,269,403,392)(195,270,404,379)(196,271,405,380), (1,134,352,329)(2,135,353,330)(3,136,354,331)(4,137,355,332)(5,138,356,333)(6,139,357,334)(7,140,358,335)(8,127,359,336)(9,128,360,323)(10,129,361,324)(11,130,362,325)(12,131,363,326)(13,132,364,327)(14,133,351,328)(15,184,149,216)(16,185,150,217)(17,186,151,218)(18,187,152,219)(19,188,153,220)(20,189,154,221)(21,190,141,222)(22,191,142,223)(23,192,143,224)(24,193,144,211)(25,194,145,212)(26,195,146,213)(27,196,147,214)(28,183,148,215)(29,58,312,432)(30,59,313,433)(31,60,314,434)(32,61,315,421)(33,62,316,422)(34,63,317,423)(35,64,318,424)(36,65,319,425)(37,66,320,426)(38,67,321,427)(39,68,322,428)(40,69,309,429)(41,70,310,430)(42,57,311,431)(43,292,416,379)(44,293,417,380)(45,294,418,381)(46,281,419,382)(47,282,420,383)(48,283,407,384)(49,284,408,385)(50,285,409,386)(51,286,410,387)(52,287,411,388)(53,288,412,389)(54,289,413,390)(55,290,414,391)(56,291,415,392)(71,162,108,96)(72,163,109,97)(73,164,110,98)(74,165,111,85)(75,166,112,86)(76,167,99,87)(77,168,100,88)(78,155,101,89)(79,156,102,90)(80,157,103,91)(81,158,104,92)(82,159,105,93)(83,160,106,94)(84,161,107,95)(113,178,341,272)(114,179,342,273)(115,180,343,274)(116,181,344,275)(117,182,345,276)(118,169,346,277)(119,170,347,278)(120,171,348,279)(121,172,349,280)(122,173,350,267)(123,174,337,268)(124,175,338,269)(125,176,339,270)(126,177,340,271)(197,406,257,243)(198,393,258,244)(199,394,259,245)(200,395,260,246)(201,396,261,247)(202,397,262,248)(203,398,263,249)(204,399,264,250)(205,400,265,251)(206,401,266,252)(207,402,253,239)(208,403,254,240)(209,404,255,241)(210,405,256,242)(225,439,377,308)(226,440,378,295)(227,441,365,296)(228,442,366,297)(229,443,367,298)(230,444,368,299)(231,445,369,300)(232,446,370,301)(233,447,371,302)(234,448,372,303)(235,435,373,304)(236,436,374,305)(237,437,375,306)(238,438,376,307)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,166)(27,167)(28,168)(29,398)(30,399)(31,400)(32,401)(33,402)(34,403)(35,404)(36,405)(37,406)(38,393)(39,394)(40,395)(41,396)(42,397)(57,262)(58,263)(59,264)(60,265)(61,266)(62,253)(63,254)(64,255)(65,256)(66,257)(67,258)(68,259)(69,260)(70,261)(71,223)(72,224)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,217)(80,218)(81,219)(82,220)(83,221)(84,222)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,141)(96,142)(97,143)(98,144)(99,196)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)(108,191)(109,192)(110,193)(111,194)(112,195)(113,295)(114,296)(115,297)(116,298)(117,299)(118,300)(119,301)(120,302)(121,303)(122,304)(123,305)(124,306)(125,307)(126,308)(127,294)(128,281)(129,282)(130,283)(131,284)(132,285)(133,286)(134,287)(135,288)(136,289)(137,290)(138,291)(139,292)(140,293)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,225)(178,226)(179,227)(180,228)(181,229)(182,230)(197,426)(198,427)(199,428)(200,429)(201,430)(202,431)(203,432)(204,433)(205,434)(206,421)(207,422)(208,423)(209,424)(210,425)(239,316)(240,317)(241,318)(242,319)(243,320)(244,321)(245,322)(246,309)(247,310)(248,311)(249,312)(250,313)(251,314)(252,315)(267,373)(268,374)(269,375)(270,376)(271,377)(272,378)(273,365)(274,366)(275,367)(276,368)(277,369)(278,370)(279,371)(280,372)(323,382)(324,383)(325,384)(326,385)(327,386)(328,387)(329,388)(330,389)(331,390)(332,391)(333,392)(334,379)(335,380)(336,381)(337,436)(338,437)(339,438)(340,439)(341,440)(342,441)(343,442)(344,443)(345,444)(346,445)(347,446)(348,447)(349,448)(350,435)(351,410)(352,411)(353,412)(354,413)(355,414)(356,415)(357,416)(358,417)(359,418)(360,419)(361,420)(362,407)(363,408)(364,409), (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), (1,59,302,95)(2,60,303,96)(3,61,304,97)(4,62,305,98)(5,63,306,85)(6,64,307,86)(7,65,308,87)(8,66,295,88)(9,67,296,89)(10,68,297,90)(11,69,298,91)(12,70,299,92)(13,57,300,93)(14,58,301,94)(15,419,198,342)(16,420,199,343)(17,407,200,344)(18,408,201,345)(19,409,202,346)(20,410,203,347)(21,411,204,348)(22,412,205,349)(23,413,206,350)(24,414,207,337)(25,415,208,338)(26,416,209,339)(27,417,210,340)(28,418,197,341)(29,328,106,370)(30,329,107,371)(31,330,108,372)(32,331,109,373)(33,332,110,374)(34,333,111,375)(35,334,112,376)(36,335,99,377)(37,336,100,378)(38,323,101,365)(39,324,102,366)(40,325,103,367)(41,326,104,368)(42,327,105,369)(43,255,125,146)(44,256,126,147)(45,257,113,148)(46,258,114,149)(47,259,115,150)(48,260,116,151)(49,261,117,152)(50,262,118,153)(51,263,119,154)(52,264,120,141)(53,265,121,142)(54,266,122,143)(55,253,123,144)(56,254,124,145)(71,234,314,135)(72,235,315,136)(73,236,316,137)(74,237,317,138)(75,238,318,139)(76,225,319,140)(77,226,320,127)(78,227,321,128)(79,228,322,129)(80,229,309,130)(81,230,310,131)(82,231,311,132)(83,232,312,133)(84,233,313,134)(155,360,427,441)(156,361,428,442)(157,362,429,443)(158,363,430,444)(159,364,431,445)(160,351,432,446)(161,352,433,447)(162,353,434,448)(163,354,421,435)(164,355,422,436)(165,356,423,437)(166,357,424,438)(167,358,425,439)(168,359,426,440)(169,248,285,220)(170,249,286,221)(171,250,287,222)(172,251,288,223)(173,252,289,224)(174,239,290,211)(175,240,291,212)(176,241,292,213)(177,242,293,214)(178,243,294,215)(179,244,281,216)(180,245,282,217)(181,246,283,218)(182,247,284,219)(183,272,406,381)(184,273,393,382)(185,274,394,383)(186,275,395,384)(187,276,396,385)(188,277,397,386)(189,278,398,387)(190,279,399,388)(191,280,400,389)(192,267,401,390)(193,268,402,391)(194,269,403,392)(195,270,404,379)(196,271,405,380), (1,134,352,329)(2,135,353,330)(3,136,354,331)(4,137,355,332)(5,138,356,333)(6,139,357,334)(7,140,358,335)(8,127,359,336)(9,128,360,323)(10,129,361,324)(11,130,362,325)(12,131,363,326)(13,132,364,327)(14,133,351,328)(15,184,149,216)(16,185,150,217)(17,186,151,218)(18,187,152,219)(19,188,153,220)(20,189,154,221)(21,190,141,222)(22,191,142,223)(23,192,143,224)(24,193,144,211)(25,194,145,212)(26,195,146,213)(27,196,147,214)(28,183,148,215)(29,58,312,432)(30,59,313,433)(31,60,314,434)(32,61,315,421)(33,62,316,422)(34,63,317,423)(35,64,318,424)(36,65,319,425)(37,66,320,426)(38,67,321,427)(39,68,322,428)(40,69,309,429)(41,70,310,430)(42,57,311,431)(43,292,416,379)(44,293,417,380)(45,294,418,381)(46,281,419,382)(47,282,420,383)(48,283,407,384)(49,284,408,385)(50,285,409,386)(51,286,410,387)(52,287,411,388)(53,288,412,389)(54,289,413,390)(55,290,414,391)(56,291,415,392)(71,162,108,96)(72,163,109,97)(73,164,110,98)(74,165,111,85)(75,166,112,86)(76,167,99,87)(77,168,100,88)(78,155,101,89)(79,156,102,90)(80,157,103,91)(81,158,104,92)(82,159,105,93)(83,160,106,94)(84,161,107,95)(113,178,341,272)(114,179,342,273)(115,180,343,274)(116,181,344,275)(117,182,345,276)(118,169,346,277)(119,170,347,278)(120,171,348,279)(121,172,349,280)(122,173,350,267)(123,174,337,268)(124,175,338,269)(125,176,339,270)(126,177,340,271)(197,406,257,243)(198,393,258,244)(199,394,259,245)(200,395,260,246)(201,396,261,247)(202,397,262,248)(203,398,263,249)(204,399,264,250)(205,400,265,251)(206,401,266,252)(207,402,253,239)(208,403,254,240)(209,404,255,241)(210,405,256,242)(225,439,377,308)(226,440,378,295)(227,441,365,296)(228,442,366,297)(229,443,367,298)(230,444,368,299)(231,445,369,300)(232,446,370,301)(233,447,371,302)(234,448,372,303)(235,435,373,304)(236,436,374,305)(237,437,375,306)(238,438,376,307) );
G=PermutationGroup([(1,52),(2,53),(3,54),(4,55),(5,56),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,161),(22,162),(23,163),(24,164),(25,165),(26,166),(27,167),(28,168),(29,398),(30,399),(31,400),(32,401),(33,402),(34,403),(35,404),(36,405),(37,406),(38,393),(39,394),(40,395),(41,396),(42,397),(57,262),(58,263),(59,264),(60,265),(61,266),(62,253),(63,254),(64,255),(65,256),(66,257),(67,258),(68,259),(69,260),(70,261),(71,223),(72,224),(73,211),(74,212),(75,213),(76,214),(77,215),(78,216),(79,217),(80,218),(81,219),(82,220),(83,221),(84,222),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,141),(96,142),(97,143),(98,144),(99,196),(100,183),(101,184),(102,185),(103,186),(104,187),(105,188),(106,189),(107,190),(108,191),(109,192),(110,193),(111,194),(112,195),(113,295),(114,296),(115,297),(116,298),(117,299),(118,300),(119,301),(120,302),(121,303),(122,304),(123,305),(124,306),(125,307),(126,308),(127,294),(128,281),(129,282),(130,283),(131,284),(132,285),(133,286),(134,287),(135,288),(136,289),(137,290),(138,291),(139,292),(140,293),(169,231),(170,232),(171,233),(172,234),(173,235),(174,236),(175,237),(176,238),(177,225),(178,226),(179,227),(180,228),(181,229),(182,230),(197,426),(198,427),(199,428),(200,429),(201,430),(202,431),(203,432),(204,433),(205,434),(206,421),(207,422),(208,423),(209,424),(210,425),(239,316),(240,317),(241,318),(242,319),(243,320),(244,321),(245,322),(246,309),(247,310),(248,311),(249,312),(250,313),(251,314),(252,315),(267,373),(268,374),(269,375),(270,376),(271,377),(272,378),(273,365),(274,366),(275,367),(276,368),(277,369),(278,370),(279,371),(280,372),(323,382),(324,383),(325,384),(326,385),(327,386),(328,387),(329,388),(330,389),(331,390),(332,391),(333,392),(334,379),(335,380),(336,381),(337,436),(338,437),(339,438),(340,439),(341,440),(342,441),(343,442),(344,443),(345,444),(346,445),(347,446),(348,447),(349,448),(350,435),(351,410),(352,411),(353,412),(354,413),(355,414),(356,415),(357,416),(358,417),(359,418),(360,419),(361,420),(362,407),(363,408),(364,409)], [(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)], [(1,59,302,95),(2,60,303,96),(3,61,304,97),(4,62,305,98),(5,63,306,85),(6,64,307,86),(7,65,308,87),(8,66,295,88),(9,67,296,89),(10,68,297,90),(11,69,298,91),(12,70,299,92),(13,57,300,93),(14,58,301,94),(15,419,198,342),(16,420,199,343),(17,407,200,344),(18,408,201,345),(19,409,202,346),(20,410,203,347),(21,411,204,348),(22,412,205,349),(23,413,206,350),(24,414,207,337),(25,415,208,338),(26,416,209,339),(27,417,210,340),(28,418,197,341),(29,328,106,370),(30,329,107,371),(31,330,108,372),(32,331,109,373),(33,332,110,374),(34,333,111,375),(35,334,112,376),(36,335,99,377),(37,336,100,378),(38,323,101,365),(39,324,102,366),(40,325,103,367),(41,326,104,368),(42,327,105,369),(43,255,125,146),(44,256,126,147),(45,257,113,148),(46,258,114,149),(47,259,115,150),(48,260,116,151),(49,261,117,152),(50,262,118,153),(51,263,119,154),(52,264,120,141),(53,265,121,142),(54,266,122,143),(55,253,123,144),(56,254,124,145),(71,234,314,135),(72,235,315,136),(73,236,316,137),(74,237,317,138),(75,238,318,139),(76,225,319,140),(77,226,320,127),(78,227,321,128),(79,228,322,129),(80,229,309,130),(81,230,310,131),(82,231,311,132),(83,232,312,133),(84,233,313,134),(155,360,427,441),(156,361,428,442),(157,362,429,443),(158,363,430,444),(159,364,431,445),(160,351,432,446),(161,352,433,447),(162,353,434,448),(163,354,421,435),(164,355,422,436),(165,356,423,437),(166,357,424,438),(167,358,425,439),(168,359,426,440),(169,248,285,220),(170,249,286,221),(171,250,287,222),(172,251,288,223),(173,252,289,224),(174,239,290,211),(175,240,291,212),(176,241,292,213),(177,242,293,214),(178,243,294,215),(179,244,281,216),(180,245,282,217),(181,246,283,218),(182,247,284,219),(183,272,406,381),(184,273,393,382),(185,274,394,383),(186,275,395,384),(187,276,396,385),(188,277,397,386),(189,278,398,387),(190,279,399,388),(191,280,400,389),(192,267,401,390),(193,268,402,391),(194,269,403,392),(195,270,404,379),(196,271,405,380)], [(1,134,352,329),(2,135,353,330),(3,136,354,331),(4,137,355,332),(5,138,356,333),(6,139,357,334),(7,140,358,335),(8,127,359,336),(9,128,360,323),(10,129,361,324),(11,130,362,325),(12,131,363,326),(13,132,364,327),(14,133,351,328),(15,184,149,216),(16,185,150,217),(17,186,151,218),(18,187,152,219),(19,188,153,220),(20,189,154,221),(21,190,141,222),(22,191,142,223),(23,192,143,224),(24,193,144,211),(25,194,145,212),(26,195,146,213),(27,196,147,214),(28,183,148,215),(29,58,312,432),(30,59,313,433),(31,60,314,434),(32,61,315,421),(33,62,316,422),(34,63,317,423),(35,64,318,424),(36,65,319,425),(37,66,320,426),(38,67,321,427),(39,68,322,428),(40,69,309,429),(41,70,310,430),(42,57,311,431),(43,292,416,379),(44,293,417,380),(45,294,418,381),(46,281,419,382),(47,282,420,383),(48,283,407,384),(49,284,408,385),(50,285,409,386),(51,286,410,387),(52,287,411,388),(53,288,412,389),(54,289,413,390),(55,290,414,391),(56,291,415,392),(71,162,108,96),(72,163,109,97),(73,164,110,98),(74,165,111,85),(75,166,112,86),(76,167,99,87),(77,168,100,88),(78,155,101,89),(79,156,102,90),(80,157,103,91),(81,158,104,92),(82,159,105,93),(83,160,106,94),(84,161,107,95),(113,178,341,272),(114,179,342,273),(115,180,343,274),(116,181,344,275),(117,182,345,276),(118,169,346,277),(119,170,347,278),(120,171,348,279),(121,172,349,280),(122,173,350,267),(123,174,337,268),(124,175,338,269),(125,176,339,270),(126,177,340,271),(197,406,257,243),(198,393,258,244),(199,394,259,245),(200,395,260,246),(201,396,261,247),(202,397,262,248),(203,398,263,249),(204,399,264,250),(205,400,265,251),(206,401,266,252),(207,402,253,239),(208,403,254,240),(209,404,255,241),(210,405,256,242),(225,439,377,308),(226,440,378,295),(227,441,365,296),(228,442,366,297),(229,443,367,298),(230,444,368,299),(231,445,369,300),(232,446,370,301),(233,447,371,302),(234,448,372,303),(235,435,373,304),(236,436,374,305),(237,437,375,306),(238,438,376,307)])
Matrix representation ►G ⊆ GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 28 | 0 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 20 | 18 |
| 0 | 0 | 0 | 18 | 9 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,1,0],[17,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,20,18,0,0,0,18,9] >;
280 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X | 7A | ··· | 7F | 14A | ··· | 14CL | 28A | ··· | 28EN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | Q8 | C7×D4 | C7×Q8 |
| kernel | C4⋊C4×C2×C14 | C14×C4⋊C4 | C23×C28 | C22×C28 | C22×C4⋊C4 | C2×C4⋊C4 | C23×C4 | C22×C4 | C22×C14 | C22×C14 | C23 | C23 |
| # reps | 1 | 12 | 3 | 16 | 6 | 72 | 18 | 96 | 4 | 4 | 24 | 24 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_2\times C_{14} % in TeX
G:=Group("C4:C4xC2xC14"); // GroupNames label
G:=SmallGroup(448,1296);
// by ID
G=gap.SmallGroup(448,1296);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,792]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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