direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C42.C2, C4.9(Q8×C14), (C2×C28).80Q8, C28.98(C2×Q8), (C2×C42).18C14, C42.88(C2×C14), C14.58(C22×Q8), C22.18(Q8×C14), (C2×C14).347C24, (C2×C28).660C23, (C4×C28).372C22, C22.21(C23×C14), C23.71(C22×C14), (C22×C14).469C23, (C22×C28).509C22, C2.4(Q8×C2×C14), (C2×C4×C28).41C2, (C14×C4⋊C4).47C2, (C2×C4⋊C4).18C14, (C2×C4).22(C7×Q8), C4⋊C4.63(C2×C14), C2.10(C14×C4○D4), C14.229(C2×C4○D4), (C2×C14).116(C2×Q8), C22.33(C7×C4○D4), (C7×C4⋊C4).386C22, (C2×C4).15(C22×C14), (C2×C14).233(C4○D4), (C22×C4).101(C2×C14), SmallGroup(448,1310)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 274 in 226 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C7, C2×C4 [×18], C2×C4 [×12], C23, C14, C14 [×6], C42 [×4], C4⋊C4 [×24], C22×C4, C22×C4 [×6], C28 [×4], C28 [×12], C2×C14, C2×C14 [×6], C2×C42, C2×C4⋊C4 [×6], C42.C2 [×8], C2×C28 [×18], C2×C28 [×12], C22×C14, C2×C42.C2, C4×C28 [×4], C7×C4⋊C4 [×24], C22×C28, C22×C28 [×6], C2×C4×C28, C14×C4⋊C4 [×6], C7×C42.C2 [×8], C14×C42.C2
Quotients:
C1, C2 [×15], C22 [×35], C7, Q8 [×4], C23 [×15], C14 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C2×C14 [×35], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C7×Q8 [×4], C22×C14 [×15], C2×C42.C2, Q8×C14 [×6], C7×C4○D4 [×4], C23×C14, C7×C42.C2 [×4], Q8×C2×C14, C14×C4○D4 [×2], C14×C42.C2
Generators and relations
G = < a,b,c,d | a14=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
►Regular action on 448 points
Generators in S448
(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)
(1 256 317 114)(2 257 318 115)(3 258 319 116)(4 259 320 117)(5 260 321 118)(6 261 322 119)(7 262 309 120)(8 263 310 121)(9 264 311 122)(10 265 312 123)(11 266 313 124)(12 253 314 125)(13 254 315 126)(14 255 316 113)(15 178 422 275)(16 179 423 276)(17 180 424 277)(18 181 425 278)(19 182 426 279)(20 169 427 280)(21 170 428 267)(22 171 429 268)(23 172 430 269)(24 173 431 270)(25 174 432 271)(26 175 433 272)(27 176 434 273)(28 177 421 274)(29 60 327 87)(30 61 328 88)(31 62 329 89)(32 63 330 90)(33 64 331 91)(34 65 332 92)(35 66 333 93)(36 67 334 94)(37 68 335 95)(38 69 336 96)(39 70 323 97)(40 57 324 98)(41 58 325 85)(42 59 326 86)(43 373 292 163)(44 374 293 164)(45 375 294 165)(46 376 281 166)(47 377 282 167)(48 378 283 168)(49 365 284 155)(50 366 285 156)(51 367 286 157)(52 368 287 158)(53 369 288 159)(54 370 289 160)(55 371 290 161)(56 372 291 162)(71 140 337 141)(72 127 338 142)(73 128 339 143)(74 129 340 144)(75 130 341 145)(76 131 342 146)(77 132 343 147)(78 133 344 148)(79 134 345 149)(80 135 346 150)(81 136 347 151)(82 137 348 152)(83 138 349 153)(84 139 350 154)(99 302 406 241)(100 303 393 242)(101 304 394 243)(102 305 395 244)(103 306 396 245)(104 307 397 246)(105 308 398 247)(106 295 399 248)(107 296 400 249)(108 297 401 250)(109 298 402 251)(110 299 403 252)(111 300 404 239)(112 301 405 240)(183 445 352 233)(184 446 353 234)(185 447 354 235)(186 448 355 236)(187 435 356 237)(188 436 357 238)(189 437 358 225)(190 438 359 226)(191 439 360 227)(192 440 361 228)(193 441 362 229)(194 442 363 230)(195 443 364 231)(196 444 351 232)(197 415 385 224)(198 416 386 211)(199 417 387 212)(200 418 388 213)(201 419 389 214)(202 420 390 215)(203 407 391 216)(204 408 392 217)(205 409 379 218)(206 410 380 219)(207 411 381 220)(208 412 382 221)(209 413 383 222)(210 414 384 223)
(1 131 430 201)(2 132 431 202)(3 133 432 203)(4 134 433 204)(5 135 434 205)(6 136 421 206)(7 137 422 207)(8 138 423 208)(9 139 424 209)(10 140 425 210)(11 127 426 197)(12 128 427 198)(13 129 428 199)(14 130 429 200)(15 381 309 152)(16 382 310 153)(17 383 311 154)(18 384 312 141)(19 385 313 142)(20 386 314 143)(21 387 315 144)(22 388 316 145)(23 389 317 146)(24 390 318 147)(25 391 319 148)(26 392 320 149)(27 379 321 150)(28 380 322 151)(29 361 250 53)(30 362 251 54)(31 363 252 55)(32 364 239 56)(33 351 240 43)(34 352 241 44)(35 353 242 45)(36 354 243 46)(37 355 244 47)(38 356 245 48)(39 357 246 49)(40 358 247 50)(41 359 248 51)(42 360 249 52)(57 225 105 366)(58 226 106 367)(59 227 107 368)(60 228 108 369)(61 229 109 370)(62 230 110 371)(63 231 111 372)(64 232 112 373)(65 233 99 374)(66 234 100 375)(67 235 101 376)(68 236 102 377)(69 237 103 378)(70 238 104 365)(71 181 223 123)(72 182 224 124)(73 169 211 125)(74 170 212 126)(75 171 213 113)(76 172 214 114)(77 173 215 115)(78 174 216 116)(79 175 217 117)(80 176 218 118)(81 177 219 119)(82 178 220 120)(83 179 221 121)(84 180 222 122)(85 438 399 157)(86 439 400 158)(87 440 401 159)(88 441 402 160)(89 442 403 161)(90 443 404 162)(91 444 405 163)(92 445 406 164)(93 446 393 165)(94 447 394 166)(95 448 395 167)(96 435 396 168)(97 436 397 155)(98 437 398 156)(183 302 293 332)(184 303 294 333)(185 304 281 334)(186 305 282 335)(187 306 283 336)(188 307 284 323)(189 308 285 324)(190 295 286 325)(191 296 287 326)(192 297 288 327)(193 298 289 328)(194 299 290 329)(195 300 291 330)(196 301 292 331)(253 339 280 416)(254 340 267 417)(255 341 268 418)(256 342 269 419)(257 343 270 420)(258 344 271 407)(259 345 272 408)(260 346 273 409)(261 347 274 410)(262 348 275 411)(263 349 276 412)(264 350 277 413)(265 337 278 414)(266 338 279 415)
(1 232 430 373)(2 233 431 374)(3 234 432 375)(4 235 433 376)(5 236 434 377)(6 237 421 378)(7 238 422 365)(8 225 423 366)(9 226 424 367)(10 227 425 368)(11 228 426 369)(12 229 427 370)(13 230 428 371)(14 231 429 372)(15 155 309 436)(16 156 310 437)(17 157 311 438)(18 158 312 439)(19 159 313 440)(20 160 314 441)(21 161 315 442)(22 162 316 443)(23 163 317 444)(24 164 318 445)(25 165 319 446)(26 166 320 447)(27 167 321 448)(28 168 322 435)(29 415 250 338)(30 416 251 339)(31 417 252 340)(32 418 239 341)(33 419 240 342)(34 420 241 343)(35 407 242 344)(36 408 243 345)(37 409 244 346)(38 410 245 347)(39 411 246 348)(40 412 247 349)(41 413 248 350)(42 414 249 337)(43 172 351 114)(44 173 352 115)(45 174 353 116)(46 175 354 117)(47 176 355 118)(48 177 356 119)(49 178 357 120)(50 179 358 121)(51 180 359 122)(52 181 360 123)(53 182 361 124)(54 169 362 125)(55 170 363 126)(56 171 364 113)(57 153 105 382)(58 154 106 383)(59 141 107 384)(60 142 108 385)(61 143 109 386)(62 144 110 387)(63 145 111 388)(64 146 112 389)(65 147 99 390)(66 148 100 391)(67 149 101 392)(68 150 102 379)(69 151 103 380)(70 152 104 381)(71 326 223 296)(72 327 224 297)(73 328 211 298)(74 329 212 299)(75 330 213 300)(76 331 214 301)(77 332 215 302)(78 333 216 303)(79 334 217 304)(80 335 218 305)(81 336 219 306)(82 323 220 307)(83 324 221 308)(84 325 222 295)(85 139 399 209)(86 140 400 210)(87 127 401 197)(88 128 402 198)(89 129 403 199)(90 130 404 200)(91 131 405 201)(92 132 406 202)(93 133 393 203)(94 134 394 204)(95 135 395 205)(96 136 396 206)(97 137 397 207)(98 138 398 208)(183 257 293 270)(184 258 294 271)(185 259 281 272)(186 260 282 273)(187 261 283 274)(188 262 284 275)(189 263 285 276)(190 264 286 277)(191 265 287 278)(192 266 288 279)(193 253 289 280)(194 254 290 267)(195 255 291 268)(196 256 292 269)
G:=sub<Sym(448)| (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), (1,256,317,114)(2,257,318,115)(3,258,319,116)(4,259,320,117)(5,260,321,118)(6,261,322,119)(7,262,309,120)(8,263,310,121)(9,264,311,122)(10,265,312,123)(11,266,313,124)(12,253,314,125)(13,254,315,126)(14,255,316,113)(15,178,422,275)(16,179,423,276)(17,180,424,277)(18,181,425,278)(19,182,426,279)(20,169,427,280)(21,170,428,267)(22,171,429,268)(23,172,430,269)(24,173,431,270)(25,174,432,271)(26,175,433,272)(27,176,434,273)(28,177,421,274)(29,60,327,87)(30,61,328,88)(31,62,329,89)(32,63,330,90)(33,64,331,91)(34,65,332,92)(35,66,333,93)(36,67,334,94)(37,68,335,95)(38,69,336,96)(39,70,323,97)(40,57,324,98)(41,58,325,85)(42,59,326,86)(43,373,292,163)(44,374,293,164)(45,375,294,165)(46,376,281,166)(47,377,282,167)(48,378,283,168)(49,365,284,155)(50,366,285,156)(51,367,286,157)(52,368,287,158)(53,369,288,159)(54,370,289,160)(55,371,290,161)(56,372,291,162)(71,140,337,141)(72,127,338,142)(73,128,339,143)(74,129,340,144)(75,130,341,145)(76,131,342,146)(77,132,343,147)(78,133,344,148)(79,134,345,149)(80,135,346,150)(81,136,347,151)(82,137,348,152)(83,138,349,153)(84,139,350,154)(99,302,406,241)(100,303,393,242)(101,304,394,243)(102,305,395,244)(103,306,396,245)(104,307,397,246)(105,308,398,247)(106,295,399,248)(107,296,400,249)(108,297,401,250)(109,298,402,251)(110,299,403,252)(111,300,404,239)(112,301,405,240)(183,445,352,233)(184,446,353,234)(185,447,354,235)(186,448,355,236)(187,435,356,237)(188,436,357,238)(189,437,358,225)(190,438,359,226)(191,439,360,227)(192,440,361,228)(193,441,362,229)(194,442,363,230)(195,443,364,231)(196,444,351,232)(197,415,385,224)(198,416,386,211)(199,417,387,212)(200,418,388,213)(201,419,389,214)(202,420,390,215)(203,407,391,216)(204,408,392,217)(205,409,379,218)(206,410,380,219)(207,411,381,220)(208,412,382,221)(209,413,383,222)(210,414,384,223), (1,131,430,201)(2,132,431,202)(3,133,432,203)(4,134,433,204)(5,135,434,205)(6,136,421,206)(7,137,422,207)(8,138,423,208)(9,139,424,209)(10,140,425,210)(11,127,426,197)(12,128,427,198)(13,129,428,199)(14,130,429,200)(15,381,309,152)(16,382,310,153)(17,383,311,154)(18,384,312,141)(19,385,313,142)(20,386,314,143)(21,387,315,144)(22,388,316,145)(23,389,317,146)(24,390,318,147)(25,391,319,148)(26,392,320,149)(27,379,321,150)(28,380,322,151)(29,361,250,53)(30,362,251,54)(31,363,252,55)(32,364,239,56)(33,351,240,43)(34,352,241,44)(35,353,242,45)(36,354,243,46)(37,355,244,47)(38,356,245,48)(39,357,246,49)(40,358,247,50)(41,359,248,51)(42,360,249,52)(57,225,105,366)(58,226,106,367)(59,227,107,368)(60,228,108,369)(61,229,109,370)(62,230,110,371)(63,231,111,372)(64,232,112,373)(65,233,99,374)(66,234,100,375)(67,235,101,376)(68,236,102,377)(69,237,103,378)(70,238,104,365)(71,181,223,123)(72,182,224,124)(73,169,211,125)(74,170,212,126)(75,171,213,113)(76,172,214,114)(77,173,215,115)(78,174,216,116)(79,175,217,117)(80,176,218,118)(81,177,219,119)(82,178,220,120)(83,179,221,121)(84,180,222,122)(85,438,399,157)(86,439,400,158)(87,440,401,159)(88,441,402,160)(89,442,403,161)(90,443,404,162)(91,444,405,163)(92,445,406,164)(93,446,393,165)(94,447,394,166)(95,448,395,167)(96,435,396,168)(97,436,397,155)(98,437,398,156)(183,302,293,332)(184,303,294,333)(185,304,281,334)(186,305,282,335)(187,306,283,336)(188,307,284,323)(189,308,285,324)(190,295,286,325)(191,296,287,326)(192,297,288,327)(193,298,289,328)(194,299,290,329)(195,300,291,330)(196,301,292,331)(253,339,280,416)(254,340,267,417)(255,341,268,418)(256,342,269,419)(257,343,270,420)(258,344,271,407)(259,345,272,408)(260,346,273,409)(261,347,274,410)(262,348,275,411)(263,349,276,412)(264,350,277,413)(265,337,278,414)(266,338,279,415), (1,232,430,373)(2,233,431,374)(3,234,432,375)(4,235,433,376)(5,236,434,377)(6,237,421,378)(7,238,422,365)(8,225,423,366)(9,226,424,367)(10,227,425,368)(11,228,426,369)(12,229,427,370)(13,230,428,371)(14,231,429,372)(15,155,309,436)(16,156,310,437)(17,157,311,438)(18,158,312,439)(19,159,313,440)(20,160,314,441)(21,161,315,442)(22,162,316,443)(23,163,317,444)(24,164,318,445)(25,165,319,446)(26,166,320,447)(27,167,321,448)(28,168,322,435)(29,415,250,338)(30,416,251,339)(31,417,252,340)(32,418,239,341)(33,419,240,342)(34,420,241,343)(35,407,242,344)(36,408,243,345)(37,409,244,346)(38,410,245,347)(39,411,246,348)(40,412,247,349)(41,413,248,350)(42,414,249,337)(43,172,351,114)(44,173,352,115)(45,174,353,116)(46,175,354,117)(47,176,355,118)(48,177,356,119)(49,178,357,120)(50,179,358,121)(51,180,359,122)(52,181,360,123)(53,182,361,124)(54,169,362,125)(55,170,363,126)(56,171,364,113)(57,153,105,382)(58,154,106,383)(59,141,107,384)(60,142,108,385)(61,143,109,386)(62,144,110,387)(63,145,111,388)(64,146,112,389)(65,147,99,390)(66,148,100,391)(67,149,101,392)(68,150,102,379)(69,151,103,380)(70,152,104,381)(71,326,223,296)(72,327,224,297)(73,328,211,298)(74,329,212,299)(75,330,213,300)(76,331,214,301)(77,332,215,302)(78,333,216,303)(79,334,217,304)(80,335,218,305)(81,336,219,306)(82,323,220,307)(83,324,221,308)(84,325,222,295)(85,139,399,209)(86,140,400,210)(87,127,401,197)(88,128,402,198)(89,129,403,199)(90,130,404,200)(91,131,405,201)(92,132,406,202)(93,133,393,203)(94,134,394,204)(95,135,395,205)(96,136,396,206)(97,137,397,207)(98,138,398,208)(183,257,293,270)(184,258,294,271)(185,259,281,272)(186,260,282,273)(187,261,283,274)(188,262,284,275)(189,263,285,276)(190,264,286,277)(191,265,287,278)(192,266,288,279)(193,253,289,280)(194,254,290,267)(195,255,291,268)(196,256,292,269)>;
G:=Group( (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), (1,256,317,114)(2,257,318,115)(3,258,319,116)(4,259,320,117)(5,260,321,118)(6,261,322,119)(7,262,309,120)(8,263,310,121)(9,264,311,122)(10,265,312,123)(11,266,313,124)(12,253,314,125)(13,254,315,126)(14,255,316,113)(15,178,422,275)(16,179,423,276)(17,180,424,277)(18,181,425,278)(19,182,426,279)(20,169,427,280)(21,170,428,267)(22,171,429,268)(23,172,430,269)(24,173,431,270)(25,174,432,271)(26,175,433,272)(27,176,434,273)(28,177,421,274)(29,60,327,87)(30,61,328,88)(31,62,329,89)(32,63,330,90)(33,64,331,91)(34,65,332,92)(35,66,333,93)(36,67,334,94)(37,68,335,95)(38,69,336,96)(39,70,323,97)(40,57,324,98)(41,58,325,85)(42,59,326,86)(43,373,292,163)(44,374,293,164)(45,375,294,165)(46,376,281,166)(47,377,282,167)(48,378,283,168)(49,365,284,155)(50,366,285,156)(51,367,286,157)(52,368,287,158)(53,369,288,159)(54,370,289,160)(55,371,290,161)(56,372,291,162)(71,140,337,141)(72,127,338,142)(73,128,339,143)(74,129,340,144)(75,130,341,145)(76,131,342,146)(77,132,343,147)(78,133,344,148)(79,134,345,149)(80,135,346,150)(81,136,347,151)(82,137,348,152)(83,138,349,153)(84,139,350,154)(99,302,406,241)(100,303,393,242)(101,304,394,243)(102,305,395,244)(103,306,396,245)(104,307,397,246)(105,308,398,247)(106,295,399,248)(107,296,400,249)(108,297,401,250)(109,298,402,251)(110,299,403,252)(111,300,404,239)(112,301,405,240)(183,445,352,233)(184,446,353,234)(185,447,354,235)(186,448,355,236)(187,435,356,237)(188,436,357,238)(189,437,358,225)(190,438,359,226)(191,439,360,227)(192,440,361,228)(193,441,362,229)(194,442,363,230)(195,443,364,231)(196,444,351,232)(197,415,385,224)(198,416,386,211)(199,417,387,212)(200,418,388,213)(201,419,389,214)(202,420,390,215)(203,407,391,216)(204,408,392,217)(205,409,379,218)(206,410,380,219)(207,411,381,220)(208,412,382,221)(209,413,383,222)(210,414,384,223), (1,131,430,201)(2,132,431,202)(3,133,432,203)(4,134,433,204)(5,135,434,205)(6,136,421,206)(7,137,422,207)(8,138,423,208)(9,139,424,209)(10,140,425,210)(11,127,426,197)(12,128,427,198)(13,129,428,199)(14,130,429,200)(15,381,309,152)(16,382,310,153)(17,383,311,154)(18,384,312,141)(19,385,313,142)(20,386,314,143)(21,387,315,144)(22,388,316,145)(23,389,317,146)(24,390,318,147)(25,391,319,148)(26,392,320,149)(27,379,321,150)(28,380,322,151)(29,361,250,53)(30,362,251,54)(31,363,252,55)(32,364,239,56)(33,351,240,43)(34,352,241,44)(35,353,242,45)(36,354,243,46)(37,355,244,47)(38,356,245,48)(39,357,246,49)(40,358,247,50)(41,359,248,51)(42,360,249,52)(57,225,105,366)(58,226,106,367)(59,227,107,368)(60,228,108,369)(61,229,109,370)(62,230,110,371)(63,231,111,372)(64,232,112,373)(65,233,99,374)(66,234,100,375)(67,235,101,376)(68,236,102,377)(69,237,103,378)(70,238,104,365)(71,181,223,123)(72,182,224,124)(73,169,211,125)(74,170,212,126)(75,171,213,113)(76,172,214,114)(77,173,215,115)(78,174,216,116)(79,175,217,117)(80,176,218,118)(81,177,219,119)(82,178,220,120)(83,179,221,121)(84,180,222,122)(85,438,399,157)(86,439,400,158)(87,440,401,159)(88,441,402,160)(89,442,403,161)(90,443,404,162)(91,444,405,163)(92,445,406,164)(93,446,393,165)(94,447,394,166)(95,448,395,167)(96,435,396,168)(97,436,397,155)(98,437,398,156)(183,302,293,332)(184,303,294,333)(185,304,281,334)(186,305,282,335)(187,306,283,336)(188,307,284,323)(189,308,285,324)(190,295,286,325)(191,296,287,326)(192,297,288,327)(193,298,289,328)(194,299,290,329)(195,300,291,330)(196,301,292,331)(253,339,280,416)(254,340,267,417)(255,341,268,418)(256,342,269,419)(257,343,270,420)(258,344,271,407)(259,345,272,408)(260,346,273,409)(261,347,274,410)(262,348,275,411)(263,349,276,412)(264,350,277,413)(265,337,278,414)(266,338,279,415), (1,232,430,373)(2,233,431,374)(3,234,432,375)(4,235,433,376)(5,236,434,377)(6,237,421,378)(7,238,422,365)(8,225,423,366)(9,226,424,367)(10,227,425,368)(11,228,426,369)(12,229,427,370)(13,230,428,371)(14,231,429,372)(15,155,309,436)(16,156,310,437)(17,157,311,438)(18,158,312,439)(19,159,313,440)(20,160,314,441)(21,161,315,442)(22,162,316,443)(23,163,317,444)(24,164,318,445)(25,165,319,446)(26,166,320,447)(27,167,321,448)(28,168,322,435)(29,415,250,338)(30,416,251,339)(31,417,252,340)(32,418,239,341)(33,419,240,342)(34,420,241,343)(35,407,242,344)(36,408,243,345)(37,409,244,346)(38,410,245,347)(39,411,246,348)(40,412,247,349)(41,413,248,350)(42,414,249,337)(43,172,351,114)(44,173,352,115)(45,174,353,116)(46,175,354,117)(47,176,355,118)(48,177,356,119)(49,178,357,120)(50,179,358,121)(51,180,359,122)(52,181,360,123)(53,182,361,124)(54,169,362,125)(55,170,363,126)(56,171,364,113)(57,153,105,382)(58,154,106,383)(59,141,107,384)(60,142,108,385)(61,143,109,386)(62,144,110,387)(63,145,111,388)(64,146,112,389)(65,147,99,390)(66,148,100,391)(67,149,101,392)(68,150,102,379)(69,151,103,380)(70,152,104,381)(71,326,223,296)(72,327,224,297)(73,328,211,298)(74,329,212,299)(75,330,213,300)(76,331,214,301)(77,332,215,302)(78,333,216,303)(79,334,217,304)(80,335,218,305)(81,336,219,306)(82,323,220,307)(83,324,221,308)(84,325,222,295)(85,139,399,209)(86,140,400,210)(87,127,401,197)(88,128,402,198)(89,129,403,199)(90,130,404,200)(91,131,405,201)(92,132,406,202)(93,133,393,203)(94,134,394,204)(95,135,395,205)(96,136,396,206)(97,137,397,207)(98,138,398,208)(183,257,293,270)(184,258,294,271)(185,259,281,272)(186,260,282,273)(187,261,283,274)(188,262,284,275)(189,263,285,276)(190,264,286,277)(191,265,287,278)(192,266,288,279)(193,253,289,280)(194,254,290,267)(195,255,291,268)(196,256,292,269) );
G=PermutationGroup([(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)], [(1,256,317,114),(2,257,318,115),(3,258,319,116),(4,259,320,117),(5,260,321,118),(6,261,322,119),(7,262,309,120),(8,263,310,121),(9,264,311,122),(10,265,312,123),(11,266,313,124),(12,253,314,125),(13,254,315,126),(14,255,316,113),(15,178,422,275),(16,179,423,276),(17,180,424,277),(18,181,425,278),(19,182,426,279),(20,169,427,280),(21,170,428,267),(22,171,429,268),(23,172,430,269),(24,173,431,270),(25,174,432,271),(26,175,433,272),(27,176,434,273),(28,177,421,274),(29,60,327,87),(30,61,328,88),(31,62,329,89),(32,63,330,90),(33,64,331,91),(34,65,332,92),(35,66,333,93),(36,67,334,94),(37,68,335,95),(38,69,336,96),(39,70,323,97),(40,57,324,98),(41,58,325,85),(42,59,326,86),(43,373,292,163),(44,374,293,164),(45,375,294,165),(46,376,281,166),(47,377,282,167),(48,378,283,168),(49,365,284,155),(50,366,285,156),(51,367,286,157),(52,368,287,158),(53,369,288,159),(54,370,289,160),(55,371,290,161),(56,372,291,162),(71,140,337,141),(72,127,338,142),(73,128,339,143),(74,129,340,144),(75,130,341,145),(76,131,342,146),(77,132,343,147),(78,133,344,148),(79,134,345,149),(80,135,346,150),(81,136,347,151),(82,137,348,152),(83,138,349,153),(84,139,350,154),(99,302,406,241),(100,303,393,242),(101,304,394,243),(102,305,395,244),(103,306,396,245),(104,307,397,246),(105,308,398,247),(106,295,399,248),(107,296,400,249),(108,297,401,250),(109,298,402,251),(110,299,403,252),(111,300,404,239),(112,301,405,240),(183,445,352,233),(184,446,353,234),(185,447,354,235),(186,448,355,236),(187,435,356,237),(188,436,357,238),(189,437,358,225),(190,438,359,226),(191,439,360,227),(192,440,361,228),(193,441,362,229),(194,442,363,230),(195,443,364,231),(196,444,351,232),(197,415,385,224),(198,416,386,211),(199,417,387,212),(200,418,388,213),(201,419,389,214),(202,420,390,215),(203,407,391,216),(204,408,392,217),(205,409,379,218),(206,410,380,219),(207,411,381,220),(208,412,382,221),(209,413,383,222),(210,414,384,223)], [(1,131,430,201),(2,132,431,202),(3,133,432,203),(4,134,433,204),(5,135,434,205),(6,136,421,206),(7,137,422,207),(8,138,423,208),(9,139,424,209),(10,140,425,210),(11,127,426,197),(12,128,427,198),(13,129,428,199),(14,130,429,200),(15,381,309,152),(16,382,310,153),(17,383,311,154),(18,384,312,141),(19,385,313,142),(20,386,314,143),(21,387,315,144),(22,388,316,145),(23,389,317,146),(24,390,318,147),(25,391,319,148),(26,392,320,149),(27,379,321,150),(28,380,322,151),(29,361,250,53),(30,362,251,54),(31,363,252,55),(32,364,239,56),(33,351,240,43),(34,352,241,44),(35,353,242,45),(36,354,243,46),(37,355,244,47),(38,356,245,48),(39,357,246,49),(40,358,247,50),(41,359,248,51),(42,360,249,52),(57,225,105,366),(58,226,106,367),(59,227,107,368),(60,228,108,369),(61,229,109,370),(62,230,110,371),(63,231,111,372),(64,232,112,373),(65,233,99,374),(66,234,100,375),(67,235,101,376),(68,236,102,377),(69,237,103,378),(70,238,104,365),(71,181,223,123),(72,182,224,124),(73,169,211,125),(74,170,212,126),(75,171,213,113),(76,172,214,114),(77,173,215,115),(78,174,216,116),(79,175,217,117),(80,176,218,118),(81,177,219,119),(82,178,220,120),(83,179,221,121),(84,180,222,122),(85,438,399,157),(86,439,400,158),(87,440,401,159),(88,441,402,160),(89,442,403,161),(90,443,404,162),(91,444,405,163),(92,445,406,164),(93,446,393,165),(94,447,394,166),(95,448,395,167),(96,435,396,168),(97,436,397,155),(98,437,398,156),(183,302,293,332),(184,303,294,333),(185,304,281,334),(186,305,282,335),(187,306,283,336),(188,307,284,323),(189,308,285,324),(190,295,286,325),(191,296,287,326),(192,297,288,327),(193,298,289,328),(194,299,290,329),(195,300,291,330),(196,301,292,331),(253,339,280,416),(254,340,267,417),(255,341,268,418),(256,342,269,419),(257,343,270,420),(258,344,271,407),(259,345,272,408),(260,346,273,409),(261,347,274,410),(262,348,275,411),(263,349,276,412),(264,350,277,413),(265,337,278,414),(266,338,279,415)], [(1,232,430,373),(2,233,431,374),(3,234,432,375),(4,235,433,376),(5,236,434,377),(6,237,421,378),(7,238,422,365),(8,225,423,366),(9,226,424,367),(10,227,425,368),(11,228,426,369),(12,229,427,370),(13,230,428,371),(14,231,429,372),(15,155,309,436),(16,156,310,437),(17,157,311,438),(18,158,312,439),(19,159,313,440),(20,160,314,441),(21,161,315,442),(22,162,316,443),(23,163,317,444),(24,164,318,445),(25,165,319,446),(26,166,320,447),(27,167,321,448),(28,168,322,435),(29,415,250,338),(30,416,251,339),(31,417,252,340),(32,418,239,341),(33,419,240,342),(34,420,241,343),(35,407,242,344),(36,408,243,345),(37,409,244,346),(38,410,245,347),(39,411,246,348),(40,412,247,349),(41,413,248,350),(42,414,249,337),(43,172,351,114),(44,173,352,115),(45,174,353,116),(46,175,354,117),(47,176,355,118),(48,177,356,119),(49,178,357,120),(50,179,358,121),(51,180,359,122),(52,181,360,123),(53,182,361,124),(54,169,362,125),(55,170,363,126),(56,171,364,113),(57,153,105,382),(58,154,106,383),(59,141,107,384),(60,142,108,385),(61,143,109,386),(62,144,110,387),(63,145,111,388),(64,146,112,389),(65,147,99,390),(66,148,100,391),(67,149,101,392),(68,150,102,379),(69,151,103,380),(70,152,104,381),(71,326,223,296),(72,327,224,297),(73,328,211,298),(74,329,212,299),(75,330,213,300),(76,331,214,301),(77,332,215,302),(78,333,216,303),(79,334,217,304),(80,335,218,305),(81,336,219,306),(82,323,220,307),(83,324,221,308),(84,325,222,295),(85,139,399,209),(86,140,400,210),(87,127,401,197),(88,128,402,198),(89,129,403,199),(90,130,404,200),(91,131,405,201),(92,132,406,202),(93,133,393,203),(94,134,394,204),(95,135,395,205),(96,136,396,206),(97,137,397,207),(98,138,398,208),(183,257,293,270),(184,258,294,271),(185,259,281,272),(186,260,282,273),(187,261,283,274),(188,262,284,275),(189,263,285,276),(190,264,286,277),(191,265,287,278),(192,266,288,279),(193,253,289,280),(194,254,290,267),(195,255,291,268),(196,256,292,269)])
Matrix representation ►G ⊆ GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 17 | 2 |
| 0 | 0 | 0 | 1 | 12 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 13 | 5 | 0 | 0 |
| 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 0 | 21 | 22 |
| 0 | 0 | 0 | 9 | 8 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,22,0,0,0,0,0,22,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,17,1,0,0,0,2,12],[28,0,0,0,0,0,13,24,0,0,0,5,16,0,0,0,0,0,21,9,0,0,0,22,8] >;
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 7A | ··· | 7F | 14A | ··· | 14AP | 28A | ··· | 28BT | 28BU | ··· | 28DP |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | Q8 | C4○D4 | C7×Q8 | C7×C4○D4 |
| kernel | C14×C42.C2 | C2×C4×C28 | C14×C4⋊C4 | C7×C42.C2 | C2×C42.C2 | C2×C42 | C2×C4⋊C4 | C42.C2 | C2×C28 | C2×C14 | C2×C4 | C22 |
| # reps | 1 | 1 | 6 | 8 | 6 | 6 | 36 | 48 | 4 | 8 | 24 | 48 |
In GAP, Magma, Sage, TeX
C_{14}\times C_4^2.C_2 % in TeX
G:=Group("C14xC4^2.C2"); // GroupNames label
G:=SmallGroup(448,1310);
// by ID
G=gap.SmallGroup(448,1310);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,1576,4790,604]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
// generators/relations
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