direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×C2.D8, (C2×C8)⋊5C28, C8⋊7(C2×C28), (C2×C56)⋊15C4, C56⋊33(C2×C4), C2.2(C14×D8), C4.2(Q8×C14), (C2×C14).54D8, C14.74(C2×D8), C28.63(C4⋊C4), (C2×C28).77Q8, C28.91(C2×Q8), C2.2(C14×Q16), (C2×C28).418D4, (C22×C8).8C14, (C2×C14).21Q16, C14.49(C2×Q16), C23.57(C7×D4), C22.13(C7×D8), C22.6(C7×Q16), C4.25(C22×C28), (C22×C56).26C2, C22.48(D4×C14), (C2×C56).420C22, (C2×C28).899C23, C28.183(C22×C4), (C22×C14).218D4, (C22×C28).588C22, C4.14(C7×C4⋊C4), C2.12(C14×C4⋊C4), C14.68(C2×C4⋊C4), (C2×C4).73(C7×D4), (C2×C4⋊C4).14C14, (C14×C4⋊C4).43C2, (C2×C4).19(C7×Q8), C4⋊C4.42(C2×C14), (C2×C8).75(C2×C14), (C2×C4).75(C2×C28), C22.21(C7×C4⋊C4), (C2×C14).64(C4⋊C4), (C2×C28).336(C2×C4), (C2×C14).624(C2×D4), (C7×C4⋊C4).363C22, (C2×C4).74(C22×C14), (C22×C4).117(C2×C14), SmallGroup(448,834)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C2.D8
G = < a,b,c,d | a14=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 194 in 130 conjugacy classes, 98 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C28, C28, C28, C2×C14, C2×C14, C2.D8, C2×C4⋊C4, C22×C8, C56, C2×C28, C2×C28, C2×C28, C22×C14, C2×C2.D8, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C22×C28, C22×C28, C7×C2.D8, C14×C4⋊C4, C22×C56, C14×C2.D8
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C2×C28, C7×D4, C7×Q8, C22×C14, C2×C2.D8, C7×C4⋊C4, C7×D8, C7×Q16, C22×C28, D4×C14, Q8×C14, C7×C2.D8, C14×C4⋊C4, C14×D8, C14×Q16, C14×C2.D8
►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)
(1 303)(2 304)(3 305)(4 306)(5 307)(6 308)(7 295)(8 296)(9 297)(10 298)(11 299)(12 300)(13 301)(14 302)(15 390)(16 391)(17 392)(18 379)(19 380)(20 381)(21 382)(22 383)(23 384)(24 385)(25 386)(26 387)(27 388)(28 389)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 148)(36 149)(37 150)(38 151)(39 152)(40 153)(41 154)(42 141)(43 195)(44 196)(45 183)(46 184)(47 185)(48 186)(49 187)(50 188)(51 189)(52 190)(53 191)(54 192)(55 193)(56 194)(57 81)(58 82)(59 83)(60 84)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(85 422)(86 423)(87 424)(88 425)(89 426)(90 427)(91 428)(92 429)(93 430)(94 431)(95 432)(96 433)(97 434)(98 421)(99 351)(100 352)(101 353)(102 354)(103 355)(104 356)(105 357)(106 358)(107 359)(108 360)(109 361)(110 362)(111 363)(112 364)(113 409)(114 410)(115 411)(116 412)(117 413)(118 414)(119 415)(120 416)(121 417)(122 418)(123 419)(124 420)(125 407)(126 408)(127 378)(128 365)(129 366)(130 367)(131 368)(132 369)(133 370)(134 371)(135 372)(136 373)(137 374)(138 375)(139 376)(140 377)(155 200)(156 201)(157 202)(158 203)(159 204)(160 205)(161 206)(162 207)(163 208)(164 209)(165 210)(166 197)(167 198)(168 199)(169 272)(170 273)(171 274)(172 275)(173 276)(174 277)(175 278)(176 279)(177 280)(178 267)(179 268)(180 269)(181 270)(182 271)(211 262)(212 263)(213 264)(214 265)(215 266)(216 253)(217 254)(218 255)(219 256)(220 257)(221 258)(222 259)(223 260)(224 261)(225 336)(226 323)(227 324)(228 325)(229 326)(230 327)(231 328)(232 329)(233 330)(234 331)(235 332)(236 333)(237 334)(238 335)(239 399)(240 400)(241 401)(242 402)(243 403)(244 404)(245 405)(246 406)(247 393)(248 394)(249 395)(250 396)(251 397)(252 398)(281 314)(282 315)(283 316)(284 317)(285 318)(286 319)(287 320)(288 321)(289 322)(290 309)(291 310)(292 311)(293 312)(294 313)(337 435)(338 436)(339 437)(340 438)(341 439)(342 440)(343 441)(344 442)(345 443)(346 444)(347 445)(348 446)(349 447)(350 448)
(1 55 428 149 274 377 362 115)(2 56 429 150 275 378 363 116)(3 43 430 151 276 365 364 117)(4 44 431 152 277 366 351 118)(5 45 432 153 278 367 352 119)(6 46 433 154 279 368 353 120)(7 47 434 141 280 369 354 121)(8 48 421 142 267 370 355 122)(9 49 422 143 268 371 356 123)(10 50 423 144 269 372 357 124)(11 51 424 145 270 373 358 125)(12 52 425 146 271 374 359 126)(13 53 426 147 272 375 360 113)(14 54 427 148 273 376 361 114)(15 157 350 290 84 220 232 402)(16 158 337 291 71 221 233 403)(17 159 338 292 72 222 234 404)(18 160 339 293 73 223 235 405)(19 161 340 294 74 224 236 406)(20 162 341 281 75 211 237 393)(21 163 342 282 76 212 238 394)(22 164 343 283 77 213 225 395)(23 165 344 284 78 214 226 396)(24 166 345 285 79 215 227 397)(25 167 346 286 80 216 228 398)(26 168 347 287 81 217 229 399)(27 155 348 288 82 218 230 400)(28 156 349 289 83 219 231 401)(29 178 133 103 418 296 186 98)(30 179 134 104 419 297 187 85)(31 180 135 105 420 298 188 86)(32 181 136 106 407 299 189 87)(33 182 137 107 408 300 190 88)(34 169 138 108 409 301 191 89)(35 170 139 109 410 302 192 90)(36 171 140 110 411 303 193 91)(37 172 127 111 412 304 194 92)(38 173 128 112 413 305 195 93)(39 174 129 99 414 306 196 94)(40 175 130 100 415 307 183 95)(41 176 131 101 416 308 184 96)(42 177 132 102 417 295 185 97)(57 254 326 239 387 199 445 320)(58 255 327 240 388 200 446 321)(59 256 328 241 389 201 447 322)(60 257 329 242 390 202 448 309)(61 258 330 243 391 203 435 310)(62 259 331 244 392 204 436 311)(63 260 332 245 379 205 437 312)(64 261 333 246 380 206 438 313)(65 262 334 247 381 207 439 314)(66 263 335 248 382 208 440 315)(67 264 336 249 383 209 441 316)(68 265 323 250 384 210 442 317)(69 266 324 251 385 197 443 318)(70 253 325 252 386 198 444 319)
(1 383 303 22)(2 384 304 23)(3 385 305 24)(4 386 306 25)(5 387 307 26)(6 388 308 27)(7 389 295 28)(8 390 296 15)(9 391 297 16)(10 392 298 17)(11 379 299 18)(12 380 300 19)(13 381 301 20)(14 382 302 21)(29 220 142 257)(30 221 143 258)(31 222 144 259)(32 223 145 260)(33 224 146 261)(34 211 147 262)(35 212 148 263)(36 213 149 264)(37 214 150 265)(38 215 151 266)(39 216 152 253)(40 217 153 254)(41 218 154 255)(42 219 141 256)(43 251 195 397)(44 252 196 398)(45 239 183 399)(46 240 184 400)(47 241 185 401)(48 242 186 402)(49 243 187 403)(50 244 188 404)(51 245 189 405)(52 246 190 406)(53 247 191 393)(54 248 192 394)(55 249 193 395)(56 250 194 396)(57 175 81 278)(58 176 82 279)(59 177 83 280)(60 178 84 267)(61 179 71 268)(62 180 72 269)(63 181 73 270)(64 182 74 271)(65 169 75 272)(66 170 76 273)(67 171 77 274)(68 172 78 275)(69 173 79 276)(70 174 80 277)(85 233 422 330)(86 234 423 331)(87 235 424 332)(88 236 425 333)(89 237 426 334)(90 238 427 335)(91 225 428 336)(92 226 429 323)(93 227 430 324)(94 228 431 325)(95 229 432 326)(96 230 433 327)(97 231 434 328)(98 232 421 329)(99 346 351 444)(100 347 352 445)(101 348 353 446)(102 349 354 447)(103 350 355 448)(104 337 356 435)(105 338 357 436)(106 339 358 437)(107 340 359 438)(108 341 360 439)(109 342 361 440)(110 343 362 441)(111 344 363 442)(112 345 364 443)(113 207 409 162)(114 208 410 163)(115 209 411 164)(116 210 412 165)(117 197 413 166)(118 198 414 167)(119 199 415 168)(120 200 416 155)(121 201 417 156)(122 202 418 157)(123 203 419 158)(124 204 420 159)(125 205 407 160)(126 206 408 161)(127 284 378 317)(128 285 365 318)(129 286 366 319)(130 287 367 320)(131 288 368 321)(132 289 369 322)(133 290 370 309)(134 291 371 310)(135 292 372 311)(136 293 373 312)(137 294 374 313)(138 281 375 314)(139 282 376 315)(140 283 377 316)
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), (1,303)(2,304)(3,305)(4,306)(5,307)(6,308)(7,295)(8,296)(9,297)(10,298)(11,299)(12,300)(13,301)(14,302)(15,390)(16,391)(17,392)(18,379)(19,380)(20,381)(21,382)(22,383)(23,384)(24,385)(25,386)(26,387)(27,388)(28,389)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,149)(37,150)(38,151)(39,152)(40,153)(41,154)(42,141)(43,195)(44,196)(45,183)(46,184)(47,185)(48,186)(49,187)(50,188)(51,189)(52,190)(53,191)(54,192)(55,193)(56,194)(57,81)(58,82)(59,83)(60,84)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(85,422)(86,423)(87,424)(88,425)(89,426)(90,427)(91,428)(92,429)(93,430)(94,431)(95,432)(96,433)(97,434)(98,421)(99,351)(100,352)(101,353)(102,354)(103,355)(104,356)(105,357)(106,358)(107,359)(108,360)(109,361)(110,362)(111,363)(112,364)(113,409)(114,410)(115,411)(116,412)(117,413)(118,414)(119,415)(120,416)(121,417)(122,418)(123,419)(124,420)(125,407)(126,408)(127,378)(128,365)(129,366)(130,367)(131,368)(132,369)(133,370)(134,371)(135,372)(136,373)(137,374)(138,375)(139,376)(140,377)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,197)(167,198)(168,199)(169,272)(170,273)(171,274)(172,275)(173,276)(174,277)(175,278)(176,279)(177,280)(178,267)(179,268)(180,269)(181,270)(182,271)(211,262)(212,263)(213,264)(214,265)(215,266)(216,253)(217,254)(218,255)(219,256)(220,257)(221,258)(222,259)(223,260)(224,261)(225,336)(226,323)(227,324)(228,325)(229,326)(230,327)(231,328)(232,329)(233,330)(234,331)(235,332)(236,333)(237,334)(238,335)(239,399)(240,400)(241,401)(242,402)(243,403)(244,404)(245,405)(246,406)(247,393)(248,394)(249,395)(250,396)(251,397)(252,398)(281,314)(282,315)(283,316)(284,317)(285,318)(286,319)(287,320)(288,321)(289,322)(290,309)(291,310)(292,311)(293,312)(294,313)(337,435)(338,436)(339,437)(340,438)(341,439)(342,440)(343,441)(344,442)(345,443)(346,444)(347,445)(348,446)(349,447)(350,448), (1,55,428,149,274,377,362,115)(2,56,429,150,275,378,363,116)(3,43,430,151,276,365,364,117)(4,44,431,152,277,366,351,118)(5,45,432,153,278,367,352,119)(6,46,433,154,279,368,353,120)(7,47,434,141,280,369,354,121)(8,48,421,142,267,370,355,122)(9,49,422,143,268,371,356,123)(10,50,423,144,269,372,357,124)(11,51,424,145,270,373,358,125)(12,52,425,146,271,374,359,126)(13,53,426,147,272,375,360,113)(14,54,427,148,273,376,361,114)(15,157,350,290,84,220,232,402)(16,158,337,291,71,221,233,403)(17,159,338,292,72,222,234,404)(18,160,339,293,73,223,235,405)(19,161,340,294,74,224,236,406)(20,162,341,281,75,211,237,393)(21,163,342,282,76,212,238,394)(22,164,343,283,77,213,225,395)(23,165,344,284,78,214,226,396)(24,166,345,285,79,215,227,397)(25,167,346,286,80,216,228,398)(26,168,347,287,81,217,229,399)(27,155,348,288,82,218,230,400)(28,156,349,289,83,219,231,401)(29,178,133,103,418,296,186,98)(30,179,134,104,419,297,187,85)(31,180,135,105,420,298,188,86)(32,181,136,106,407,299,189,87)(33,182,137,107,408,300,190,88)(34,169,138,108,409,301,191,89)(35,170,139,109,410,302,192,90)(36,171,140,110,411,303,193,91)(37,172,127,111,412,304,194,92)(38,173,128,112,413,305,195,93)(39,174,129,99,414,306,196,94)(40,175,130,100,415,307,183,95)(41,176,131,101,416,308,184,96)(42,177,132,102,417,295,185,97)(57,254,326,239,387,199,445,320)(58,255,327,240,388,200,446,321)(59,256,328,241,389,201,447,322)(60,257,329,242,390,202,448,309)(61,258,330,243,391,203,435,310)(62,259,331,244,392,204,436,311)(63,260,332,245,379,205,437,312)(64,261,333,246,380,206,438,313)(65,262,334,247,381,207,439,314)(66,263,335,248,382,208,440,315)(67,264,336,249,383,209,441,316)(68,265,323,250,384,210,442,317)(69,266,324,251,385,197,443,318)(70,253,325,252,386,198,444,319), (1,383,303,22)(2,384,304,23)(3,385,305,24)(4,386,306,25)(5,387,307,26)(6,388,308,27)(7,389,295,28)(8,390,296,15)(9,391,297,16)(10,392,298,17)(11,379,299,18)(12,380,300,19)(13,381,301,20)(14,382,302,21)(29,220,142,257)(30,221,143,258)(31,222,144,259)(32,223,145,260)(33,224,146,261)(34,211,147,262)(35,212,148,263)(36,213,149,264)(37,214,150,265)(38,215,151,266)(39,216,152,253)(40,217,153,254)(41,218,154,255)(42,219,141,256)(43,251,195,397)(44,252,196,398)(45,239,183,399)(46,240,184,400)(47,241,185,401)(48,242,186,402)(49,243,187,403)(50,244,188,404)(51,245,189,405)(52,246,190,406)(53,247,191,393)(54,248,192,394)(55,249,193,395)(56,250,194,396)(57,175,81,278)(58,176,82,279)(59,177,83,280)(60,178,84,267)(61,179,71,268)(62,180,72,269)(63,181,73,270)(64,182,74,271)(65,169,75,272)(66,170,76,273)(67,171,77,274)(68,172,78,275)(69,173,79,276)(70,174,80,277)(85,233,422,330)(86,234,423,331)(87,235,424,332)(88,236,425,333)(89,237,426,334)(90,238,427,335)(91,225,428,336)(92,226,429,323)(93,227,430,324)(94,228,431,325)(95,229,432,326)(96,230,433,327)(97,231,434,328)(98,232,421,329)(99,346,351,444)(100,347,352,445)(101,348,353,446)(102,349,354,447)(103,350,355,448)(104,337,356,435)(105,338,357,436)(106,339,358,437)(107,340,359,438)(108,341,360,439)(109,342,361,440)(110,343,362,441)(111,344,363,442)(112,345,364,443)(113,207,409,162)(114,208,410,163)(115,209,411,164)(116,210,412,165)(117,197,413,166)(118,198,414,167)(119,199,415,168)(120,200,416,155)(121,201,417,156)(122,202,418,157)(123,203,419,158)(124,204,420,159)(125,205,407,160)(126,206,408,161)(127,284,378,317)(128,285,365,318)(129,286,366,319)(130,287,367,320)(131,288,368,321)(132,289,369,322)(133,290,370,309)(134,291,371,310)(135,292,372,311)(136,293,373,312)(137,294,374,313)(138,281,375,314)(139,282,376,315)(140,283,377,316)>;
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), (1,303)(2,304)(3,305)(4,306)(5,307)(6,308)(7,295)(8,296)(9,297)(10,298)(11,299)(12,300)(13,301)(14,302)(15,390)(16,391)(17,392)(18,379)(19,380)(20,381)(21,382)(22,383)(23,384)(24,385)(25,386)(26,387)(27,388)(28,389)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,149)(37,150)(38,151)(39,152)(40,153)(41,154)(42,141)(43,195)(44,196)(45,183)(46,184)(47,185)(48,186)(49,187)(50,188)(51,189)(52,190)(53,191)(54,192)(55,193)(56,194)(57,81)(58,82)(59,83)(60,84)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(85,422)(86,423)(87,424)(88,425)(89,426)(90,427)(91,428)(92,429)(93,430)(94,431)(95,432)(96,433)(97,434)(98,421)(99,351)(100,352)(101,353)(102,354)(103,355)(104,356)(105,357)(106,358)(107,359)(108,360)(109,361)(110,362)(111,363)(112,364)(113,409)(114,410)(115,411)(116,412)(117,413)(118,414)(119,415)(120,416)(121,417)(122,418)(123,419)(124,420)(125,407)(126,408)(127,378)(128,365)(129,366)(130,367)(131,368)(132,369)(133,370)(134,371)(135,372)(136,373)(137,374)(138,375)(139,376)(140,377)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,197)(167,198)(168,199)(169,272)(170,273)(171,274)(172,275)(173,276)(174,277)(175,278)(176,279)(177,280)(178,267)(179,268)(180,269)(181,270)(182,271)(211,262)(212,263)(213,264)(214,265)(215,266)(216,253)(217,254)(218,255)(219,256)(220,257)(221,258)(222,259)(223,260)(224,261)(225,336)(226,323)(227,324)(228,325)(229,326)(230,327)(231,328)(232,329)(233,330)(234,331)(235,332)(236,333)(237,334)(238,335)(239,399)(240,400)(241,401)(242,402)(243,403)(244,404)(245,405)(246,406)(247,393)(248,394)(249,395)(250,396)(251,397)(252,398)(281,314)(282,315)(283,316)(284,317)(285,318)(286,319)(287,320)(288,321)(289,322)(290,309)(291,310)(292,311)(293,312)(294,313)(337,435)(338,436)(339,437)(340,438)(341,439)(342,440)(343,441)(344,442)(345,443)(346,444)(347,445)(348,446)(349,447)(350,448), (1,55,428,149,274,377,362,115)(2,56,429,150,275,378,363,116)(3,43,430,151,276,365,364,117)(4,44,431,152,277,366,351,118)(5,45,432,153,278,367,352,119)(6,46,433,154,279,368,353,120)(7,47,434,141,280,369,354,121)(8,48,421,142,267,370,355,122)(9,49,422,143,268,371,356,123)(10,50,423,144,269,372,357,124)(11,51,424,145,270,373,358,125)(12,52,425,146,271,374,359,126)(13,53,426,147,272,375,360,113)(14,54,427,148,273,376,361,114)(15,157,350,290,84,220,232,402)(16,158,337,291,71,221,233,403)(17,159,338,292,72,222,234,404)(18,160,339,293,73,223,235,405)(19,161,340,294,74,224,236,406)(20,162,341,281,75,211,237,393)(21,163,342,282,76,212,238,394)(22,164,343,283,77,213,225,395)(23,165,344,284,78,214,226,396)(24,166,345,285,79,215,227,397)(25,167,346,286,80,216,228,398)(26,168,347,287,81,217,229,399)(27,155,348,288,82,218,230,400)(28,156,349,289,83,219,231,401)(29,178,133,103,418,296,186,98)(30,179,134,104,419,297,187,85)(31,180,135,105,420,298,188,86)(32,181,136,106,407,299,189,87)(33,182,137,107,408,300,190,88)(34,169,138,108,409,301,191,89)(35,170,139,109,410,302,192,90)(36,171,140,110,411,303,193,91)(37,172,127,111,412,304,194,92)(38,173,128,112,413,305,195,93)(39,174,129,99,414,306,196,94)(40,175,130,100,415,307,183,95)(41,176,131,101,416,308,184,96)(42,177,132,102,417,295,185,97)(57,254,326,239,387,199,445,320)(58,255,327,240,388,200,446,321)(59,256,328,241,389,201,447,322)(60,257,329,242,390,202,448,309)(61,258,330,243,391,203,435,310)(62,259,331,244,392,204,436,311)(63,260,332,245,379,205,437,312)(64,261,333,246,380,206,438,313)(65,262,334,247,381,207,439,314)(66,263,335,248,382,208,440,315)(67,264,336,249,383,209,441,316)(68,265,323,250,384,210,442,317)(69,266,324,251,385,197,443,318)(70,253,325,252,386,198,444,319), (1,383,303,22)(2,384,304,23)(3,385,305,24)(4,386,306,25)(5,387,307,26)(6,388,308,27)(7,389,295,28)(8,390,296,15)(9,391,297,16)(10,392,298,17)(11,379,299,18)(12,380,300,19)(13,381,301,20)(14,382,302,21)(29,220,142,257)(30,221,143,258)(31,222,144,259)(32,223,145,260)(33,224,146,261)(34,211,147,262)(35,212,148,263)(36,213,149,264)(37,214,150,265)(38,215,151,266)(39,216,152,253)(40,217,153,254)(41,218,154,255)(42,219,141,256)(43,251,195,397)(44,252,196,398)(45,239,183,399)(46,240,184,400)(47,241,185,401)(48,242,186,402)(49,243,187,403)(50,244,188,404)(51,245,189,405)(52,246,190,406)(53,247,191,393)(54,248,192,394)(55,249,193,395)(56,250,194,396)(57,175,81,278)(58,176,82,279)(59,177,83,280)(60,178,84,267)(61,179,71,268)(62,180,72,269)(63,181,73,270)(64,182,74,271)(65,169,75,272)(66,170,76,273)(67,171,77,274)(68,172,78,275)(69,173,79,276)(70,174,80,277)(85,233,422,330)(86,234,423,331)(87,235,424,332)(88,236,425,333)(89,237,426,334)(90,238,427,335)(91,225,428,336)(92,226,429,323)(93,227,430,324)(94,228,431,325)(95,229,432,326)(96,230,433,327)(97,231,434,328)(98,232,421,329)(99,346,351,444)(100,347,352,445)(101,348,353,446)(102,349,354,447)(103,350,355,448)(104,337,356,435)(105,338,357,436)(106,339,358,437)(107,340,359,438)(108,341,360,439)(109,342,361,440)(110,343,362,441)(111,344,363,442)(112,345,364,443)(113,207,409,162)(114,208,410,163)(115,209,411,164)(116,210,412,165)(117,197,413,166)(118,198,414,167)(119,199,415,168)(120,200,416,155)(121,201,417,156)(122,202,418,157)(123,203,419,158)(124,204,420,159)(125,205,407,160)(126,206,408,161)(127,284,378,317)(128,285,365,318)(129,286,366,319)(130,287,367,320)(131,288,368,321)(132,289,369,322)(133,290,370,309)(134,291,371,310)(135,292,372,311)(136,293,373,312)(137,294,374,313)(138,281,375,314)(139,282,376,315)(140,283,377,316) );
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)], [(1,303),(2,304),(3,305),(4,306),(5,307),(6,308),(7,295),(8,296),(9,297),(10,298),(11,299),(12,300),(13,301),(14,302),(15,390),(16,391),(17,392),(18,379),(19,380),(20,381),(21,382),(22,383),(23,384),(24,385),(25,386),(26,387),(27,388),(28,389),(29,142),(30,143),(31,144),(32,145),(33,146),(34,147),(35,148),(36,149),(37,150),(38,151),(39,152),(40,153),(41,154),(42,141),(43,195),(44,196),(45,183),(46,184),(47,185),(48,186),(49,187),(50,188),(51,189),(52,190),(53,191),(54,192),(55,193),(56,194),(57,81),(58,82),(59,83),(60,84),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(85,422),(86,423),(87,424),(88,425),(89,426),(90,427),(91,428),(92,429),(93,430),(94,431),(95,432),(96,433),(97,434),(98,421),(99,351),(100,352),(101,353),(102,354),(103,355),(104,356),(105,357),(106,358),(107,359),(108,360),(109,361),(110,362),(111,363),(112,364),(113,409),(114,410),(115,411),(116,412),(117,413),(118,414),(119,415),(120,416),(121,417),(122,418),(123,419),(124,420),(125,407),(126,408),(127,378),(128,365),(129,366),(130,367),(131,368),(132,369),(133,370),(134,371),(135,372),(136,373),(137,374),(138,375),(139,376),(140,377),(155,200),(156,201),(157,202),(158,203),(159,204),(160,205),(161,206),(162,207),(163,208),(164,209),(165,210),(166,197),(167,198),(168,199),(169,272),(170,273),(171,274),(172,275),(173,276),(174,277),(175,278),(176,279),(177,280),(178,267),(179,268),(180,269),(181,270),(182,271),(211,262),(212,263),(213,264),(214,265),(215,266),(216,253),(217,254),(218,255),(219,256),(220,257),(221,258),(222,259),(223,260),(224,261),(225,336),(226,323),(227,324),(228,325),(229,326),(230,327),(231,328),(232,329),(233,330),(234,331),(235,332),(236,333),(237,334),(238,335),(239,399),(240,400),(241,401),(242,402),(243,403),(244,404),(245,405),(246,406),(247,393),(248,394),(249,395),(250,396),(251,397),(252,398),(281,314),(282,315),(283,316),(284,317),(285,318),(286,319),(287,320),(288,321),(289,322),(290,309),(291,310),(292,311),(293,312),(294,313),(337,435),(338,436),(339,437),(340,438),(341,439),(342,440),(343,441),(344,442),(345,443),(346,444),(347,445),(348,446),(349,447),(350,448)], [(1,55,428,149,274,377,362,115),(2,56,429,150,275,378,363,116),(3,43,430,151,276,365,364,117),(4,44,431,152,277,366,351,118),(5,45,432,153,278,367,352,119),(6,46,433,154,279,368,353,120),(7,47,434,141,280,369,354,121),(8,48,421,142,267,370,355,122),(9,49,422,143,268,371,356,123),(10,50,423,144,269,372,357,124),(11,51,424,145,270,373,358,125),(12,52,425,146,271,374,359,126),(13,53,426,147,272,375,360,113),(14,54,427,148,273,376,361,114),(15,157,350,290,84,220,232,402),(16,158,337,291,71,221,233,403),(17,159,338,292,72,222,234,404),(18,160,339,293,73,223,235,405),(19,161,340,294,74,224,236,406),(20,162,341,281,75,211,237,393),(21,163,342,282,76,212,238,394),(22,164,343,283,77,213,225,395),(23,165,344,284,78,214,226,396),(24,166,345,285,79,215,227,397),(25,167,346,286,80,216,228,398),(26,168,347,287,81,217,229,399),(27,155,348,288,82,218,230,400),(28,156,349,289,83,219,231,401),(29,178,133,103,418,296,186,98),(30,179,134,104,419,297,187,85),(31,180,135,105,420,298,188,86),(32,181,136,106,407,299,189,87),(33,182,137,107,408,300,190,88),(34,169,138,108,409,301,191,89),(35,170,139,109,410,302,192,90),(36,171,140,110,411,303,193,91),(37,172,127,111,412,304,194,92),(38,173,128,112,413,305,195,93),(39,174,129,99,414,306,196,94),(40,175,130,100,415,307,183,95),(41,176,131,101,416,308,184,96),(42,177,132,102,417,295,185,97),(57,254,326,239,387,199,445,320),(58,255,327,240,388,200,446,321),(59,256,328,241,389,201,447,322),(60,257,329,242,390,202,448,309),(61,258,330,243,391,203,435,310),(62,259,331,244,392,204,436,311),(63,260,332,245,379,205,437,312),(64,261,333,246,380,206,438,313),(65,262,334,247,381,207,439,314),(66,263,335,248,382,208,440,315),(67,264,336,249,383,209,441,316),(68,265,323,250,384,210,442,317),(69,266,324,251,385,197,443,318),(70,253,325,252,386,198,444,319)], [(1,383,303,22),(2,384,304,23),(3,385,305,24),(4,386,306,25),(5,387,307,26),(6,388,308,27),(7,389,295,28),(8,390,296,15),(9,391,297,16),(10,392,298,17),(11,379,299,18),(12,380,300,19),(13,381,301,20),(14,382,302,21),(29,220,142,257),(30,221,143,258),(31,222,144,259),(32,223,145,260),(33,224,146,261),(34,211,147,262),(35,212,148,263),(36,213,149,264),(37,214,150,265),(38,215,151,266),(39,216,152,253),(40,217,153,254),(41,218,154,255),(42,219,141,256),(43,251,195,397),(44,252,196,398),(45,239,183,399),(46,240,184,400),(47,241,185,401),(48,242,186,402),(49,243,187,403),(50,244,188,404),(51,245,189,405),(52,246,190,406),(53,247,191,393),(54,248,192,394),(55,249,193,395),(56,250,194,396),(57,175,81,278),(58,176,82,279),(59,177,83,280),(60,178,84,267),(61,179,71,268),(62,180,72,269),(63,181,73,270),(64,182,74,271),(65,169,75,272),(66,170,76,273),(67,171,77,274),(68,172,78,275),(69,173,79,276),(70,174,80,277),(85,233,422,330),(86,234,423,331),(87,235,424,332),(88,236,425,333),(89,237,426,334),(90,238,427,335),(91,225,428,336),(92,226,429,323),(93,227,430,324),(94,228,431,325),(95,229,432,326),(96,230,433,327),(97,231,434,328),(98,232,421,329),(99,346,351,444),(100,347,352,445),(101,348,353,446),(102,349,354,447),(103,350,355,448),(104,337,356,435),(105,338,357,436),(106,339,358,437),(107,340,359,438),(108,341,360,439),(109,342,361,440),(110,343,362,441),(111,344,363,442),(112,345,364,443),(113,207,409,162),(114,208,410,163),(115,209,411,164),(116,210,412,165),(117,197,413,166),(118,198,414,167),(119,199,415,168),(120,200,416,155),(121,201,417,156),(122,202,418,157),(123,203,419,158),(124,204,420,159),(125,205,407,160),(126,206,408,161),(127,284,378,317),(128,285,365,318),(129,286,366,319),(130,287,367,320),(131,288,368,321),(132,289,369,322),(133,290,370,309),(134,291,371,310),(135,292,372,311),(136,293,373,312),(137,294,374,313),(138,281,375,314),(139,282,376,315),(140,283,377,316)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28BT | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | D4 | Q8 | D4 | D8 | Q16 | C7×D4 | C7×Q8 | C7×D4 | C7×D8 | C7×Q16 |
| kernel | C14×C2.D8 | C7×C2.D8 | C14×C4⋊C4 | C22×C56 | C2×C56 | C2×C2.D8 | C2.D8 | C2×C4⋊C4 | C22×C8 | C2×C8 | C2×C28 | C2×C28 | C22×C14 | C2×C14 | C2×C14 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 6 | 24 | 12 | 6 | 48 | 1 | 2 | 1 | 4 | 4 | 6 | 12 | 6 | 24 | 24 |
Matrix representation of C14×C2.D8 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 31 | 51 |
| 15 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 111 | 28 |
| 0 | 0 | 12 | 2 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,7,0,0,0,0,7],[112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,31,0,0,51,51],[15,0,0,0,0,1,0,0,0,0,111,12,0,0,28,2] >;
C14×C2.D8 in GAP, Magma, Sage, TeX
C_{14}\times C_2.D_8 % in TeX
G:=Group("C14xC2.D8"); // GroupNames label
G:=SmallGroup(448,834);
// by ID
G=gap.SmallGroup(448,834);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1968,9804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^8=1,d^2=b,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