metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.2(C4×Q8), Dic7⋊C4⋊2C4, C14.19(C4×D4), C2.2(C28⋊Q8), (C2×C28).33Q8, Dic7⋊1(C4⋊C4), C14.10(C4⋊Q8), (C2×Dic7).9Q8, C2.5(C4×Dic14), C22.51(D4×D7), (C22×C4).6D14, C22.10(Q8×D7), C14.1(C4⋊D4), (C2×C4).20Dic14, C14.2(C22⋊Q8), C2.1(D14⋊D4), (C2×Dic7).126D4, C2.1(Dic7.Q8), C14.5(C42.C2), C2.C42.8D7, C2.5(Dic7⋊4D4), C22.26(C4○D28), C22.13(C2×Dic14), C23.244(C22×D7), C22.26(D4⋊2D7), C14.C42.31C2, (C22×C14).272C23, (C22×C28).327C22, C2.2(C22⋊Dic14), C7⋊1(C23.65C23), (C22×Dic7).2C22, C2.5(D7×C4⋊C4), C14.1(C2×C4⋊C4), (C2×C4).23(C4×D7), C22.81(C2×C4×D7), (C2×C28).31(C2×C4), (C2×C4⋊Dic7).2C2, (C2×C14).57(C2×Q8), (C2×C4×Dic7).20C2, (C2×C14).188(C2×D4), (C2×Dic7⋊C4).2C2, (C2×C14).38(C22×C4), (C2×Dic7).40(C2×C4), (C2×C14).121(C4○D4), (C7×C2.C42).15C2, SmallGroup(448,181)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.(C4×Q8)
G = < a,b,c,d | a14=b4=c4=1, d2=c2, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=a7b, dcd-1=a7c-1 >
Subgroups: 668 in 170 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, Dic7, Dic7, C28, C2×C14, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.65C23, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C14.(C4×Q8)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, Dic14, C4×D7, C22×D7, C23.65C23, C2×Dic14, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, Q8×D7, C4×Dic14, C22⋊Dic14, Dic7⋊4D4, D14⋊D4, C28⋊Q8, Dic7.Q8, D7×C4⋊C4, C14.(C4×Q8)
►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 88 29 57)(2 89 30 58)(3 90 31 59)(4 91 32 60)(5 92 33 61)(6 93 34 62)(7 94 35 63)(8 95 36 64)(9 96 37 65)(10 97 38 66)(11 98 39 67)(12 85 40 68)(13 86 41 69)(14 87 42 70)(15 101 54 82)(16 102 55 83)(17 103 56 84)(18 104 43 71)(19 105 44 72)(20 106 45 73)(21 107 46 74)(22 108 47 75)(23 109 48 76)(24 110 49 77)(25 111 50 78)(26 112 51 79)(27 99 52 80)(28 100 53 81)(113 207 150 176)(114 208 151 177)(115 209 152 178)(116 210 153 179)(117 197 154 180)(118 198 141 181)(119 199 142 182)(120 200 143 169)(121 201 144 170)(122 202 145 171)(123 203 146 172)(124 204 147 173)(125 205 148 174)(126 206 149 175)(127 218 158 185)(128 219 159 186)(129 220 160 187)(130 221 161 188)(131 222 162 189)(132 223 163 190)(133 224 164 191)(134 211 165 192)(135 212 166 193)(136 213 167 194)(137 214 168 195)(138 215 155 196)(139 216 156 183)(140 217 157 184)(225 313 258 294)(226 314 259 281)(227 315 260 282)(228 316 261 283)(229 317 262 284)(230 318 263 285)(231 319 264 286)(232 320 265 287)(233 321 266 288)(234 322 253 289)(235 309 254 290)(236 310 255 291)(237 311 256 292)(238 312 257 293)(239 334 280 304)(240 335 267 305)(241 336 268 306)(242 323 269 307)(243 324 270 308)(244 325 271 295)(245 326 272 296)(246 327 273 297)(247 328 274 298)(248 329 275 299)(249 330 276 300)(250 331 277 301)(251 332 278 302)(252 333 279 303)(337 422 378 396)(338 423 365 397)(339 424 366 398)(340 425 367 399)(341 426 368 400)(342 427 369 401)(343 428 370 402)(344 429 371 403)(345 430 372 404)(346 431 373 405)(347 432 374 406)(348 433 375 393)(349 434 376 394)(350 421 377 395)(351 437 380 415)(352 438 381 416)(353 439 382 417)(354 440 383 418)(355 441 384 419)(356 442 385 420)(357 443 386 407)(358 444 387 408)(359 445 388 409)(360 446 389 410)(361 447 390 411)(362 448 391 412)(363 435 392 413)(364 436 379 414)
(1 371 20 379)(2 370 21 392)(3 369 22 391)(4 368 23 390)(5 367 24 389)(6 366 25 388)(7 365 26 387)(8 378 27 386)(9 377 28 385)(10 376 15 384)(11 375 16 383)(12 374 17 382)(13 373 18 381)(14 372 19 380)(29 344 45 364)(30 343 46 363)(31 342 47 362)(32 341 48 361)(33 340 49 360)(34 339 50 359)(35 338 51 358)(36 337 52 357)(37 350 53 356)(38 349 54 355)(39 348 55 354)(40 347 56 353)(41 346 43 352)(42 345 44 351)(57 422 73 443)(58 421 74 442)(59 434 75 441)(60 433 76 440)(61 432 77 439)(62 431 78 438)(63 430 79 437)(64 429 80 436)(65 428 81 435)(66 427 82 448)(67 426 83 447)(68 425 84 446)(69 424 71 445)(70 423 72 444)(85 399 103 410)(86 398 104 409)(87 397 105 408)(88 396 106 407)(89 395 107 420)(90 394 108 419)(91 393 109 418)(92 406 110 417)(93 405 111 416)(94 404 112 415)(95 403 99 414)(96 402 100 413)(97 401 101 412)(98 400 102 411)(113 269 131 263)(114 268 132 262)(115 267 133 261)(116 280 134 260)(117 279 135 259)(118 278 136 258)(119 277 137 257)(120 276 138 256)(121 275 139 255)(122 274 140 254)(123 273 127 253)(124 272 128 266)(125 271 129 265)(126 270 130 264)(141 251 167 225)(142 250 168 238)(143 249 155 237)(144 248 156 236)(145 247 157 235)(146 246 158 234)(147 245 159 233)(148 244 160 232)(149 243 161 231)(150 242 162 230)(151 241 163 229)(152 240 164 228)(153 239 165 227)(154 252 166 226)(169 323 196 318)(170 336 183 317)(171 335 184 316)(172 334 185 315)(173 333 186 314)(174 332 187 313)(175 331 188 312)(176 330 189 311)(177 329 190 310)(178 328 191 309)(179 327 192 322)(180 326 193 321)(181 325 194 320)(182 324 195 319)(197 296 212 288)(198 295 213 287)(199 308 214 286)(200 307 215 285)(201 306 216 284)(202 305 217 283)(203 304 218 282)(204 303 219 281)(205 302 220 294)(206 301 221 293)(207 300 222 292)(208 299 223 291)(209 298 224 290)(210 297 211 289)
(1 165 20 153)(2 166 21 154)(3 167 22 141)(4 168 23 142)(5 155 24 143)(6 156 25 144)(7 157 26 145)(8 158 27 146)(9 159 28 147)(10 160 15 148)(11 161 16 149)(12 162 17 150)(13 163 18 151)(14 164 19 152)(29 134 45 116)(30 135 46 117)(31 136 47 118)(32 137 48 119)(33 138 49 120)(34 139 50 121)(35 140 51 122)(36 127 52 123)(37 128 53 124)(38 129 54 125)(39 130 55 126)(40 131 56 113)(41 132 43 114)(42 133 44 115)(57 218 73 203)(58 219 74 204)(59 220 75 205)(60 221 76 206)(61 222 77 207)(62 223 78 208)(63 224 79 209)(64 211 80 210)(65 212 81 197)(66 213 82 198)(67 214 83 199)(68 215 84 200)(69 216 71 201)(70 217 72 202)(85 196 103 169)(86 183 104 170)(87 184 105 171)(88 185 106 172)(89 186 107 173)(90 187 108 174)(91 188 109 175)(92 189 110 176)(93 190 111 177)(94 191 112 178)(95 192 99 179)(96 193 100 180)(97 194 101 181)(98 195 102 182)(225 376 251 384)(226 377 252 385)(227 378 239 386)(228 365 240 387)(229 366 241 388)(230 367 242 389)(231 368 243 390)(232 369 244 391)(233 370 245 392)(234 371 246 379)(235 372 247 380)(236 373 248 381)(237 374 249 382)(238 375 250 383)(253 344 273 364)(254 345 274 351)(255 346 275 352)(256 347 276 353)(257 348 277 354)(258 349 278 355)(259 350 279 356)(260 337 280 357)(261 338 267 358)(262 339 268 359)(263 340 269 360)(264 341 270 361)(265 342 271 362)(266 343 272 363)(281 428 303 435)(282 429 304 436)(283 430 305 437)(284 431 306 438)(285 432 307 439)(286 433 308 440)(287 434 295 441)(288 421 296 442)(289 422 297 443)(290 423 298 444)(291 424 299 445)(292 425 300 446)(293 426 301 447)(294 427 302 448)(309 397 328 408)(310 398 329 409)(311 399 330 410)(312 400 331 411)(313 401 332 412)(314 402 333 413)(315 403 334 414)(316 404 335 415)(317 405 336 416)(318 406 323 417)(319 393 324 418)(320 394 325 419)(321 395 326 420)(322 396 327 407)
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), (1,88,29,57)(2,89,30,58)(3,90,31,59)(4,91,32,60)(5,92,33,61)(6,93,34,62)(7,94,35,63)(8,95,36,64)(9,96,37,65)(10,97,38,66)(11,98,39,67)(12,85,40,68)(13,86,41,69)(14,87,42,70)(15,101,54,82)(16,102,55,83)(17,103,56,84)(18,104,43,71)(19,105,44,72)(20,106,45,73)(21,107,46,74)(22,108,47,75)(23,109,48,76)(24,110,49,77)(25,111,50,78)(26,112,51,79)(27,99,52,80)(28,100,53,81)(113,207,150,176)(114,208,151,177)(115,209,152,178)(116,210,153,179)(117,197,154,180)(118,198,141,181)(119,199,142,182)(120,200,143,169)(121,201,144,170)(122,202,145,171)(123,203,146,172)(124,204,147,173)(125,205,148,174)(126,206,149,175)(127,218,158,185)(128,219,159,186)(129,220,160,187)(130,221,161,188)(131,222,162,189)(132,223,163,190)(133,224,164,191)(134,211,165,192)(135,212,166,193)(136,213,167,194)(137,214,168,195)(138,215,155,196)(139,216,156,183)(140,217,157,184)(225,313,258,294)(226,314,259,281)(227,315,260,282)(228,316,261,283)(229,317,262,284)(230,318,263,285)(231,319,264,286)(232,320,265,287)(233,321,266,288)(234,322,253,289)(235,309,254,290)(236,310,255,291)(237,311,256,292)(238,312,257,293)(239,334,280,304)(240,335,267,305)(241,336,268,306)(242,323,269,307)(243,324,270,308)(244,325,271,295)(245,326,272,296)(246,327,273,297)(247,328,274,298)(248,329,275,299)(249,330,276,300)(250,331,277,301)(251,332,278,302)(252,333,279,303)(337,422,378,396)(338,423,365,397)(339,424,366,398)(340,425,367,399)(341,426,368,400)(342,427,369,401)(343,428,370,402)(344,429,371,403)(345,430,372,404)(346,431,373,405)(347,432,374,406)(348,433,375,393)(349,434,376,394)(350,421,377,395)(351,437,380,415)(352,438,381,416)(353,439,382,417)(354,440,383,418)(355,441,384,419)(356,442,385,420)(357,443,386,407)(358,444,387,408)(359,445,388,409)(360,446,389,410)(361,447,390,411)(362,448,391,412)(363,435,392,413)(364,436,379,414), (1,371,20,379)(2,370,21,392)(3,369,22,391)(4,368,23,390)(5,367,24,389)(6,366,25,388)(7,365,26,387)(8,378,27,386)(9,377,28,385)(10,376,15,384)(11,375,16,383)(12,374,17,382)(13,373,18,381)(14,372,19,380)(29,344,45,364)(30,343,46,363)(31,342,47,362)(32,341,48,361)(33,340,49,360)(34,339,50,359)(35,338,51,358)(36,337,52,357)(37,350,53,356)(38,349,54,355)(39,348,55,354)(40,347,56,353)(41,346,43,352)(42,345,44,351)(57,422,73,443)(58,421,74,442)(59,434,75,441)(60,433,76,440)(61,432,77,439)(62,431,78,438)(63,430,79,437)(64,429,80,436)(65,428,81,435)(66,427,82,448)(67,426,83,447)(68,425,84,446)(69,424,71,445)(70,423,72,444)(85,399,103,410)(86,398,104,409)(87,397,105,408)(88,396,106,407)(89,395,107,420)(90,394,108,419)(91,393,109,418)(92,406,110,417)(93,405,111,416)(94,404,112,415)(95,403,99,414)(96,402,100,413)(97,401,101,412)(98,400,102,411)(113,269,131,263)(114,268,132,262)(115,267,133,261)(116,280,134,260)(117,279,135,259)(118,278,136,258)(119,277,137,257)(120,276,138,256)(121,275,139,255)(122,274,140,254)(123,273,127,253)(124,272,128,266)(125,271,129,265)(126,270,130,264)(141,251,167,225)(142,250,168,238)(143,249,155,237)(144,248,156,236)(145,247,157,235)(146,246,158,234)(147,245,159,233)(148,244,160,232)(149,243,161,231)(150,242,162,230)(151,241,163,229)(152,240,164,228)(153,239,165,227)(154,252,166,226)(169,323,196,318)(170,336,183,317)(171,335,184,316)(172,334,185,315)(173,333,186,314)(174,332,187,313)(175,331,188,312)(176,330,189,311)(177,329,190,310)(178,328,191,309)(179,327,192,322)(180,326,193,321)(181,325,194,320)(182,324,195,319)(197,296,212,288)(198,295,213,287)(199,308,214,286)(200,307,215,285)(201,306,216,284)(202,305,217,283)(203,304,218,282)(204,303,219,281)(205,302,220,294)(206,301,221,293)(207,300,222,292)(208,299,223,291)(209,298,224,290)(210,297,211,289), (1,165,20,153)(2,166,21,154)(3,167,22,141)(4,168,23,142)(5,155,24,143)(6,156,25,144)(7,157,26,145)(8,158,27,146)(9,159,28,147)(10,160,15,148)(11,161,16,149)(12,162,17,150)(13,163,18,151)(14,164,19,152)(29,134,45,116)(30,135,46,117)(31,136,47,118)(32,137,48,119)(33,138,49,120)(34,139,50,121)(35,140,51,122)(36,127,52,123)(37,128,53,124)(38,129,54,125)(39,130,55,126)(40,131,56,113)(41,132,43,114)(42,133,44,115)(57,218,73,203)(58,219,74,204)(59,220,75,205)(60,221,76,206)(61,222,77,207)(62,223,78,208)(63,224,79,209)(64,211,80,210)(65,212,81,197)(66,213,82,198)(67,214,83,199)(68,215,84,200)(69,216,71,201)(70,217,72,202)(85,196,103,169)(86,183,104,170)(87,184,105,171)(88,185,106,172)(89,186,107,173)(90,187,108,174)(91,188,109,175)(92,189,110,176)(93,190,111,177)(94,191,112,178)(95,192,99,179)(96,193,100,180)(97,194,101,181)(98,195,102,182)(225,376,251,384)(226,377,252,385)(227,378,239,386)(228,365,240,387)(229,366,241,388)(230,367,242,389)(231,368,243,390)(232,369,244,391)(233,370,245,392)(234,371,246,379)(235,372,247,380)(236,373,248,381)(237,374,249,382)(238,375,250,383)(253,344,273,364)(254,345,274,351)(255,346,275,352)(256,347,276,353)(257,348,277,354)(258,349,278,355)(259,350,279,356)(260,337,280,357)(261,338,267,358)(262,339,268,359)(263,340,269,360)(264,341,270,361)(265,342,271,362)(266,343,272,363)(281,428,303,435)(282,429,304,436)(283,430,305,437)(284,431,306,438)(285,432,307,439)(286,433,308,440)(287,434,295,441)(288,421,296,442)(289,422,297,443)(290,423,298,444)(291,424,299,445)(292,425,300,446)(293,426,301,447)(294,427,302,448)(309,397,328,408)(310,398,329,409)(311,399,330,410)(312,400,331,411)(313,401,332,412)(314,402,333,413)(315,403,334,414)(316,404,335,415)(317,405,336,416)(318,406,323,417)(319,393,324,418)(320,394,325,419)(321,395,326,420)(322,396,327,407)>;
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), (1,88,29,57)(2,89,30,58)(3,90,31,59)(4,91,32,60)(5,92,33,61)(6,93,34,62)(7,94,35,63)(8,95,36,64)(9,96,37,65)(10,97,38,66)(11,98,39,67)(12,85,40,68)(13,86,41,69)(14,87,42,70)(15,101,54,82)(16,102,55,83)(17,103,56,84)(18,104,43,71)(19,105,44,72)(20,106,45,73)(21,107,46,74)(22,108,47,75)(23,109,48,76)(24,110,49,77)(25,111,50,78)(26,112,51,79)(27,99,52,80)(28,100,53,81)(113,207,150,176)(114,208,151,177)(115,209,152,178)(116,210,153,179)(117,197,154,180)(118,198,141,181)(119,199,142,182)(120,200,143,169)(121,201,144,170)(122,202,145,171)(123,203,146,172)(124,204,147,173)(125,205,148,174)(126,206,149,175)(127,218,158,185)(128,219,159,186)(129,220,160,187)(130,221,161,188)(131,222,162,189)(132,223,163,190)(133,224,164,191)(134,211,165,192)(135,212,166,193)(136,213,167,194)(137,214,168,195)(138,215,155,196)(139,216,156,183)(140,217,157,184)(225,313,258,294)(226,314,259,281)(227,315,260,282)(228,316,261,283)(229,317,262,284)(230,318,263,285)(231,319,264,286)(232,320,265,287)(233,321,266,288)(234,322,253,289)(235,309,254,290)(236,310,255,291)(237,311,256,292)(238,312,257,293)(239,334,280,304)(240,335,267,305)(241,336,268,306)(242,323,269,307)(243,324,270,308)(244,325,271,295)(245,326,272,296)(246,327,273,297)(247,328,274,298)(248,329,275,299)(249,330,276,300)(250,331,277,301)(251,332,278,302)(252,333,279,303)(337,422,378,396)(338,423,365,397)(339,424,366,398)(340,425,367,399)(341,426,368,400)(342,427,369,401)(343,428,370,402)(344,429,371,403)(345,430,372,404)(346,431,373,405)(347,432,374,406)(348,433,375,393)(349,434,376,394)(350,421,377,395)(351,437,380,415)(352,438,381,416)(353,439,382,417)(354,440,383,418)(355,441,384,419)(356,442,385,420)(357,443,386,407)(358,444,387,408)(359,445,388,409)(360,446,389,410)(361,447,390,411)(362,448,391,412)(363,435,392,413)(364,436,379,414), (1,371,20,379)(2,370,21,392)(3,369,22,391)(4,368,23,390)(5,367,24,389)(6,366,25,388)(7,365,26,387)(8,378,27,386)(9,377,28,385)(10,376,15,384)(11,375,16,383)(12,374,17,382)(13,373,18,381)(14,372,19,380)(29,344,45,364)(30,343,46,363)(31,342,47,362)(32,341,48,361)(33,340,49,360)(34,339,50,359)(35,338,51,358)(36,337,52,357)(37,350,53,356)(38,349,54,355)(39,348,55,354)(40,347,56,353)(41,346,43,352)(42,345,44,351)(57,422,73,443)(58,421,74,442)(59,434,75,441)(60,433,76,440)(61,432,77,439)(62,431,78,438)(63,430,79,437)(64,429,80,436)(65,428,81,435)(66,427,82,448)(67,426,83,447)(68,425,84,446)(69,424,71,445)(70,423,72,444)(85,399,103,410)(86,398,104,409)(87,397,105,408)(88,396,106,407)(89,395,107,420)(90,394,108,419)(91,393,109,418)(92,406,110,417)(93,405,111,416)(94,404,112,415)(95,403,99,414)(96,402,100,413)(97,401,101,412)(98,400,102,411)(113,269,131,263)(114,268,132,262)(115,267,133,261)(116,280,134,260)(117,279,135,259)(118,278,136,258)(119,277,137,257)(120,276,138,256)(121,275,139,255)(122,274,140,254)(123,273,127,253)(124,272,128,266)(125,271,129,265)(126,270,130,264)(141,251,167,225)(142,250,168,238)(143,249,155,237)(144,248,156,236)(145,247,157,235)(146,246,158,234)(147,245,159,233)(148,244,160,232)(149,243,161,231)(150,242,162,230)(151,241,163,229)(152,240,164,228)(153,239,165,227)(154,252,166,226)(169,323,196,318)(170,336,183,317)(171,335,184,316)(172,334,185,315)(173,333,186,314)(174,332,187,313)(175,331,188,312)(176,330,189,311)(177,329,190,310)(178,328,191,309)(179,327,192,322)(180,326,193,321)(181,325,194,320)(182,324,195,319)(197,296,212,288)(198,295,213,287)(199,308,214,286)(200,307,215,285)(201,306,216,284)(202,305,217,283)(203,304,218,282)(204,303,219,281)(205,302,220,294)(206,301,221,293)(207,300,222,292)(208,299,223,291)(209,298,224,290)(210,297,211,289), (1,165,20,153)(2,166,21,154)(3,167,22,141)(4,168,23,142)(5,155,24,143)(6,156,25,144)(7,157,26,145)(8,158,27,146)(9,159,28,147)(10,160,15,148)(11,161,16,149)(12,162,17,150)(13,163,18,151)(14,164,19,152)(29,134,45,116)(30,135,46,117)(31,136,47,118)(32,137,48,119)(33,138,49,120)(34,139,50,121)(35,140,51,122)(36,127,52,123)(37,128,53,124)(38,129,54,125)(39,130,55,126)(40,131,56,113)(41,132,43,114)(42,133,44,115)(57,218,73,203)(58,219,74,204)(59,220,75,205)(60,221,76,206)(61,222,77,207)(62,223,78,208)(63,224,79,209)(64,211,80,210)(65,212,81,197)(66,213,82,198)(67,214,83,199)(68,215,84,200)(69,216,71,201)(70,217,72,202)(85,196,103,169)(86,183,104,170)(87,184,105,171)(88,185,106,172)(89,186,107,173)(90,187,108,174)(91,188,109,175)(92,189,110,176)(93,190,111,177)(94,191,112,178)(95,192,99,179)(96,193,100,180)(97,194,101,181)(98,195,102,182)(225,376,251,384)(226,377,252,385)(227,378,239,386)(228,365,240,387)(229,366,241,388)(230,367,242,389)(231,368,243,390)(232,369,244,391)(233,370,245,392)(234,371,246,379)(235,372,247,380)(236,373,248,381)(237,374,249,382)(238,375,250,383)(253,344,273,364)(254,345,274,351)(255,346,275,352)(256,347,276,353)(257,348,277,354)(258,349,278,355)(259,350,279,356)(260,337,280,357)(261,338,267,358)(262,339,268,359)(263,340,269,360)(264,341,270,361)(265,342,271,362)(266,343,272,363)(281,428,303,435)(282,429,304,436)(283,430,305,437)(284,431,306,438)(285,432,307,439)(286,433,308,440)(287,434,295,441)(288,421,296,442)(289,422,297,443)(290,423,298,444)(291,424,299,445)(292,425,300,446)(293,426,301,447)(294,427,302,448)(309,397,328,408)(310,398,329,409)(311,399,330,410)(312,400,331,411)(313,401,332,412)(314,402,333,413)(315,403,334,414)(316,404,335,415)(317,405,336,416)(318,406,323,417)(319,393,324,418)(320,394,325,419)(321,395,326,420)(322,396,327,407) );
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)], [(1,88,29,57),(2,89,30,58),(3,90,31,59),(4,91,32,60),(5,92,33,61),(6,93,34,62),(7,94,35,63),(8,95,36,64),(9,96,37,65),(10,97,38,66),(11,98,39,67),(12,85,40,68),(13,86,41,69),(14,87,42,70),(15,101,54,82),(16,102,55,83),(17,103,56,84),(18,104,43,71),(19,105,44,72),(20,106,45,73),(21,107,46,74),(22,108,47,75),(23,109,48,76),(24,110,49,77),(25,111,50,78),(26,112,51,79),(27,99,52,80),(28,100,53,81),(113,207,150,176),(114,208,151,177),(115,209,152,178),(116,210,153,179),(117,197,154,180),(118,198,141,181),(119,199,142,182),(120,200,143,169),(121,201,144,170),(122,202,145,171),(123,203,146,172),(124,204,147,173),(125,205,148,174),(126,206,149,175),(127,218,158,185),(128,219,159,186),(129,220,160,187),(130,221,161,188),(131,222,162,189),(132,223,163,190),(133,224,164,191),(134,211,165,192),(135,212,166,193),(136,213,167,194),(137,214,168,195),(138,215,155,196),(139,216,156,183),(140,217,157,184),(225,313,258,294),(226,314,259,281),(227,315,260,282),(228,316,261,283),(229,317,262,284),(230,318,263,285),(231,319,264,286),(232,320,265,287),(233,321,266,288),(234,322,253,289),(235,309,254,290),(236,310,255,291),(237,311,256,292),(238,312,257,293),(239,334,280,304),(240,335,267,305),(241,336,268,306),(242,323,269,307),(243,324,270,308),(244,325,271,295),(245,326,272,296),(246,327,273,297),(247,328,274,298),(248,329,275,299),(249,330,276,300),(250,331,277,301),(251,332,278,302),(252,333,279,303),(337,422,378,396),(338,423,365,397),(339,424,366,398),(340,425,367,399),(341,426,368,400),(342,427,369,401),(343,428,370,402),(344,429,371,403),(345,430,372,404),(346,431,373,405),(347,432,374,406),(348,433,375,393),(349,434,376,394),(350,421,377,395),(351,437,380,415),(352,438,381,416),(353,439,382,417),(354,440,383,418),(355,441,384,419),(356,442,385,420),(357,443,386,407),(358,444,387,408),(359,445,388,409),(360,446,389,410),(361,447,390,411),(362,448,391,412),(363,435,392,413),(364,436,379,414)], [(1,371,20,379),(2,370,21,392),(3,369,22,391),(4,368,23,390),(5,367,24,389),(6,366,25,388),(7,365,26,387),(8,378,27,386),(9,377,28,385),(10,376,15,384),(11,375,16,383),(12,374,17,382),(13,373,18,381),(14,372,19,380),(29,344,45,364),(30,343,46,363),(31,342,47,362),(32,341,48,361),(33,340,49,360),(34,339,50,359),(35,338,51,358),(36,337,52,357),(37,350,53,356),(38,349,54,355),(39,348,55,354),(40,347,56,353),(41,346,43,352),(42,345,44,351),(57,422,73,443),(58,421,74,442),(59,434,75,441),(60,433,76,440),(61,432,77,439),(62,431,78,438),(63,430,79,437),(64,429,80,436),(65,428,81,435),(66,427,82,448),(67,426,83,447),(68,425,84,446),(69,424,71,445),(70,423,72,444),(85,399,103,410),(86,398,104,409),(87,397,105,408),(88,396,106,407),(89,395,107,420),(90,394,108,419),(91,393,109,418),(92,406,110,417),(93,405,111,416),(94,404,112,415),(95,403,99,414),(96,402,100,413),(97,401,101,412),(98,400,102,411),(113,269,131,263),(114,268,132,262),(115,267,133,261),(116,280,134,260),(117,279,135,259),(118,278,136,258),(119,277,137,257),(120,276,138,256),(121,275,139,255),(122,274,140,254),(123,273,127,253),(124,272,128,266),(125,271,129,265),(126,270,130,264),(141,251,167,225),(142,250,168,238),(143,249,155,237),(144,248,156,236),(145,247,157,235),(146,246,158,234),(147,245,159,233),(148,244,160,232),(149,243,161,231),(150,242,162,230),(151,241,163,229),(152,240,164,228),(153,239,165,227),(154,252,166,226),(169,323,196,318),(170,336,183,317),(171,335,184,316),(172,334,185,315),(173,333,186,314),(174,332,187,313),(175,331,188,312),(176,330,189,311),(177,329,190,310),(178,328,191,309),(179,327,192,322),(180,326,193,321),(181,325,194,320),(182,324,195,319),(197,296,212,288),(198,295,213,287),(199,308,214,286),(200,307,215,285),(201,306,216,284),(202,305,217,283),(203,304,218,282),(204,303,219,281),(205,302,220,294),(206,301,221,293),(207,300,222,292),(208,299,223,291),(209,298,224,290),(210,297,211,289)], [(1,165,20,153),(2,166,21,154),(3,167,22,141),(4,168,23,142),(5,155,24,143),(6,156,25,144),(7,157,26,145),(8,158,27,146),(9,159,28,147),(10,160,15,148),(11,161,16,149),(12,162,17,150),(13,163,18,151),(14,164,19,152),(29,134,45,116),(30,135,46,117),(31,136,47,118),(32,137,48,119),(33,138,49,120),(34,139,50,121),(35,140,51,122),(36,127,52,123),(37,128,53,124),(38,129,54,125),(39,130,55,126),(40,131,56,113),(41,132,43,114),(42,133,44,115),(57,218,73,203),(58,219,74,204),(59,220,75,205),(60,221,76,206),(61,222,77,207),(62,223,78,208),(63,224,79,209),(64,211,80,210),(65,212,81,197),(66,213,82,198),(67,214,83,199),(68,215,84,200),(69,216,71,201),(70,217,72,202),(85,196,103,169),(86,183,104,170),(87,184,105,171),(88,185,106,172),(89,186,107,173),(90,187,108,174),(91,188,109,175),(92,189,110,176),(93,190,111,177),(94,191,112,178),(95,192,99,179),(96,193,100,180),(97,194,101,181),(98,195,102,182),(225,376,251,384),(226,377,252,385),(227,378,239,386),(228,365,240,387),(229,366,241,388),(230,367,242,389),(231,368,243,390),(232,369,244,391),(233,370,245,392),(234,371,246,379),(235,372,247,380),(236,373,248,381),(237,374,249,382),(238,375,250,383),(253,344,273,364),(254,345,274,351),(255,346,275,352),(256,347,276,353),(257,348,277,354),(258,349,278,355),(259,350,279,356),(260,337,280,357),(261,338,267,358),(262,339,268,359),(263,340,269,360),(264,341,270,361),(265,342,271,362),(266,343,272,363),(281,428,303,435),(282,429,304,436),(283,430,305,437),(284,431,306,438),(285,432,307,439),(286,433,308,440),(287,434,295,441),(288,421,296,442),(289,422,297,443),(290,423,298,444),(291,424,299,445),(292,425,300,446),(293,426,301,447),(294,427,302,448),(309,397,328,408),(310,398,329,409),(311,399,330,410),(312,400,331,411),(313,401,332,412),(314,402,333,413),(315,403,334,414),(316,404,335,415),(317,405,336,416),(318,406,323,417),(319,393,324,418),(320,394,325,419),(321,395,326,420),(322,396,327,407)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4×D7 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 |
| kernel | C14.(C4×Q8) | C14.C42 | C7×C2.C42 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | Dic7⋊C4 | C2×Dic7 | C2×Dic7 | C2×C28 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 8 | 4 | 2 | 2 | 3 | 4 | 9 | 12 | 12 | 12 | 6 | 3 | 3 |
Matrix representation of C14.(C4×Q8) ►in GL6(𝔽29)
| 4 | 10 | 0 | 0 | 0 | 0 |
| 14 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 27 |
| 0 | 0 | 0 | 0 | 28 | 17 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 0 |
| 0 | 0 | 18 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 13 |
| 0 | 0 | 0 | 0 | 15 | 26 |
| 16 | 6 | 0 | 0 | 0 | 0 |
| 20 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 5 | 0 | 0 |
| 0 | 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 28 |
| 0 | 0 | 0 | 0 | 21 | 14 |
G:=sub<GL(6,GF(29))| [4,14,0,0,0,0,10,28,0,0,0,0,0,0,25,4,0,0,0,0,25,11,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,12,28,0,0,0,0,27,17],[12,0,0,0,0,0,4,17,0,0,0,0,0,0,2,18,0,0,0,0,11,27,0,0,0,0,0,0,3,15,0,0,0,0,13,26],[16,20,0,0,0,0,6,13,0,0,0,0,0,0,13,24,0,0,0,0,5,16,0,0,0,0,0,0,15,21,0,0,0,0,28,14] >;
C14.(C4×Q8) in GAP, Magma, Sage, TeX
C_{14}.(C_4\times Q_8) % in TeX
G:=Group("C14.(C4xQ8)"); // GroupNames label
G:=SmallGroup(448,181);
// by ID
G=gap.SmallGroup(448,181);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,344,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^7*b,d*c*d^-1=a^7*c^-1>;
// generators/relations