Copied to
clipboard

G = C14×C2.C42  order 448 = 26·7

Direct product of C14 and C2.C42

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C14×C2.C42, (C22×C4)⋊5C28, (C22×C28)⋊11C4, (C23×C28).4C2, C22.8(C4×C28), (C23×C4).3C14, C23.53(C7×D4), C23.10(C7×Q8), C22.7(Q8×C14), C24.38(C2×C14), C14.29(C2×C42), (C2×C14).30C42, C23.36(C2×C28), C22.26(D4×C14), (C22×C14).29Q8, (C22×C14).214D4, C23.48(C22×C14), C22.12(C22×C28), (C23×C14).118C22, (C22×C14).439C23, (C22×C28).486C22, C2.1(C2×C4×C28), (C2×C4)⋊9(C2×C28), C2.1(C14×C4⋊C4), (C2×C28)⋊35(C2×C4), C14.57(C2×C4⋊C4), C2.1(C14×C22⋊C4), C22.18(C7×C4⋊C4), (C2×C14).99(C2×Q8), (C2×C14).61(C4⋊C4), (C2×C14).593(C2×D4), C14.89(C2×C22⋊C4), (C22×C4).80(C2×C14), C22.31(C7×C22⋊C4), (C2×C14).211(C22×C4), (C22×C14).145(C2×C4), (C2×C14).133(C22⋊C4), SmallGroup(448,783)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C14×C2.C42
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C7×C2.C42 — C14×C2.C42
Lower central
C1 — C2 — C14×C2.C42
Upper central
C1 — C23×C14 — C14×C2.C42

Generators and relations for C14×C2.C42
 G = < a,b,c,d | a14=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 450 in 330 conjugacy classes, 210 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, C14, C14, C22×C4, C22×C4, C24, C28, C2×C14, C2×C14, C2.C42, C23×C4, C2×C28, C2×C28, C22×C14, C22×C14, C2×C2.C42, C22×C28, C22×C28, C23×C14, C7×C2.C42, C23×C28, C14×C2.C42
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C28, C7×D4, C7×Q8, C22×C14, C2×C2.C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C2.C42, C2×C4×C28, C14×C22⋊C4, C14×C4⋊C4, C14×C2.C42

Smallest permutation representation of C14×C2.C42
►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)

(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 113)(11 114)(12 115)(13 116)(14 117)(15 111)(16 112)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 241)(30 242)(31 243)(32 244)(33 245)(34 246)(35 247)(36 248)(37 249)(38 250)(39 251)(40 252)(41 239)(42 240)(43 192)(44 193)(45 194)(46 195)(47 196)(48 183)(49 184)(50 185)(51 186)(52 187)(53 188)(54 189)(55 190)(56 191)(57 349)(58 350)(59 337)(60 338)(61 339)(62 340)(63 341)(64 342)(65 343)(66 344)(67 345)(68 346)(69 347)(70 348)(71 180)(72 181)(73 182)(74 169)(75 170)(76 171)(77 172)(78 173)(79 174)(80 175)(81 176)(82 177)(83 178)(84 179)(85 319)(86 320)(87 321)(88 322)(89 309)(90 310)(91 311)(92 312)(93 313)(94 314)(95 315)(96 316)(97 317)(98 318)(127 434)(128 421)(129 422)(130 423)(131 424)(132 425)(133 426)(134 427)(135 428)(136 429)(137 430)(138 431)(139 432)(140 433)(141 197)(142 198)(143 199)(144 200)(145 201)(146 202)(147 203)(148 204)(149 205)(150 206)(151 207)(152 208)(153 209)(154 210)(155 366)(156 367)(157 368)(158 369)(159 370)(160 371)(161 372)(162 373)(163 374)(164 375)(165 376)(166 377)(167 378)(168 365)(211 232)(212 233)(213 234)(214 235)(215 236)(216 237)(217 238)(218 225)(219 226)(220 227)(221 228)(222 229)(223 230)(224 231)(253 381)(254 382)(255 383)(256 384)(257 385)(258 386)(259 387)(260 388)(261 389)(262 390)(263 391)(264 392)(265 379)(266 380)(267 435)(268 436)(269 437)(270 438)(271 439)(272 440)(273 441)(274 442)(275 443)(276 444)(277 445)(278 446)(279 447)(280 448)(281 414)(282 415)(283 416)(284 417)(285 418)(286 419)(287 420)(288 407)(289 408)(290 409)(291 410)(292 411)(293 412)(294 413)(295 354)(296 355)(297 356)(298 357)(299 358)(300 359)(301 360)(302 361)(303 362)(304 363)(305 364)(306 351)(307 352)(308 353)(323 396)(324 397)(325 398)(326 399)(327 400)(328 401)(329 402)(330 403)(331 404)(332 405)(333 406)(334 393)(335 394)(336 395)

(1 221 398 156)(2 222 399 157)(3 223 400 158)(4 224 401 159)(5 211 402 160)(6 212 403 161)(7 213 404 162)(8 214 405 163)(9 215 406 164)(10 216 393 165)(11 217 394 166)(12 218 395 167)(13 219 396 168)(14 220 397 155)(15 201 384 239)(16 202 385 240)(17 203 386 241)(18 204 387 242)(19 205 388 243)(20 206 389 244)(21 207 390 245)(22 208 391 246)(23 209 392 247)(24 210 379 248)(25 197 380 249)(26 198 381 250)(27 199 382 251)(28 200 383 252)(29 99 147 258)(30 100 148 259)(31 101 149 260)(32 102 150 261)(33 103 151 262)(34 104 152 263)(35 105 153 264)(36 106 154 265)(37 107 141 266)(38 108 142 253)(39 109 143 254)(40 110 144 255)(41 111 145 256)(42 112 146 257)(43 301 433 285)(44 302 434 286)(45 303 421 287)(46 304 422 288)(47 305 423 289)(48 306 424 290)(49 307 425 291)(50 308 426 292)(51 295 427 293)(52 296 428 294)(53 297 429 281)(54 298 430 282)(55 299 431 283)(56 300 432 284)(57 74 271 321)(58 75 272 322)(59 76 273 309)(60 77 274 310)(61 78 275 311)(62 79 276 312)(63 80 277 313)(64 81 278 314)(65 82 279 315)(66 83 280 316)(67 84 267 317)(68 71 268 318)(69 72 269 319)(70 73 270 320)(85 347 181 437)(86 348 182 438)(87 349 169 439)(88 350 170 440)(89 337 171 441)(90 338 172 442)(91 339 173 443)(92 340 174 444)(93 341 175 445)(94 342 176 446)(95 343 177 447)(96 344 178 448)(97 345 179 435)(98 346 180 436)(113 237 334 376)(114 238 335 377)(115 225 336 378)(116 226 323 365)(117 227 324 366)(118 228 325 367)(119 229 326 368)(120 230 327 369)(121 231 328 370)(122 232 329 371)(123 233 330 372)(124 234 331 373)(125 235 332 374)(126 236 333 375)(127 419 193 361)(128 420 194 362)(129 407 195 363)(130 408 196 364)(131 409 183 351)(132 410 184 352)(133 411 185 353)(134 412 186 354)(135 413 187 355)(136 414 188 356)(137 415 189 357)(138 416 190 358)(139 417 191 359)(140 418 192 360)

(1 240 284 77)(2 241 285 78)(3 242 286 79)(4 243 287 80)(5 244 288 81)(6 245 289 82)(7 246 290 83)(8 247 291 84)(9 248 292 71)(10 249 293 72)(11 250 294 73)(12 251 281 74)(13 252 282 75)(14 239 283 76)(15 190 273 227)(16 191 274 228)(17 192 275 229)(18 193 276 230)(19 194 277 231)(20 195 278 232)(21 196 279 233)(22 183 280 234)(23 184 267 235)(24 185 268 236)(25 186 269 237)(26 187 270 238)(27 188 271 225)(28 189 272 226)(29 418 173 119)(30 419 174 120)(31 420 175 121)(32 407 176 122)(33 408 177 123)(34 409 178 124)(35 410 179 125)(36 411 180 126)(37 412 181 113)(38 413 182 114)(39 414 169 115)(40 415 170 116)(41 416 171 117)(42 417 172 118)(43 443 222 99)(44 444 223 100)(45 445 224 101)(46 446 211 102)(47 447 212 103)(48 448 213 104)(49 435 214 105)(50 436 215 106)(51 437 216 107)(52 438 217 108)(53 439 218 109)(54 440 219 110)(55 441 220 111)(56 442 221 112)(57 378 382 136)(58 365 383 137)(59 366 384 138)(60 367 385 139)(61 368 386 140)(62 369 387 127)(63 370 388 128)(64 371 389 129)(65 372 390 130)(66 373 391 131)(67 374 392 132)(68 375 379 133)(69 376 380 134)(70 377 381 135)(85 334 141 354)(86 335 142 355)(87 336 143 356)(88 323 144 357)(89 324 145 358)(90 325 146 359)(91 326 147 360)(92 327 148 361)(93 328 149 362)(94 329 150 363)(95 330 151 364)(96 331 152 351)(97 332 153 352)(98 333 154 353)(155 256 431 337)(156 257 432 338)(157 258 433 339)(158 259 434 340)(159 260 421 341)(160 261 422 342)(161 262 423 343)(162 263 424 344)(163 264 425 345)(164 265 426 346)(165 266 427 347)(166 253 428 348)(167 254 429 349)(168 255 430 350)(197 295 319 393)(198 296 320 394)(199 297 321 395)(200 298 322 396)(201 299 309 397)(202 300 310 398)(203 301 311 399)(204 302 312 400)(205 303 313 401)(206 304 314 402)(207 305 315 403)(208 306 316 404)(209 307 317 405)(210 308 318 406)
 
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), (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,113)(11,114)(12,115)(13,116)(14,117)(15,111)(16,112)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,241)(30,242)(31,243)(32,244)(33,245)(34,246)(35,247)(36,248)(37,249)(38,250)(39,251)(40,252)(41,239)(42,240)(43,192)(44,193)(45,194)(46,195)(47,196)(48,183)(49,184)(50,185)(51,186)(52,187)(53,188)(54,189)(55,190)(56,191)(57,349)(58,350)(59,337)(60,338)(61,339)(62,340)(63,341)(64,342)(65,343)(66,344)(67,345)(68,346)(69,347)(70,348)(71,180)(72,181)(73,182)(74,169)(75,170)(76,171)(77,172)(78,173)(79,174)(80,175)(81,176)(82,177)(83,178)(84,179)(85,319)(86,320)(87,321)(88,322)(89,309)(90,310)(91,311)(92,312)(93,313)(94,314)(95,315)(96,316)(97,317)(98,318)(127,434)(128,421)(129,422)(130,423)(131,424)(132,425)(133,426)(134,427)(135,428)(136,429)(137,430)(138,431)(139,432)(140,433)(141,197)(142,198)(143,199)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,366)(156,367)(157,368)(158,369)(159,370)(160,371)(161,372)(162,373)(163,374)(164,375)(165,376)(166,377)(167,378)(168,365)(211,232)(212,233)(213,234)(214,235)(215,236)(216,237)(217,238)(218,225)(219,226)(220,227)(221,228)(222,229)(223,230)(224,231)(253,381)(254,382)(255,383)(256,384)(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)(264,392)(265,379)(266,380)(267,435)(268,436)(269,437)(270,438)(271,439)(272,440)(273,441)(274,442)(275,443)(276,444)(277,445)(278,446)(279,447)(280,448)(281,414)(282,415)(283,416)(284,417)(285,418)(286,419)(287,420)(288,407)(289,408)(290,409)(291,410)(292,411)(293,412)(294,413)(295,354)(296,355)(297,356)(298,357)(299,358)(300,359)(301,360)(302,361)(303,362)(304,363)(305,364)(306,351)(307,352)(308,353)(323,396)(324,397)(325,398)(326,399)(327,400)(328,401)(329,402)(330,403)(331,404)(332,405)(333,406)(334,393)(335,394)(336,395), (1,221,398,156)(2,222,399,157)(3,223,400,158)(4,224,401,159)(5,211,402,160)(6,212,403,161)(7,213,404,162)(8,214,405,163)(9,215,406,164)(10,216,393,165)(11,217,394,166)(12,218,395,167)(13,219,396,168)(14,220,397,155)(15,201,384,239)(16,202,385,240)(17,203,386,241)(18,204,387,242)(19,205,388,243)(20,206,389,244)(21,207,390,245)(22,208,391,246)(23,209,392,247)(24,210,379,248)(25,197,380,249)(26,198,381,250)(27,199,382,251)(28,200,383,252)(29,99,147,258)(30,100,148,259)(31,101,149,260)(32,102,150,261)(33,103,151,262)(34,104,152,263)(35,105,153,264)(36,106,154,265)(37,107,141,266)(38,108,142,253)(39,109,143,254)(40,110,144,255)(41,111,145,256)(42,112,146,257)(43,301,433,285)(44,302,434,286)(45,303,421,287)(46,304,422,288)(47,305,423,289)(48,306,424,290)(49,307,425,291)(50,308,426,292)(51,295,427,293)(52,296,428,294)(53,297,429,281)(54,298,430,282)(55,299,431,283)(56,300,432,284)(57,74,271,321)(58,75,272,322)(59,76,273,309)(60,77,274,310)(61,78,275,311)(62,79,276,312)(63,80,277,313)(64,81,278,314)(65,82,279,315)(66,83,280,316)(67,84,267,317)(68,71,268,318)(69,72,269,319)(70,73,270,320)(85,347,181,437)(86,348,182,438)(87,349,169,439)(88,350,170,440)(89,337,171,441)(90,338,172,442)(91,339,173,443)(92,340,174,444)(93,341,175,445)(94,342,176,446)(95,343,177,447)(96,344,178,448)(97,345,179,435)(98,346,180,436)(113,237,334,376)(114,238,335,377)(115,225,336,378)(116,226,323,365)(117,227,324,366)(118,228,325,367)(119,229,326,368)(120,230,327,369)(121,231,328,370)(122,232,329,371)(123,233,330,372)(124,234,331,373)(125,235,332,374)(126,236,333,375)(127,419,193,361)(128,420,194,362)(129,407,195,363)(130,408,196,364)(131,409,183,351)(132,410,184,352)(133,411,185,353)(134,412,186,354)(135,413,187,355)(136,414,188,356)(137,415,189,357)(138,416,190,358)(139,417,191,359)(140,418,192,360), (1,240,284,77)(2,241,285,78)(3,242,286,79)(4,243,287,80)(5,244,288,81)(6,245,289,82)(7,246,290,83)(8,247,291,84)(9,248,292,71)(10,249,293,72)(11,250,294,73)(12,251,281,74)(13,252,282,75)(14,239,283,76)(15,190,273,227)(16,191,274,228)(17,192,275,229)(18,193,276,230)(19,194,277,231)(20,195,278,232)(21,196,279,233)(22,183,280,234)(23,184,267,235)(24,185,268,236)(25,186,269,237)(26,187,270,238)(27,188,271,225)(28,189,272,226)(29,418,173,119)(30,419,174,120)(31,420,175,121)(32,407,176,122)(33,408,177,123)(34,409,178,124)(35,410,179,125)(36,411,180,126)(37,412,181,113)(38,413,182,114)(39,414,169,115)(40,415,170,116)(41,416,171,117)(42,417,172,118)(43,443,222,99)(44,444,223,100)(45,445,224,101)(46,446,211,102)(47,447,212,103)(48,448,213,104)(49,435,214,105)(50,436,215,106)(51,437,216,107)(52,438,217,108)(53,439,218,109)(54,440,219,110)(55,441,220,111)(56,442,221,112)(57,378,382,136)(58,365,383,137)(59,366,384,138)(60,367,385,139)(61,368,386,140)(62,369,387,127)(63,370,388,128)(64,371,389,129)(65,372,390,130)(66,373,391,131)(67,374,392,132)(68,375,379,133)(69,376,380,134)(70,377,381,135)(85,334,141,354)(86,335,142,355)(87,336,143,356)(88,323,144,357)(89,324,145,358)(90,325,146,359)(91,326,147,360)(92,327,148,361)(93,328,149,362)(94,329,150,363)(95,330,151,364)(96,331,152,351)(97,332,153,352)(98,333,154,353)(155,256,431,337)(156,257,432,338)(157,258,433,339)(158,259,434,340)(159,260,421,341)(160,261,422,342)(161,262,423,343)(162,263,424,344)(163,264,425,345)(164,265,426,346)(165,266,427,347)(166,253,428,348)(167,254,429,349)(168,255,430,350)(197,295,319,393)(198,296,320,394)(199,297,321,395)(200,298,322,396)(201,299,309,397)(202,300,310,398)(203,301,311,399)(204,302,312,400)(205,303,313,401)(206,304,314,402)(207,305,315,403)(208,306,316,404)(209,307,317,405)(210,308,318,406)>;
 
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), (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,113)(11,114)(12,115)(13,116)(14,117)(15,111)(16,112)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,241)(30,242)(31,243)(32,244)(33,245)(34,246)(35,247)(36,248)(37,249)(38,250)(39,251)(40,252)(41,239)(42,240)(43,192)(44,193)(45,194)(46,195)(47,196)(48,183)(49,184)(50,185)(51,186)(52,187)(53,188)(54,189)(55,190)(56,191)(57,349)(58,350)(59,337)(60,338)(61,339)(62,340)(63,341)(64,342)(65,343)(66,344)(67,345)(68,346)(69,347)(70,348)(71,180)(72,181)(73,182)(74,169)(75,170)(76,171)(77,172)(78,173)(79,174)(80,175)(81,176)(82,177)(83,178)(84,179)(85,319)(86,320)(87,321)(88,322)(89,309)(90,310)(91,311)(92,312)(93,313)(94,314)(95,315)(96,316)(97,317)(98,318)(127,434)(128,421)(129,422)(130,423)(131,424)(132,425)(133,426)(134,427)(135,428)(136,429)(137,430)(138,431)(139,432)(140,433)(141,197)(142,198)(143,199)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,366)(156,367)(157,368)(158,369)(159,370)(160,371)(161,372)(162,373)(163,374)(164,375)(165,376)(166,377)(167,378)(168,365)(211,232)(212,233)(213,234)(214,235)(215,236)(216,237)(217,238)(218,225)(219,226)(220,227)(221,228)(222,229)(223,230)(224,231)(253,381)(254,382)(255,383)(256,384)(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)(264,392)(265,379)(266,380)(267,435)(268,436)(269,437)(270,438)(271,439)(272,440)(273,441)(274,442)(275,443)(276,444)(277,445)(278,446)(279,447)(280,448)(281,414)(282,415)(283,416)(284,417)(285,418)(286,419)(287,420)(288,407)(289,408)(290,409)(291,410)(292,411)(293,412)(294,413)(295,354)(296,355)(297,356)(298,357)(299,358)(300,359)(301,360)(302,361)(303,362)(304,363)(305,364)(306,351)(307,352)(308,353)(323,396)(324,397)(325,398)(326,399)(327,400)(328,401)(329,402)(330,403)(331,404)(332,405)(333,406)(334,393)(335,394)(336,395), (1,221,398,156)(2,222,399,157)(3,223,400,158)(4,224,401,159)(5,211,402,160)(6,212,403,161)(7,213,404,162)(8,214,405,163)(9,215,406,164)(10,216,393,165)(11,217,394,166)(12,218,395,167)(13,219,396,168)(14,220,397,155)(15,201,384,239)(16,202,385,240)(17,203,386,241)(18,204,387,242)(19,205,388,243)(20,206,389,244)(21,207,390,245)(22,208,391,246)(23,209,392,247)(24,210,379,248)(25,197,380,249)(26,198,381,250)(27,199,382,251)(28,200,383,252)(29,99,147,258)(30,100,148,259)(31,101,149,260)(32,102,150,261)(33,103,151,262)(34,104,152,263)(35,105,153,264)(36,106,154,265)(37,107,141,266)(38,108,142,253)(39,109,143,254)(40,110,144,255)(41,111,145,256)(42,112,146,257)(43,301,433,285)(44,302,434,286)(45,303,421,287)(46,304,422,288)(47,305,423,289)(48,306,424,290)(49,307,425,291)(50,308,426,292)(51,295,427,293)(52,296,428,294)(53,297,429,281)(54,298,430,282)(55,299,431,283)(56,300,432,284)(57,74,271,321)(58,75,272,322)(59,76,273,309)(60,77,274,310)(61,78,275,311)(62,79,276,312)(63,80,277,313)(64,81,278,314)(65,82,279,315)(66,83,280,316)(67,84,267,317)(68,71,268,318)(69,72,269,319)(70,73,270,320)(85,347,181,437)(86,348,182,438)(87,349,169,439)(88,350,170,440)(89,337,171,441)(90,338,172,442)(91,339,173,443)(92,340,174,444)(93,341,175,445)(94,342,176,446)(95,343,177,447)(96,344,178,448)(97,345,179,435)(98,346,180,436)(113,237,334,376)(114,238,335,377)(115,225,336,378)(116,226,323,365)(117,227,324,366)(118,228,325,367)(119,229,326,368)(120,230,327,369)(121,231,328,370)(122,232,329,371)(123,233,330,372)(124,234,331,373)(125,235,332,374)(126,236,333,375)(127,419,193,361)(128,420,194,362)(129,407,195,363)(130,408,196,364)(131,409,183,351)(132,410,184,352)(133,411,185,353)(134,412,186,354)(135,413,187,355)(136,414,188,356)(137,415,189,357)(138,416,190,358)(139,417,191,359)(140,418,192,360), (1,240,284,77)(2,241,285,78)(3,242,286,79)(4,243,287,80)(5,244,288,81)(6,245,289,82)(7,246,290,83)(8,247,291,84)(9,248,292,71)(10,249,293,72)(11,250,294,73)(12,251,281,74)(13,252,282,75)(14,239,283,76)(15,190,273,227)(16,191,274,228)(17,192,275,229)(18,193,276,230)(19,194,277,231)(20,195,278,232)(21,196,279,233)(22,183,280,234)(23,184,267,235)(24,185,268,236)(25,186,269,237)(26,187,270,238)(27,188,271,225)(28,189,272,226)(29,418,173,119)(30,419,174,120)(31,420,175,121)(32,407,176,122)(33,408,177,123)(34,409,178,124)(35,410,179,125)(36,411,180,126)(37,412,181,113)(38,413,182,114)(39,414,169,115)(40,415,170,116)(41,416,171,117)(42,417,172,118)(43,443,222,99)(44,444,223,100)(45,445,224,101)(46,446,211,102)(47,447,212,103)(48,448,213,104)(49,435,214,105)(50,436,215,106)(51,437,216,107)(52,438,217,108)(53,439,218,109)(54,440,219,110)(55,441,220,111)(56,442,221,112)(57,378,382,136)(58,365,383,137)(59,366,384,138)(60,367,385,139)(61,368,386,140)(62,369,387,127)(63,370,388,128)(64,371,389,129)(65,372,390,130)(66,373,391,131)(67,374,392,132)(68,375,379,133)(69,376,380,134)(70,377,381,135)(85,334,141,354)(86,335,142,355)(87,336,143,356)(88,323,144,357)(89,324,145,358)(90,325,146,359)(91,326,147,360)(92,327,148,361)(93,328,149,362)(94,329,150,363)(95,330,151,364)(96,331,152,351)(97,332,153,352)(98,333,154,353)(155,256,431,337)(156,257,432,338)(157,258,433,339)(158,259,434,340)(159,260,421,341)(160,261,422,342)(161,262,423,343)(162,263,424,344)(163,264,425,345)(164,265,426,346)(165,266,427,347)(166,253,428,348)(167,254,429,349)(168,255,430,350)(197,295,319,393)(198,296,320,394)(199,297,321,395)(200,298,322,396)(201,299,309,397)(202,300,310,398)(203,301,311,399)(204,302,312,400)(205,303,313,401)(206,304,314,402)(207,305,315,403)(208,306,316,404)(209,307,317,405)(210,308,318,406) );
 
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)], [(1,118),(2,119),(3,120),(4,121),(5,122),(6,123),(7,124),(8,125),(9,126),(10,113),(11,114),(12,115),(13,116),(14,117),(15,111),(16,112),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,241),(30,242),(31,243),(32,244),(33,245),(34,246),(35,247),(36,248),(37,249),(38,250),(39,251),(40,252),(41,239),(42,240),(43,192),(44,193),(45,194),(46,195),(47,196),(48,183),(49,184),(50,185),(51,186),(52,187),(53,188),(54,189),(55,190),(56,191),(57,349),(58,350),(59,337),(60,338),(61,339),(62,340),(63,341),(64,342),(65,343),(66,344),(67,345),(68,346),(69,347),(70,348),(71,180),(72,181),(73,182),(74,169),(75,170),(76,171),(77,172),(78,173),(79,174),(80,175),(81,176),(82,177),(83,178),(84,179),(85,319),(86,320),(87,321),(88,322),(89,309),(90,310),(91,311),(92,312),(93,313),(94,314),(95,315),(96,316),(97,317),(98,318),(127,434),(128,421),(129,422),(130,423),(131,424),(132,425),(133,426),(134,427),(135,428),(136,429),(137,430),(138,431),(139,432),(140,433),(141,197),(142,198),(143,199),(144,200),(145,201),(146,202),(147,203),(148,204),(149,205),(150,206),(151,207),(152,208),(153,209),(154,210),(155,366),(156,367),(157,368),(158,369),(159,370),(160,371),(161,372),(162,373),(163,374),(164,375),(165,376),(166,377),(167,378),(168,365),(211,232),(212,233),(213,234),(214,235),(215,236),(216,237),(217,238),(218,225),(219,226),(220,227),(221,228),(222,229),(223,230),(224,231),(253,381),(254,382),(255,383),(256,384),(257,385),(258,386),(259,387),(260,388),(261,389),(262,390),(263,391),(264,392),(265,379),(266,380),(267,435),(268,436),(269,437),(270,438),(271,439),(272,440),(273,441),(274,442),(275,443),(276,444),(277,445),(278,446),(279,447),(280,448),(281,414),(282,415),(283,416),(284,417),(285,418),(286,419),(287,420),(288,407),(289,408),(290,409),(291,410),(292,411),(293,412),(294,413),(295,354),(296,355),(297,356),(298,357),(299,358),(300,359),(301,360),(302,361),(303,362),(304,363),(305,364),(306,351),(307,352),(308,353),(323,396),(324,397),(325,398),(326,399),(327,400),(328,401),(329,402),(330,403),(331,404),(332,405),(333,406),(334,393),(335,394),(336,395)], [(1,221,398,156),(2,222,399,157),(3,223,400,158),(4,224,401,159),(5,211,402,160),(6,212,403,161),(7,213,404,162),(8,214,405,163),(9,215,406,164),(10,216,393,165),(11,217,394,166),(12,218,395,167),(13,219,396,168),(14,220,397,155),(15,201,384,239),(16,202,385,240),(17,203,386,241),(18,204,387,242),(19,205,388,243),(20,206,389,244),(21,207,390,245),(22,208,391,246),(23,209,392,247),(24,210,379,248),(25,197,380,249),(26,198,381,250),(27,199,382,251),(28,200,383,252),(29,99,147,258),(30,100,148,259),(31,101,149,260),(32,102,150,261),(33,103,151,262),(34,104,152,263),(35,105,153,264),(36,106,154,265),(37,107,141,266),(38,108,142,253),(39,109,143,254),(40,110,144,255),(41,111,145,256),(42,112,146,257),(43,301,433,285),(44,302,434,286),(45,303,421,287),(46,304,422,288),(47,305,423,289),(48,306,424,290),(49,307,425,291),(50,308,426,292),(51,295,427,293),(52,296,428,294),(53,297,429,281),(54,298,430,282),(55,299,431,283),(56,300,432,284),(57,74,271,321),(58,75,272,322),(59,76,273,309),(60,77,274,310),(61,78,275,311),(62,79,276,312),(63,80,277,313),(64,81,278,314),(65,82,279,315),(66,83,280,316),(67,84,267,317),(68,71,268,318),(69,72,269,319),(70,73,270,320),(85,347,181,437),(86,348,182,438),(87,349,169,439),(88,350,170,440),(89,337,171,441),(90,338,172,442),(91,339,173,443),(92,340,174,444),(93,341,175,445),(94,342,176,446),(95,343,177,447),(96,344,178,448),(97,345,179,435),(98,346,180,436),(113,237,334,376),(114,238,335,377),(115,225,336,378),(116,226,323,365),(117,227,324,366),(118,228,325,367),(119,229,326,368),(120,230,327,369),(121,231,328,370),(122,232,329,371),(123,233,330,372),(124,234,331,373),(125,235,332,374),(126,236,333,375),(127,419,193,361),(128,420,194,362),(129,407,195,363),(130,408,196,364),(131,409,183,351),(132,410,184,352),(133,411,185,353),(134,412,186,354),(135,413,187,355),(136,414,188,356),(137,415,189,357),(138,416,190,358),(139,417,191,359),(140,418,192,360)], [(1,240,284,77),(2,241,285,78),(3,242,286,79),(4,243,287,80),(5,244,288,81),(6,245,289,82),(7,246,290,83),(8,247,291,84),(9,248,292,71),(10,249,293,72),(11,250,294,73),(12,251,281,74),(13,252,282,75),(14,239,283,76),(15,190,273,227),(16,191,274,228),(17,192,275,229),(18,193,276,230),(19,194,277,231),(20,195,278,232),(21,196,279,233),(22,183,280,234),(23,184,267,235),(24,185,268,236),(25,186,269,237),(26,187,270,238),(27,188,271,225),(28,189,272,226),(29,418,173,119),(30,419,174,120),(31,420,175,121),(32,407,176,122),(33,408,177,123),(34,409,178,124),(35,410,179,125),(36,411,180,126),(37,412,181,113),(38,413,182,114),(39,414,169,115),(40,415,170,116),(41,416,171,117),(42,417,172,118),(43,443,222,99),(44,444,223,100),(45,445,224,101),(46,446,211,102),(47,447,212,103),(48,448,213,104),(49,435,214,105),(50,436,215,106),(51,437,216,107),(52,438,217,108),(53,439,218,109),(54,440,219,110),(55,441,220,111),(56,442,221,112),(57,378,382,136),(58,365,383,137),(59,366,384,138),(60,367,385,139),(61,368,386,140),(62,369,387,127),(63,370,388,128),(64,371,389,129),(65,372,390,130),(66,373,391,131),(67,374,392,132),(68,375,379,133),(69,376,380,134),(70,377,381,135),(85,334,141,354),(86,335,142,355),(87,336,143,356),(88,323,144,357),(89,324,145,358),(90,325,146,359),(91,326,147,360),(92,327,148,361),(93,328,149,362),(94,329,150,363),(95,330,151,364),(96,331,152,351),(97,332,153,352),(98,333,154,353),(155,256,431,337),(156,257,432,338),(157,258,433,339),(158,259,434,340),(159,260,421,341),(160,261,422,342),(161,262,423,343),(162,263,424,344),(163,264,425,345),(164,265,426,346),(165,266,427,347),(166,253,428,348),(167,254,429,349),(168,255,430,350),(197,295,319,393),(198,296,320,394),(199,297,321,395),(200,298,322,396),(201,299,309,397),(202,300,310,398),(203,301,311,399),(204,302,312,400),(205,303,313,401),(206,304,314,402),(207,305,315,403),(208,306,316,404),(209,307,317,405),(210,308,318,406)]])
 

280 conjugacy classes

class 1 2A···2O4A···4X7A···7F14A···14CL28A···28EN
order12···24···47···714···1428···28
size11···12···21···11···12···2

280 irreducible representations

dim111111112222
type++++-
imageC1C2C2C4C7C14C14C28D4Q8C7×D4C7×Q8
kernelC14×C2.C42C7×C2.C42C23×C28C22×C28C2×C2.C42C2.C42C23×C4C22×C4C22×C14C22×C14C23C23
# reps1432462418144623612

Matrix representation of C14×C2.C42 ►in GL5(𝔽29)

280000
028000
002800
00050
00005
,
10000
01000
00100
000280
000028
,
120000
017000
00100
0002014
000159
,
170000
01000
00100
00001
00010

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[12,0,0,0,0,0,17,0,0,0,0,0,1,0,0,0,0,0,20,15,0,0,0,14,9],[17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
 

C14×C2.C42 in GAP, Magma, Sage, TeX 

C_{14}\times C_2.C_4^2
 
% in TeX
 
G:=Group("C14xC2.C4^2");
 
// GroupNames label
 
G:=SmallGroup(448,783);
 
// by ID
 
G=gap.SmallGroup(448,783);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^14=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁