Jelly, 37 bytes

ØẠiⱮ“Uu½Ḥ]æṃ<Ȥḍ+`$ỊtUḂeḂ»
³ÆRṁ¢ż¢*/€P

Try it online!

Encodes the number as a list of exponents to primes, probably could be optimized with cleverer builtins

ØẠiⱮ“Uu½Ḥ]æṃ<Ȥḍ+`$ỊtUḂeḂ»
    “Uu½Ḥ]æṃ<Ȥḍ+`$ỊtUḂeḂ» the string "tTIFBCAAAAA A A A  A"
  iⱮ                      find indices of each character in
ØẠ                        the alphabet in both cases "A..Za..z"

³ÆRṁ¢ż¢*/€P
 ÆR         all primes below
³           100
   ṁ        shaped like
    ¢       the above line
     ż      zipped with
      ¢     the above line
       */€  exponent for each pair
          P product

Jelly, 37 30 bytes

ØẠiⱮ“tTIFBC”;⁽<(B¤
³ÆRṁ¢ż¢*/€P

Encodes the number as a list of exponents to primes, probably could be optimized with cleverer builtins

ØẠiⱮ“tTIFBC”;⁽<(B¤
    “tTIFBC”       the string "tTIFBC"
  iⱮ               find indices of each character in
ØẠ                 the alphabet in both cases "A..Za..z"
             ⁽<( ¤ the number 16041
                B¤ converted to binary
            ;      append

³ÆRṁ¢ż¢*/€P
 ÆR         all primes below
³           100
   ṁ        shaped like
    ¢       the above line
     ż      zipped with
      ¢     the above line
       */€  exponent for each pair
          P product

-7 bytes by encoding the final 0/1 sequence as binary

Try it online!