OrePolynomial[{{-t^2 + 52*t^4 - 624*t^6 + 1728*t^8, {3}}, 
  {-5*t + 4*t^2 + 348*t^3 - 4080*t^5 - 576*t^6 + 12096*t^7, {2}}, 
  {4*(-1 + 4*t + 129*t^2 + 24*t^3 - 1380*t^4 - 576*t^5 + 4320*t^6), {1}}, 
  {4*(2 + 33*t + 24*t^2 - 276*t^3 - 288*t^4 + 864*t^5), {0}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
