Dwane Campbell has constructed a 5x5x5x5 pantriagonal magic cube. I have used the same method to construct a 7x7x7x7 pantriagonal magic cube.
You need a 7x7 magic square which is symmetric, but not panmagic. Produce 2 grids consisting of the 7x7 magic square and the 48 shifted versions of the 7x7 magic square.
See for grids, the result and the check (if all numbers are in the magic cube and addition of the numbers give the right magic sum) the download below.