The 4-dimensional, i.e. $3 \times 3 \times 3 \times 3$, equivalent of the Rubik's cube has 8 three-dimensional sides, each of which consists of $3^3 = 27$ three-dimensional colored "squares". Of these 27 "squares", the one in the center is fixed. Thus in total our 4-dimensional Rubik's cube has $8 \cdot 26 = 208$ movable "squares", and the group of its sequences of moves under composition embeds therefore into ${\rm S}_{208}$.
Now just as the usual 3-dimensional Rubik's cube has 2 kinds of movable cubies (edge stones and corner stones), our 4-dimensional Rubik's cube has 3 distinct kinds of movable cubies:
The corner stones, of which there are $2^4 = 16$.
Each corner stone has 4 visible colored "squares".
The edge stones, of which there are $2 \cdot 12 + 8 = 32$.
Each edge stone has 3 visible colored "squares".
The face stones, of which there are $2 \cdot 6 + 12 = 24$.
Each face stone has 2 visible colored "squares".
Therefore, analogous to the usual 3-dimensional case, we can finally say that our 4-dimensional Rubik's cube group embeds into the following direct product of wreath products: $$ {\rm C}_4 \wr {\rm S}_{16} \times {\rm C}_3 \wr {\rm S}_{32} \times {\rm C}_2 \wr {\rm S}_{24}. $$$$ {\rm A}_4 \wr {\rm S}_{16} \times {\rm S}_3 \wr {\rm S}_{32} \times {\rm C}_2 \wr {\rm S}_{24}. $$