Does a graph exist such that its vertices can be distributed among "boxes" so that conditions are satisfied?

Question

I want to a graph $G$ on $52$ vertices, call them $x_1 := (a_1, b_1), \ldots, x_{52} = (a_{52}, b_{52})$, where $G := \{(x_i, x_j): x_i \text{ and } x_j \text{ share a common element}\}$ such that the vertices can be distributed among 1365 "boxes" under the following conditions:

1. Each box has at least 1 vertex.
2. No box has more than 5 vertices.
3. Each vertex is in exactly 66 of the boxes.
4. If $x_i, x_j$ share an edge in $G$, they occur in 15 of the same boxes, otherwise they occur in 1 of the same boxes.