The incidence matrix of the real encasing polytope (hex) is

  8 ♦  6 | 3 3 3 3 |  8
----+----+---------+---
  2 | 24 | 1 1 1 1 |  4
----+----+---------+---
  3 |  3 | 8 * * * |  2 (r)
  3 |  3 | * 8 * * |  2 (y)
  3 |  3 | * * 8 * |  2 (g)
  3 |  3 | * * * 8 |  2 (b)
----+----+---------+---
♦ 4 |  6 | 1 1 1 1 | 16

The vertex figure (oct) can be considered as trigonal antiprism, having the bases red (r), and additionally the lateral up-edges of the girthing zickzack colored red too. Being rolled onto an other face, the same can be done for yellow (y), green (g), and blue (b), thus coloring all edges and faces exactly once. - Being a vertex figure, the oct-coloring of edges represents the colors of vertex incident faces of hex, whereas that of oct faces represents that of the opposite faces of the vertex incident cells (tet). Thus all these vertex incident tet are 4-colored. Further, this scheme can be extended to all cells of hex consistently.

The incidence matrix of the complex polytope thus is

8 | 3
--+--
3 | 8

Generators

R0 : permutes the vertices of some (red) triangle, say span( (1,0,0,0), (0,1,0,0), (0,0,1,0) ) 
R1 : permutes the (red) triangles of some vertex, say (1,0,0,0)

     /  0  0  1  0 \        / 1  0  0  0 \
     |  1  0  0  0 |        | 0  0 -1  0 |
R0 = |  0  1  0  0 | , R1 = | 0  0  0 -1 |
     \  0  0  0  1 /        \ 0  1  0  0 /

R03  =  1                        (+1+2+3)(+4)(-1-2-3)(-4) : permutes both, vertices and triangles *)
R13  =  1                        (+1)(+2+4-3)(-1)(-2-4+3) : permutes both, vertices and triangles *)
R0 * R1 * R0  =  R1 * R0 * R1    = (+1+4-1-4)(+2-3-2+3)

*) triangles of the other colors will be permuted across their color-subsets