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

 24 ♦  2  2  2  2 |  3  3  3  3 |  6
----+-------------+-------------+---
  2 | 24  *  *  * |  1  1  1  0 |  3 (b)
  2 |  * 24  *  * |  1  1  0  1 |  3 (g)
  2 |  *  * 24  * |  1  0  1  1 |  3 (y)
  2 |  *  *  * 24 |  0  1  1  1 |  3 (r)
----+-------------+-------------+---
  3 |  1  1  1  0 | 24  *  *  * |  2 (r)
  3 |  1  1  0  1 |  * 24  *  * |  2 (y)
  3 |  1  0  1  1 |  *  * 24  * |  2 (g)
  3 |  0  1  1  1 |  *  *  * 24 |  2 (b)
----+-------------+-------------+---
♦ 6 |  3  3  3  3 |  2  2  2  2 | 24

The oct cells are to be colored, with opposite faces alike, in 4 colors. This scheme can be extended to all cells of ico consistently. 3 such oct and thus differently colored triangles will be incident at each edge. Therefore each edge gets that missing 4th color. - The vertex figure is a cube. The edges of this cube correspond to the vertex-incident triangles of the ico. So those 12 edges each will be colored too accordingly. 3 in each color. Those 3 are the alternating ones of the Petrie polygon of that cube.

The incidence matrix of the complex polytope thus is

24 |  3
---+---
 3 | 24

Generators

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

     /  0  0  1  0 \                  / 1 -1  1  1 \
     |  1  0  0  0 |                  | 1 -1 -1 -1 |
R0 = |  0  1  0  0 | ,   R1 = (1/2) * | 1  1  1 -1 |
     \  0  0  0  1 /                  \ 1  1 -1  1 /

R03  =  1                                            / 1 -1  1 -1 \
R13  =  1                                            | 1  1 -1 -1 |
R0 * R1 * R0 * R1  =  R1 * R0 * R1 * R0    = (1/2) * |-1  1  1 -1 |
                                                     \ 1  1  1  1 /