The incidence matrix of the encasing polytope (ico) is

o3x3o *b3o

. . .    . | 24 ♦  8 |  4  4  4 | 2 2 2
-----------+----+----+----------+------
. x .    . |  2 | 96 |  1  1  1 | 1 1 1
-----------+----+----+----------+------
o3x .    . |  3 |  3 | 32  *  * | 1 1 0 (r)
. x3o    . |  3 |  3 |  * 32  * | 1 0 1 (g)
. x . *b3o |  3 |  3 |  *  * 32 | 0 1 1 (b)
-----------+----+----+----------+------
o3x3o    . ♦  6 | 12 |  4  4  0 | 8 * * (b)
o3x . *b3o ♦  6 | 12 |  4  0  4 | * 8 * (g)
. x3o *b3o ♦  6 | 12 |  0  4  4 | * * 8 (r)

Color the triangles of 3 oct each alternatingly in red (r) and green (g), in (r) and blue (b), resp. (g) and (b). Take these 3 types of oct (each being called by the not-used face-color) to tile the ico. The matching rule here obviously being given by the colors of the triangles. - This corresoponds in the edges of the vertex figure (cube) being colored in the sense of the general briquet, x-edges in (r), say, y-edges in (g), and z-edges in (b).

The incidence matrix of the abstract polytope thus is

24 |  4
---+---
 3 | 32

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 (0,0,1,1) according to the cycle
 ((1,0,0,1), (0,1,0,1), (0,0,1,1)) → 
 ((-1,0,0,1), (0,-1,0,1), (0,0,1,1)) → 
 ((0,-1,1,0), (-1,0,1,0), (0,0,1,1)) → 
 ((0,1,1,0), (1,0,1,0), (0,0,1,1)) → 
 ((1,0,0,1), (0,1,0,1), (0,0,1,1))

     /  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
R14  =  1
R0 * R1 * R0 * R1 * R0 * R1 * R0 * R1 * R0 * R1 * R0 * R1  =  R1 * R0 * R1 * R0 * R1 * R0 * R1 * R0 * R1 * R0 * R1 * R0  =  1

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 (0,0,1,1) according to the cycle
 ((1,0,0,1), (0,1,0,1), (0,0,1,1)) → 
 ((0,-1,0,1), (-1,0,0,1), (0,0,1,1)) → 
 ((0,-1,1,0), (-1,0,1,0), (0,0,1,1)) → 
 ((1,0,1,0), (0,1,1,0), (0,0,1,1)) → 
 ((1,0,0,1), (0,1,0,1), (0,0,1,1))

     /  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
R1'4 =  1
R0 * R1' * R0 * R1' * R0 * R1' * R0 * R1'  =  R1' * R0 * R1' * R0 * R1' * R0 * R1' * R0  =  -1

Here R1' is just the same as R1 in cycling through the same set of triangles, only that any second one will be flipped. The R0 are identical in both instances. According to (R0,R1) this abstract polytope should be described as 3{12}4, according to (R0,R1') this (same) abstract polytope should be described as 3{8}4.

All the given generators are not representable as unitary matrices. Therefore these representations of this abstract polytope do not qualify as complex polytopes.