The incidence matrix of the real encasing polytope is

The incidence matrix of the complex polytope thus is

24 |  4
---+---
 4 | 24

Generators

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

R04  =  1                                  (permutes the vertices of inscribed hex and tes separately)
R14  =  1                                  (mixes up these sets)
R0 * R1 * R0 * R1  =  R1 * R0 * R1 * R0    (...) ???