The real encasing polytope is the n-gonal dihedral n-choron, looking like the completely incident overlay of n n-gons.

The incidence matrix of the complex polytope thus is

n | n
--+--
n | n

Generators

with e  = exp(2πi/n),   E  = exp(2πi k/n),
where 1 < k < n and k not divisor of n

R0 = / e 0 \ ,  R1 = / 1 0 \
     \ 0 1 /         \ 0 E /

R0n  =  1                                  (rotation through vertices)
R1n  =  1                                  (polygons, slightly bended into orthogonal space, are thus permuted)
R0 * R1  =  R1 * R0