Let n be a natural number larger than two. Let D2n=⟨r,s:rn=s2=e,srs=rn−1⟩ be the Dihedral group, and κ an
n-dimensional unitary representation of D2n acting in Cn as
follows. (κ(r)v)(j)=v((j−1)modn) and (κ(s)v)(j)=v((n−j)modn)
for v∈Cn. For any representation which is unitarily equivalent to
κ, we prove that when n is prime there exists a Zariski open subset
E of Cn such that for any vector v∈E, any subset of
cardinality n of the orbit of v under the action of this representation is
a basis for Cn. However, when n is even there is no vector in
Cn which satisfies this property. As a result, we derive that if
n is prime, for almost every (with respect to Lebesgue measure) vector v in
Cn the Γ-orbit of v is a frame which is maximally
robust to erasures. We also consider the case where τ is equivalent to an
irreducible unitary representation of the Dihedral group acting in a vector
space Hτ∈{C,C2} and we
provide conditions under which it is possible to find a vector
v∈Hτ such that τ(Γ)v has the Haar
property