  We study certain types of Fuchsian groups of the first kind denoted by
$R(N)$, which coincide with the Fricke groups or the arithmetic Hecke triangle
groups of low levels. We find all elliptic points and cusps of $R(p)$ for a
prime $p$, and prove that there is a one-to-one correspondence between the set
of equivalence classes of elliptic points of $R(p)$ and the imaginary quadratic
class group. We also find the explicit formula of the Fourier expansion of the
non-holomorphic Eisenstein series for $R(N)$ and study their analytic
properties. These non-holomorphic Eisenstein series together with cusp forms
provide a basis for the space of polyharmonic Maass forms for $R(N)$.
