  Although the theory of discrete Painlev\'e (dP) equations is rather young,
more and more examples of such equations appear in interesting and important
applications. Thus, it is essential to be able to recognize these equations, to
be able to identify their type, and to see where they belong in the
classification scheme. The definite classification scheme for dP equations was
proposed by H. Sakai, who used geometric ideas to identify 22 different classes
of these equations. However, in a major contrast with the theory of ordinary
differential Painlev\'e equations, there are infinitely many non-equivalent
discrete equations in each class. Thus, there is no general form for a dP
equation in each class, although some nice canonical examples in each equation
class are known. The main objective of this paper is to illustrate that, in
addition to providing the classification scheme, the geometric ideas of Sakai
give us a powerful tool to study dP equations. We consider a very complicated
example of a dP equation that describes a simple Schlesinger transformation of
a Fuchsian system and we show how this equation can be identified with a much
simpler canonical example of the dP equation of the same type and moreover, we
give an explicit change of coordinates transforming one equation into the
other. Among our main tools are the birational representation of the affine
Weyl symmetry group of the equation and the period map. Even though we focus on
a concrete example, the techniques that we use are general and can be easily
adapted to other examples.
