  Basic modules of McLain groups $M=M(\Lambda,\leq, R)$ are defined and
investigated. These are (possibly infinite dimensional) analogues of Andr\'e's
supercharacters of $U_n(q)$. The ring $R$ need not be finite or commutative and
the field underlying our representations is essentially arbitrary: we deal with
all characteristics, prime or zero, on an equal basis. The set $\Lambda$,
totally ordered by $\leq$, is allowed to be infinite. We show that distinct
basic modules are disjoint, determine the dimension of the endomorphism algebra
of a basic module, find when a basic module is irreducible, and exhibit a full
decomposition of a basic module as direct sum of irreducible submodules,
including their multiplicities. Several examples of this decomposition are
presented, and a criterion for a basic module to be multiplicity-free is given.
In general, not every irreducible module of a McLain group is a constituent of
a basic module.
