  The representations of the Schr\"odinger group in one space dimension are
explicitly constructed in the basis of the harmonic oscillator states. These
representations are seen to involve matrix orthogonal polynomials in a discrete
variable that have Charlier and Meixner polynomials as building blocks. The
underlying Lie-theoretic framework allows for a systematic derivation of the
structural formulas (recurrence relations, difference equations, Rodrigues'
formula etc.) that these matrix orthogonal polynomials satisfy.
