  We study moments of characteristic polynomials of truncated Haar distributed
matrices from the three classical compact groups O(N), U(N) and Sp(2N). For
finite matrix size we calculate the moments in terms of hypergeometric
functions of matrix argument and give explicit integral representations
highlighting the duality between the moment and the matrix size as well as the
duality between the orthogonal and symplectic cases. Asymptotic expansions in
strong and weak non-unitarity regimes are obtained. Using the connection to
matrix hypergeometric functions, we establish limit theorems for the
log-modulus of the characteristic polynomial evaluated on the unit circle.
