  For a countable group G and a multiplier c on G with values in the circle, we
study the property of G having a unitary projective c-representation which is
both irreducible and projectively faithful. We show that this property is
equivalent to G being the quotient of an appropriate group by its centre. A
criterion is given in terms of the minisocle of G. Several examples are
described to show the existence of various behaviours.
