Finite quotients of 3-manifold groups

It is well-known that for any finite group G, there exists a closed 3-manifold M with G as a quotient of the fundamental group of M. However, we can ask more detailed questions about the possible finite quotients of 3-manifold groups, e.g. for G and H_1,…,H_n finite groups, does there exist a 3-manifold group with G as a quotient but no H_i as a quotient?  We answer all such questions.  To prove non-existence, we prove new parity properties of the fundamental
groups of 3-manifolds.  To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the fundamental group of a random 3-manifold, in the sense of Dunfield-Thurston. This is joint work with Will Sawin.