We introduce a new product for permutation groups. It takes as input two permutation groups, M and N, and produces an infinite group M [X] N which carries many of the permutational properties of M. Under mild conditions on M and N the group M [X] N is simple. As a permutational product, its most significant property is the following: M [X] N is primitive if and only if M is primitive but not regular, and N is transitive. Despite this remarkable similarity with the wreath product in product action, M [X] N and M Wr N are thoroughly dissimilar. The product provides a general way to build exotic examples of non-discrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups. We use this to solve a well-known open problem from topological group theory, by obtaining the first construction of uncountably many pairwise non-isomorphic simple topological groups that are totally disconnected, locally compact, compactly generated and non-discrete. The groups we construct all contain the same compact open subgroup. To build the product, we describe a group U(M,N) that acts on an edge-transitive biregular tree T. This group has a natural universal property and is analogous to the iconic universal group construction of M. Burger and S. Mozes for locally finite regular trees.

Updated to match version published in Duke Math. J. Volume 166, Number 15 (2017), 2965-2999