It is a generalization of the Siegel modular group, and has the same relation to polarized abelian varieties that the Siegel modular group has to principally polarized abelian varieties.
It is the group of automorphisms of Z2n preserving a non-degenerate skew symmetric form.
Paramodular groups were introduced by Conforto (1952) and named by Shimura (1958, section 8).
This section describes how to represent it as a subgroup of Sp4(Q) with entries that are not necessarily integers.
The fact that this matrix is symplectic forces some further congruence conditions, so in fact the paramodular group consists of the symplectic matrices of the form The paramodular group in this case is generated by matrices of the forms for integers x, y, and z.
which gives similar results except that the rows and columns get permuted; for example, the paramodular group then consists of the symplectic matrices of the form