The subject of this paper is a family of network topologies which are intended to be practical enough to be implemented in a parallel computer using contemporary technology. These network topologies offer an alternative to the three-dimensional mesh or toroid and are most likely to be useful for a network with between 512 and 16,384 nodes. The topologies, which are called supertoroids, are Cayley graphs of groups. The groups are semidirect products of two cyclic groups, one of order (c k) and the other of order (c^{2} l). The resulting Cayley graph has (c^{3} k l) vertices and is of degree 4. The main results of this paper are two theorems. The first states that the diameter of the supertoroidal network with (c^{3} k l) nodes is (\lfloor c l / 2\rfloor+\lfloor c k / 2\rfloor) if (c \geq 8). The second states that the average distance is asymptotically (in c) equal to ((\lfloor c l / 2\rfloor+\lfloor c k / 2\rfloor) / 2). Because of these facts, the supertoroidal network has latency which is lower than the latency of a three-dimensional toroid of the same size. Although the proofs of these theorems are a tour-de-force of arithmetic in the underlying group, they are based upon exploitation of parallelism of paths in a Cayley graph. โฒ95 Academic Press, Lrc.