m coprime to 2n. Alternatively, the fundamental group has presentation for coprime integers m, n with m ≥ 1, n ≥ 2. (The n here equals the previous n, and the m here is 2k-1 times the previous m.) We continue with the latter presentation. This group is a metacyclic group of order 4mn with abelianization of order 4m (so m and n are both determined by this group). The element y generates a cyclic normal subgroup of order 2n, and the element x has order 4m. The center is cyclic of order 2m and is generated by x2, and the quotient by the center is the dihedral group of order 2n. When m = 1 this group is a binary dihedral or dicyclic group. The simplest example is m = 1, n = 2, when π1(M) is the quaternion group of order 8. Prism manifolds are uniquely determined by their fundamental groups: if a closed 3-manifold has the same fundamental group as a prism manifold M, it is homeomorphic to M. Prism manifolds can be represented as Seifert fiber spaces in two ways. Tetrahedral case The fundamental group is a product of a cyclic group of order m with a group having presentation for integers k, m with k ≥ 1, m ≥ 1 and m coprime to 6. Alternatively, the fundamental group has presentation for an odd integer m ≥ 1. (The m here is 3k-1 times the previous m.) We continue with the latter presentation. This group has order 24m. The elements x and y generate a normal subgroup isomorphic to the quaternion group of order 8. The center is cyclic of order 2m. It is generated by the elements z3 and x2 = y2, and the quotient by the center is the tetrahedral group, equivalently, the alternating group A4. When m = 1 this group is the binary tetrahedral group. These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 3. Octahedral case The fundamental group is a product of a cyclic group of order m coprime to 6 with the binary octahedral group (of order 48) which has the presentation These manifolds are uniquely determined by their fundamental groups. They can all be represented in