  We demonstrate how a 3-manifold, a Heegaard diagram, and a group presentation
can each be interpreted as a pair of signed permutations in the symmetric group
$S_d.$ We demonstrate the power of permutation data in programming and discuss
an algorithm we have developed that takes the permutation data as input and
determines whether the data represents a closed 3-manifold. We therefore have
an invariant of groups, that is given any group presentation, we can determine
if that fixed presentation presents a closed 3-manifold.
  (The proposed techniques begin with a pair of signed permutations and builds
a finite group presentation. The finite group presentation results in a finite
class of associated 3-manifolds. Notice that a negative answer only implies the
fixed presentation does not result in a closed 3-manifold under this
construction, but says nothing about an isomorphic form of the group
presentation.)
