After calculating the group extensions of Z+Z by Z, with constant action, I am curious whether the groups have any more natural form than I found. I mean extension of Z+Z by Z in this sense, as a sequence of groups where E need not be commutative: 0 --> Z --> E --> Z+Z --> 0 and the kernel is in the center of E. The form I found is parametrized by the integers this way: For any integer c, the group E_c has triples of integers (i,,j,k) as elements and the multiplication rule is coordinate-wise addition plus an extra bit in the first coordinate. (i,,j,k).(q,r,s) = ( (i+j+c.(kr)), j+r, k+s) When c=0 this is commutative and is just the coproduct Z+Z+Z. In any group E_c, the element (c,0,0) is the commutator of (0,0,1) and (0,1,0). The Baer sum of extensions corresponds to addition of the parameters c as integers. So I understand the group of extensions. Of course I understood it before I calculated it, since it is the second cohomology group of the torus. That is why I tried the algebraic calculation. But is there a natural way to think about each group E_c, for non-zero values of c? Do these groups appear in any other natural way? thanks, colin