Extensions of Z+Z by Z

[Colin McLarty <cxm7@po.cwru.edu>]

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