On Tue, 23 Sep 2003, John Conway wrote:
[I "learned" the groups of order 16 by heart long ago, though now I may be a bit shaky. The standard reference for them is a table in the back of Coxeter and Moser, but they describe them in ways that don't tell you the structure, so it's better to use ATLAS-type notations that do.]
Since writing that a few minutes ago, I've reconstructed them. They are the abelian ones: 16, 8x2, 4x4, 4x2^2, 2^4, the dihedral and quaternionic ones D16, Q16, 2xD8, 2xQ8, 2D8=2Q8, the split extensions 8 :^3 2, 8 :^5 2, 4 :^3 4, and the last group (4,4|2,2). In general D2n = < a,b | 1 = a^n = b^2, a^b = a^-1 >, Q4n = < a,b | 1 = a^2n, b^2 = a^n, a^b = a^-1 >, while 2D4n = 2Q4n is obtained from either of these by adjoining a new central square root of a^n. Finally m :^k n = < a,b | 1 = a^m = b^n, a^b = a^k > and (m,n|p,q) = < a,b | 1 = a^m = b^n = (ab)^p = (ab^-1)^q >. [I'm a little bit scared that maybe I've missed one out and two of the above are isomorphic, but I don't think so.] JHC