We study simple graph operations that create larger spherical latin trades from smaller ones. In particular, we analyse the effect of these operations on the canonical groups of the trades. We then apply our results to answer several open problems about canonical and minimal groups recently posed by Cavenagh and Wanless and by Blackburn and McCourt.

A family of trades is constructed that exhibits the fastest known
asymptotic growth for |A*_{W}| with respect to the size
|W|. It achieves |A*_{W}|=r^{|W|(1+o(1))}, where
r is approximately 1.329. Another family is shown to embed minimally into
groups of exponentially increasing order; previously exponential
growth had only been observed for canonical groups.

Questions about the rank (size of a smallest generating set) of canonical and minimal groups are also resolved. The rank of any minimal group is shown to be at most logarithmic in the size of the trade, resulting in an example where the canonical group grows much faster than any minimal group, with regards to both order and rank.