  The Cayley table representation of a group uses $\mathcal{O}(n^2)$ words for
a group of order $n$ and answers multiplication queries in time
$\mathcal{O}(1)$. It is interesting to ask if there is a $o(n^2)$ space
representation of groups that still has $\mathcal{O}(1)$ query-time. We show
that for any $\delta$, $\frac{1}{\log n} \le \delta \le 1$, there is an
$\mathcal{O}(\frac{n^{1 +\delta}}{\delta})$ space representation for groups of
order $n$ with $\mathcal{O}(\frac{1}{\delta})$ query-time.
  We also show that for Z-groups, simple groups and several group classes
defined in terms of semidirect product, there are linear space representations
with at most logarithmic query-time.
  Farzan and Munro (ISSAC'06) defined a model for group representation and gave
a succinct data structure for abelian groups with constant query-time. They
asked if their result can be extended to categorically larger group classes. We
construct data structures in their model for Hamiltonian groups and some other
classes of groups with constant query-time.
