  This article deals with the study of cactus groups from a combinatorial point
of view. These groups have been gaining prominence lately in various domains of
mathematics, amongst which are their relations with well-known groups such as
braid groups, diagram groups, to name a few. We compute a minimal presentation
for cactus groups in terms of generators and non-redundant relations. We also
construct homomorphisms of these groups onto certain finite groups, which leads
to results about finite quotients of cactus groups. More precisely, we prove
that all (infinite) dihedral groups appear as quotients of cactus groups. We
also investigate the lower central series and its consecutive quotients. While
there are already known established similarities with braid groups, we deduce a
considerable disparity between the two groups.
