  In the classical setting, a convex polytope is said to be semiregular if its
facets are regular and its symmetry group is transitive on vertices. This paper
studies semiregular abstract polytopes, which have abstract regular facets,
still with combinatorial automorphism group transitive on vertices. We analyze
the structure of the automorphism group, focusing in particular on polytopes
with two kinds of regular facets occurring in an "alternating" fashion. In
particular we use group amalgamations to prove that given two compatible
n-polytopes P and Q, there exists a universal abstract semiregular
(n+1)-polytope which is obtained by "freely" assembling alternate copies of P
and Q. We also employ modular reduction techniques to construct finite
semiregular polytopes from reflection groups over finite fields.
