Bertoni et al.  introduced in Lect. Notes Comput. Sci.2710 (2003) 1–20 a new model of 1-way quantum finite automaton (1qfa) called 1qfa with control language (1qfc). This model, whose recognizing power is exactly the class of regular languages, generalizes main models of 1qfa's proposed in the literature. Here, we investigate some properties of 1qfc's. In particular, we provide algorithms for constructing 1qfc's accepting the inverse homomorphic images and quotients of languages accepted by 1qfc's. Moreover, we give instances of binary regular languages on which 1qfc's are proved to be more succinct (i.e. , to have less states) than the corresponding classical (deterministic) automata.