  In this paper, we extend the notion of diskcyclicity and disk transitivity of
a single operator to a subset of $\mathcal{B}(X)$. We establish a diskcyclicity
criterion and we give the relationship between this criterion and the
diskcyclicity. As applications, we study the diskcyclicty of $C_0$-semigroups
and $C$-regularized groups of operators. We show that a diskcyclic
$C_0$-semigroup exists on a complex topological vector space $X$ if and only if
dim$(X)=1$ or dim$(X)=\infty$ and we prove that diskcyclicity and disk
transitivity of a $C_0$-semigroups and $C$-regularized groups are equivalent.
