Examples of non-quasicommutative semigroups‎ ‎decomposed into unions of groups

Decomposability of an algebraic structure into a union of its‎ ‎sub-structures have been studied by many authors for groups‎, ‎rings‎ ‎and non-group semigroups since 1926‎. ‎A sub-class of non-group‎ ‎semigroups is the well known quasicommutative semigroups where it‎ ‎is known that a regular quasicommutative semigroup is decomposable‎ ‎into a union of groups‎. ‎The converse of this result is a natural‎ ‎question‎. ‎Obviously‎, ‎if a semigroup $S$ is decomposable into a‎ ‎union of groups then $S$ is regular so‎, ‎the aim of this paper is‎ ‎to give examples of non-quasicommutative semigroups which are‎ ‎decomposable into the disjoint unions of groups‎. ‎Our examples are‎ ‎two infinite classes of finite semigroups‎.