## Not all the semigroups <PSL(2, 8), a>\PSL(2, 8) are idempotent generated.
##
## GAP log file

gap> n:=9;; G:=PrimitiveGroup(9,8);; a:=Transformation( [ 1, 2, 3, 5, 4, 5, 4, 4, 5 ] );;
gap> gens:=List(GeneratorsOfGroup(G), x-> AsTransformation(x, n));;
gap> S:=Semigroup(Concatenation(gens, [a]));;
gap> Size(S);
4655169
gap> I:=Idempotents(S, 5);;
gap> T:=Semigroup(I);
<semigroup with 756 generators>
gap> Size(T)=Size(S)-Size(G);
false