## Not all the semigroups <PGL(2, 7), a>\PGL(2, 7) are idempotent generated.
##
## GAP log file

gap> n:=8;; G:=PrimitiveGroup(8,5);; a:=Transformation( [ 6, 2, 3, 4, 6, 6, 6, 6 ] );;
gap> gens:=List(GeneratorsOfGroup(G), x-> AsTransformation(x, n));;
gap> S:=Semigroup(Concatenation(gens, [a]));;
gap> Size(S);
108984
gap> I:=Idempotents(S, 4);;
gap> T:=Semigroup(I);
<semigroup with 112 generators>
gap> Size(T)=Size(S)-Size(G);
false