How to find presentation of finitely presented monoid on Gap with given conditions

Question

Let G be monoid generated by x,y,z,e with given relations :

relations := [ [x^245, e], [y^245, e], [z^245, e], [x^28, y^19*z^9],
[y^31, x^15*z^17], [z^42, x^12*y^7], [x*y, y*x], [x*z, z*x], 
[y*z, z*y]]; 

How to get the elements of G on the format x^a*y^b*z^c, such that

1. x<28,y<31,z<42 and
2. if x^a*y^b*z^c is element of the set also x^(a-1)*y^b*z^c, x*^a*y^(b-1)*z^c and x^a*y^b*z^(c-1) must be elements of that set.

I tried to solve considering first conditon but do not know how to define on the code the second condition

gap> F := FreeSemigroup("id","x","y","z");
<free semigroup on the generators [ id, x, y, z ]>
gap> id := F.1;; x := F.2;; y := F.3;; z := F.4;;
gap> rels:= [ [x^245, e], [y^245, e], [z^245, e], [x^28, y^19*z^9], [y^31, x^15*z^17], [z^42, x^12*y^7], [x*y, y*x], [x*z, z*x], [y*z, z*y]];;
gap> S := F/rels;
<fp semigroup on the generators [ id, x, y, z ]>
gap> x := S.2; y := S.3; z := S.4;;
x
y
gap> List(Cartesian([1..27],[1..30],[1..41]),t->x^t[1]*y^t[2]*z^t[3]);;
gap> elms := List(Cartesian([1..27],[1..30],[1..41]),t->x^t[1]*y^t[2]*z^t[3]);;
gap> elms := Concatenation(elms,List(Cartesian([1..30],[1..41]),t->y^t[1]*z^t[2]));;
gap> elms := Concatenation(elms,List(Cartesian([1..27],[1..41]),t->x^t[1]*z^t[2]));;
gap> elms := Concatenation(elms,List(Cartesian([1..27],[1..30]),t->x^t[1]*y^t[2]));;
gap> cls := EquivalenceClasses(elms,\=);;
gap> Length(cls);
245

Answer 1

You have a group (every generator is invertible) that is abelian (generators commute).

Considering the relator matrix $$ \begin{pmatrix}% 245&0&0\\% 0&245&0\\% 0&0&245\\% 28&-19&-9\\% -15&31&-17\\% -12&-7&42\\% \end{pmatrix} $$ we get (using t:=SmithNormalFormIntegerMatTransforms(m)) a Smith normal form t.normal of $$ \begin{pmatrix}% 1&0&0\\% 0&1&0\\% 0&0&245\\% 0&0&0\\% 0&0&0\\% 0&0&0\\% \end{pmatrix} $$ with column transformations t.coltrans $$ \begin{pmatrix}% 1&0&-91\\% 0&1&-83\\% 0&0&1\\% \end{pmatrix} $$ This shows that the group is cyclic, generated by $z$, and that $x=z^{-91}$ and $y=z^{-83}$.