The question is very small - I am trying to understand an example of a group, but its presentation looks incomplete.
In a paper, Charles Ford (1970) mentions following example of a group.
Let $G$ be a group with following relations: $$ A_1^7=1, \,\,\,\, A_2^{13}=1, \,\,\,\, A_1A_2=A_2A_1,\\ X_1^3=Z, \,\,\,\, X_2^3=Z, \,\,\,\, Z^9=1, \,\,\,\, Z \mbox{ is central in } G\\ X_1^{-1}A_1X_1=A_1^2, \,\,\,\, X_2^{-1}A_2X_2=A_2^3 , \\ A_1X_2=X_2A_1, \,\,\,\, A_2X_1=X_1A_2. $$
My question is very simple: are there no relation between $X_1$ and $X_2$?
I think, it should be mentioned, whether they commute or not. For example, if we mention $[X_1,X_2]=Z$, then we get one group, and if we had mentioned $[X_1,X_2]=1$, then the group will be different (non-isomorphic) from previous.
If anyone has came across this paper or work of Ford, can one tell what relation between $X_1$ and $X_2$ should be taken?
The author tries to give example of a (finite) group with some non-trivial Schur index of a representation.