How do I work with the character tables of Weyl groups in Sage to compute restrictions to parabolic subgroups?

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The question is essentially what is in the title. To be more concrete, let's start with the Weyl group of type $E_6$. This contains a parabolic subgroup of type $D_5$. I know how to look at the character tables of these groups using sage: for instance

W=WeylGroup(["E",6]); ctE6=W.character_table(); ctE6

and I can do the same thing with $D_5$, of course. Also, I can realize $D_5$ as the subgroup of $W$ generated by five of the simple reflections. The problem is that I don't know how to get Sage to tell me which elements are in the conjugacy class (for example) labelled 6b in the E6 table---all I know about this class is that it consists of elements of order 6. For my purposes, I really need to know which elements these are, in matrix form, so that I can restrict the characters to $D_5$ and expand the result there in terms of irreducible $D_5$ characters.