Does Dixon's algorithm for calculating character tables always order
classes the same way that "ConjugacyClasses" does?

Suppose I have a permutation representation of a group and I use "CharTable"
to calculate the character table which GAP does using Dixon's algorithm.
If I then use "ConjugacyClasses" to get (what else!) the conjugacy classes.
Is it *guaranteed* that the classes are sorted in the same order in both
cases. My guess, and my hope, is that they are. In fact I have been relying
on this fact for some calculations, is this dangerous practice?

My other question is a long shot but here it goes :

I have a matrix group generated by M1,M2,M3,and M4. <M1,M2,M3> generate
a subgroup for which I found a permutation representation <p1,p2,p3>.
<M1,M2,M4> generate another subgroup for which I can also get a
permutation reprersentation <q1,q2,q4>. The p's may or may not be of
the same degree as the q's. I can't get a permutation rep for the entire
group directly because it's a huge group, so I'm hoping to approach it
using the reps of the two subgroups. Are there any facilities in GAP that
can help? Thanks.

Jacob Hirbawi
Jcbhrb@CERF.net