Distinguish between conjugacy class 4A and 4B (ATLAS).

Question

Is any one aware of how to distinguish between the conjugacy classes 4A and 4B in the group M$_{12}$ in GAP. Both centralizers have an order of 32 and I can't seem to find there being any difference between them?

Thank you.

Answer 1

In GAP,

gap> c:=CharacterTable("M12");
gap> AutomorphismsOfTable(c);
Group([ (14,15), (6,7)(11,12) ])
gap> ClassNames(c){[6,7,11,12,14,15]};
[ "4a", "4b", "8a", "8b", "11a", "11b" ]

returns a group of those class permutations (6/7 are classes 4A/B) that can be undone by a character permutation. (In a small case as this you could even do so by hand). Thus as far as ordinary representation theory is concerned there is no way how you can distinguish classes 4A and 4B (with the caveat that identifying one also makes a decision for classes 8A/B) and you can decide arbitrarily.

The reason for this is that the classes fuse together under an outer automorphism of $M_{12}$.

(In theory (but not here) there could be classes that are indistinguishable under ordinary representations, but become distinguished under modular representations; unless you really start looking at modular representations this level of subtlety will be of no concern.)