Identifying conjugacy classes in GAP

Question

I would like to know which column of the character table of $SL(2,\mathbb{Z}/q\mathbb{Z})$ that GAP produces with the command Display(CharacterTable(SL(2,q))); corresponds to the conjugacy class of the element $\left(\begin{smallmatrix}1 & 1 \\ 0 & 1\end{smallmatrix}\right)$.

Can someone tell me the GAP command to achieve this?

I've seen the function IdentificationOfConjugacyClasses, which gives the bijection between conjugacy classes of a group and conjugacy class as stored in a character table, but I can't work out even how to ask which conjugacy class of a group contains a given element.

Answer 1

It seems that if we construct the character table using CharacterTable(SL(2,q)); then IdentificationOfConjugacyClasses is always just the identity. Thus

Position(ConjugacyClasses(SL(2,q)),ConjugacyClass(SL(2,q),[[Z(p)^0,Z(p)^0],[0*Z(p),Z(p)^0]]));

is all that's needed here. The GAP manual tells us that 0*Z(p) is the additive identity, and Z(p)^0 is the multiplicative identity.

(I'm a bit confused by why we're meant to write 0*Z(p) and Z(p)^0 rather than 0*Z(q) and Z(q)^0, but it seems to work.)

Answer 2

In a case like this, where you actually compute the character table from the group, it is easiest to keep the group as an object and use its conjugacy classes to identify in which class an element lies. Taking your example:

gap> q:=11;;g:=SL(2,q); 
SL(2,11)
gap> c:=CharacterTable(g);
CharacterTable( SL(2,11) )

Now take the list of conjugacy classes, create the element, and test in which class the element lies, using IdentificationOfConjugacyClasses to deal with any potential reordering.

gap> cl:=ConjugacyClasses(g);;  # double semicolon: Do not print the list
gap> id:=IdentificationOfConjugacyClasses(c);
[ 1 .. 15 ]
gap> elm:=[[1,1],[0,1]]*One(GF(q));
[ [ Z(11)^0, Z(11)^0 ], [ 0*Z(11), Z(11)^0 ] ]
gap> pos:=id[PositionProperty(cl,x->elm in x)];
6
gap> ClassNames(c)[pos];
"11b"

So it is (in this case -- the arrangement and naming of classes might differ between different runs of GAP or if the group is constructed a second time) class number 6, named 11B.