TITLE: Conjugacy classes of the Ree groups
QUESTION [5 upvotes]: Dear all,
Consider the finite simple Ree groups $G=^2\hspace{-1mm}G_2(3^{2n+1})$ where $n$ is a positive integer. I would like to know the orders of conjugacy class representatives of $G$ and from there to know the number of conjugacy classes of $3$-singular elements of $G$ (an element is 3-singular if its order is divisible by 3). This is probably well-known but I couldn't find it somewhere in the literature.
Thank you in advance!

REPLY [7 votes]: I think there is enough information in Chapter III of Ward's paper to determine the classes of 3-singular elements, and the paragraph numbers below refer to that).
Let $P \in {\rm Syl}_3(G)$. Then $|P|=q^3$, with upper and lower central series of $P$  both $1 < C < P_1 < P$ (using Ward's notation), where $|C|=q$, $P_1$ is elementary abelian of order $q^2$, and $P/P_1$ is elementary abelian.
The normalizer of $P$ is a semidirect product of $P$ with a cyclic group of order $q-1$, which acts fixed-point-freely on $C$ and on $P/P_1$, but acts as a cycle of order $(q-1)/2$ on the middle layer $P_1/P$. The element of order 2 in this cyclic group has centralizer $C_2 \times L_2(q)$.
The $q-1$ elements in $C \setminus \{1\}$ form a single class. The $q^2-q$ elements of $P_1 \setminus C$ (which have order 3) split into two classes (paragraph 7). These elements centralize an element of order 2, and there are also two classes of elements of order 6. Finally, the $q^3-q^2$ elements in $P \setminus P_1$ split into three classes of elements of order 9 (paragraphs 9,10,11).