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!



Conjugacy classes of the Ree groups
To amplify Geoff's brief comment, the most standard source for details about (most of) the ordinary characters of a Ree group of type $G\_2$ (specified by an odd power of $3$ at least $27$ which we call $q$ here but is sometimes written $q^2$) is the old paper by H.N. Ward based on his thesis. This is now freely available online
[here](http://www.ams.org/journals/tran/1966-121-01/S0002-9947-1966-0197587-8/home.html?pagingLink=%3Ca+href%3D%22%2Fepubsearch%2Fservlet%2FPubSearch%3Fco1%3Dand%26co2%3Dand%26co3%3Dand%26endmo%3D00%26f1%3Dmsc%26f2%3Dtitle%26f3%3Danywhere%26f4%3Dauthor%26pubname%3Done%26sendit22%3DSearch%26sperpage%3D30%26ssort%3Dd%26startmo%3D00%26v4%3Dward%26onepub%3Dtran%26startRec%3D1%22%3E).


Note that Ward leaves aside the more complicated Ree groups of type $F\_4$ in characteristic 2. Instead he starts with certain known properties of the $G\_2$ Ree groups (which define for him a group of "Ree type") and then deduces a great deal about the group including its simplicity.


From Ward's organization of the ordinary irreducible characters you can obtain the number of these (which I guess is $q+8$), equal to the number of conjugacy classes. On the other hand, Steinberg showed that the number of semisimple ($3$-regular) classes is just $q$. This correlates with the fact that the $3$-modular irreducible representations are obtained by restricting certain irreducible rational representations of the ambient algebraic group, whose highest weights are easily specified. (His original results are in *Nagoya J. Math.* in 1963, but are also included in his 1967-68 Yale lectures 
[here](http://www.math.ucla.edu/~rst/).)


While Steinberg's method is probably optimal, I'm not sure whether there are better methods by now to enumerate all conjugacy classes than the somewhat indirect ones used by Ward. (At any rate, it would be interesting to derive the ordinary characters systematically using the later methods of Deligne-Lusztig.)


