How many involutions are there in $O\_7(11)$ and $PSp\_6(11)$ respectively? (Note that the sizes of the two groups mentioned here are the same.)



On the number of involutions in some groups
Let $q$ be an odd prime power and say that $q\equiv\epsilon\pmod4$, where $\epsilon=\pm1$. 


Using Geoff's suggestion I calculated the number of involutions in $\Omega\_7(q)$ (the simple group; called $O\_7(q)$ in the Atlas) to be
$$
\frac12 q^5(q^4+q^2+1)(q+\epsilon) + \frac12 q^6(q^4+q^2+1)(q^2+1) + \frac12 q^3(q^3+\epsilon).$$


The summands give the cardinalities of the three conjugacy classes. 


In $PSp\_6(q)$, the number of involutions is
$$q^4(q^4+q^2+1)+\frac12 q^6(q^3+\epsilon)(q^2+1)(q+\epsilon),$$
there being just two conjugacy classes. 


Given the centralizer information I cited in my comment above, it is straightforward but laborious to work this out, or to work out $\Omega\_{2n+1}(q)$ and $PSp\_{2n}(q)$ for any $n>2$. One can easily trip up getting the right power of $2$ in each summand. 


I did check the answers against the Atlas for $q=3$. 


