Thanks to the fibrations


\begin{align\*}
SO(n) \to SO(n+1) &\to S^n\\
SU(n) \to SU(n+1) &\to S^{2n+1}\\
Sp(n) \to Sp(n+1) &\to S^{4n+3}
\end{align\*}


we know that 


\begin{align\*}
\pi\_i(SO(n)) \cong \pi\_i(SO(n+1)) \cong \pi\_i(SO), \quad i &\leq n-2\\
\pi\_i(SU(n)) \cong \pi\_i(SU(n+1)) \cong \pi\_i(SU), \quad i &\leq 2n - 1 = (2n+1) - 2\\
\pi\_i(Sp(n)) \cong \pi\_i(Sp(n+1)) \cong \pi\_i(Sp), \quad i &\leq 4n+1 = (4n + 3) - 2.
\end{align\*}


These values of $i$ are known as the stable range. So the first unstable groups are $\pi\_{n-1}(SO(n))$, $\pi\_{2n}(SU(n))$, and $\pi\_{4n+2}(Sp(n))$ respectively.


I was able to find $\pi\_{n-1}(SO(n))$ for $1 \leq n \leq 16$ by combining the tables on [the nLab page for the orthogonal group](https://ncatlab.org/nlab/show/orthogonal+group) and appendix A, section 6, part VII of the *Encyclopedic Dictionary of Mathematics*. The groups are


$$0, \mathbb{Z}, 0, \mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}\_2, \mathbb{Z}, 0, \mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}\_2\oplus\mathbb{Z}\_2, \mathbb{Z}\oplus\mathbb{Z}\_2, \mathbb{Z}\_2, \mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}\_2, \mathbb{Z}, \mathbb{Z}\_2, \mathbb{Z}\oplus\mathbb{Z}.$$


There doesn't seem to be any pattern here, so I guess that there is no general result for $\pi\_{n-1}(SO(n))$. (*Feel free to correct me if I'm wrong*.) I just noticed that every second term contains a copy of $\mathbb{Z}$, while every fourth term contains two copies.


The case of $SU(n)$ is completely different: in *The space of loops on a Lie group*, Bott proved, among other things, that $\pi\_{2n}(SU(n)) \cong \mathbb{Z}\_{n!}$, see Theorem 5. 


Again consulting the Encyclopedic Dictionary of Mathematics, I was able to find $\pi\_{4n+2}(Sp(n))$ for $n = 1, 2, 3$. The groups are $\mathbb{Z}\_{12}$, $\mathbb{Z}\_{120}$, and $\mathbb{Z}\_{10080}$. This seems to suggest that this case is more similar to $SU(n)$ than $SO(n)$, so one might hope there is a Bott-type result.



> 
> Is there an analogue of Bott's result for $Sp(n)$? That is, is there some increasing function $f : \mathbb{N} \to \mathbb{N}$ such that $\pi\_{4n+2}(Sp(n)) \cong \mathbb{Z}\_{f(n)}$?
> 
> 
> 


OEIS has no sequences beginning $12, 120, 10080$, so I have no guess what $f(n)$ could be. It is interesting to note that $12 \mid 120$ and $120 \mid 10080$ which is another similarity with the $SU(n)$ case. 


Of course, three groups is not much to go on, so this may be a completely misguided guess. Some questions that would be nice to answer before seriously hoping for such a result are:


* Is $\pi\_{4n+2}(Sp(n))$ always cyclic?
* Is $\pi\_{4n+2}(Sp(n))$ always finite?
* Is $|\pi\_{4n+2}(Sp(n))|$ increasing in $n$?


Any information regarding these three questions would also be interesting to know.


Falling short of answering any of these questions, have any more of these groups (namely $\pi\_{18}(Sp(4)), \pi\_{22}(Sp(5)), \dots$) been computed?




---


**Update:** I added the sequence $|\pi\_{4n+2}(Sp(n))|$ to the OEIS: [A301898](https://oeis.org/A301898).


Also, the answer to the question I asked was also in the Encyclopedic Dictionary of Mathematics on page 1746.



The first unstable homotopy group of $Sp(n)$
The answer appears to be in the paper [Homotopy groups of symplectic groups](https://doi.org/10.1215/kjm/1250524819) by Mimura and Toda. They claim the calculation was already in a paper of [Harris](https://www.ams.org/journals/tran/1963-106-01/S0002-9947-1963-0143216-6/S0002-9947-1963-0143216-6.pdf), but that was stated in terms of a symmetric space and it's not immediately obvious to me how to translate into information about the groups.


They state that the group is $\mathbb Z\_{(2n+1)!}$ if $n$ is even and $\mathbb Z\_{(2n+1)! \cdot 2}$ if $n$ is odd, which agrees with your data.


