The 23-rd and 24-th homotopy groups of the n-th rotation group

We denote by ${\pi }_k(R_n)$ the $k$-th homotopy group of the $n$-th rotation group $R_n$ and ${\pi }_k(R_n:2)$ the 2-primary components of it. We determine the group structures of ${\pi }_k(R_n:2)$ for $k=23$ and $24$ by use of the fibration $R_{n+1}{S}^n$. The method is based on Toda’s composition methods.