Miyakawa has proved that the maximal set $Pol\big(\begin{smallmatrix}0&1&2&1&0&1&2\\0&1&2&0&1&2&1\end{smallmatrix}\big)$ of functions of $P_3$ is divided into 54 classes of functions. In this paper we show that each of these classes contains a symmetric function. Thus classes of bases and $S$-bases of the set coincide.