On spectrum of medial $T_2$-quasigroups

There exist medial $T_2$-quasigroups of any order of the form
$$ 2^{k_1}3^{k_2}5^{k_3}11^{k_4}17^{k_5}23^{k_6}53^{k_7}59^{k_8}83^{k_9}101^{k_{10}}p_1^{\alpha_1}p_2^{\alpha_2}\dots p_m^{\alpha_m}, $$
where $k_1\geq2$, $k_2,\dots,k_{10}\geq1$, $p_i$ are prime numbers of the form $6t+1$, $\alpha_i \in\mathbb N$, $i\in\{1,\dots,m\}$. Some other results on $T_2$-quasigroups are given.