A supercharacter theory for Sylow $p$-subgroups of the Steinberg triality groups

We determine a supercharacter theory for the matrix Sylow $p$-subgroup ${}^3{D}_4^{syl}(q^3)$ of the Steinberg triality group ${}^3{D}_4(q^3)$, and establish the supercharacter table of ${}^3{D}_4^{syl}(q^3)$.