The main objective of this work is to show how to construct a ternary $\mathbb{Z}_3$-graded Clifford algebra on two generators~[2] by using a group algebra of an extra-special $3$-group $G$ of order $27.$ The approach used is an extension of the method implemented to classify $\mathbb{Z}_2$-graded Clifford algebras as images of group algebras of Salingaros $2$-groups~[1]. We will show how non-equivalent irreducible representations of the $\mathbb{Z}_3$-graded Clifford algebra are determined by two distinct irreducible characters of~$G$ of degree~$3.$ We will comment how this approach may yield a complete classification of the ternary Clifford algebras and their irreducible representations reminiscent of the classification with Periodicity of Eight of the $\mathbb{Z}_2$-graded Clifford algebras. We will also comment on extending this approach to defining $p$-ary Clifford algebras, their irreducible representations, and classification on the basis of extra-special $p$-groups and their central products for a prime $p \geq 5.$\\ \noindent \textbf{Keywords:} $3$-group, central product, Clifford algebra, cyclic group, elementary abelian group, extra-special group, $\mathbb{Z}_3$-graded algebra, graded algebra morphism, group algebra, homogeneous ideal, irreducible character, quotient algebra, ternary Clifford algebra \begin{center} References \end{center} \small \noindent [1] R.~Ab\l amowicz, M.~Varahagiri, and A.~M.~Walley: \textit{A Classification of Clifford Algebras as Images of Group Algebras of Salingaros Vee Groups}. Adv. Appl. Clifford Algebras~\textbf{28}, 38 (2018). https://doi.org/10.1007/s00006-018-0854-y\newline %%%%%%%%%%%%%%%%%%%%%%%%%%%%% [2] P.~Cerejeiras and M.~B.~Vajiac, \textit{Ternary Clifford Algebras}, ISAAC 12, Aveiro, July 29--August 2, 2019. Topical Collection of AACA edited by S.~Bernstein, U.~Kaehler, I.~Sabadini, and F.~Sommen (submitted April 2020).