Let $\Gamma = (\Omega,E)$ be a strongly-regular graph with adjacency matrix $A_1$, and let $A_2$ be the adjacency matrix of its complement. For any vertex $\omega\in \Omega$, we define $E_{ 0,\omega}^*$ $E_{ 1,\omega}^*$ and $E_{ 2,\omega}^*$ to be respectively the diagonal matrices whose main diagonal is the row corresponding to $\omega$ in the matrices $I, A_1$, and $A_2$. The Terwilliger algebra of $\Gamma$ with respect to the vertex $\omega\in \Omega$ is the subalgebra $T_\omega = \left\langle I,A_1,A_2,E_{ 0,\omega}^*,E_{ 1,\omega}^*,E_{ 2,\omega}^* \right\rangle$ of the complex matrix algebra $\operatorname{ M_{ |\Omega|}}(\mathbb{ C})$. The algebra $T_\omega$ contains the subspace $T_{ 0,\omega} = \operatorname{ Span}\left\{ E_{ i,\omega}^*A_jE_{ k,\omega}^*: 0\leq i,j,k\leq 2 \right\}$. In addition, if $G = \Aut{ \Gamma}$, then $T_\omega$ is a subalgebra of the centralizer algebra $\tilde{ T}_\omega = \End{ G_\omega}{ \mathbb{ C}^\Omega}$. The strongly-regular graph $\Gamma=(\Omega,E)$ is triply transitive if $\Gamma$ is vertex transitive and $T_{ 0,\omega} = T_\omega = \tilde{ T}_\omega$, for any $\omega \in \Omega$. In this paper, we classify all triply transitive strongly-regular graphs that are not isomorphic to the collinearity graph of the polar space $O_{ 6}^-(q)$, where $q$ is a prime power, or the affine polar graph $\vo_{ 2m}^\varepsilon(2)$, where $m\geq 1$ and $\varepsilon = \pm 1$.
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