The Terwilliger algebras of the group association schemes of three metacyclic groups

Abstract

For any finite group $G$, the Terwilliger algebra $T\left(G\right)$ of the group association scheme satisfies the following inclusions: $T_0\left(G\right)\subseteq T\left(G\right)\subseteq \tilde{T}\left(G\right)$, where $T_0\left(G\right)$ is a specific vector space and $\tilde{T}\left(G\right)$ is the centralizer algebra of the permutation representation of $G$ induced by the action of conjugation. The group $G$ is said to be triply transitive if $T_0\left(G\right)=\tilde{T}\left(G\right)$. In this paper, we determine the dimensions of $T_0\left(G\right)$ and $\tilde{T}\left(G\right)$ for $G$ being $T_{n,k}=〈a,b\mid a^{2n}=1,a^n=b^2,ba{b}^{-1}=a^k〉$, $C_n⋊C_p$ and $C_p⋊C_n$, and show that $T_{n,k},C_n⋊C_2$ and $C_3⋊C_{2n}$ are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of $T_{n,k}$, $C_n⋊C_p$ and $C_p⋊C_n$ when they are triply transitive.