Extensions of some $2$-groups

Abstract

Let $H$ be a $2$-group with faithful irreducible characters all which are algebraically conjugate to each other, and $\phi$ be any faithful irreducible character of $H$. We are interested in $2$-group $G$ with the normal subgroup $H$ such that induced character $\phi^G$ is irreducible. For example, for $2$-groups $H$ that are the cyclic groups, the dihedral groups $D_n$ and the generalized quaternion groups $Q_n$, all of such $2$-groups $G$ was determined ([3]-[5]). In paticular, we showed that such a $2$-group $G$ for $H=D_n$ or $Q_n$ is uniquely determined. Let $G_t(D_n)$ and $G_t(Q_n)$ be those $2$-groups, respectively. The purpose of this paper is to determine all $2$-groups $G$ for $H=G_t(D_n)$ and $G_t(Q_n)$ and faithful irreducible characters $\phi$ of $H$. In this paper we determine the character tables of $G_t(D_n)$ and $G_t(Q_n)$ in order to show that these groups have faithful irreducible characters all which are algebraically conjugate to each other. As result it is shown that these $2$-groups have identical character tables.