Finite group $G$ with $\exp(G)=2^{n-1}$

Question

Let $G$ be a finite non abelian group of order $2^n$ and exponent $2^{n-1}$. What can we say about $G$ ? Does $G$ isomorphic either to the Dihedral group $D_{2^n}$ or to the generalized Quaternion group $Q_{2^n}$ ?

Answer 1

There are four families of examples, which are most easily defined by presentations:

1. Dihedral groups $\langle x,y \mid x^{2^{n-1}} = y^2 = 1, y^{-1}xy=x^{-1} \rangle$ for $n \ge 3$;

2. (Generalized) quaternion groups: $\langle x,y \mid x^{2^{n-1}} = 1, y^2 = x^{2^{n-2}}, y^{-1}xy=x^{-1} \rangle$ for $n \ge 3$;

3. Semidihedral groups: $\langle x,y \mid x^{2^{n-1}}= y^2 = 1, y^{-1}xy=x^{2^{n-2}-1} \rangle$ for $n \ge 4$;

4. Sometimes called modular groups: $\langle x,y \mid x^{2^{n-1}}= y^2 = 1, y^{-1}xy=x^{2^{n-2}+1} \rangle$ for $n \ge 4$.