On certain 2-extensions of $\mathbb{Q}$ unramified at 2 and $\infty$

Abstract

Based on the method of Boston and Leedham-Green et al. for computing the Galois groups of tamely ramified $p$-extensions of number fields, this paper gives a large family of triples of odd prime numbers such that the maximal totally real $2$-extension of the rationals unramified outside the three prime numbers has the Galois group of order $512$ and derived length $3$. This family is characterized arithmetically, and the explicit presentation of the Galois group by generators and relations is also determined completely.