We study a certain family of determinantal quintic hypersurfaces in $${\mathbb{P}^{4}}$$ P 4 whose singularities are similar to the well-studied Barth–Nieto quintic. Smooth Calabi–Yau threefolds with Hodge numbers (h 1,1,h 2,1) = (52, 2) are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi–Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi–Yau threefolds in $${\mathbb{P}^{4}\times\mathbb{P}^{4}}$$ P 4 × P 4 which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over $${\mathbb{P}^{2}}$$ P 2 and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over $${\mathbb{P}^{2}}$$ P 2 completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.