Minimal volume product of three dimensional convex bodies invariant under certain groups of order four

We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under two kinds of discrete subgroups of $O(3)$ of order four. We also characterize the convex bodies with the minimal volume product in each case. This provides new partial results of the non-symmetric version of Mahler’s conjecture in the three dimensional case.