Simply transitive optimal ball packings for the orientable crystallographic groups of the cubic system

In this paper we look for densest ball packings of Euclidean space E3to given symmetry groups. We restrict our investigation to the 13 orientable (orientation preserving) crystallographic groups of cubic system, and we search for only those packings where the group acts simply transitively on the balls. In order to find the centre of a ball and its radius we will apply an algorithm and the corresponding computer program, which was developed by the second author. In the list of our results we will give the coordinates of the ball centre and the radius, moreover, we will compute the density of the optimal packing and display the corresponding D-V cell for each space group above.