for n = 1, 2, 3, 4, or 6. Cn and Dn for n = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups, and the four groups with n = 1 and 2, give also rise to 7 frieze groups. For each of the wallpaper groups p1, p2, p3, p4, p6, the image under p of all isometry groups (i.e. the "projections" onto E(2) / T or O(2) ) are all equal to the corresponding Cn; also two frieze groups correspond to C1 and C2. The isometry groups of p6m are each mapped to one of the point groups of type D6. For the other 11 wallpaper groups, each isometry group is mapped to one of the point groups of the types D1, D2, D3, or D4. Also five frieze groups correspond to D1 and D2. For a given hexagonal translation lattice there are two different groups D3, giving rise to P31m and p3m1. For each of the types D1, D2, and D4 the distinction between the 3, 4, and 2 wallpaper groups, respectively, is determined by the translation vector associated with each reflection in the group: since isometries are in the same coset regardless of translational components, a reflection and a glide reflection with the same mirror are in the same coset. Thus, isometry groups of e.g. type p4m and p4g are both mapped to point groups of type D4. For a given isometry group, the conjugates of a translation in the group by the elements of the group generate a translation group (a lattice)—that is a subgroup of the isometry group that only depends on the translation we started with, and the point group associated with the isometry group. This is because the conjugate of the translation by a glide reflection is the same as by the corresponding reflection: the translation vector is reflected. If the isometry group contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. If, in the case of a discrete isometry group containing a translation, we apply this for a translation of minimum length, then, considering the vector difference of translations in two adjacent directions, it follows that n ≤ 6, and for odd n that 2n ≤ 6, hence n = 1, 2, 3,