Matching extensions of cyclic groups

In this work we show that the generalized dihedral and quaternion groups are particular cases of extension of cyclic groups. For both classes of groups, Bali and Selvakumaran found matching maps with specific APSK signal sets. Considering generalized extension of cyclic groups we find other groups matched to signal sets, and also we show that this consideration improves the Squared Euclidean Distance of the matching of the dihedral and quaternion groups.