For positive integers d and n, the vector space Hd(x1, x2, … , xn) of homogeneous polynomials of degree d is a representation of the symmetric group Sn acting by permutation of variables. Regarding this as a representation for the dihedral subgroup Dn, we prove a formula for the dimension of all the isotypical subrepresentations. Our formula is simpler than the existing one found by Zamani and Babaei (Bull. Iranian Math. Soc.40(4) (2014) 863–874). By varying the degrees d we compute the generating functions for these dimensions. Further, our formula leads us naturally to a specific supercharacter theory of Dn. It turns out to be a ∗ -product of a specific supercharacter theory studied in depth by Fowler et al. (The Ramanujan Journal (2014)), with the unique supercharacter theory of a group of order 2. © 2020, Indian Academy of Sciences.