Abstract In this article, we study rational representations of groups of order p 5 p^5, where 𝑝 is an odd prime. For a 𝑝-group 𝐺 and an irreducible complex character 𝜒 of 𝐺, the construction of an irreducible rational matrix representation of 𝐺 affording the character Ω (χ) () is equivalent to determining a pair (H, ψ) (H, ), with 𝐻 a subgroup of 𝐺 and 𝜓 a linear character of 𝐻 such that ψ G = χ ^G= and Q (ψ) = Q (χ) Q () =Q (), where Ω (χ) = m Q (χ) ∑ σ ∈ Gal (Q (χ) / Q) χ σ () =mₐ () (ₐ () /ₐ) ^ and m Q (χ) mₐ () denotes the Schur index of 𝜒 over ℚ. For each inequivalent irreducib
Choudhary et al. (Mon,) studied this question.