The Hecke algebra of $G(r,1,n)$, also known as Ariki-Koike algebra, is isomorphic to a cyclotomic quiver Hecke algebra $\mathrm{R}$, whose underlying quiver is given by copies of cyclic quivers. Using a particular automorphism of the quiver and restricting the previous isomorphism to the Hecke algebra of $G(r,p,n)$, we find a presentation of the corresponding subalgebra of $\mathrm{R}$. The presentation we get looks like the presentation of $\mathrm{R}$.