Abstract: We are principally interested on determinant calculation and on eigenvalues with multiplicities determination of the regular representation of group algebra element. We first briefly review the representation of group algebra and its connection with arrangements of hyperplanes. After that, we expose the determinant calculations of elements of the group algebra of the symmetric group and of the hyperoctahedral group. We also expose the eigenvalues with multiplicities determination of an element of the group algebra of the symmetric group. Then, we finish with the presentation of an open problem which is the eigenvalues with multiplicities determination of an element of the group algebra of the hyperoctahedral group.