A number of problems in reactor analysis require the determination of the second largest reactor eigenvalue. If one limits himself to a one-velocity description of neutron diffusion, this eigenvalue and the corresponding eigenfunction may be determined by familiar methods. When (as is almost universally the case) one must consider more than one energy group of neutrons, the neutron diffusion equations are no longer self-adjoint and the customary analysis yields information only about the eigenfunction of largest eigenvalue. In the present work the symmetry properties of reactor eigenfunctions have been applied to the calculation of the first few reactor eigenvalues. Each reactor has geometrical symmetry elements that enable one to define what is known as the symmetry group of the reactor, and the transformations of the reactor under the elements of this group enable one to determine the degeneracy and symmetry properties of the reactor eigenfunctions. After a detailed review of the necessary group theoretical fundamentals, the eigenfunctions of a reactor with a trigonal control element are investigated and the adaptation of an existing diffusion theory code to the computation of higher reactor eigenvalues discussed.