Character 1: Dimension: 1 Multiplicity: 1, i.e. the following SALC is an eigen vector of the Hamiltonian $H$: $$ b_1=\tfrac1{\sqrt6}(e_1+e_2+e_3+e_4+e_5+e_6),\\ $$ Character 6: Dimension: 2 Multiplicity: 1, i.e. the following SALCs are eigen vectors for $H$: \begin{eqnarray*} b_2&=&\tfrac1{2}(-e_1+e_3-e_4+e_6),\\ b_3&=&\tfrac1{\sqrt{12}}(e_1+e_3+e_4+e_6)+\tfrac1{\sqrt3}(-e_2-e_5) \end{eqnarray*} Character 10: Dimension: 3 Multiplicity: 1 i.e. the following SALCs are eigen vectors for $H$: \begin{eqnarray*} b_4&=&\tfrac1{\sqrt2}(-e_1+e_3),\\ b_5&=&\tfrac1{\sqrt2}(-e_2+e_4),\\ b_6&=&\tfrac1{\sqrt2}(-e_3+e_6) \end{eqnarray*} Maximal number of distinct eigen values of $H$: 3. Thus $H$ is given by: $$ Hb_1=\l_1b_1,\quad Hb_2=\l_2b_2,\quad Hb_3=\l_2b_3,\quad Hb_4=\l_3b_4,\quad Hb_5=\l_3b_5,\quad Hb_6=\l_3b_6~. $$