This discrete group has three inequivalent one-dimensional representations and one three-dimensional irreducible representation, 3 ; being the Clebsch-Gordan series of their non-trivial products In the basis where S is diagonal and the decomposition of 3$_{x}$ × 3$_{y}$ reads with 3$_{x}$ = ( x$_{1}$, x$_{2}$, x$_{3}$ ) and 3$_{y}$ = ( y$_{1}$, y$_{2}$, y$_{3}$ ). The ten SO (5) generators can be written in the fundamental representation ( 5 ) 2. SO (5) generators in the fundamental representation 1 : S = 1 , T = 1 , 1 ′ : S = 1 , T = e i 2 π/ 3 = ω, 1 ′′ : S = 1 , T = e i 4 π/ 3 = ω $^{2}$, (A2) 1 ′ × 1 ′ = 1 ′′ , 1 ′ × 1 ′′ = 1 , 1 ′′ × 1 ′′ = 1 ′ , 1 ′ × 3 = 3 , 1 ′′ × 3 = 3 , 3$_{x}$ × 3$_{y}$ = 3$_{1}$ + 3$_{2}$ + 1 + 1 ′ + 1 ′′ . (A3) S =    1 0 0 0 − 1 0 0 0 − 1    , T =    0 1 0 0 0 1 1 0 0    , (A4) 1 = x$_{1}$y$_{1}$ + x$_{2}$y$_{2}$ + x$_{3}$y$_{3}$, 1 ′ = x$_{1}$y$_{1}$ + ω $^{2}$x$_{2}$ y$_{2}$ + ωx$_{3}$y$_{3}$, 1 ′′ = x$_{1}$y$_{1}$ + ωx$_{2}$y$_{2}$ + ω $^{2}$x$_{3}$ y$_{3}$, (A5) 3$_{1}$ = ( x$_{2}$y$_{3}$, x$_{3}$y$_{1}$, x$_{1}$y$_{2}$ ) , 3$_{2}$ = ( x$_{3}$y$_{2}$, x$_{1}$y$_{3}$, x$_{2}$y$_{1}$ ) , T a L,ij = − i 2 [ 1 2 ϵ abc ( δ b $_{i}$δ c j − δ b $_{j}$δ c i ) + ( δ a i δ 4 j − δ a $_{j}$δ 4 i ) ] , a = 1 , 2 , 3 , T a R,ij = − i 2 1 2 ϵ abc δ b $_{i}$δ c j − δ b $_{j}$δ c i − δ a i δ 4 j − δ a $_{j}$δ 4 i , a = 1 , 2 , 3 , (A6) T ˆ a C,ij = − i √$_{2}$ [ δ ˆ a i δ 5 j − δ ˆ a $_{j}$δ 5 i ] , ˆ a = 1 , 2 , 3 , 4 . 32