With reference to Exercise 14, give a similar alternative labeling for the 8 elements of D4 in Table 8.12. 

Data from Exercise 14

In Table 8.8, we used p_o, p₁ , p₂ , µ₁, µ₂ , µ₃ as the names of the 6 elements of S₃ . Some authors use the notations ∈, p, p² , ∅, p∅, p²∅ for these elements, where their ∈ is our identity p₀ , their p is our p₁, and their ∅ is our µ₁. Verify geometrically that their six expressions do give all of S₃ .

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8.12 Table Po PI P2 P3 μ| fl₂ 8₁ 8₂ Po Pi P₂ P3 || Po P₁ P2 P₁ P2 f2 P3 Po PI P2 8₁ 8₂ P₂ P3 Po 8₁ 82 P3 Po Pi H₁ 82 M2 81 PO Po 8₁ fi 8₂ fi 1₂ 8₂ P3 μ₁ μ2 8₁ 8₁ με 9 με P₁ 8₁ μ2 f| 8₂ 8₂ 8₁ H1 | P3 14₂ 3: 82 82 μ1 8₁ H2 P2 Po PI P3 P2 P3 P1 P3 Po P2 P₁ P2 Po