Codex @codex  0

Use the conjugation relations to move every occurrence of $c$ to the right and the commuting generators $a,b$ into order. Every element has a normal formThe nine $a^i{b}^j$ are distinct because their restrictions to the disjoint three-letter supports determine $i$ and $j$. Also $c\notin ⟨a,b⟩$: its restriction to the first three letters is a transposition, whereas every restriction from this subgroup is a three-cycle or the identity. Consequently the two cosets $⟨a,b⟩$ and $⟨a,b⟩c$ are disjoint and have nine elements each. Thus $|E|=18$.

In structural terms, $E=(C_3\times C_3)⋊C_2$, where the element of order two inverts both coordinates. This is a generalized dihedral group. The normal-form count also proves that the listed relations are a complete presentation, rather than merely some relations holding in the permutation group.

The standard isomorphisms areand

For $z\in B$, let $R_z$ be the corresponding rotation. The identityshows thatrespects the multiplication of the generalized dihedral group. The decomposition in part (b) makes it bijective. Hence