= Solution
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 form
$$
\boxed{a^i b^j c^e,\qquad0\le i,j<3,\quad e\in\{0,1\}}.
$$
The nine $a^i b^j$ are distinct because their restrictions to the disjoint three-letter supports determine $i$ and $j$. Also $c\notin\langle a,b\rangle$: 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 $\langle a,b\rangle$ and $\langle a,b\rangle c$ are disjoint and have nine elements each. Thus $|E|=18$.
In structural terms, $E=(C_3\times C_3)\rtimes C_2$, 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>.
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