= Solution
Write the generators additively in the <abelianization>. The relators give
$$
x=2a,\qquad y=3b,\qquad c=-a-b,
$$
and
$$
7c+x+y=0.
$$
After substitution, the last relation becomes
$$
5a+4b=0.
$$
The commutator relators disappear automatically, so
$$
G^{\mathrm{ab}}\cong\mathbb Z^2/\langle(5,4)\rangle\cong\mathbb Z,
$$
because $\gcd(5,4)=1$. Explicitly, an isomorphism to $\mathbb Z$ sends
$$
a\longmapsto4,\quad b\longmapsto-5,\quad
c\longmapsto1,\quad x\longmapsto8,\quad y\longmapsto-15.
$$
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