  A group is $\frac32$-generated if every non-trivial element is part of a
generating pair. In 2019, Donoven and Harper showed that many Thompson groups
are $\frac32$-generated and posed five questions. The first of these is whether
there exists a 2-generated group with every proper quotient cyclic that is not
$\frac32$-generated. This is a natural question given the significant work in
proving that no finite group has this property, but we show that there is such
an infinite group. The groups we consider are a family of finite index
subgroups of the Houghton group $\text{FSym}(\mathbb{Z})\rtimes\mathbb{Z}$. We
then show that the first two groups in our family are $\frac32$-generated, and
investigate the related notion of spread for these groups. We are able to show
that they have finite spread which is greater than 2. These are therefore the
first infinite groups to be shown to have finite positive spread, and the first
to be shown to have spread greater than 2 (other than $\mathbb{Z}$ and the
Tarski monsters, which have infinite spread).
