  The modular group $\operatorname{PSL}_2(\mathbb{Z})$ acts on the upper-half
plane $\mathbb{HP}$ with quotient the modular orbifold, uniformized by the
function $\mathfrak{j} \colon \mathbb{HP}\to \mathbb{C}$. We first show that
second derived subgroup $\operatorname{PSL}_2(\mathbb{Z})''$ corresponds to a
$\mathbb{Z}^2\rtimes \mathbb{Z}/6$ Galois cover of the modular orbifold by a
hexpunctured plane, uniformized by the hexponential map $\operatorname{hexp}
\colon \mathbb{HP} \to \mathbb{C} \setminus (\omega_0\mathbb{Z}[j])$, which is
a primitive of $C\eta^4$ where $\omega_0\in i\mathbb{R}$ and $C\in \mathbb{R}$
are explicit constants and $\eta$ is Dedekind eta function. We describe the
values of the cusp-compactification $\partial \operatorname{hexp}\colon
\mathbb{QP}^1\to \omega_0 \mathbb{Z}[j]$. After defining the
radial-compactification $\operatorname{Shexp} \colon \mathscr{R} \to
\mathbb{R}/(2\pi\mathbb{Z})$, we construct a simple section
$\operatorname{InSh} \colon \mathbb{R}/(2\pi\mathbb{Z}) \to \mathscr{S}
\bmod{\operatorname{PSL}_2(\mathbb{Z})'}$ where $\mathscr{S} \subset
\mathbb{RP}^1$ is a set of numbers whose continued fraction expansions arise
from Sturmian sequences, which contains the set $\mathscr{M}$ of Markov
quadratic irrationals as those numbers arising from periodic Sturmian
sequences. We will show that the values of $\operatorname{InSh}$ are either
Markov quadratic irrationals or transcendental. Finally we provide a continued
fraction expansion for $\operatorname{hexp}$, and discuss its monodromy.
