s,5}\},\nonumber\\
v_{11}=z_1&=&\{\overline{\phi}^{1,\dots,4}\},\nonumber\\
v_{12}=z_2&=&\{\overline{\phi}^{5,\dots,8}\},\nonumber\\
v_{13}=\alpha&=&\{\overline{\psi}^{4,5},\overline{\phi}^{1,2}\},\nonumber\\
v_{14}=\beta&=&\{\overline{\psi}^{4,5}=\textstyle\frac{1}{2},
\overline{\phi}^{1,...,6}=\textstyle\frac{1}{2}\}.
\nonumber\end{aligned}$$ The basis vector [**1**]{} generates models with $SO(44)$ gauge group from the Neveu–Schwarz sector. The vector $S$ produces ${N} = 4$ space–time supersymmetry. The vectors $e_{1}$,$\dots$,$e_{6}$ break the $SO(44)$ gauge group to $SO(32) \times U(1)^6$ and preserve the ${N = 4}$ space–time supersymmetry. The $e_i$ basis vectors correspond to all the possible symmetric shifts of the six internal bosonic coordinates. The basis vectors $b_1$ and $b_2$ correspond to ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ orbifold twists and break ${N} = 4$ space–time supersymmetry to $N=1$. Additionally, they reduce the rank of gauge group by breaking the $U(1)^6$ symmetry. Combined with the projections of the basis vectors $z_{1}$ and $z_{2}$ the $SO(32)$ gauge group is reduced to $SO(10) \times U(1)^3 \times SO(8)_1\times SO(8)_2$, where $SO(10) \times U(1)^3$ and $SO(8)_1\times SO(8)_2$ correspond to the observable and hidden gauge groups, respectively. The combined projection of the basis vectors $\alpha$ and $\beta$ breaks the $SO(10)$ GUT symmetry to $SU(4) \times SU(2) \times U(1)$, where $\alpha$ is identical to the basis vector used in the classification of the Pati–Salam models, and hence breaks the $SO(10)$ symmetry to $SO(6)\times SO(4)$ and finally using the $\beta$ basis vector with fractional boundary conditions reduces the $SO(10)$ gauge symmetry to $SU(4) \times SU(2) \times U(1)$.

The String Spectrum {#analysisspec}
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The space–time vector bosons that are obtained from the Neveu–Schwarz (NS) sector and that survive the GGSO projections, defined by the basis vectors in (\[basis\]) generate the observable and hidden gauge groups given by: $$\b