$
\def\MA{{\frac{1}{2}}}
\def\MB{{\frac{1}{2 \sqrt{3}}}}
\def\MC{{\frac{\sqrt{2}}{\sqrt{3}}}}
\def\MD{{\frac{5}{6}}}
\def\ME{{\frac{\sqrt{2}}{3}}}
\def\MF{{\frac{1}{3}}}
\def\MG{{\frac{\sqrt{3}}{2}}}
\def\MH{{\frac{2\sqrt{2}}{3}}}
\def\MI{{\frac{1}{\sqrt{3}}}}
\def\MJ{{\frac{2}{3}}}
\def\MK{{\frac{1}{6}}}
$
Tetrahedron
Initial vertex: $V_1=\left[\begin{matrix}0\\0\\1\end{matrix}\right]$
Transforms for vertex generation:
$\tilde{T}_i \in \left\{
\left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right],
\left[\begin{matrix}-\MA&-\MB&-\MC\\ \MB&\MD&-\ME\\ \MC&-\ME&-\MF\end{matrix}\right],
\left[\begin{matrix}-\MA&-\MG&0\\ \MG&-\MA&0\\0&0&1\end{matrix}\right]
\right\}$
Vertexes:
$T_2 V_1=\left[\begin{matrix}-\MC\\-\ME\\-\MF\end{matrix}\right]=V_2$
$T_2 V_2=\left[\begin{matrix}\MC\\-\ME\\-\MF\end{matrix}\right]=V_3$
$T_3 V_3=\left[\begin{matrix}0\\ \MH\\-\MF\end{matrix}\right]=V_4$
All Transforms:
$T_2 T_2=\left[\begin{matrix}-\MA&\MB&\MC\\-\MB&\MD&-\ME\\-\MC&-\ME&-\MF\end{matrix}\right]=T_4$
$T_3 T_2=\left[\begin{matrix}0&-\MI&\MC\\-\MI&-\MJ&-\ME\\ \MC&-\ME&-\MF\end{matrix}\right]=T_5$
$T_3 T_4=\left[\begin{matrix}\MA&-\MG&0\\-\MB&-\MK&\MH\\-\MC&-\ME&-\MF\end{matrix}\right]=T_6$
$T_2 T_5=\left[\begin{matrix}-\MA&\MG&0\\-\MG&-\MA&0\\0&0&1\end{matrix}\right]=T_7$
$T_3 T_5=\left[\begin{matrix}\MA&\MG&0\\ \MB&-\MK&\MH\\ \MC&-\ME&-\MF\end{matrix}\right]=T_8$
$T_3 T_6=\left[\begin{matrix}0&\MI&-\MC\\ \MI&-\MJ&-\ME\\-\MC&-\ME&-\MF\end{matrix}\right]=T_9$
$T_2 T_7=\left[\begin{matrix}\MA&-\MB&-\MC\\-\MG&-\MK&-\ME\\0&\MH&-\MF\end{matrix}\right]=T_{10}$
$T_2 T_8=\left[\begin{matrix}-{1}&0&0\\0&\MF&\MH\\0&\MH&-\MF\end{matrix}\right]=T_{11}$
$T_2 T_9=\left[\begin{matrix}\MA&\MB&\MC\\ \MG&-\MK&-\ME\\0&\MH&-\MF\end{matrix}\right]=T_{12}$
relabeled vertexes as {1, 4, 3, 2}
Vertexes as column vectors:
$V=\left[\begin{matrix}0&0&\MC&-\MC\\0&\MH&-\ME&-\ME\\1&-\MF&-\MF&-\MF\end{matrix}\right]$
Vertex inner products:
$V^T V=\left[\begin{matrix}0&0&1\\0&\MH&-\MF\\ \MC&-\ME&-\MF\\-\MC&-\ME&-\MF\end{matrix}\right]
\left[\begin{matrix}0&0&\MC&-\MC\\0&\MH&-\ME&-\ME\\1&-\MF&-\MF&-\MF\end{matrix}\right]
= \left[\begin{matrix}1&-\MF&-\MF&-\MF\\-\MF&1&-\MF&-\MF\\-\MF&-\MF&1&-\MF\\-\MF&-\MF&-\MF&1\end{matrix}\right]$
Table of $T_i\cdot V_j=V_k$:
V1 V2 V3 V4
T1 V1 V2 V3 V4
T2 V2 V3 V1 V4
T3 V1 V3 V4 V2
T4 V3 V1 V2 V4
T5 V3 V4 V1 V2
T6 V4 V1 V3 V2
T7 V1 V4 V2 V3
T8 V4 V2 V1 V3
T9 V2 V1 V4 V3
T10 V2 V4 V3 V1
T11 V4 V3 V2 V1
T12 V3 V2 V4 V1
Table of $T_i\cdot T_j=T_k$:
T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
T1 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
T2 T2 T4 T9 T1 T7 T8 T10 T11 T12 T5 T6 T3
T3 T3 T5 T7 T6 T8 T9 T1 T2 T4 T12 T10 T11
T4 T4 T1 T12 T2 T10 T11 T5 T6 T3 T7 T8 T9
T5 T5 T6 T4 T3 T1 T2 T12 T10 T11 T8 T9 T7
T6 T6 T3 T11 T5 T12 T10 T8 T9 T7 T1 T2 T4
T7 T7 T8 T1 T9 T2 T4 T3 T5 T6 T11 T12 T10
T8 T8 T9 T6 T7 T3 T5 T11 T12 T10 T2 T4 T1
T9 T9 T7 T10 T8 T11 T12 T2 T4 T1 T3 T5 T6
T10 T10 T11 T2 T12 T4 T1 T9 T7 T8 T6 T3 T5
T11 T11 T12 T8 T10 T9 T7 T6 T3 T5 T4 T1 T2
T12 T12 T10 T5 T11 T6 T3 T4 T1 T2 T9 T7 T8