$ \def\MA{{\frac{1}{\sqrt{3}}}} \def\MB{{\frac{1}{3}}} $

Cube

Initial vertex: $V_1=\left[\begin{matrix}\MA\\ \MA\\ \MA\end{matrix}\right]$

Transforms for vertex generation:

$\tilde{T}_i \in \left\{ \left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right], \left[\begin{matrix}1&0&0\\0&0&-1\\0&1&0\end{matrix}\right], \left[\begin{matrix}0&0&1\\0&1&0\\-1&0&0\end{matrix}\right] \right\}$

Vertexes:

$T_2 V_1=\left[\begin{matrix}\MA\\-\MA\\ \MA\end{matrix}\right]=V_2$
$T_2 V_2=\left[\begin{matrix}\MA\\-\MA\\-\MA\end{matrix}\right]=V_3$
$T_2 V_3=\left[\begin{matrix}\MA\\ \MA\\-\MA\end{matrix}\right]=V_4$
$T_3 V_4=\left[\begin{matrix}-\MA\\ \MA\\-\MA\end{matrix}\right]=V_5$
$T_2 V_5=\left[\begin{matrix}-\MA\\ \MA\\ \MA\end{matrix}\right]=V_6$
$T_2 V_6=\left[\begin{matrix}-\MA\\-\MA\\ \MA\end{matrix}\right]=V_7$
$T_2 V_7=\left[\begin{matrix}-\MA\\-\MA\\-\MA\end{matrix}\right]=V_8$

All Transforms:

$T_2 T_2=\left[\begin{matrix}1&0&0\\0&-1&0\\0&0&-1\end{matrix}\right]=T_4$
$T_3 T_2=\left[\begin{matrix}0&1&0\\0&0&-1\\-1&0&0\end{matrix}\right]=T_5$
$T_2 T_4=\left[\begin{matrix}1&0&0\\0&0&1\\0&-1&0\end{matrix}\right]=T_6$
$T_3 T_4=\left[\begin{matrix}0&0&-1\\0&-1&0\\-1&0&0\end{matrix}\right]=T_7$
$T_2 T_5=\left[\begin{matrix}0&1&0\\1&0&0\\0&0&-1\end{matrix}\right]=T_8$
$T_3 T_5=\left[\begin{matrix}-1&0&0\\0&0&-1\\0&-1&0\end{matrix}\right]=T_9$
$T_3 T_6=\left[\begin{matrix}0&-1&0\\0&0&1\\-1&0&0\end{matrix}\right]=T_{10}$
$T_2 T_7=\left[\begin{matrix}0&0&-1\\1&0&0\\0&-1&0\end{matrix}\right]=T_{11}$
$T_3 T_7=\left[\begin{matrix}-1&0&0\\0&-1&0\\0&0&1\end{matrix}\right]=T_{12}$
$T_2 T_8=\left[\begin{matrix}0&1&0\\0&0&1\\1&0&0\end{matrix}\right]=T_{13}$
$T_2 T_9=\left[\begin{matrix}-1&0&0\\0&1&0\\0&0&-1\end{matrix}\right]=T_{14}$
$T_3 T_9=\left[\begin{matrix}0&-1&0\\0&0&-1\\1&0&0\end{matrix}\right]=T_{15}$
$T_2 T_{10}=\left[\begin{matrix}0&-1&0\\1&0&0\\0&0&1\end{matrix}\right]=T_{16}$
$T_3 T_{10}=\left[\begin{matrix}-1&0&0\\0&0&1\\0&1&0\end{matrix}\right]=T_{17}$
$T_2 T_{11}=\left[\begin{matrix}0&0&-1\\0&1&0\\1&0&0\end{matrix}\right]=T_{18}$
$T_3 T_{12}=\left[\begin{matrix}0&0&1\\0&-1&0\\1&0&0\end{matrix}\right]=T_{19}$
$T_2 T_{13}=\left[\begin{matrix}0&1&0\\-1&0&0\\0&0&1\end{matrix}\right]=T_{20}$
$T_2 T_{15}=\left[\begin{matrix}0&-1&0\\-1&0&0\\0&0&-1\end{matrix}\right]=T_{21}$
$T_3 T_{16}=\left[\begin{matrix}0&0&1\\1&0&0\\0&1&0\end{matrix}\right]=T_{22}$
$T_2 T_{18}=\left[\begin{matrix}0&0&-1\\-1&0&0\\0&1&0\end{matrix}\right]=T_{23}$
$T_2 T_{19}=\left[\begin{matrix}0&0&1\\-1&0&0\\0&-1&0\end{matrix}\right]=T_{24}$


relabeled vertexes as {1, 6, 2, 4, 7, 5, 3, 8}

Vertexes as column vectors:

$V=\left[\begin{matrix} \MA&-\MA&\MA&\MA&-\MA&-\MA&\MA&-\MA\\ \MA&\MA&-\MA&\MA&-\MA&\MA&-\MA&-\MA\\ \MA&\MA&\MA&-\MA&\MA&-\MA&-\MA&-\MA \end{matrix}\right]$

Vertex inner products:

$V^T V=\left[\begin{matrix} \MA&\MA&\MA\\ -\MA&\MA&\MA\\ \MA&-\MA&\MA\\ \MA&\MA&-\MA\\ -\MA&-\MA&\MA\\ -\MA&\MA&-\MA\\ \MA&-\MA&-\MA\\ -\MA&-\MA&-\MA \end{matrix}\right] \left[\begin{matrix} \MA&-\MA&\MA&\MA&-\MA&-\MA&\MA&-\MA\\ \MA&\MA&-\MA&\MA&-\MA&\MA&-\MA&-\MA\\ \MA&\MA&\MA&-\MA&\MA&-\MA&-\MA&-\MA \end{matrix}\right] =\left[\begin{matrix} 1&\MB&\MB&\MB&-1&-1&-1&-1\\ \MB&1&-1&-1&\MB&\MB&-1&-1\\ \MB&-1&1&-1&\MB&-1&\MB&-1\\ \MB&-1&-1&1&-1&\MB&\MB&-1\\ -1&\MB&\MB&-1&1&-1&-1&\MB\\ -1&\MB&-1&\MB&-1&1&-1&\MB\\ -1&-1&\MB&\MB&-1&-1&1&\MB\\ -1&-1&-1&-1&\MB&\MB&\MB&1 \end{matrix}\right]$

Table of $T_i\cdot V_j=V_k$:
    V1 V2 V3 V4 V5 V6 V7 V8 
T1  V1 V2 V3 V4 V5 V6 V7 V8 
T2  V2 V3 V4 V1 V6 V7 V8 V5 
T3  V4 V3 V8 V5 V6 V1 V2 V7 
T4  V3 V4 V1 V2 V7 V8 V5 V6 
T5  V3 V8 V5 V4 V1 V2 V7 V6 
T6  V4 V1 V2 V3 V8 V5 V6 V7 
T7  V8 V5 V4 V3 V2 V7 V6 V1 
T8  V4 V5 V6 V1 V2 V3 V8 V7 
T9  V8 V7 V6 V5 V4 V3 V2 V1 
T10 V5 V4 V3 V8 V7 V6 V1 V2 
T11 V5 V6 V1 V4 V3 V8 V7 V2 
T12 V7 V6 V5 V8 V3 V2 V1 V4 
T13 V1 V6 V7 V2 V3 V4 V5 V8 
T14 V5 V8 V7 V6 V1 V4 V3 V2 
T15 V7 V2 V1 V6 V5 V8 V3 V4 
T16 V6 V1 V4 V5 V8 V7 V2 V3 
T17 V6 V5 V8 V7 V2 V1 V4 V3 
T18 V6 V7 V2 V1 V4 V5 V8 V3 
T19 V2 V1 V6 V7 V8 V3 V4 V5 
T20 V2 V7 V8 V3 V4 V1 V6 V5 
T21 V8 V3 V2 V7 V6 V5 V4 V1 
T22 V1 V4 V5 V6 V7 V2 V3 V8 
T23 V7 V8 V3 V2 V1 V6 V5 V4 
T24 V3 V2 V7 V8 V5 V4 V1 V6


Table of $T_i \cdot T_j = T_k$: 
    T1  T2  T3  T4  T5  T6  T7  T8  T9  T10 T11 T12 T13 T14 T15 T16 T17 T18 T19 T20 T21 T22 T23 T24
T1  T1  T2  T3  T4  T5  T6  T7  T8  T9  T10 T11 T12 T13 T14 T15 T16 T17 T18 T19 T20 T21 T22 T23 T24
T2  T2  T4  T22 T6  T8  T1  T11 T13 T14 T16 T18 T9  T20 T17 T21 T15 T12 T23 T24 T5  T10 T19 T7  T3 
T3  T3  T5  T14 T7  T9  T10 T12 T11 T15 T17 T16 T19 T6  T18 T2  T22 T13 T1  T4  T24 T23 T8  T20 T21
T4  T4  T6  T19 T1  T13 T2  T18 T20 T17 T15 T23 T14 T5  T12 T10 T21 T9  T7  T3  T8  T16 T24 T11 T22
T5  T5  T7  T8  T10 T11 T3  T16 T6  T18 T22 T1  T15 T24 T13 T23 T2  T19 T20 T21 T9  T17 T4  T12 T14
T6  T6  T1  T24 T2  T20 T4  T23 T5  T12 T21 T7  T17 T8  T9  T16 T10 T14 T11 T22 T13 T15 T3  T18 T19
T7  T7  T10 T4  T3  T6  T5  T1  T24 T13 T2  T20 T18 T9  T19 T17 T23 T15 T12 T14 T11 T22 T21 T16 T8 
T8  T8  T11 T13 T16 T18 T22 T15 T1  T23 T19 T2  T21 T3  T20 T7  T4  T24 T5  T10 T14 T12 T6  T9  T17
T9  T9  T12 T11 T17 T16 T14 T22 T10 T1  T8  T3  T2  T21 T6  T20 T5  T4  T24 T23 T15 T13 T7  T19 T18
T10 T10 T3  T21 T5  T24 T7  T20 T9  T19 T23 T12 T13 T11 T15 T22 T17 T18 T16 T8  T6  T2  T14 T1  T4 
T11 T11 T16 T6  T22 T1  T8  T2  T3  T20 T4  T5  T23 T14 T24 T12 T7  T21 T9  T17 T18 T19 T10 T15 T13
T12 T12 T17 T7  T14 T10 T9  T3  T21 T6  T5  T24 T1  T15 T4  T13 T20 T2  T19 T18 T16 T8  T23 T22 T11
T13 T13 T18 T20 T15 T23 T19 T21 T2  T7  T24 T4  T10 T22 T5  T11 T6  T3  T8  T16 T17 T9  T1  T14 T12
T14 T14 T9  T18 T12 T15 T17 T19 T16 T2  T13 T22 T4  T10 T1  T5  T8  T6  T3  T7  T21 T20 T11 T24 T23
T15 T15 T19 T16 T13 T22 T18 T8  T17 T3  T11 T14 T5  T23 T10 T24 T9  T7  T21 T20 T2  T6  T12 T4  T1 
T16 T16 T22 T10 T8  T3  T11 T5  T14 T24 T7  T9  T20 T18 T21 T19 T12 T23 T15 T13 T1  T4  T17 T2  T6 
T17 T17 T14 T23 T9  T21 T12 T24 T15 T4  T20 T19 T6  T16 T2  T8  T13 T1  T22 T11 T10 T5  T18 T3  T7 
T18 T18 T15 T1  T19 T2  T13 T4  T22 T5  T6  T8  T7  T17 T3  T9  T11 T10 T14 T12 T23 T24 T16 T21 T20
T19 T19 T13 T12 T18 T17 T15 T14 T23 T10 T9  T21 T3  T2  T7  T6  T24 T5  T4  T1  T22 T11 T20 T8  T16
T20 T20 T23 T5  T21 T7  T24 T10 T4  T11 T3  T6  T16 T19 T8  T18 T1  T22 T13 T15 T12 T14 T2  T17 T9 
T21 T21 T24 T15 T20 T19 T23 T13 T12 T22 T18 T17 T8  T7  T16 T3  T14 T11 T10 T5  T4  T1  T9  T6  T2 
T22 T22 T8  T17 T11 T14 T16 T9  T18 T21 T12 T15 T24 T1  T23 T4  T19 T20 T2  T6  T3  T7  T13 T5  T10
T23 T23 T21 T2  T24 T4  T20 T6  T19 T8  T1  T13 T11 T12 T22 T14 T18 T16 T17 T9  T7  T3  T15 T10 T5 
T24 T24 T20 T9  T23 T12 T21 T17 T7  T16 T14 T10 T22 T4  T11 T1  T3  T8  T6  T2  T19 T18 T5  T13 T15