Octahedron

Initial vertex: ${{ v} _1} = {\left[\begin{array}{c} 1\\ 0\\ 0\end{array}\right]}$

Transforms for vertex generation:

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

${{{{{ T} _2}} {{{ V} _3}}} = {\left[\begin{array}{c} 0\\ 0\\ 1\end{array}\right]}} = {{ V} _2}$
${{{{{ T} _2}} {{{ V} _2}}} = {\left[\begin{array}{c} 0\\ 1\\ 0\end{array}\right]}} = {{ V} _3}$
${{{{{ T} _2}} {{{ V} _4}}} = {\left[\begin{array}{c} 0\\ -{1}\\ 0\end{array}\right]}} = {{ V} _4}$
${{{{{ T} _3}} {{{ V} _1}}} = {\left[\begin{array}{c} 0\\ 0\\ -{1}\end{array}\right]}} = {{ V} _5}$
${{{{{ T} _3}} {{{ V} _2}}} = {\left[\begin{array}{c} -{1}\\ 0\\ 0\end{array}\right]}} = {{ V} _6}$
${{{{{ T} _2}} {{{ T} _2}}} = {\left[\begin{array}{ccc} 1& 0& 0\\ 0& -{1}& 0\\ 0& 0& -{1}\end{array}\right]}} = {{ T} _4}$
${{{{{ T} _3}} {{{ T} _2}}} = {\left[\begin{array}{ccc} 0& 1& 0\\ 0& 0& -{1}\\ -{1}& 0& 0\end{array}\right]}} = {{ T} _5}$
${{{{{ T} _2}} {{{ T} _4}}} = {\left[\begin{array}{ccc} 1& 0& 0\\ 0& 0& 1\\ 0& -{1}& 0\end{array}\right]}} = {{ T} _6}$
${{{{{ T} _3}} {{{ T} _4}}} = {\left[\begin{array}{ccc} 0& 0& -{1}\\ 0& -{1}& 0\\ -{1}& 0& 0\end{array}\right]}} = {{ T} _7}$
${{{{{ T} _2}} {{{ T} _5}}} = {\left[\begin{array}{ccc} 0& 1& 0\\ 1& 0& 0\\ 0& 0& -{1}\end{array}\right]}} = {{ T} _8}$
${{{{{ T} _3}} {{{ T} _5}}} = {\left[\begin{array}{ccc} -{1}& 0& 0\\ 0& 0& -{1}\\ 0& -{1}& 0\end{array}\right]}} = {{ T} _9}$
${{{{{ T} _3}} {{{ T} _6}}} = {\left[\begin{array}{ccc} 0& -{1}& 0\\ 0& 0& 1\\ -{1}& 0& 0\end{array}\right]}} = {{{ T} _1} _0}$
${{{{{ T} _2}} {{{ T} _7}}} = {\left[\begin{array}{ccc} 0& 0& -{1}\\ 1& 0& 0\\ 0& -{1}& 0\end{array}\right]}} = {{{ T} _1} _1}$
${{{{{ T} _3}} {{{ T} _7}}} = {\left[\begin{array}{ccc} -{1}& 0& 0\\ 0& -{1}& 0\\ 0& 0& 1\end{array}\right]}} = {{{ T} _1} _2}$
${{{{{ T} _2}} {{{ T} _8}}} = {\left[\begin{array}{ccc} 0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]}} = {{{ T} _1} _3}$
${{{{{ T} _2}} {{{ T} _9}}} = {\left[\begin{array}{ccc} -{1}& 0& 0\\ 0& 1& 0\\ 0& 0& -{1}\end{array}\right]}} = {{{ T} _1} _4}$
${{{{{ T} _3}} {{{ T} _9}}} = {\left[\begin{array}{ccc} 0& -{1}& 0\\ 0& 0& -{1}\\ 1& 0& 0\end{array}\right]}} = {{{ T} _1} _5}$
${{{{{ T} _2}} {{{{ T} _1} _0}}} = {\left[\begin{array}{ccc} 0& -{1}& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]}} = {{{ T} _1} _6}$
${{{{{ T} _3}} {{{{ T} _1} _0}}} = {\left[\begin{array}{ccc} -{1}& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]}} = {{{ T} _1} _7}$
${{{{{ T} _2}} {{{{ T} _1} _1}}} = {\left[\begin{array}{ccc} 0& 0& -{1}\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]}} = {{{ T} _1} _8}$
${{{{{ T} _3}} {{{{ T} _1} _2}}} = {\left[\begin{array}{ccc} 0& 0& 1\\ 0& -{1}& 0\\ 1& 0& 0\end{array}\right]}} = {{{ T} _1} _9}$
${{{{{ T} _2}} {{{{ T} _1} _3}}} = {\left[\begin{array}{ccc} 0& 1& 0\\ -{1}& 0& 0\\ 0& 0& 1\end{array}\right]}} = {{{ T} _2} _0}$
${{{{{ T} _2}} {{{{ T} _1} _5}}} = {\left[\begin{array}{ccc} 0& -{1}& 0\\ -{1}& 0& 0\\ 0& 0& -{1}\end{array}\right]}} = {{{ T} _2} _1}$
${{{{{ T} _3}} {{{{ T} _1} _6}}} = {\left[\begin{array}{ccc} 0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]}} = {{{ T} _2} _2}$
${{{{{ T} _2}} {{{{ T} _1} _8}}} = {\left[\begin{array}{ccc} 0& 0& -{1}\\ -{1}& 0& 0\\ 0& 1& 0\end{array}\right]}} = {{{ T} _2} _3}$
${{{{{ T} _2}} {{{{ T} _1} _9}}} = {\left[\begin{array}{ccc} 0& 0& 1\\ -{1}& 0& 0\\ 0& -{1}& 0\end{array}\right]}} = {{{ T} _2} _4}$
Vertexes as column vectors:

${V} = {\left[\begin{array}{cccccc} 1& 0& 0& 0& 0& -{1}\\ 0& 0& 1& -{1}& 0& 0\\ 0& 1& 0& 0& -{1}& 0\end{array}\right]}$

Vertex inner products:

${{{{{ V} ^T}} {{V}}} = {{{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ 0& -{1}& 0\\ 0& 0& -{1}\\ -{1}& 0& 0\end{array}\right]}} {{\left[\begin{array}{cccccc} 1& 0& 0& 0& 0& -{1}\\ 0& 0& 1& -{1}& 0& 0\\ 0& 1& 0& 0& -{1}& 0\end{array}\right]}}}} = {\left[\begin{array}{cccccc} 1& 0& 0& 0& 0& -{1}\\ 0& 1& 0& 0& -{1}& 0\\ 0& 0& 1& -{1}& 0& 0\\ 0& 0& -{1}& 1& 0& 0\\ 0& -{1}& 0& 0& 1& 0\\ -{1}& 0& 0& 0& 0& 1\end{array}\right]}$

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


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