$
\def\MA{{\frac{1}{2}}}
\def\MB{{\frac{1+\sqrt{5}}{4}}}
\def\MC{{\frac{1-\sqrt{5}}{4}}}
\def\MD{{\frac{-1+\sqrt{5}}{4}}}
$
600-cell
Initial vertex: $V_1=\left[\begin{matrix}0\\0\\0\\1\end{matrix}\right]$
Transforms for vertex generation:
$\tilde{T}_i\in\left\{
\left[\begin{matrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}\right],
\left[\begin{matrix}-\MA&-\MA&-\MA&-\MA\\ \MA&\MA&-\MA&-\MA\\ \MA&-\MA&\MA&-\MA\\ \MA&-\MA&-\MA&\MA\end{matrix}\right],
\left[\begin{matrix}\MB&-\MA&\MC&0\\ \MA&\MB&0&\MD\\ \MD&0&\MB&-\MA\\0&\MC&\MA&\MB\end{matrix}\right],
\left[\begin{matrix}\MD&-\MB&-\MA&0\\ \MB&\MA&\MC&0\\ \MA&\MC&\MB&0\\0&0&0&1\end{matrix}\right]
\right\}$
Vertexes:
$T_2 V_1=\left[\begin{matrix}-\MA\\-\MA\\-\MA\\ \MA\end{matrix}\right]=V_2$
$T_2 V_2=\left[\begin{matrix}\MA\\-\MA\\-\MA\\ \MA\end{matrix}\right]=V_3$
$T_3 V_3=\left[\begin{matrix}\MB\\0\\-\MA\\ \MD\end{matrix}\right]=V_4$
$T_2 V_4=\left[\begin{matrix}\MC\\ \MA\\0\\ \MB\end{matrix}\right]=V_5$
$T_2 V_5=\left[\begin{matrix}-\MA\\ \MC\\-\MB\\0\end{matrix}\right]=V_6$
$T_3 V_6=\left[\begin{matrix}0\\-\MA\\-\MB\\ \MC\end{matrix}\right]=V_7$
$T_2 V_7=\left[\begin{matrix}\MB\\ \MD\\0\\ \MA\end{matrix}\right]=V_8$
$T_2 V_8=\left[\begin{matrix}-\MB\\ \MD\\0\\ \MA\end{matrix}\right]=V_9$
$T_3 V_9=\left[\begin{matrix}-\MB\\0\\-\MA\\ \MD\end{matrix}\right]=V_{10}$
$T_2 V_{10}=\left[\begin{matrix}\MA\\ \MC\\-\MB\\0\end{matrix}\right]=V_{11}$
$T_2 V_{11}=\left[\begin{matrix}\MD\\ \MA\\0\\ \MB\end{matrix}\right]=V_{12}$
$T_3 V_{12}=\left[\begin{matrix}0\\ \MB\\ \MC\\ \MA\end{matrix}\right]=V_{13}$
$T_2 V_{13}=\left[\begin{matrix}-\MA\\ \MD\\-\MB\\0\end{matrix}\right]=V_{14}$
$T_2 V_{14}=\left[\begin{matrix}\MA\\ \MD\\-\MB\\0\end{matrix}\right]=V_{15}$
$T_3 V_{15}=\left[\begin{matrix}\MA\\ \MA\\-\MA\\-\MA\end{matrix}\right]=V_{16}$
$T_2 V_{16}=\left[\begin{matrix}0\\1\\0\\0\end{matrix}\right]=V_{17}$
$T_2 V_{17}=\left[\begin{matrix}-\MA\\ \MA\\-\MA\\-\MA\end{matrix}\right]=V_{18}$
$T_3 V_{18}=\left[\begin{matrix}-\MA\\0\\ \MC\\-\MB\end{matrix}\right]=V_{19}$
$T_2 V_{19}=\left[\begin{matrix}\MB\\ \MD\\0\\-\MA\end{matrix}\right]=V_{20}$
$T_2 V_{20}=\left[\begin{matrix}\MC\\ \MB\\ \MA\\0\end{matrix}\right]=V_{21}$
$T_3 V_{21}=\left[\begin{matrix}-\MB\\ \MA\\ \MD\\0\end{matrix}\right]=V_{22}$
$T_2 V_{22}=\left[\begin{matrix}0\\ \MC\\-\MA\\-\MB\end{matrix}\right]=V_{23}$
$T_2 V_{23}=\left[\begin{matrix}\MB\\ \MA\\ \MD\\0\end{matrix}\right]=V_{24}$
$T_3 V_{24}=\left[\begin{matrix}\MD\\ \MB\\ \MA\\0\end{matrix}\right]=V_{25}$
$T_2 V_{25}=\left[\begin{matrix}-\MB\\ \MD\\0\\-\MA\end{matrix}\right]=V_{26}$
$T_2 V_{26}=\left[\begin{matrix}\MA\\0\\ \MC\\-\MB\end{matrix}\right]=V_{27}$
$T_3 V_{27}=\left[\begin{matrix}\MA\\0\\ \MD\\-\MB\end{matrix}\right]=V_{28}$
$T_2 V_{28}=\left[\begin{matrix}0\\ \MA\\ \MB\\ \MC\end{matrix}\right]=V_{29}$
$T_2 V_{29}=\left[\begin{matrix}-\MA\\0\\ \MD\\-\MB\end{matrix}\right]=V_{30}$
$T_3 V_{30}=\left[\begin{matrix}-\MA\\-\MA\\ \MA\\-\MA\end{matrix}\right]=V_{31}$
$T_2 V_{31}=\left[\begin{matrix}\MA\\-\MA\\ \MA\\-\MA\end{matrix}\right]=V_{32}$
$T_2 V_{32}=\left[\begin{matrix}0\\0\\1\\0\end{matrix}\right]=V_{33}$
$T_3 V_{33}=\left[\begin{matrix}\MC\\0\\ \MB\\ \MA\end{matrix}\right]=V_{34}$
$T_2 V_{34}=\left[\begin{matrix}-\MA\\-\MB\\0\\ \MC\end{matrix}\right]=V_{35}$
$T_2 V_{35}=\left[\begin{matrix}\MB\\-\MA\\ \MD\\0\end{matrix}\right]=V_{36}$
$T_3 V_{36}=\left[\begin{matrix}\MB\\0\\ \MA\\ \MD\end{matrix}\right]=V_{37}$
$T_2 V_{37}=\left[\begin{matrix}-\MB\\0\\ \MA\\ \MD\end{matrix}\right]=V_{38}$
$T_2 V_{38}=\left[\begin{matrix}0\\-\MB\\ \MC\\-\MA\end{matrix}\right]=V_{39}$
$T_3 V_{39}=\left[\begin{matrix}\MA\\-\MB\\0\\ \MC\end{matrix}\right]=V_{40}$
$T_2 V_{40}=\left[\begin{matrix}\MD\\0\\ \MB\\ \MA\end{matrix}\right]=V_{41}$
$T_2 V_{41}=\left[\begin{matrix}-\MB\\-\MA\\ \MD\\0\end{matrix}\right]=V_{42}$
$T_3 V_{42}=\left[\begin{matrix}-\MA\\-\MB\\0\\ \MD\end{matrix}\right]=V_{43}$
$T_2 V_{43}=\left[\begin{matrix}\MA\\-\MB\\0\\ \MD\end{matrix}\right]=V_{44}$
$T_2 V_{44}=\left[\begin{matrix}0\\ \MC\\ \MA\\ \MB\end{matrix}\right]=V_{45}$
$T_3 V_{44}=\left[\begin{matrix}\MB\\ \MC\\0\\ \MA\end{matrix}\right]=V_{46}$
$T_2 V_{46}=\left[\begin{matrix}-\MA\\0\\ \MD\\ \MB\end{matrix}\right]=V_{47}$
$T_2 V_{47}=\left[\begin{matrix}\MC\\-\MB\\-\MA\\0\end{matrix}\right]=V_{48}$
$T_3 V_{48}=\left[\begin{matrix}\MD\\-\MB\\-\MA\\0\end{matrix}\right]=V_{49}$
$T_2 V_{49}=\left[\begin{matrix}\MA\\0\\ \MD\\ \MB\end{matrix}\right]=V_{50}$
$T_2 V_{50}=\left[\begin{matrix}-\MB\\ \MC\\0\\ \MA\end{matrix}\right]=V_{51}$
$T_4 V_{51}=\left[\begin{matrix}0\\-\MB\\ \MC\\ \MA\end{matrix}\right]=V_{52}$
$T_2 V_{52}=\left[\begin{matrix}\MD\\-\MA\\0\\ \MB\end{matrix}\right]=V_{53}$
$T_2 V_{53}=\left[\begin{matrix}\MC\\-\MA\\0\\ \MB\end{matrix}\right]=V_{54}$
$T_3 V_{54}=\left[\begin{matrix}0\\ \MC\\-\MA\\ \MB\end{matrix}\right]=V_{55}$
$T_3 V_{55}=\left[\begin{matrix}\MD\\0\\-\MB\\ \MA\end{matrix}\right]=V_{56}$
$T_2 V_{56}=\left[\begin{matrix}0\\ \MD\\-\MA\\ \MB\end{matrix}\right]=V_{57}$
$T_2 V_{57}=\left[\begin{matrix}\MC\\0\\-\MB\\ \MA\end{matrix}\right]=V_{58}$
$T_3 V_{58}=\left[\begin{matrix}0\\0\\-1\\0\end{matrix}\right]=V_{59}$
$T_2 V_{59}=\left[\begin{matrix}\MA\\ \MA\\-\MA\\ \MA\end{matrix}\right]=V_{60}$
$T_2 V_{60}=\left[\begin{matrix}-\MA\\ \MA\\-\MA\\ \MA\end{matrix}\right]=V_{61}$
$T_3 V_{60}=\left[\begin{matrix}\MD\\ \MB\\-\MA\\0\end{matrix}\right]=V_{62}$
$T_2 V_{62}=\left[\begin{matrix}\MC\\ \MB\\-\MA\\0\end{matrix}\right]=V_{63}$
$T_2 V_{63}=\left[\begin{matrix}0\\ \MA\\-\MB\\ \MC\end{matrix}\right]=V_{64}$
$T_3 V_{64}=\left[\begin{matrix}0\\ \MD\\-\MA\\-\MB\end{matrix}\right]=V_{65}$
$T_2 V_{65}=\left[\begin{matrix}\MA\\ \MB\\0\\ \MC\end{matrix}\right]=V_{66}$
$T_2 V_{66}=\left[\begin{matrix}-\MA\\ \MB\\0\\ \MC\end{matrix}\right]=V_{67}$
$T_4 V_{67}=\left[\begin{matrix}-\MB\\0\\-\MA\\ \MC\end{matrix}\right]=V_{68}$
$T_2 V_{68}=\left[\begin{matrix}\MB\\0\\-\MA\\ \MC\end{matrix}\right]=V_{69}$
$T_2 V_{69}=\left[\begin{matrix}0\\ \MB\\ \MD\\ \MA\end{matrix}\right]=V_{70}$
$T_3 V_{70}=\left[\begin{matrix}-\MA\\ \MB\\0\\ \MD\end{matrix}\right]=V_{71}$
$T_2 V_{71}=\left[\begin{matrix}\MC\\0\\-\MB\\-\MA\end{matrix}\right]=V_{72}$
$T_2 V_{72}=\left[\begin{matrix}\MB\\ \MA\\ \MC\\0\end{matrix}\right]=V_{73}$
$T_4 V_{72}=\left[\begin{matrix}\MD\\0\\-\MB\\-\MA\end{matrix}\right]=V_{74}$
$T_2 V_{74}=\left[\begin{matrix}\MA\\ \MB\\0\\ \MD\end{matrix}\right]=V_{75}$
$T_2 V_{75}=\left[\begin{matrix}-\MB\\ \MA\\ \MC\\0\end{matrix}\right]=V_{76}$
$T_3 V_{68}=\left[\begin{matrix}-\MA\\-\MA\\-\MA\\-\MA\end{matrix}\right]=V_{77}$
$T_2 V_{77}=\left[\begin{matrix}1\\0\\0\\0\end{matrix}\right]=V_{78}$
$T_2 V_{78}=\left[\begin{matrix}-\MA\\ \MA\\ \MA\\ \MA\end{matrix}\right]=V_{79}$
$T_4 V_{77}=\left[\begin{matrix}\MA\\-\MA\\-\MA\\-\MA\end{matrix}\right]=V_{80}$
$T_2 V_{80}=\left[\begin{matrix}\MA\\ \MA\\ \MA\\ \MA\end{matrix}\right]=V_{81}$
$T_2 V_{81}=\left[\begin{matrix}-1\\0\\0\\0\end{matrix}\right]=V_{82}$
$T_3 V_{82}=\left[\begin{matrix}-\MB\\-\MA\\ \MC\\0\end{matrix}\right]=V_{83}$
$T_2 V_{83}=\left[\begin{matrix}\MB\\-\MA\\ \MC\\0\end{matrix}\right]=V_{84}$
$T_2 V_{84}=\left[\begin{matrix}0\\ \MD\\ \MA\\ \MB\end{matrix}\right]=V_{85}$
$T_3 V_{80}=\left[\begin{matrix}\MB\\ \MC\\0\\-\MA\end{matrix}\right]=V_{86}$
$T_2 V_{86}=\left[\begin{matrix}0\\ \MA\\ \MB\\ \MD\end{matrix}\right]=V_{87}$
$T_2 V_{87}=\left[\begin{matrix}-\MB\\ \MC\\0\\-\MA\end{matrix}\right]=V_{88}$
$T_3 V_{86}=\left[\begin{matrix}\MB\\0\\ \MA\\ \MC\end{matrix}\right]=V_{89}$
$T_2 V_{89}=\left[\begin{matrix}-\MA\\ \MD\\ \MB\\0\end{matrix}\right]=V_{90}$
$T_2 V_{90}=\left[\begin{matrix}\MC\\-\MA\\0\\-\MB\end{matrix}\right]=V_{91}$
$T_3 V_{91}=\left[\begin{matrix}0\\-\MB\\ \MD\\-\MA\end{matrix}\right]=V_{92}$
$T_2 V_{92}=\left[\begin{matrix}\MA\\ \MC\\ \MB\\0\end{matrix}\right]=V_{93}$
$T_2 V_{93}=\left[\begin{matrix}-\MA\\ \MC\\ \MB\\0\end{matrix}\right]=V_{94}$
$T_3 V_{94}=\left[\begin{matrix}-\MA\\-\MA\\ \MA\\ \MA\end{matrix}\right]=V_{95}$
$T_2 V_{95}=\left[\begin{matrix}0\\-1\\0\\0\end{matrix}\right]=V_{96}$
$T_2 V_{96}=\left[\begin{matrix}\MA\\-\MA\\ \MA\\ \MA\end{matrix}\right]=V_{97}$
$T_4 V_{95}=\left[\begin{matrix}0\\-\MB\\ \MD\\ \MA\end{matrix}\right]=V_{98}$
$T_4 V_{94}=\left[\begin{matrix}\MC\\-\MB\\ \MA\\0\end{matrix}\right]=V_{99}$
$T_2 V_{99}=\left[\begin{matrix}\MD\\-\MB\\ \MA\\0\end{matrix}\right]=V_{100}$
$T_2 V_{100}=\left[\begin{matrix}0\\-\MA\\ \MB\\ \MD\end{matrix}\right]=V_{101}$
$T_4 V_{91}=\left[\begin{matrix}\MD\\-\MA\\0\\-\MB\end{matrix}\right]=V_{102}$
$T_2 V_{102}=\left[\begin{matrix}\MA\\ \MD\\ \MB\\0\end{matrix}\right]=V_{103}$
$T_2 V_{103}=\left[\begin{matrix}-\MB\\0\\ \MA\\ \MC\end{matrix}\right]=V_{104}$
$T_4 V_{86}=\left[\begin{matrix}\MA\\ \MA\\ \MA\\-\MA\end{matrix}\right]=V_{105}$
$T_2 V_{105}=\left[\begin{matrix}-\MA\\ \MA\\ \MA\\-\MA\end{matrix}\right]=V_{106}$
$T_2 V_{106}=\left[\begin{matrix}0\\0\\0\\-1\end{matrix}\right]=V_{107}$
$T_3 V_{107}=\left[\begin{matrix}0\\ \MC\\ \MA\\-\MB\end{matrix}\right]=V_{108}$
$T_2 V_{108}=\left[\begin{matrix}\MD\\0\\ \MB\\-\MA\end{matrix}\right]=V_{109}$
$T_2 V_{109}=\left[\begin{matrix}\MC\\0\\ \MB\\-\MA\end{matrix}\right]=V_{110}$
$T_3 V_{108}=\left[\begin{matrix}0\\-\MA\\ \MB\\ \MC\end{matrix}\right]=V_{111}$
$T_3 V_{66}=\left[\begin{matrix}0\\ \MB\\ \MD\\-\MA\end{matrix}\right]=V_{112}$
$T_2 V_{112}=\left[\begin{matrix}\MC\\ \MA\\0\\-\MB\end{matrix}\right]=V_{113}$
$T_2 V_{113}=\left[\begin{matrix}\MD\\ \MA\\0\\-\MB\end{matrix}\right]=V_{114}$
$T_3 V_{114}=\left[\begin{matrix}0\\ \MD\\ \MA\\-\MB\end{matrix}\right]=V_{115}$
$T_3 V_{62}=\left[\begin{matrix}0\\ \MB\\ \MC\\-\MA\end{matrix}\right]=V_{116}$
$T_3 V_{57}=\left[\begin{matrix}0\\ \MA\\-\MB\\ \MD\end{matrix}\right]=V_{117}$
$T_4 V_{55}=\left[\begin{matrix}\MA\\0\\ \MC\\ \MB\end{matrix}\right]=V_{118}$
$T_2 V_{118}=\left[\begin{matrix}-\MA\\0\\ \MC\\ \MB\end{matrix}\right]=V_{119}$
$T_2 V_{119}=\left[\begin{matrix}0\\-\MA\\-\MB\\ \MD\end{matrix}\right]=V_{120}$