I am looking for matrix representation of Tits group $^2F_4(2)'$ of size 17 971 200. Atlas of finite groups offer several matrix representations but they are not embedded in U(27). They are in GL(27, C).
I am looking really for embedding of $^2F_4(2)'$ in $E_6$ compact Lie group. I tried to guess the embedding but no luck so far. I have come to idea that if I have this group generators in U(27) then I will find them in $E_6$ really.
Atsuyama has defined embedding of EIII symmetric space into E6 Lie group by mapping point in EIII to "reflection". The formula for such reflection can be found in Atsuyma paper. I am hoping to find 1755 points in EIII in which reflections would be 2A conjugacy class in $^2F_4(2)'$.
Motivation The motivation for my research is following. Daniel Allcock has written to me that "There seems to be a friendship between $^3D_4(2)$ and $Co_0$ even though neither contains the other". That friendship can be expressed as mapping 819 reflections from 2A conjugacy class in $^3D_4(2)$ embedded in F4 into elements of 2A conjugacy class of $Co_0$.
One could think that there might be such friendship between $^2F_4(2)'$ embedded in E6 Lie group and some sporadic group X.