2 math formatting

Hello,

I am looking for matrix representation of Tits group 2F4(2)' $^2F_4(2)'$ 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 2F4(2)' $^2F_4(2)'$ in E6 $E_6$ 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 E6 $E_6$ 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 2F4(2)'.$^2F_4(2)'$.

Motivation The motivation for my research is following. Daniel Allcock has written to me that "There seems to be a friendship between 3D4(2) $^3D_4(2)$ and Co0 $Co_0$ even though neither contains the other". That friendship can be expressed as mapping 819 reflections from 2A conjugacy class in 3D4(2) $^3D_4(2)$ embedded in F4 into elements of 2A conjugacy class of Co0.$Co_0$.

One could think that there might be such friendship between 2F4(2)' $^2F_4(2)'$ embedded in E6 Lie group and some sporadic group X.

Regards, Marek

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# Matrix representation of 2F4(2)' in unitary U(27)

Hello,

I am looking for matrix representation of Tits group 2F4(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 2F4(2)' in E6 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 E6 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 2F4(2)'.

Motivation The motivation for my research is following. Daniel Allcock has written to me that "There seems to be a friendship between 3D4(2) and Co0 even though neither contains the other". That friendship can be expressed as mapping 819 reflections from 2A conjugacy class in 3D4(2) embedded in F4 into elements of 2A conjugacy class of Co0.

One could think that there might be such friendship between 2F4(2)' embedded in E6 Lie group and some sporadic group X.

Regards, Marek