I am searching for (two) presentations of the group $\mathbf{PGL}\_3(\mathbb{F}\_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra relations, of course], in two separate cases:


REP. 1: $(ab)^4 = 1, (bc)^4 = 1, (ac)^2 = 1$;


REP. 2: $(ab)^3 = (bc)^3 = (ac)^3 = 1$.


Can anyone help me out here ? (Thanks !!)



Presentations of $\mathbf{PGL}_3(\mathbb{F}_2)$ by three involutions
This cannot be done. Let $G\_1$ and $G\_2$ be the groups
$$G\_1 = \langle a,b,c | a^2 = b^2 = c^2 = (ab)^4 = (bc)^4 = (ac)^2 \rangle$$
$$G\_2 = \langle a,b,c | a^2 = b^2 = c^2 = (ab)^3 = (bc)^3 = (ac)^3 \rangle.$$


You are looking for surjections from the $G\_j$ onto $PGL\_3(\mathbb{F}\_2)$, which is the simple group of order $168$.


Now, $G\_1$ and $G\_2$ are Coxeter groups and, in fact, they are two of the affine Coxeter groups -- types $\tilde{B}\_2$ and $\tilde{A}\_2$ respectively. They are known to have the alternate descriptions $D\_6 \ltimes \mathbb{Z}^2$ and $D\_8 \ltimes \mathbb{Z}^2$, where $D\_{2m}$ is the dihedral group of order $2m$.


In particular, these two groups are solvable. However, $PGL\_3(\mathbb{F}\_2)$ is not solvable, so it cannot be a quotient of a solvable group.


