On genetic codes of certain groups with 3-transpositions

Coxeter groups have numerous applications in mathematics and beyond, and B. Fischer's 3-transposition groups underly the internal geometric analysis in the theory of finite (simple) groups. The intersection of these classes of groups consists of finite Weyl groups $W(A_n)\simeq S_{n+1}$, $W(D_n)$, and $W(E_n)$ for $n=6,7,8$, simple finite-dimensional algebras, and Lie groups. In previous papers by A. I. Sozutov, A. A. Kuznetsov, and the author, systems $S$ of generating transvections (3-transpositions) of groups $Sp_{2m}(2)$ and $O^\pm_{2m}(2)$ were found such that the graphs $\Gamma(S)$ are trees. A set $\{\Gamma_n\}$, $n\geq m$, of nested graphs is called an $E$-series if these graphs are trees, contain the subgraph $E_6$, and their subgraphs with vertices $m,m+1,\ldots,n$ are simple chains. In the present paper, we find genetic codes of the groups $Sp_{2m}(2)$ and $O^\pm_{2m}(2)$, $8\leq 2m \leq 20$; these codes are close to the genetic codes of some Coxeter groups. Our main hypothesis is the following: the groups $Sp_{2m}(2)$ and $O^\pm_{2m}(2)$ (cases (ii)–(iii) in Fischer's theorem) can be obtained from the corresponding infinite Coxeter groups with the use of one or two additional relations of the form $w^2=1$. The graphs $I_n$ considered in this paper contain the subgraph $E_6$ and comprise an $E$-series of nested graphs $\{I_n\,\mid\,n=7, 8,\ldots\}$, in which the subgraph $I_n\setminus E_6$ is a simple chain. We prove that the isomorphisms $X(I_{4k+1})\simeq Sp_{4k}(2)\times Z_2$ and $X(I_{2m})\simeq O^\pm_{2m}(2)$ (the sign $\pm$ depends on $m$) hold for the groups $X(I_n)$ obtained from the Coxeter groups $G(I_n)$ by imposing an additional relation $(s_4^ts_7)^2=1$, where $t=s_3s_2s_1s_5s_6s_3s_2s_5s_3s_4$, if $n=4k +\delta$ ($\delta=0,1,2$). The proof uses the Todd–Coxeter algorithm from the GAP system.