²E₆(q) - Knowlpedia

For a prime power $q$ , ${}^2E_6(q)$ is the central quotient of the Steinberg fixed-point group

{}^2E_6(q)=\mathbf G_{E_6,\mathrm{sc}}^F/Z(\mathbf G_{E_6,\mathrm{sc}}^F),\qquad F=\gamma F_q.

The algebraic group is split and simply connected over $\overline{\mathbb F}_p$ , where $q$ is a power of $p$ . The pinned diagram automorphism $\gamma$ has order $2$ and is the nontrivial symmetry of the $E_6$ diagram. It commutes with coordinatewise Frobenius $F_q$ , so $F^2=F_{q^2}$ . Multiplication is multiplication of cosets of the finite center.

Order and simplicity

Its order is

|{}^2E_6(q)|=\frac{q^{36}(q^{12}-1)(q^9+1)(q^8-1)(q^6-1)(q^5+1)(q^2-1)}{\gcd(3,q+1)}.

It is a nonabelian simple group for every admitted parameter.

Field and center convention

Here $q$ names the graph–Frobenius parameter; some sources write $q^2$ in the group name instead. The absolute rank is $6$ . The finite center of the simply connected fixed-point group has order $\gcd(3,q+1)$ , which is divided out.

References

Robert Steinberg, Lectures on Chevalley Groups (Yale, 1967), §11, especially Theorems 34–35, PDF pp. 193–196; notation for field and twisting parameters differs.
B. Akbari, ODs-characterization of some low-dimensional finite classical groups, Table 2, p. 85: order column only; its additional Restrictions concern a separate invariant.