E₇(q) - Knowlpedia

For a prime power $q\geq 2$ , $E_7(q)$ is the finite Chevalley central quotient of type $E_7$ :

E_7(q)=\mathbf G_{E_7,\mathrm{sc}}(\mathbb F_{q})/Z\bigl(\mathbf G_{E_7,\mathrm{sc}}(\mathbb F_{q})\bigr).

Here $\mathbf G_{E_7,\mathrm{sc}}$ is the split simply connected algebraic group of the displayed root-system type. Its finite points are fixed by coordinatewise $q$ -power Frobenius, and multiplication is multiplication of cosets of their finite center.

Order and simplicity

Its order is

|E_7(q)|=\frac{q^{63}(q^{2}-1)(q^{6}-1)(q^{8}-1)(q^{10}-1)(q^{12}-1)(q^{14}-1)(q^{18}-1)}{\gcd(2,q-1)}.

It is a nonabelian simple group for every admitted prime power q.

Rank and central quotient

The absolute rank is $7$ . The finite center of the simply connected group has order $\gcd(2,q-1)$ ; the notation here always denotes the quotient by that center. There are no omitted small-field simplicity exceptions for this type.

References

Robert Steinberg, Lectures on Chevalley Groups (Yale, 1967), §3 construction; §4 Theorem 5, PDF pp. 52–54; §9 Theorems 24–25, PDF pp. 136–138.
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.