G₂(2) - Knowlpedia

For $q=2$ , $G_2(2)$ is the finite Chevalley central quotient of type $G_2$ :

G_2(2)=\mathbf G_{G_2,\mathrm{sc}}(\mathbb F_{2})/Z\bigl(\mathbf G_{G_2,\mathrm{sc}}(\mathbb F_{2})\bigr).

Here $\mathbf G_{G_2,\mathrm{sc}}$ 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

|G_2(2)|=12096.

It is not simple: its commutator subgroup has index two and is isomorphic to PSU(3,3). The exact order is 12,096.

Rank and central quotient

The absolute rank is $2$ . The finite center of the simply connected group has order $1$ ; the notation here always denotes the quotient by that center. The value $q=2$ defines a group of order 12,096, but only its derived subgroup $G_2(2)'$ is simple, of order 6,048. The simple-family tile therefore starts at $q=3$ .

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.
ATLAS of Finite Group Representations: U3(3) and G2(2) derived group, Heading and Standard generators: U3(3) = G2(2)′ has order 6048; U3(3):2 = G2(2).