²G₂(3) - Knowlpedia

For $q=3=3^{2m+1}$ , where $m=0$ , ${}^2G_2(3)$ is the Steinberg fixed-point group

{}^2G_2(3)=\mathbf G_{G_2,\mathrm{sc}}^F,\qquad F=\tau F_{3^m},\qquad \tau^2=F_3.

Here $\mathbf G_{G_2,\mathrm{sc}}$ is the split simply connected algebraic group of type $G_2$ over $\overline{\mathbb F}_3$ , and $\tau$ is its exceptional endomorphism interchanging the long and short root data. It commutes with Frobenius. Thus $F^2=F_q$ ; the group consists of all $g$ with $F(g)=g$ , under inherited multiplication. Its center is trivial.

Order and simplicity

Its order is

|{}^2G_2(3)|=1512.

It is not simple: its commutator subgroup is PSL(2,8), of index three. The exact order is 1,512.

Parameter convention

The absolute root-system rank is $2$ , and the exponent of $3$ in $q$ must be odd. The notation does not define this family over arbitrary prime powers.

The prime-field boundary

At $m=0$ , the group ${}^2G_2(3)$ has order 1,512. Its commutator subgroup has index three and is isomorphic to $\operatorname{PSL}_2(8)$ , of order 504. The full group at this parameter is excluded from the simple-family tile.

References

Robert A. Wilson, On the simple groups of Suzuki and Ree (24 April 2010), Introduction; §5.1 orders, §5.2 simplicity, and §5.3 the groups over the prime fields, pp. 24–30.
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.