²G₂(3) derived subgroup

${}^2G_2(3)'$ is the commutator subgroup of ${}^2G_2(3)$ :

{}^2G_2(3)'=\langle aba^{-1}b^{-1}:a,b\in {}^2G_2(3)\rangle.

Its multiplication is inherited from the specified finite group. This definition uses the full group at its exceptional small parameter before taking the commutator subgroup.

Order and simplicity

Its order is

|{}^2G_2(3)'|=504.

It is a nonabelian simple group. The exact order is 504.

Identification and index

It is a normal subgroup of index $3$ in ${}^2G_2(3)$ . There is an exceptional isomorphism ${}^2G_2(3)'\cong\operatorname{PSL}_2(8)$ .

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.