G₂(2) derived subgroup

$G_2(2)'$ is the commutator subgroup of $G_2(2)$ :

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

|G_2(2)'|=6048.

It is a nonabelian simple group. The exact order is 6,048.

Identification and index

It is a normal subgroup of index $2$ in $G_2(2)$ . There is an exceptional isomorphism $G_2(2)'\cong\operatorname{PSU}_3(3)$ .

References

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).
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.