³D₄(q) - Knowlpedia

For a prime power $q$ , ${}^3D_4(q)$ is the central quotient of the Steinberg fixed-point group

{}^3D_4(q)=\mathbf G_{D_4,\mathrm{sc}}^F/Z(\mathbf G_{D_4,\mathrm{sc}}^F),\qquad F=\gamma F_q.

The algebraic group is split and simply connected over $\overline{\mathbb F}_p$ , where $q$ is a power of $p$ . The pinned diagram automorphism $\gamma$ has order $3$ and cyclically permutes the three outer nodes of the $D_4$ diagram. It commutes with coordinatewise Frobenius $F_q$ , so $F^3=F_{q^3}$ . Multiplication is multiplication of cosets of the finite center.

Order and simplicity

Its order is

|{}^3D_4(q)|=q^{12}(q^8+q^4+1)(q^6-1)(q^2-1).

It is a nonabelian simple group for every admitted parameter.

Field and center convention

Here $q$ names the graph–Frobenius parameter; some sources write $q^3$ in the group name instead. The absolute rank is $4$ . The finite fixed-point group has trivial center.

References

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.