PSL(3,4) - Knowlpedia

$\operatorname{PSL}_{3}(4)$ is the following finite group, with $n=3$ and $q=4$ :

\operatorname{PSL}_{3}(4)=\operatorname{SL}_{3}(\mathbb F_{4})/\{\lambda I:\lambda\in\mathbb F_{4}^\times,\ \lambda^{3}=1\}, \qquad \operatorname{SL}_{3}(\mathbb F_{4})=\{A\in M_{3}(\mathbb F_{4}):\det A=1\}.

This is the projective special linear group; multiplication is multiplication of scalar cosets. Here $\mathbb F_{4}$ is the field with $4$ elements.

Order and simplicity

Its order is

|\operatorname{PSL}_{3}(4)|=20160.

It is a nonabelian simple group. The exact order is 20,160.

Parameter convention

Type $A_{n-1}$ uses matrix size $n$ , so the Lie rank is $n-1$ . The simple family omits $\operatorname{PSL}_2(2)\cong S_3$ and $\operatorname{PSL}_2(3)\cong A_4$ .

Small-group identification

This group and $\operatorname{PSL}_4(2)\cong A_8$ both have order 20,160, but are not isomorphic. Equal order is not sufficient evidence for a group isomorphism.

References

Peter J. Cameron, Notes on Classical Groups, §§2.4–2.5, pp. 19–21; Theorem 2.11.
Peter J. Cameron, Notes on Classical Groups, §6, pp. 64–74, especially Theorems 6.3 and 6.6; §7 introduction, p. 75.