PSp(4,3) - Knowlpedia

For $n=2$ and $q=3$ , $\operatorname{PSp}_{4}(3)$ is the finite group

\operatorname{PSp}_{4}(3)=\operatorname{Sp}_{4}(3)/\{\lambda I:\lambda\in\mathbb F_{3}^\times,\lambda^2=1\}, \quad \operatorname{Sp}_{4}(3)=\{A\in\operatorname{GL}_{4}(\mathbb F_{3}):A^{\mathsf T}JA=J\},

where $\mathbb F_{3}$ is the field with $3$ elements. The matrix $J=\begin{pmatrix}0&I_{2}\\-I_{2}&0\end{pmatrix}$ gives the bilinear form $B(v,w)=v^{\mathsf T}Jw$ , which is alternating, meaning $B(v,v)=0$ for every $v$ , and nondegenerate. Multiplication is multiplication of scalar cosets.

Order and simplicity

Its order is

|\operatorname{PSp}_{4}(3)|=25920.

It is a nonabelian simple group. The exact order is 25,920.

Parameter convention

Type $C_n$ has rank $n$ and matrix dimension $2n$ . The center has size $\gcd(2,q-1)$ , so the projective quotient does nothing when $q$ is even. Alternating means $B(v,v)=0$ , also in characteristic two.

Small-group identification

There is an exceptional isomorphism $\operatorname{PSp}_4(3)\cong\operatorname{PSU}_4(2)$ .

References

Peter J. Cameron, Notes on Classical Groups, §§4.2–4.3, pp. 49–54; the Sp(4,2) action on six quadratic forms, p. 51.
Peter J. Cameron, Notes on Classical Groups, §§5.2–5.3, pp. 59–63; PSU(2,q), PSU(3,2), and the PSU(4,2) identification.