PSp(4,2) - Knowlpedia

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

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

where $\mathbb F_{2}$ is the field with $2$ 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}(2)|=720.

It is not simple: PSp(4,2) is isomorphic to S₆. The exact order is 720.

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

The action on the six quadratic forms of minus type with this polar form identifies $\operatorname{Sp}_4(2)=\operatorname{PSp}_4(2)$ with $S_6$ . Its commutator subgroup is $A_6$ , of index two.

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.