PSp(4,2) derived subgroup

$\operatorname{PSp}_4(2)'$ is the commutator subgroup of $\operatorname{PSp}_{4}(2)$ :

\operatorname{PSp}_4(2)'=\langle aba^{-1}b^{-1}:a,b\in \operatorname{PSp}_{4}(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

|\operatorname{PSp}_4(2)'|=360.

It is a nonabelian simple group. The exact order is 360.

Identification and index

It is a normal subgroup of index $2$ in $\operatorname{PSp}_{4}(2)$ . Under $\operatorname{PSp}_4(2)\cong S_6$ , its derived subgroup corresponds to $A_6$ .

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.