Double cover of symmetric group

Definition
The term double cover of symmetric group is used for a stem extension where the base normal subgroup is cyclic group:Z2 and the quotient group is an symmetric group of finite degree.

A double cover exists for the symmetric group $$S_n$$ only when $$n \ge 4$$. Further, for each $$n$$, there are two possibilities for the double cover, the "+" type and the "-" type.

If we consider the group cohomology of symmetric groups and in particular the second cohomology group for trivial group action $$H^2(S_n;\mathbb{Z}_2)$$ for $$n \ge 4$$, this second cohomology group is isomorphic to the Klein four-group, and two of the three nontrivial elements of the group correspond to these double covers. The third nontrivial element corresponds to a group extension where the alternating group in the quotient is kept intact and the odd permutations square to the non-identity central element.

In all cases, both double covers are Schur covering groups for the corresponding symmetric group, because the Schur multiplier is precisely cyclic group:Z2.

Presentation for "-" type double cover
This group, denoted $$2 \cdot S_n^-$$, has a presentation with generating set of size $$n$$ given by $$z,s_1,s_2,\dots,s_{n-1}$$. The idea is that under the surjective map to $$S_n$$, $$z$$ maps to the identity and the relations collapse to the Coxeter presentation of $$S_n$$. The subgroup $$\langle z \rangle$$ is the base cyclic group:Z2. The relations (here $$e$$ denotes the identity element):

Presentation for "+" type double cover
This group, denoted $$2 \cdot S_n^+$$, has a presentation with generating set of size $$n$$ given by $$z,s_1,s_2,\dots,s_{n-1}$$. The idea is that under the surjective map to $$S_n$$, $$z$$ maps to the identity and the relations collapse to the Coxeter presentation of $$S_n$$. The subgroup $$\langle z \rangle$$ is the base cyclic group:Z2. The relations (here $$e$$ denotes the identity element):