# Double Covers of the Alternating and Symmetric Groups

These functions construct representations of the groups $2A_n$ and $2S_n$ as matrix groups over a field ${\bf F}_{p}$ or (usually) ${\bf F}_{p^2}$ for an odd prime $p$. Their dimension is $2^d$ with $d = (n-1)/2,\, (n-2)/2$, or $(n-3)/2$ depending on $n \bmod 6$. The method used is described in [[Maas, 2009](../../references.md#cite-maas)].

## `DoubleCoverSymmetricGroup(n: parameters): RngIntElt -> GrpMat`

```magma
Sign          : MonStgElt                    Default: "-"
Characteristic: RngIntElt                    Default: 3
```

The double cover $2S_n$ of the symmetric groups $S_n$ of degree $n$ for $n \ge 4$ represented as a matrix group over the field ${\bf F}_{p}$ or (usually) ${\bf F}_{p^2}$, where $p$, which defaults to $3$, can be specified as `Characteristic`.

There are two isomorphism classes of groups with this structure, which are distinguished by the optional parameter `Sign`, which can be positive, (`Sign` = “+” or “plus”), or (default) negative (`Sign` = “-” or “minus”). The inverse images of the Coxeter generators of $S_n$ have order $2$ when `Sign` is “+” and $4$ when `Sign` is “-“.

## `DoubleCoverAlternatingGroup(n: parameters): RngIntElt -> GrpMat`

```magma
Characteristic: RngIntElt                    Default: 3
```

The double cover $2A_n$ of the symmetric groups $A_n$ of degree $n$ for $n \ge 4$ represented as a matrix group over the field ${\bf F}_{p}$ or (usually) ${\bf F}_{p^2}$, where $p$, which defaults to $3$, can be specified as `Characteristic`.