Covering groups of the alternating and symmetric groups

When the homomorphism is understood, the group D is often called the Schur cover or Darstellungsgruppe.

The symmetric group of degree n ≥ 4 has Schur covers of order 2⋅n!

The alternating group of degree n has one isomorphism class of Schur cover, which has order n!

Schur covers can be described by means of generators and relations.

The symmetric group Sn has a presentation on n − 1 generators ti for i = 1, 2, ..., n − 1 and relations These relations can be used to describe two non-isomorphic covers of the symmetric group.

The presentation of 2⋅S−n becomes particularly simple in this form: (titj)mij = z, and zz = 1.

For the symmetric group, the Schur multiplier vanishes for n ≤ 3, and is the cyclic group of order 2 for n ≥ 4: The double covers can be constructed as spin (respectively, pin) covers of faithful, irreducible, linear representations of An and Sn.

and the exceptional double cover of the group of Lie type G2(4).

[citation needed] For low dimensions there are exceptional isomorphisms with the map from a special linear group over a finite field to the projective special linear group.

For n = 3, the symmetric group is SL(2, 2) ≅ PSL(2, 2) and is its own Schur cover.

The Schur cover of PGL(2, 5) is contained in GL(2, 25) – as before, 25 = 52, so this extends the scalars.

For n = 6, the double cover of the alternating group is given by SL(2, 9) → PSL(2, 9) ≅ A6.

While PGL(2, 9) is contained in the automorphism group PΓL(2, 9) of PSL(2, 9) ≅ A6, PGL(2, 9) is not isomorphic to S6, and its Schur covers (which are double covers) are not contained in nor a quotient of GL(2, 9).

The Schur covers of the symmetric group S6 itself have no faithful representations as a subgroup of GL(d, 9) for d ≤ 3.

Schur covers of finite perfect groups are superperfect, that is both their first and second integral homology vanish.