It is well-known that the symmetric group S4 has two Schur covering groups, S4-tilde and S4-hat. There are explicit presentations for both groups, and we know that S4-hat is isomorphic to GL(2,3). Question: Is there an alternative, explicit description/realization of S4-tilde?

The lowest degree d for which S4-tilde can be realized as a transitive permutation group is d=16. Is there a concrete (geometric?) construction of this permutation action? (other than via coset action...)

According to wiki, S4-tilde can be embedded inside GL(2,9). What is a good way to "see" this (geometric?) action? (transitive, I presume?)

Last but not least: Is there a "geometric" way to think about/realize the two Schur covers of S_n for general n? And what the six-fold covers of A_n for n=6,7?



1 Answer 1


What you call $\tilde{S}_{4}$ is also known as the binary octahedral group. Probably the easiest way to construct it is to take the natural matrix representation of $G = {\rm GL}(2,3)$ and leave the elements of ${\rm SL}(2,3)$ as they are, but replace the extra generator $\left(\begin{array}{clcr} 1 & 0\\0& -1 \end{array} \right)$ by the scalar multiple $\left(\begin{array}{clcr} i & 0\\0& -i \end{array} \right),$ where $i$ is a primitive $4$-th root of unity in ${\rm GF}(9).$ This does not change $G/Z(G)$, but the new group $\tilde{G} \subseteq {\rm SL}(2,9)$ has a generalized quaternion Sylow $2$-subgroup with only one element of order $2$.

Later edit: The general picture for double covers of $S_{n}$ (as constructed by Schur himself) is similar. In one, transpositions become involutions in the double cover, and in the other, they become elements of order $4$.

  • $\begingroup$ Thanks for the information about double covers of S_n! What about concrete realization of the six-fold covers of A_n for n=6,7? $\endgroup$
    – W Sao
    Jan 28, 2015 at 22:27
  • $\begingroup$ The triple cover of $A_{6}$ has a very well-known complex faithful representation of degree $3.$ the double coverof $A_{7}$ has a nice $4$-dimensional complex representation of degree $4$ which "lifts" a $4$−dimensional representation as a subgroup of ${\rm SL}(4,2) \cong A_{8}.$ The character tables of these $6$-fold covers can be found in the "Atlas of Finite Groups", for example. $\endgroup$ Jan 28, 2015 at 23:08
  • $\begingroup$ Geoff: Your math mode took over after a while in the first comment, and both comments need editing. (It's easiest to copy and then delete each, then paste the copy in a new comment box and edit as needed.) $\endgroup$ Jan 29, 2015 at 16:37

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