The Octahedral Group

The Octahedral Group ${S}_{4}$

The group ${S}_{4}$ can be represented in several different ways. Some of these are given in the following table.

Set Operation
permutations of $4$ elements composition of permutations
rotations preserving a cube compositions of rotations
rotations preserving an octahedron composition of rotations

The complete multiplication table for ${S}_{4}$ is $24 × 24$ matrix. This matrix is too big to include here. In the following tables we shall use one-line notation to represent the permutations in ${S}_{4}$.

Center Abelian Conjugacy classes
$Z\left({S}_{4}\right)=\left\{1\right\}$ No ${𝒞}_{\left({1}^{4}\right)}=\left\{\left(1234\right)\right\}$
${𝒞}_{\left(2{1}^{2}\right)}=\left\{\left(2134\right),\left(3214\right),\left(4231\right),\left(1324\right),\left(1432\right),\left(1243\right)\right\}$
${𝒞}_{\left({2}^{2}\right)}=\left\{\left(2134\right),\left(3412\right),\left(4321\right)\right\}$
${𝒞}_{\left(31\right)}=\left\{\left(3124\right),\left(4132\right),\left(4213\right),\left(1423\right),\left(2314\right),\left(2431\right),\left(3241\right),\left(1342\right)\right\}$
${𝒞}_{\left(31\right)}=\left\{\left(4123\right),\left(3142\right),\left(2413\right),\left(4312\right),\left(2341\right),\left(3421\right)\right\}$

There are more than $30$ subgroups of the group ${S}_{4}$. We shall not give a list of all the subgroups ans we shall not give a subgroup lattice here. The following table lists only the normal subgroups of ${S}_{4}$.

Subgroups ${H}_{i}$ Structure Index Normal Quotient group
${N}_{0}={S}_{4}$ ${N}_{0}={S}_{4}$ $\left[{S}_{4}:{S}_{4}\right]=1$ Yes ${S}_{4}/{H}_{0}\cong ⟨1⟩$
${N}_{1}={A}_{4}$ ${N}_{1}={A}_{4}$ $\left[{S}_{4}:{A}_{4}\right]=2$ Yes ${S}_{4}/{A}_{4}\cong {\mu }_{2}$
${N}_{2}=\left\{\left(1234\right),\left(2143\right),\left(3412\right),\left(4321\right)\right\}$ ${N}_{2}\cong {\mu }_{2} × {\mu }_{2}$ $\left[{S}_{4}:{H}_{2}\right]=6$ Yes ${S}_{4}/{N}_{2}\cong {S}_{3}$
${N}_{3}=\left\{\left(1234\right)\right\}$ ${N}_{3}\cong ⟨1⟩$ $\left\{{S}_{4}:{N}_{3}\right\}=24$ Yes ${S}_{4}/⟨1⟩\cong {S}_{4}$

The following table gives two useful presentations of the octahedral group ${S}_{4}$.

Generators Relations Realization
$S,T$ ${S}^{4}={T}^{2}={\left(ST\right)}^{3}=1$ $S=\left(4123\right),T=\left(4231\right)$
${s}_{1},{s}_{2},{s}_{3}$ ${s}_{1}^{2}={s}_{2}^{2}={s}_{3}^{2}=1$ ${s}_{1}=\left(2134\right)$
${s}_{1}{s}_{2}{s}_{1}={s}_{2}{s}_{1}{s}_{2}$ ${s}_{2}=\left(1321\right)$
${s}_{2}{s}_{3}{s}_{2}={s}_{3}{s}_{2}{s}_{3}$ ${s}_{3}=\left(1243\right)$

In the following table ${s}_{1}=\left(2134\right),$ ${s}_{2}=\left(1324\right),$ ${s}_{3}=\left(1243\right)$ denote the simple transpositions in the group ${S}_{4}$. These simple transpositions generate ${S}_{4}$. Note also that the homomorphism labelled ${\phi }_{\left({1}^{4}\right)}$ is the sign homomorphism $ϵ$ of the symmetric group ${S}_{4}$.

Homomorphism Kernel
$\begin{array}{rrcc}\varphi :& {S}_{4}& \to & {S}_{3}\\ & {s}_{1}& ↦& \left(213\right)\\ & {s}_{2}& ↦& \left(132\right)\\ & {s}_{3}& ↦& \left(213\right)\end{array}$ $ker \varphi ={N}_{2}$
$\begin{array}{rrcc}{\varphi }_{\left(4\right)}:& {S}_{4}& \to & ⟨1⟩\\ & {s}_{1}& ↦& 1\\ & {s}_{2}& ↦& 1\\ & {s}_{3}& ↦& 1\end{array}$ $ker {\varphi }_{\left(4\right)}={S}_{4}$
$\begin{array}{rrcc}{\varphi }_{\left({1}^{4}\right)}:& {S}_{4}& \to & {\mu }_{2}\\ & {s}_{1}& ↦& -1\\ & {s}_{2}& ↦& -1\\ & {s}_{3}& ↦& -1\end{array}$ $ker {\varphi }_{\left({1}^{4}\right)}={A}_{4}$
$\begin{array}{rrcc}{\varphi }_{\left(2{1}^{2}\right)}:& {S}_{4}& \to & {GL}_{3}\\ & {s}_{1}& ↦& \left(\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right)\\ & {s}_{2}& ↦& \left(\begin{array}{ccc}-1& 0& 0\\ 0& 1/2& 3/2\\ 0& 1/2& -1/2\end{array}\right)\\ & {s}_{3}& ↦& \left(\begin{array}{ccc}1/3& 4/3& 0\\ 2/3& -1/3& 0\\ 0& 0& -1\end{array}\right)\end{array}$ $ker {\varphi }_{\left(2{1}^{2}\right)}=⟨1⟩$
$\begin{array}{rrcc}{\varphi }_{\left(31\right)}:& {S}_{4}& \to & {GL}_{3}\\ & {s}_{1}& ↦& \left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\\ & {s}_{2}& ↦& \left(\begin{array}{ccc}1/2& 3/2& 0\\ 1/2& -1/2& 0\\ 0& 0& 1\end{array}\right)\\ & {s}_{3}& ↦& \left(\begin{array}{ccc}1& 0& 0\\ 0& 1/3& 4/3\\ 0& 2/3& -1/3\end{array}\right)\end{array}$ $ker {\varphi }_{\left(31\right)}=⟨1⟩$
$\begin{array}{rrcc}{\varphi }_{\left(22\right)}:& {S}_{4}& \to & {GL}_{2}\\ & {s}_{1}& ↦& \left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\\ & {s}_{2}& ↦& \left(\begin{array}{cc}1/2& 3/2\\ 1/2& -1/2\end{array}\right)\\ & {s}_{3}& ↦& \left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\end{array}$ $ker {\varphi }_{\left(22\right)}={N}_{2}$

References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)