The Octahedral Group $S_4$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 10 February 2011

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 $24\hspace{0.17em}\times \hspace{0.17em}24$ 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	${𝒞}_{\left(1^4\right)}=\left\{\left(1234\right)\right\}$
		${𝒞}_{\left(21^2\right)}=\left\{\left(2134\right),\left(3214\right),\left(4231\right),\left(1324\right),\left(1432\right),\left(1243\right)\right\}$
		${𝒞}_{\left(2^2\right)}=\left\{\left(2134\right),\left(3412\right),\left(4321\right)\right\}$
		${𝒞}_{\left(31\right)}=\left\{\left(3124\right),\left(4132\right),\left(4213\right),\left(1423\right),\left(2314\right),\left(2431\right),\left(3241\right),\left(1342\right)\right\}$
		${𝒞}_{\left(31\right)}=\left\{\left(4123\right),\left(3142\right),\left(2413\right),\left(4312\right),\left(2341\right),\left(3421\right)\right\}$

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	$S_4/A_4\cong {\mu }_2$
$N_2=\left\{\left(1234\right),\left(2143\right),\left(3412\right),\left(4321\right)\right\}$	$N_2\cong {\mu }_2\hspace{0.17em}\times \hspace{0.17em}{\mu }_2$	$\left[S_4:H_2\right]=6$	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
$\begin{array}{rrcc}\varphi :& S_4& \to & S_3\\ & s_1& \mapsto & \left(213\right)\\ & s_2& \mapsto & \left(132\right)\\ & s_3& \mapsto & \left(213\right)\end{array}$	$ker\hspace{0.17em}\varphi =N_2$
$\begin{array}{rrcc}{\varphi }_{\left(4\right)}:& S_4& \to & ⟨1⟩\\ & s_1& \mapsto & 1\\ & s_2& \mapsto & 1\\ & s_3& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\varphi }_{\left(4\right)}=S_4$
$\begin{array}{rrcc}{\varphi }_{\left(1^4\right)}:& S_4& \to & {\mu }_2\\ & s_1& \mapsto & -1\\ & s_2& \mapsto & -1\\ & s_3& \mapsto & -1\end{array}$	$ker\hspace{0.17em}{\varphi }_{\left(1^4\right)}=A_4$
$\begin{array}{rrcc}{\varphi }_{\left(21^2\right)}:& S_4& \to & {GL}_3\\ & s_1& \mapsto & \left(\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right)\\ & s_2& \mapsto & \left(\begin{array}{ccc}-1& 0& 0\\ 0& 1/2& 3/2\\ 0& 1/2& -1/2\end{array}\right)\\ & s_3& \mapsto & \left(\begin{array}{ccc}1/3& 4/3& 0\\ 2/3& -1/3& 0\\ 0& 0& -1\end{array}\right)\end{array}$	$ker\hspace{0.17em}{\varphi }_{\left(21^2\right)}=⟨1⟩$
$\begin{array}{rrcc}{\varphi }_{\left(31\right)}:& S_4& \to & {GL}_3\\ & s_1& \mapsto & \left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\\ & s_2& \mapsto & \left(\begin{array}{ccc}1/2& 3/2& 0\\ 1/2& -1/2& 0\\ 0& 0& 1\end{array}\right)\\ & s_3& \mapsto & \left(\begin{array}{ccc}1& 0& 0\\ 0& 1/3& 4/3\\ 0& 2/3& -1/3\end{array}\right)\end{array}$	$ker\hspace{0.17em}{\varphi }_{\left(31\right)}=⟨1⟩$
$\begin{array}{rrcc}{\varphi }_{\left(22\right)}:& S_4& \to & {GL}_2\\ & s_1& \mapsto & \left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\\ & s_2& \mapsto & \left(\begin{array}{cc}1/2& 3/2\\ 1/2& -1/2\end{array}\right)\\ & s_3& \mapsto & \left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\end{array}$	$ker\hspace{0.17em}{\varphi }_{\left(22\right)}=N_2$

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)

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