The Octahedral Group S4
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 S4
The group S4 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 S4 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 S4.
| Center | Abelian | Conjugacy classes |
|
ZS4=1
|
No
|
𝒞14=1234
|
|
|
|
𝒞212=2134,3214,4231,1324,1432,1243
|
|
|
|
𝒞22=2134,3412,4321
|
|
|
|
𝒞31=3124,4132,4213,1423,2314,2431,3241,1342
|
|
|
|
𝒞31=4123,3142,2413,4312,2341,3421
|
There are more than 30 subgroups of the group S4. 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 S4.
| Subgroups Hi |
Structure |
Index |
Normal |
Quotient group |
|
N0=S4
|
N0=S4
|
S4:S4=1
|
Yes
|
S4/H0≅⟨1⟩
|
|
N1=A4
|
N1=A4
|
S4:A4=2
|
Yes
|
S4/A4≅μ2
|
|
N2=1234,2143,3412,4321
|
N2≅μ2 × μ2
|
S4:H2=6
|
Yes
|
S4/N2≅S3
|
|
N3=1234
|
N3≅⟨1⟩
|
S4:N3=24
|
Yes
|
S4/⟨1⟩≅S4
|
The following table gives two useful presentations of the octahedral group S4.
| Generators |
Relations |
Realization |
|
S,T
|
S4=T2=ST3=1
|
S=4123,T=4231
|
|
s1,s2,s3
|
s12=s22=s32=1
|
s1=2134
|
|
|
|
s1s2s1=s2s1s2
|
s2=1321
|
|
|
s2s3s2=s3s2s3
|
s3=1243
|
In the following table
s1=2134,
s2=1324,
s3=1243
denote the simple transpositions in the group S4. These simple transpositions generate
S4. Note also that the homomorphism labelled
φ14
is the sign homomorphism ϵ of the symmetric group S4.
| Homomorphism |
Kernel |
|
ϕ:S4→S3s1↦213s2↦132s3↦213
|
ker ϕ=N2
|
|
ϕ4:S4→⟨1⟩s1↦1s2↦1s3↦1
|
ker ϕ4=S4
|
|
ϕ14:S4→μ2s1↦-1s2↦-1s3↦-1
|
ker ϕ14=A4
|
|
ϕ212:S4→GL3s1↦-1000-10001s2↦-10001/23/201/2-1/2s3↦1/34/302/3-1/3000-1
|
ker ϕ212=⟨1⟩
|
|
ϕ31:S4→GL3s1↦-100010001s2↦1/23/201/2-1/20001s3↦10001/34/302/3-1/3
|
ker ϕ31=⟨1⟩
|
|
ϕ22:S4→GL2s1↦-1001s2↦1/23/21/2-1/2s3↦-1001
|
ker ϕ22=N2
|
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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