a) Determine the class equation of the octahedral group.
b) This group contains two proper normal subgroups. Find them, show that they are normal, and show that there are no others.
a) The octahedral group O has the following six conjugacy classes:
C1: The identity
C2: The six 90-degree rotations
C3: The eight 120-degree rotations
C4: The three 180-degree rotations about a four-fold axis
C5: The six 180-degree rotations about a two-fold axis
C6: The 24 reflections.
Therefore the class equation of O is
48 = 1 + 6 + 8 + 3 + 6 + 24.
b) The proper normal subgroups of O are the subgroup ...
We derive the class equation for the octahedral group and determine its normal subgroups.