Binary octahedral group

The group 2O has a presentation given by

${\displaystyle \langle r,s,t\mid r^{2}=s^{3}=t^{4}=rst\rangle }$

or equivalently,

${\displaystyle \langle s,t\mid (st)^{2}=s^{3}=t^{4}\rangle .}$

Quaternion generators with these relations are given by

${\displaystyle r={\tfrac {1}{\sqrt {2}}}(i+j)\qquad s={\tfrac {1}{2}}(1+i+j+k)\qquad t={\tfrac {1}{\sqrt {2}}}(1+i),}$

with ${\displaystyle r^{2}=s^{3}=t^{4}=rst=-1.}$

The binary octahedral group 2O=⟨2,3,4⟩ order 48, has 3 primary subgroups: 2T=⟨2,3,3⟩, index 2, Q16=⟨2,2,4⟩ index 3, and Q12=⟨2,2,3⟩ index 4.
⟨l,m,n⟩=binary polyhedral group
⟨p⟩≃Z_2p, (p)≃Z_p (cyclic groups)

The binary tetrahedral group, 2T, consisting of the 24 Hurwitz units, forms a normal subgroup of index 2. The quaternion group, Q8, consisting of the 8 Lipschitz units forms a normal subgroup of 2O of index 6. The quotient group is isomorphic to S₃ (the symmetric group on 3 letters). These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2O.

The generalized quaternion group, Q16, also forms a subgroup of 2O, index 3. This subgroup is self-normalizing so its conjugacy class has 3 members. There are also isomorphic copies of the binary dihedral groups Q8 and Q12 in 2O.

All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).[1]