Binary octahedral group

Definition
The binary octahedral group can be defined in the following equivalent ways:

$$\! \langle a,b,c \mid a^4 = b^3 = c^2 = abc\rangle$$ We denote the element $$a^4 = b^3 = c^2$$ as $$z$$. This element has order two.
 * 1) It is the defining ingredient::Schur cover of symmetric group:S4 of type "-" which in symbols means it is the group $$2 \cdot S_4^-$$.
 * 2) It is a binary von Dyck group with parameters $$(2,3,4)$$, i.e., it has the presentation: