NrPolyhedralSubgroups( tbl, c1, c2, c3 )
returns the number and isomorphism type of polyhedral subgroups of the group with character table tbl which are generated by an element g of class c1 and an element h of class c2 with the property that the product gh lies in class c3.
gap> NrPolyhedralSubgroups(L3_2, 2, 2, 4); rec( number := 21, type := "D8" )
According to~cite[p.~233]NPP84 the number of polyhedral subgroups of isomorphism type V_4, D_{2n}, A_4, S_4 and A_5 can be derived from the class multiplication coefficient (see ClassMultCoeffCharTable) and the number of Galois conjugates of a class (see ClassOrbitCharTable).
Note that the classes c1, c2 and c3 in the parameter list must be ordered according to the order of the elements in these classes.
GAP 3.4.4