GAP Manual: 48.36. NrPolyhedralSubgroups

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 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.