Symmetries of Geometrical Objects in Space

Question

The symmetries of the platonic solids form groups of order 12, 24 and 60. My question is that, are these groups $A_4$ , $S_4$ and $A_5$ ? I also want to know any other object - regular or irregular with its group of symmetries Can anybody help plz

Answer 1

Every geometrical object has group of symmetries - trivial group if the object has no symmetry otherthan rotation by $0^0$. Let start from plan, the regular n-gons in the plan generate the famous dihedral groups $D_n$, but the irregular planer shapes also have symmetries - the alphabets A, B, K, D, U have one line symmetry, F & G don't have any while H has two lines if summaries. The cyclic groups $C_3$ and $C_12$ are the examples of the groups symmetries of Fan and Clock in space.