Example: 
Dihedral-3 Group

Clear["Global`*"]

Find the matrix representation

id={{ 1, 0}
   ,{ 0, 1}};
  
r={{ Cos[th], -Sin[th] }
  ,{ Sin[th],  Cos[th]}}  /.{th->2 Pi/3};
  
f={{-1, 0}
  ,{ 0, 1}};

r //MatrixForm

a) Find the order of {1,r, r^2 ...}

Table[ MatrixPower[r,n] ,{n,0,3}] //N //MatrixForm

b) Find the order of {1,f, f^2 ...}

Table[ MatrixPower[f,n] ,{n,0,2}] //MatrixForm

d) Combine G1={1,r,r^2} and G2={1,f}

group1={id,r,r.r};
group2={id,f};

Outer[Dot, group1,group2,1] //N[#,3]& //MatrixForm

e) Do r & f commute?

r.f //MatrixForm

f.r //MatrixForm

r.f-f.r //MatrixForm