(*^
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Example: 
Dihedral-3 Group
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = subsection; inactive; preserveAspect]
Find the matrix representation
:[font = input; preserveAspect]
id={{ 1, 0}
   ,{ 0, 1}};
  
r={{ Cos[th], -Sin[th] }
  ,{ Sin[th],  Cos[th]}}  /.{th->2 Pi/3};
  
f={{-1, 0}
  ,{ 0, 1}};
:[font = input; preserveAspect; startGroup]
r //MatrixForm
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1/2, 3^(1/2)/2}, {-3^(1/2)/2, -1/2}}]
;[o]
  1        Sqrt[3]
-(-)       -------
  2           2

-Sqrt[3]     1
--------   -(-)
   2         2
:[font = subsection; inactive; preserveAspect; startGroup]
a) Find the order of {1,r, r^2 ...}
:[font = input; preserveAspect; startGroup]
Table[ MatrixPower[r,n] ,{n,0,3}] //N //MatrixForm
:[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup; endGroup]
MatrixForm[{{{1., 0}, {0, 1.}}, {{-0.5, -0.8660254037844386468}, 
 
    {0.8660254037844386468, -0.5}}, {{-0.5, 0.8660254037844386468}, 
 
    {-0.8660254037844386468, -0.5}}, {{1., 0}, {0, 1.}}}]
;[o]
1.          0
0           1.

-0.5        0.866025
-0.866025   -0.5

-0.5        -0.866025
0.866025    -0.5

1.          0
0           1.
:[font = subsection; inactive; preserveAspect; startGroup]
b) Find the order of {1,f, f^2 ...}
:[font = input; preserveAspect; startGroup]
Table[ MatrixPower[f,n] ,{n,0,2}] //MatrixForm
:[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup; endGroup]
MatrixForm[{{{1, 0}, {0, 1}}, {{-1, 0}, {0, 1}}, {{1, 0}, {0, 1}}}]
;[o]
1    0
0    1

-1   0
0    1

1    0
0    1
:[font = subsection; inactive; preserveAspect]
d) Combine G1={1,r,r^2} and G2={1,f}
:[font = input; preserveAspect]
group1={id,r,r.r};
group2={id,f};

:[font = input; preserveAspect; startGroup]
Outer[Dot, group1,group2,1] //N[#,3]& //MatrixForm
:[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup]
MatrixForm[{{{{1., 0}, {0, 1.}}, {{-1., 0}, {0, 1.}}}, 
 
   {{{-0.5, -0.8660254037844386468}, {0.8660254037844386468, -0.5}}, 
 
    {{0.5, -0.8660254037844386468}, {-0.8660254037844386468, -0.5}}}, 
 
   {{{-0.5, 0.8660254037844386468}, {-0.8660254037844386468, -0.5}}, 
 
    {{0.5, 0.8660254037844386468}, {0.8660254037844386468, -0.5}}}}]
;[o]
1.     0        -1.    0
0      1.       0      1.

-0.5   -0.866   0.5    -0.866
0.866  -0.5     -0.866 -0.5

-0.5   0.866    0.5    0.866
-0.866 -0.5     0.866  -0.5
:[font = subsection; inactive; preserveAspect; startGroup]
e) Do r & f commute?
:[font = input; preserveAspect; startGroup]
r.f //MatrixForm
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1/2, -3^(1/2)/2}, {-3^(1/2)/2, -1/2}}]
;[o]
1          -Sqrt[3]
-          --------
2             2

-Sqrt[3]     1
--------   -(-)
   2         2
:[font = input; preserveAspect; startGroup]
f.r //MatrixForm
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1/2, 3^(1/2)/2}, {3^(1/2)/2, -1/2}}]
;[o]
1         Sqrt[3]
-         -------
2            2

Sqrt[3]     1
-------   -(-)
   2        2
:[font = input; preserveAspect; startGroup]
r.f-f.r //MatrixForm
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup]
MatrixForm[{{0, -3^(1/2)}, {-3^(1/2), 0}}]
;[o]
0          -Sqrt[3]

-Sqrt[3]   0
^*)