Conjugacy classes of loop

Question

If we consider the symmetry group H of rotations and reflections of 3 equidistant points on a circular loop then the group elemetns in cycle notation are: 

 

e: (1)(2)(3)

a: (123)

b: (132)

c: (12)(3)

d: (13)(2)

f: (23)(1)

 

but what is the multiplication table for the group and also the conjugacy classes and subgroups of H?

 

Please show me in steps if possible. Thank you

Answer 1

        *             e: (1)(2)(3)        a: (123)           b: (132)           c: (12)(3)         d: (13)(2)        f: (23)(1)

 

e: (1)(2)(3)      e: (1)(2)(3)        a: (123)           b: (132)           c: (12)(3)         d: (13)(2)        f: (23)(1)

a: (123)           a: (123)             b: (132)           e: (1)(2)(3)      f: (23)(1)          c: (12)(3)        d: (13)(2)

b: (132)           b: (132)             e: (1)(2)(3)      a: (123)           d: (13)(2)          f: (23)(1)        c: (12)(3)

c: (12)(3)         c: (12)(3)           d: (13)(2)        f: (23)(1)         e: (1)(2)(3)        a: (123)          b: (132)

 

d: (13)(2)        d: (13)(2)            f: (23)(1)        c: (12)(3)         b: (132)             e: (1)(2)(3)     a: (123)

f: (23)(1)         f: (23)(1)            c: (12)(3)        d: (13)(2)        a: (123)              b: (132)          e: (1)(2)(3)

 

 

Every row should have one of each element.

Every column should have one of each element.

e * h  =  h * e  = h  where h∈H.