List of all elements of the Weyl group of type $C_3$.

Question

What is the list of all elements of the Weyl group of type $C_3$ in terms of simple refletions $s_1, s_2, s_3$? There are 48 elements in the group. Thank you very much.

Answer 1

gap> F := FreeGroup("r","s","t");
gap> r := F.1; s:=F.2; t:=F.3;
gap> G := F/[r^2,s^2,t^2,(r*t)^2,(r*s)^3,(s*t)^4];
<fp group on the generators [ r, s, t ]>
gap> Order(G);
48
gap> StructureDescription(G);
"C2 x S4"
gap> Elements(G);
[ <identity ...>, s, r*s*r*t*s*r*t*s*t, s*r,
  r*s*r*t*s*r*t*s, s*t*s*t, r*s*t*s*r*t*s*t, r,
  r*s*t*s*r*t*s, t*s*t, r*t*s*r*t*s*t, t, r*s*t*s*r, r*s,
  r*t*s*r*t*s, r*s*r*t*s*t, r*s*t*s*r*t, s*r*t*s*r*t*s*t,
  s*t, r*s*r*t*s*r, r*s*r, s*r*t*s*r*t*s, r*s*t*s*t,
  r*s*r*t*s*r*t, s*t*s*r*t*s*t, s*r*t, t*s*r, s*t*s,
  s*t*s*r*t*s, r*t*s*t, t*s*r*t, t*s*r*t*s*t, r*t,
  s*t*s*r, t*s, t*s*r*t*s, s*r*t*s*t, s*t*s*r*t, r*s*t,
  s*r*t*s*r, r*s*r*t*s, s*r*t*s*r*t, r*s*r*t, r*t*s*r,
  r*s*t*s, r*t*s*r*t, r*t*s, s*r*t*s ]

Answer 2

Besides to the above codes generating the group, you may consult GAP to write the elements in a reduced form as follows:

gap> SetReducedMultiplication(G);;
     AssignGeneratorVariables(G);;
     S:=Subgroup(G,[G.1,G.2,G.3]);;
     Elements(S);
[ <identity ...>, s^-1, r^-1*s^-1*r^-1*t^-1*s^-1*r^-1*t^-1*s^-1*t^-1, 
s^-1*r^-1, r^-1*s^-1*r^-1*t^-1*s^-1*r^-1*t^-1*s^-1, s^-1*t^-1*s^-1*t^-1, 
r^-1*s^-1*t^-1*s^-1*r^-1*t^-1*s^-1*t^-1, r^-1, 
r^-1*s^-1*t^-1*s^-1*r^-1*t^-1*s^-1, t^-1*s^-1*t^-1, 
r^-1*t^-1*s^-1*r^-1*t^-1*s^-1*t^-1, t^-1, r^-1*s^-1*t^-1*s^-1*r^-1, 
r^-1*s^-1, r^-1*t^-1*s^-1*r^-1*t^-1*s^-1, r^-1*s^-1*r^-1*t^-1*s^-1*t^-1, 
r^-1*s^-1*t^-1*s^-1*r^-1*t^-1, s^-1*r^-1*t^-1*s^-1*r^-1*t^-1*s^-1*t^-1, 
s^-1*t^-1, r^-1*s^-1*r^-1*t^-1*s^-1*r^-1, r^-1*s^-1*r^-1, 
s^-1*r^-1*t^-1*s^-1*r^-1*t^-1*s^-1, r^-1*s^-1*t^-1*s^-1*t^-1, 
r^-1*s^-1*r^-1*t^-1*s^-1*r^-1*t^-1, s^-1*t^-1*s^-1*r^-1*t^-1*s^-1*t^-1, 
s^-1*r^-1*t^-1, t^-1*s^-1*r^-1, s^-1*t^-1*s^-1, 
s^-1*t^-1*s^-1*r^-1*t^-1*s^-1, r^-1*t^-1*s^-1*t^-1, t^-1*s^-1*r^-1*t^-1, 
t^-1*s^-1*r^-1*t^-1*s^-1*t^-1, r^-1*t^-1, s^-1*t^-1*s^-1*r^-1, t^-1*s^-1, 
t^-1*s^-1*r^-1*t^-1*s^-1, s^-1*r^-1*t^-1*s^-1*t^-1, 
s^-1*t^-1*s^-1*r^-1*t^-1, r^-1*s^-1*t^-1, s^-1*r^-1*t^-1*s^-1*r^-1, 
r^-1*s^-1*r^-1*t^-1*s^-1, s^-1*r^-1*t^-1*s^-1*r^-1*t^-1, 
r^-1*s^-1*r^-1*t^-1, r^-1*t^-1*s^-1*r^-1, r^-1*s^-1*t^-1*s^-1, 
r^-1*t^-1*s^-1*r^-1*t^-1, r^-1*t^-1*s^-1, s^-1*r^-1*t^-1*s^-1 ]