In MAGMA, I input the following:

G:=SmallGroup(20,3);
G;

E:=[xx:xx in G];
S:=[E[6],E[7],E[13],E[20]];
S;

S[1]^2;

S[2]^2;

S[3]*S[4];

This gives the output:

GrpPC : G of order 20 = 2^2 * 5

PC-Relations:
G.1^2 = G.2,
G.2^2 = Id(G),
G.3^5 = Id(G),
G.3^G.1 = G.3^2,
G.3^G.2 = G.3^4

[ G.2, G.2 * G.3, G.1 * G.3^2, G.1 * G.2 * G.3^4 ]

Id(G)

Id(G)

Id(G)

I input this group G and set S to GAP:

gap> F:=FreeGroup( 3 ,"G");
gap> rels:=[F.1^2*F.2^(-1),F.2^2,F.3^5,F.1*F.3*F.1^(-1)*F.3^(-2),F.2*F.3*F.2^(-1)*F.3^(-4)];;

gap> G:=F/rels;; S:=[G.2,G.2*G.3,G.1*G.3^2,G.1*G.2*G.3^4];;

gap> S[1]^2=Identity(G);

true

gap> S[2]^2=Identity(G);

true

gap> S[3]*S[4]=Identity(G);

false

The last statement indicates that S is somehow no longer inverse-closed. Can someone help me understand what's going on here? I would like to have a form of G in terms of the generators and relators, and S in terms of the elements of G; such that I can reproduce (G,S) somewhere other than MAGMA.