I have computed a semidirect product, $s$ of $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ as below and have drawn a Cayley graph for $s$ with respect to a generating set $S$. But I wanted to compare the result I got by the CayleyGraph command, with the result I get when I multiply the elements of $s$ representing vertices of the Cayley graph by the elements in the generating set, according to the definition of the Cayley graph (there is an edge from $g$ to $gt$ for $g \in s, t \in S$).
gap> LogTo("C:/Users/Desktop/Test1/Ndoc15.txt");
gap> Read("C:/Users/Desktop/Test1/UndirectedGeneratingSets.gap");
gap> LoadPackage("grape");;
gap> Z3:=Group((1,2,3));
Group([ (1,2,3) ])
gap> Z3gen:=GeneratorsOfGroup(Z3)[1];
(1,2,3)
gap> Z5:=Group((1,2,3,4,5));
Group([ (1,2,3,4,5) ])
gap> h:=DirectProduct(Z5,Z5);
Group([ (1,2,3,4,5), (6,7,8,9,10) ])
gap> Auts:=AutomorphismGroup(h);
<group with 4 generators>
gap> AutsOfOrd3:=Filtered(Elements(Auts),elt->Order(elt)=3);;
gap> ExHom:=GroupHomomorphismByImages(Z3,Auts,[Z3gen],[AutsOfOrd3[1]]);
[ (1,2,3) ] -> [ [ (1,2,3,4,5), (6,7,8,9,10) ] -> [ (1,5,4,3,2)(6,10,9,8,7),
(1,2,3,4,5) ] ]
gap> s:=SemidirectProduct(Z3,ExHom,h);
Group([ (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)
(17,18,23), (1,6,11,16,21)(2,7,12,17,22)
(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25), (1,2,3,4,5)(6,7,8,9,10)
(11,12,13,14,15)(16,17,18,19,20)
(21,22,23,24,25) ])
gap> S:=IrredUndirGenSetsUpToAut(s);
[ [ (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,
15)(16,17,18,19,20)(21,22,23,24,25) ],
[ (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23), (1,2,7)(3,12,25)(4,17,19)(5,22,13)(6,21,
8)(9,11,20)(10,16,14)(18,24,23) ] ]
gap> S1:=S[1];
[ (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,
15)(16,17,18,19,20)(21,22,23,24,25) ]
gap> Order(S1[1]);
3
gap> Order(S1[2]);
5
gap> X1:=CayleyGraph(s,S1);
rec( adjacencies := [ [ 2, 3, 4, 13 ] ], group := <permutation group of size 75 with 3 generators>, isGraph := true,
isSimple := true, names := [ (), (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23),
(2,25,6)(3,19,11)(4,13,16)(5,7,21)(8,20,10)(9,14,15)(12,22,24)(17,23,18), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,
15)(16,17,18,19,20)(21,22,23,24,25), (1,2,7)(3,12,25)(4,17,19)(5,22,13)(6,21,8)(9,11,20)(10,16,14)(18,24,23),
(1,2,21)(3,20,6)(4,14,11)(5,8,16)(7,22,25)(9,15,10)(12,23,19)(13,17,24), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,
14)(16,18,20,17,19)(21,23,25,22,24), (1,3,13)(2,8,7)(4,18,25)(5,23,19)(6,22,14)(9,12,21)(10,17,20)(11,16,15),
(1,3,16)(2,22,21)(4,15,6)(5,9,11)(7,23,20)(8,17,25)(12,24,14)(13,18,19), (1,4,2,5,3)(6,9,7,10,8)(11,14,12,15,
13)(16,19,17,20,18)(21,24,22,25,23), (1,4,19)(2,9,13)(3,14,7)(5,24,25)(6,23,20)(10,18,21)(11,17,16)(12,22,15),
(1,4,11)(2,23,16)(3,17,21)(5,10,6)(7,24,15)(8,18,20)(9,12,25)(13,19,14), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,
12)(16,20,19,18,17)(21,25,24,23,22), (1,5,25)(2,10,19)(3,15,13)(4,20,7)(6,24,21)(8,9,14)(11,18,22)(12,23,16),
(1,5,6)(2,24,11)(3,18,16)(4,12,21)(7,25,10)(8,19,15)(9,13,20)(17,22,23), (1,6,7)(2,5,12)(3,24,17)(4,18,22)(8,25,
11)(9,19,16)(10,13,21)(14,20,15), (1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25),
(1,6,5)(2,11,24)(3,16,18)(4,21,12)(7,10,25)(8,15,19)(9,20,13)(17,23,22), (1,7,2)(3,25,12)(4,19,17)(5,13,22)(6,8,
21)(9,20,11)(10,14,16)(18,23,24), (1,7,13,19,25)(2,8,14,20,21)(3,9,15,16,22)(4,10,11,17,23)(5,6,12,18,24),
(1,7,6)(2,12,5)(3,17,24)(4,22,18)(8,11,25)(9,16,19)(10,21,13)(14,15,20), (1,8,22)(3,21,7)(4,20,12)(5,14,17)(6,9,
16)(10,15,11)(13,23,25)(18,24,19), (1,8,15,17,24)(2,9,11,18,25)(3,10,12,19,21)(4,6,13,20,22)(5,7,14,16,23),
(1,8,12)(2,13,6)(3,18,5)(4,23,24)(9,17,25)(10,22,19)(11,21,14)(15,16,20), (1,9,17)(2,3,22)(4,16,7)(5,15,12)(6,
10,11)(8,23,21)(13,24,20)(14,18,25), (1,9,12,20,23)(2,10,13,16,24)(3,6,14,17,25)(4,7,15,18,21)(5,8,11,19,22),
(1,9,18)(2,14,12)(3,19,6)(4,24,5)(7,8,13)(10,23,25)(11,22,20)(15,17,21), (1,10,12)(2,4,17)(3,23,22)(5,11,7)(8,
24,16)(9,18,21)(13,25,15)(14,19,20), (1,10,14,18,22)(2,6,15,19,23)(3,7,11,20,24)(4,8,12,16,25)(5,9,13,17,21),
(1,10,24)(2,15,18)(3,20,12)(4,25,6)(7,9,19)(8,14,13)(11,23,21)(16,17,22), (1,11,13)(2,10,18)(3,4,23)(5,17,8)(6,
12,7)(9,24,22)(14,25,16)(15,19,21), (1,11,21,6,16)(2,12,22,7,17)(3,13,23,8,18)(4,14,24,9,19)(5,15,25,10,20),
(1,11,4)(2,16,23)(3,21,17)(5,6,10)(7,15,24)(8,20,18)(9,25,12)(13,14,19), (1,12,8)(2,6,13)(3,5,18)(4,24,23)(9,25,
17)(10,19,22)(11,14,21)(15,20,16), (1,12,23,9,20)(2,13,24,10,16)(3,14,25,6,17)(4,15,21,7,18)(5,11,22,8,19),
(1,12,10)(2,17,4)(3,22,23)(5,7,11)(8,16,24)(9,21,18)(13,15,25)(14,20,19), (1,13,3)(2,7,8)(4,25,18)(5,19,23)(6,
14,22)(9,21,12)(10,20,17)(11,15,16), (1,13,25,7,19)(2,14,21,8,20)(3,15,22,9,16)(4,11,23,10,17)(5,12,24,6,18),
(1,13,11)(2,18,10)(3,23,4)(5,8,17)(6,7,12)(9,22,24)(14,16,25)(15,21,19), (1,14,23)(2,8,3)(4,21,13)(5,20,18)(6,
15,17)(7,9,22)(10,16,12)(19,24,25), (1,14,22,10,18)(2,15,23,6,19)(3,11,24,7,20)(4,12,25,8,16)(5,13,21,9,17),
(1,14,17)(2,19,11)(3,24,10)(5,9,23)(6,8,18)(7,13,12)(15,22,25)(16,21,20), (1,15,18)(2,9,23)(4,22,8)(5,16,13)(6,
11,12)(7,10,17)(14,24,21)(19,25,20), (1,15,24,8,17)(2,11,25,9,18)(3,12,21,10,19)(4,13,22,6,20)(5,14,23,7,16),
(1,15,23)(2,20,17)(3,25,11)(4,5,10)(6,9,24)(7,14,18)(8,19,12)(16,22,21), (1,16,19)(2,15,24)(3,9,4)(5,22,14)(6,
17,13)(7,11,18)(8,10,23)(20,25,21), (1,16,6,21,11)(2,17,7,22,12)(3,18,8,23,13)(4,19,9,24,14)(5,20,10,25,15),
(1,16,3)(2,21,22)(4,6,15)(5,11,9)(7,20,23)(8,25,17)(12,14,24)(13,19,18), (1,17,14)(2,11,19)(3,10,24)(5,23,9)(6,
18,8)(7,12,13)(15,25,22)(16,20,21), (1,17,8,24,15)(2,18,9,25,11)(3,19,10,21,12)(4,20,6,22,13)(5,16,7,23,14),
(1,17,9)(2,22,3)(4,7,16)(5,12,15)(6,11,10)(8,21,23)(13,20,24)(14,25,18), (1,18,9)(2,12,14)(3,6,19)(4,5,24)(7,13,
8)(10,25,23)(11,20,22)(15,21,17), (1,18,10,22,14)(2,19,6,23,15)(3,20,7,24,11)(4,16,8,25,12)(5,17,9,21,13),
(1,18,15)(2,23,9)(4,8,22)(5,13,16)(6,12,11)(7,17,10)(14,21,24)(19,20,25), (1,19,4)(2,13,9)(3,7,14)(5,25,24)(6,
20,23)(10,21,18)(11,16,17)(12,15,22), (1,19,7,25,13)(2,20,8,21,14)(3,16,9,22,15)(4,17,10,23,11)(5,18,6,24,12),
(1,19,16)(2,24,15)(3,4,9)(5,14,22)(6,13,17)(7,18,11)(8,23,10)(20,21,25), (1,20,24)(2,14,4)(3,8,9)(5,21,19)(6,16,
18)(7,15,23)(10,22,13)(11,17,12), (1,20,9,23,12)(2,16,10,24,13)(3,17,6,25,14)(4,18,7,21,15)(5,19,8,22,11),
(1,20,22)(2,25,16)(3,5,15)(4,10,9)(6,14,23)(7,19,17)(8,24,11)(12,13,18), (1,21,2)(3,6,20)(4,11,14)(5,16,8)(7,25,
22)(9,10,15)(12,19,23)(13,24,17), (1,21,25)(2,20,5)(3,14,10)(4,8,15)(6,22,19)(7,16,24)(11,23,13)(12,17,18),
(1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,8)(4,24,19,14,9)(5,25,20,15,10), (1,22,8)(3,7,21)(4,12,20)(5,17,14)(6,
16,9)(10,11,15)(13,25,23)(18,19,24), (1,22,20)(2,16,25)(3,15,5)(4,9,10)(6,23,14)(7,17,19)(8,11,24)(12,18,13),
(1,22,18,14,10)(2,23,19,15,6)(3,24,20,11,7)(4,25,16,12,8)(5,21,17,13,9), (1,23,14)(2,3,8)(4,13,21)(5,18,20)(6,
17,15)(7,22,9)(10,12,16)(19,25,24), (1,23,15)(2,17,20)(3,11,25)(4,10,5)(6,24,9)(7,18,14)(8,12,19)(16,21,22),
(1,23,20,12,9)(2,24,16,13,10)(3,25,17,14,6)(4,21,18,15,7)(5,22,19,11,8), (1,24,20)(2,4,14)(3,9,8)(5,19,21)(6,18,
16)(7,23,15)(10,13,22)(11,12,17), (1,24,10)(2,18,15)(3,12,20)(4,6,25)(7,19,9)(8,13,14)(11,21,23)(16,22,17),
(1,24,17,15,8)(2,25,18,11,9)(3,21,19,12,10)(4,22,20,13,6)(5,23,16,14,7), (1,25,21)(2,5,20)(3,10,14)(4,15,8)(6,
19,22)(7,24,16)(11,13,23)(12,18,17), (1,25,5)(2,19,10)(3,13,15)(4,7,20)(6,21,24)(8,14,9)(11,22,18)(12,16,23),
(1,25,19,13,7)(2,21,20,14,8)(3,22,16,15,9)(4,23,17,11,10)(5,24,18,12,6) ], order := 75,
representatives := [ 1 ], schreierVector := [ -1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3,
3, 3, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 3, 1, 3, 1, 3, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 1, 1, 1,
3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 1, 1, 1 ] )
gap> A:=VertexNames(X1);
[ (), (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23), (2,25,6)(3,19,11)(4,13,16)(5,7,21)(8,
20,10)(9,14,15)(12,22,24)(17,23,18), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25),
(1,2,7)(3,12,25)(4,17,19)(5,22,13)(6,21,8)(9,11,20)(10,16,14)(18,24,23), (1,2,21)(3,20,6)(4,14,11)(5,8,16)(7,22,
25)(9,15,10)(12,23,19)(13,17,24), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24),
(1,3,13)(2,8,7)(4,18,25)(5,23,19)(6,22,14)(9,12,21)(10,17,20)(11,16,15), (1,3,16)(2,22,21)(4,15,6)(5,9,11)(7,23,
20)(8,17,25)(12,24,14)(13,18,19), (1,4,2,5,3)(6,9,7,10,8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23),
(1,4,19)(2,9,13)(3,14,7)(5,24,25)(6,23,20)(10,18,21)(11,17,16)(12,22,15), (1,4,11)(2,23,16)(3,17,21)(5,10,6)(7,24,
15)(8,18,20)(9,12,25)(13,19,14), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22),
(1,5,25)(2,10,19)(3,15,13)(4,20,7)(6,24,21)(8,9,14)(11,18,22)(12,23,16), (1,5,6)(2,24,11)(3,18,16)(4,12,21)(7,25,
10)(8,19,15)(9,13,20)(17,22,23), (1,6,7)(2,5,12)(3,24,17)(4,18,22)(8,25,11)(9,19,16)(10,13,21)(14,20,15),
(1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25), (1,6,5)(2,11,24)(3,16,18)(4,21,12)(7,10,
25)(8,15,19)(9,20,13)(17,23,22), (1,7,2)(3,25,12)(4,19,17)(5,13,22)(6,8,21)(9,20,11)(10,14,16)(18,23,24),
(1,7,13,19,25)(2,8,14,20,21)(3,9,15,16,22)(4,10,11,17,23)(5,6,12,18,24), (1,7,6)(2,12,5)(3,17,24)(4,22,18)(8,11,
25)(9,16,19)(10,21,13)(14,15,20), (1,8,22)(3,21,7)(4,20,12)(5,14,17)(6,9,16)(10,15,11)(13,23,25)(18,24,19),
(1,8,15,17,24)(2,9,11,18,25)(3,10,12,19,21)(4,6,13,20,22)(5,7,14,16,23), (1,8,12)(2,13,6)(3,18,5)(4,23,24)(9,17,
25)(10,22,19)(11,21,14)(15,16,20), (1,9,17)(2,3,22)(4,16,7)(5,15,12)(6,10,11)(8,23,21)(13,24,20)(14,18,25),
(1,9,12,20,23)(2,10,13,16,24)(3,6,14,17,25)(4,7,15,18,21)(5,8,11,19,22), (1,9,18)(2,14,12)(3,19,6)(4,24,5)(7,8,
13)(10,23,25)(11,22,20)(15,17,21), (1,10,12)(2,4,17)(3,23,22)(5,11,7)(8,24,16)(9,18,21)(13,25,15)(14,19,20),
(1,10,14,18,22)(2,6,15,19,23)(3,7,11,20,24)(4,8,12,16,25)(5,9,13,17,21), (1,10,24)(2,15,18)(3,20,12)(4,25,6)(7,9,
19)(8,14,13)(11,23,21)(16,17,22), (1,11,13)(2,10,18)(3,4,23)(5,17,8)(6,12,7)(9,24,22)(14,25,16)(15,19,21),
(1,11,21,6,16)(2,12,22,7,17)(3,13,23,8,18)(4,14,24,9,19)(5,15,25,10,20), (1,11,4)(2,16,23)(3,21,17)(5,6,10)(7,15,
24)(8,20,18)(9,25,12)(13,14,19), (1,12,8)(2,6,13)(3,5,18)(4,24,23)(9,25,17)(10,19,22)(11,14,21)(15,20,16),
(1,12,23,9,20)(2,13,24,10,16)(3,14,25,6,17)(4,15,21,7,18)(5,11,22,8,19), (1,12,10)(2,17,4)(3,22,23)(5,7,11)(8,16,
24)(9,21,18)(13,15,25)(14,20,19), (1,13,3)(2,7,8)(4,25,18)(5,19,23)(6,14,22)(9,21,12)(10,20,17)(11,15,16),
(1,13,25,7,19)(2,14,21,8,20)(3,15,22,9,16)(4,11,23,10,17)(5,12,24,6,18), (1,13,11)(2,18,10)(3,23,4)(5,8,17)(6,7,
12)(9,22,24)(14,16,25)(15,21,19), (1,14,23)(2,8,3)(4,21,13)(5,20,18)(6,15,17)(7,9,22)(10,16,12)(19,24,25),
(1,14,22,10,18)(2,15,23,6,19)(3,11,24,7,20)(4,12,25,8,16)(5,13,21,9,17), (1,14,17)(2,19,11)(3,24,10)(5,9,23)(6,8,
18)(7,13,12)(15,22,25)(16,21,20), (1,15,18)(2,9,23)(4,22,8)(5,16,13)(6,11,12)(7,10,17)(14,24,21)(19,25,20),
(1,15,24,8,17)(2,11,25,9,18)(3,12,21,10,19)(4,13,22,6,20)(5,14,23,7,16), (1,15,23)(2,20,17)(3,25,11)(4,5,10)(6,9,
24)(7,14,18)(8,19,12)(16,22,21), (1,16,19)(2,15,24)(3,9,4)(5,22,14)(6,17,13)(7,11,18)(8,10,23)(20,25,21),
(1,16,6,21,11)(2,17,7,22,12)(3,18,8,23,13)(4,19,9,24,14)(5,20,10,25,15), (1,16,3)(2,21,22)(4,6,15)(5,11,9)(7,20,
23)(8,25,17)(12,14,24)(13,19,18), (1,17,14)(2,11,19)(3,10,24)(5,23,9)(6,18,8)(7,12,13)(15,25,22)(16,20,21),
(1,17,8,24,15)(2,18,9,25,11)(3,19,10,21,12)(4,20,6,22,13)(5,16,7,23,14), (1,17,9)(2,22,3)(4,7,16)(5,12,15)(6,11,
10)(8,21,23)(13,20,24)(14,25,18), (1,18,9)(2,12,14)(3,6,19)(4,5,24)(7,13,8)(10,25,23)(11,20,22)(15,21,17),
(1,18,10,22,14)(2,19,6,23,15)(3,20,7,24,11)(4,16,8,25,12)(5,17,9,21,13), (1,18,15)(2,23,9)(4,8,22)(5,13,16)(6,12,
11)(7,17,10)(14,21,24)(19,20,25), (1,19,4)(2,13,9)(3,7,14)(5,25,24)(6,20,23)(10,21,18)(11,16,17)(12,15,22),
(1,19,7,25,13)(2,20,8,21,14)(3,16,9,22,15)(4,17,10,23,11)(5,18,6,24,12), (1,19,16)(2,24,15)(3,4,9)(5,14,22)(6,13,
17)(7,18,11)(8,23,10)(20,21,25), (1,20,24)(2,14,4)(3,8,9)(5,21,19)(6,16,18)(7,15,23)(10,22,13)(11,17,12),
(1,20,9,23,12)(2,16,10,24,13)(3,17,6,25,14)(4,18,7,21,15)(5,19,8,22,11), (1,20,22)(2,25,16)(3,5,15)(4,10,9)(6,14,
23)(7,19,17)(8,24,11)(12,13,18), (1,21,2)(3,6,20)(4,11,14)(5,16,8)(7,25,22)(9,10,15)(12,19,23)(13,24,17),
(1,21,25)(2,20,5)(3,14,10)(4,8,15)(6,22,19)(7,16,24)(11,23,13)(12,17,18), (1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,
8)(4,24,19,14,9)(5,25,20,15,10), (1,22,8)(3,7,21)(4,12,20)(5,17,14)(6,16,9)(10,11,15)(13,25,23)(18,19,24),
(1,22,20)(2,16,25)(3,15,5)(4,9,10)(6,23,14)(7,17,19)(8,11,24)(12,18,13), (1,22,18,14,10)(2,23,19,15,6)(3,24,20,11,
7)(4,25,16,12,8)(5,21,17,13,9), (1,23,14)(2,3,8)(4,13,21)(5,18,20)(6,17,15)(7,22,9)(10,12,16)(19,25,24),
(1,23,15)(2,17,20)(3,11,25)(4,10,5)(6,24,9)(7,18,14)(8,12,19)(16,21,22), (1,23,20,12,9)(2,24,16,13,10)(3,25,17,14,
6)(4,21,18,15,7)(5,22,19,11,8), (1,24,20)(2,4,14)(3,9,8)(5,19,21)(6,18,16)(7,23,15)(10,13,22)(11,12,17),
(1,24,10)(2,18,15)(3,12,20)(4,6,25)(7,19,9)(8,13,14)(11,21,23)(16,22,17), (1,24,17,15,8)(2,25,18,11,9)(3,21,19,12,
10)(4,22,20,13,6)(5,23,16,14,7), (1,25,21)(2,5,20)(3,10,14)(4,15,8)(6,19,22)(7,24,16)(11,13,23)(12,18,17),
(1,25,5)(2,19,10)(3,13,15)(4,7,20)(6,21,24)(8,14,9)(11,22,18)(12,16,23), (1,25,19,13,7)(2,21,20,14,8)(3,22,16,15,
9)(4,23,17,11,10)(5,24,18,12,6) ]
gap> B1:=A[1]*S1[1];
(2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23)
gap> B2:=B1*S1[2];
(1,2,7)(3,12,25)(4,17,19)(5,22,13)(6,21,8)(9,11,20)(10,16,14)(18,24,23)
gap> C:=[B1,B2];
[ (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23),
(1,2,7)(3,12,25)(4,17,19)(5,22,13)(6,21,
8)(9,11,20)(10,16,14)(18,24,23) ]
gap> C1:=[A[2],A[18]];
[ (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23),
(1,6,5)(2,11,24)(3,16,18)(4,21,12)(7,10,
25)(8,15,19)(9,20,13)(17,23,22) ]
gap> UndirectedEdges(X1);
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 13 ], [ 2, 3 ], [ 2, 18 ], [ 2, 61 ], [ 3, 19 ], [ 3, 74 ], [ 4, 5 ], [ 4, 6 ],
[ 4, 7 ], [ 5, 6 ], [ 5, 21 ], [ 5, 64 ], [ 6, 22 ], [ 6, 62 ], [ 7, 8 ], [ 7, 9 ], [ 7, 10 ], [ 8, 9 ], [ 8, 24 ],
[ 8, 67 ], [ 9, 25 ], [ 9, 65 ], [ 10, 11 ], [ 10, 12 ], [ 10, 13 ], [ 11, 12 ], [ 11, 27 ], [ 11, 70 ],
[ 12, 28 ], [ 12, 68 ], [ 13, 14 ], [ 13, 15 ], [ 14, 15 ], [ 14, 30 ], [ 14, 73 ], [ 15, 16 ], [ 15, 71 ],
[ 16, 17 ], [ 16, 18 ], [ 16, 34 ], [ 17, 18 ], [ 17, 20 ], [ 17, 29 ], [ 18, 33 ], [ 19, 20 ], [ 19, 21 ],
[ 19, 37 ], [ 20, 21 ], [ 20, 23 ], [ 21, 36 ], [ 22, 23 ], [ 22, 24 ], [ 22, 40 ], [ 23, 24 ], [ 23, 26 ],
[ 24, 39 ], [ 25, 26 ], [ 25, 27 ], [ 25, 43 ], [ 26, 27 ], [ 26, 29 ], [ 27, 42 ], [ 28, 29 ], [ 28, 30 ],
[ 28, 31 ], [ 29, 30 ], [ 30, 45 ], [ 31, 32 ], [ 31, 33 ], [ 31, 49 ], [ 32, 33 ], [ 32, 35 ], [ 32, 44 ],
[ 33, 48 ], [ 34, 35 ], [ 34, 36 ], [ 34, 52 ], [ 35, 36 ], [ 35, 38 ], [ 36, 51 ], [ 37, 38 ], [ 37, 39 ],
[ 37, 55 ], [ 38, 39 ], [ 38, 41 ], [ 39, 54 ], [ 40, 41 ], [ 40, 42 ], [ 40, 58 ], [ 41, 42 ], [ 41, 44 ],
[ 42, 57 ], [ 43, 44 ], [ 43, 45 ], [ 43, 46 ], [ 44, 45 ], [ 45, 60 ], [ 46, 47 ], [ 46, 48 ], [ 46, 65 ],
[ 47, 48 ], [ 47, 50 ], [ 47, 59 ], [ 48, 61 ], [ 49, 50 ], [ 49, 51 ], [ 49, 68 ], [ 50, 51 ], [ 50, 53 ],
[ 51, 64 ], [ 52, 53 ], [ 52, 54 ], [ 52, 71 ], [ 53, 54 ], [ 53, 56 ], [ 54, 67 ], [ 55, 56 ], [ 55, 57 ],
[ 55, 74 ], [ 56, 57 ], [ 56, 59 ], [ 57, 70 ], [ 58, 59 ], [ 58, 60 ], [ 58, 62 ], [ 59, 60 ], [ 60, 73 ],
[ 61, 62 ], [ 61, 63 ], [ 62, 63 ], [ 63, 66 ], [ 63, 75 ], [ 64, 65 ], [ 64, 66 ], [ 65, 66 ], [ 66, 69 ],
[ 67, 68 ], [ 67, 69 ], [ 68, 69 ], [ 69, 72 ], [ 70, 71 ], [ 70, 72 ], [ 71, 72 ], [ 72, 75 ], [ 73, 74 ],
[ 73, 75 ], [ 74, 75 ] ]
Then when I compute A[1]*S1[1] which is the vertex number 2 in Cayley graph, and then multiply it again by S1[2], I get the value of the vertex number 5 instead of vertex number 18, which should be the correct one according to the list of Undirected edges.
My question is why am I getting an edge from 2 to 5 instead of 2 to 18? Is there something missing when I take the multiplication of elements of the semidirect product?
Please help me with this question.
Definition of Cayley graph:
Let $G$ be a group and $S \subseteq G$ be a generating set of $G$. The Cayley digraph of $G$ with respect to $S$, $X=\overrightarrow{Cay}(G,\: S)$ is a graph whose vertices are the elements of $G$ and there is an edge from $g$ to $gs$ whenever $g \in G$ and $s \in S$. The Cayley graph, $X=Cay(G,\: S)$ is the undirected graph whose vertices are the elements of $G$ and there is an edge from $g$ to $gs$ and from $g$ to $gs^{-1}$ whenever $g \in G$ and $s \in S$.
Thanks a lot in advance.