Do the two presentations below, $$G=\langle d,v \mid dv^2d=vdv, dv^3d=v^2 \rangle$$ and $$\langle r,s,t \mid r^2=s^3=t^5=rst \rangle = \langle s,t \mid (st)^2=s^3=t^5 \rangle,$$ define the same group?
Motivation: I am working on Poincaré homology sphere $X$, constructed by identifying the opposite faces of a dodecahedron using the minimal clockwise twist to line up the faces. I was able to verify that its homology groups are the same as the 3-sphere, and now I would like to compute its fundamental group. Using van Kampen theorem, I found the first presentation for $\pi_1(X)$; however, I did not succeed in identifying it with the binary icosahedral group (hoping I computed correctly the fundamental group), given by the second presentation.
Nota Bene: Using a mathematical software, I checked that $G$ has order $120$.