Smallest group which is $(2,5,5)$-generated.

Question

A finite group $G$ is said to be $(2,5,5)$-generated if there are elements $x,y,z \in G$ with $|x| = 2, |y| = |z| = 5$ so that $G = \langle x, y, z \rangle$ and $xyz = 1$. An example of $(2,5,5)$-generated group is a non-abelian group of order $80$, which is solvable. Are there any example of groups of smaller size?

Answer 1

$A_5$ of order $60$ is $(2,5,5)$-generated with, for example, $x=(1, 2)(3, 4)$, $y=(1, 4, 2, 3, 5)$, $z=(1, 5, 4, 2, 3)$ (that's composing left to right; swap $y$ and $z$ if you prefer to compose right to left).

Since a $(2,5,5)$-group has order divisible by $10$ and does not have a normal Sylow $5$-subgroup, 60 is the smallest possible order.

All other groups of order 60 are solvable and have a normal Sylow $5$-subgroup, so $80$ is the smallest possible order of a solvable $(2,5,5)$-generated group.