deficiency zero groups in which prime power of generators are central

The infinite family of groups defined by the presentation gp = ⟨x, y|x^p = y^p , xyx^my^n = 1⟩, in which p is a prime in {2, 3, 5} and m, n ∈ n0, will be considered and finite and infinite groups in the family will be determined. for the primes p = 2, 3 the group gp is finite and for p = 5, the group is finite if and only if m ≡ n ≡ 1 (mod 5) is not the case.