Infinite splitting in the syzygies of quaternionic groups

Let $${F} \, = \, ( { {→ } ^{∂ ₙ₊₁}} {F}_n { {→ } ∂ _n} {F}ₙ₋₁{ {→ } ^{∂ ₙ₋₁}} { {→ } ∂ _1} {F}_0 → {R} → 0)$$ F = ( ⋯ → ∂ n + 1 F n → ∂ n F n - 1 → ∂ n - 1 ⋯ ⋯ → ∂ 1 F 0 → R → 0 ) be a free resolution over the group ring $${R}[Φ ]$$ R [ Φ ] where $${R}$$ R is commutative and $$Φ $$ Φ is finite. The $$nᵗʰ$$ n th syzygy $$Ω _n^{{R}[Φ ]}$$ Ω n R [ Φ ] is the stable class of $$Im(∂ _n)$$ Im ( ∂ n ) and has a tree structure with roots which do not extend infinitely downwards. We show that $$Ω _3^{{R}[Q₈ₚ]}$$ Ω 3 R [ Q 8 p ] has infinitely many isomorphically distinct modules at the minimal level when $$\,{R} = {Z}[C_∞ ]$$ R = Z [ C ∞ ] is the integral group ring of the infinite cyclic