We construct locally homogeneous six-dimensional nearly Kähler manifolds as quotients of homogeneous nearly Kähler manifolds M by freely acting finite subgroups of Aut 0 (M). We show that non-trivial such groups do only exists if M = S 3 × S 3 . In that case, we classify all freely acting subgroups of Aut 0 (M) = SU(2) × SU(2) × SU(2) of the form A × B, where A ⊂ SU(2) × SU(2) and B ⊂ SU(2).

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We give a systematic way to construct almost conjugate pairs of finite subgroups of Spin(2n + 1) and Pin(n) for n ∈ N sufficiently large. As a geometric application, we give an infinite family of pairs M dn 1 and M dn 2 of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions d n > 6. We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension.