| Summary: | The quintic family must be the most studied family of Calabi-Yau threefolds.Particularly symmetric members of this family are known to admit quotientsby freely acting symmetries isomorphic toZ5×Z5. The correspondingquotient manifolds may themselves be symmetric. That is, they may admitsymmetries that descend from the symmetries that the manifold enjoysbefore the quotient is taken. The formalism for identifying these symmetrieswas given a long time ago by Witten and instances of these symmetricquotients were given also, for the familyP7[2,2,2,2], by Goodman andWitten. We rework this calculation here, with the benefit of computerassistance, and provide a complete classification. Our motivation is largely todevelop methods that apply also to the analysis of quotients of other CICYmanifolds, whose symmetries have been classified recently. For theZ5×Z5quotients of the quintic family, our list contains families of smooth manifoldswith symmetryZ4,Dic3and Dic5, families of singular manifolds with fourconifold points, with symmetryZ6andQ8, and rigid manifolds, each with atleast a curve of singularities, and symmetryZ10. We intend to return to thecomputation of the symmetries of the quotients of other CICYs elsewhere.
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