Geodesic
Double quadrics with large automorphism groups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 167-190

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We classify nodal Fano threefolds that are double covers of smooth quadrics branched over intersections with quartics and are acted on by finite simple non-abelian groups. We also study their rationality. 
@article{TM_2016_294_a9,
     author = {Victor V. Przyjalkowski and Constantin A. Shramov},
     title = {Double quadrics with large automorphism groups},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {167--190},
     publisher = {mathdoc},
     volume = {294},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_294_a9/}
}
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Victor V. Przyjalkowski; Constantin A. Shramov. Double quadrics with large automorphism groups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 167-190. http://geodesic.mathdoc.fr/item/TM_2016_294_a9/