Geodesic
Rationality of fields of invariants of some four-dimensional linear groups, and an equivariant construction related to the Segre cubic
Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 169-183

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Let $G\subset SL(4)$ be a finite primitive linear group. We prove that if $G$ contains a normal subgroup of order 32 then the quotient variety $\mathbf P^3/G$ is birationally isomorphic to $X/G$, where $X$ is the Segre cubic. We also prove the rationality of $\mathbf P^3/G$ for a large class of such groups (in particular, solvable groups). 
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     author = {I. Ya. Kolpakov-Miroshnichenko and Yu. G. Prokhorov},
     title = {Rationality of fields of invariants of some four-dimensional linear groups, and an equivariant construction related to the {Segre} cubic},
     journal = {Sbornik. Mathematics},
     pages = {169--183},
     publisher = {mathdoc},
     volume = {74},
     number = {1},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1993_74_1_a12/}
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I. Ya. Kolpakov-Miroshnichenko; Yu. G. Prokhorov. Rationality of fields of invariants of some four-dimensional linear groups, and an equivariant construction related to the Segre cubic. Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 169-183. http://geodesic.mathdoc.fr/item/SM_1993_74_1_a12/