It is known that the moduli space of smooth Fano–Mukai fourfolds $$V_{18}$$ V18 of genus 10 has dimension one. We show that any such fourfold is a completion of $${\mathbb {C}}^4$$ C4 in two different ways. Up to isomorphism, there is a unique fourfold $$V_{18}^{{\mathrm {s}}}$$ V18s acted upon by $${\mathrm{SL}}_2({\mathbb {C}})$$ SL2(C) . The group is a semidirect product . Furthermore, $$V_{18}^{{\mathrm {s}}}$$ V18s is a $${\mathrm{GL}}_2({\mathbb {C}})$$ GL2(C) -equivariant completion of $${\mathbb {C}}^4$$ C4 , and as well of $${\mathrm{GL}}_2({\mathbb {C}})$$ GL2(C) . The restriction of the $${\mathrm{GL}}_2({\mathbb {C}})$$ GL2(C) -action on $$V_{18}^{{\mathrm {s}}}$$ V18s to yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold $$V_{18}^{\mathrm a}$$ V18a such that the group is a semidirect product . For a Fano–Mukai fourfold $$V_{18}$$ V18 isomorphic neither to $$V_{18}^{{\mathrm {s}}}$$ V18s , nor to $$V_{18}^{\mathrm a}$$ V18a , the group is a semidirect product of $$({{\mathbb {G}}}_{\mathrm {m}})^2$$ (Gm)2 and a finite cyclic group whose order is a factor of 6. Besides, we establish that the affine cone over any polarized Fano–Mukai variety $$V_{18}$$ V18 is flexible in codimension one, and flexible if $$V_{18}=V_{18}^{{\mathrm {s}}}$$ V18=V18s .