Distance-regular graph with intersection array $\{27, 20, 7; 1, 4, 21\}$ does not exist

In the class of distance-regular graphs of diameter $3$ there are $5$ intersection arrays of graphs with at most $28$ vertices and noninteger eigenvalue. These arrays are $\{18, 14, 5; 1, 2, 144\}$, $\{18, 15, 9; 1, 1, 10\}$, $\{21, 16, 10; 1, 2, 12\}$, $\{24, 21, 3; 1, 3, 18\}$, and $\{27, 20, 7; 1, 4, 21\}$. Automorphisms of graphs with intersection arrays $\{18, 15, 9; 1, 1, 10\}$ and $\{24, 21, 3; 1, 3, 18\}$ were found earlier by A. A. Makhnev and D. V. Paduchikh. In this paper, it is proved that a graph with the intersection array $\{27, 20, 7; 1, 4, 21\}$ does not exist.