A quartic vertex-transitive graph is a quartic graph that is vertex transitive. Read and Wilson (1998, pp. 164-166) enumerate all connected quartic vertex-transitive graphs on 19 and fewer nodes, some of which are illustrated above.
The quartic symmetric graphs are a special case of the quartic vertex-transitive graphs (i.e., those that are also edge-transitive).
Classes of connected quartic vertex-transitive graphs include the antiprism graphs. Specific
cases are summarized in the following table. In particular, Qt31 can be constructed
as the distance-3 graph distance graph of
the Heawood graph or as the Levi
graph of the biplane on 7 points (DistanceRegular.org).
It is also a distance-regular graph with
intersection array 
that is also distance-transitive.
| vertices | id | graph |
| 5 | Qt1 | pentatope graph  |
| 6 | Qt2 | octahedral graph  |
| 8 | Qt5 | (2,4)-rook graph  |
| 8 | Qt6 | complete
bipartite graph  |
| 9 | Qt9 | generalized quadrangle  |
| 10 | Qt11 | crown
graph  |
| 12 | Qt20 | cuboctahedral graph |
| 14 | Qt31 | Heawood graph distance-3 graph distance graph |
| 15 | Qt39 | Petersen line graph  |
| 16 | Qt51 | tesseract graph  |
| 18 | Qt66 | Bouwer graph  |
Incidence Graph of Biplane on 7 Points."