Smallest semisymmetric cubic graphs for five Goldschmidt types

Determine the smallest connected finite semisymmetric cubic graphs of types 2^2, 2^3, 3, 4, and 5.

Background

The paper catalogs finite connected cubic edge-transitive graphs according to the Goldschmidt and Djoković–Miller amalgam types of their automorphism groups. In the semisymmetric case, the authors provide smallest known examples for several types and identify types for which no example occurs among graphs of order at most 10000.

For semisymmetric types 2^2, 2^3, 3, 4, and 5, the paper reports either no examples up to order 10000 or only substantially larger examples, but does not determine the smallest possible graph. The unresolved task is therefore to establish the minimum order and corresponding graph for each of these five types.