For s≥5, every connected (G,s)-geodesic transitive graph of girth 2s-2 or 2s-1 either has a normal quotient of the same girth, or is the Foster graph covering the Tutte 8-cage; in the quasiprimitive case G is almost simple.
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Finite $s$-geodesic transitive graphs under certain girths
For s≥5, every connected (G,s)-geodesic transitive graph of girth 2s-2 or 2s-1 either has a normal quotient of the same girth, or is the Foster graph covering the Tutte 8-cage; in the quasiprimitive case G is almost simple.