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Can Georges Graph

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(or any other nonhamiltonian bicubic graph ) be embedded on an oriented surface of genus -2, i.e. a double torus?

If it helps, it would have $F=E+\chi-V=75-2-50=23$ faces...

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A double torus is a sphere with two handles and two holes, so its genus (which is the number of holes, see, for instance, p. 133 in “Chromatic Graph Theory” by Gary Chartrand and Ping Zhang, CRC Press, 2009) is $2$. My search at the House of Graphs of bicubic non-hamiltonian graphs of genus at most two provided the following three graphs: #6923, #27548, and #27678.

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