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More specifically:
When were the symmetries of polygons/solids first presented as groups in Cayley tables?

Textbooks often use the symmetries of polygons/solids to introduce group theory, however, I have found little historical material on the early discoveries relating groups and geometry.

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    $\begingroup$ Have you looked into the work of Klein on group theory, particularly the Erlangen program? $\endgroup$
    – KCd
    Commented May 8, 2016 at 0:22

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The earliest applications of groups in geometry are dealing with continuous groups (Lie, Klein 1872). You seem to be talking about discrete groups. Their earliest application in a geometric subject was in crystallography in the 1879-1900. See "Crystallographic groups" in Wikipedia.

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