Abstract regular polytopes of $G_2(q)$ in even characteristic

For every even prime power $q>2$, we explicitly construct a rank five string C-group representation of the Chevalley simple group $G_2(q)$ using its natural seven-dimensional matrix representation. We also give a census for the string C-group representations of $G_2(q)$ of ranks four and above for every prime power $3\le q\le 11$; providing computational evidence that rank five string C-group representations of $G_2(q)$ are rare and that this construction is exceptional among the currently known examples.