Sentence:  A <WSD>"flag</WSD>" of "P" is a sequence ("F<sub>0</sub>, F<sub>1</sub>, ..., F<sub>m</sub>") of faces of "P" such that dim "F<sub>i</sub>" = "i" for each "i" and "F<sub>i</sub>" is a side of "F"<sub>"i"+1</sub> for each "i" < "m"....A "regular polytope" in "X" is a polytope "P" in "X" whose group of symmetries in <"P"> acts transitively on its <WSD>flags</WSD>. 
Meaning of the word marked with <WSD> tags:  A sequence of faces of a given polytope, one of each dimension up to that of the polytope (formally, though in practice not always explicitly, including the null face and the polytope itself), such that each face in the sequence is part of the next-higher dimension face.