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One classic way to depict the effect of a plane linear transformation is to see what it does to a stylized drawing of a cat's face. First, a utility to transform points and a function to reflect across the vertical axis: pointQ[p_] := VectorQ[p, NumberQ] && Length[p] == 2 image[T_, object_] := object /. pt_?pointQ :> T[pt] reflect[p_?PointQ] ...


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Something like this? g = FiniteGroupData[{"DihedralGroup", 4}, "CayleyGraph"]; g1 = SetProperty[g, VertexLabels -> "Name"]; l = PropertyValue[{g1, #}, VertexCoordinates] & /@ VertexList[g1]; m = {{2, 1}, {-1, 2}}; Framed@Row[{g1, SetProperty[g1, VertexCoordinates -> (m.# & /@ l)]}]



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