By Bass's theorem, it has a polynomial growth rate of order 4. One can generate any element through  If one takes "a, b, c" in Z/"p" Z for an odd prime "p", then one has the Heisenberg group modulo p. It is a group of order "p"3 with generators "x,y" and relations:  Analogues of Heisenberg groups over "finite" fields of odd prime order "p" are called extra special groups, or more properly, extra special groups of exponent "p". More generally, if the derived subgroup of a group "G" is contained in the center "Z" of "G", then the map from "G/Z" × "G/Z" → "Z" is a skew-symmetric bilinear operator on abelian groups. However, requiring that "G/Z" to be a finite vector space requires the Frattini subgroup of "G" to be contained in the center, and requiring that "Z" be a one-dimensional vector space over Z/"p" Z requires that "Z" have order "p", so if "G" is not abelian, then "G" is extra special. If "G" is extra special but does not have exponent "p", then the general construction below applied to the symplectic vector space "G/Z" does not yield a group isomorphic to "G". The Heisenberg group modulo 2 is of order 8 and is isomorphic to the dihedral group D4 (the symmetries of a square). Observe that if  Then  and  The elements "x" and "y" correspond to reflections (with 45° between them), whereas "xy" and "yx" correspond to rotations by 90°. The other reflections are "xyx" and "yxy", and rotation by 180° is "xyxy" (="yxyx"). The Lie algebra formula_16 of the Heisenberg group formula_1 (over the real numbers) is the space of formula_18 matrices of the form with formula_20. The following three elements form a basis for formula_16: The basis elements satisfy the commutation relations: The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics: where formula_25 is the position operator, formula_26 is the momentum operator, and formula_27 is Planck's constant.