Using the substitution T = R₁P we come to an algebraic equivalent of the previous presentation of the group mg - a new presentation of the same group:

Instead of the asymmetric figure, which under the action of the group mg gives the frieze pattern, by considering the orbit of the closure of a fundamental region of the group mg we obtain the corresponding frieze tiling. The fundamental region of the group mg and all other frieze symmetry groups, is unbounded and allows the variation of all boundaries which do not belong to reflection lines. Figure 1.21 shows two of these possibilities.

The Cayley diagram of the group mg is derived as the orbit of a point in general position with respect to the group mg. Instead of a direct mutual linking of all vertexes (i.e. orbit points) and obtaining the complete graph, we can, aiming for simplification, link only the homologous points of the group generators. By denoting with the broken oriented line the glide reflection P, and with the dotted non-oriented line the reflection R₁, we get the Cayley diagram which corresponds to the first presentation of the group mg (Figure 1.22a).

By an analogous procedure we come to the graph which corresponds to its second presentation with the generator set {R₁,T}, where a half-turn is indicated with the dot-dash line (Figure 1.22b).

Let us note also, that the defining relations can be read off directly from the graph of the group. Each cycle, i.e. closed path in which the beginning point coincides with the endpoint, corresponds to a relation between the elements of the group and vice versa. Cayley diagrams (graphs of the groups) may also very efficiently serve to determine the subgroups of the given symmetry group. Namely, every connected subgraph of the given graph satisfying the following condition determines a certain subgroup of the group discussed, and vice versa. The condition in question is: an element (transformation) is included in the subgraph either wherever it occurs, or not at all (i.e. it is deleted). Of course, to be able to determine all the subgroups of a given group, it is necessary to use its complete graph as the basis for defining the subgraphs.

Since in the group mg there are indirect isometries, this group does not give enantiomorphic modifications. For the groups consisting only of direct symmetries, the enantiomorphic modifications can be obtained by applying the "left" (e.g., b) and "right" (d) form of an elementary asymmetric figure. For example, for the group 11, generated by a translation X, this results in the enantiomorphic friezes: bbbbbbbbbbbbbbbbbbbbbbbbbbbbb and dddddddddddddddddddddddddddddd. The translation axis l of the group mg is non-polar, because there exists an indirect transformation, the reflection R₁ for which the relation R₁(l) = -l holds. Rotations of the order 2 in the group mg are polar because each circle c drawn around the center of rotation of the order 2 is invariant only with respect to this rotation and to the identity transformation E, so that the group C₂ ( 2) (generated by the half-turn T) of transformations preserving the circle c invariant, a rosette subgroup C₂ (2) of the group mg, consists of direct transformations. Besides the rosette subgroups C₂ ( 2), the group mg has also the rosette subgroups D₁ (m), namely the one generated by the reflection R₁, or by its conjugates.

The group mg contains as subgroups the following symmetry groups of friezes: p1 (11) generated by the translation X = P², p1g (1g) generated by the glide reflection P, pm1 (m1) generated by the translation X and the reflection R₁, and itself. Besides the list of all frieze groups, subgroups of the group mg, the table of the minimal indexes of subgroups of the given group points out the possible desymmetrizations which lead to this subgroup. In particular, considering the use of antisymmetry and color- symmetry desymmetrizations, from this table we can see that antisymmetry desymmetrizations of group mg result in the subgroups of the index 2: 1g, 12 and m1. This can be achieved by a black-white coloring (or, e.g., 1-2 indexing) according to the laws of antisymmetry, using the following systems of (anti)generators: {P,e₁R₁} or {e₁R₁,e₁T} for obtaining the antisymmetry desymmetrization mg/1g; {e₁P,R₁} or {R₁,e₁T} for obtaining the antisymmetry desymmetrization mg/m1; {e₁P,e₁R₁} or {e₁R₁,T} for obtaining the antisymmetry desymmetrization mg/12, where e₁ = (12), i.e. the group of color permutations P_N = P₂ = C₂ (Figure 1.23).

The junior antisymmetry groups obtained can be understood also as adequate visual interpretations of the symmetry groups of bands G₃₂₁ - as the Weber diagrams of the symmetry groups of bands p2₁11, pm11 and p112 respectively. In this case the alternation of colors white-black is understood in the sense "above-under" the invariant plane of the frieze, i.e. as the identification of the antiidentity transformation e₁ with the plane reflection in the invariant plane of the group mg. The seven generating symmetry groups of friezes G₂₁, seven senior antisymmetry groups and seventeen junior antisymmetry groups correspond to the 31 groups of symmetry of bands, offering complete information on their presentations and structures.

Using N = 4 colors and the system of colored generators {c₁P,c₂R₁} or {c₂R₁,c₁c₂T}, we get the color-symmetry desymmetrization mg/11, where c₁ = (12)(34) and c₂ = (13)(24); hence, the group of color permutations is P_N = P₄ = C₂×C₂ = D₂ (Figure 1.24).

In all the antisymmetry and color-symmetry desymmetrizations mentioned, for which the group P_N is regular, the subgroup H derived by the desymmetrization is a normal subgroup of the group mg (1g, m1, 12, 12). Because of this, complete information on the antisymmetry or colored symmetry group, i.e. on the corresponding desymmetrization, is given by the number N and by the group/subgroup symbol G/H. The next case of coloring with N = 3 colors, the irregular group P_N and the subgroup H which is not a normal subgroup of the group G, demands the symbols G/H/H₁. In this case, besides the number N, the group of colored symmetry G^*, i.e. the corresponding color-symmetry desymmetrization is uniquely defined by the generating group G, the stationary subgroup H of G^*, which maintains every individual index (color) unchanged and its symmetry subgroup H₁ which is the final result of the color-symmetry desymmetrization. The index of the subgroup H in the group G is equal to N and the product of the index of the subgroup H₁ in group H and the number N is equal to the order of the group of color permutations P_N, i.e. [G:H] = N, [H:H₁] = N₁, and the order of the group P_N is NN₁.

As an example of the irregular case we can use the color-symmetry desymmetrization of the group mg obtained by N = 3 colors, i.e. by the system of colored generators: {c₁P,c₂R₁} or {c₂R₁,c₁c₂T}, which results in the color-symmetry desymmetrization mg/mg/1g, where c₁ = (123), c₂ = (23), P_N = P₃ = D₃ and [mg: mg]=3, [mg:1g]=2. This color-symmetry desymmetrization mg/mg/1g, N = 3 is shown on Figure 1.25a, while the stationary subgroup H (mg) which maintains each individual index (color) unchanged is singled out on Figure 1.25b. All cases of subgroups which are not normal subgroups of the given group are denoted in the tables of (minimal) indexes of subgroups in groups by italic indexes (e.g., [mg:mg]=3).

In terms of construction, for frieze group mg we can also distinguish the rosettal method of construction - the multiplication of a rosette with the symmetry group C₂ (2) (generated by the half-turn T) or D₁ ( m) (generated by the reflection R₁) by the glide reflection P (Figure 1.26a, b). Like all other symmetry groups of friezes, the group mg is the subgroup of the maximal symmetry group of friezes mm generated by reflections. Since it is the normal subgroup of the index 2, the antisymmetry desymmetrization of the generating group mm with a set of generators {X,R,R₁} or {R,R₁,R₂} where X is the translation, R the reflection in translation axis line, and R₁, R₂ reflections with reflection lines perpendicular to the translation axis, can be used.

By means of the system of (anti)generators: {e₁X,e₁R,R₁} = {e₁X, e₁R,e₁R₁} or {e₁R,R₁,e₁R₂} the antisymmetry desymmetrization mm/mg is obtained (Figure 1.27), where e₁ = (12), P_N = P₂ = C₂, [mm:mg] = 2.

Many visual properties of the group mg, e.g., a relative constructional and visual simplicity of corresponding friezes conditioned by a high degree of symmetry, specific balance of the stationariness conditioned by the presence of reflections, by the non-polarity of the glide reflection axis, by the absence of enantiomorphism, and the dynamism conditioned by the presence of glide reflection and by polar, oriented rotations, are the direct consequences of the algebraic-geometric characteristics mentioned. Also, the different possibilities that the group mg offers, e.g., the possibilities for antisymmetry and color-symmetry desymmetrizations, the ways of varying the form of the fundamental region, construction possibilities etc., become evident after the analysis of this symmetry group of friezes from the point of view of the theory of symmetry.