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Let $ R $ be a ring with identity. The commuting graph of $ R $ is the graph associated to $ R $ whose vertices are non-central elements in $ R $, and distinct vertices $ A $ and $ B $ are adjacent if and only if $ AB = BA $. In this paper, we completely determine the automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $, where $ \mathbb{Z}_{p^{s}} $ is the ring of integers modulo $ p^{s} $, $ p $ is a prime and $ s $ is a positive integer.

In this paper, we classified Hadamard 2-(51, 25, 12) designs having a non-abelian automorphism group of order 10, i.e., Frob10 ≌ Z5: Z2, where a subgroup isomorphic to Z2 fixed setwise all the Z5-orbits of the sets of points and blocks. Furthermore, we classified 2-(59, 29, 14) designs having a non-abelian automorphism group of order 14, i.e., Frob14 ≌ Z7: Z2, where a subgroup isomorphic to Z2 fixed setwise all the Z7-orbits of the sets of points and blocks. We also showed that there was no Hadamard 2-(59, 29, 14) design with automorphism group Frob21 ≌ Z7: Z3, where a subgroup of order 3 fixed setwise all the Z7-orbits of points and blocks. Additionally, we used Hadamard 2-(59, 29, 14) designs obtained to construct ternary linear codes and linear codes over the finite field of order 5. We constructed ternary self-dual codes and self-dual codes over the finite field of order 5 from the corresponding Hadamard matrices of order 60. [ABSTRACT FROM AUTHOR]

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

In this paper we settle the question of whether a finite-dimensional vector space V over F p , with p an odd prime, and the family of all the k-sets of elements of V summing up to a given element x, form a 1- (v , k , λ 1) or a 2- (v , k , λ 2) block design, and, in either case, we find a closed form for λ i and characterize the automorphism group. The question is discussed also in the case where the elements of the k-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The "twin case" p = 2 , which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author. [ABSTRACT FROM AUTHOR]

This paper contributes to the classification of flag-transitive symmetric 2- (v , k , λ) designs with λ prime. We investigate the structure of flag-transitive, point-quasiprimitive automorphism groups (G) of such 2-designs by applying the classification of quasiprimitive permutation groups. It is shown that the automorphism groups (G) have either an abelian socle or a non-abelian simple socle. Moreover, according to the classification of finite simple groups, we demonstrate that point-quasiprimitivity implies point-primitivity of G, except when the socle of G is P S L n (q) . [ABSTRACT FROM AUTHOR]

This paper presents a criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves and describes its application via a novel example of a plane curve with non-collinear Galois points. In addition, this paper presents a new characterisation of the Fermat curve in terms of non-collinear Galois points. [ABSTRACT FROM AUTHOR]

Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, $q$ -ary optimal locally repairable codes from subcodes of Reed-Solomon codes given by Tamo and Barg have length upper bounded by $q$. Recently, it was shown through extension of construction by Tamo and Barg that length of $q$ -ary optimal locally repairable codes can be $q+1$ by Jin et al.. Surprisingly it was shown by Barg et al. that, unlike classical MDS codes, $q$ -ary optimal locally repairable codes could have length bigger than $q+1$. Thus, it becomes an interesting and challenging problem to construct $q$ -ary optimal locally repairable codes of length bigger than $q+1$. In this paper, we make use of rich algebraic structures of elliptic curves to construct a family of $q$ -ary optimal locally repairable codes of length up to $q+2\sqrt {q}$. It turns out that locality of our codes can be as big as 23 and distance can be linear in length. [ABSTRACT FROM AUTHOR]

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]

Locally repairable codes with more than one recovering set are demanded in the application to distributed storage. For each failure node (or disk), it is desired to have as many recovering sets as possible. In this paper, we make use of automorphisms of rational function fields to construct locally repairable codes with multiple recovering sets. Although we focus on two recovering sets, our construction can be easily generalized to the case of multiple recovering sets. In particular, we obtain a class of locally repairable codes with minimum distance only 1 less than the upper bound. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a connected graph with the vertex-set V and the edgeset E. The subdivision graph S(G) of the graph G is obtained from G by adding a vertex in the middle of every edge of G. In this paper, we investigate some properties of the graphs S (G) and L(S (G)), where L(S (G)) is the line graph of S (G). We will see that S (G) and L(S (G)) inherit some properties of G. For instance, we show that if G Cn, then Aut(G) = Aut(L(S(G))) (as abstract groups), where Cn is the cycle of order n. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new counting polynomial, namely the orbit polynomial. It is well-known that this polynomial has a unique positive zero δ in the interval [0, 1]. The aim of this paper is to study the specific properties of this polynomial and then determine the location of this root for several classes of complex networks to compare with other graphical measures. Additionally, we compare the unique positive zero measure with several well-known centrality graph measures. [ABSTRACT FROM AUTHOR]

Let S be a finite thick generalized quadrangle, and suppose that G is an automorphism group of S . If G acts primitively on both the points and lines of S , then it is known that G must be almost simple. In this paper, we show that if the socle of G is PSL (2 , q) with q ≥ 4 , then q = 9 and S is the unique generalized quadrangle of order 2. [ABSTRACT FROM AUTHOR]

This paper contributes to the classification of flag-transitive 2-designs. Let D be a 2- (v , k , λ) design with prime square replication number. Suppose that G is a flag-transitive, point-primitive automorphism group of D with socle an alternating group. We prove that D is the unique 2-(10, 6, 5) design with G = A l t (6) or Sym(6). [ABSTRACT FROM AUTHOR]

A graph is said to be symmetric if its full automorphism group is transitive on its arcs. In this paper, a complete classification of pentavalent symmetric graphs of order 8 p 2 is given for each prime p. It is shown that a connected pentavalent symmetric graph of order 8 p 2 exists if and only if p = 2 or 3, and there are four such graphs up to isomorphism. [ABSTRACT FROM AUTHOR]

Let G be a group and AutΦ(G) denote the group of all automorphisms of G centralizing G/Φ(G) elementwise. In this paper, we give a necessary and sufficient condition on a finite p-group G for the group AutΦ(G) to be elementary abelian. [ABSTRACT FROM AUTHOR]

The Tutte polynomial is an isomorphism invariant of graphs that generalizes the chromatic and the flow polynomials. This two-variable polynomial with integral coefficients is known to carry important information about the properties of the graph. It has been used to prove long-standing conjectures in knot theory. Furthermore, it is related to the Potts and Ising models in statistical physics. The purpose of this paper is to study the interaction between the Tutte polynomial and graph symmetries. More precisely, we prove that if the automorphism group of the graph G contains an element of prime order p, then the coefficients of the Tutte polynomial of G satisfy certain necessary conditions. [ABSTRACT FROM AUTHOR]

It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, Aut(Fn), has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of Aut(Fn) for n ≥ 4 where kth powers of all transvections act trivially for some fixed k. Thus, for some values of k, we deduce that the quotient of Aut(Fn) by the subgroup generated by kth powers of transvections contains nonabelian free groups. This expands on results of Malestein and Putman and of Bridson and Vogtmann. [ABSTRACT FROM AUTHOR]

In this paper, we determine all dihedral subgroups in $PGL_{2}(\mathbb {F}_{q})$ , where $q=2^{l}$ and $l$ is a positive integer; and construct binary expurgated and extended Goppa codes with dihedral automorphism groups. Moreover, it is easy to obtain binary quasi-cyclic expurgated or extended Goppa codes. [ABSTRACT FROM AUTHOR]

Let X be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic p. Let g and γ be the genus and p-rank of X , respectively. The influence of g and γ on the automorphism group A u t (X) of X is well-known in the literature. If g ≥ 2 then A u t (X) is a finite group, and unless X is the so-called Hermitian curve, its order is upper bounded by a polynomial in g of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth's bound of degree 3 in g up to few exceptions, all having p-rank zero. In this paper a further refinement of Henn's result is proposed. First, we prove that if an algebraic curve of genus g ≥ 2 has more than 336 g 2 automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame, that is, the action of the group is completely known. Finally when | A u t (X) | ≥ 900 g 2 sufficient conditions for X to have p-rank zero are provided. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of regular graphs, which is related to the Grassmann graph. This class of graphs is called the doubled Grassman graph. The Grassmann graph is the class of graphs, which is defined similar to the Johnson graph. This time, however, we are concerned with the subspaces of a vector space, rather than with the subsets of a set. The doubled Grassmann graph is constructed from the Grassmann graph. This graph, was discovered by Biggs and Gardiner. In this paper, we determine the full automorphism group of the doubled Grassmann graph. For determining the automorphism group of the doubled Grassmann graph, we will use the methods used by Mirafzal (Proc. Indian Acad. Sci. (Math Sci.)129(3) (2019), Art. 34, 8). [ABSTRACT FROM AUTHOR]

In this paper, we have found the automorphism group of the Zappa-Szép product of two groups fixing a subgroup. We have computed these automorphisms for a subgroup of order p2 of a group G which is the Zappa-Szép product of two cyclic groups in which one is of order p2 and other is of order m. [ABSTRACT FROM AUTHOR]

All regular two-graphs having up to 36 vertices are known, and the first open case is the enumeration of two-graphs on 38 vertices. It is known that there are at least 191 regular two-graphs on 38 vertices and at least 18 regular two-graphs on 42 vertices. The number of descendants of these two-graphs is 6760 and 120, respectively. In this paper, we classify strongly regular graphs with parameters (41, 20, 9, 10) having nontrivial automorphisms and show that there are exactly 7152 such graphs. We enumerate all regular two-graphs on 38 and 42 vertices with at least one descendant whose full automorphism group is nontrivial and establish that there are at least 194 regular two-graphs on 38 vertices and at least 752 regular two-graphs on 42 vertices. Furthermore, we construct descendants with a trivial automorphism group of the newly constructed two-graphs and increase the number of known strongly regular graphs with parameters (37, 18, 8, 9) and (41, 20, 9, 10) to 6802 and 18439m respectively. This significantly increases the number of known strongly regular graphs with parameters (41, 20, 9, 10). [ABSTRACT FROM AUTHOR]

In this paper, we study the flag-transitive automorphism groups of 2-designs and prove that if G is a flag-transitive automorphism group of a 2-design D with r > λ (k - 3) , then G is a primitive permutation group of affine, almost simple type or product type. Moreover, it generalizes the above result to the case that r > (r , λ) (k - 3) . [ABSTRACT FROM AUTHOR]

The main result of the paper consists in the proof of the fact that for any germ of a real analytic hypersurface in complex 3-space the following alternative (dimension conjecture) takes place: either the dimension of the group of holomorphic symmetries of the germ is at most the dimension of that of a nondegenerate hyperquadric (the latter equals 15), or the group is infinite-dimensional. We also discuss mistakes found in A. Ershova’s paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of cubic graphs Γ(m), called Generalized Pappus graphs, where m ≥ 3. We compute the automorphism group of Γ(m) and characterize when it is a Cayley graph. [ABSTRACT FROM AUTHOR]

Let q = 2 m . The projective general linear group PGL (2 , q) acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over GF (2 h) that are invariant under PGL (2 , q) are trivial codes: the repetition code, the whole space GF (2 h) 2 m + 1 , and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all 3 - (q + 1 , k , λ) designs that are invariant under PGL (2 , q) are determined. The second objective is to present two infinite families of cyclic codes over GF (2 m) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL (2 , q) , therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters [ q + 1 , q - 3 , 4 ] q , where q = 2 m , and m ≥ 4 is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3- (q + 1 , 4 , 2) design. A code from the second family has parameters [ q + 1 , 4 , q - 4 ] q , q = 2 m , m ≥ 4 even, and the minimum weight codewords support a 3- (q + 1 , q - 4 , (q - 4) (q - 5) (q - 6) / 60) design, whose complementary 3- (q + 1 , 5 , 1) design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over GF (q) that can support a 3- (q + 1 , q - 4 , (q - 4) (q - 5) (q - 6) / 60) design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound. [ABSTRACT FROM AUTHOR]

Let G be a group. An automorphism α of the group G is called a central automorphism, if x-1xα ∈ Z(G) for all x ∈ G. Let Lc(G) be the central kernel of G, that is, the set of elements of G fixed by all central automorphisms of G and AutLc (G) denote the group of all central automorphisms of G fixing G=Lc(G) element-wise. In the present paper, we investigate the properties of such automorphisms. Moreover, a full classification of p-groups G of order at most p5 where AutLc (G) = Inn(G) is also given. [ABSTRACT FROM AUTHOR]

In this paper we consider variants of the power set and the lattice of subspaces and study automorphism groups of these variants. We obtain irreducible generating sets for variants of subsets of a finite set lattice and subspaces of a finite vector space lattice. We prove that automorphism group of the variant of subsets of a finite set lattice is a wreath product of two symmetric permutation groups such as first of this groups acts on subsets. The automorphism group of the variant of the subspace of a finite vector space lattice is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the automorphism group of subspaces lattice and the second is defined by the certain set of symmetric groups. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a graph with the vertex-set V and the edge-set E. Let N(v) denote the set of neighbors of the vertex v of G. The graph G is called irreducible whenever for every v , w ∈ V if v ≠ w , then N (v) ≠ N (w). In this paper, we present a method for finding automorphism groups of connected bipartite irreducible graphs. Then, by our method, we determine automorphism groups of some classes of connected bipartite irreducible graphs, including a class of graphs which are derived from Grassmann graphs. Let a 0 be a fixed positive integer. We show that if G is a connected non-bipartite irreducible graph such that c (v , w) = | N (v) ∩ N (w) | = a 0 when v, w are adjacent, whereas c (v , w) ≠ a 0 , when v, w are not adjacent, then G is a stable graph, that is, the automorphism group of the bipartite double cover of G is isomorphic with the group Aut (G) × Z 2 . Finally, we show that the Johnson graph J(n, k) is a stable graph. [ABSTRACT FROM AUTHOR]

A graph is symmetric if its automorphism group is transitive on the arc set of the graph. In this paper, we give a complete classification of connected pentavalent symmetric graphs of order 18p, for each prime p. It is shown that, such graphs there exist if and only if p = 2, 7 or 19, and up to isomorphism, there are only four such graphs. [ABSTRACT FROM AUTHOR]

Abstract In a recent paper (Mérida-Angulo and Thas, 2017), the authors introduced a map F which associates a Deitmar constructible set (which is defined over the field with one element, denoted by F 1) with any given loose graph Γ. By base extension, a constructible set X k = F (Γ) ⊗ F 1 k over any field k arises. In the present paper, we will show that all these mappings are functors, and we will use this fact to study automorphism groups of the constructible sets X k. Several automorphism groups are considered: combinatorial, topological and also groups induced by automorphisms of the ambient projective space. When Γ is a finite tree, we will give a precise description of the combinatorial and projective groups, amongst other results. [ABSTRACT FROM AUTHOR]

In this paper, we determine the automorphism groups of the hyperelliptic modular curves X1(N), and determine explicit forms for the actions of all automorphisms on certain defining equations of X1(N). [ABSTRACT FROM AUTHOR]

In this paper, we show that if an automorphism α of an abelian torsion group, which is in fact a direct sum of its p -components, has the weak extension property then α = π id A p + ρ , where p is a prime number, π is an invertible p -adic number and ρ ∈ Hom ( A p , A p 1 ) with A p 1 is the first Ulm subgroup of the p -component A p of A . [ABSTRACT FROM AUTHOR]

A Z2Z4 -linear Hadamard code of length \alpha +2\beta =2^t is a binary Hadamard code, which is the Gray map image of a Z2Z4 -additive code with \alpha binary coordinates and \beta quaternary coordinates. It is known that there are exactly \lfloor \frac t-12 \rfloor and \lfloor \frac t2 \rfloor nonequivalent Z2Z4 -linear Hadamard codes of length 2^t , with $\alpha =0$ and \alpha \not =0$ , respectively, for all t\geq 3$ . In this paper, it is shown that each \alpha =0$ is equivalent to a Z2Z4 -linear Hadamard code with \alpha \not =0 nonequivalent Z2Z4 -linear Hadamard codes of length 2^t . Moreover, the order of the monomial automorphism group for the Z2Z4 -additive Hadamard codes and the permutation automorphism group of the corresponding Z2Z4 -linear Hadamard codes are given. [ABSTRACT FROM AUTHOR]

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of PSL2(q). A quasi-platonic action of a group G on a closed Riemann S surface is a conformal action for which S/G is a sphere and S → S/G is branched over {0, 1, ∞}. The unit interval in S/G may be lifted to a dessin d'enfant D, an embedded bipartite graph in S. The dessin forms the edges and vertices of a tiling on Sψ by dihedrally symmetric polygons, generalizing the idea of a platonic solid. Each automorphism ψ in the absolute Galois group determines a transform S by transforming the coefficients of the defining equations of S. The transform defines a possibly new quasi-platonic action and a transformed dessin Dψ. Here, in this paper, we describe the quasi-platonic actions of PSL2(q) and the action of the absolute Galois group on PSL2(q) actions. The second paper discusses the quasi-platonic actions constructed from symmetries (reflections) and the resulting triangular tiling that refines the dessin d'enfant. In particular, the number of ovals and the separation properties of the mirrors of a symmetry are determined. [ABSTRACT FROM AUTHOR]

Let G be a group and AutΦ(G) denote the group of all automorphisms of G centralizing G/Φ(G) elementwise. In this paper, we characterize the finite p-groups G with cyclic Frattini subgroup for which |AutΦ(G):Inn(G)|=p. [ABSTRACT FROM AUTHOR]

In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ℂ. The automorphism group of such a surface β: B → ℙ, denoted by Au t( B), consists of all biholomorphic maps on the complex manifold B. The group Au t( B) is isomorphic to the semi-direct product MW( B) ⋊ Aut ( B) of the Mordell-Weil group MW( B) (the group of sections of B), and the subgroup Aut ( B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW( B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut ( B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section. [ABSTRACT FROM AUTHOR]

In this paper, some designs from the primitive permutation representations of the groups PSL2(81) and PSL2(89) are constructed using the Key-Moori Method 1. We determine the automorphism groups of all the obtained designs and prove that the groups PSL2(81), PSL2(81).2, PSL2(81):2, PGL2(81), PSL2(81), PGL2(81), PSL2(89) and PSL2(89):2 appear as the automorphism groups of these constructed designs. [ABSTRACT FROM AUTHOR]

In this paper, we classified Hadamard $ 2 $-$ (51, 25, 12) $ designs having a non-abelian automorphism group of order 10, i.e., $ {\rm{Frob}}_{10} \cong Z_5:Z_2 $, where a subgroup isomorphic to $ Z_2 $ fixed setwise all the $ Z_5 $-orbits of the sets of points and blocks. Furthermore, we classified $ 2 $-$ (59, 29, 14) $ designs having a non-abelian automorphism group of order 14, i.e., $ {\rm{Frob}}_{14} \cong Z_7:Z_2 $, where a subgroup isomorphic to $ Z_2 $ fixed setwise all the $ Z_7 $-orbits of the sets of points and blocks. We also showed that there was no Hadamard $ 2 $-$ (59, 29, 14) $ design with automorphism group $ {\rm{Frob}}_{21} \cong Z_7:Z_3 $, where a subgroup of order 3 fixed setwise all the $ Z_7 $-orbits of points and blocks. Additionally, we used Hadamard $ 2 $-$ (59, 29, 14) $ designs obtained to construct ternary linear codes and linear codes over the finite field of order 5. We constructed ternary self-dual codes and self-dual codes over the finite field of order 5 from the corresponding Hadamard matrices of order 60.