SQU Journal for Science, 2019, 24(2), 129-138 DOI:10.24200/squjs.vol24iss2pp129-138 Sultan Qaboos University 129 Connected Components of the Hurwitz Space for the Symmetric Group of Degree 7 Haval M. Mohammed Salih Soran University, Faculty of Science, Mathematics Department-Kawa St, Soran, Erbil, Iraq; University of Raparian, College of Basic Education, Department of Mathematic, Ranya, Kurdistan Region of Iraq. Email: haval.mahammed@soran.edu.iq. ABSTRACT: The Hurwitz space ℋr in(G) is the space of genus g = 0 covers of the Riemann sphere ℙ1 with r branch points and the monodromy group G. Let G be the symmetric group S7. In this paper, we enumerate the connected components of ℋr in(S7). Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of ℋr in(S7). This work gives us the complete classification of primitive genus zero symmetric group of degree seven. Keywords: Monodromy Groups; Braid Orbits; Connected Components. 7من الدرجة زمرة تناظريةفضاء هيوريتز للمكونات المتصلة لال فال م. محمد صالحه ℋr,gالفضاء الهوروتيزي :صلخمال A (G) هي نوع من غالف في الفضاء الريمانيℙ1 مع نقاط تفرع𝑟 والزمرة المونودرومية. G حيث𝐺 زمرة تناظرية 𝑆7 المكونات المتصلة ل ددع، ن. في هذا البحثℋr in(S7) . ستخدم االدوات الحاسوبية باالعتماد على نظام الجبر الحاسوبي نطريقتنا فيGAP وحزمة MAPCLASS المكونات المتصلة ب على للعثورℋr in(S7) من الدرجة السابعة. للزمرة تناظريةتصنيفات الجنس الصفرية يعطنا.هذا العمل المكونات المتصلة. و مدارات بريد ،المنودروميةة زمر: مفتاحيةالكلمات ال 1. Introduction et us start this section by the following definition: A primitive genus 𝑔 system is a triple (𝐺, Ω, (𝑥1, … , 𝑥𝑟 )) where Ω is a finite set of size 𝑛 and 𝐺 is a primitive subgroup of 𝑆𝑛 such that 𝐺 =< 𝑥1, … , 𝑥𝑟 > (1) ∏ 𝑥𝑖 𝑟 𝑖=1 = 1 (2) 2(𝑛 + 𝑔 − 1) = ∑ 𝑖𝑛𝑑 𝑥𝑖 𝑟 𝑖=1 (3) where 𝑥𝑖 ∈ 𝐺\{1}. For 𝑥 ∈ 𝐺 define 𝑖𝑛𝑑 𝑥 = 𝑛 − 𝑜𝑟𝑏 (𝑥), 𝐹𝑖𝑥 𝑥 = {𝑤 ∈ Ω |𝑥𝑤 = 𝑤}, 𝑓(𝑥) = |𝐹𝑖𝑥 𝑥| and 𝑜𝑟𝑏(𝑥) = 1 𝑑 ∑ 𝑓(𝑥 𝑖 )𝑑−1𝑖=0 , where 𝑑 is the order of 𝑥 in 𝑆𝑛 . These conditions (1), (2) and (3) are equivalent to the existence of the branched cover 𝜇: 𝑋 → ℙ1, where 𝑋 is a Riemann surface of genus 𝑔. The number of holes is called the genus, and μ is a meromorphic function where ℙ1 = ℂ⋃{∞} is the Riemann sphere. Let 𝐶𝑖 be a non-trivial conjugacy class of 𝑥𝑖 . Then the set 𝐶 = {𝐶1, … , 𝐶𝑟 } in 𝐺 is called the ramification type of the cover 𝜇. Note that the trivial conjugacy class contains only the identity element. In this paper, we classify primitive genus 0 systems for 𝑆7. It is clear that there are seven primitive groups of degree 7. In [1], we classified all those groups except 𝑆7. Now we are going to classify the group 𝑆7 by using the computer algebra system GAP. All together give the complete classification of primitive genus 0 groups of degree 7. L mailto:haval.mahammed@soran.edu.iq MOHAMMED SALIH, H.M. 130 Braid orbits can be interpreted as saying interesting things about components of the moduli space of curves ℳ𝑔 [3] and equivalence classes of branched covers of the Riemann sphere ℙ1. The full details of the following results and concepts can be found in [1-11]. Let 𝐶1, … , 𝐶𝑟 be non-trivial conjugacy classes of a finite group 𝐺. The set of generating systems (𝑥1, … , 𝑥𝑟 ) of 𝐺 with 𝑥1 … 𝑥𝑟 = 1 and such that there is a permutation 𝜋 ∈ 𝑆𝑟 with 𝑥𝑖 ∈ 𝑆𝜋(𝑖) for 𝑖 = 1, … , 𝑟 is called a Nielsen class and denoted by 𝒩(C), where 𝐶 = (𝐶1, … , 𝐶𝑟 ). Each Nielsen class is the disjoint union of braid orbits, which are defined as the smallest subsets of the Nielsen class closed under the braid operations (𝑥1, … , 𝑥𝑟 ) 𝑄𝑖 = (𝑥1, … , 𝑥𝑖+1, 𝑥𝑖+1 −1 𝑥𝑖 𝑥𝑖+1, … , 𝑥𝑟 ) (4) for 𝑖 = 1, … , 𝑟. We denote by Or, the space of subsets of ℂ of cardinality r. Definition 1.1 Let 𝐵 ∈ 𝑂𝑟 and 𝑏0 ∈ ℙ 1 ∖ 𝐵. We call a map 𝜑: 𝜋1(ℙ 1 ∖ 𝐵, 𝑏0) → 𝐺 admissible if it is a surjective homomorphsim, and 𝜑(𝜃𝑏 ) ≠ 1 for each 𝑏 ∈ 𝐵. Here 𝜃𝑏 is the conjugacy class of 𝜋1(ℙ 1 ∖ 𝐵, 𝑏0). Definition 1.2 Let 𝐵 ∈ 𝑂𝑟 and 𝜑: 𝜋1(ℙ 1 ∖ 𝐵, ∞) → 𝐺 be admissible. Then we say that two pairs (B, φ) and (B̅, φ̅) are A-equivalent if and only if 𝐵 = �̅� and �̅� = 𝑎 ∘ 𝜑 for some a ∈ A. Let [B, φ]A denote the A-equivalence class of (𝐵, 𝜑). The set of equivalence classes [𝐵, 𝜑]𝐴 is denoted by ℋ𝑟 𝐴 (𝐺) and is called the Hurwitz space of 𝐺-covers. Lemma 1.3 Let 𝐶 be a fixed ramifcation type in G, and the subset ℋ𝑟 𝑖𝑛 (𝐶) of ℋ𝑟 𝑖𝑛 (𝐺) consist of all [𝐵, ∅]𝐴 with 𝐵 = {𝑏1, … , 𝑏𝑟 }, ∅: 𝜋1(ℙ 1\𝐵, ∞) → 𝐺 and ∅(𝜃𝑏𝑖 ) ∈ 𝐶𝑖 for i = 1, … , r. Then ℋ𝑟 𝐴(𝐶) is a union of connected components in ℋr A(G). Under the bijection from Lemma 2.2, the fiber in ℋ𝑟 𝐴 (𝐶) over B0 corresponds the set 𝒩 A(C). This yields a one to one correspondence between components of ℋ𝑟 𝐴 (𝐶) and the braid orbits on 𝒩 𝐴(𝐶). In particular, ℋ𝑟 𝑖𝑛 (𝐶) is connected if and only if there is only one braid orbit. Proof. For a proof see [2]. 2. Computing Indexes and Labeling Conjugacy Classes In this paper, we discuss two methods for computing index as follows: Method one (Via Fixed Points) Let G be a group acting on a finite set Ω of size 𝑛. If 𝑥 ∈ 𝐺, define the index of x by 𝑖𝑛𝑑 𝑥 = 𝑛 − 𝑜𝑟𝑏 (𝑥) where 𝑜𝑟𝑏 (𝑥) is the number of orbits of < 𝑥 > on Ω. Also 𝐹𝑖𝑥 𝑥 = {𝜔 ∈ 𝛺 |𝑥𝜔 = 𝜔}, 𝑓(𝑥) = |𝐹𝑖𝑥 𝑥|. Furthermore, 𝑜𝑟𝑏 (𝑥) = 1 𝑑 ∑ 𝑓(𝑥 𝑖 )𝑑−1𝑖=0 where x has order d.We discussed this method in detail in [2]. Method two (Via Cycle Types) As we know that 𝑖𝑛𝑑 𝑥𝑖 is the minimal number of 2-cycles needed to express 𝑥𝑖 as a product. We will label the fourteen nontrivial conjugacy classes of S7 as ATLAS notation by: Table 1. Non trivial conjugacy classes of 𝑺𝟕. Type Conjugacy class Ind 2A (1,2)𝑆7 1 2B (1,2)(3,4)𝑆7 2 2C (1,2)(3,4)(5,6)𝑆7 3 3A (1,2,3)𝑆7 2 3B (1,2,3)(4,5,6)𝑆7 4 4A (1,2,3,4)𝑆7 3 4B (1,2,3,4)(5,6)𝑆7 4 5A (1,2,3,4,5)𝑆7 4 6A (1,2,3)(4,5)(6,7)𝑆7 4 6B (1,2,3)(4,5)𝑆7 3 6C (1,2,3,4,5,6)𝑆7 5 7A (1,2,3,4,5,6,7)𝑆7 6 10A (1,2,3,4,5)(6,7)𝑆7 5 12A (1,2,3,4)(5,6,7)𝑆7 5 CONNECTED COMPONENTS OF THE HURWITZ SPACE 131 3. Algorithm To achieve connected components of ℋ𝑟 𝑖𝑛 (𝐺), we need to perform the following steps: Step 1: Select the primitive group 𝑆7 by using the GAP code [7] : Primitive Group ( 7, 7 ). Step 2: Find all ramification types that satisfy equation (3) for given 𝑆7, degree 7 and genus 0. Step 3: Remove those types which have zero structure constant from the character table of 𝑆7 via the following equation. 𝑛(𝐶1, … , 𝐶𝑘 ) = |𝐶1||𝐶2|…|𝐶𝑘| |𝐺| ∑ 𝜒(𝑥1)𝜒(𝑥2)…𝜒(𝑥𝑘) 𝜒(1)𝑘−2𝜒∈𝐼𝑟𝑟(𝐺) (5) With equation (5), we compute the number of k-tuples (𝑥1, … , 𝑥𝑘 ) of elements 𝑥𝑖 in the conjugacy class 𝐶𝑖 of a group 𝑆7 such that 𝑥1𝑥2 … 𝑥𝑘 = 1. In other words, we remove those types which don’t satisfy equation (2). Step 4: For the remaining types, that pass equation (1) which are called generating types. Step 5: For the generating types, compute braid orbits by using MAPCLASS package. Now we perform the above steps by using the program described in [12], but with a few modifications to it. That is, we remove the condition of affine type in that program. In this paper we only consider primitive groups. 4. Results In this paper, we use the algorithm which is presented in section 3 to compute braid orbits on Nielsen class. An application of the algorithm is the classification of the primitive genus zero systems for 𝑆7. That is, we find the connected components ℋr in (C) of S7-curves X, such that g(X/S_7) = 0. In our situation, the computation shows that there are exactly 1071 braid orbits of primitive genus 0 systems of degree 7. The degree and the number of the branch points are given in Tables 2 and 3. The detail of the Table 1 exists in [10]. Table 2. Primitive Genus Zero Systems: Number of Components. Degree # Group Iso types #RTs # comp’s r = 3 # comp’s r = 4 # comp’s r = 5 # comp’s r = 6 # comp’s Total 7 5 154 179 61 67 10 317 Table 3. Primitive Genus Zero Systems: Number of Components. Degree # Group Iso types #RTs # comp’s r = 3 # comp’s r = 4 # comp’s r = 5 # comp’s r = 6 # comp’s r =7,8,9,10,11,12 # comp’s Total 7 1 632 171 183 172 113 61,31,14,6,2,1 754 Theorem 4.1 Up to isomorphism, there exist exactly 6 primitive genus zero groups of degree seven. The corresponding primitive genus zero groups are enumerated in Table 4 and Tables 2-3. Lemma 4.2 The Hurwitz spaces, ℋ𝑟 𝑖𝑛 (𝐶) are connected if 𝐺 = 𝑆7 and 𝑟 ≥ 5. Proof. It follows from the fact that the Nielsen classes 𝒩(𝐶) are the disjoint union of braid orbits but we have only one braid orbit for 𝑆7 and 𝑟 ≥ 5. From Lemma 1.3, we obtain that the Hurwitz spaces ℋ𝑟 𝑖𝑛 (𝐶) are connected. Lemma 4.3 The Hurwitz spaces, ℋ𝑟 𝑖𝑛 (𝐶) are disconnected if 𝐺 = 𝑆7 and 𝑟 ≤ 4. Proof. Since we have at least two braid orbits for some type 𝐶 for 𝑟 ≤ 4 and 𝐺 = 𝑆7 and the Nielsen classes 𝒩(C) are the disjoint union of braid orbits. From Lemma 1.3, we obtain that the Hurwitz spaces ℋ𝑟 𝑖𝑛 (𝐶) are disconnected. MOHAMMED SALIH, H.M. 132 Table 4. Primitive genus zero systems of 𝑺𝟕. Ramification Type Number of orbits Length of largest orbit Ramification Type Number of orbits Length of largest orbit (4A,5A,6C) (4A,5A,10A) (4A,5A,12A) (4A,4B,6C) (4A,4B,10A) (4A,4B,12A) (4A,4A,7A) (4A,3B,6C) (4A,3B,10A) (4A,3B,12A) (4A,6A,6C) (4A,6A,10A) (4A,6A,12A) (6B,5A,6C) (6B,5A,10A) (6B,5A,12A) (6B,4B,6C) (6B,4B,10A) (6B,4B,12A) (6B,4A,7A) (6B,3B,6C) (6B,3B,10A) (6B,3B,12A) (6B,6A,6C) (6B,6A,10A) (6B,6A,12A) (6B,6B,6C) (3A,6C,6C) (3A,10A,6C) (3A,10A,10A) (3A,12A,6C) (3A,12A,10A) (3A,12A,12A) (2C,5A,6C) (2C,5A,10A) (2C,5A,12A) (2C,4B,6C) (2C,4B,10A) (2C,4B,12A) (2C,4A,7A) (2C,3B,10A) (2C,3B,12A) (2C,6A,6C) (2C,6A,10A) (2C,6A,12A) (2C,6B,7A) (2B,6C,6C) (2B,10A,6C) (2B,10A,10A) (2C,12A,6C) (2C,12A,10A) 1 2 3 4 4 4 1 2 3 1 3 1 2 6 7 5 12 9 9 3 6 4 4 6 4 3 9 1 2 1 2 2 1 3 1 2 4 3 2 1 1 1 1 1 1 3 4 3 3 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (3A,6B,4A,4B) (3A,6B,4A,3B) (3A,6B,4A,6A) (3A,6B,6B,5A) (3A,6B,6B,4B) (3A,6B,6B,3B) (3A,6B,6B,6A) (3A,3A,4A,6C) (3A,3A,4A,10A) (3A,3A,4A,12A) (3A,3A,6B,6C) (3A,3A,6B,10A) (3A,3A,6B,12A) (3A,3A,2C,6C) (3A,3A,2C,10A) (3A,3A,2C,12A) (3A,2C,4A,5A) (3A,2C,4A,4B) (3A,2C,4A,3B) (3A,2C,4A,6A) (3A,2C,6B,5A) (3A,2C,6B,4B) (3A,2C,6B,3B) (3A,2C,6B,6A) (3A,2C,2C,5A) (3A,2C,2C,4B) (3A,2C,2C,3B) (3A,2C,2C,6A) (2C,4A,4A,4A) (2C,6B,4A,4A) (2C,6B,6B,4A) (2C,6B,6B,6B) (2C,2C,4A,4A) (2C,2C,6B,4A) (2C,2C,6B,6B) (2C,2C,2C,4A) (2C,2C,2C,6B) (2B,4A,4A,5A) (2B,4A,4A,4B) (2B,4A,4A,3B) (2B,4A,4A,6A) (2B,6B,4A,5A) (2B,6B,4A,4B) (2B,6B,4A,3B) (2B,6B,4A,6A) (2B,6B,6B,5A) (2B,6B,6B,4B) (2B,6B,6B,3B) (2B,6B,6B,6A) (2B,3A,4A,6C) (2B,3A,4A,10A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 141 64 69 210 348 168 153 12 20 18 48 50 44 20 10 16 30 42 27 13 75 105 48 39 15 32 13 11 32 88 188 102 16 56 84 16 36 40 96 52 44 150 248 118 114 395 620 282 247 33 30 (2C,12A,12A) (2A,6C,7A) (2A,10A,7A) (2A,12A,7A) (4A,4A,4A,4A) 2 1 1 1 1 1 1 1 1 16 (2B,3A,4A,12A) (2B,3A,6B,6C) (2B,3A,6B,10A) (2B,3A,6B,12A) (2B,3A,2C,6C) 1 1 1 1 1 34 102 85 75 35 CONNECTED COMPONENTS OF THE HURWITZ SPACE 133 (6B,4A,4A,4A) (6B,6B,4A,4A) (6B,6B,6B,4A) (6B,6B,6B,6B) (3A,4A,4A,5A) (3A,4A,4A,4B) (3A,4A,4A,3B) (3A,4A,4A,6A) (3A,6B,4A,5A) (2B,2C,2C,5A) (2B,2C,2C,4B) (2B,2C,2C,3B) (2B,2C,2C,6A) (2B,2B,4A,6C) (2B,2B,4A,10A) (2B,2B,4A,12A) (2B,2B,6B,6C) (2B,2B,6B,10A) (2B,2B,6B,12A) (2B,2B,2C,6C) (2B,2B,2C,10A) (2B,2B,2C,12A) (2A,4A,5A,5A) (2A,4A,4B,5A) (2A,4A,4B,4B) (2A,4A,4A,6C) (2A,4A,4A,10A) (2A,4A,4A,12A) (2A,4A,3B,6C) (2A,4A,3B,10A) (2A,4A,3B,12A) (2A,4A,6A,5A) (2A,4A,6A,4B) (2A,4A,6A,3B) (2A,4A,6A,6A) (2A,6B,5A,5A) (2A,6B,4B,5A) (2A,6B,4B,4B) (2A,6B,4A,6C) (2A,6B,4A,10A) (2A,6B,4A,12A) (2A,6B,3B,5A) (2A,6B,3B,4B) (2A,6B,3B,3B) (2A,6B,6A,5A) (2A,6B,6A,4B) (2A,6B,6A,3B) (2A,6B,6A,6A) (2A,6B,6B,6C) (2A,6B,6B,10A) (2A,6B,6B,12A) 1 2 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 72 176 640 1008 15 44 23 34 65 35 48 22 18 72 60 52 198 140 124 60 40 36 12 39 80 12 16 20 23 38 12 26 34 21 10 62 119 188 48 49 41 54 90 42 59 77 36 28 144 112 100 (2B,3A,2C,10A) (2B,3A,2C,12A) (2B,2C,4A,5A) (2B,2C,4A,4B) (2B,2C,4A,3B) (2B,2C,4A,6A) (2B,2C,6B,5A) (2B,2C,6B,4B) (2B,2C,6B,3B) (2B,2C,6B,6A) (2A,2C,3B,4B) (2A,2C,3B,3B) (2A,2C,6A,5A) (2A,2C,6A,4B) (2A,2C,6A,3B) (2A,2C,6A,6A) (2A,2C,6B,6C) (2A,2C,6B,10A) (2A,2C,6B,12A) (2A,2C,2C,6C) (2A,2C,2C,10A) (2A,2C,2C,12A) (2A,2B,5A,6C) (2A,2B,5A,10A) (2A,2B,5A,12A) (2A,2B,4B,6C) (2A,2B,4B,10A) (2A,2B,4B,12A) (2A,2B,4A,7A) (2A,2B,3B,6C) (2A,2B,3B,10A) (2A,2B,3B,12A) (2A,2B,6A,6C) (2A,2B,6A,10A) (2A,2B,6A,12A) (2A,2B,6B,7A) (2A,2B,2C,7A) (2A,2A,6C,6C) (2A,2A,10A,6C) (2A,2A,10A,10A) (2A,2A,5A,7A) (2A,2A,12A,6C) (2A,2A,12A,10A) (2A,2A,12A,12A) (2A,2A,4B,7A) (2A,2A,3B,7A) (2A,2A,6A,7A) (2B,2B,3A,6N,4A) (3A,3A,3A,4A,4A) (3A,3A,3A,6B,4A) (3A,3A,3A,6B,6B) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 25 22 50 80 36 28 125 176 84 65 24 12 11 20 9 8 48 31 29 12 10 8 33 28 27 60 47 42 14 30 21 19 27 20 17 42 14 12 12 10 7 12 10 8 14 7 7 2396 163 606 1827 (2A,2B,3A,2C,3B) (2A,2B,3A,2C,6A) (2A,2B,2C,4A,4A) (2A,2B,2C,6B,4A) (2A,2B,2C,6B,6B) (2A,2B,2C,2C,4A) (2A,2B,2C,2C,6B) (2A,2B,2C,2C,2C) (2A,2B,2B,4A,5A) (2A,2B,2B,4A,4B) (2A,2B,2B,4A,3B) 1 1 1 1 1 1 1 1 1 1 1 249 197 400 976 2256 276 612 168 760 1264 600 (2A,2A,2A,2C,7A) (2A,2A,3A,3A,7A) (2A,2A,3A,2C,6C) (2A,2A,3A,2C,10A) (2A,2A,3A,2C,12A) (2A,2A,2C,4A,5A) (2A,2A,2C,4A,4B) (2A,2A,2C,4A,3B) (2A,2A,2C,4A,6A) (2A,2A,2C,6B,5A) (2A,2A,2C,6B,4B) 1 1 1 1 1 1 1 1 1 1 1 49 49 132 85 92 196 296 156 100 486 684 MOHAMMED SALIH, H.M. 134 (2A,2B,2B,4A,6A) (2A,2B,2B,6B,5A) (2A,2B,2B,6B,4B) (2A,2B,2B,6B,3B) (2A,2B,2B,6B,6A) (2A,2B,2B,2C,5A) (2A,2B,2B,2C,4B) (2A,2B,2B,2C,3B) (2A,2B,2B,2C,6A) (2A,2B,2B,3A,6C) (2A,2B,2B,3A,10A) (2A,2B,2B,3A,12A) (2A,2B,2B,2B,6C) (2A,2B,2B,2B,10A) (2A,2B,2B,2B,12A) (2A,2A,4A,4A,5A) (2A,2A,4A,4A,4B) (2A,2A,4A,4A,3B) (2A,2A,4A,4A,6A) (2A,2A,6B,4A,5A) (2A,2A,6B,4A,4B) (2A,2A,6B,4A,3B) (2A,2A,6B,4A,6A) (2A,2A,2B,2C,10A) (2A,2A,2B,2C,12A) (2A,2A,2B,2B,7A) (2A,2A,2A,5A,6C) (2A,2A,2A,5A,10A) (2A,2A,2A,5A,12A) (2A,2A,2A,4B,6C) (2A,2A,2A,4B,10A) (2A,3A,3A,3A,3A,4A) (2A,3A,3A,3A,3A,6B) (2A,3A,3A,3A,3A,2C) (2A,2B,3A,3A,3A,4A) (2A,2B,3A,3A,3A,6B) (2A,2B,3A,3A,3A,2C) (2A,2B,2B,3A,3A,4A) (2A,2B,2B,3A,3A,6B) (2A,2B,2B,3A,3A,2C) (2A,2B,2B,2B,3A,4A) (2A,2B,2B,2B,3A,6B) (2A,2B,2B,2B,3A,2C) (2A,2B,2B,2B,2B,4A) (2A,2B,2B,2B,2B,6B) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 540 2115 3080 1248 1223 615 880 414 330 513 405 364 972 690 612 131 336 182 190 519 948 438 447 160 144 196 108 105 105 216 180 1456 4788 1872 3350 9441 2967 6608 17164 5174 12316 30099 8658 21824 51336 (2A,2A,2C,6B,3B) (2A,2A,2C,6B,6A) (2A,2A,2C,2C,5A) (2A,2A,2C,2C,4B) (2A,2A,2C,2C,3B) (2A,2A,2C,2C,6A) (2A,2A,2B,5A,5A) (2A,2A,2B,4B,5A) (2A,2A,2B,4A,6C) (2A,2A,2B,4B,4B) (2A,2A,2B,4A,10A) (2A,2A,2B,4A,12A) (2A,2A,2B,3B,5A) (2A,2A,2B,3B,4B) (2A,2A,2B,3B,3B) (2A,2A,2B,6B,6C) (2A,2A,2B,6B,10A) (2A,2A,2B,6B,12A) (2A,2A,2B,6A,5A) (2A,2A,2B,6A,4B) (2A,2A,2B,6A,3B) (2A,2A,2B,6A,6A) (2A,2A,2B,3A,7A) (2A,2A,2B,2C,6C) (2A,2A,2A,2B,4A,3B) (2A,2A,2A,2B,4A,6A) (2A,2A,2A,2B,6B,5A) (2A,2A,2A,2B,6B,4B) (2A,2A,2A,2B,6B,3B) (2A,2A,2A,2B,6B,6A) (2A,2A,2A,3A,6B,5A) (2A,2A,2A,3A,6B,4B) (2A,2A,2A,3A,6B,3B) (2A,2A,2A,3A,6B,6A) (2A,2A,2A,4A,4A,4A) (2A,2A,2A,6B,4A,4A) (2A,2A,2A,6B,6B,4A) (2A,2A,2A,6B,6B,6B) (2A,2A,2A,2C,4A,4A) (2A,2A,2A,2C,6B,4A) (2A,2A,2A,2C,6B,6B) (2A,2A,2A,2C,2C,4A) (2A,2A,2A,2C,2C,6B) (2A,2A,2A,2C,2C,2C) (2A,2A,2A,2B,3A,6C) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 324 252 120 192 84 72 328 592 252 952 215 206 291 444 204 720 545 487 265 388 183 148 98 228 2304 2076 7530 11796 5508 4818 3900 6720 3132 2991 1296 4428 12000 29802 1560 3828 8772 992 2376 672 1836 (2A,3A,5A,6C) (2A,3A,5A,10A) (2A,3A,5A,12A) (2A,3A,4B,6C) (2A,3A,4B,10A) (2A,3A,4B,12A) (2A,3A,4A,7A) (2A,3A,3B,6C) (2A,3A,3B,10A) (2A,3A,3B,12A) (2A,3A,6A,6C) (2A,3A,6A,10A) (2A,3A,6A,12A) (2A,3A,6B,7A) (2A,3A,2C,7A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 16 17 30 27 27 7 15 16 10 18 10 12 21 7 (3A,3A,3A,2C,4A) (3A,3A,3A,2C,6B) (3A,3A,3A,2C,2C) (2B,3A,3A,4A,4A) (2B,3A,3A,6B,4A) (2B,3A,3A,2C,4B) (2B,3A,3A,2C,6B) (2B,3A,3A,2C,2C) (2B,2B,3A,4A,4A) (2B,2B,3A,6B,6B) (2B,2B,3A,2C,4A) (2B,2B,3A,2C,4A) (2B,2B,3A,2C,2C) (2B,2B,2B,4A,4A) (2B,2B,2B,6B,4A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 272 612 133 418 1285 412 1038 286 876 5935 748 1721 474 1712 4296 CONNECTED COMPONENTS OF THE HURWITZ SPACE 135 (2A,2C,5A,5A) (2A,2C,4B,5A) (2A,2C,4B,4B) (2A,2C,4A,6C) (2A,2C,4A,10A) (2A,2C,4A,12A) (2A,2C,3B,5A) (2A,3A,3A,6B,5A) (2A,3A,3A,6B,4B) (2A,3A,3A,6B,3B) (2A,3A,3A,6B,6A) (2A,3A,3A,3A,6C) (2A,3A,3A,3A,10A) (2A,3A,3A,3A,12A) (2A,3A,3A,2C,5A) (2A,3A,3A,2C,12A) (2A,3A,3A,2C,3B) (2A,3A,3A,2C,6A) (2A,3A,2C,4A,4A) (2A,3A,2C,6B,4A) (2A,3A,2C,6B,6B) (2A,3A,2C,2C,4A) (2A,3A,2C,2C,6B) (2A,3A,2C,2C,2C) (2A,2B,4A,4A,4A) (2A,2B,6B,4A,4A) (2A,2B,6B,6B,4A) (2A,2B,3A,4A,5A) (2A,2B,6B,6B,6B) (2A,2B,3A,4A,4B) (2A,2B,3A,4A,3B) (2A,2B,3A,4A,6A) (2A,2B,3A,6B,5A) (2A,2B,3A,6B,4B) (2A,2B,3A,6B,3B) (2A,2B,3A,6B,6A) (2A,2B,3A,3A,6C) (2A,2B,3A,3A,10A) (2A,2B,3A,3A,12A) (2A,2B,3A,2C,5A) (2A,2B,3A,2C,4B) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 26 38 56 20 11 14 21 525 996 462 468 108 135 120 210 300 168 105 238 587 1335 136 366 108 384 1232 3196 355 7752 690 349 315 1085 1763 828 754 252 220 220 350 536 (2B,2B,2B,6B,6B) (2B,2B,2B,2C,4A) (2B,2B,2B,2C,6B) (2B,2B,2B,2C,2C) (2A,3A,4A,4A,4A) (2A,3A,6B,4A,4A) (2A,3A,6B,6B,4A) (2A,3A,6B,6B,6B) (2A,3A,3A,4A,5A) (2A,3A,3A,4A,4B) (2A,3A,3A,4A,3B) (2A,3A,3A,4A,6A) (2A,2A,6B,6B,5A) (2A,2A,6B,6B,4B) (2A,2A,6B,6B,3B) (2A,2A,6B,6B,6A) (2A,2A,3A,6A,5A) (2A,2A,3A,6A,4B) (2A,2A,3A,6A,3B) (2A,2A,3A,6A,6A) (2A,2A,3A,5A,5A) (2A,2A,3A,4B,5A) (2A,2A,3A,4B,4B) (2A,2A,3A,4A,6C) (2A,2A,3A,4A,10A) (2A,2A,3A,4A,12A) (2A,2A,3A,3B,5A) (2A,2A,3A,3B,4B) (2A,2A,3A,3B,3B) (2A,2A,2A,4B,12A) (2A,2A,2A,4A,7A) (2A,2A,2A,3B,6C) (2A,2A,2A,3B,10A) (2A,2A,3A,6B,6C) (2A,2A,3A,6B,6A) (2A,2A,3A,6B,12A) (2A,2A,2A,3B,12A) (2A,2A,2A,6A,6C) (2A,2A,2A,6A,10A) (2A,2A,2A,6A,12A) (2A,2A,2A,6B,7A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10128 1248 2880 748 156 589 1782 4455 145 350 172 216 1267 2352 1098 975 170 236 126 82 131 302 548 82 120 123 156 264 105 168 49 108 90 360 325 286 72 108 75 7 147 (2A,2B,2B,2B,2B,2C) (2A,2A,3A,3A,4A,4A) (2A,2A,3A,3A,6B,4A) (2A,2A,3A,3A,6B,6B) (2A,2A,3A,3A,3A,5A) (2A,2A,3A,3A,3A,4B) (2A,2A,3A,3A,3A,3B) (2A,2A,3A,3A,3A,6A) (2A,2A,2B,3A,3A,5A) (2A,2A,2B,3A,3A,4B) (2A,2A,2B,3A,3A,3B) (2A,2A,2B,3A,3A,6A) (2A,2A,2B,2B,3A,5A) (2A,2A,2B,2B,3A,4B) (2A,2A,2B,2B,3A,3B) (2A,2A,2B,2B,3A,6A) (2A,2A,3A,3A,2C,4A) (2A,2A,3A,3A,2C,6B) (2A,2A,3A,3A,2C,2C) (2A,2A,2B,3A,4A,4A) (2A,2A,2B,3A,2C,4A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14448 1398 4588 12600 1260 2640 1260 1422 2795 4952 2418 2197 5530 8896 4191 3671 1688 4044 1020 3150 2824 (2A,2A,2A,2B,3A,10A) (2A,2A,2A,2B,3A,12A) (2A,2A,2A,2B,2C,5A) (2A,2A,2A,2B,2C,4B) (2A,2A,2A,2B,2C,3B) (2A,2A,2A,2B,2C,6A) (2A,2A,2A,2B,2B,6C) (2A,2A,2A,2B,2B,10A) (2A,2A,2A,2B,2B,12A) (2A,2A,2A,2A,4A,6C) (2A,2A,2A,2A,4A,10A) (2A,2A,2A,2A,4A,12A) (2A,2A,2A,2A,5A,5A) (2A,2A,2A,2A,6B,6C) (2A,2A,2A,2A,6B,10A) (2A,2A,2A,2A,6B,12A) (2A,2A,2A,2A,3A,7A) (2A,2A,2A,2A,2C,6C) (2A,2A,2A,2A,2C,10A) (2A,2A,2A,2A,2C,12A) (2A,2A,2A,2A,4A,7A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1500 1402 2340 3456 1620 1296 3564 2650 2327 864 800 784 1110 2592 2100 1872 343 864 600 576 686 MOHAMMED SALIH, H.M. 136 (2A,2A,2B,3A,2C,6B) (2A,2A,2B,3A,2C,2C) 1 1 6744 1860 (2A,2A,2A,2A,4B,5A) (2A,2A,2A,2A,4B,4B) (2A,2A,2A,2A,6A,5A) 1 1 1 2160 3648 1050 (2A,2A,2B,2B,4A,4A) (2A,2A,2B,2B,6B,4A) (2A,2A,2B,2B,6B,6B) (2A,2A,2B,2B,2C,4A) (2A,2A,2B,2B,2C,6B) (2A,2A,2B,2B,2C,2C) (2A,2A,2A,3A,3A,6C) (2A,2A,2A,3A,3A,10A) (2A,2A,2A,3A,3A,12A) (2A,2A,2B,2B,2B,5A) (2A,2A,2B,2B,2B,4B) (2A,2A,2B,2B,2B,3B) (2A,2A,2B,2B,2B,6A) (2A,2A,2A,3A,4A,5A) (2A,2A,2A,2B,4A,4B) (2A,2A,2A,2A,2A,4A,5A) (2A,2A,2A,2A,2A,4A,4B) (2A,2A,2A,2A,2A,4A,3B) (2A,2A,2A,2A,2A,4A,6A) (2A,2A,2A,2A,2A,6B,5A) (2A,2A,2A,2A,2A,6B,4B) (2A,2A,2A,2A,2A,6B,3B) (2A,2A,2A,2A,2A,6B,6A) (2A,2A,2A,2A,2A,2C,5A) (2A,2A,2A,2A,2A,2C,4B) (2A,2A,2A,2A,2A,2C,3B) (2A,2A,2A,2A,2A,2C,6A) (2A,2A,2A,2A,2A,2B,6C) (2A,2A,2A,2A,2A,2B,10A) (2A,2A,2A,2A,2A,2B,12A) (2A,2A,2A,2A,2A,2A,7A) (2A,2A,2A,2A,2B,2B,5A) (2A,2A,2A,2A,2B,2B,4B) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6320 11628 39288 4880 11268 3080 864 850 808 10215 15408 7182 6066 1205 4744 9500 17600 8640 8160 27750 45120 21060 19080 9000 13440 8480 5040 12960 10000 9120 2401 38300 59264 (2A,2A,2A,2A,6A,4B) (2A,2A,2A,2A,6A,3B) (2A,2A,2A,2A,6A,6A) (2A,2A,2A,2A,3B,5A) (2A,2A,2A,2A,3B,4B) (2A,2A,2A,2A,3B,3B) (2A,2A,2B,2B,3A,3A,3A) (2A,2A,2A,3A,4A,4B) (2A,2A,2A,3A,4A,3B) (2A,2A,2A,3A,4A,6A) (2A,2A,2A,3A,2C,5A) (2A,2A,2A,3A,2C,4B) (2A,2A,2A,3A,2C,3B) (2A,2A,2A,3A,2C,6A) (2A,2A,2A,2B,4A,5A) (2A,2A,2B,2B,2B,2B,2B) (2A,2A,23A,3A,3A,3A,3A) (2A,2A,2B,3A,3A,3A,3A) (2A,2A,2A,2A,2A,2A,2B,5A) (2A,2A,2A,2A,2A,2A,2B,4B) (2A,2A,2A,2A,2A,2A,2B,3B) (2A,2A,2A,2A,2A,2A,2B,6A) (2A,2A,2A,2A,2A,2A,3A,5A) (2A,2A,2A,2A,2A,2A,3A,4B) (2A,2A,2A,2A,2A,2A,3A,3B) (2A,2A,2A,2A,2A,2A,3A,6A) (2A,2A,2A,2A,3A,3A,3A,3A) (2A,2A,2A,2A,2B,3A,3A,3A) (2A,2A,2A,2A,2B,2B,3A,3A) (2A,2A,2A,2A,2B,2B,2B,3A) (2A,2A,2A,2A,2B,2B,2B,2B) (2A,2A,2A,2A,2A,3A,3A,4A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1536 756 756 1080 1728 750 48859 2516 1260 1290 1365 2040 1026 744 2710 260848 12207 25860 142500 226560 106920 92880 73125 126720 60750 56160 92880 183252 336372 593244 1016904 87840 CONNECTED COMPONENTS OF THE HURWITZ SPACE 137 Table 5. Primitive genus zero systems of 𝑺𝟕. (2A,2A,2A,2A,2B,2B,3B) (2A,2A,2A,2A,2B,2B,6A) (2A,2A,2A,2A,2B,3A,5A) (2A,2A,2A,2A,2B,3A,4B) (2A,2A,2A,2A,2B,3A,3B) (2A,2A,2A,2A,2B,3A,6A) (2A,2A,2A,2A,3A,3A,5A) (2A,2A,2A,2A,3A,3A,4B) (2A,2A,2A,2A,3A,3A,3B) (2A,2A,2A,2A,3A,3A,6A) (2A,2A,2A,2A,2A,3A,6C) (2A,2A,2A,2A,2A,3A,10A) (2A,2A,2A,2A,2A,3A,12A) (2A,2A,2A,2A,2B,2C,4A) (2A,2A,2A,2A,2B,2C,6B) (2A,2A,2A,2A,2B,2C,2C) (2A,2A,2A,2A,3A,2C,4A) (2A,2A,2A,2A,3A,2C,6B) (2A,2A,2A,2A,3A,2C,2C) (2A,2A,2A,2A,2B,4A,4A) (2A,2A,2A,2A,2A,6B,4A) (2A,2A,2A,2A,2B,6B,6B) (2A,2A,2A,2A,3A,4A,4A) (2A,2A,2A,2A,3A,6B,4A) (2A,2A,2A,2A,3A,6B,6B) (2A,2A,2A,2B,2B,2B,4A) (2A,2A,2A,2B,2B,2B,6B) (2A,2A,2A,2B,2B,2B,2C) (2A,2A,2A,2B,2B,3A,4A) (2A,2A,2A,2B,2B,3A,6B) (2A,2A,2A,2B,2B,3A,2C) (2A,2A,2A,2B,3A,3A,4A) (2A,2A,2A,2B,3A,3A,6B) (2A,2A,2A,2B,3A,3A,2C) (2A,2A,2B,2B,2B,2B,3A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 27756 23760 20250 33696 16062 14304 9825 18432 8910 8796 6480 5625 5280 18816 44064 12096 11040 26448 7056 23128 61836 151428 11036 33096 86070 83536 82632 56616 46296 115674 33876 24320 65100 19908 153498 (2A,2A,2A,2A,2A,3A,3A,6B) (2A,2A,2A,2A,2A,3A,3A,2C) (2A,2A,2A,2A,2A,2B,3A,4A) (2A,2A,2A,2A,2A,2B,3A,6B) (2A,2A,2A,2A,2A,2B,3A,2C) (2A,2A,2A,2A,2A,2B,2B,4A) (2A,2A,2A,2A,2A,2B,2B,6B) (2A,2A,2A,2A,2A,2B,2B,2C) (2A,2A,2A,2A,2A,2A,4A,4A) (2A,2A,2A,2A,2A,2A,6B,4A) (2A,2A,2A,2A,2A,2A,6B,6B) (2A,2A,2A,2A,2A,2A,2C,4A) (2A,2A,2A,2A,2A,2A,2C,6B) (2A,2A,2A,2A,2A,2A,2C,2C) (2A,2A,2A,2A,2A,2A,2A,6C) (2A,2A,2A,2A,2A,2A,2A,10A) (2A,2A,2A,2A,2A,2A,2A,12A) (2A,2A,2A,2A,2A,2A,3A,3A,3A) (2A,2A,2A,2A,2A,2A,2B,3A,3A) (2A,2A,2A,2A,2A,2A,2B,2B,3A) (2A,2A,2A,2A,2A,2A,2B,2B,2B) (2A,2A,2A,2A,2A,2A,2A,3A,4A) (2A,2A,2A,2A,2A,2A,2A,3A,6B) (2A,2A,2A,2A,2A,2A,2A,3A,2C) (2A,2A,2A,2A,2A,2A,2A,2B,4A) (2A,2A,2A,2A,2A,2A,2A,2B,6B) (2A,2A,2A,2A,2A,2A,2A,2B,2C) (2A,2A,2A,2A,2A,2A,2A,2A,5A) (2A,2A,2A,2A,2A,2A,2A,2A,10A) (2A,2A,2A,2A,2A,2A,2A,2A,3B) (2A,2A,2A,2A,2A,2A,2A,2A,6A) (2A,2A,2A,3A,3A,3A,4A) (2A,2A,2A,3A,3A,3A,6B) (2A,2A,2A,3A,3A,3A,2C) (2A,2A,2B,2B,2B,3A,3A) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 244020 78000 172920 442720 131760 317920 772200 221760 83360 231840 581040 72960 172800 47040 46656 37500 43560 676830 1275960 2283120 3954720 638400 1682100 514080 1202880 2978640 866880 525000 880160 408240 362880 11656 34695 11910 88254 (2A,2A,2A,2A,2A,2A,2A,2A,3A,3A) (2A,2A,2A,2A,2A,2A,2A,2A,2B,3A) (2A,2A,2A,2A,2A,2A,2A,2A,2B,2B) (2A,2A,2A,2A,2A,2A,2A,2A,2A,4A) (2A,2A,2A,2A,2A,2A,2A,2A,2A,6B) (2A,2A,2A,2A,2A,2A,2A,2A,2A,2C) (2A,2A,2A,2A,2A,2A,2A,2A,2A,2A,3A) (2A,2A,2A,2A,2A,2A,2A,2A,2A,2A,2B) (2A,2A,2A,2A,2A,2A,2A,2A,2A,2A,2A,2A) 1 1 1 1 1 1 1 1 1 4798080 8749440 15355040 4515840 11430720 3386880 33339600 59270400 228191040 Conclusion Here, we compute braid orbits on Nielsen class with the aid of the computer algebra system GAP and MAPCLASS package. A result of the algorithm is that it gives the complete classification of the symmetric group 𝑆7 up to braid actions and diagonal conjugations. The computation shows that there are exactly 754 braid orbits of S7. As a consequence of Lemma 1.3, we find the connected components ℋ𝑟 𝑖𝑛 (𝑆7) of 𝑆7-curves 𝑋, such that 𝑔 = 0. So we have 754 connected components ℋ𝑟 𝑖𝑛 (𝑆7) of the symmetric group of degree seven. Acknowledgments The author would like thank the anonymous referees, whose comments and suggestions helped to improve the manuscript. MOHAMMED SALIH, H.M. 138 References 1. Volklein, H. Groups as Galois groups An Introduction to Cambridge Studies in Advanced Mathematics, volume 53, Cambridge University Press, Cambridge, 1996. 2. Mohammed Salih, H. Finite Groups of Small Genus. 2014, Ph.D. Thesis, University of Birmingham. 3. Michael, D.F. and Volklein, H. The inverse Galois problem and rational points on moduli spaces. Mathematical Annual, 1991, 290(4), 771-800. 4. 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