�
EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 36, Num. 1 (2021), 51 – 62

doi:10.17398/2605-5686.36.1.51

Available online June 9, 2021

Hurwitz components of groups with socle PSL(3, q)

H.M. Mohammed Salih

Department of Mathematics, Faculty of Science, Soran University

Kawa St. Soran, Erbil, Iraq

havalmahmood07@gmail.com

Received January 20, 2021 Presented by A. Turull
Accepted May 10, 2021

Abstract: For a finite group G, the Hurwitz space Hinr,g(G) is the space of genus g covers of the
Riemann sphere P1 with r branch points and the monodromy group G. In this paper, we give a
complete list of some almost simple groups of Lie rank two. That is, we assume that G is a primitive

almost simple groups of Lie rank two. Under this assumption we determine the braid orbits on the

suitable Nielsen classes, which is equivalent to finding connected components in Hinr,g(G).

Key words: Genus zero systems, Braid orbits, Connected components.

MSC (2020): 20B15

1. Introduction

Let Ω be a finite set of order n and G be a transitive subgroup of Sn such
that

G = 〈x1,x2, . . . ,xr〉 , (1.1)
r∏
i=1

xi = 1 , xi ∈ G# = G\{1} , i = 1, . . . ,r , (1.2)

r∑
i=1

ind xi = 2(n + g − 1) , (1.3)

where ind xi is the minimal number of 2-cycles needed to express xi as a
product. We call G a group of genus g and the triple (G, Ω,〈x1,x2, . . . ,xr〉) a
genus g system. These conditions correspond to the existence of an n sheeted
branched covering of Riemann surface X of genus g with r-branch points and
monodromy group G [9].

In [9], Guralnick and Thompson have conjectured that the set E∗(g) of
possible isomorphism classes of composition factors of simple groups which

ISSN: 0213-8743 (print), 2605-5686 (online)

c© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.36.1.51
mailto:havalmahmood07@gmail.com
https://publicaciones.unex.es/index.php/EM
https://creativecommons.org/licenses/by-nc/3.0/


52 h.m. mohammed salih

are neither cyclic nor alternating, is finite for all g ≥ 0. Furthermore, they
have observed that the conjecture reduces to the consideration of the system
(G, Ω,〈x1,x2, . . . ,xr〉) where G is primitive on Ω. A useful reference for more
details is [9]. The primitive permutation representations of finite groups are
determined by their maximal subgroups whose structure has been described
by Aschbacher and O’Nan-Scott Theorem [3].

Proposition 1.1. ([3]) Suppose that G is a finite group and M is a max-
imal subgroup of G such that ⋂

g∈G
Mg = 1.

Let S be a minimal normal subgroup of G, let L be a minimal normal subgroup
of S, and let ∆ = {L = L1,L2, . . . ,Lm} be the set of the G-conjugates of L.
Then L is simple, S = 〈L1, . . . ,Lr〉, G = MS and furthermore either

(A) L is of prime order p;
or L is non abelian simple group and one of the following hold:

(B) F∗(G) = S ×R, where S ∼= R and M ∩S = 1;
(C1) F∗(G) = S and M ∩S = 1;
(C2) F∗(G) = S and M ∩S 6= 1 = M ∩L;
(C3) F∗(G) = S and M ∩ S = M1 × M2 × ···× Mt, where Mi = M ∩ Li,

1 ≤ i ≤ t.

As far as we know (see [14, 10, 11, 12]), there are four types of classification
of genus g system as follows:

1. Up to signature.

2. Up to ramification type.

3. Up to the braid action and diagonal conjugation by Aut(G).

4. Up to the braid action and diagonal conjugation by Inn(G).

The weakest classification is up to signature (that is 1.) and the strongest
one is up to the braid action and diagonal conjugation by Inn(G) (that is 4.),
because it includes all 1, 2 and 3.

In [14, 15, 9, 1, 5, 4, 2], they have classified these cases (A), (B), (C1),
(C2), (C3) up to signatures for genus zero. In [11, 12], they have produced a



hurwitz components of groups with socle PSL(3, q) 53

complete list of affine primitive genus 0, 1 and 2 groups up to the braid action
and diagonal conjugation by Inn(G).

A group G is said to be almost simple if it contains a non-abelian simple
group S and S ≤ G ≤ Aut(S). In [10], Kong works on almost simple groups
of type projective special linear group PSL(3,q). Let G be a group such that
PSL(3,q) ≤ G ≤PΓL(3,q) where PΓL(3,q) is the projective semilinear group.
G acts on points in the natural module, that is the set of projective points
of 2-dimensional projective geometry PG(2,q). She gave a complete list for
some almost simple groups of Lie rank 2 up to ramification type in her PhD
thesis for a genus 0, 1 and 2 system.

In this paper, we consider almost simple groups of Lie rank 2 for genus
zero and classify them up to the braid action and diagonal conjugation
by Inn(G).

The equivalence classes of G-covers X of P1 with r branched points are
called a Hurwitz space and denoted by Hinr,g(G) where in denotes an inner
automorphism of G. Note that X is a Riemann surface of genus g.

Let Ci be the conjugacy class of xi. Then the multi set of non trivial
conjugacy classes C = {C1, . . . ,Cr} in G is called the ramification type of
the G-covers X. For any r-tuple (x1, . . . ,xr) gives a ramification type C̄
with xi ∈ Ci for i = 1, . . . ,r. Let C̄ be a fixed ramification type, then the
subset Hinr (G,C̄) of Hinr (G) consists of all [P,φ] with admissible surjective
map φ: π(P1\P,p) → G sends the conjugacy class

∑
pi

to the conjugacy class

Ci for i = 1, . . . ,r. It is a union of connected components in Hinr (G).
In this paper, we study the Hurwitz space Hinr (G). In particular we focus

on the subset Hinr (G,C̄) of Hinr (G). We try to find the connected components
Hr(G,C̄) of G-curves X of genus 0 such that g(X/G) = g(P1) = 0. To do this,
one needs to find corresponding braid orbits. The our main result is Theorem
1.2, which gives the complete classification of primitive genus 0 systems of
almost simple group of Lie rank two.

Theorem 1.2. Up to isomorphism, there exist exactly seven primitive
genus zero groups with socle PSL(3,q) for some q, 3 ≤ q ≤ 13. The corre-
sponding primitive genus zero groups are enumerated in Table 2 and
Table 3.

In our situation, the computation shows that there is exactly 514 braid
orbits of primitive genus 0 systems for some almost simple groups of Lie rank
two. The degree and the number of the branch points are given in Table 1.



54 h.m. mohammed salih

Table 1: Primitive Genus Zero Systems: Number of Components

Degree

Number of
Group Iso.

Types

Number of
Ramification

Types

Number of
Components

r = 3

Number of
Comp.
r = 4

Number of
Comp.
r = 5

Number of
Comp.

Total

13 1 45 93 14 5 112

21 4 76 204 26 4 234

31 1 15 92 2 - 94

57 1 8 72 2 - 74

Totals 7 144 461 44 9 514

This paper consists of four sections as follows. Section 2 sets out some no-
tation and results that will be needed throughout the paper. We then discuss
the relationship between connected components of Hurwitz spaces and braid
orbits on Nielsen classes. In Section 3, we describe our methodology which
will be used to obtain the ramification types and braid orbits. Furthermore,
we give a particular example to explain this methodology. Finally, several
results are given about Hurwitz spaces.

2. Braid action on Nielsen classes

We begin this section with a formal definition of the Artin braid group.

Definition 2.1. For r ≥ 2, the Artin braid group Br is generated by r−1
elements σ1,σ2, . . . ,σr−1 that satisfy the following relations: σiσj = σjσi for
all i,j = 1, 2, . . . ,r − 1 with |i − j| ≥ 2, and σiσi+1σi = σi+1σiσi+1 for
i = 1, 2, . . . ,r − 2. These relations are known as the braid relations.

The braid σi acts on generating tuples x = (x1, . . . ,xr) of a finite group G
with

∏r
i=1 xi = 1 as follows:

(x1, . . . ,xi,xi+1, . . . ,xr)σi = (x1, . . . ,xi+1,x
−1
i+1xixi+1, . . . ,xr) (2.1)

for i = 1, . . . ,r − 1. The braid orbit of x is the smallest set of tuples which
contains x and is closed under the operations (2.1).

Applying φ: π(P1 \P,p) → G to the canonical generators of π1(P1 \P,p)
gives the generators of a product one generating tuple in G that is, φ(λi) = xi.
We define

�r(G) =

{
(x1, . . . ,xr) : G = 〈x1, . . . ,xr〉 ,

r∏
i=1

xi = 1 , xi ∈ G# , i = 1, . . . ,r
}
.



hurwitz components of groups with socle PSL(3, q) 55

Let A ≤ Aut(G). Then the subgroup A acts on �r(G) via sending (x1, . . . ,xr)
to (a(x1), . . . ,a(xr)), for a ∈ A, which is known as the diagonal conjugation.
This action commutes with the operations (2.1). Thus A permutes the braid
orbits. If A = Inn(G), then it leaves each braid orbit invariant [16]. Let
�inr (G) = �r(G)/ Inn(G).

For a ramification type C̄, we define the subset N(C̄) =
{

(x1, . . . ,xr) :
G = 〈x1, . . . ,xr〉,

∏r
i=1 xi = 1, ∃σ ∈ Sn such that xi ∈ Ciσ for all i

}
which is

called the Nielsen class of C̄.

The topology on HAr (G) is well defined. Let Or be the set of all r-tuples
of distinct elements in P1, equipped with the product topology [16].

For the remaining of this section, we collect few results which will be used
to explain the relationship between the braid orbits and their corresponding
covers.

Lemma 2.2. ([16]) The map ΨA : HAr (G) → Or, ΨA([P,φ])) = P , is
covering.

The fundamental group π1(Or,P0) = Br acts on Ψ
−1
A (P0) where P0 =

{1, . . . ,r} is the base point in Or via path lifting where the fiber is

Ψ−1A (P0) =
{

[P0,φ]A : φ : π1(P1 \P0,∞) → G is admissible }.

This φ gives as product one generating tuple (x1, . . . ,xr) of G.

Lemma 2.3. ([16]) We obtain a bijection Ψ−1A (P0) → �
A
r (G) by sending

[P0,φ]A to the generators (x1, . . . ,xr) where xi = φ([γi]) for i = 1, . . . ,r.

The image NA(C̄) of N(C̄) in �Ar (G) is the union of braid orbits. If ΨA
in Lemma 2.2 restricts to a connected component H of HAr (G), then Lemma
2.3 implies that the fiber in H over P0 corresponds to the set NA(C̄).

Proposition 2.4. ([16]) Let C̄ be a fixed ramification type in G, and
the subset HAr (G,C̄) of HAr (G) consists of all [B,φ]A with B = {b1, . . . ,br},
φ: π1(P1 \P,∞) → G and φ(θbi )) ∈ Ci for i = 1, . . . ,r. Then H

A
r (G,C̄) is a

union of connected components in HAr (G). Under the bijection from Lemma
2.3, the fiber in HAr (G,C̄) over B0 corresponds the set NA(C̄). This yields
a one to one correspondence between components of HAr (C) and the braid
orbits on NA(C̄). In particular, Hinr (G,C) is connected if and only if there is
only one braid orbit.



56 h.m. mohammed salih

The following Riemann Existence Theorem tells us there is a one to one
correspondence between the equivalence classes of product one generating tu-
ples (x1, . . . ,xr) of G and the equivalence classes of G-covers of type C̄ such
that xi ∈ Ci for i = 1, . . . ,r.

Proposition 2.5. ([8]) Let G be a finite group and C̄ = {C1, . . . ,Cr} be
a ramification type. Then there exists a G-cover of type C̄ if and only if there
exists a generating tuple (x1, . . . ,xr) of G with

∏r
i=1 xi = 1 and xi ∈ Ci, for

i = 1, . . . ,r.

Definition 2.6. ([8]) Two generating tuples are braid equivalent if they
lie in the same orbit under the group generated by the braid action and diag-
onal conjugation by Inn(G).

This means that if two generating tuples lie in the same braid orbit under
either the diagonal conjugation or the braid action, then the corresponding
covers are equivalent by Riemann’s Existence Theorem.

Definition 2.7. Two coverings µ1 : X1 → P1 and µ2 : X2 → P1 are equiv-
alent if there exists a homeomorphism α: X1 → X2 with µ2α = µ1.

As a consequence we have the following result.

Proposition 2.8. ([16]) Two generating tuples are braid equivalent if
and only if their corresponding covers are equivalent.

To answer whether or not Hr(G,C̄) is connected which is still an open
problem, both computationally and theoretically. The MAPCLASS package
of James, Magaard, Shpectorov and Volklein, is designed to perform braid
orbit computations for a given finite group and given type. Few results were
known about it such as in [11] and [13].

3. Methodology and example: Listing primitive genus zero
systems

The theory introduced in the previous section provides reformation of the
geometric problem into the language of permutation groups. This leads us
to work with permutation groups rather than with G-covers (see Proposition
2.5). The following method shows that the existence primitive genus 0 system
for a given group G and type C̄, and then computing braid orbits on the set



hurwitz components of groups with socle PSL(3, q) 57

of Nielsen class NA(C̄). Proposition 2.4 yields a one to one correspondence
between the braid orbits on NA(C̄) and connected components of HAr (G,C̄).
Now we can decide whether or not HAr (G,C̄) connected, when G is a primitive
almost simple groups of Lie rank two and given type C̄.

We are presenting our computations in Tables 2 and 3. To obtain these
tables we needed to do the following steps:

• We extract all primitive permutation group G by using the GAP function
AllPrimitiveGroups(DegreeOperation,n).

• For every almost simple group G, compute the conjugacy class repre-
sentatives and permutation indices on n points.

• For given n,g and G we use the GAP function RestrictedPartions to
compute all possible ramification types satisfying the Riemann-Hurwitz
formula.

• Compute the character table of G if possible and remove those types
which have zero structure constant.

• For each of the remaining types of length greater than or equal to 4, we
use MAPCLASS package to compute braid orbits, especially by using
the function GeneratingMCOrbits(G,0,tuple). For tuples of length 3
determine braid orbits via double cosets [8].

• We use the same rules for labeling and ordering conjugacy classes of G
as in [13].

This will be done by both the proof in algebraic topology and calculations
of GAP (Groups, Algorithms, Programming) software. Also genus 0 generat-
ing tuples for almost simple groups of type PSL(3,q) on their other primitive
actions and genus 0 are given.

The next example show that how to compute the ramification types and
braid orbits for the group PSL(3, 3).

Example 3.1. Suppose that G = PSL(3, 3) and |Ω| = n = 3
3−1
3−1 = 13.

gap> a:=AllPrimitiveGroups(DegreeOperation,13);

[ C(13), D(2*13), 13:3, 13:4, 13:6, AGL(1, 13), L(3, 3), A(13),

S(13) ]

gap> List(a,x->ONanScottType(x));

[ "1", "1", "1", "1", "1", "1", "2", "2", "2" ]

gap> LoadPackage("mapclass");;

gap> Read("qu1.g");



58 h.m. mohammed salih

gap> CheckingTheGroup(k);

gap> k:=a[7];

L(3, 3)

gap> CheckingTheGroup(k);

gap> gt:=GeneratingType(k,13,0);

Checking the ramification type 66 with 0 remaining

[ [ 7, 8, 8 ], [ 7, 7, 8 ], [ 7, 7, 7 ], [ 6, 8, 5 ],

[ 6, 8, 4 ], [ 6, 3, 5 ], [ 6, 3, 4 ], [ 3, 8, 8 ],

[ 3, 7, 8 ], [ 3, 7, 7 ], [ 3, 3, 8 ], [ 3, 3, 7 ],

[ 3, 3, 3 ], [ 2, 8, 12 ], [ 2, 8, 11 ], [ 2, 8, 10],

[ 2, 8, 9 ], [ 2, 7, 12 ], [ 2, 7, 11 ], [ 2, 7, 10 ],

[ 2, 7, 9 ], [ 2, 6, 6, 8 ], [ 2, 6, 6, 3 ], [ 2, 4, 5 ],

[ 2, 3, 12 ], [ 2, 3, 11 ], [ 2, 3, 10 ], [ 2, 3, 9 ],

[ 2, 2, 8, 8 ], [ 2, 2, 7, 8 ], [ 2, 2, 7, 7 ],

[ 2, 2, 6, 5 ], [ 2, 2, 6, 4 ], [ 2, 2, 3, 8 ],

[ 2, 2, 3, 7 ], [ 2, 2, 3, 3 ], [ 2, 2, 2, 12 ],

[ 2, 2, 2, 11 ], [ 2, 2, 2, 10 ], [ 2, 2, 2, 9 ],

[ 2, 2, 2, 6, 6 ], [ 2, 2, 2, 2, 8 ], [ 2, 2, 2, 2, 7 ],

[ 2, 2, 2, 2, 3 ], [ 2, 2, 2, 2, 2, 2 ] ]

gap> Length(gt);

45

We can pick one of the generating tuple t and compute braid orbits as
follows:

gap> t:=List(gt[45],x->CC[x]);

[ (2,13)(3,10)(4,9)(5,6), (2,13)(3,10)(4,9)(5,6),

(2,13)(3,10)(4,9)(5,6), (2,13)(3,10)(4,9)(5,6),

(2,13)(3,10)(4,9)(5,6), (2,13)(3,10)(4,9)(5,6) ]

gap> orb:=GeneratingMCOrbits(k,0,t);;

Total Number of Tuples: 183980160

Collecting 20 generating tuples .. done

Cleaning done; 20 random tuples remaining

Orbit1:

Length=32760

Generating Tuple =[ ( 2,11)( 4,12)( 7,10)( 9,13),

( 1, 9)( 4, 7)( 6,10)( 8,12), ( 1, 6)( 2, 3)( 7,13)(10,12),

( 1, 9)( 2, 8)( 5, 7)(10,11), ( 2,10)( 5, 7)( 6,12)( 8,11),

( 3,11)( 4,10)( 7, 9)(12,13) ]

Centralizer size=1

0 tuples remaining

Cleaning a list of 20 tuples

Random Tuples Remaining: 0

Cleaning done; 0 random tuples remaining

Computation complete : 1 orbits found.



hurwitz components of groups with socle PSL(3, q) 59

4. Results

In this section, we present some results which related to the connectedness
of the Hurwitz space for some almost simple groups of Lie rank two for genus
zero.

Proposition 4.1. If r ≥ 4 and G = PSL(3,q) where q = 3, 5, then
Hinr (G,C̄) is connected.

Proof. Since we have just one braid orbit for all types C̄ and the Nielsen
classes N(C̄) are the disjoint union of braid orbits. From Proposition 2.4, we
obtain that the Hurwitz spaces Hinr (G,C) are disconnected.

Proposition 4.2. If G = PSL(3, 4).2 or G = PSL(3,q) where q = 4, 9,
then Hinr (G,C̄) is disconnected.

Proof. Since we have at least two braid orbits for some type C̄ and the
Nielsen classes N(C̄) are the disjoint union of braid orbits. From Proposition
2.4, we obtain that the Hurwitz spaces Hinr (G,C) are disconnected.

The proof of the following is analogous to the proof of Proposition 4.1.

Proposition 4.3. If G = PGL(3,q) where q = 4, 7, then Hinr (G,C) is
connected.

Proposition 4.4. If G = PΓL(3, 4) where r ≥ 4, then Hinr (G,C) is con-
nected.

Acknowledgements

I would like thank to the referee for careful reading of the article and
detailed report including suggestions and comments; and I appreciate
his/her effort on reviewing the article.

References

[1] M. Aschbacher, On conjectures of Guralnick and Thompson, J. Algebra
135 (2) (1990), 277 – 343.

[2] M. Aschbacher, R. Guralnick, K. Magaard, Rank 3 permutation
characters and primitive groups of low genus, preprint.



60 h.m. mohammed salih

[3] M. Aschbacher, L. Scott, Maximal subgroups of finite groups, J. Algebra
92 (1) (1985), 44 – 80.

[4] D. Frohardt, R. Guralnick, K. Magaard, Genus 2 point actions of
classical groups, preprint.

[5] D. Frohardt, R. Guralnick, K. Magaard, Genus 0 actions of groups
of Lie rank 1, in “Arithmetic Fundamental Groups and Noncommutative Al-
gebra”, Proceedings of Symposia in Pure Mathematics, 70, AMS, Providence,
Rhode Island, 2002, 449 – 483.

[6] D. Frohardt, K. Magaard, Composition factors of monodromy groups,
Ann. of Math. 154 (2) (2001), 327 – 345.

[7] The GAP Group, GAP – Groups, Algorithms, and Programming, Version
4.6.2, 2013. http://www.gap-system.org

[8] W. Gehao, “Genus Zero Systems for Primitive Groups of Affine Type”, PhD
Thesis, University of Birmingham, 2011.

[9] R. M. Guralnick, J. G. Thompson, Finite groups of genus zero, J.
Algebra 131 (1) (1990), 303 – 341.

[10] X. Kong, Genus 0, 1, 2 actions of some almost simple groups of lie rank 2,
PhD Thesis, Wayne State University, 2011.

[11] K. Magaard, S. Shpectorov, G. Wang, Generating sets of affine groups
of low genus, in “Computational Algebraic and Analytic Geometry”, Con-
temp. Math., 572, AMS, Providence, Rhode Island, 2012, 173 – 192.

[12] H. Mohammed Salih, “Finite Groups of Small Genus”, PhD Thesis, Uni-
versity of Birmingham, 2014.

[13] H. Mohammed Salih, Connected components of affine primitive permuta-
tion groups, J. Algebra 561 (2020), 355 – 373.

[14] M. G. Neubauer, “On Solvable Monodromy Groups of Fixed Genus”, PhD
Thesis, University of Southern California, 1989.

[15] T. Shih, A note on groups of genus zero, Comm. Algebra 19 (10) (1991),
2813 – 2826.

[16] H. Völklein, “Groups as Galois Groups”, Cambridge Studies in Advanced
Mathematics, 53, Cambridge University Press, 1996.

5. Appendix

Note that N.O means number of orbits, L.O means largest length of the
orbit and GZS means Genus 0 System. The following Tables represent almost
simple groups of Lie rank two.

http://www.gap-system.org


hurwitz components of groups with socle PSL(3, q) 61

Table 2: Part1: GZSs for Almost Simple Groups of Lie Rank Two

group ramification type N.O L.O ramification type N.O L.O

(3E,3B,6B) 18 1 (3E,3D,6B) 8 1

(3E,3C,6A) 8 1 (3B,6B,5A) 1 1

(3B,6B,5B) 1 1 (3B,3C,5A) 1 1

(3B,3C,5B) 1 1 (3A,6A,5A) 1 1

(3A,6A,5B) 1 1 (3A,3D,5A) 1 1

PGL(3, 4) (2A,6B,15A) 1 1 (3A,3D,5B) 1 1

(2A,6B,15C) 1 1 (2A,6A,15D) 1 1

(2A,6A,15B) 1 1 (2A,3D,15D) 1 1

(2A,3C,15A) 1 1 (2A,3D,15B) 1 1

(2A,3C,15C) 1 1

(2A,2A,3B,6B) 1 24 (2A,2A,3B,3C) 1 18

(2A,2A,3A,6A) 1 24 (2A,2A,3A,3D) 1 18

Table 3: Part1: GZSs for Almost Simple Groups of Lie Rank Two

group ramification type N.O L.O ramification type N.O L.O

(6A,3B,3B) 8 1 (6A,6A,3B) 12 1

(6A,6A,6A) 8 1 (2A,8A,8B) 1 1

(3A,3B,8A) 1 1 (3A,3B,8B) 1 1

(3A,4A,8A) 1 1 (3A,4B,8B) 6 1

(4A,3B,3B) 8 1 (4A,6A,3B) 8 1

(4A,6A,6A) 8 1 (4A,4A,3B) 12 1

(4A,4A,6A) 6 1 (4A,4A,4A) 1 1

(2A,3B,13A) 1 1 (2A,3B,13B) 1 1

(2A,3B,13C) 1 1 (2A,3B,13D) 1 1

(2A,6A,13A) 1 1 (2A,6A,13B) 1 1

(2A,6A,13C) 1 1 (2A,6A,13D) 1 1

(2A,4A,13A) 1 1 (2A,4A,13B) 1 1

PSL(3, 3) (2A,4A,13C) 1 1 (2A,4A,13D) 1 1

(2A,3A,3A,4A) 1 12 (2A,3A,3A,3B) 1 12

(2A,2A,4A,4A) 1 124 (2A,2A,3B,4A) 1 120

(2A,2A,3B,3B) 1 108 (2A,2A,4A,6A) 1 144

(2A,2A,3B,6A) 1 144 (2A,2A,6A,6A) 1 132

(2A,2A,3A,8A) 1 8 (2A,2A,2A,8B) 1 8

(2A,2A,2A,13A) 1 13 (2A,2A,2A,13B) 1 13

(2A,2A,2A,13C) 1 13 (2A,2A,2A,13D) 1 13

(2A,2A,2A,3A,3A) 1 120 (2A,2A,2A,2A,4A) 1 2016

(2A,2A,2A,2A,3B) 1 1944 (2A,2A,2A,2A,6A) 1 2160

(2A,2A,2A,2A,2A) 1 32760



62 h.m. mohammed salih

Table 3 (continued): Part1: GZSs for Almost Simple Groups of Lie Rank Two

group ramification type N.O L.O ramification type N.O L.O

(2A,4C,7B) 2 1 (2A,4C,7A) 2 1

(2A,4B,7B) 2 1 (2A,4B,7A) 2 1

(2A,4A,7B) 2 1 (2A,4A,7A) 2 1

PSL(3, 4) (2A,5A,5B) 6 1 (3A,4B,4C) 8 1

(3A,4A,4C) 8 1 (3A,4A,4B) 8 1

(3A,3A,5B) 12 1 (3A,3A,5A) 12 1

(2A,2A,2A,2A,2A) 2 756 (2A,2A,2A,5B) 2 30

(2A,8A,5B) 1 1 (2A,8A,10A) 1 1

(2A,8B,5B) 1 1 (2A,8B,10A) 1 1

(2A,6A,8A) 1 1 (2A,6A,8B) 1 1

PSL(3, 5) (2A,3A,24B) 1 1 (2A,3A,24A) 1 1

(2A,3A,24C) 1 1 (2A,3A,24D) 1 1

(4C,4C,4C) 28 1 (3A,4C,4C) 26 1

(3A,3A,4C) 28 1

(2A,2A,2A,8A) 1 32 (2A,2A,2A,8B) 1 32

(2A,2A,2A,4A) 2 180 (3A,3A,4A) 48 1

PSL(3, 7) (2A,4A,8B) 6 1 (2A,4A,8A) 6 1

(2A,4A,7B) 2 1 (2A,4A,7A) 2 1

(2A,4A,7C) 2 1 (2A,4A,14A) 6 1

(4B,4B,4C) 8 1 (3A,4B,6A) 10 1

(2B,4B,14A) 1 1 (2B,4B,14B) 1 1

(2A,4C,14A) 1 1 (2A,4C,14B) 1 1

PΣL(3, 4) (2A,6A,7A) 2 1 (2A,6A,7B) 2 1

(2B,6A,8A) 8 1 (2A,5A,8A) 2 1

(2A,2B,3A,6A) 1 42 (2A,2B,2B,8A) 2 16

(2A,2A,3A,4C) 1 64 (2A,2A,2B,7A) 1 7

(2A,2A,2B,7B) 1 7

(2B,4B,21A) 1 1 (2B,4B,21B) 1 1

(2B,6A,14A) 3 1 (2B,6A,14B) 3 1

(2B,3B,21A) 1 1 (2B,3B,21B) 1 1

(2B,6B,15A) 2 1 (2B,6B,15B) 2 1

(4B,4B,6A) 12 1 (4B,4B,3B) 14 1

PΓL(3, 4) (3A,6B,6B) 12 1

(2B,2B,3C,3B) 1 58 (2B,2B,3C,6A) 1 156

(2B,2B,3A,5A) 1 20 (2B,2B,4B,4B) 1 192

(2B,2B,2B,14A) 1 28 (2B,2B,2B,14B) 1 28

(2B,2B,2A,15B) 1 264 (2B,2B,2A,15A) 1 10

(2B,2A,3A,6B) 1 54

(2B,2B,2B,2B,3A) 1 1824 (2B,2B,2A,2A,3A) 1 192


	Introduction
	Braid action on Nielsen classes
	Methodology and example: Listing primitive genus zero systems
	Results
	Appendix