Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 5 (2012) 371–382
Two atlases of abstract chiral polytopes for
small groups
Michael I. Hartley
DownUnder GeoSolutions, 76 Kings Park Rd, West Perth, 6005, Australia
Isabel Hubard
Instituto de Matemáticas, Universidad Nacional Autónoma de México
Ciudad Universitaria, México D.F. 04510
Dimitri Leemans
Department of Mathematics, University of Auckland
Private Bag 92019, Auckland 1142, New Zealand
Received 28 February 2011, accepted 24 February 2012, published online 8 June 2012
Abstract
We construct chiral abstract polytopes in two different ways. Firstly we seek them as
quotients of regular polytopes arising from the Atlas of Small Regular Polytopes (http:
//www.abstract-polytopes.com/atlas/); the resulting atlas of chiral poly-
topes atlas is available on the website http://www.abstract-polytopes.com/
chiral/. Secondly, for each almost simple group Γ such that S ≤ Γ ≤ Aut(S) where
S is a simple group and Γ is a group of order less than 900,000 listed in the Atlas of
Finite Groups, we give, up to isomorphism, the number of abstract chiral polytopes on
which Γ acts regularly. The results have been obtained using a series of MAGMA pro-
grams. All these polytopes are made available on the third author’s website, at http:
//math.auckland.ac.nz/˜dleemans/CHIRAL/.
Keywords: Abstract chiral polytopes, almost simple groups.
Math. Subj. Class.: 52B15, 20F65
E-mail addresses: mikeh@dugeo.com (Michael I. Hartley), hubard@matem.unam.mx (Isabel Hubard),
d.leemans@auckland.ac.nz (Dimitri Leemans)
Copyright c© 2012 DMFA Slovenije
372 Ars Math. Contemp. 5 (2012) 371–382
1 Introduction
Abstract polytopes are combinatorial structures that generalize the face lattice of convex
polytopes. Abstract polytopes of rank 3 are essentially maps (2-embeddings of maps on
surfaces); while every rank 3 polytope is indeed a map, the converse is not true; however,
checking if a map is indeed a polytope or not is not a difficult task (it amounts to checking
the so-called diamond condition for polytopes, defined in Section 2).
There are several approaches to classifying maps (and hypermaps) with high degree of
symmetry. For instance, one can classify all regular or chiral maps of a given genus. Atlases
of that kind have been built essentially by Conder and various collaborators ([2, 3]). One
could also classify maps by their underlying graphs (see for example [6, 7, 8, 15]) or by
their automorphism groups.
The latter approach has also been used to classify regular abstract polytopes. There are
three atlases of regular polytopes; all of them classify the polytopes by their automorphism
groups. The first of these atlases, [11], contains information about all regular polytopes
with automorphism group of size n, where n is at most 2000, and not equal to 1024 or
1536. The second atlas, [18], contains all regular polytopes whose automorphism group is
an almost simple group Γ such that S ≤ Γ ≤ Aut(S), where S is a simple group and Γ is
a group of order less than 1 million appearing in the Atlas of Finite Groups ([4]). The third
atlas [14] extends the second atlas to sporadic groups.
The aim of this paper is to produce atlases of chiral abstract polytopes. In building these
atlases we take two different approaches. The results of each appear in different websites.
On the one hand, we find all the chiral polytopes for which the minimal regular cover falls
into the Atlas of Small Regular Polytopes [11]. In particular, the minimal regular cover
of all such chiral polytopes has at most 2,000 flags. On the other hand, we find chiral
polytopes with automorphism group isomorphic to a small almost simple group, in the
spirit of the second atlas described above.
The paper is organized as follows. Section 2 reviews the basics of abstract polytopes.
In Section 3, we give the basics of abstract chiral polytopes. We also explain how to get
chiral polytopes as quotients of regular polytopes. In Section 4, we describe the information
available on the two websites mentioned in the abstract.
2 Basic notions
We start by reviewing the basic theory of abstract polytopes and regular polytopes. For
details, we refer the reader to [19].
An (abstract) polytope of rank n or an n-polytope is a partially ordered set P endowed
with a strictly monotone rank function having range {−1, . . . , n}. The elements of P are
called faces. For 0 ≤ j < n, a face of rank j is often called a j-face and the faces of rank
0, 1 and n−1 are usually called the vertices, edges and facets of the polytope, respectively.
We shall ask that P have a smallest face F−1, and a greatest face Fn (called the improper
faces of P), and that each flag (that is, maximal chain of the order) of P contain exactly
n + 2 faces. Two flags are said to be adjacent if they differ by exactly one face, they are
j-adjacent, if the rank of the face they differ on is precisely j. We also require that P be
strongly flag-connected, that is, any two flags Φ,Ψ ∈ F(P) can be joined by a sequence
of flags Φ = Φ0,Φ1, . . . ,Φk = Ψ such that each two successive flags Φi−1 and Φi are
adjacent with Φ ∩ Ψ ⊆ Φi for all i. Finally, we require the diamond condition, namely,
whenever F ≤ G, with rank(F ) = j − 1 and rank(G) = j + 1, there are exactly two faces
M. I. Hartley et al.: Two atlases of abstract chiral polytopes for small groups 373
H of rank j such that F ≤ H ≤ G.
The diamond condition implies that given a flag Φ of P , for each i ∈ {0, . . . , n − 1}
there exists a unique flag Φi which is i-adjacent flag to Φ.
Given two faces F and G of a polytope P such that F ≤ G, the section G/F of P is
the set of faces {H ∈ P |F ≤ H ≤ G}, with the induced partial order. If F0 is a vertex,
then the section Fn/F0 is called the vertex-figure of F0. Note that every section G/F of a
polytope P is also a polytope and that rank(G/F ) = rank(G)− rank(F )− 1.
Let P andQ be two n-polytopes. An isomorphism from P toQ is a bijection γ : P →
Q such that γ and γ−1 preserve the order. An anti-isomorphism δ : P → Q is a bijection
reversing the order, in which case P andQ are said to be duals of each other, and the usual
convention is to denote Q by P∗. (Note that (P∗)∗ ∼= P .) An isomorphism from P onto
itself is called an automorphism of P . The set of all automorphisms of P forms a group,
its automorphism group, denoted by Γ(P). It is not difficult to see that Γ(P) acts freely on
F(P), the set of all flags of P . An anti-isomorphism from P to itself is called a duality.
When a duality of P exists, P is said to be self-dual. Note that the set of all dualities is not
a group, as the product of two dualities is in fact an automorphism. However, the dualities
and automorphisms of P together do form the extended group of P , denoted by Γ̄(P).
A polytope P is said to be regular if Γ(P) is transitive on the flags of P . The auto-
morphism group of a regular polytope P is generated by n involutions ρ0, ρ1, . . . , ρn−1,
such that each ρi maps a given (base) flag Φ to the i-adjacent flag, Φi. These distinguished
generators satisfy (among others) the relations
(ρiρj)
pij =  for 0 ≤ i ≤ j ≤ n− 1, (2.1)
where the symbol  denotes the identity element of Γ(P), pii = 1 for all i, and pji = pij ≥
2 whenever |i−j| = 1, and pij = 2 otherwise. Letting pi = pi−1,i = pi,i−1 for 1 ≤ i < n,
we say that P has Schläfli type {p1, . . . , pn−1}.
Furthermore, the generators ρi for Γ(P) satisfy an additional condition, often called
the intersection property, namely
〈 ρi | i ∈ I 〉∩〈 ρi | i ∈ J 〉 = 〈 ρi | i ∈ I∩J 〉 for every I, J ⊆ {0, 1, . . . , n−1}. (2.2)
Conversely, if Γ is a group generated by elements ρ0, ρ1, . . . , ρn−1 which satisfy the rela-
tions (2.1) and condition (2.2), then there exists a polytope P with Γ(P) ∼= Γ. For more
details on this correspondence, we refer to [19].
3 Chiral polytopes
In this section we define the basic properties of chiral polytopes. We state some relations
between chiral and regular polytopes. For details see [20] and [21]. Finally, in a subsection,
we explain how to get chiral polytopes as quotients of regular polytopes.
Every regular polytope P has a rotation subgroup Γ+(P) of Γ(P) generated by
σi := ρi−1ρi, i = 1, 2, . . . , n− 1.
These σi satisfy at least the relations
σpii =  for 1 ≤ i ≤ n− 1, (3.1)
(σiσi+1 . . . σj)
2 =  for 1 ≤ i < j ≤ n− 1. (3.2)
374 Ars Math. Contemp. 5 (2012) 371–382
Here again {p1, p2, . . . , pn−1} is the Schläfli type of P . Note that Γ+(P) has index at most
two in Γ(P). A regular n-polytope P is called directly or orientably regular if Γ+(P) has
index two in Γ(P).
An n-polytope P with base flag Φ is called chiral if it is not regular, but there ex-
ist automorphisms σ1, σ2, . . . , σn−1 such that each σi fixes all faces in Φ different from
(Φ)i−1 and (Φ)i, and cyclically permutes consecutive i-faces of P in the rank 2 section
(Φ)i+1/(Φ)i−2 of P . (By (Φ)i we mean here the i-face of Φ.) Such automorphisms gen-
erate Γ(P) and are called the distinguished generators of Γ(P) with respect to Φ. These
distinguished generators satisfy the relations (3.1) and (3.2), and {p1, . . . , pn−1} is again
said to be the Schläfli type ofP . Note that chiral polytopes occur in pairs of enantiomorphic
forms, with one being the ‘mirror image’ of the other. In fact, one enantiomorphic form of
a polytope is associated with a base flag Φ and the other with any of the flags adjacent to it.
Furthermore, if σ1, . . . , σn−1 are the distinguished generators of a chiral polytope P with
respect to a base flag Φ, then the distinguished generators of P with respect to Φ0 (i.e. the
enantiomorphic form) are σ−11 , σ
2
1σ2, σ3, . . . , σn−1.
In a similar way as for the regular case, the distinguished generators of the automor-
phism group of a chiral polytope satisfy an intersection condition, arising from considering
the stabilizers of the chains ΦJ̄ := {(Φ)j | j /∈ J}, for each J ⊆ {0, 1, . . . , n− 1}.
In order to state this intersection condition we first define the “half-turns” in Γ(P) to
be the involutions τi,j := σi . . . σj , for 1 ≤ i < j ≤ n − 1. Furthermore, for each
i ∈ {1, . . . n− 1}, we define τi,i := σi and τ0,j = τi,n := , the identity element of Γ(P).
Then the stabilizer in Γ(P) of the chain ΦJ̄ (with J ⊆ {0, . . . , n− 1}) is the subgroup
ΓJ := 〈τi,j | i ≤ j; i− 1, j ∈ J〉.
Hence, the intersection condition for Γ, stated in terms of these half-turns, is given by
ΓI ∩ ΓJ = ΓI∩J , for all I, J ⊆ {0, . . . , n− 1}. (3.3)
Conversely, if Γ is any group generated by elements σ1, σ2, . . . , σn−1 which satisfy
the relations (3.1) and (3.2), as well as the intersection condition (3.3), then there exists a
polytope P of rank n which is either directly regular or chiral. The Schläfli type of P is
{p1, . . . , pn−1}, where pi is the order of σi (for 1 ≤ i < n); and Γ(P) ∼= Γ if P is chiral,
or Γ+(P) ∼= Γ if P is directly regular.
Moreover, P is directly regular if and only if there exists an involutory group automor-
phism ρ : Γ→ Γ such that
ρ(σ1) = σ
−1
1 , ρ(σ2) = σ
2
1σ2, ρ(σi) = σi for 3 ≤ i ≤ n− 1 (3.4)
(or in other words, acting like conjugation by the generator ρ0 in the orientably regular
case). That is, P is chiral whenever no such automorphism exists.
Hence, to know whether or not a given a group Γ is the automorphism group of a
chiral polytope, one would have to check if Γ can be generated by elements σ1, . . . , σn−1
satisfying relations (3.1) and (3.2), as well as the intersection condition (3.3). In addition
one would have to check that there exists no group automorphism as in (3.4).
A self-dual polytope P is said to be properly self-dual if there exists a duality δ of P
mapping a base flag to a flag in the same orbit. Clearly, δ must preserve the flag orbits. If no
such δ exits, we say that P is improperly self-dual. Hence if P is a properly (improperly)
self-dual chiral polytope, then every duality of P preserves (interchanges) the two flag
orbits.
M. I. Hartley et al.: Two atlases of abstract chiral polytopes for small groups 375
3.1 Chiral polytopes as quotients of regular ones
Another way to identify chiral polytopes is to seek them as quotients of regular cover-
ing polytopes. In [9] it is shown that any polytope Q may be written in the form P/N
for some regular polytope P and some subgroup N of the automorphism group of P .
These subgroupsN of Γ(P) = 〈ρ0, . . . , ρn−1〉 satisfy a rather technical condition, and are
called semisparse subgroups. The automorphism group of Q may be written (see [10]) as
NΓ(P)(N )/N , where NΓ(P)(N ) is the normaliser of N in Γ(P). Note that the number of
flags ofQ is |Γ(P) : N|. If we are dealing with finite groups, we can conclude immediately
from the definition of a chiral polytope :
Theorem 3.1. The polytope Q = P/N is chiral if and only if NΓ(P)(N ) has index 2 in
Γ(P), and contains none of the generators ρ0, . . . , ρn−1.
This suggests an algorithm for searching for chiral quotients of a given regular polytope
P with automorphism group Γ(P) = 〈ρ0, . . . , ρn−1〉.
• Find all normal subgroups G of Γ(P) of index 2 which do not contain any of the ρi.
• For each such group G, find all the normal subgroups N of G which are not normal
in Γ(P). This is sufficient and necessary to ensure that G is the normaliser for N in
Γ(P).
• Ignore any suchN that are not semisparse in Γ(P). This ensures that we only retain
chiral polytopes, and ignore other combinatorial structures.
• If it is desired to find only chiral quotients Q of P for which P is the minimal
covering polytope whose automorphism group acts on Q via the flag action, then
ignore any such N for which CoreΓ(P)(N ) = ∩w∈Γ(P)Nw is not the trivial group.
• AnyN still retained will be such thatP/N is a chiral polytopeQwith automorphism
group G/N . In fact, this algorithm produces exactly two such N for each Q, since
it finds separately N and N ρ0 for each G. These duplicates are easy to identify and
remove from the list.
4 The Atlases
As we mentioned before, we constructed two different atlases, using different approaches.
One atlas, The Atlas Of Chiral Polytopes With Small Regular Covers [12], contains
information about all chiral polytopes whose regular covers have automorphism group of
order at most 2000, but not 1024 or 1536. To construct it, the algorithm described in Section
3.1 was tried on every polytope in the Atlas of Small Regular Polytopes [11].
The other atlas, The Atlas of Chiral polytopes for Small Almost Simple Groups [13],
was built by writing MAGMA [1] programs that classify, up to isomorphism, ordered tuples
of generators of a given group Γ that satisfy conditions (3.1), (3.2) and (3.3). Our program
then tells us whether such generators correspond to chiral or orientably regular polytopes.
This program was run on all almost simple groups Γ such that S ≤ Γ ≤ Aut(S) where S
is a simple group appearing in the Atlas of Finite Groups by Conway et al [4] and Γ is of
order less than 1 million .
The groups analysed are subdivided into six families, namely
• Sporadic groups and their automorphism groups;
376 Ars Math. Contemp. 5 (2012) 371–382
P Aut(P) Aut(Q)
{3, 3, 8} ∗ 768b ((Q8 × 2) o 2) o S4 Q8 o S4
{3, 6, 9} ∗ 972a 33 o (D9 × 2) (33 o 3) o 2
{3, 6, 18} ∗ 1944a (33 o (D9 × 2))× 2 ((33 o 3) o 2)× 2
{6, 6, 9} ∗ 1944a (33 o (D9 × 2))× 2 ((33 o 3) o 2)× 2
Table 1: Rank 4 chiral quotients Q of small regular polytopes
• Alternating groups and their automorphism groups;
• PSL(2, q) groups and their automorphism groups;
• Other linear groups and their automorphism groups;
• Unitary groups and their automorphism groups;
• Suzuki groups and their automorphism groups.
5 Results
As expected, the first atlas of chiral polytopes, those obtained as quotient of regular poly-
topes, gave us fewer examples of chiral polytopes that the one built with almost simple
groups.
In the first atlas, in total, 56 chiral polytopes were discovered, 48 of rank 3, and 8 of
rank 4. Note that a polytope and its dual are counted as two polytopes in these totals. One
of each dual pair of the rank 4 polytopes is outlined in Table 1, and the entire collection
may be perused online (see [12]).
The polytopes in the last two rows of this table do indeed have isomorphic rotation
groups and isomorphic automorphism groups for their regular covers. The vertex figures
of type {6, 9} and {6, 18} are in turn chiral and appear on the website.
The results of the Atlas of Small Chiral Polytopes for Small Almost Simple Groups are
summarised in the Tables 2 to 7. The Tables are all organised as follows. For a group G,
we give its automorphism group Aut(G), its order (#G), the number of polytopes that G
acts on regularly up to isomorphism, and the number of polytopes G acts on chirally up to
isomorphism. These latter two numbers are sometimes split in several numbers. When we
write x = x1 + x2 + ... + xn, it means there are x1 (resp. x2, . . . xn) polytopes of rank 3
(resp. 4, . . . , n+ 2). Otherwise, it means all polytopes found are of rank three.
On the website, for each group, a list of all polytopes found is available, sorted by rank
and by Schläfli symbols, as well as a MAGMA file containing the involutions generating the
corresponding groups.
6 Some observations on the results
One of the reasons to build such atlases is to try to get insight into the question of whether
are more or fewer regular polytopes than chiral polytopes. Of course, the answer to this
question may depend on tones measure.
For instance, in Marston Conder’s website (http://www.math.auckland.ac.
nz/˜conder/), we find maps classified by genus. It turns out that there are 3378 ori-
entable regular maps of genus less than 102, 862 non-orientable regular maps of genus less
M. I. Hartley et al.: Two atlases of abstract chiral polytopes for small groups 377
G Aut(G) #G #Regular #Chiral
M11 M11 7920 0 66
M12 M12 : 2 95040 67 = 40 + 27 184 = 118 + 64 + 2
M12 : 2 M12 : 2 190080 502 = 416 + 86 700 = 608 + 92
J1 J1 175560 300 = 296 + 4 1096 = 1056 + 40
M22 M22 : 2 443510 0 242
M22 : 2 M22 : 2 887040 375 = 252 + 123 1506 = 1442 + 64
J2 J2 : 2 604800 292 = 261 + 31 986 = 888 + 98
Total 1536 4780
Table 2: Sporadic groups and their automorphism groups
than 203 and 594 chiral orientably-regular maps of genus less than 102. So in terms of
(small) genus, it seems that maps are more often regular than chiral.
Here we measure our results in terms of number of polytopes, up to isomorphism, for a
given group. We say that a group is more chiral than regular if it has more chiral polytopes
than regular polytopes. The Atlas of Regular Polytopes for Small Groups [18] gave, up
to isomorphism and duality, 5265 regular polytopes in total. We computed here that there
are, up to isomorphism, 9205 such polytopes, when Aut(J2) is not taken into account. We
found, up to isomorphism, 17114 chiral polytopes in this atlas. Of course, chiral polytopes
come in pairs of enantiomorphic forms. Therefore, if we decide to count one chiral poly-
tope and its enantiomorphic form as one, we get here 17114/2 = 8557 polytopes. What
clearly appears from Tables 2 to 7 is that some families of almost simple groups are more
chiral than others. For instance, for sporadic groups, we get 4780 chiral polytopes, against
1536 regular polytopes. On the other hand, we get 2050 chiral polytopes and 3904 regular
polytopes for almost simple groups of PSL(2, q) type. But for the latter family, there are
506 chiral polytopes of rank at least 4 and only 51 regular polytopes of rank at least 4.
Unitary groups and Suzuki groups also seem to be much more chiral than regular.
7 Acknowledgements
We gratefully acknowledge financial support of the “Fonds David et Alice Van Buuren”,
the “Communauté Française de Belgique, Action de Recherche Concertée”, and “IACOD
– UNAM (Grant IA101911) Mexico” for this project. We also thank Marston Conder for
interesting discussions during the writing of this paper.
References
[1] W. Bosma, C. Cannon and C. Playoust, The Magma algebra system I: The user language, J.
Symbolic Comput. 24 (1997), 235–265.
[2] M. Conder, Regular maps and hypermaps of Euler characteristic −1 to −200, J. Combin.
Theory Ser. B 99 (2009), 455–459.
[3] M. Conder and P. Dobcsanyi, Determination of all regular maps of small genus, J. Combin.
Theory Ser. B 81 (2001), 224–242.
[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R.A. Wilson, Atlas of Finite Groups,
Oxford U. P., Oxford, 1985.
378 Ars Math. Contemp. 5 (2012) 371–382
G
A
ut(G
)
#G
#R
egular
#C
hiral
A
lt(5)
S
y
m
(5)
60
3
0
S
y
m
(5)
S
y
m
(5)
120
8
=
7+1
6
=
0
+
6
A
lt(6)
P
Γ
L
(2
,9)
360
0
0
P
G
L
(2
,9)
P
Γ
L
(2
,9)
720
24
2
=
0
+
2
S
y
m
(6)
=
P
Σ
L
(2
,9)
P
Γ
L
(2
,9)
720
11
=
3
+
7
+
1
4
=
2
+
0
+
2
M
1
0
P
Γ
L
(2
,9)
720
0
0
P
Γ
L
(2
,9)
P
Γ
L
(2
,9)
1440
21
28
=
8
+
20
A
lt(7)
S
y
m
(7)
2520
0
0
S
y
m
(7)
S
y
m
(7)
5040
81
=
64
+
14
+
2
+
1
102
=
50
+
52
A
lt(8)
S
y
m
(8)
20160
0
14
S
y
m
(8)
S
y
m
(8)
40320
220
=
126
+
71
+
20
+
2
+
1
238
=
182
+
48
+
8
A
lt(9)
S
y
m
(9)
181440
84
=
73
+
11
348
=
270
+
78
S
y
m
(9)
S
y
m
(9)
362880
352
=
249
+
73
+
14
+
968
=
836
+
132
13
+
2
+
1
Total
804
1702
Table
3:A
lternating
groups
and
theirautom
orphism
groups
M. I. Hartley et al.: Two atlases of abstract chiral polytopes for small groups 379
G Aut(G) #G #Regular #Chiral
Alt(5) = PSL(2, 4) = PSL(2, 5) Sym(5) 60 3 0
Sym(5) = PGL(2, 5) Sym(5) 120 8 = 7+1 6 = 0 + 6
PSL(3, 2) = PSL(2, 7) PΓL(2, 7) 168 0 0
PGL(2, 7) = PΓL(2, 7) PΓL(2, 7) 336 28 10 = 0 + 10
Alt(6) = PSL(2, 9) PΓL(2, 9) 360 0 0
PGL(2, 9) PΓL(2, 9) 720 24 2 = 0 + 2
Sym(6) = PΣL(2, 9) PΓL(2, 9) 720 11 = 3 + 7 + 1 4 = 2 + 0 + 2
M10 PΓL(2, 9) 720 0 0
PΓL(2, 9) PΓL(2, 9) 1440 21 28 = 8 + 20
PSL(2, 8) PΓL(2, 8) 504 14 2 = 0 + 2
PΓL(2, 8) PΓL(2, 8) 1512 0 28
PSL(2, 11) PGL(2, 11) 660 5 = 4+1 0
PGL(2, 11) PGL(2, 11) 1320 78 24 = 0 + 24
PSL(2, 13) PGL(2, 13) 1092 19 6 = 0 + 6
PGL(2, 13) PGL(2, 13) 2184 111 14 = 0 + 14
PSL(2, 17) PGL(2, 17) 2448 30 10 = 0 + 10
PGL(2, 17) PGL(2, 17) 4896 208 8 = 0 + 8
PSL(2, 19) PGL(2, 19) 3420 31 = 30+1 4 = 0 + 4
PGL(2, 19) PGL(2, 19) 6840 268 28 = 0 + 28
PSL(2, 16) PΓL(2, 16) 4080 51 2 = 0 + 2
PSL(2, 16) : 2 PΓL(2, 16) 8160 46 = 39 + 7 48 = 32 + 16
PΓL(2, 16) PΓL(2, 16) 16320 0 122
PSL(2, 23) PGL(2, 23) 6072 52 0
PGL(2, 23) PGL(2, 23) 12144 408 10 = 0 + 10
PSL(2, 25) PΓL(2, 25) 7800 30 2 = 0 + 2
PGL(2, 25) PΓL(2, 25) 15600 240 6 = 0 + 6
PΣL(2, 25) PΓL(2, 25) 15600 88 = 61 + 27 62 = 38 + 24
PSL(2, 25).2 PΓL(2, 25) 7800 0 30
PΓL(2, 25) PΓL(2, 25) 31200 117 152 = 108 + 44
PSL(2, 27) PΓL(2, 27) 9828 27 0
PGL(2, 27) PΓL(2, 27) 19656 190 4 = 0 + 4
PΣL(2, 27) PΓL(2, 27) 29484 0 108
PΓL(2, 27) PΓL(2, 27) 58968 0 324
PSL(2, 29) PGL(2, 29) 12180 93 10 = 0 + 10
PGL(2, 29) PGL(2, 29) 24360 655 26 = 0 + 26
PSL(2, 31) PGL(2, 31) 14880 96 6 = 0 + 6
PGL(2, 31) PGL(2, 31) 29760 766 46 = 0 + 46
PSL(2, 32) PΓL(2, 32) 32736 186 6 = 0 + 6
PΓL(2, 32) PΓL(2, 32) 163680 0 744
Total 3904 2050
Table 4: PSL(2, q) groups and their automorphism groups
380 Ars Math. Contemp. 5 (2012) 371–382
G
A
ut(G
)
#G
#R
egular
#C
hiral
P
S
L
(3,2
)
=
P
S
L
(2
,7)
P
Γ
L
(2
,7)
168
0
0
P
G
L
(2,7)
=
P
Γ
L
(2
,7)
P
Γ
L
(2
,7)
336
28
10
=
0
+
10
P
S
L
(3
,3)
P
S
L
(3,3)
:
2
5616
0
0
P
S
L
(3
,3)
:
2
P
S
L
(3,3)
:
2
11232
125
=
124+1
168
=
136
+
32
P
S
L
(3
,4)
P
S
L
(3
,4).D
1
2
20160
0
0
P
S
L
(3
,4).2
1
P
S
L
(3
,4).D
1
2
40320
6
28
=
24
+
4
P
S
L
(3
,4
).3
=
P
G
L
(3,4)
P
S
L
(3
,4).D
1
2
60480
0
0
P
S
L
(3
,4).3.2
3
P
S
L
(3
,4).D
1
2
120960
103
=
99
+
4
262
=
224
+
38
P
S
L
(3
,4
).3.2
2
=
P
Γ
L
(3,4)
P
S
L
(3
,4).D
1
2
120960
0
24
P
S
L
(3
,4).6
P
S
L
(3
,4).D
1
2
120960
0
60
P
S
L
(3
,4).D
1
2
P
S
L
(3
,4).D
1
2
241920
233
=
195
+
32
+
6
392
=
296
+
88
+
8
P
S
L
(3
,4).2
3
P
S
L
(3
,4).2
2
40320
96
=
79
+
17
138
=
102
+
36
P
S
L
(3
,4
).2
2
=
P
Σ
L
(3
,4)
P
S
L
(3
,4).2
2
40320
0
30
=
26
+
4
P
S
L
(3
,4).2
2
P
S
L
(3
,4).2
2
80640
288
=
171
+
117
216
=
164
+
52
P
S
L
(3
,5)
P
S
L
(3,5)
:
2
372000
0
2
=
0
+
2
P
S
L
(3
,5)
:
2
P
S
L
(3,5)
:
2
744000
967
=
964
+
3
2668
=
2494
+
174
Total
1840
3998
Table
5:O
therlineargroups
and
theirautom
orphism
groups
M. I. Hartley et al.: Two atlases of abstract chiral polytopes for small groups 381
G Aut(G) #G #Regular #Chiral
PSU(3, 3) PΓL(3, 3) 6048 0 0
PΓU(3, 3) PΓU(3, 3) 12096 60 = 48 + 12 166 = 146 + 20
PSU(4, 2) PΓU(4, 2) 25920 0 26
PΓU(4, 2) PΓU(4, 2) 51840 276 = 162 + 96 + 18 370 = 270 + 100
PSU(3, 4) PΓU(3, 4) 62400 0 0
PSU(3, 4) : 2 PΓU(3, 4) 124800 153 = 150 + 3 418 = 376 + 42
PΓU(3, 4) PΓU(3, 4) 249600 0 526
PSU(3, 5) PΓU(3, 5) 126000 0 0
PGU(3, 5) PΓU(3, 5) 378000 0 0
PΓU(3, 5) PΓU(3, 5) 756000 488 = 468 + 20 1754 = 1580 + 174
PΣU(3, 5) PΣU(3, 5) 252000 225 = 204 + 21 962 = 834 + 120 + 8
Total 1202 4222
Table 6: Unitary groups and their automorphism groups
G Aut(G) #G #Regular #Chiral
Sz(8) Sz(8) : 3 29120 14 128
Sz(8) : 3 Sz(8) : 3 87360 0 284
Total 14 412
Table 7: Suzuki groups and their automorphism groups
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