Polytopes Derived from Sporadic Simple Groups

Abstract

In this article, certain of the sporadic simple groups are analysed, and the polytopes having these groups as automorphism groups are characterised. The sporadic groups considered include all with order less than 4030387201, that is, all up to and including the order of the Held group. Four of these simple groups yield no polytopes, and the highest ranked polytopes are four rank 5 polytopes each from the Higman-Sims group, and the Mathieu group M24.

Full text

POLYTOPES DERIVED FROM SPORADIC SIMPLE
GROUPS
MICHAEL I. HARTLEY AND ALEXANDER HULPKE
Abstract. In this article, certain of the sporadic simple groups are
analysed, and the polytopes having these groups as automorphism groups
are characterised. The sporadic groups considered include all with or-
der less than 4030387201, that is, all up to and including the order of
the Held group. Four of these simple groups yield no polytopes, and
the highest ranked polytopes are four rank 5 polytopes each from the
Higman-Sims group, and the Mathieu group M24.
1. Introduction
The finite simple groups are the building blocks of finite group theory.
Most fall into a few infinite families of groups, but there are 26 (or 27 if
the Tits group 2F4(2)′is counted also) which these infinite families do not
include. These sporadic simple groups range in size from the Mathieu group
M11 of order 7920, to the Monster group Mof order approximately 8×1053.
One key to the study of these groups is identifying some geometric structure
on which they act. This provides intuitive insight into the structure of the
group. Abstract regular polytopes are combinatorial structures that have
their roots deep in geometry, and so potentially also lend themselves to this
purpose. Furthermore, if a polytope is found that has a sporadic group as
its automorphism group, this gives (in theory) a presentation of the sporadic
group over some generating involutions.
In [7], a simple algorithm was given to find all polytopes acted on by a
given abstract group. In this article, that algorithm is applied to various
sporadic simple groups, and the polytopes for each group are enumerated.
For certain of the groups, this confirms results obtained previously. For
others, the results are new. In particular, this article gives new polytopes
acted on by the Mathieu group M24, the Higman-Sims group H S, the Janko
group J3, the McLaughlin group M cL and the Held group H e.
The algorithm, as stated in [7], is as follows.
Date: (date1), and in revised form (date2).
2000 Mathematics Subject Classification. 51M20, 20F65, 52B15.
Key words and phrases. Abstract Polytopes, Sporadic Simple Groups, Mathieu Group,
Janko Group, Tits Group, Higman-Sims Group, Held Group, McLaughlin Group.
1

2 MICHAEL I. HARTLEY AND ALEXANDER HULPKE
•Let Sbe the set of all involutions of Γ, and let Tbe a set consisting
of one representative of each orbit of the action of K= Aut(Γ) on
S. Let S1be a set of 1-tuples corresponding to the elements of T.
•Given Sk, a collection of k-tuples of elements of S, construct Sk+1
as follows. For each tuple tin Sk, let Ktbe the stabilizer of tin K.
Let Tbe a set consisting of one representative of each orbit of the
action of Kton S. For each u∈T, if appending uto tyields a set of
involutions that has a string diagram and satisfies the intersection
property, add it to Pk+1 or Sk+1 respectively as these involutions
generate Γ or a proper subgroup of Γ.
•Stop when some Snis empty.
•Applying the coset poset construction to each t∈Pkyields the
polytopes of rank kwhose automorphism groups are Γ.
A naive application of this algorithm works well for smaller groups. For
larger groups (in particular J3and He), even storing the set Sis too much for
a modern computer’s memory. The changes needed to make the algorithm
work for these groups is explained in Section 6.
The canonical reference for abstract regular polytopes is [18]. For con-
venience, the basic theory is outlined here. An abstract n-polytope is a
partially ordered set Pwith unique minimal and maximal elements, and
an order-preserving rank function from Ponto {−1,0,...,n −1, n}. This
poset corresponds to the face-lattice of the polytope in the classical theory.
The flags of Pare the maximal totally ordered subsets of P, which must
have size n+ 1. For any flag Φ and each i∈ {0,...,n−1}, we require that
there exist a unique flag Φ′of Pdiffering from Φ only by an element of rank
i. This allows the definition of exchange maps φithat map each Φ to the
corresponding Φ′. A further condition on an abstract polytope is that the
group generated by the exchange maps acts transitively on the set of flags
of the polytope. The term “exchange maps” is due to Gordon Williams (see
[20]).
Another action on the flags of the polytope may be defined, that is, the
action of the polytope’s automorphism group. The polytope is called regular
if this action too is transitive on the set of flags. A group generated by invo-
lutions (a ggi) is a group Wwith a specified generating set hs0,...,sn−1iof
involutions. If the group has a string diagram, that is, if (sisj)2= 1 when-
ever i6=j, j ±1, it is called a string ggi, or sggi for short. If it also satisfies
the intersection property, that is, hsi:i∈Ii ∩ hsi:i∈Ji=hsi:i∈I∩Ji,
it is called a string C-group. String Coxeter groups are examples of string
C-groups. The most important result regarding abstract regular polytopes
is that there is a one-to-one correspondence between regular polytopes and
string C-groups. See Chapter 2 of [18], in particular Corollary 2E13.
This correspondence is not complex. Let Wbe a string C-group, and let
Hi=hsj:j6=ii. A unique polytope P(W) is obtained by letting P(W) be
the poset {P−1, Pn} ∪ {uHi:u∈W, 0≤i≤n−1}, with unique maximal

POLYTOPES DERIVED FROM SPORADIC SIMPLE GROUPS 3
and minimal elements Pnand P−1, and with uHi≤vHjif and only if
i≤jand uHi∩vHjis nonempty. The polytope P(W) is regular, and
its automorphism group is isomorphic to W. Conversely, if Pis a regular
polytope, let Wbe its automorphism group. Choose and fix one flag Ψ of
P, and for 0 ≤i≤n−1, let sibe the element of Wmapping Ψ to Ψφi.
Then the sigenerate Wand make it a string C-group. Furthermore, the
polytope constructed from Wwill be isomorphic to P. Writing W= Γ,
and Aut(P) = Γ(P) (as some authors do), the relationship between regular
polytopes and string C-groups may be written succintly as P(Γ(P)) ∼
=P
and Γ(P(Γ)) ∼
=Γ.
It follows that if a simple group does act on a polytope as its automor-
phism group, the group’s structure is truly bound up in the structure of the
polytope.
Before moving on, a few more basic concepts will be outlined. A section
of a polytope is a subset of the form F/G ={x:G≤x≤F}. The sections
of a polytope are also polytopes. The faces are sections of the form F/P−1,
and the cofaces those of the form Pn/G. The rank of a polytope is n, where
the size of the flags is n+ 1, and the rank of an element Fof a polytope
is the rank of the corresponding face. If, for sections F /G of rank 2, the
isomorphism type of the section depends only on the ranks of Fand G
and not on Fand Gthemselves, then Pis said to be equivelar, and has a
well-defined Schl¨afli symbol {p1,...,pn−1}where F/G is a pi-gon whenever
rank F−2 = i= rank G+ 1. Note that all regular polytopes are equivelar.
The rank n−1 faces and cofaces are called the facets and vertex figures
respectively.
A structure reversing bijection from a polytope Pto a polytope Qis called
aduality, with Pbeing the dual of Q(and vice-versa). If in fact P∼
=Q,
then Pis said to be self-dual.
If there exists a structure-preserving surjection from a polytope Pto
another polytope Q, then Pis said to cover (or be a cover for Q), and Q
is a quotient of P. Every polytope is covered by a regular polytope (see
[5]). Furthermore, if there exists a polytope with certain regular facets and
vertex figures, then there exists a universal such polytope which covers all
others (see Theorem 4A2 of [18], and Theorem 2.5 of [6]).
As demonstrated earlier, a regular polytope is isomorphic to a coset geom-
etry of its automorphism group, using the cosets Hi=hsj:j6=ii. For this
reason, the polytopes for many smaller simple groups have been classified as
the geometries for the groups were classified. It should be noted, however,
that most such classifications fail to distinguish polytopes from their duals,
since a polytope and its dual are regarded as identical geometries. More im-
portantly, if a classification of geometries for a particular group is restricted
to subcategory of geometries that does not encapsulate polytopes, it will fail
to completely classify all polytopes related to the group.

4 MICHAEL I. HARTLEY AND ALEXANDER HULPKE
In particular, in [1], [2], [13], [12], [14], [16] and [15] the residually weakly
primitive geometries of various sporadic simple groups were analysed. Al-
though that work had the potential to discover some polytopes for the groups
studied, they could not have classified these polytopes. In point of fact, they
did not discover polytopes for these groups.
In [17] (originally appearing as [19]), most almost simple groups of order
106or less were studied, and the polytopes for each such group were enu-
merated. In particular, [17] completely enumerated the abstract polytopes
whose automorphism groups are the sporadic simple groups M11,M12 ,M22,
J1and J2.
The main contribution of this article is therefore threefold.
•It classifies the polytopes related to the Mathieu groups M23 and
M24, the Tits group, the Higman-Sims group, the third Janko group
J3, the McLaughlin group and the Held group.
•It confirms the results of [17].
•It also serves to distinguish the polytopes discovered in [17] from
their duals.
The structure of this article is simple. Section 2 discusses some of the
more interesting polytopes that arise from the previously published work.
Section 3 desribes the polytopes arising from the Mathieu group M23 , Sec-
tion 4 describes the polytopes for the Tits group 2F4(2)′, Section 5 for the
Higman-Sims, Section 6 for J3, Section 7 for M24 Section 8 for the McLaugh-
lin group McL, and Section 9 for the Held group He. This article provides
some summary information for the polytopes discovered. For example, Ta-
ble 1 summarises the numbers of polytopes for each group, with the num-
bers of self-dual polytopes indicated in brackets (unless obvious). Far more
detailed information is given in the auxiliary information for this article,
[8], including (amongst other things) generating sets for the automorphism
groups of every polytopes discovered.
2. Interesting Polytopes.
This section describes the polytopes arising from the sporadic simple
groups examined in [17].
The Mathieu groups M11 and M22 are not automorphism groups of any
polytopes. M12 is more interesting. It has 40 rank 3 polytopes, which are
summarised in Table 2. The entry in a row labeled pand a column labeled
qis then number of polytopes of type {p, q}discovered. This format also is
adopted for similar tables elsewhere in the article.
Six of the rank 3 polytopes found are self-dual. M12 has one self-dual
rank 4 polytope, of type {6,3,6}, a finite quotient of the infinite universal
{{6,3}(4,0),{3,6}(4,0)}={{6,3}8,{3,6}8}.
The remaining rank 4 polytopes (excluding duals) of M12 are of types
{{3,5}5,{5,4|5}},{3,6,4}(facet {3,6}(3,0)), {3,6,6}(facet {3,6}8), {3,8,4},
{3,10,4},{4,5,5}(two, both with vertex figures {5,5}5, one with facets

POLYTOPES DERIVED FROM SPORADIC SIMPLE GROUPS 5
Group Order Rank 3 Rank 4 Rank 5 Rank ≥6
M11 7920 0 0 0 0
M12 95040 40 (6) 27 (1) 0 0
J1175560 296(0) 4(0) 0 0
M22 443520 0 0 0 0
J2=HJ 604800 261(13) 31(3) 0 0
M23 10200960 0 0 0 0
2F4(2)′17971200 468(20) 57(5) 0 0
HS 44352000 465(39) 111(3) 4(0) 0
J350232960 584(22) 2(2) 0 0
M24 244823040 946(34) 310(0) 4(0) 0
McL 898128000 0 0 0 0
He 4030387200 2292(84) 145(7) 0 0
Table 1. Polytopes for Sporadic Groups.
5 6 8 10
5 0 1 1 0
6 1 3(1) 6 2
8 1 6 6(4) 5
10 0 2 5 1(1)
Table 2. Rank 3 polytopes for the Mathieu group M12 .
{4,5|5}), {4,5,6}(facets {4,5}6), {4,6,5}(two examples), {4,6,6}(facets
{4,6}5), {4,8,4}and {{5,3}5,{3,6}8}.
The rank 4 polytopes of J1have been studied in some detail. There
are two dual pairs. One has type {5,3,5}with dodecahedral facets and
hemi-icosahedral vertex figures (and its dual). It became important in the
construction of the universal polytope of this type, which has group J1×
L2(19). See [9] for more information. The other dual pair has type {5,6,5},
and is a pair of universal polytopes, that is, they have no proper cover with
the same facet and vertex figure types. The vertex figures (of one of the
dual pair) are the unique map Lof type {6,5}with group of order 660. The
rank 3 polytopes of J1are summarised in Table 3. In [10] it is shown how
the polytope of type {5,3,5}may be used as a basis to construct all of the
rank 4 thin residually connected geometries of the Janko group J1.
The second Janko group J2is the automorphism group of 296 rank 3
polytopes, summarised in Table 4, and of 31 rank 4 polytopes. Of the
rank 4 polytopes, 3 are self-dual, of type {5,6,5},{{6,3}10,{3,6}10}and
{{10,3}6,{3,10}6}. The remaining rank 4 polytopes are of type {3,5,5},
{3,8,3},{3,8,4},{3,8,5}(three examples), {{5,3}5,{3,6}10},{4,8,5}(two
examples), {5,5,10},{5,8,5}(two examples), {6,3,10}and their duals.

6 MICHAEL I. HARTLEY AND ALEXANDER HULPKE
3 5 6 7 10 11 15 19
3 0 0 0 1 2 1 2 4
5 0 2(0) 0 2 4 0 4 8
6 0 0 0 1 2 3 4 6
7 1 2 1 2(0) 4 3 8 9
10 2 4 2 4 4(0) 6 8 14
11 1 0 3 3 6 2(0) 6 9
15 2 4 4 8 8 6 6(0) 18
19 4 8 6 9 14 9 18 22(0)
Table 3. Rank 3 polytopes for the Janko group J1.
3 4 5 6 7 8 10 12 15
3 0 0 0 0 0 0 0 1 1
4 0 0 0 0 1 0 0 0 2
5 0 0 0 3 3 2 6 1 3
6 0 0 3 3(1) 3 3 12 4 6
7 0 1 3 3 2(2) 2 9 2 6
8 0 0 2 3 2 3(1) 8 0 4
10 0 0 6 12 9 8 30(2) 8 13
12 1 0 1 4 2 0 8 1(1) 3
15 1 2 3 6 6 4 13 3 10(6)
Table 4. Rank 3 polytopes for the Janko group J2.
The remaining sections consider one by one the sporadic simple groups
not covered in [17], in order of increasing size, and detail the polytopes of
which they act automorphically.
3. The Mathieu group M23 of order 10200960 = 27.32.5.7.11.23.
There are no polytopes which have the Mathieu group M23 as their au-
tomorphism group.
4. The Tits group 2F4(2)′of order 17971200 = 211.33.52.13.
The Tits group is the automorphism group of 468 rank 3 polytopes and
57 rank 4 polytopes. The rank 3 polytopes are summarised in Table 5.
Amongst the rank 4 polytopes are 5 self-dual polytopes, and 26 dual pairs.
The self-dual polytopes have facets of type {4,4}(5,0) with group of order 200,
{5,5}with group of order 720, {4,6}of order 1440 (a quotient of {4,6}8),
{5,5}of order 5120, and {8,6}of order 11232. For the first three of these,
this description gives enough information to completely identify the facet
(using [7] for example).

POLYTOPES DERIVED FROM SPORADIC SIMPLE GROUPS 7
3 4 5 6 8 10 12 13
3 0 0 0 0 0 1 0 1
4 0 0 0 0 4 5 5 6
5 0 0 0 2 4 2 8 7
6 0 0 2 0 6 6 10 12
8 0 4 4 6 22(4) 24 18 22
10 1 5 2 6 24 21(5) 12 12
12 0 5 8 10 18 12 27(5) 22
13 1 6 7 12 22 12 22 20(6)
Table 5. Rank 3 polytopes for the Tits group 2F4(2)′.
3 4 5 6 7 8 10 12 15
3 0 0 0 0 0 0 0 0 1
4 0 0 0 0 2 0 2 0 7
5 0 0 0 0 2 4 0 2 2
6 0 0 0 2(2) 5 11 7 6 6
7 0 2 2 5 9(5) 15 11 5 11
8 0 0 4 11 15 31(9) 24 16 23
10 0 2 0 7 11 24 10(6) 10 14
12 0 0 2 6 5 16 10 7(3) 8
15 1 7 2 6 11 23 14 8 18(14)
Table 6. Rank 3 polytopes for the Higman-Sims group.
5. The Higman-Sims group HS of order 44352000 = 29.32.53.7.11.
The Higman-Sims group acts on 465 rank 3 polytopes, 111 rank 4 poly-
topes, and four rank 5 polytopes. The rank 3 polytopes are summarised in
Table 6.
Of the 111 rank 4 polytopes of the Higman-Sims group, only three are
self-dual. The facets of these are
•type {8,3}, a quotient of {8,3}8, with group of order 336,
•type {5,5}, self-dual, with group of order 600, and
•type {4,5}, with group of order 1920.
The rank 5 polytopes of the Higman-Sims group form two dual pairs, of
types {3,8,5,3}and {5,8,5,3}and their duals. The vertex figures of these
are identical, their group is a non-solvable non-simple group of order 7680,
and centre of order 2. The group {s′
0, s′
1, s′
2, s′
3, s′
4}of the {5,8,5,3}may be
constructed from the group {s0, s1, s2, s3, s4}of the {3,8,5,3}by allowing
s′
i=sifor i6= 1, and s′
i=siωwhere ωis the generator of the centre of
hs1, s2, s3, s4i. The facets of these polytopes have groups of order 252000,
the latter being PΣU3(5), that is, the index 3 subgroup of the automorphism
group of the Unitary group U3(5). In fact, there are two polytopes of type

8 MICHAEL I. HARTLEY AND ALEXANDER HULPKE
{3,8,5}which have PΣU3(5) as their automorphism group. The one in
question here is the one whose facets {3,8}have groups of order 720.
6. The Janko group J3of order 50232960 = 27.35.5.17.19.
As mentioned earlier, the algorithm given in [7] could not be directly
applied to the larger groups considered here, since it requires too much
information to be stored. Two changes were introduced to enable the poly-
topes of J3and He to be found, one in the way involutions were stored, the
other in the way the algorithm orders its operations.
When GAP is asked to compute the right transversal of a group, it stores
it in a compact manner that requires much less memory than would the list of
elements of the right transversal. The involutions of J3may be stored as the
index of elements of the right transversal of the normaliser in J3of a single
involution. It is easy (and efficient) to convert an index to an involution.
The algorithm does not need to convert involutions to indexes, although if it
did, this, too, could be made efficient. The automorphism group of J3(that
is, J3itself) can be represented as as a group permuting these indices, which
in turn means that the stabilisers needed by the algorithm also require less
memory.
For the Held group, there are two conjugacy classes of involutions, so
there are two right transversals that need to be indexed.
The second change to the way the algorithm works is to ensure it op-
erates in a “depth first” rather than a “breadth first” manner. As stated
in [7], the algorithm will begin by finding S1, then find the whole of S2
before calculating S3, then S4and so on. For large groups, these sets can
become extremely large, and overwhelm the computer’s memory before the
algorithm can complete. To overcome this problem, each Skis computed
only partially, and then a corresponding part of Sk+1 is analysed before the
next part of Skis computed.
So, to analyse t= (s1,...,sk)∈Sk,
•find the stabiliser Ktof tin K= Aut(Γ).
•for each orbit Ω of the action of Kton S, find a representative u∈Ω.
•if appending uto tdoes not yield a set of involutions that has a
string diagram and satisfies the intersection property, then move to
the next orbit.
•otherwise, if {s1,...,sk, u}generates Γ, output (s1,...,sk, u) and
move to the next orbit.
•if {s1,...,sk, u}generates a proper subgroup of Γ, then analyse
(s1,...,sk, u) recursively.
•when all orbits Ω have been tested, the analysis of tis complete.
All polytopes for a given group may therefore be found by analysing in turn
each element (s1) of S1.

POLYTOPES DERIVED FROM SPORADIC SIMPLE GROUPS 9
3 4 5 6 8 9 10 12 15 17
3 0 0 0 0 0 0 1 2 2 2
4 0 0 0 0 0 2 1 0 1 0
5 0 0 0 1 4 13 6 5 8 8
6 0 0 1 1(1) 2 8 1 2 4 1
8 0 0 4 2 0 11 6 2 8 5
9 0 2 13 8 11 29(7) 18 16 33 23
10 1 1 6 1 6 18 9(3) 5 12 14
12 2 0 5 2 2 16 5 6(4) 8 8
15 2 1 8 4 8 33 12 8 14(4) 14
17 2 0 8 1 5 23 14 8 14 11(3)
Table 7. Rank 3 polytopes for the Janko group J3.
This method was applied to the Janko group J3and to the Held group
He. The results for J3are summarised below, and for He may be found in
Section He.
The third Janko group J3acts on 584 rank 3 polytopes, and only two
rank 4 polytopes. The two rank 4 polytopes are not a dual pair, but are
self-dual of types {5,9,5}and {3,17,3}. The facets of these are polytopes
whose automorphism groups are the projective special linear groups L2(19)
of order 3420 and L2(16) of order 4080 respectively. The rank 3 polytopes
are summarised in Table 7.
7. The Mathieu group M24 of order 244823040 = 210 .33.5.7.11.23.
In contrast to M11,M22 and M23, the Mathieu group M24 is the auto-
morphism group of many polytopes, 946 of rank 3, 310 of rank 4, and four
of rank 5. The four of rank 5 come as two dual pairs, represented by Schl¨afli
type {3,3,10,4}and {4,3,10,4}and their duals. As for the rank 5 poly-
topes of the Higman-Sims group, the vertex figures of these are isomorphic,
however, in this case the group of these vertex figures is centreless. The
latter group is in fact M22 : 2, which has three polytopes of type {3,10,4}.
The one in question here is the one for which the vertex figure {10,4}has
group of order 320, with centre hωiof order 2. (This {10,4}is in fact a
quotient of {10,4}5.)
The group hs′
0, s′
1, s′
2, s′
3, s′
4iof the {4,3,10,4}may be constructed from
the group hs0, s1, s2, s3, s4iof the {3,3,10,4}by allowing s′
i=sifor i6= 0,
and letting s′
0=s0ω, where ωis the centre of the group of the coface of type
{10,4}
The rank 3 polytopes are summarised in Table 8. Interestingly, of the 310
rank 4 polytopes, not one is self-dual.

10 MICHAEL I. HARTLEY AND ALEXANDER HULPKE
4 5 6 8 10 11 12
4 0 0 0 2 4 14 14
5 0 0 2 0 2 6 10
6 0 2 12(0) 12 16 33 53
8 2 0 12 4(2) 12 10 30
10 4 2 16 12 12(4) 28 54
11 14 6 33 10 28 36(14) 73
12 14 10 53 30 54 73 132(14)
Table 8. Rank 3 polytopes for the Mathieu group M24 .
3 4 5 6 7 8 10 12 15 17 21
3 0 0 0 0 0 0 0 3 0 5 8
4 0 0 0 0 0 1 3 14 3 16 28
5 0 0 0 0 0 1 0 6 2 3 6
6 0 0 0 5(1) 2 8 13 53 15 45 61
7 0 0 0 2 0 3 3 5 4 8 7
8 0 1 1 8 3 4(2) 8 21 11 28 27
10 0 3 0 13 3 8 7(3) 47 9 27 39
12 3 14 6 53 5 21 47 121(21) 35 94 118
15 0 3 2 15 4 11 9 35 13(3) 36 42
17 5 16 3 45 8 28 27 94 36 82(22) 104
21 8 28 6 61 7 27 39 118 42 104 116(32)
Table 9. Rank 3 polytopes for the Held group.
8. The McLaughlin group McL of order 898128000 = 27.36.53.7.11.
There are no polytopes having the McLaughlin group as their automor-
phism group.
9. The Held group He of order 4030387200 = 210 .33.52.73.17.
The Held group is the automorphism group of many polytopes, no less
than 2292 of rank 3, and 145 of rank 4. The rank 3 polytopes are summarised
in Table 9. Amongst the rank 4 polytopes, seven are self-dual, of type
{4,6,4},{4,7,4},{5,6,5},{6,3,6},{6,6,6}(two examples), and {6,7,6}.
The {6,3,6}is a quotient of the infinite universal {{6,3}(6,0),{3,6}(6,0) }.
Four of the remaining rank 4 polytopes (two dual pairs) have simple
groups as the groups of both the facets and their vertex figures. They are
two of type {3,15,4}and their duals. The facets of type {3,15}have as
their group the projective special linear group L2(16). The vertex figures of
type {15,4}have group the symplectic group S4(4).

POLYTOPES DERIVED FROM SPORADIC SIMPLE GROUPS 11
10. Summary and Conclusions.
A web page providing more information on all these polytopes is accessi-
ble via [8]. The reader will find there tables giving more detailed information
about the polytopes discovered, as well as gzipped GAP [3] files containing,
for each polytope, a generating set of its automorphism group as a permu-
tation group.
The authors would like to thank Dimitri Leemans for helpful comments
on early drafts of this paper.
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4. Gottschalk, H., Leemans, D. “The residually Weakly Primitive Geometries of the
Janko Group J1”, in Pasini, A. et al (eds) “Groups and Geometries”, 65–79
(Birkh¨auser, 1998).
5. M. I. Hartley, “All Polytopes are Quotients, and Isomorphic Polytopes are Quotients
by Conjugate Subgroups”, Discrete Comput. Geom. 21, 289–298 (1999).
6. M. I. Hartley, “Simpler Tests for Semisparse Subgroups”, Ann. Combin. 10 343–352,
(2006)
7. M. I. Hartley, “An Atlas of Small Regular Abstract Polytopes”, Periodica Mathemat-
ica Hungarica 53 149–156 (2006)
8. M. I. Hartley, “Polytopes Derived from Sporadic Simple Groups : Auxiliary Informa-
tion”, http://www.abstract-polytopes.com/sporpolys/ (2006)
9. M. I. Hartley, D. Leemans, “Quotients of a Locally Projective Polytope of Type
{5,3,5}”, Mathematische Zeitschrift, 247 663–674 (2004).
10. M. I. Hartley, D. Leemans, “On the Thin Regular Geometries of Rank Four for the
Janko Group J1”, Innovations in Incidence Geometry, 1181–190 (2005).
11. Leemans, D., “An Atlas of Regular Thin Geometries for Small Groups” Mathematics
of Computation 68 1631–1647 (1999).
12. Leemans, D., “The Residually Weakly Primitive Geometries of J2”, Note di Mathe-
matica 21 77–81 (2002).
13. Leemans, D., “The Residually Weakly Primitive Geometries of M22”, Designs, Codes,
Cryptography, 29 177–178 (2003).
14. Leemans, D., “The Residually Weakly Primitive Geometries of M23”, Atti Sem. Mat.
Fis. Univ. Modena, LI I 313–316 (2004).
15. Leemans, D., “The Residually Weakly Primitive Geometries of J3”, Experimental
Mathematics 13 429–433 (2004).
16. Leemans, D., “The Residually Weakly Primitive Geometries of H S”, Australas. J. of
Combin. 33 231–236 (2005).
17. Leemans, D.; Vauthier, L., “An Atlas of Abstract Regular Polytopes for Small
Groups”, Aequationes Math., 72 313–320 (2006)
18. P. McMullen, E. Schulte, “Abstract Regular Polytopes” (Cambridge University Press,
2002).
19. L. Vauthier, “Construction Algorithmique de Polytopes R´eguliers Abstraits”, MSc
Thesis, Universit´e Libre de Bruxelles, (2005).

12 MICHAEL I. HARTLEY AND ALEXANDER HULPKE
20. G. Williams, “Petrie Schemes”, Canadian J. Math 57, 844–870 (2005).
(Michael I. Hartley) DownUnder Geosolutions, 80 Churchill Ave, Subiaco,
6008, Australia
E-mail address, Michael I. Hartley: mikeh@dugeo.com
(Alexander Hulpke) Department of Mathematics, Colorado State University,
1874 Campus Delivery, Fort Collins, CO 80526, USA
E-mail address, Alexander Hulpke: hulpke@math.colostate.edu