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ARS MATHEMATICA CONTEMPORANEA 2 (2009) 163–171
Polytopes with groups of type PGL2(q)
Dimitri Leemans ∗
Université Libre de Bruxelles, Département de Mathématiques – C.P. 216
Boulevard du Triomphe, B-1050 Bruxelles
Egon Schulte †
Northeastern University, Department of Mathematics
360 Huntington Avenue, Boston, MA 02115, USA
Received 19 March 2009, accepted 13 August 2009, published online 30 September 2009
Abstract
There exists just one regular polytope of rank larger than 3 whose full automorphism
group is a projective general linear group PGL2(q), for some prime-power q. This poly-
tope is the 4-simplex and the corresponding group is PGL2(5) ∼= S5.
Keywords: Projective general linear groups, abstract regular polytopes.
Math. Subj. Class.: 20G40, 52B11
1 Introduction
In an earlier paper [13] we determined the projective special linear groups L2(q) (often
denoted PSL2(q)), for q a prime power, which occur as full automorphism groups of
abstract regular polytopes of rank 4 or higher. We found that the only groups possible
are L2(11) and L2(19), associated with two locally projective regular polytopes of rank
4, namely Grünbaum’s 11-cell of type {3, 5, 3} and Coxeter’s 57-cell of type {5, 3, 5},
respectively (see [4, 5, 9]).
In the present paper we investigate the projective general linear groups PGL2(q), for
q a prime power. As for L2(q), these groups cannot be the full automorphism groups of
abstract regular polytopes of rank 5 or higher. Moreover, we shall see that if PGL2(q) is the
full automorphism group of an abstract regular polytope of rank 4, then necessarily q = 5
and the polytope is the 4-simplex with group PGL2(5) ∼= S5. This is in stark contrast
to the situation in rank 3, a phenomenon already observed for the groups L2(q). The
∗Supported by the “Communauté Française de Belgique – Actions de Recherche Concertées”.
†Supported by NSA-grant H98230-07-1-0005.
E-mail addresses: dleemans@ulb.ac.be (Dimitri Leemans), schulte@neu.edu (Egon Schulte)
Copyright c© 2009 DMFA Slovenije
164 Ars Math. Contemp. 2 (2009) 163–171
exceptional case is based on the sporadic isomorphism between PGL2(5) and S5. There
is a wealth of regular polyhedra or maps on surfaces with groups (or rotation subgroups)
isomorphic to PGL2(q) or L2(q) (see, for example, McMullen, Monson & Weiss [17],
Glover & Sjerve [8], Conder [3], McMullen [16], Monson & Weiss [19]). In particular it
is known that PGL2(q) is the automorphism group of a regular polyhedron for any q > 2
(see Sjerve & Cherkassoff [22]); the same is true for L2(q) when q 6= 2, 3, 7 or 9.
2 Basic notions
For general background on (abstract) regular polytopes and C-groups we refer to McMullen
& Schulte [18].
A polytope P is a ranked partially ordered set whose elements are called faces. A
polytope P of rank n has faces of ranks −1, 0, . . . , n; the faces of ranks 0, 1 or n − 1
are also called vertices, edges or facets, respectively. In particular, P has a smallest and a
largest face, of rank −1 and n, respectively. A flag of P is a maximal subset of pairwise
incident faces of P . Each flag of P contains n+ 2 faces, one for each rank. In addition to
being locally and globally connected (in a well-defined sense), P is thin; that is, for every
flag and every j = 0, . . . , n−1, there is precisely one other (j-adjacent) flag with the same
faces except the j-face. A polytope of rank 3 is a polyhedron. A polytope P is regular if its
(combinatorial automorphism) group Γ(P) is transitive on the flags. If Γ(P) has exactly
two orbits on the flags such that adjacent flags are in distinct orbits, then P is said to be
chiral.
The groups of regular polytopes are string C-groups, and vice versa. A C-group of rank
n is a group G generated by pairwise distinct involutions ρ0, . . . , ρn−1 which satisfy the
following intersection property:
〈ρj | j ∈ J〉 ∩ 〈ρj | j ∈ K〉 = 〈ρj | j ∈ J ∩K〉 (for J,K ⊆ {0, . . . , n− 1}).
Moreover, G, or rather (G, {ρ0, . . . , ρn−1}), is a string C-group if the underlying Coxeter
diagram is a string diagram; that is, if the generators satisfy the relations
(ρjρk)2 = 1 (for 0 ≤ j < k − 1 ≤ n− 2).
Each string C-group G determines (uniquely) a regular n-polytope P with automorphism
group G. The i-faces of P are the right cosets of the distinguished subgroup Gi := 〈ρj |
j 6= i〉 for each i = 0, 1, . . . , n − 1, and two faces are incident just when they intersect as
cosets; formally, we must adjoin two copies of G itself, as the (unique) (−1)- and n-faces
of P . Conversely, the group Γ(P) of a regular n-polytope P is a string C-group, whose
generators ρj map a fixed, or base, flag Φ of P to the j-adjacent flag Φj (differing from Φ
in the j-face).
We denote by D2n a dihedral group of order 2n and by Zn a cyclic group of order n.
3 The subgroup structure of PGL2(q)
Our first proposition reviews the subgroup structure of L2(q) and may be found in Dick-
son [7] (but was first obtained in papers by Moore [20] and Wiman [23]); see also Hup-
pert [10, Ch. II, §8] for a weaker version, and Kantor [12] for interesting historical infor-
mation about these groups.
D. Leemans and E. Schulte: Polytopes with groups of type PGL2(q) 165
Proposition 3.1. The group L2(q) of order q(q
2−1)
(2,q−1) , where q = p
r with p a prime and r a
positive integer, contains only the following subgroups:
1. Elementary abelian subgroups of order q, denoted by Eq .
2. Cyclic subgroups Zd, for all divisors d of
(q±1)
(2,q−1) .
3. q(q
2−1)
2d(2,q−1) dihedral subgroups groups D2d, for all divisors d of
(q±1)
(2,q−1) with d > 2.
The number of conjugacy classes of these subgroups is 1 if (q±1)d(2,q−1) is odd, and 2 if
it is even.
4. For q odd, q(q
2−1)
12(2,q−1) dihedral subgroups D4 (Klein 4-groups). The number of conju-
gacy classes of these subgroups is 1 if q ≡ ±3(8) and 2 if q ≡ ±1(8). For q even,
the subgroups D4 are listed under family (5).
5. Elementary abelian subgroups of order ps, denoted by Eps , for all natural number s
such that 1 ≤ s ≤ r − 1.
6. Subgroups Eps : Zh, each a semidirect product of an elementary abelian subgroup
Eps and a cyclic subgroup Zh, for all natural numbers s such that 1 ≤ s ≤ r and all
divisors h of p
k−1
(2,1,1) , where k = (r, s) and (2, 1, 1) is defined as 2, 1 or 1 according
as p > 2 and rk is even, p > 2 and
r
k is odd, or p = 2.
7. For q odd or q = 4m, alternating groups A4, of order 12.
8. For q ≡ ±1(8), symmetric groups S4, of order 24.
9. For q ≡ ±1(5) or q = 4m, alternating groups A5, of order 60. For q ≡ 0(5), the
groups A5 are listed under family (10).
10. q(q
2−1)
pw(p2w−1) groups L2(p
w), for all divisors w of r. The number of conjugacy classes
of these subgroups is 2, 1 or 1 according as p > 2 and rw is even, p > 2 and
r
w is
odd, or p = 2.
11. Groups PGL2(pw), for all w such that 2w is a divisor of r.
Observe here that when q is even, PGL2(q) ∼= L2(q), so that family (11) of Proposi-
tion 3.1 really is a subfamily of family (10) in that case.
We frequently require properties of the subgroup lattice of PGL2(q). Two key obser-
vations employed in our proofs are that PGL2(q) can be viewed as a subgroup of L2(q2)
and that PGL2(q) has a unique subgroup isomorphic to L2(q). This allows to extract the
list of subgroups of PGL2(q) from the list of subgroups of L2(q2), leading to the follow-
ing result taken from Cameron, Omidi & Tayfeh-Rezaie [1, §3] (see also Moore [20]). If
q = pr, we let  = ±1 be determined by q ≡  mod 4.
Proposition 3.2. The subgroups of PGL2(q), where q = pr with p an odd prime and r a
positive integer, are as follows.
1. Two conjugacy classes of cyclic subgroups Z2: one class consisting of q(q + )/2
subgroups Z2 contained in the subgroup L2(q), the other of q(q − )/2 subgroups
Z2 not contained in L2(q).
2. One conjugacy class of q(q ∓ )/2 cyclic subgroups Zd, where d | q ±  and d > 2.
166 Ars Math. Contemp. 2 (2009) 163–171
3. Two conjugacy classes of dihedral subgroups D4: one class consisting of q(q2 −
1)/24 subgroups D4 contained in the subgroup L2(q), the other of q(q2 − 1)/8
subgroups D4 not contained in L2(q).
4. Two conjugacy classes of dihedral subgroups D2d, where d | q±2 and d > 2: one
class consisting of q(q2 − 1)/(4d) subgroups D2d contained in the subgroup L2(q),
the other of q(q2 − 1)/(4d) subgroups D2d not contained in L2(q).
5. One conjugacy class of q(q2 − 1)/(2d) dihedral subgroups D2d, where (q ± )/d is
an odd integer and d > 2.
6. q(q2 − 1)/24 subgroups isomorphic to A4, q(q2 − 1)/24 subgroups isomorphic to
S4, and q(q2 − 1)/60 subgroups isomorphic to A5 when q ≡ ±1(10). There is only
one conjugacy class of each of these types of subgroups, and all subgroups lie in the
subgroup L2(q) except for those of type S4 when q ≡ ±3(8).
7. One conjugacy class of pr(p2r−1)/(pm(p2m−1)) subgroups L2(pm), wherem | r.
8. The subgroups PGL(2, pm) where m | r.
9. A semidirect product of the elementary abelian group of order pm with m ≤ r and
the cyclic group of order d with d | q − 1 and d | pm − 1.
Note in particular that L2(q) and PGL2(q) do have dihedral subgroups D2p, even
though they are not explicitly listed among the dihedral groups in items (3,4) of Proposi-
tion 3.1 or items (3,4,5) of Proposition 3.2, respectively; however, they do show up among
the groups in items (6) or (9) of the respective Propositions.
On the other hand, L2(q) and PGL2(q) do not have subgroups isomorphic toD4p. This
makes it impossible for the subgroup G23 = 〈ρ0, ρ1〉 (respectively G01 = 〈ρ2, ρ3〉) in the
next sections to be isomorphic to D2p because otherwise, G2 = 〈ρ0, ρ1, ρ3〉 (respectively
G1 = 〈ρ0, ρ2, ρ3〉) would have to be isomorphic to D4p. Observe that this is what permits
us to consider only dihedral subgroups of the maximal dihedral subgroups in the proof of
Lemma 7 of [13] and should have been pointed out in [13]. A similar comment applies to
Lemma 4.6.
4 PGL2(q) acting on polytopes
In this section, we assume that G is a group isomorphic to PGL2(q), with q = pr. The
prime p = 2 is special: in fact, if p = 2 then PGL2(q) ∼= L2(q) and we may simply appeal
to our paper [13] about polytopes with groups L2(q) and exclude this possibility. Thus we
may assume that p is odd.
We first establish an analogue of Theorem 2 in [14] eliminating polytopes of high ranks.
Lemma 4.1. If PGL2(q) is the full automorphism group of a regular polytope P , then the
rank of P is at most 4.
Proof. Suppose P is a regular polytope of rank n ≥ 4 with group G := Γ(P) = 〈ρ0, . . . ,
ρn−1〉. Then the subgroup 〈ρ0, ρ2, ρ3, . . . , ρn−1〉 of Γ(P) is a subgroup of the centralizer
CG(ρ0) of ρ0 inG. By Proposition 3.2, there are two conjugacy classes of cyclic subgroups
Z2, of respective lengths q(q − 1)/2 and q(q + 1)/2. If G acts by conjugation on the
conjugacy class containing ρ0, then clearly this action is transitive and the stabilizer of ρ0
is its normalizer inG. But the normalizer of an involution is just its centralizer, and its index
in G is just the number of involutions in the conjugacy class. Therefore, the centralizer has
D. Leemans and E. Schulte: Polytopes with groups of type PGL2(q) 167
index q(q− 1)/2 or q(q+ 1)/2 in G and thus it has order 2(q+ 1) or 2(q− 1). Therefore,
the rank is at most 4.
Alternatively we could have argued as follows. The subgroup 〈ρ0, ρ1, ρ3, ρ4〉 of Γ(P) is
of the form D2k×D2l for some k, l. Inspection of the list of subgroups of PGL2(q) given
in Proposition 3.2 then immediately shows that this cannot occur. While this argument is
quick, the previous proof seems to have the advantage of potentially wider applicability to
other types of groups.
We next investigate the possibility for a regular polytope of rank 4 to have a group of
type PGL2(q). (As mentioned before, the case of rank 3 has already been settled in [22].)
Thus we assume that (G, {ρ0, . . . , ρ3}) is a string C-group of type {t, l, s} (that is, the
orders of ρ0ρ1, ρ1ρ2 and ρ2ρ3 in G are t, l and s, respectively). Clearly, t, l, s ≥ 3, since
G is not a direct product of two non-trivial groups. As before we let
Gi = 〈ρj | j 6= i〉 (for i = 0, . . . , 3).
If P is the regular 4-polytope with group G, then G0 must be the group of a vertex-figure
and G3 the group of a facet of P . By Proposition 3.2, the only possible types of subgroups
for G0 and G3 are S4, A5, the (Frobenius) groups of family (9) of Proposition 3.2, or
L2(q′) or PGL2(q′) for some q′. Dihedral subgroups or A4 cannot occur because they
are not irreducible rank 3 C-groups (see Lemma 2 in [13]). The following sequence of
lemmas is aimed at eliminating the groups of family (9) as well as the subgroups L2(q′)
and PGL2(q′) as possibilities.
We require the following basic facts about the groups PGL2(q), which follow directly
from similar properties for the projective special linear groups (see Lemma 3 in [13]); bear
in mind here that PGL2(q) is a subgroup of L2(q2). The centre of a nonabelian subgroup
of G must have order at most 2. Moreover, if H is a nonabelian subgroup of G whose
centre has size 2, then (q is odd and) H must be a dihedral group.
Lemma 4.2. The orders t of ρ0ρ1 and s of ρ2ρ3 must be odd.
Proof. If t is even, then the centre of the nonabelian subgroup
G2 = Z2 ×D2t ∼= (Z2 × Z2)×Dt
is too large. Hence t, and similarly s, must be odd. 2
Our next two lemmas deal with subgroups of type L2(q′) of PGL2(q).
Lemma 4.3. Every subgroup L2(q′) of PGL2(q) lies in the unique subgroup L2(q) of
PGL2(q).
Proof: This is implicit in the count of subgroups of type L2(q′) for PGL2(q) and
L2(q). In fact, there are
q(q2−1)
q′(q′2−1) subgroups L2(q
′) in PGL2(q) (see Proposition 3.2,
item (7)) and the same number of subgroups L2(q′) in L2(q) (see Proposition 3.1, item
(10)).
Lemma 4.4. Let H and K be two subgroups of type L2(q′) in PGL2(q), with q′m = q
for some positive integer m. Then H ∩K cannot contain a dihedral group D2k with k > 2
and k a divisor of (q
′±1)
2 .
168 Ars Math. Contemp. 2 (2009) 163–171
Proof. Since H and K are subgroups of L2(q), we may appeal to a similar such statement
about subgroups of type L2(q′) in L2(q) established in Lemma 6 of [13]. Lemma 6 itself
is slightly weaker than the present lemma, but its proof actually establishes the stronger
version presented here. For the convenience of the reader we repeat the argument here.
(Note that the stronger version forL2(q) leads to some simplifications in the proofs of [13].)
Let k > 2, and let k be a divisor of q
′±1
2 . From Proposition 3.1 we know that
• in L2(q), there are q(q
2−1)
q′(q′2−1) subgroups isomorphic to L2(q
′) and q(q
2−1)
4k subgroups
isomorphic to D2k;
• in L2(q′), there are q
′(q′2−1)
4k subgroups isomorphic to D2k.
Let n := (q
′−1)
2 if k |
(q′−1)
2 and n :=
(q′+1)
2 if k |
(q′+1)
2 . By Proposition 3.1, there
are q(q
2−1)
4n subgroups D2n in L2(q). Each subgroup D2n contains
n
k subgroups D2k.
Therefore, each subgroup D2k is contained in exactly one subgroup D2n. The same kind
of arguments show that each D2n is contained in exactly one L2(q′). Therefore, each
subgroup D2k of L2(q) (with k > 2 and k a divisor of
(q′±1)
2 ) is contained in a subgroup
L2(q′) of L2(q), and the number of subgroups L2(q′) containing a given subgroup D2k is
precisely one. Now the lemma follows.
Observe that the same result may be proven in exactly the same way (with Proposi-
tion 3.1 replaced by Proposition 3.2) if G is L2(q).
Lemma 4.5. The subgroups G0 and G3 of G cannot be isomorphic to a group of family
(9) of Proposition 3.2.
Proof. The involutions in a subgroup Epm : Zd of family (9) are exactly the elements of
the form ϕψ with ϕ ∈ Epm and ψ the involution in the cyclic factor Zd; recall here that
Epm : Zd is a subgroup of Eq : Zq−1 ∼= AGL1(q). The product of two such involutions
is necessarily of order p or trivial. If G3 (say) is isomorphic to Epm : Zd, then this forces
ρ0ρ1 to have order p. But then 〈ρ0, ρ1, ρ3〉 is isomorphic to D4p which is impossible as we
already mentioned earlier.
Lemma 4.6. The subgroupsG0 andG3 ofG cannot be isomorphic toL2(q′), with q′m = q
for some positive integer m.
Proof. Suppose that G3 (say) is isomorphic to L2(q′). Then G3 and its conjugate G
ρ3
3 by
ρ3 are two subgroups of type L2(q′) containing the dihedral group G23 := 〈ρ0, ρ1〉. By
Lemma 4.4, this dihedral group cannot be a group D2k with k > 2 and k a divisor of
(q′±1)
2 . So the only possibility left here is to have G23
∼= D2p, with p the prime divisor of
q, which cannot occur.
Note here that G3 and G
ρ3
3 really are distinct subgroups. In fact, suppose that G3 =
Gρ33 . Then, since ρ0, ρ1, ρ2 ∈ G3, each generator ρj of G must normalize G3 and hence
G3 must be normal in G. But G3 is a subgroup of the simple group L2(q), so it must also
be normal in L2(q) and hence coincide with L2(q). This forces the underlying polytope to
have only two facets, which is impossible since s > 2.
Next we eliminate the subgroups PGL2(q′) as facet and vertex-figure groups.
D. Leemans and E. Schulte: Polytopes with groups of type PGL2(q) 169
Lemma 4.7. The subgroups G0 and G3 of G cannot be isomorphic to PGL2(q′), with
q′m = q for some positive integer m > 1.
Proof. Suppose to the contrary thatG3 ∼= PGL2(q′) (say), with q′m = q andm > 1. Then
the two subgroups G3 and G
ρ3
3 are isomorphic to PGL2(q
′), and D := G3 ∩Gρ33 contains
the dihedral group D2t := 〈ρ0, ρ1〉 of order 2t (≥ 6). The latter group is centralized by ρ3.
Note that t 6= p.
Now, G ∼= PGL2(q) can be embedded in a group K ∼= PGL2(q2). This group K
contains a subgroup I of index 2 isomorphic to L2(q2), and this subgroup I itself contains
G. Moreover, G3 (resp. G
ρ3
3 ) is contained in a subgroup I3 (resp. I
ρ3
3 ) isomorphic to
L2(q′2), and I3 and I
ρ3
3 are both contained in I . Since I3 contains the element ρ0ρ1 of order
t, we must have t | (q
′2±1)
2 , by Proposition 3.1, item (2), applied to L2(q
′2). Therefore,
Lemma 4.4 forces I3 to be equal to I
ρ3
3 ; in other words, ρ3 must normalize I3 in K. Hence
〈G3, ρ3〉 = G ∼= PGL2(q) is a subgroup of the normalizer NK(I3) of I3 in K, and
this normalizer is isomorphic to PGL2(q′2). (Note for the latter that a supergroup of a
subgroup L2(q′2) of PGL2(q) is necessarily of the form L2(q′k) or PGL2(q′k) for some
k; the original subgroup is normal in the supergroup only when k = 2.) Therefore m is
at most 2, and hence m = 2 and q = q′2. Now, (ρ2ρ3)s = 1 and s is odd, so ρ2 and
ρ3 are conjugate in G. But by Proposition 3.2, item (1), applied to G ∼= PGL2(q), since
ρ2 is contained in I3 ∼= L2(q), so is ρ3. It follows that 〈ρ0, ρ1, ρ2, ρ3〉 = L2(q) < G, a
contradiction.
Finally, then, we obtain our main theorem which was first conjectured to be true based
on computations for the atlas of polytopes [14].
Theorem 4.8. If PGL2(q) is the full automorphism group of a regular polytope P of rank
n ≥ 4, then n = 4, q = 5, and P is the 4-simplex {3, 3, 3}. Rephrased in terms of C-
groups, if (G, {ρ0, . . . , ρn−1}) is a string C-group of rank n ≥ 4 isomorphic to PGL2(q),
then n = 4, q = 5, and G ∼= S5 (occurring in its natural representation as the group of the
4-simplex).
Proof. Lemmas 4.5, 4.6 and 4.7 (as well as Lemma 4.1) reduce the possible types of
subgroups of G that can occur as G0 and G3 to only two kinds in each case, namely S4 and
A5. We show that this leaves only one possibility.
It is well-known (and straightforward to check — see [6]) that the only rank 3 polytopes
with group S4 are the tetrahedron {3, 3} (= {3, 3}4), the hemi-octahedron {3, 4}3, and the
hemi-cube {4, 3}3; and those with group A5 are the hemi-icosahedron {3, 5}5, the hemi-
dodecahedron {5, 3}5, and the great dodecahedron {5, 5}3 (= {5, 52}). By [13], Table 1,
we readily see that, when their automorphism groups are taken in pairs to form the vertex-
figure group G0 and facet group G3 of the group G of a regular rank 4 polytope, then
G can only be isomorphic to a group PGL2(q) if both G0 and G3 are groups S4 in its
natural representation as the group of a tetrahedron {3, 3}, that is, if G itself is the group
PGL2(q) ∼= S5 in its natural representation as the group of the 4-simplex {3, 3, 3}.
Our main result can be rephrased by saying that the 4-simplex is the only abstract poly-
tope of rank 4 (or higher) on which a group of type PGL2(q) can admit a faithful transitive
action on the flags. It is quite remarkable that no similar result can hold for actions with
170 Ars Math. Contemp. 2 (2009) 163–171
two flag orbits. In fact, there are many chiral 4-polytopes whose automorphism group is
of type PGL2(q). (Recall here that a polytope is chiral if it has two orbits on the flags
such that any two adjacent flags are in distinct orbits.) For example, if p is a prime with
p ≡ 5 (8) and b, c are positive integers with p = b2 + c2, then there exists a chiral 4-
polytope P of type {4, 4, 3} with toroidal facets {4, 4}(b,c) and with group isomorphic to
PGL2(p) (see [21, p.239,240]). For p = 5 and p = 13 these are the universal chiral poly-
topes {{4, 4}(2,1), {4, 3}} and {{4, 4}(3,2), {4, 3}} with groups PGL2(5) and PGL2(13),
respectively (see also [2]). Similar examples also exist for other Schläfli symbols.
As observed in [21] (see also Section 6 of [13]), if p is a prime with p ≡ 3(4), then there
are regular 4-polytopes whose rotation (even) subgroup is isomorphic to L2(p2) and has
index 2 in the full group. Then, by Theorem 4.8, these polytopes must have a group differ-
ent from PGL2(p2). In fact, given an odd square prime power q, the outer automorphism
group of L2(q) contains a Klein group D4 = {1, a, b, c}, such that the extension L2(q)〈a〉
of L2(q) is isomorphic to PGL2(q) and such that b is induced by the field automorphism
of order 2. Then the extension L2(q)〈c〉 is sharply 3-transitive on the points of the projec-
tive line, as is PGL2(q). There is a third subgroup of index 2 in the automorphism group
Aut(L2(q)), namely the extension L2(q)〈b〉 usually denoted by PΣL2(q). This group is
not sharply 3-transitive on the points of the projective line.
For q = 9, PΣL2(9) ∼= S6, which is known to occur as group of regular polytopes of
ranks 4 and 5 (see [14]). The group L2(9)〈c〉 is isomorphic to the Mathieu group M10 and
is known to be a group that cannot be generated by involutions. Hence it is not the group
of a regular polytope.
For q = 25, a computer search produces, up to isomorphism, 17 rank four polytopes
for PΣL2(25), with Schläfli symbols {3, 3, 6}, {3, 4, 5}, {3, 5, 3}, {3, 6, 3}, {3, 6, 4},
{3, 6, 5}, {3, 6, 6}, {4, 3, 5}, {4, 3, 6}, {4, 5, 4}, {4, 5, 5}, {4, 6, 4}, {5, 3, 6}, {5, 4, 5},
{5, 5, 6}, {5, 6, 5} and {6, 4, 6}. Again, the group L2(25)〈c〉 has no polytope.
Conjecture 4.9. Let q be an odd square prime power, and let L2(q)〈c〉 and L2(q)〈b〉 =
PΣL2(q) be as defined above. Then,
(a) L2(q)〈c〉 is not the group of a regular polytope;
(b) L2(q)〈b〉 is the group of a regular polytope of rank 4.
We remark that groups of type L2(q) or PGL2(q) also occur quite frequently as quo-
tients of certain hyperbolic Coxeter groups (or their rotation subgroups) with non-string
diagrams (see [15], [11]).
5 Acknowledgements
The authors thank the referees and Daniel Pellicer for very useful comments on the prelim-
inary version of this paper.
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