Sporadic and Exceptional

We study the web of correspondences linking the exceptional Lie algebras $E_{8,7,6}$ and the sporadic simple groups Monster, Baby and the largest Fischer group. This is done via the investigation of classical enumerative problems on del Pezzo surfaces in relation to the cusps of certain subgroups of $PSL(2,R)$ for the relevant McKay-Thompson series in Generalized Moonshine. We also study Conway's sporadic group, as well as its association with the Horrocks-Mumford bundle.


Introduction
The classification of mathematical structures often confronts the dichotomy between the regular and the exceptional wherein the former generically organizes into some infinite families whilst the latter tantalizes with what at first may seem an eclectic collage but out of whose initial disparity emerges striking order.The earliest and perhaps most well-known example is that of the classification of symmetries in R 3 , the regulars are the infinite families of the cyclic and dihedral groups of the regular n-gon and the exceptionals are the symmetry groups of the five Platonic solids.Similarly, the classification of Lie algebras gave us the Dynkin diagrams of the infinite families of classical groups which are the familiar isometries of vector spaces; in addition, there are the five exceptional diagrams.Indeed, the relation between the R 3 symmetries and the simply laced cases of the (affine) Lie algebras has come to be known as the McKay Correspondence, and which now occupies a cornerstone of modern algebraic geometry and representation theory.
Another significant example is the classification of finite simple groups which, after decades of arduous work, is by now complete.The regulars here are the infinite families of cyclic groups of prime order, alternating groups, as well as the classical Lie groups over finite fields, while the exceptionals are known as the 26 sporadic groups, the largest of which is the Monster, of tremendous size.Here, the second author's old observation that 196884 = 196883 + 1, relating the Monster and the elliptic j-function, prompted the field of Moonshine [1,2].
These above illustrative cases exemplify how astute recognition of exceptional structures can help to unravel new mathematics of profound depth.Of particular curiosity is the less well-known fact-in parallel to the above identity-that the constant term of the j-invariant, viz., 744, satisfies 744 = 3 × 248.The number 248 is, of course, the dimension of the adjoint of the largest exceptional algebra E 8 .In fact, that the cube root of j should encode the representations of E 8 was settled [3] long before the final proof of the Moonshine conjectures [4,5].This relationship between the largest sporadic group and the largest exceptional algebra would connect the McKay Correspondence to Moonshine and thereby weave another beautiful thread into the panoramic tapestry of mathematics.Over the years, there have been various generalizations of Monstrous Moonshine (cf.[6,7]) in mathematics and, more recently, in physics [8][9][10][11][12].Of the vast literature, we will focus on relating the exceptional algebras and the sporadics, review the pertinent background and present a new set of correspondences, which though numerologically intriguing, remain fundamentally mysterious and await further exploration.To guide the reader, we summarise the web of inter-connections in Figure 1 as well as the ensuing list which point to the relevant sections: • McKay Correspondence between E 6,7,8 Dynkin diagrams and (binary extended) symmetries of Platonic solids in (31) and (32); • Dimensions of fundamental representations of E 6,7,8 in (37), number of lines (respectively bitangents and tritangents) as well as (−1)-curves in classical geometry of del Pezzo surfaces and curves thereon in Table 1; • Classes of involutions in M and the nodes of the E 8 Dynkin diagram in (48), similarly, involutory classes in 2.B and E 7 in (49), and 3-transposition classes of 3.Fi ′ 24 and E 6 in (50); • Cusp number sums of the invariant groups of McKay-Thompson series for M, 2.B, 3.Fi ′ 24 and the enumerative geometries for the del Pezzo surfaces dP 3,2,1 in Observations 1, 2, and 3; • Conway's group Co 1 and the Horrocks-Mumford bundle in Observation 5.
The outline of the paper is as follows.We begin with setting the notation for the various subjects upon which we will touch, as well as reviewing the rudiments in as a self-contained manner as possible in §1, before delving into our correspondences in §2.We end with a digression on the Horrocks-Mumford bundle in §3.

Rudiments and Nomenclature
First let us set the notation, and refresh the reader's mind on the characteristics of the various dramatis personae.
1.1 PSL(2, Z) and PSL(2, R) We will denote the modular group as The two important subgroups of Γ for our purposes are the congruence subgroups: Cusps : The action of Γ on the upper-half plane H = {z : Im(z) > 0} by linear-fractional transformation z → az+b cz+d is fundamental.In addition to H , of particular interest is the set of cusps Q ∪ {∞} on the real axis boundary.This is the set of rational points which are invariant under the linear-fractional maps in Γ and are the only points on the real axis which should be adjoined to H when considering the action of Γ.Thus, we will generally speak of the extended upper half plane For any subgroup Θ of Γ (including itself), we can define the set of cusps as An important fact about congruence subgroups Θ is that C(Θ) is finite 1 and we will henceforth call the number of elements in C(Θ) the cusp number.Indeed, with respect to the full modular group, for any two points in the extended upper half-plane 2 x, y ∈ Q ∪ {∞}, there exists a γ ∈ Γ such that γ(x) = y.Thus, the cusp number of the full modular group is 1, i.e., |C(Γ)| = 1.
For the congruence groups, the index of where Modular Curves: By adjoining appropriate cusps which serve as compactification points, we can form the quotient Θ\H * of the extended upper half plane by subgroups Θ of Γ.The first classical result 3 , dating to Klein and Dedekind, is that for the full modular group Γ, this quotient is the Riemann sphere: In general, we can quotient H * by a congruence subgroup and obtain a compact Riemann surface, dubbed the modular curve.
What subgroups Θ also have X(Θ) ≃ P 1 ?This so-called genus zero property is important in several contexts and subgroups possessing it are quite rare [13,14].For example, that there are only 33 (finite-index) torsion-free genus zero subgroups was the classification of [15] and the physical interpretation, the subject of [9,16].
We can write the elements of this normalizer more explicitly [1,17] as follows.Let h be the largest divisor 4 of 24 for which h 2 divides N and N = nh = kh 2 , then 5 The above may seem a little difficult to use and present.In order to more conveniently describe it, as will be later needed extensively in the context of Moonshine, we need some further nomenclature.An important subgroup of Γ 0 (N) + is generated by the so-called Fricke involution 0 −1 N 0 taking z → − 1 Nz .This extends Γ 0 (N) to a group 6 called the Fricke group inside Γ 0 (N) + , and in which the former is of index 2.More generally, we have the following • There are the so-called Atkin-Lehner involutions [17] which are matrices W e of the form (here we adhere to the notation of [1,18,19]) in Γ 0 (N): where the || symbol denotes the Hall divisor, i.e., e||N means e|N and gcd(e, N e ) = 1.We see that the Fricke involution is a special case of W e=1 .The set W e forms a coset of Γ 0 (N) in Γ 0 (N) + and satisfies the relations: W 2 e = I and • For h|n and F h := h 0 0 1 , and define The quantities w m are then the Atkin-Lehner involutions for this group Γ 0 (n|h).
• We now introduce the short-hand notation which has become standard to the literature since [1] (as always, ⟨x With the above notation, the key result is the theorem [17] that Indeed, in our shorthand "N|" simply denotes Γ 0 (N) and furthermore, following [1], we use "1" to denote the full modular group Γ = PSL(2, Z).
Principal Moduli: The central analytic object, again a classical realization dating at least to Klein, is the j-invariant, which is the "only" meromorphic function defined on the upper-half plane invariant under the full modular group; by "only" we mean that all invariant functions are rational functions in j(z).Thus the modular action of Γ on H fixes the field C( j) of rational functions of j.Other than a simple pole at i∞, j(z) is the only holomorphic function invariant under Γ once we fix the normalization j(exp( 2πi Writing the nome q := exp(πiz), we obtain the famous Fourier expansion of j(q) as Now, the pole at z = i∞ (i.e., for q = 0) is explicit and all the coefficients are positive integers.In the ensuing we will often make use of the normalized form where the constant 744 has been set to 0, this is habitually denoted as j M , the subscript will become clear in the following section.Furthermore, we could divide by 1728 to ensure ramification only at 0,1 and ∞.To clarify our convention, we adhere to the following 6 For primes p, using the Fricke involution, we can generate Γ 0 (p) + quite simply as: The j-invariant is a special case of a hauptmodul, or principal modulus.For genus zero subgroups Θ ⊂ Γ, the modular functions, i.e., the field of functions invariant under Θ, is generated by a single function, much like the aforementioned case of the full modular group Γ = PSL(2, Z) where the j-invariant generates C( j).For higher genera, two or more functions are needed to generate the invariants, and there is not as nice a notion of a unique canonical choice 7 .There is, however, at least a notion of replicable functions for higher genus and the reader can consult the work of Smith [21].
In general, a principal modulus for an arbitrary subgroup Θ of Γ can be seen as an isomorphism from Θ\H to C, normalized so that its Fourier series starts as q −1 + O(1).Now, for any genus, up to conjugation there are only a finite number [22] of subgroups of PSL(2, R) and for our case of genus 0, there are [23] precisely 6484.Of these, 616 infinite series have integer q-expansion coefficients (cf.[24] for some recent work beyond genus 0 as well as a classic work on why genus 0 in [25]).

The Monster
Of the many fascinating properties of the Monster sporadic group, we will make particular use of the following, in the context of the Moonshine Conjectures [1], some initial computations which were settled in [2,26] and much of which later proven in [4,5].
Supersingular Primes: The order of the Monster is The observation of Ogg [27] was that a prime p such that Γ 0 (p) + \H is genus zero if and only if p is one appearing in the above list.To this day, though the Moonshine conjectures [1] have been proven [4,5], this earliest observation on Moonshine remains unexplained and Borcherds' proof does not actually contain a direct explanation of the appearance of these particular primes.Thus, Ogg's prize of a bottle of Jack Daniels is yet to be collected [28].These primes 8 are called supersingular and in summary, they are the 15 primes obeying the following equivalent definitions: • The modular curve X 0 (p) + = Γ 0 (p) + \H is genus zero (where Γ 0 (p) + is the congruence group adjoining the Fricke involution as defined earlier); • The terminology "supersingular" coincides with that in the theory of elliptic curves for a reason: every supersingular elliptic curve over F p r can be in fact defined just over the subfield F p .The simplest working definition of such a curve is that, in Legendre form y 2 = x(x − 1)(x − λ ), we have that (cf.[29]) the Hasse invariant • In [30], it was noticed that the Hecke Conjecture is true for Γ 0 (p) only for these 15 primes9 .
• The second author later observed10 that only the trivial cusp-form for the symplectic group Γ p as classified in [31] exists when p is one of the above 15 primes; • Recently, Duncan and Ono [28] point out that the McKay-Thompson series (which we will introduce in detail in (26)) are encoded by precisely the j-invariants of supersingular elliptic curves.
It is interesting that for Γ 0 (N) itself, one has X 0 (N) = Γ 0 (N)\H being genus zero precisely for 15 values of N, namely These values have recently [10] been pointed out to match the Coxeter numbers of root systems of A-type Niemeier Lattices (cf.Eq 2.20 therein).
These are the dimension (degree) of the irreducible representations ρ 1 , ρ 196883 , ρ 21296876 , etc. Now, consider an infinite-dimensional representation of M with The second equality is remarkable and is part of the key results of Moonshine, relating finite groups to modular groups.Now, a character of an element, g ∈ M, is rational if g is conjugate to its inverse.Of the 194 conjugacy classes of M, the only irrational characters are 22 which are complex quadratic valued.If we replace an irreducible representation, R, by the sum with its dual, i.e., R ⊕ R, and remove duplicate representations and classes, this yields the rational character table of M, on which we now focus.The vector of dimensions of irreducible representations is the first column of the character table which makes the above sum into the j-function.We can perform a similar sum for all the 172 rational conjugacy classes and obtain a generating function for each as (26) where are the characters (indeed, characters, being traces of finite matrices, are defined over conjugacy classes) of these representations, called head characters H n (g) and h n (g) = H n (g).This is the McKay-Thompson series.Indeed, for g = id, T id (q) is the above (normalized) j-function, or j M in our notation.As we will see in §2, two conjugacy classes with centralizers of different sizes (27A and 27B) yield the same McKay-Thompson series, hence only 171 (including the identity) are candidates for Moonshine.The reader is referred to OEIS where the integer coefficients of these 172 series are presented [33], of particular interest to us are sequences A000521, A045478, A007246, and A007243 in the said encyclopaedia.
There are further numerological mysteries [34] surrounding M. In addition to the supersingular primes mentioned above, the fact that of the 171 integer characters, some of the associated McKay-Thompson series are linearly dependent over Z, and that subsequently there are 163 Z-independent McKay-Thompson series for the Monster (cf.pp 310 and 317 of [1]).This is intriguing: it is well known that 163 is the largest of the Heegner numbers.We recall that there are precisely 9 of such numbers: Heegner = {1, 2, 3, 7, 11, 19, 43, 67, 163} .
These numbers H have the distinction that the imaginary quadratic field Q( √ −H) has class number one, i.e., the associated ring of integers has unique factorization.
A more recent observation of the second author (cf.an account in [35]) is in E 8 × E 8 heterotic string theory compactification on a K3 surface which is dual to F-theory compactification on a Calabi-Yau threefold elliptically fibred over a complex surface B. In the extremal case where one of the E 8 gauge groups is completely broken, the base surface B has Picard number exactly 194.These and ever-increasing number of observations continue to intrigue us [6,7,[10][11][12][35][36][37][38][39][40].

Monstrous Moonshine
The key result of Moonshine is that the McKay-Thompson series defined in (26), for each conjugacy class of an element g in the Monster, has the following property: Theorem 1 (Moonshine) The q-series T g (q) is the normalized generator of a genus zero function field arising from a group between Γ 0 (N) and its normalizer Γ 0 (N) + in PSL(2, R).
The integer N can be determined in several equivalent ways [1].Let F (g) be precisely the elements of PSL(2, R) which fix T g .That is, as J(q) is the modular invariant of Γ, T g (q) is the invariant of F (g)), then • N is the level of F (g); N is the smallest integer so that the group element sending z → z Nz+1 is in F (g); • If n is the period of the (conjugacy class of) group element g in M (i.e., the smallest integer n such that g n = id), then The "Euler characteristic" (to be detailed later) and cusp number of all the F (g) for the 194 conjugacy classes of the Monster were calculated in [1].More recently, a generalization was performed where the following set ∆ := {G : genus(G) = 0, Γ 0 (m) ⊆ G ⊆ Γ 0 (m) + } for some integer m is analysed [41,42].Here G is some modular subgroup residing between the congruence subgroup Γ 0 (m) and its normalizer Γ 0 (m) + in PSL(2, R) and genus(G) is the genus of the Riemann surface G\H .Setting m = nh 2 where h is the largest divisor of 24 such that h 2 |m, the number of distinct pairs (n, h) is 419.

Exceptional Affine Lie Algebras
The Cartan-Killing classification of simple Lie algebras is a triumph of late 19 th century mathematics.Of the Dynkin diagrams, the simply-laced ones consist only of single bonds and fall into an ADE pattern: the two infinite series A n ≃ sl n+1 (C) and D n ≃ so 2n (C), and the exceptionals E 6,7,8 , the Dynkin diagrams of which are: To each of the above is associated a Platonic solid(s), where the famous five group themselves exactly into 3 classes in the sense that the cube and the octahedron, as well as the dodecahedron and the icosahedron are graph duals and share the same symmetry group, whereas the tetrahedron is self-dual: The finite group of symmetries of each solid can be read off from the Dynkin diagram as follows.Associate generators R, S, T to the extremalities of the Dynkin diagram and the order thereof is equal to the length of legs between it and the central trivalent node, plus 1: Incidentally, We see that in each case |G| divided by the order of R, S, T gives a triple, viz., (6,4,4), (12,8,6) and (30,20,12).We recognize these as the number of (edges, vertices/faces, faces/vertices) of the corresponding solids.The relationship 11 between the Lie groups and the Platonic solids was made more striking in [44,45], where by taking the double covers of the groups in (30) -thus making them discrete subgroups of SU(2), rather than SO(3) in which the rotational symmetries of the solids are visualized -and distinguishing the fundamental 2-dimensional complex irreducible representation R.These binary groups [25] will have orders twice those in (30): Note that the only difference in the presentation is that we remove the condition = I in the definitions.Subsequently, the operator R⊗ gives the decomposition over the irreducible representations {R i }: Remarkably, the a i j matrices are precisely the adjacency matrices 12 of the affine Dynkin diagrams of the extended or affine semisimple Lie algebra of ADE type (the gauge theory implication of this is discussed in [47]); this is the McKay Correspondence.
11 There is another intriguing observation of Kostant [43].The group PSL(n, q) of uni-determinant n × n matrices over the finite field of q elements is a finite group; it acts non-trivially on the projective space P n (F q ) which has q n −1 q−1 elements.However, it is rare that it acts only on a strict subset of these elements.For q a prime, this happens only for PSL(2, q) at q = 2, 3, 5, 7, 11 and the group acts non-trivially on q points only.Of these five, only when p = 5, 7, 11 is PSL(2, q) a simple finite group and in fact does not act non-trivially on fewer than q points (a fact known to Galois).Remarkably, PSL(2, 5) ≃ A 4 × set Z 5 , PSL(2, 7) ≃ S 4 × set Z 7 , and PSL(2, 11) ≃ A 5 × set Z 11 , where the notation × set is to emphasize that it is not a group product (after all, these groups are simple).We see the emergence of E 6,7,8 here from the factors A 4 , S 4 and A 5 .
12 These adjacency matrices are also exactly those with maximum eigenvalue two [46].
In this correspondence, each node of the affine Dynkin diagram is associated to an irreducible representation of the corresponding group: the dimension of the irreducible representation precisely matches the dual Coxeter labels, which are the expansion coefficients a ∨ i of the normalized highest root θ into the basis {α (i)∨ } of simple coroots: 2 Being dimensions of irreducible representations, the sum of squares of these labels will be precisely the orders of the associated binary groups: 24, 48 and 120.The Dynkin diagrams are (with the affine node in white) Algebro-geometrically, the binary discrete subgroups of SU( 2) in ( 31) furnish affine models for K3 surfaces as orbifolds of the form C 2 /Γ and are called du Val singularities [48][49][50].They can be described as affine equations in C[x, y, z] as follows: du Val Defining Eq Deg(x, y, z) E 6 x 2 + y 3 + z 4 = 0 (6, 4, 3) E 7 x 2 + y 3 + yz 3 = 0 (9, 6, 4) E 8 x 2 + y 3 + z 5 = 0 (30, 20, 12) In the above, Deg(x, y, z) means a weight which we can assign to the variables (x, y, z) respectively so that the equations become homogeneous 13 , of degree respectively 12,18,60.
Finally, the numbers relevant to us come from the Lie algebras themselves 14 .We recall that the dimensions of the fundamental representations of our (ordinary non-affine) exceptional algebra are dim dim

Classical Enumerative Geometry
Parallel to the aforementioned Lie algebras, another set of celebrated C19th mathematics comes from enumerative geometry; these will also be of concern to us (see Hitchin's lectures [53]).In particular, three counting problems distinguish themselves to us.We will use the notation that [n|a 1 , a 2 , . . ., a k ] means the (not necessarily complete) intersection of k polynomials of degree a 1 , . . ., a k respectively in P n .

Jacobi (1850):
The quartic curve in P 2 has exactly 28 bitangents.We recall that bitangents are lines tangent to a curve at 2 different points; indeed, by degree count using Bézout, starting at degree 4, curves can have such bitangents.Of course, there are always an infinite number of secants and tangents.Such a curve can be realized as [2|4] and is a Riemann surface of genus 1 2 (4 − 1)(4 − 2) = 3 by adjunction [54].Clebsch (1863): The canonical sextic curve of genus 4 has exactly 120 tritangent planes (i.e., planes which are tangent to the curve at precisely 3 points).This curve can be realized as [4|1, 2, 3], i.e., the intersection of a line, a quadric and a cubic in Fermat form in the 5 homogeneous coordinates of P 4 , giving us the so-called Bring's curve [55,56].
Specifically, Bring's curves can be realized as the Fermat cubic, sliced by the Fermat quadric, and then the hyperplane, in the homogeneous coordinates of P 4 : These classic results may at first seem esoteric.However, our attention is drawn to the numbers 27, 56 = 28 • 2 and 360 = 120 • 3.This is not a coincidence and is well-understood in terms of del Pezzo surfaces (cf.[57]).Now, it is well known that the second homology H 2 (dP d ; Z) of a del Pezzo surface of degree d is generated by the hyperplane class H on the P 2 , as 13 With the exception of E 7 , on comparing with (30) we could see the remnants of the R, S, T generators and their relations with the number of (edges, faces/vertices, vertices/faces) for each of the solids as well as the order of the symmetry groups.To get from the solids to these polynomials in a quick way, q.v., Baez's short introduction to ADE theory in http://math.ucr.edu/home/baez/ADE.html. 14The reader is referred also to the so-called Arnol'd Trinities [51], a mysterious web of correspondences involving, inter alia, E 6,7,8 and R, C, H; therein is nice recasting of 24, 48, 120 in terms of the real projective plane.Moreover, the numbers 120 and 2 • 248 emerge in the context of 2-local subgroups of M and B [52].
well as the 9 − d exceptional blow-up P 1 -curve classes.The intersection matrix of these d + 1 classes is the Cartan matrix of the affine E 9−d algebra and whence the adjacency matrix of the associated Dynkin diagram (cf.(32)).
Indeed, [3|3] is birational to P 2 blown up at 6 generic points [54], furnishing a del Pezzo surface dP 3 of degree 3, and there are 27 lines passing through these blow-up points appropriately as (−1)-curves (curves whose self-intersection equals −1).Likewise, dP 2 , the del Pezzo surface of degree 2, has 56 (−1)-curves.The linear system of the anti-canonical divisor of dP 2 maps to P 2 branched over [2|4] and the 56 curves pair to the 28 bitangents.Finally, dP 3 , has 240 (−1)-curves.The linear system of its anti-canonical divisor maps to P 2 branched over [4|1, 2, 3], with these 240 curves pairing to the 120 tritangents (NB. the order of the binary icosahedral group is 120).Furthermore, we can explicitly see the Weyl groups of the root system of the respective Lie algebras are the automorphism groups of the aforementioned geometric objects [53,58].
Of interest to us also, since we are touching on the subject of bitangents, is the theta characteristic of an algebraic curve X.We recall [57,59] that this is an element ϑ ∈ Pic(X), the Picard group of line bundles on X, which squares to the canonical bundle: ϑ ⊗2 = ω X .It is even/odd according to whether the number of global sections h 0 (X, ϑ ) is even/odd.We have that the number of theta characteristics on X of genus g is The total number 2 2g is, incidentally, the number of points of the Jacobian Jac(X) defined over the finite field F 2 .Importantly, the number of bitangent planes to a curve X is precisely that of odd theta characteristics in (39).In summary, for the above three classical enumerative problems and in relation to the del Pezzo surfaces, we collect the relevant facts in Table 1.

Geometry Configuration
Table 1.The correspondences between 3 classical enumerative geometrical problems and the exceptional Lie algebras.The geometry, [n|a 1 , a 2 , . . ., a k ] means the intersection of k polynomials of degrees a 1 , . . ., a k respectively in P n .W (g) means the Weyl group of the root system of the Lie algebra g, which here is equal to the automorphism group Aut(C ) of the configuration of lines and tangents in the geometry.

Correspondences
Let us first examine the 194 conjugacy classes of M in more detail.In the standard notation of ATLAS [60], the classes are recorded as 'nX' where n is the period of an element in the class and X is a capital letter indexing the classes of period n, ascending alphabetically according to increasing sizes of the centralizer.
As mentioned earlier, this gives the rational character table of size 194 − 22 = 172.With 9 further linear relations amongst the McKay-Thompson series (cf.p310 of [1]), we have the column rank of 172 − 9 = 163, the largest Heegner number.

Desire for Adjacency
The second author's initial observation was that the j-function not only encodes the irreducible representations of M but also that j(q) 15 In GAP [61], these can readily be found using the ClassOrbit( ) command for CharacterTable("M").

DOI
This was in fact the first matter to be settled [3]: the unique level-1 highest-weight representation of the affine Kac-Moody algebra E 8 has graded dimension encoded by j(q) 1 3 .One should also be mindful 16 of the fact that the theta-series for the , the Eisenstein series, so that we have where η(q) is the Dedekind eta-function and ∆(q) is the Ramanujan delta-function.If we were to multiply any two elements of 2A, the resulting element can be in one of only 9 conjugacy classes, viz., 1A, 2A, 3A, 4A, 5A, 6A, 4B, 2B, 3C.The second author then noticed that we have seen these numbers before [36,63]!Glancing back at E 8 in (33), we see that they are precisely the (dual Coxeter) labels of the 9 nodes in the affine Dynkin diagram, which we also know to be the dimensions of the irreducible representations of the binary icosahedral group by [44,45].That is, we have The edges, i.e., the meaning of adjacency in analogy with (32), however, have no clear interpretation in this correspondence which still awaits clarification [36][37][38][39].One difficulty is that the hauptmoduln for the different nodes (classes) here are of different modular levels.A recent work [38] recasts this observation solely in terms of the properties of PSL(2, R).

The Baby and E 7
An important subgroup of the Monster is the affectionately named Baby Monster, B, of order 2 41 • 3 13 Its double cover, 2.B, is the centralizer of class 2A in M. Indeed, comparing with Table 3, we see that the order of 2.B is the size of the associated centralizer of class 2A.
The observation in (48) was generalized by [39] to relate B to E 7 using the explicit embedding of the vertex algebra.In summary, we have the product of two involution classes of the Baby falling into 8 classes whose orders coincide with the dual Coxeter numbers of affine In the above the class names are those of B so that in the double cover 2.B some of these split into different conjugacy classes.In particular, whereas we have (1a, 2b, 3a) on one branch in the diagram, the (1a, 2b, 3a) on the other branch are these classes in B multiplied by the centre Z/2Z element when lifting to 2.B.In other words, one could read the above diagram modulo the Z/2Z centre of 2.B.Furthermore, one could fold the diagram according to the Z 2 symmetry and thereby associate B to the Dynkin diagram of F 4 .

Fischer's Fi ′
24 and E 6 Another important subgroup of M is the largest of the simple Fischer groups, Fi ′ 24 (sometimes denoted as F 3+ ) , of order 24 is the centralizer of class 3A in M. Indeed, on comparing with Table 3, we see that the order of 3.Fi ′ 24 is |M| divided by the size of class 3A.Here, the analogue of (48) was again generalized by [39] for Fi 24 , the double cover of Fi ′ 24 .In particular, the involution classes multiply to only 7 classes which correspond to the affine E 6 labels: As above, the class names are those of Fi ′ 24 and could split up in the triple cover.In particular while on one branch we have classes (1a, 2a) and these on the other two branches are these multiplied by the Z/3Z centre of 3.Fi ′ 24 .Likewise, one could fold according to the Z 3 symmetry and associate Fi ′ 24 itself to the Dynkin diagram of G 2 .

Cusp Numbers
We now strengthen this correspondence of with a further series of observations.Recalling our definitions in §1.2.1, in [1], the cusp number C of the fixing group F (g) associated to the class of g is computed.Moreover, the "Euler characteristic" of F (g) is also computed; this is the integer D such that 2π 3D is the area of the fundamental domain of F (g).For reference, we record the quadruple: (1) class name, (2) cusp number C, (3) indicator D for the area of fundamental domain, and (4) normalizer group Γ 0 (N) + ⊂ PSL(2, R) in the notation of Eq. ( 14) for the 194 conjugacy classes of M, reproduced from Table 2 of [1]; this is presented in Table 2.Note that the D is a multiple of C. As is customary, the Galois conjugates have been grouped together -e.g., classes 23A and 23B are combined in 23AB -because, as aforementioned, they have the same McKay-Thompson series.For completeness we also tally the occurrences of the cusp numbers within the 172 rational conjugacy classes: We are finally ready to state the first of our key observations, which was in fact made by the second author a number of years ago.The goal of the remainder of this paper will be to generalize this observation in various contexts.
Observation 1 For the Monster, we have the following sums for the cusp numbers C g over the 172 rational conjugacy classes: We remark that these independent classes have distinct McKay-Thompson series, except for 27A and 27B, which share the same Hauptmodul.The 360 we recall, from Table 1, is thrice 120, which is the number of tritangent planes to Bring's curve.We will generalize this to a wider context of groups shortly.Furthermore, in analogy to Bring's sextic curve from (38), there is the octavic of Fricke [64] of genus 9, the Fermat [4|1, 2, 4] defined as The number of tritangent planes on F is precisely 2048 = 2 • 1024, twice the sum of square of the cusps.Furthermore, in light of (39), the numbers of odd and even theta characteristics on a curve of genus 4, as is the case with Bring's curve, are respectively 2 4−1 (2 4 − 1) = 120 and 2 4−1 (2 4 + 1) = 136, for a total of 2 8 .The number of odd theta characteristics is precisely the number of bitangents.

Cusp Character
Let us now consider the full length 194 vector of the cusp numbers, without considering the linear dependencies.The centralizer Z(c) attached to each of the 194 conjugacy classes c of the Monster can be found in [60] and also in Table 2a of [1].For reference, we give their size (in prime-factorized form) together with the class names, in Table 3.We see, for example, that the size of the centralizer Z(1A) for the identity class 1A, is |M|.In general, we have for each of the 194 conjugacy classes c.Indeed, we have the orthonormality condition for any character table T iγ := χ i (c γ ) of a finite group G with i = 1, 2, . . ., n indexing the irreducible representations and γ = 1, 2, . . ., n indexing the conjugacy classes.The condition states that the weighted table is unitary: Less succinctly, the above is customarily presented as the following relations Row Orthogonality: Column Orthogonality: We can then use row orthogonality to invert this to obtain Since we are summing over the algebraic conjugate representations, this ensures that a j are integers, as required.The above are generalities, which we can certainly apply to the Monster.For instance, the multiplicity coefficients for its centralizing representation begin with a j = 194, 203334, 21397838 . . .However, let us now consider the vector of cusp 12 • 3); (16B, 2 13 ); (16C, 2 13 ); (17A, Is this a character of a representation?
Let us expand as above, i.e., Inverting using (53), we obtain We find the 194 coefficients and see that they are all positive integers!Due to their sizes, we present only the first few: The first number, 2 3 • 7 2 = 392 is simply the sum over the cusps numbers, exceeding our familiar 360 by 32 because we are summing over those corresponding to non-rational classes as well.That all integer coefficients are positive is non-trivial here.We thus have: The vector v γ is the character of a representation which we shall call the cusp representation.
This representation appears to be significant.

The Baby and E 7 again
Given that Moonshine has been extended to other groups, even at the very inception of the Monster [1,65], it is only natural to speculate whether other sporadics closely related to M give generalizations of Observation 1, and in particular, ones Now, there are 247 conjugacy classes of the group 2.B (for the baby B herself, there are 184 conjugacy classes.)and we will name them according to GAP's database, which is also in accord with standard literature [61].The names are also in the form of -and not to be confused with the McKay-Thompson series -number followed by lower-case letter.Here, the numbers go from 1 to 110 and the letters go from "a" to "x" variously.Using this and the above notation for the McKay-Thompson series for the associated modular subgroup, we can combine the tables of [66,67] and Tables 2 and 3 of [23] to obtain all the cusp numbers of these 247 classes.This is presented in Table 4 for the reader's convenience.
As with the case of the Monster, we remove the duplicates where different conjugacy classes correspond to the same McKay-Thompson series in Table 4, which gives us 207 independent classes over which we can, much as before, sum the cusp numbers as well as their squares.We arrive at Observation 2 For 2.B, we have the following sums for the cusp numbers C g over the 207 conjugacy classes with distinct McKay-Thompson series: Again, examining Table 1, the 448 is 2 3 times 56, the dimension of the fundamental representation of E 7 .Likewise, it is a simple (power of 2) factor of 28, which is the number of bitangent lines for E 7 .
We make two further remarks.First, there is a total (with repeats) of 106 classes in 2.B which are Monstrous (having capital letters) and the sum over cusps C g is ∑ g∈M C g (2.B) = 266 = 2 × 133.
We recognize 133 as the complex dimension (and likewise 266 as the real dimension) of E 7 .Furthermore, if we only take the rational classes, of which there is a total of 226 (so indeed there are some of these which share the same McKay-Thompson series), the cusp sum becomes simply 2 9 = 512.

Cusp Character
As in §2.2.1, we can weight the 247 cusp numbers by the sizes of the centralizers of the 247 conjugacy classes for 2.B.Again, we can expand, according to (58), this vector as a linear combination of the dimensions of the irreducible representations.Remarkably, we find the coefficients to be non-negative integers,signifying that the weighted cusp numbers form the character of some non-trivial representation.For reference, the first few expansion coefficients, in the order of the conjugacy classes presented in Table 4 and in prime factorized form, are Despite some formidable-looking primes, there is indeed a cusp representation for 2.B.

Fischer's Group
Having related the baby to E 7 in our context, as discussed in §2.1.3and §2.1.2,the next natural group to consider is Fi ′ 24 , the largest of Fischer's sporadic groups.Now, its triple cover 3.Fi ′ 24 corresponds to class 3A of the Monster whose McKay-Thompson series is = q −1 + 783q + 8672q 2 + 65367q 3 + . . .
which indeed encodes the dimensions of the irreducible representations of 3.Fi ′ 24 , viz., 1, 8671, 57477 . . .There are 256 conjugacy classes of 3.Fi ′ 24 in total which again, in standard GAP notation, are labeled as 1a, 2a, . . ., 105b.Amongst these 108 come from Fi ′ 24 in an obvious way while the remaining appear as pairs of conjugates under the Z 3 -action; these 108 classes are called essential in [67].Of course, some essentials embed into the 256 as singlets and have no Galois orbits of size 3. Generalized Moonshine for 3.Fi ′ 24 was studied in [67] where all the McKay-Thompson series for the 108 essentials were explicitly constructed.The orbit classes in the full 256 have McKay-Thompson series being multiplied by q 1/3 and q 2/3 and are in some sense not new.
We present, in Table 5, the orbit class structure of the classes, corresponding McKay-Thompson series and the associated cusp number, as a triple in the usual notation {nx, mX, c} where (nx) is either a singlet or a triplet of class names depending how an essential class (which is denoted by the first entry) embeds into the full group, mX is Norton's notation for the series and c is the cusp number.Indeed, there will be 108 entries (and on expanding the orbits, the total number of classes is 256).Now, as before, we extract the classes with unique McKay-Thompson series.Of the 108 essentials, we see that 83 are distinct, therefrom, likewise in the full group, there will be 213 out of the 256.However, we are confronted, for the first time, with irrational McKay-Thompson series due to the multiplication of q ±1/3 , which we could either interpret as being new or not, and we will make the sum in both cases for comparison: Observation 3 For 3.Fi ′ 24 , we have the following sums for the cusp numbers C g over the 213 rational conjugacy classes which have distinct McKay-Thompson series: Had we not considered the irrational McKay-Thompson series as distinct but the same as the essential ones, we are then effectively working over the group Fi ′ 24 , in which case, we have the cusp sums over the 83 distinct classes being For the 256 classes of the group 3.Fi ′ 24 , each is a triple {{mx}, nX, c} where mx is the standard class-name in GAP notation, either as a singlet or as a triplet, nX is the identifier for the McKay-Thompson series for the class in the notation of [68] and c is the cusp number of the associated modular subgroup.The triplet {mx} is organized according to the Z 3 Galois orbit of one of the 108 "essential" classes of Fi ′ 24 which share the same McKay-Thompson series.

Cusp Character
As above, we can weight the 256 cusp numbers by the sizes of the centralizers of the 256 conjugacy classes for 3.Fi ′ 24 .We can expand, according to (58), the multiplicity coefficients in terms of the dimension of the irreducible representations.Once again, we find that, interestingly, these are all non-negative integers so there is indeed a cusp representation for 3.Fi ′ 24 .The first few coefficients, in the order of the conjugacy classes presented in Table 5, are

Conway's Group
Other than the Baby, perhaps the closest sporadic groups to the Monster is Conway's group Co 1 , which is associated to class 2B of M by a cover of order 2 1+24 .The sporadic simple group Co 1 , of order 2  After [65], there has been a host of activity to study [18,19,[72][73][74] Moonshine for Co 0 as well as Co 1 , of which we will employ the most recent results in the last reference.As far back as the earliest results of [72], it was realized that the McKay-Thompson series are given explicitly as products and quotients of Dedekind eta-functions whose arguments are appropriate powers of the nome q; these are so-called eta-quotients [75][76][77].For the Mathieu group M 24 over which there has been extensive activity [10,11], all the McKay-Thompson series can be written entirely as eta-products.
The invariance groups 19 of the classes of Co 0 and Co 1 are tabulated in the Appendix of [74], in the original notation of [1].Unsurprisingly, there are many cases which are not Monstrous McKay-Thompson series, viz., those which are of the form nX with capital "X" in Norton's notation.Unfortunately, the new ones are not given in the standard nx and n ∼ x notation as in our 18 In analogy to (47), the theta-series for the Leech lattice 1−q 2m − ∆(q 2 ) . 19Along a parallel vein, we can examine the second part of [18,19], where the relevant results for the square-free case for Co 0 are presented in the table under the section entitled "Genus 0 groups" at the end.Importantly, the moonshine (modular) group where N is square-free and in the notation of [1] is given in column 2, for each of the classes of Co 0 in comparison with those of the Monster.There is a total of 41 square-free conjugacy classes of Co 0 , corresponding to the following classes of the Monster: 2B, 3B, 5B, 6B, 6C, 6D, 6E, 6E, 6E, 7B, 10D, 10B, 10C, 10E, 10E, 10E, 13B, 14C, 14B, 15C, 15B, 21D, 21B, 22B, 26B, 30F, 30D, 30A, 30C, 30G, 30G, 30G, 33A, 35B, 39CD, 42D, 42B, 46AB, 66B, 70B, 78BC .
In the above, other than direct reference to the where the notation n a means η(q n ) a .Since we can readily find the q-expansions for these, we determine which McKay-Thompson series they are and thus the corresponding class and moonshine group from Table 2.We find that these are, respectively, the series for the classes 6E, 6E, 10E, 10E, 30G, 30G.Checking against the cusp numbers for the Monstrous classes in the above, we readily find that the sum over the cusp numbers is 100 and that of their squares, 272.
Removing repeats, such as precisely the 6 classes of the eta-quotients above , there are 35 classes and the cusp sum now becomes 76 and the square sum, 176.
previous cases so extracting their cusp numbers is not immediate.We leave this exercise to the full study of all the cusp-sums for all the sporadic groups with Moonshine to a future work.For now, we remark that for those classes of the  20 we have that

Genus Zero
Finally, it is worth returning to the genus zero principle congruence groups themselves, independently of any particular groups.Consider the 15 (not necessarily prime) numbers N in (22) for which the genus of Γ(N) is zero.We can first use the formula (6)  Observation 4 For the 15 genus 0 groups Γ 0 (N) above, the sum over their cusp numbers is 56, and the sum over their squares is 266.
These are respectively the dimension of the fundamental representation and the real dimension of E 7 .

The Horrocks-Mumford Bundle: A Digression
Having addressed Conway's group in our cusp-sporadic correspondence, let us conclude with a parting digression on another context in which the Conway group and the curves of Bring and Fricke arise in relation to a classical geometric object.
Let us investigate the problem of vector bundles on projective spaces, a central subject in algebraic geometry.Horrocks and Mumford famously constructed their rank 2 indecomposable bundle on P 4 , which is the only known such an example [78].An excellent account, with historical context, is given in [79].From this later reference we summarize the following key points about vector bundles on P n (by vector bundles we henceforth mean algebraic, holomorphic, complex vector bundles over projective varieties): n = 1 Grothendieck's theorem [80] guarantees that any vector bundle E of rank r on P 1 splits completely into a direct sum of line bundles as E = r i=1 O P 1 (a i ).n = 2 Wu's theorem [81] states that isomorphism classes of C 2 -vector bundles E on P 2 are classified by the first and second Chern classes.n = 3 Atiyah-Rees [82] proves that every C 2 -bundle on P 3 admits an algebraic structure.
n ≥ 6 Hartshorne [84] conjectures that every rank 2 bundle on P n≥6 splits (this in particular implies that smooth projective varieties X ⊂ P n for dim(X) > 2 3 n are complete intersections).
One can see that the higher the dimension of the projective space the less is known.In dimension 4, the bundle of Horrocks-Mumford [78] is essentially the only non-trivial rank 2 bundle [85], which fits well within our realm of exceptional/sporadic objects.There are several equivalent constructions [79] and we will follow the so-called monad construction (cf. a very explicit computational-geometric description in [86]).Consider the (non-exact) complex of vector bundles where T is the tangent bundle on P 4 and p and q are respectively injective and surjective maps of bundles which will be specified in §I.1.The Horrocks-Mumford bundle is simply the cohomology of the above complex: The total Chern class of the bundle, in terms of the hyperplane class H of P 4 , is Subsequently, one can use Riemann-Roch to obtain the Hilbert polynomial [86] as i.e., χ(F HM ⊗ O P 4 ((n − 5)H)) = 1 12 (n2 − 1)(n 2 − 24), which is the equation at the end of §2 of [78].We can obtain the Hilbert Series as the generating function of global sections.Using Kodaira vanishing, (72) can be used for h 0 (F HM (nH)) for all n ∈ Z >2 , viz., h 0 (F HM (nH)) = χ(F HM ⊗ O P 4 ((n − 2)H)) for n > 2 while for n = 0, 1, 2, h 0 (F HM (nH)) = 0, 0, 4 respectively: We point out that while in the original paper [78], F HM was defined with (69), often in later literature [85,86] the dual twisted by O( 2) is defined as the Horrocks-Mumford bundle, i.e., as the cohomology of the complex so that F ′ = ker(q ′ )/ Im(p ′ ), where T * is now the cotangent bundle of P 4 .In this definition, c(F ′ ) = 1 − H + 4H 2 .

Symmetry Groups
It was shown in [78] that the group G HM of symmetries 21 on this bundle is the normalizer of the Heisenberg group H (5) within SL(5; Q(ω 5 )).Here, we recall that SL(5; Q(ω 5 )) is the special linear group in dimension 5 defined over the cyclotomic field of Q extended by the primitive 5-th root ω 5 of unity.Furthermore, H (5) is the order 125 extra-special group which is the non-Abelian central extension of the Abelian group Z 5 × Z 5 : In fact, G HM is a semi-direct product of H (5) with the binary icosahedral group E 8 = SL(2, 5), of order 120.The Heisenberg group itself acts on the standard basis e i of C 5 as: with subscripts on coordinates defined modulo 5.The generators 22 and presentation of G HM can be computed using [61] (we record these here because most of the literature is not explicit about these).The natural 5-dimensional complex faithful representation gives G HM as a 2-generated group: We see that while both semi-direct factors require 3 generators, the full group needs only 2. Calling the two generators above as f 1 and f 2 , the presentation of the group is simply For reference, we include the character table -both the ordinary and the rational, as well as the modular versions -of G HM in Appendix I.It is interesting to note that the modular Brauer character table, defined over F 5 , has 5 irreducible representations and 26 conjugacy classes, most of which have 0 in the Brauer table, except five of them.This gives essentially a 5 × 5 character table and thus it behooves us to consider the McKay quiver [44,45].Now, because G HM naturally embeds into SL(2, Z), it is expedient to take the fundamental 2 representation, as in the ADE case of (32), and decompose 2 ⊗ r i = 5 j=1 a i j r j .We readily find by checking, for example, decompositions such as 2 ⊗ 2 = 1 ⊕ 3 whose character is (4, 4, 1, 0, 1) = (1, 1, 1, 1, 1) + (3, 3, 0, −1, 0).The adjacency matrix a i j and the accompanying McKay quiver are as follows: In the quiver, every undirected edge is a pair of arrows in opposing directions.The eigenvalues of this adjacency matrix are ±2, ±1, 0.

The HM Quintic Calabi-Yau Threefold
The space of H (5)-invariant quintics arose from [78] as invariant sections of the bundle O P 4 (5) and has been widely studied since, especially in the context of heterotic string compactifications [87,88].We recall that this is a 6-dimensional space of quintics in the projective coordinates [x 0 : . . .: where as always the subscripts on the coordinates are defined mod 5.In particular, the well-studied Fermat quintic ∑ i x 5 i and its Schoen [89] cousin ∑ i x 5 i + ψx 0 x 1 x 2 x 3 x 4 are both illustrative examples.In general, linear combinations of the above 6 quintics are called the Horrocks-Mumford quintic X HM in P 4 .
We emphasize that we are considering linear invariants here.When working in projective space, here P 4 , we need only consider projective invariants.For the Heisenberg group, it suffices to consider Z 5 × Z 5 , which is the quotient of H (5) by its centre Z 5 .Therefore, in the literature, the HM-quintic is traditionally called the Z 5 × Z 5 quintic-quotient [87].Incidentally, we notice that both Molien series have palindromic numerators.This means that geometrically, considering them as the Hilbert series of the affine varieties C 5 /H (5) and C 5 /G HM respectively, these quotients are affine (singular) Calabi-Yau [90,91].This is consistent with the factor that all our explicit matrix generators, and hence all group elements, have unit determinant, and thus H (5) and G HM are discrete finite subgroups of SU (5).Therefore, our quotients are local Calabi-Yau 5-fold orbifolds [47].
Defining Equations for X HM : In terms of the section of the bundle F HM , it was shown in [78] that if s 1 = (s 11 , s 12 ) and s 2 = (s 21 , s 22 ) are generic sections, here written as 2-vectors because F HM is rank 2, then Horrocks-Mumford quintics can be written as s 11 s 22 − s 12 s 21 = 0 (87) and thus have nodal singularities: there are in fact 100 of them.The form above suggests that X HM might be determinantal varieties.This is indeed the case [92].Defining the matrices M y (x) i j := y 3(i− j) x 3(i+ j) , L y (z) i j := y i− j z 2i− j (88) for y j projective coordinates on P 4 [y j ] and z j projective coordinates on P 4 [z j ] , we have (note that M y (x)z = L y (z)x with x and z treated as column vectors) that {det M y (x) = 0} ⊂ P 4 [x i ] , {det L y (z) = 0} ⊂ P 4 are Horrocks-Mumford quintics.It was shown, incidentally, that a particular blow-up of a HM quintic has (the Mellin transform of) its L-function being the unique weight 4, level 55 modular form [92].Moreover, in light of the sextic of Bring in (38) and the octavic of Fricke in (51) in the context of Observation 1, it is natural to consider the decimic on the Fermat Calabi-Yau, viz., of genus 16.It would be interesting to find out how many tritangent planes (the number of bitangents, by (39), is 2 15 (2 16 − 1)) there are on D, a classical though somewhat tedious exercise which should be performed.

Embedding into Conway's Group
We can immediately check that of all the sporadic groups only Co 1 , the first Conway group and HN, the Harada-Norton group, contain conjugacy classes whose centralizer is of order 15000.These are class 5C of Co 1 and classes 5C and 5D of HN.The natural question is then whether the centralizer is precisely G HM .
Using [61], we can readily check by direct computation 23 that this is indeed so for Co 1 and that the two cases for HN are not.We conclude therefore Observation 5 The Horrocks-Mumford group G HM is the centralizer for exactly one conjugacy class of precisely one sporadic group: namely class 5C of Co 1 .
Parting Words: We have taken the reader on a journey, along two parallel paths: the McKay Correspondence, linking finite groups to Lie groups, and the Moonshine Conjecture, linking modular forms to sporadic groups.These more recent observations connecting the two correspondences, via enumerative geometry, would constitute a new paradigm of inter-relations.We hope that the main body of text has now clarified the summary of the various links in Figure 1 presented at the outset.Of course, everything here is based on numerology.But numerology could be a powerful tool, as the second author has found over the decades.

Figure 1 .
Figure 1.The web of correspondences amongst the exceptional Lie algebras and sporadic groups, via some classical enumerative geometry and modular subgroups.The notation [3|3] etc. for the classical geometries is explained in the beginning of §1.4.

x 4 ]
of P 4 , H 0 (P 4 , O P 4 (5)) H (5) =span ∑ [32]a subgroup Γ p of the symplectic group as Γ Thus defined, a non-trivial cusp (modular) form for Γ p exists if p > 71 or if p ∈ {37, 43, 53, 61, 67}.The complement of these primes is the set of 15 primes which appear in(21), dividing the order of the Monster, the same as the Ogg list; a clarification of this is in[32].There are 194 (linear, ordinary) irreducible representations of M, and 194 conjugacy classes.This constitutes a standard 194 × 194 character table, the first column of which is the vector of the dimensions of the irreducible representations, starting with {1, 196883, 21296876, 842609326, 18538750076, . ..} .

Table 2 .
The quadruple consisting of (1) class name, (2) cusp number C, (3) indicator D for the area of fundamental domain, and (4) normalizer group Γ 0 (N) + ⊂ PSL(2, R) in the notation of Eq. (14) for the 172 rational conjugacy classes of Monster group.Now, consider the list of centralizer sizes |Z γ | = |G|/|c γ | for γ = 1, 2, . . ., n.This is the character of a reducible representation, which we call the centralizing representation R Z

Table 4 .
[68]the 247 classes of the group 2.B, each is a triple {mx, nX, c} where mx is the class-name in GAP notation, nX is the identifier for the McKay-Thompson series for the class in the notation of[68]and c is the cusp number of the associated modular subgroup.Of these 247 classes, there are 207 distinct McKay-Thompson series. 21 • 39• 54• 72• 11 • 13 • 23 is itself the Z 2 -quotient of the non-simple group Co 0 , which is the automorphism group of the famous Leech lattice18, the unique even self-dual lattice in 24-dimensions with no vectors of norm two.It is worth recalling that Co 0 = 2.Co 1 has 167 conjugacy classes while Co 1 has 101.
Table, there are 6 which have no moonshine group labeled explicitly, corresponding to the eta-quotients Conway group which do have Monstrous McKay-Thompson series, the sums are ∑ g C g = 165 = 3 • 5 • 11 and ∑ g C 2 g = 615 = 3 • 5 • 41.In fact, [74] does more, and lists certain twisted McKay-Thompson series for Co 0 and Co 1 , which are, in fact, all Monstrous (cf Table 2 in cit.ibid.), in which case we have 80 distinct (Monstrous) McKay-Thompson series and checking their associated cusp numbers