Third Homology of some Sporadic Finite Groups
We compute the integral third homology of most of the sporadic finite simple groups and of their central extensions.
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| Цитувати: | Third Homology of some Sporadic Finite Groups / T. Johnson-Freyd, D. Treumann // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 44 назв. — англ. |
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| author | Johnson-Freyd, T. Treumann, D. |
| author_facet | Johnson-Freyd, T. Treumann, D. |
| citation_txt | Third Homology of some Sporadic Finite Groups / T. Johnson-Freyd, D. Treumann // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 44 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We compute the integral third homology of most of the sporadic finite simple groups and of their central extensions.
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| first_indexed | 2025-12-07T21:24:51Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 059, 38 pages
Third Homology of some Sporadic Finite Groups
Theo JOHNSON-FREYD † and David TREUMANN ‡
† Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada
E-mail: theojf@pitp.ca
‡ Department of Mathematics, Boston College, Boston, Massachusetts, USA
E-mail: treumann@bc.edu
Received September 30, 2018, in final form August 06, 2019; Published online August 10, 2019
https://doi.org/10.3842/SIGMA.2019.059
Abstract. We compute the integral third homology of most of the sporadic finite simple
groups and of their central extensions.
Key words: sporadic groups; group cohomology
2010 Mathematics Subject Classification: 20D08; 20J06
1 Introduction
In this paper we compute the third homology of some of the sporadic simple groups, and of their
central extensions. For many of these groups we are able to name elements (characteristic classes)
that generate H4(G;Z), the Pontryagin dual of H3(G). In the following table we write n.G for
the Schur covering of the sporadic group G – for a sporadic simple group, the covering is always
by a cyclic group n = H2(G) – and have left empty spaces where G = n.G.
M11 M12 M22 M23 M24
n = H2(G) 1 2 12 1 1
H3(G) 8 2× 24 1 1 12
H3(n.G) 8× 24 24
HS J2 Co1 Co2 Co3 McL Suz
H2(G) 2 2 2 1 1 3 6
H3(G) 2× 2 30 12 4 6 1 4
H3(n.G) 2× 8 120 24 1 24
J1 O′N J3 Ru J4 Ly
H2(G) 1 3 3 2 1 1
H3(G) 30 8 15 ? 1 1
H3(n.G) 8 3× 15 ?
He HN Th Fi22 Fi23 Fi′24 B M
H2(G) 1 1 1 6 1 3 2 1
H3(G) 12 ? ? 1 ? ? ? 24× [≤4]
H3(n.G) 3× [≤4] ? ?
An expression like “a× b” is short for Z/a⊕Z/b. Question marks in the table denote groups
for which we do not know the answer, and “[≤4]” denotes an unknown, possibly trivial, group
This paper is a contribution to the Special Issue on Moonshine and String Theory. The full collection is
available at https://www.emis.de/journals/SIGMA/moonshine.html
mailto:theojf@pitp.ca
treumann@bc.edu
https://doi.org/10.3842/SIGMA.2019.059
https://www.emis.de/journals/SIGMA/moonshine.html
2 T. Johnson-Freyd and D. Treumann
of order dividing 4. Further partial results for the groups HN, Th, Fi23, and Fi′24 are listed in
Section 8.
Only some entries in the table are original. The Schur multiplier row (the first row in the
table) was computed over many years, partly in service of the classification of finite simple
groups, and is available in the ATLAS [6]. With F2-coefficients, the entire cohomology rings of
many of the smaller sporadic groups are listed in [2], and at large primes the cohomology rings
of many sporadic groups are computed in [37, 38]. The Mathieu entries are reviewed in [14].
Significantly, H3(M24) was first computed in that paper using Graham Ellis’s software package
“HAP”, which we have found can also determine H3(G) for G ∈ {HS, 2HS, J2, 2J2, J1, J3,McL}
using the permutation models given in the ATLAS. For the larger groups G, although HAP
cannot calculate H3(G) on its own, it played an essential role in our calculations, as did the
“Cohomolo” package by Derek Holt.
1.1 Motivation
If G is a compact simple Lie group, or a finite cover of a compact simple Lie group, the
cohomology of its classifying space can be complicated at small primes but one always has
H4(BG;Z) ∼= Z; see [24] for a proof and some discussion of its role in conformal field the-
ory. In unpublished work [23], Jesper Grodal has shown that, with finitely many exceptions,
H4(G;Z) ∼= Z/
(
q2 − 1
)
whenever G is a simple finite group which arises as the Fq-points of
a split and simply connected algebraic group over Fq. Part of our motivation has been to see
whether we could discern any patterns in H4(G;Z) when G is sporadic.
We have also been inspired by the idea that 3-cocycles G × G × G → U(1) (when G is
finite, these represent classes in H4(G;Z)) can explain and predict some features of moonshine
[4, 16, 17, 18]. Such a cocycle can arise as the gauge anomaly of a G-action on a conformal field
theory. Even in the newer examples of moonshine where no conformal-field-theoretic explanation
is known, there are some numerical hints about this cocycle. For example, Duncan–Mertens–Ono
have used our calculations to explore a cocycle in their “O’Nan moonshine” [13, Section 3].
To some extent these hints can be pursued in an elementary way in pure group theory. If s
and t are a pair of commuting elements in a finite group G, we may define the following infinite
group:
Γ(s, t) :=
{((
a b
c d
)
, g
)
∈ SL2(Z)×G
∣∣∣∣ gsg−1 = satb and gtg−1 = sctd
}
.
It is the fundamental group of one of the components of the moduli stack of pairs (E, T ), where E
is an elliptic curve and T is a G-torsor over E. If there is a natural family of McKay–Thompson
series attached to G, one expects that their modularity properties (and more ambiguously, their
mock modularity properties) can be expressed in terms of a holomorphic line bundle on this
space, or equivalently in terms of a Γ(s, t)-equivariant line bundle on the upper-half plane. The
topological types of such line bundles are parametrized by the finite group H2(Γ(s, t);Z), which
is the target of a transgression map H4(G;Z)→ H2(Γ(s, t);Z) [18, Section 2].
2 Preliminaries
2.1 Notation
We will generally follow the ATLAS naming conventions for finite groups. We will write
both “Z/n” and plain “n” for the cyclic group of order n. When q is a prime power, we will
occasionally use “q” to denote the finite field Fq of that order. Physicists typically denote the
cyclic group of order n by Zn. Following mathematics conventions, we will instead reserve Zp,
where p is prime, for the ring of p-adic integers.
Third Homology of some Sporadic Finite Groups 3
We will write “N.J” or “NJ” for an extension with normal subgroup N and quotient J .
Extensions that are known to split are written with a colon “N : J”, and extensions which are
known not to split are written with a raised dot “N · J”. The name “pn”, where p is prime,
denotes an elementary abelian group of that order, and if n is even then “p1+n” denotes an
extraspecial group of that order. (There are two such extraspecial groups, called “p1+n± ”.)
We diverge from the ATLAS in the names for orthogonal groups. The group called “On(q)”
in the ATLAS is not the n× n orthogonal group over Fq. Rather, the ATLAS uses “On(q)” for
the simple subquotient of the orthogonal group. To avoid confusion, we will follow Dieudonné
and write “Ωn(q)” for this simple group. We will care only about the case when n ≥ 5 is odd –
when n is even, there are two orthogonal groups, called Ω±n (q). When n ≥ 5 and q is odd, Ωn(q)
is the commutator subgroup of SOn(Fq) = Ωn(q) : 2, and is the image of Spinn(Fq) = 2.Ωn(q)
in SOn(Fq), and is the kernel of the “spinor norm” SOn(Fq)→ F×q /{squares} ∼= Z/2.
Conjugacy classes of order n are named na, nb, nc, and so on. For simple groups the
conjugacy classes are ordered by size of the centralizer (from largest to smallest). In all cases
we follow GAP’s character table library, which includes a copy of the ATLAS character tables,
for the names of conjugacy classes. The online version of the ATLAS [42] includes a number of
irreducible modular representations. (We henceforth adopt the standard abbreviation “irrep” for
“irreducible representation”.) These are typically assigned letters “a”, “b”, etc., to distinguish
irreps of the same dimension and characteristic.
If G is a finite group, the names “H∗(G)” and “H∗(G)” always refer to group (co)homology,
or equivalently the space (co)homology of the classifying space BG of G. When G is a Lie
group, we will explicitly write H∗(BG) and H∗(BG) to avoid confusion with the (co)homology
of the underlying manifold of G. Cohomology groups of G with (twisted) coefficients in A are
denoted H∗(G;A). We sometimes abbreviate H∗(G;Z) by just H∗(G). All homology groups in
this paper are with Z-coefficients.
2.2 General methods
In this section and the next we review some standard techniques in group cohomology, which
we return to repeatedly in the following sections. These techniques are by no means due to us –
we employed them successfully in [28] to calculate the cohomology of Conway’s largest sporadic
group, and find in this paper that they also apply to most of the other sporadic groups. These
techniques are designed to understand the cohomology groups of a finite group G and not, say,
to compute explicit resolutions of Z over Z[G].
The first technique is to compute the p-primary part of H4(G;Z), which we denote by
H4(G;Z)(p), one prime at a time. An upper bound for the p-primary part is provided by the
following lemma [3, Section XII.8]:
Lemma 2.1. Let G be a finite group and let S ⊆ G be a subgroup that contains a Sylow p-
subgroup for some prime p. The restriction map α 7→ α|S : Hk(G;Z)(p) → Hk(S;Z)(p) is an
injection onto a direct summand.
Lemma 2.1, together with some basic properties (which we review in some detail in Sec-
tion 3.1) of H4(Z/p;Z) and H4(Z/p×Z/p;Z), allows us to dispose of many of the larger primes,
at least for sporadic groups:
Lemma 2.2. Let p be a prime and let G be a finite group with strictly fewer than (p − 1)/2
conjugacy classes of order p. If the p-Sylow subgroup of G is isomorphic to Z/p or to Z/p×Z/p,
then the p-part of H4(G;Z) vanishes.
If p ≥ 5 and G is a sporadic simple group whose p-Sylow has order p or p2, then one
sees by inspecting the tables of conjugacy classes that the criterion applies unless p = 5 and
G ∈ {J2, Suz}. We will see in Lemma 6.10 that the 5-part of H4(Suz;Z) vanishes as well.
4 T. Johnson-Freyd and D. Treumann
Proof. In any group with fewer than (p−1)/2 conjugacy classes of order p, the cyclic subgroups
C ⊂ G of that order have the following property: there is a generator h ∈ C and an element
x ∈ G such that xhx−1 = ha, where a is neither 1 nor −1 mod p. Conjugation by such an x
acts trivially on H•(G;Z) but nontrivially on H4(C;Z) – indeed it scales a nontrivial element
t ∈ H2(H;Z) ∼= Z/p to at and the cup-square of that nontrivial element t2 ∈ H4(H;Z) ∼= Z/p
to a2t2. It follows that the image of the restriction map H4(G;Z)→ H4(C;Z) is zero, for every
order p-subgroup C ⊂ G.
Let H be a p-Sylow subgroup of G, and consider the subgroup X ⊂ H4(H;Z) that vanishes
on every order-p subgroup C ⊂ H. The discussion above shows that the image of the restriction
map H4(G;Z)→ H4(H;Z) lies in X, and by Lemma 2.1, this restriction map is an injection on
the p-primary part of H4(G;Z). When p is odd and H is an elementary abelian p-group of rank
at most two, H4(H;Z) ∼= Sym2(H∗) (see Lemma 3.1), and so H4(H;Z) is detected on cyclic
subgroups, i.e., X = 0. �
In many cases not covered by Lemma 2.2, there is a maximal subgroup S ⊆ G that contains
a p-Sylow, and that has shape S = E.J where E is either an elementary abelian or an extraspecial
p-group. (See [44] for a survey of maximal subgroups of finite groups.) Sometimes we know
H4(J ;Z), either by induction or by computer. The Lyndon–Hochschild–Serre (LHS) spectral
sequence (detailed for example in [41, Section 6.8])
Eij2 = Hi
(
J ; Hj(E;Z)
)
=⇒ Hi+j(S;Z)
gives an upper bound for H4(S;Z), and therefore for H4(G;Z)(p), in terms of H4(J ;Z), which
we assume is known by earlier computations, together with the cohomology groups with twisted
coefficients
H0
(
J ; H4(E;Z)
)
, H1
(
J ; H3(E;Z)
)
, H2
(
J ; H2(E;Z)
)
.
The contribution from H3
(
J ; H1(E;Z)
)
is zero, since H1(E;Z) = 0 for every finite E. We
describe the groups Hj(E;Z) for j = 2, 3, 4 as Aut(E)-modules in Section 3. We used extensively
Derek Holt’s software package “Cohomolo” to determine the groups H1(J ;−) and H2(J ;−), but
sometimes the following vanishing criterion can be employed instead:
Lemma 2.3. Suppose that the center Z(J) has order prime to p and acts on Hj(E;Z) through
a nontrivial character Z(J)→ F×p . Then Hi
(
J ; Hj(E;Z)
)
= 0 for all i.
Proof. The statement is vacuous when p = 2, and so we assume p is odd for the remainder of
the proof. Let Zp[J ] denote the group ring of J with coefficients in the p-adic integers Zp. For
j > 0, Hj(E;Z) is a finite p-group, so Hi
(
J ; Hj(E;Z)
) ∼= ExtiZp[J ]
(
Zp, H
j(E,Z)
)
when Zp is
given the trivial J-action.
Let χ be the composite of the character Z(J) → F×p with the Teichmuller isomorphism
F×p
∼= Z×p [tor], where Z×p [tor] ⊆ Z×p denotes the torsion subgroup, and let
e =
1
|Z(J)|
∑
z∈Z(J)
χ(z)−1z
be the corresponding central idempotent in Zp[J ], so that Zp[J ] = eZp[J ] × (1 − e)Zp[J ] as
rings. Since χ is nontrivial there is a projective resolution P• → Zp of the trivial J-module with
Pm = (1 − e)Pm for every m. It follows that ExtiZp[J ]
(Zp,M) = 0, for all i, whenever M is
a Zp[J ]-module with M = eM . �
Third Homology of some Sporadic Finite Groups 5
The LHS spectral sequence allows us a comparison between the cohomology of a group and
of its Schur cover. Let G be a finite group with n ⊆ H2(G) such that H1(G;Z/n) = 0. Then the
corresponding central extension nG is unique up to isomorphism. Consider the LHS spectral
sequence for this extension. Since the extension is central, G acts trivially on n and so on
Hj(n;Z), and so we have an isomorphism of bigraded rings
Eij2
∼= Hi(G; Hj(n;Z)) ∼= Hi(G;Z[y]/(ny)),
where y has bidegree (i, j) = (0, 2); see, e.g., [33, Section II.8] and [25, Section II.5]. Using that
H1(G;Z/n) = 0, in total degree ≤ 5 the E2 page reads:
0
(Z/n)y2 0
0 0 0
(Z/n)y 0 H2(G;Z/n) H2(G;Z/n)
0 0 0 0 0
Z 0 H2(G;Z) H3(G;Z) H4(G;Z) H5(G;Z)
It follows that the pullback H4(G;Z)→ H4(nG;Z) is an injection. (Such pullbacks are examples
of edge maps, described for example in [41, Section 6.8.2].)
Let us focus on the case when n is a power of a prime p, and restrict to p-parts. Then
H1(G)(p) = H2(G;Z)(p) = 0. If furthermore H2(G)(p) is cyclic, then H2(G;Z/n) ∼= Z/n, and we
have the E2 page
0
(Z/n)y2 0
0 0 0
(Z/n)y 0 Z/n H3(G;Z/n)
0 0 0 0 0
Z 0 0 H3(G)(p) H4(G)(p) H5(G)(p)
The universal coefficient theorem describes H3(G;Z/n) as an extension
H3(G;Z/n) =
[
H3(G)(p) ⊗ (Z/n)y
]
.hom(H3(G),Z/n).
The d2 differential vanishes for degree reasons, and so Eij3 = Eij2 .
The extension nG → G splits when pulled back along itself, which implies that pullback
H3(G)(p) → H3(nG)(p) has kernel of order n, forcing the differential d3 : (Z/n)y → H3(G)(p) to
be an inclusion. The Leibniz rule then determines d3
(
y2
)
. If for instance H2(G)(p) is cyclic
of order N , then so is H3(G)(p); calling its generator “x”, we have d3y = (N/n)x and d3y
2 =
(2N/n)xy, where xy is the generator of the submodule Z/n ∼= H3(G)(p)⊗ (Z/n)y ⊂ H2(G;Z/n).
All together we learn:
Lemma 2.4. Let G be a finite group.
If p is an odd prime such that H1(G)(p) = 0 and H2(G)(p) = p, then the pullback map
H4(G;Z) → H4(pG;Z) is an injection with cokernel of order dividing p, and all classes in
H4(pG;Z) restrict trivially to the central p ⊆ pG.
If H1(G)(2) = 0 and H2(G)(2) is (nontrivial and) cyclic, then the pullback H4(G;Z) →
H4(2G;Z) is an injection with cokernel of order dividing 4, and if the cokernel has order 4
then there are classes in H4(2G;Z) with nontrivial restriction to the central 2 ⊂ 2G.
If H1(G)(2) = 0 and H2(G)(2) = 4, then the pullback H4(G;Z) → H4(4G;Z) is an injection
with cokernel of order dividing 8; again equality forces there to exist a class in H4(4G;Z) with
nontrivial restriction to the central 4 ⊆ 4G, and all classes in H4(4G;Z) vanish when restricted
to the central 2 ⊂ 4 ⊂ 4G.
6 T. Johnson-Freyd and D. Treumann
Lemma 2.5. Let p be an odd prime such that H1(G)(p) = 0 and H2(G)(p) = p. Let pG denote
a nonsplit central extension of G by the group Z/p. Suppose that a p-Sylow S ⊆ G also has
H1(S;Z/p) = 0, and that the central extension pG, when restricted to S, is nonsplit. Then the
pullback map H4(pG;Z)→ H4(pS;Z) induces an injection
coker
(
H4(G;Z)→ H4(pG;Z)
)
↪→ coker
(
H4(S;Z)→ H4(pS;Z)
)
.
This injection is an isomorphism if S contains the p-Sylow of G.
Proof. By Lemma 2.4, coker
(
H4(G;Z) → H4(pG;Z)
)
and coker
(
H4(S;Z) → H4(pS;Z)
)
are
each either trivial or of order p. We need only to show that if coker
(
H4(G;Z)→ H4(pG;Z)
)
=
Z/p, then coker
(
H4(S;Z)→ H4(pS;Z)
)
= Z/p.
Consider spectral sequence for the extension pG → G discussed before Lemma 2.4: we see
that coker
(
H4(G;Z) → H4(pG;Z)
)
= p if and only if the d3 : E22
3 → E50
3 vanishes. Let α ∈
H2(G;Z/p) ∼= E22
3 denote the generator classifying the extension pG. Then d3 : α 7→ Bock
(
α2
)
,
where Bock: H4(G;Z/p)→ H5(G;Z) denotes the integral Bockstein. This can be confirmed by
comparing the spectral sequence for H∗(pG;Z) with the one for H∗(pG;Z/p).
But then Bock
(
(α|S)2
)
= Bock
(
α2
)
|S also vanishes, and so coker
(
H4(S;Z) → H4(pS;Z)
)
= p by the spectral sequence for the extension pS. Conversely, assuming S contains the p-Sylow
in G, if Bock
(
α2
)
|S = 0, then Bock
(
α2
)
= 0 by Lemma 2.1. �
As we have mentioned, each page of the LHS spectral sequence provides an upper bound
for H4(G)(p). We can improve this upper bound whenever we can show that the images of the
two maps
H4(J ;Z)→ H4(S;Z)← H4(G;Z)
have trivial intersection. We can often prove this by restricting generators of H4(J ;Z) and
H4(G;Z) to cyclic subgroups and showing that no class in H4(S;Z) can simultaneously enjoy
the restrictions mandated by both H4(J ;Z) and H4(G;Z). For these calculations, we rely on
GAP’s character table library, which includes a copy of the ATLAS and, provided it contains
the subgroup S, knows how conjugacy classes fuse along the maps S → G and S → J .
2.3 Characteristic classes
With the improved upper bound in hand, the last step is to give a lower bound for H4(G;Z). In
almost all cases these come from the characteristic class of a representation V : G→ K, where K
is a Lie group. Usually we can take K = SU(N) or Spin(N), for which the characteristic classes
are called, respectively, the second Chern class c2 and the first fractional Pontryagin class p1
2 . In
two cases these “classical” characteristic classes c2 and p1
2 are not strong enough, and we appeal
to the Lie groups K = E6 and E8. For some of the Monster sections, it is not possible for Lie-
group-valued representations to give a strong enough lower bound, and we instead appeal to the
construction of [27] to provide a “monstrous characteristic class” of a representation of G in M.
We now review the story of c2 and p1
2 . See also [36] for a detailed treatment of characteristic
classes of finite groups. Suppose N ≥ 2. Then H4(BU(N);Z) ∼= Z2, with standard genera-
tors the square of the first Chern class c21 and the second Chern class c2. The first of these
restricts trivially along SU(N) ⊂ U(N), and so vanishes when restricted to any finite simple
group G; but if V : G → U(N) is an N -dimensional representation, then c2(V ) ∈ H4(G;Z) is
a potentially-interesting class. Similarly, provided N ≥ 5, the generators of H4(BSO(N);Z) ∼= Z
and H4(BSpin(N);Z) ∼= Z are called the first Pontryagin class p1 and the first fractional Pon-
tryagin class p1
2 . Like the symbol p12 suggests, the pullback H4(BSO(N);Z)→ H4(BSpin(N);Z)
along the double cover sends p1 to 2× p1
2 . There are also maps between SU(N) and SO(N) and
Third Homology of some Sporadic Finite Groups 7
Spin(2N) which either complexify a real representation or produce the underlying real repre-
sentation of a complex representation. The characteristic classes restrict along these maps as
SU(N)→ Spin(2N), SO(N)→ SU(N),
−c2 ← p1
2 , −p1 ← c2.
These classes are stable in the sense that they are preserved along the standard inclusions
SU(N) ⊆ SU(N + 1) and Spin(N) ⊆ Spin(N + 1). When N = 4, H4(BSpin(4);Z) is not genera-
ted by p1
2 , but that class is still defined by restricting along the standard inclusion into Spin(N)
for N large.
To show that the Chern class c2(V ) of an N -dimensional representation V : G → U(N) has
large order, it often suffices to restrict it to a cyclic subgroup 〈g〉 ⊂ G. If g has order n, then
H4(〈g〉;Z) ∼= Z[t]/(nt), where the degree-2 generator t is defined as the first Chern class c1(C1) of
the one-dimensional representation Cζ : g 7→ ζ = exp(2πi/n) ∈ U(1). The other 1-dimensional
representations of 〈g〉 are its tensor powers Cζm = C⊗mζ : g 7→ ζm, and c1(Cζm) = mt and
c2(Cζm) = 0. A higher-degree representation splits over 〈g〉 as a sum of 1-dimensional repre-
sentations. The Whitney sum formula says that for any group G and representations V , W , we
have
c1(V ⊕W ) = c1(V ) + c1(W ) ∈ H2(G;Z),
c2(V ⊕W ) = c2(V ) + c2(W ) + c1(V )c1(W ) ∈ H4(G;Z).
In particular, if V |〈g〉 =
⊕
kC
mk
ζ , then
c2(V )|〈g〉 =
∑
k<k′
mkmk′t
2 ∈ H4(〈g〉;Z).
The Chern classes are traditionally organized into a total Chern class of mixed degree c(V ) =
1 +
∑
i≥1
ci(V ) ∈ H•(G;Z). The full Whitney sum formula then says that c(V ⊕W ) = c(V )c(W );
for the one-dimensional representations of a cyclic group, c(Cζm) = 1 + mt; and the above
formula is the coefficient on t2 of c(V )|〈g〉 =
∏
k(1 +mkt).
A representation V : G→ SU(N) is called real if it factors, up to SU(N)-conjugacy, through
SO(N), i.e., if the representation preserves a nondegenerate symmetric bilinear form. For irreps,
this occurs if and only if the Frobenius–Schur indicator of V is +1. (A representation with
indicator −1 is called quaternionic and factors through a symplectic group.) Frobenius–Schur
indicators are quick to compute from a character table for G; they are listed in the ATLAS and
easily accessed in GAP. A real representation V : G → SO(N) is Spin if it factors (aka lifts)
through Spin(N); a choice of factorization is also called a spin structure. This occurs if and only
if the second Stiefel–Whitney class w2(V ) ∈ H2(G;Z/2) vanishes. This happens automatically
if G is a Schur cover of a simple group, as then H2(G;Z/2) = 0.
Given a real representation V : G → SO(N) with complexification V ⊗ C : G → SU(N),
the classes p1(V ) and c2(V ⊗C) agree up to sign, and so the calculation can proceed as above.
Calculating p1
2 (V ) for V : G→ Spin(N) can be harder. If V factored through W : G→ SU(N/2),
then the calculation would be easy, as then p1
2 (V ) = −c2(W ). In the cases of interest, this does
not occur for the whole representation V but does occur for its restriction V |〈g〉 to a cyclic
subgroup.
The spin structure for a real representation V : G → SO(N), if it exists, typically is not
unique. Rather, the choices form a torsor for H1(G;Z/2) = hom(G;Z/2) (so in particular
the lift is unique for quasisimple groups). Even though the lift is typically not unique, the
class p1
2 (V ), if it exists, depends only on the complex representation V : G→ SU(N) (since the
factorization through SO(N) is unique):
8 T. Johnson-Freyd and D. Treumann
Lemma 2.6. Suppose V1, V2 : G→ Spin(N) are two spin structures on the same real represen-
tation V : G→ SO(N). Then p1
2 (V1) = p1
2 (V2) ∈ H4(G;Z).
See [28, Section 1.4] for an explanation of Lemma 2.6 in terms of the “string obstruction”.
Proof. The reason that spin structures form a torsor for H1(G;Z/2) is the following. Let
c ∈ Spin(N) denote the nontrivial element in ker(Spin(N)→ SO(N)). There is a group homo-
morphism
α : Z/2× Spin(N)→ Spin(N), (i, g) 7→ cig,
covering the standard projection Z/2× SO(N)→ SO(N). Given V1 and V2 as above, there is a
unique map φ : G→ Z/2 such that
V2 = α ◦ (φ, V1).
Let π : Z/2× Spin(N)→ Spin(N) denote the standard projection. Then V1 = π ◦ (φ, V1). In
particular, it suffices to show that the pullbacks of p12 along the two maps α, π : Z/2×Spin(N)→
Spin(N) agree. But H4
(
B(Z/2 × Spin(N))
)
= H4(Z/2) ⊕ H4(BSpin(N)) by the Künneth
formula, and
π∗ p12 =
(
0, p12
)
∈ H4(Z/2)⊕H4(BSpin(N)),
α∗ p12 =
(p1
2 |〈c〉,
p1
2
)
∈ H4(Z/2)⊕H4(BSpin(N)),
so it suffices to show that p1
2 has trivial restriction to Z/2 ∼= 〈c〉 = ker(Spin(N)→ SO(N)).
Suppose that K is a compact connected Lie group with maximal torus T ⊆ K, and write
L = hom(T,U(1)) = H2(BT ) for its weight lattice. Then the restriction map H4(BK) →
H4(BT ) = Sym2(L) is an injection. When K = SO(N), there is a natural identification L ∼= ZN .
Writing e1, . . . , eN for the standard basis, we have p1 =
∑
i e
2
i . The weight lattice L′ of Spin(N)
is the extension of L through the element s = 1
2
∑
i ei. Working in Sym2 L′, we have
p1
2 = 2s2 −
∑
i<j
eiej .
But ei and 2s are in L and so restrict trivially to 〈c〉, and so p1
2 also restricts trivially. �
3 Elementary abelian and extraspecial p-groups
3.1 Elementary abelian groups
Lemma 3.1. Let E = pn be an elementary abelian p-group and let E∗ := Hom(E,µp), where µp
denotes the group of pth roots of unity in C∗.
1. If p = 2, we have isomorphisms of GL(E)-modules
H2(E;Z) = E∗, H3(E;Z) = Alt2(E∗), H4(E;Z) = E∗.Alt2(E∗).Alt3(E∗),
where the last group on the right denotes a filtered GL(E)-module whose subquotients
are E∗, Alt2(E∗), and Alt3(E∗). The submodule E∗.Alt2(E∗) is GL(E)-isomorphic to
Sym2(E∗).
2. If p is odd, we have isomorphisms of GL(E)-modules
H2(E;Z) = E∗, H3(E;Z) = Alt2(E∗), H4(E;Z) = Sym2(E∗)⊕Alt3(E∗).
Third Homology of some Sporadic Finite Groups 9
Proof. See [30, Proposition 2.2] or [28, Lemma 4.4]. �
If V is an elementary abelian p-group, we regard it as an Fp-vector space in the obvious
way. We may identify E∗ with the usual dual Fp-vector space to E by fixing at the outset an
isomorphism µp ∼= Z/p. We use Symn(V ) and Altn(V ) for the symmetric and exterior powers
of V ; recall in positive characteristic these are defined as quotients of V ⊗n in the following way:
• Symn(V ) := H0
(
Sn;V ⊗n
)
are the coinvariants of V ⊗n by the symmetric group action
• Altn(V ) is the quotient of V ⊗n by the subspace spanned by tensors with a repeated
tensorand (tensors v1 ⊗ · · · ⊗ vn with vi = vj for some i 6= j).
Though Symn(E∗) and Symn(E)∗ are not isomorphic as GL(E)-modules if p ≤ n (instead the
dual of Symn(E∗) is the space of divided powers of E), let us record:
Lemma 3.2. If p is a prime and E is an Fp-vector space, there is an isomorphism
Altn(E∗) ∼= Altn(E)∗
of GL(E)-modules.
Proof. The pairing V ⊗n ⊗ (V ∗)⊗n → Z/p given by
〈v1 ⊗ · · · ⊗ vn, w1 ⊗ · · · ⊗ wn〉 =
∑
σ∈Sn
(−1)σ〈v1, wσ(1)〉 · · · 〈vn, wσ(n)〉,
where (−1)σ denotes the sign of the permutation σ, is GL(V )-equivariant and descends to
a perfect pairing between Altn(V ) and Altn(V ∗). �
3.2 Extraspecial p-groups for p odd
If p is prime, E = pn is an elementary abelian p-group and ω is a function E × E → Z/p, we
define a multiplication on the set of formal monomials of the form zitu (where i ∈ Z/p and
u ∈ E) by the formula(
zitu
)(
zjtv
)
:= zi+j+ω(u,v)tu+v.
If ω is bilinear, this multiplication is associative, z0t0 is a two-sided unit, and z−i+ω(u,u)t−u is the
two-sided inverse to zitu: we defined a group that we denote by (p.E)ω. The groups associated
to (p.E)ω and (p.E)ω′ are isomorphic if ω−ω′ can be written as j(u+ v)− j(u)− j(v) for some
function j : E → Z/p – in particular if p is odd then ω(u, v) and
1
2(ω(u, v)− ω(v, u)) = ω(u, v)− 1
2(ω(u+ v, u+ v)− ω(u, u)− ω(v, v))
determine isomorphic groups, so when p is odd we may as well assume that ω ∈ Alt2(E∗) is
skew-symmetric. The center contains z, and if p is odd it is generated by z if and only if ω
is nondegenerate; in that case n = 2m and (p.E)ω = p1+2m is a copy of the extraspecial p-
group of exponent p. (The extraspecial group of exponent p2 comes from a non-bilinear cocycle
ω : E × E → Fp. The extraspecial groups of order 21+2m will be treated in Section 3.3; the
group (p.E)ω that we have defined is always elementary abelian when p = 2).
The automorphism group of p1+2m is E : GSp(E,ω), where E acts by inner automorphisms
zitu 7→ zi+2ω(v,u)tu and
GSp2m(E,ω) = {(g, a) | g : E → E, a ∈ GL1(Fp), ω(gu, gv) = aω(u, v)}
acts by (g, a)(zitu) = zaitgu. The scalar a = a(g) is determined by g.
10 T. Johnson-Freyd and D. Treumann
Let Lω ⊆ Alt2(E∗) denote the line spanned by ω. It is a one-dimensional GSp-submodule by
construction, and we write Lnω for its nth tensor power. Note L−1ω = L∗ω. If ω is nondegenerate
then E ⊗ Lω ∼= E∗ as GSp-modules, via the map which sends u ⊗ ω to the functional ω(u,−).
Provided p is odd, we have a splitting
Alt2(E∗) = Lω ⊕Alt2(E∗)ω,
where Alt2(E∗)ω is the kernel of the projection Alt2(E∗) ∼= Alt2(E ⊗ Lω) ∼= Alt2(E∗)∗ ⊗ L2
ω →
L−1ω ⊗ L2
ω dual to the inclusion Lω → Alt2(E∗).
If m ≥ 2 we also have an inclusion E∗ ⊗ Lω → Alt3(E∗) sending f ∈ E∗ to f ∧ ω.
Lemma 3.3. Let p be an odd prime, let E = p2m be an elementary p-group and let ω ∈ Alt2(E∗)
be a nondegenerate symplectic form. Then if m ≥ 2,
H2
(
p1+2m;Z
) ∼= E∗, H3
(
p1+2m;Z
) ∼= Alt2(E∗)ω,
as GSp2m-modules. If m ≥ 3,
H4
(
p1+2m;Z
) ∼= Sym2(E∗)⊕Alt3(E∗)/(E∗ ⊗ Lω),
while if m = 2,
H4
(
p1+4;Z
) ∼= Sym2(E∗).
(
Alt2(E∗)ω ⊗ Lω
)
,
a possibly nontrivial extension of Alt2(E∗)ω by Sym2(E∗).
Proof. We consider the action of GSp on the LHS spectral sequence
Hs(E; Ht(p))⇒ Hs+t(p.E).
We have H2(p) = Lω and H4(p) = L2
ω in the left s = 0 column. The bottom t = 0 row is
computed in Lemma 3.1. To compute the t = 2 row, recall that, provided p is odd, H•(E;Fp)
is the graded-commutative Fp-algebra generated by a copy of E∗ in degree 1 and a second copy
of E∗ in degree 2; in particular:
H1(E;Fp) ∼= E∗, H2(E;Fp) ∼= Alt2(E∗)⊕ E∗,
H3(E;Fp) ∼= Alt3(E∗)⊕ (E∗ ⊗ E∗) ∼= Alt3(E∗)⊕Alt2(E∗)⊕ Sym2(E∗).
All together, we have on the E2-page:
L2
ω
0 0 0
Lω E∗ ⊗ Lω (Alt2(E∗)⊕ E∗)⊗ Lω Alt2(E)⊗ Lω ⊕ · · ·
0 0 0 0 0
Z 0 E∗ Alt2(E∗) Sym2(E∗)⊕Alt3(E∗)
The d2 differential vanishes and the d3 differentials Lω → Alt2(E∗), E∗ ⊗ Lω → Alt3(E∗),
and L2
ω → Alt2(E∗)⊗Lω are the injections discussed above. Indeed, the LHS spectral sequence
is constructed so that d3 sends the generator ω ∈ Lω to the extension class ω ∈ Alt2(E∗), and
so it sends ω2 ∈ L2
ω to 2ω d3ω. The claim for E∗ ⊗Lω → Alt3(E∗) follows from comparing with
the Fp-cohomology.
It remains to understand d3 :
(
Alt2(E∗)⊕E∗
)
⊗Lω → H5(E). We claim that this map is an
injection when m ≥ 3, and that when m = 2 its kernel is Alt2(E∗)ω ⊗Lω ⊆ Alt2(E∗). Note also
Third Homology of some Sporadic Finite Groups 11
that when m = 2, the map E∗ ⊗ Lω → Alt3(E∗) is an isomorphism. In this range of degrees,
the sequence stabilizes after page 4, and so on the E∞ page we see
0
0 0 0
0 0 Alt2(E∗)ω ⊗ Lω
0 0 0 0 0
Z 0 E∗ Alt2(E∗)ω Sym2(E∗)
if m = 2 and
0
0 0 0
0 0 0
0 0 0 0 0
Z 0 E∗ Alt2(E∗)ω Sym2(E∗)⊕Alt3(E∗)/(E∗ ⊗ Lω)
if m ≥ 3. �
3.3 Extraspecial 2-groups
If E is an elementary abelian 2-group then any central extension 2.E is determined up to
isomorphism by the function
Q : E → F2, Q(v) =
{
1 if the lifts of v in 2.E have order 4,
0 otherwise,
which is a quadratic form. It is not usually possible to write the multiplication explicitly in
terms of Q – indeed if Q is nondegenerate and E has rank 6 or more the orthogonal group of Q
(which we denote by O(Q)) does not act on 2.E [21]. But O(Q) still acts on the cohomology
of 2.E.
The LHS spectral sequence begins:
2
0 0 0
2 E∗ Sym2(E∗)
0 0 0 0 0
Z 0 E∗ Alt2(E∗) E∗.Alt2(E∗).Alt3(E∗)
We first wish to describe the d3 differential. To do so, recall first that H•(E) injects into
H•(E;F2) ∼= Sym•(E∗) as the subalgebra in the kernel of the derivation Sq1 : Sym•(E∗) →
Sym•+1(E∗). Identifying H•(E) with its image in H•(E;F2), the d3 differential sends f ∈
Ei22
∼= Symi(E∗) to Sq1(fQ) ∈ Symi+3(E∗). In particular, it sends the generator of the 2 in
degree (0, 2) to Sq1(Q) ∈ Sym3(E∗). The image of Sq1 : Sym2(E∗) to Sym3(E∗) is isomorphic to
Alt2(E∗), and under this isomorphism Sq1 takes Q to its underlying alternating form BQ(x, y) =
Q(x+ y)−Q(x)−Q(y).
Let us suppose that Q is nondegenerate and E = 22m. Then in particular BQ 6= 0, so that
d3 : 2 → Alt2(E∗) is an injection. Let f ∈ E∗ in degree (1, 2) and consider d3(f) = Sq1(fQ) =
f2Q + f Sq1(Q). Since Sym•(E∗) has no zero-divisors, if f 6= 0 but d3(f) = 0, then we must
have Sq1(Q) = fQ. This cannot happen when m ≥ 2, and so d3 : E∗ → E∗.Alt2(E∗).Alt3(E∗)
is an injection in this case. (When m = 1, it is an injection when Q has Arf invariant −1 and
is not an injection when Q has Art invariant +1.) Thus, provided m ≥ 2, we find
H1(2.E) ∼= E∗, H2(2.E) = Alt2(E∗)/BQ.
12 T. Johnson-Freyd and D. Treumann
The d3 differential emitted by the Sym2(E∗) in degree (2, 2) always has kernel – Q itself –
and nothing more provided m ≥ 2. Finally, if m ≥ 3, then d5 : E04
5 → E50
5 is nonzero, and
the E∞ page looks like
0
0 0
0 0 Q
0 0 0 0
Z 0 E∗ Alt2(E∗)/BQ X
with
X ∼=
(
E∗.Alt2(E∗).Alt3(E∗)
)
/E∗.
This can be simplified slightly. The inclusion E∗ → E∗.Alt2(E∗).Alt3(E∗), sending f 7→
Sq1(fQ), does not land within the E∗.Alt2(E∗) ∼= Sym2(E∗) submodule, and so the composition
E∗ → E∗.Alt2(E∗).Alt3(E∗)→ Alt3(E∗) is nonzero. But E∗ is simple as an O(Q)-module, and
so this map E∗ → Alt3(E∗) is an injection. (It sends f 7→ f ∧BQ.) Thus we can write
X ∼= E∗.Alt2(E∗).
(
Alt3(E∗)/E∗
)
.
All together, provided m ≥ 3,
H4(2.E) ∼=
(
E∗.Alt2(E∗).Alt3(E∗)/E∗
)
.2.
The group X is elementary abelian, although the extensions written above do not split O(Q)-
equivariantly. The group H4(2.E) is not elementary abelian; it is isomorphic to (Z/2)n × (Z/4)
for n = dim(X)− 1 =
(
m
2
)
+
(
m
3
)
− 1 when m ≥ 3.
Finally, when m = 2, whether d5 : E04
5 → E50
5 vanishes or not depends on the Arf invariant
of Q. Indeed,
H4
(
21+4
+
)
= X.4 ∼= 29 × 8, H4
(
21+4
−
)
= X.2 ∼= 29 × 4.
(Both cases are extensions of X = E∗.Alt2(E∗).Alt3(E∗)/E∗ ∼= 210.)
4 Dempwolff groups, Chevalley groups
and their exotic Schur covers
4.1 Dempwolff and Alperin groups
In [10], Dempwolff determined that there were no nontrivial extensions of GLn(F2) by its defining
representation on 2n, unless n ≤ 5. Conversely, nontrivial extensions exist for n = 3, 4, 5; up to
isomorphism there is a unique group which can serve as the extension, which we will call
23 ·GL3(F2), 24 ·GL4(F2), 25 ·GL5(F2).
The largest of these is studied in [9], though not proved to exist until [34, 39]. A similar group
is the nonsplit Alperin-type group
43 ·GL3(F2).
Lemma 4.1. If n = 3, 4, 5, then H3(GLn(F2)) = Z/12. Furthermore,
1) H3
(
23 ·GL3(F2)
) ∼= Z/2⊕ Z/8⊕ Z/3;
Third Homology of some Sporadic Finite Groups 13
2) H3
(
24 ·GL4(F2)
) ∼= Z/2⊕ Z/4⊕ Z/3;
3) H3
(
25 ·GL5(F2)
) ∼= Z/8⊕ Z/3;
4) H3
(
43 ·GL3(F2)
) ∼= (Z/2)2 ⊕ Z/8⊕ Z/3.
Proof. HAP can handle all of these groups except the largest 25 · GL5(F2). (In Derek Holt’s
library of perfect groups, available in GAP, 23 ·GL3(F2) is PerfectGroup(1344, 2), 24 ·GL4(F2) is
PerfectGroup(322560, 5), and 43 ·GL3(F2) is PerfectGroup(10752, 4). One may call these groups
by number, have GAP find faithful permutation representations for them, and then feed those
permutation groups to HAP – no further human involvement is needed.)
We will obtain H3
(
25 ·GL5(F2)
) ∼= H4
(
25 ·GL5(F2)
)
from the LHS spectral sequence. Using
the description from Lemma 3.1 of the GL5(F2)-module structure on H≤4(25), together with
Cohomolo, we find the E2 page of that spectral sequence is
0
0 0 0
0 0 Z/2
0 0 0 0 0
Z 0 0 0 Z/12
The E22
2 entry here is the Dempwolff–Thompson–Smith computation H2
(
GL5;
(
25
)∗)
= Z/2,
and confirmed by Cohomolo.
To complete the proof of (3), it suffices to give an element of H4
(
25 · GL5(2)
)
whose order
is divisible by 8. There is a famous embedding, due to [22], of 25 · GL5(2) into the compact
Lie group E8. Let us write e for the generator of H4(BE8). We will prove that the restriction
e|25·GL5(2) is such an element.
For the remainder of the proof, let V denote the 248-dimensional adjoint representation
of E8. The dual Coxeter number of E8 is h∨ = 30. For any simple simply connected Lie
group G, the dual Coxeter number measures the ratio of the fractional Pontryagin class of the
adjoint representation of G with the generator of H4(BG):
p1
2 (adj) = h∨ ∈ Z ∼= H4(BG).
In particular, c2(V ) = −60e. Since 60 is divisible by 4, to show that the order of e|25·GL5(2) is
divisible by 8, it suffices to show that the order c2(V )|25·GL5(2) is divisible by 2.
We will do so by finding a binary dihedral group 2D8 ⊆ 25 · GL5(2) such that c2(V )|2D8
is nonzero. To find such a group, we look inside the normalizer of an order-8 element. There
are three conjugacy classes of elements of order 8 in 25 · GL5(2). The normalizer of class 8c
is SmallGroup(64, 151) in the GAP library. It can be built directly in GAP: the ATLASRep
package includes a copy of 25 ·GL5(2) as a permutation group on 7440 points; GAP can compute
orders of centralizers and normalizers, and so in particular can identify class 8c; then GAP can
build the normalizer of an element of conjugacy class 8c as a subgroup of 25 · GL5(2). There
are four conjugacy classes of order-8 elements in SmallGroup(64, 151), and GAP checks that all
four merge in 25 ·GL5(2) to conjugacy class 8c.
Finally, SmallGroup(64, 151) contains a copy of the binary dihedral group 2D8 of order 16.
Since 2D8 is a finite subgroup of SU(2), its cohomology is easy to compute: in particular,
H4(2D8) is cyclic of order |2D8| = 16 and is generated by c2 of the “defining” two-dimensional
representation. As in [28, Section 6], let us index the irreducible representations:
V1
V6
V0
V4 V5
V2
V3
14 T. Johnson-Freyd and D. Treumann
In particular, V0 is the trivial representation, V6 is the “defining” two-dimensional irrep, V5 is
the other faithful irrep, V4 is the two-dimensional real irrep of D8, and V1, V2, and V3 are the
nontrivial one-dimensional irreps.
Character table constraints provide a unique fusion map 2D8 → 25 · GL5(2) sending the
elements of order 8 to conjugacy class 8c. Along this map, the 248-dimensional irrep V of
25 ·GL5(2) decomposes as
V |2D8 = 15V0 ⊕ 15V1 ⊕ 15V2 ⊕ 15V3 ⊕ 30V4 ⊕ 32V5 ⊕ 32V6.
Lemma 6.1 of [28] gives a formula for the second Chern class of any representation of 2D8 in
which the representations V2 and V3 appear with the same coefficient. That formula is
c2
(⊕
niVi
)
= 4n4 + 9n5 + n6 (mod 16), if n1 = n2,
where we have identified H4(2D8) = Z/16 by identifying 1 ∈ Z/16 with c2(V6). Applying this
formula to the 248-dimensional representation V gives
c2(V )|2D8 = 8 (mod 16).
In particular, c2(V ) is nonzero in H4(2D8). As explained above, this implies that H4
(
25 ·GL5(2)
)
contains an element of order divisible by 8 (namely, the restriction of the generator of H4(BE8)),
and so must be isomorphic to Z/24. �
4.2 A few exotic Chevalley groups
For the most part, any central extension of a finite Chevalley group G(Fq) is the group of Fq-
points of a central extension of the algebraic group G. In particular if G is of simply connected
type then the multiplier H2(G(Fq)) is usually zero. The finitely many exceptions were classified
by Steinberg and Griess. Many of these exotic central extensions occur as centralizers in the
sporadic groups.
Lemma 4.2. H3(Sp6(F2)) = Z/2⊕ Z/4⊕ Z/3 and H3(2 · Sp6(F2)) = Z/2⊕ Z/8⊕ Z/3.
Proof. We computed these using HAP. The computation of H3(Sp6(F2)) is fast, but computing
H3(2 ·Sp6(F2)) took many hours. Two of the faithful permutation representations of 2 ·Sp6(F2)
have degrees 240 and 276 (the latter coming from the embedding 2 ·Sp6(F2) ⊆ Co3). Our laptop
computer ran out of memory running HAP on the degree 240 model, and gave the above output
after six hours for the degree 276 model. �
Lemma 4.3. We have
H3(G2(2)) = Z/2⊕ Z/8⊕ Z/3, H3(G2(3)) = Z/8⊕ Z/3, H3(G2(5)) = Z/8⊕ Z/3.
Jesper Grodal has shown that H4(G2(Fq)) is cyclic of order q2− 1 if q = pr with either p or r
sufficiently large [23]. The computations in the lemma show that this holds also for q = 5, but
not q = 3 or q = 2.
Proof. We computed G2(2) and G2(3) with HAP. The order of G2(5) is 26.33.56.7.31. The proof
of Lemma 2.2 applies to this group – for p = 7 and 31, there are strictly fewer than (p − 1)/2
conjugacy classes of order p – and so we must compute H4(G2(5))(p) for p = 2, 3, and 5.
The 2-Sylow in G2(5) is contained in the nonsplit extension 23 · GL3(2) whose cohomology,
per Lemma 4.1(1), is H4
(
23 ·GL3(2)
)
(2)
= 2× 8. According to [31], for q = 1 (mod 4),
H1(G2(q);F2) ∼= H2(G2(q);F2) ∼= 0, H3(G2(q);F2) ∼= H4(G2(q);F2) ∼= F2,
Third Homology of some Sporadic Finite Groups 15
Sq1 = 0: H3(G2(q);F2)→ H4(G2(q);F2).
It follows that H4(G2(q))(2) is cyclic of order at least 4. Setting q = 5 and recalling Lemma 2.1,
we therefore find that H4(G2(5))(2) is a cyclic direct summand of H4
(
23 ·GL3(2)
)
(2)
= 2× 8 of
order at least 4, and so H4(G2(5))(2) = 8.
The 3-Sylow in G2(5) is contained in a maximal subgroup of shape U3(3) : 2. HAP computes
H3(U3(3) : 2) = 2 × 8 × 3. Conjugacy class 3b ∈ G2(5) acts on the 124-dimensional irrep with
trace 1, and so c2(124-dim rep)|〈3b〉 6= 0. It follows that H4(G2(5))(3) = 3.
The 5-Sylow in G2(5) is contained in a maximal subgroup of shape 51+4 : GL2(5). The central
4 ⊆ GL2(5) acts on all of 54 with the same faithful central character. It therefore acts with
nontrivial central characters on Hj(51+4) for j ∈ {1, 2, 3, 4}, and so Hi
(
GL2(5),Hj
(
51+4
))
= 0
for these j by Lemma 2.3. Since H4(GL2(5)) = 4 × 8 × 3 has no five part, we find that
H4
(
51+4 : GL2(5)
)
(5)
, and hence also H4(G2(5))(5), vanishes. �
Recall from Section 2.1 that Ωn(q) denotes the simple subquotient of the orthogonal group
On(Fq), and that when n ≥ 5 and q is odd, Ωn(q) is of index 2 in SOn(Fq). We will use the
names Spinn(q) and 2.Ωn(q) interchangeably.
Lemma 4.4. H3(Ω7(3)) ∼= Z/4 and H3(2.Ω7(3)) ∼= Z/8.
Proof. The criterion in Lemma 2.2 applies for the primes p ≥ 5. The 2-Sylow is contained in
Sp6(F2), giving an upper bound of H4(Sp6(F2))(2) = 2× 4 for H3(Ω7(3)), and an upper bound
of H4(2Sp6(F2))(2) = 2× 8 for H3(Spin7(3)), both from Lemma 4.2.
Let V denote the 105-dimensional representation of Ω7(3). It is a real representation. (In-
deed, all representations of Ω7(3) are real except for the two dual complex representations of
degree 1560.) Conjugacy class 4a ∈ Ω7(3) acts on V with trace −5. Its square, conjugacy
class 2b, acts with trace 5. It follows that 4a acts with spectrum (+1)25(−1)30(i)25(−i)25, and
so the total Chern class of V |〈4a〉 is
c(V )|〈4a〉 = (1 + 2t)30(1 + t)25(1− t)25 = 1− t2 + · · · .
In particular, c2(V )|〈4a〉 has order 4, giving a lower bound of 4 to the order of c2(V ) ∈ H4(Ω7(3))
and a lower bound of 8 to the order of p1
2 (V ) ∈ H4(2.Ω7(3)).
To show that H3(Spin7(3))(2) is exactly Z/8 (which implies in turn that H3(Ω7(3))(2) is
exactly Z/4) it suffices to give a class in H4(2.Sp6(F2))(2) not in the image of restriction
H3(Spin7(3))(2) → H4(2.Sp6(F2))(2). We claim that the fractional Pontryagin class of the 15-
dimensional irrep of Sp6(F2) is such a class. (This representation is not Spin over Sp6(F2),
but is Spin over 2Sp6(F2). We will henceforth call its fractional Pontryagin class p1
2 (15) ∈
H4(2Sp6(F2)).) To prove this, we consider the conjugacy classes 2b and 2d in Sp6(F2). They
act on the 15-dimensional irrep with traces 7 and −1 respectively; equivalently, 2b acts with
spectrum 111(−1)4 whereas 2d acts with spectrum 17(−1)8. These two classes lift with order
2 to 2Sp6(2). The fractional Pontryagin classes are therefore p1
2 (15)|2b = 1 ∈ H4(〈2b〉) ∼= Z/2
and p1
2 (15)|2d = 0. But 2b and 2d both fuse to class 2c ∈ Ω7(3). It follows that p1
2 (15) ∈
H4(2.Sp6(F2)) is not the restriction of any class in H4(Spin7(3)).
It remains to handle the prime p = 3. In general, the p-Sylow in a characteristic-p group of Lie
type is the nilpotent subgroup, and so is contained in any parabolic. We will use two maximal
parabolics of the algebraic group SO7, corresponding to the Dynkin subdiagrams B2 ⊆ B3 and
A1 ×A1 ⊆ B3. These lead to two maximal subgroups of Ω7(3) that contain the 3-Sylow:
35 : SO5(F3), 31+6
+ : (2A4 ×A4).2.
16 T. Johnson-Freyd and D. Treumann
There is one more maximal subgroup of Ω7(3) containing the 3-Sylow, corresponding to the
Dynkin diagram inclusion A2 ⊆ B3, which we will not use in the present proof, but will use in
the proof of Corollary 4.5.
The spectral sequence for 35 : SO5(F3) has E2 page:
3
0 0 3
0 3 0
0 0 0 0 0
Z 0 0 2 22 × 4× 3
The bottom line was computed in HAP, and the middle entries in Cohomolo. The entry E04
2 = 3
corresponds to the symmetric pairing on 35.
We claim that the maps H4(Ω7(3))(3) → H4
(
35 : SO5(F3)
)
(3)
and H4(SO5(F3))(3) → H4
(
35 :
SO5(F3)
)
have trivial intersection. To see this, first note that SO5(F3) ∼= Weyl(E6) has a 6-
dimensional irrep, on which the conjugacy class 3c acts with trace 3. It follows that
c2(6-dim irrep)|〈3c〉 6= 0.
But 3c ∈ SO5(F3) has among its preimages in 35 : SO5(F3) one which fuses to class 3b ∈ Ω7(3),
and 3b also meets 35 ⊆ 35 : SO5(F3). It follows that c2(6-dim irrep) ∈ H4(SO5(F3)), when
pulled back along 35 : SO5(F3) → SO5(F3), distinguishes conjugate-in-Ω7(3) elements, and so
is not the restriction of a class in H4(Ω7(3)).
Since H4(Ω7(3))(3) ⊆ H4
(
35 : SO5(F3)
)
and the latter is an extension of a quotient of
H4(SO5(F3)) and a subspace of H0
(
SO5(F3); H4
(
35
))
, and since H4(Ω7(3))(3) does not meet
H4(SO5(F3)), the restriction map H4(Ω7(3))(3) → H0
(
SO5(F3); H4
(
35
))
must be an injection.
The order-3 conjugacy classes in Ω7(3) that meet 35 ⊂ 35 : SO5(F3) are classes 3a, 3b, and 3c.
Specifically, the intersection of conjugacy class 3a and 35 ∼= F5
3 consists of the nonzero vectors
of norm 0, and the intersections of 3b and 3c with 35 are the vectors of norm ±1. (These are
the three nontrivial SO5(F3)-orbits in F5
3.) The nonzero classes in H0
(
SO5(F3); H4
(
35
)) ∼= Z/3
corresponds to the symmetric pairing and its negation, and so restrict trivially to 〈3a〉 but
nontrivially to 〈3b〉 and 〈3c〉. In particular the restriction map H4(Ω7(3))(3) → H4(〈3b〉) is an
injection.
The other maximal subgroup we consider is the one of shape 31+6
+ : (2A4 × A4).2. It is the
normalizer of conjugacy class 3a. GAP can work with Ω7(3) by using its faithful degree-351
permutation representation, and find this subgroup. In particular, GAP finds that the action of
(2A4 ×A4).2 on 36 is generated by the following three matrices
. 1 . . 2 2
2 . . 2 . 1
1 1 . 1 2 2
. 2 2 . . .
. . 1 1 . 2
2 . 2 1 2 2
,
. . 2 2 . 1
. 1 2 . 1 1
. 2 . . 2 .
1 . 1 2 . 1
. 2 2 . . .
. 2 . . 1 .
,
2 . 2 1 . 2
. . 2 . 2 2
. . . . 1 .
2 . 1 . . .
. 1 2 . 1 1
. 2 . . 1 .
.
Recall from Lemma 3.3 that H4
(
31+6
) ∼= Sym2
(
36
)
⊕
(
Alt3
(
36
)
/36
)
. The group (2A4 ×
A4).2 contains the matrix −1, and which acts by −1 on
(
Alt3
(
36
)
/36
)
. Furthermore, the
representation 36 is not symmetrically self-dual. (In fact it is not self-dual: its antisymmetric
pairing changes by a sign under the odd elements of (2A4 × A4).2.) It follows that H0
(
(2A4 ×
A4).2; H4
(
31+6
))
= 0.
But this means in particular that the restriction map H4(Ω7(3))→ H4
(
31+6
)
vanishes. Con-
jugacy class 3b ∈ Ω7(3) meets 31+6. It follows that the restriction H4(Ω7(3)) → H4(〈3b〉)
Third Homology of some Sporadic Finite Groups 17
is the zero map. But we showed above that H4(Ω7(3))(3) → H4(〈3b〉) is an injection. So
H4(Ω7(3))(3) = 0. �
The Chevalley group Ω7(3) has an exceptional cover: its multiplier is 6, whereas the multiplier
of Ω7(q) is generically 2 = π1(SO(7,C)).
Corollary 4.5. H3(3.Ω7(3)) ∼= Z/12 and H3(6.Ω7(3)) ∼= Z/24.
Proof. We must calculate H4(3.Ω7(3))(3) ∼= H4(6.Ω7(3))(3). It either vanishes or is Z/3 by
Lemmas 2.4 and 4.4.
We used two of the three maximal parabolics of Ω7(3) in the proof of Lemma 4.4; for this
calculation we will use the third one, of shape 33+3 : SL(3, 3). For the remainder of the proof we
will call this subgroup S. Since S contains the 3-Sylow, the extension 3.Ω7(3) restricts to a non-
trivial central extension 3.S. One can show, for instance by running a LHS spectral sequence,
that H1(S; 3) = 0 and H2(S; 3) = 3. In particular, there is a unique nonsplit extension 3.S up
to isomorphism. (The two nonzero classes in H2(S; 3) are related by the outer automorphism
of Z/3.)
By Lemma 4.4, H4(Ω7(3))(3) = 0, and so H4(3.Ω7(3))(3) = coker
(
H4(Ω7(3))→ H4(3.Ω7(3))
)
.
This is in turn isomorphic to coker
(
H4(S)→ H4(3.S)
)
by Lemma 2.5.
The smallest complex representations of Ω7(3) and 3.Ω7(3) have dimensions 78 and 27 re-
spectively, equal to the smallest representations of the simple Lie group Eadj
6 (C) and its sim-
ply connected cover Esc
6 = 3.Eadj
6 . However, Ω7(3) does not preserve the Lie bracket on the
78-dimensional representation. It does preserve a lattice, and preserves the Lie bracket “mod-
ulo 2”: in fact, Ω7(3) embeds into the twisted Chevalley group 2E6(2) ⊆ Eadj
6 (F4). The finite
subgroups of the Lie group Eadj
6 (C) not already contained in a smaller Lie group were classified
in [5]. In particular, Eadj
6 (C) contains a subgroup isomorphic to S, lifting to the nonsplit exten-
sion 3.S ⊆ Esc
6 (C). (Presumably S is precisely the intersection of Eadj
6 (C) and Ω7(3) in their
common 78-dimensional representation.)
Both H4
(
BEadj
6
)
and H4
(
BEsc
6
)
are infinite cyclic, but the restriction map H4
(
BEadj
6
)
→
H4
(
BEsc
6
)
is not an isomorphism: its cokernel has order 3. By Lemma 2.5, this forces the
inclusion H4(S)→ H4(3.S) to have cokernel of order 3. �
The Chevalley group G2(3) has an exceptional multiplier of order 3.
Corollary 4.6. H3(3.G2(3)) has order 72.
Proof. The 7-dimensional representation of G2 provides an inclusion G2(3) ⊆ Ω7(3), and the
exceptional triple cover of Ω7(3) restricts to the exceptional triple cover of G2(3). Lemmas 2.4
and 2.5 then force the inclusion H4(G2(3))→ H4(3.G2(3)) to have cokernel of order 3. �
5 Mathieu groups
The low-degree homology groups of all Mathieu groups can be computed in HAP, and are listed
in [14], where details of HAP’s implementation are discussed. That paper was the first to
compute H3(M24), and was able to compute up to H4 exactly for all Mathieu groups, and H5
exactly for all Mathieu groups except M24, for which the 2-part was left ambiguous. In [16] it
is shown that the restriction map H4(M24) → H4(〈12b〉) is an isomorphism, and that H4(M24)
is generated by the “gauge anomaly” of “M24 moonshine”. In [28, Theorems 5.1 and 5.2] we
gave direct proofs of the results H4(M23) = 0 and H4(M24) = 12 following the method outlined
in Section 2.2, and we also recognized that the generator of H4(M24) from [16] is more simply
described as the fractional Pontryagin class of the defining degree-24 permutation representation.
We remark that the same holds for M11:
18 T. Johnson-Freyd and D. Treumann
Proposition 5.1. H4(M11) ∼= Z/8 is generated by the fractional Pontryagin class of the defining
degree-11 permutation representation.
Proof. Let Perm denote the permutation representation of M11. The two conjugacy classes of
order 8 in M11 have the same spectrum on Perm: they act by diag
(
1, 1, 1, ζ, i, ζ3,−1,−1, ζ−3,−i,
ζ−1
)
, where ζ = exp(2πi/8). Let t ∈ H2(Z/8) denote a generator of H•(Z/8) = Z[t]/(8t). The
total Chern class of Perm, restricted to a cyclic group of order 8, is therefore
c(Perm)|〈8a〉 = 13(1− t)(1− 2t)(1− 3t)(1− 4t)2(1 + 3t)(1 + 2t)(1 + t) = 1 + 2t2 + · · · .
In particular, c2(Perm) has order divisible by 4. But Perm is a real and (since H2(M11) vanishes)
therefore Spin representation, and so p1
2 (Perm) has order divisible by 8. �
The Schur cover of M12 is studied in [4]; they compute H4(2M12) = 82 × 3 with HAP,
and show that the map H4(2M12) →
∏
g∈2M12
H4(〈g〉) has kernel of order 2. To fully describe
H4(2M12) requires moving slightly beyond cyclic groups, and also requires some notation. Let
Perm denote (a choice of either) degree-12 permutation representation of M12, and write V12 for
the unique 12-dimensional faithful irrep of 2M12. Then V12 is real, and hence spin, as a 2M12-
module (since 2M12 has no central extensions). Perm ⊗R is not Spin as an M12-module, but
is automatically Spin as a 2M12-module, since H2(2M12;Z/2) = 0. Write p1
2 (Perm) and p1
2 (V12)
for their fractional Pontryagin classes. The group 2M12 has two conjugacy classes of elements of
order 3: class 3b acts on Perm with cycle structure 34. There are also four conjugacy classes of
elements of order 8. Classes 8a and 8b differ by the central element and act on Perm with cycle
structure 122181; classes 8c and 8d differ by the central element and act with cycle structure 4181.
Finally, there is a unique conjugacy class of quaternion subgroups Q8 ⊆ 2M12 in which the center
of Q8 maps to the center of 2M12.
Proposition 5.2. H4(2M12) is spanned by the classes p1
2 (Perm) and p1
2 (V12). The restriction
map
H4(2M12)→ H4(Q8)×H4(〈8a〉)×H4(〈8c〉)×H4(〈3b〉) ∼= 83 × 3
is an injection.
We remark that the outer automorphism of 2M12 switches the two degree-12 permutation
representations and also switches 8ab with 8cd.
Proof. We choose the following generators of H4(Q8) and H4(〈8a〉) ∼= H4(〈8c〉) ∼= H4(C8)
and H4(〈3b〉) ∼= H4(C3): the generator of H4(Q8) is the fractional Pontryagin class of the 4-
dimensional real representation (equal to the negative second Chern class of the 2-dimensional
complex irrep); if n divides 24, we take the unique generator of H4(Cn) which is a cup square
(it is unique by what Conway and Norton call “the defining property of 24” [7]).
It is straightforward to compute the images of p1
2 (Perm) and p1
2 (V12) to H4(Q8)×H4(〈8a〉)×
H4(〈8c〉)×H4(〈3b〉). They are
p1
2 (Perm) 7→ (3, 1, 1,−1),
p1
2 (V12) 7→ (4,−1, 1,−1).
But (3, 1, 1,−1) and (4,−1, 1,−1) together generate a subgroup isomorphic to 82×3 ∼= H4(2M12)
inside 83 × 3. �
The covers of M22 are not directly computable by HAP, since they do not have sufficiently
small permutation representations.
Third Homology of some Sporadic Finite Groups 19
Proposition 5.3. The covers of M22 have the following third homology groups:
H3(2M22) = 4, H3(3M22) = 3, H3(4M22) = 8,
H3(6M22) = 12, H3(12M22) = 24.
Proof. Given Lemma 2.4 together with the computer computation H3(M22) = 0, it suffices
to give lower bounds H4(2M22) ≥ 4, H4(4M22) ≥ 8, and H4(3M22) ≥ 3. The first two can
be handled simultaneously as follows. 2M22 has a unique faithful 210-dimensional irrep V .
Conjugacy class 4c ∈ 2M22 acts on V with spectrum 150(−1)50i55(−i)55. Let t ∈ H2(〈4c〉)
denote a generator. Then
c2(V )|〈4c〉 = t2 ∈ H4(〈4c〉)
has order 4. This gives the lower bound for H4(2M22). The representation V is real, but it is not
Spin as a 2M22-module (since, indeed, c2(V ) is not divisible by 2). It is, however, Spin as a 4M22-
module. Since H4(2M22) → H4(4M22) is an injection, c2(V )|4M22 has order divisible by 4, so
p1
2 (V ) ∈ H4(4M22) has order divisible by 8. This provides the lower bound for H3(4M22). Finally,
for 3M22, we may use either 21-dimensional faithful representation W . Element 3c ∈ 3M22 acts
on W with trace 0, and so c2(W )|〈3c〉 = −21
3 t
2 6= 0 ∈ H4(〈3c〉). Thus c2(W ) has order divisible
by 3 in H4(3M22). �
6 Leech lattice groups
6.1 Higman–Sims group
The smallest faithful permutation representations of the Higman–Sims group HS and its double
cover 2HS have degrees 100 and 704 respectively. Using these representations, we find that HAP
can compute H3(HS) and H3(2HS) without further human assistance:
H3(HS) ∼= (Z/2)2, H3(2HS) ∼= Z/2× Z/8.
Since H4(G) ⊆ H4(2G) has index 4, the latter must contain elements with nontrivial restric-
tion to the central 2 ⊆ 2G by Lemma 2.4. It is not hard to check that all complex representa-
tions V of 2HS have c2(V )|2 = 0, and all real representations have p1
2 (V )|2 = 0. In particular,
we do not know generators for H4(2HS) ∼= Z/2× Z/8.
6.2 Janko group 2
The smallest permutation representations of Janko’s second group J2 (also called the Hall–Janko
group HJ) and of its double cover 2J2 have degrees 100 and 200 respectively, and HAP computes:
H3(J2) ∼= Z/30, H3(2J2) ∼= Z/120.
We record some finer information in this section: in particular we show that H4(2J2;Z)
is generated by the Chern class of either six-dimension irreducible representation of 2J2, and
the outer automorphism of 2J2 acts by multiplication by 49. For this, we will compute the
Chern classes of the irreducible representations of SL(2, 5), which is a subgroup of 2J2 in two
non-conjugate ways.
Let π be a two-dimensional irreducible representation of C = SL(2, 5). Then π is faithful and
has trivial determinant, and we can use the McKay correspondence to parametrize the other
irreps of G, by nodes in the extended Dynkin diagram of type E8. That parametrization is
1,πS2(π)S3(π)S4(π)S5(π)
S2(π◦)
π ⊗ π◦π◦
20 T. Johnson-Freyd and D. Treumann
where we have written Sn(−) as short for the nth symmetric power of a representations, and π◦
for the image of π under a nontrivial outer automorphism of G.
Lemma 6.1. H4(SL(2, 5);Z) ∼= Z/120. If π is an irreducible representation of dimension 2 then
c := c2(π) generates H4(G;Z), and the Chern classes of the remaining irreducibles are
0.c4c10c20c35c
76c
100c49c
Proof. The integer cohomology ring of BSU(2) is isomorphic to Z[c], where c in degree 4 is c2
of the tautological representation V of SU(2), and the integer cohomology ring of BU(1) is
isomorphic to Z[b] where b in degree 2 denotes the first Chern class of the tautological one-
dimensional representation. The restriction of the nth symmetric power of V to a maximal
torus U(1) ⊂ SU(2) splits as a sum of 1-dimensional representations of weights
n, (n− 2), (n− 4), . . . , (−n+ 2), (−n).
Thus, the total Chern class of Sn(V ) can be written as
ct(S
n(V )) :=
(
1 + c2(S
n(V ))t2 + c4(S
n)t4 + · · ·
)
= (1 + nbt)(1 + (n− 2)bt) · · · (1− nbt).
In particular
c2(V ) = −b2, c2
(
S2(V )
)
= −4b2, c2
(
S3(V )
)
= −10b2,
c2
(
S4(V )
)
= −20b2, c2
(
S5(V )
)
= −35b2.
This explains six of the nine Chern classes reported in the Lemma. It remains to compute c2(π
◦),
c2
(
S2(π◦)
)
and c2(π ⊗ π◦). We will need a concrete description of the outer automorphism
of SL(2, 5).
Let o : SL(2, 5)→ SL(2, 5) denote the conjugation by the diagonal matrix(
2
1
)
∈ GL(2, 5).
Then π◦ is the composite of π with o. We claim that o acts as multiplication by 49 on
H4(SL(2, 5),Z). To see this, make the following observations:
1. There are six subgroups of order 5 in SL(2, 5). All of them are conjugate to each other
and two of them are preserved by o. Writing H5 for either one of these two o-fixed cyclic
subgroups, the action of o on H2(H5;Z) ∼= Z/5 is multiplication by a primitive 4th root
of 1 (in F5) – the action on H4(H5;Z) = H2(H5;Z)⊗2 is therefore by −1.
2. There are five subgroups of order 24 in SL(2, 5), all of them conjugate to each other.
Exactly one of them – call it H24 – is preserved by o. The action of o on H24 coincides
with the conjugation action of an element x ∈ SU(2), in particular o acts as the identity
on H4(H24;Z) ∼= Z/24.
The restriction maps H4(SL(2, 5);Z) → H4(H5;Z) ⊕ H4(H24;Z) is an isomorphism. The num-
ber 49 arises as the unique solution to 49 = −1 (mod 5) and 49 = 1 (mod 24). It follows that
c2(π
◦) = 49c and c2(S
2(π◦)) = 49 · 4c = 76c. To compute c2(π ⊗ π◦), we note that π ⊗ π◦
is isomorphic to its conjugate by o, so its restriction to H4(H5;Z) vanishes, while π ⊗ π◦ and
1 + Sym2(π) have the same restriction to H24; the number 100 arises as the unique solution to
100 = 0 (mod 5) and 100 = 4 (mod 24). �
Third Homology of some Sporadic Finite Groups 21
Proposition 6.2. H4(2J2;Z) ∼= Z/120 is generated by c2(V ), where V is either 6-dimensional
irrep of 2J2. For one (but not the other) of the two conjugacy classes of SL(2, 5)-subgroups of 2J2,
the restriction map H4(2J2;Z) → H4(SL(2, 5);Z) is an isomorphism. The outer automorphism
of 2J2 acts by multiplication by 49 on H4(2J2;Z).
Proof. Let V and V ′ denote the two 6-dimensional irreps of 2J2. They are exchanged by the
outer automorphism of 2J2. Let us write SL(2, 5)a ⊆ 2J2 and SL(2, 5)b ⊆ 2J2 for representatives
of the two conjugacy classes of SL(2, 5)-subgroups. The representations V and V ′ restrict as
V |SL(2,5)a ∼= π ⊕ 2π◦, V ′|SL(2,5)a ∼= π◦ ⊕ 2π,
V |SL(2,5)b ∼= π ⊕ S3(π), V ′|SL(2,5)b ∼= π◦ ⊕ S3(π).
It follows that the outer automorphism of 2J2 restricts along either embedding SL(2, 5) ⊆ 2J2
to the outer automorphism of SL(2, 5). Moreover, c2(π ⊕ 2π◦) = c2(π) + 2c2(π
◦) = 99c, which
has order 40 in Z/120, but c2
(
π ⊕ S3(π)
)
= c2(π) + c2(S
4(π)) = 11c has order 120. Thus
H4(2J2;Z)→ H4(SL(2, 5)b;Z) ∼= Z/120 is surjective, and hence an isomorphism given the HAP
computation. Since the outer automorphism acts by multiplication by 49 on H4(SL(2, 5)b;Z), it
must also act by multiplication by 49 on H4(2J2;Z). �
6.3 Conway groups
In [28, Theorems 0.1 and 5.3] we showed that
H4(Co1) ∼= Z/12, H4(2.Co1) ∼= Z/24,
and that these groups are generated by the fractional Pontryagin classes of the 276- and 24-
dimensional representations, respectively. Let us denote the 24-dimensional real representation
of 2.Co1 by the name Leech. The second and third Conway groups Co2 and Co3 are sub-
groups of 2.Co1, and so Leech restricts to representations of each (where it splits as a trivial
representation plus a 23-dimensional irrep).
Theorem 6.3. H4(Co2) ∼= Z/4 is generated by the restriction of p1
2 (Leech).
Proof. In [28, Theorem 7.1] we gave a formula for p1
2 (Leech)|〈g〉 for all elements g ∈ 2.Co1 in
terms of the Frame shape of g in the Leech representation. (Introduced by Frame in [15] to study
the E8 Weyl group, Frame shapes encode the characteristic polynomials of lattice-preserving
orthogonal matrices.) The conjugacy class 4g ∈ Co2 has Frame shape 46, and so p1
2 (Leech)
restricts with order 4 to this conjugacy class. This gives the lower bound H4(Co2) ≥ 4.
For the upper bound, Lemma 2.2 handles the primes ≥ 7. For the primes 3 and 5, we note
that Co2 contains a subgroup isomorphic to McL, which in turn contains the 3- and 5-Sylows.
Since H4(McL) = 0 (see Section 6.4), we learn that H4(Co2)(p) = 0 for p odd.
It remains to give an upper bound for the 2-part of H4(Co2). The 2-Sylow in Co2 is contained
in a subgroup isomorphic to 210 : (M22 : 2), where E = 210 is an irreducible (M22 : 2)-module
over F2. The subgroup E contains elements with Frame shape 212. By [28, Theorem 7.1],
p1
2 (Leech) restricts nontrivially to such an element, and so p1
2 (Leech)|E ∈ H0
(
M22 : 2; H4(E)
)
is nonzero. There are two irreducible 10-dimensional (M22 : 2)-modules over F2, which we will
call Va and Vb, where the letters “a” and “b” match the notation in [42]. They enjoy
H0(M22 : 2;Va) = H0
(
M22 : 2; Alt2(Va)
)
= H0
(
M22 : 2; Alt3(Va)
)
= 0,
H0(M22 : 2;Vb) = H0
(
M22 : 2; Alt2(Vb)
)
= 0, H0
(
M22 : 2; Alt3(Vb)
) ∼= Z/2.
Since H4(E) ∼= E∗.Alt2(E∗).Alt3(E∗) by Lemma 3.1, the only way for H4(E) to have a nontrivial
(M22 : 2)-fixed point is if E∗ ∼= Vb.
22 T. Johnson-Freyd and D. Treumann
With this isomorphism in hand, we can compute the E2 page of the LHS spectral sequence
for E : (M22 : 2):
2
0 2 22
0 0 0
0 0 0 0 0
Z 0 2 2 22
The bottom row is computed by HAP, and the middle rows by Cohomolo. The dashed line
reminds that the extension E : (M22 : 2) splits, and so H4(M22 : 2) is a direct summand of
H4(E : (M22 : 2)).
To complete the proof, it suffices to show that H4(Co2) → H4(E : M22 : 2) and H4(M22 :
2) → H4(E : M22 : 2) have trivial intersection. There are three conjugacy classes of order 2 in
M22 : 2, with cycle structures 1628, 1827, and 211 in the degree-22 permutation representation.
Together, these three classes detect H4(M22 : 2): if α ∈ H4(M22 : 2) is nonzero, then there is an
element g ∈ M22 : 2 of order 2 such that α|〈g〉 6= 0. (Indeed, the images of H4(2)→ H4(M22 : 2)
and H4(M24) → H4(M22 : 2) are transverse, and one can quickly compute the restrictions of
their images to the three elements of order 2.)
Given α 6= 0 ∈ H4(M22 : 2), choose g ∈ M22 : 2 of order 2 such that α|〈g〉 6= 0. Choose also
an order-2 lift g̃ of g in E : M22 : 2, and let α̃ ∈ H4(E : M22 : 2) denote the pullback of α. Then
α̃|〈g̃〉 = α|〈g〉 6= 0. But Co2 has only three conjugacy classes of order 2, distinguished by their
traces on Leech, and all three classes meet E. Since α̃|E = 0, we find that α̃ takes different
values on conjugate-in-Co2 elements, and so cannot be the restriction of a class in H4(Co2). This
completes the proof that H4(Co2) ∼= Z/4. �
Theorem 6.4. H4(Co3) ∼= Z/6 is generated by the restriction of p1
2 (Leech).
Proof. The conjugacy class 6e ∈ Co3 has Frame shape 64 in the Leech representation. It
follows from [28, Theorem 7.1] that p1
2 (Leech)|〈6e〉 has order 6, giving the claimed lower bound
H4(Co3) ≥ 6. Lemma 2.2 handles the primes ≥ 7, and Co3 contains a copy of McL, which
contains the 5-Sylow.
The 3-Sylow in Co3 is contained in a subgroup of shape 311 : (2 × M11). There are two
irreducible 11-dimensional representations of M11 over F3, dual to each other. They lead to
LHS spectral sequences with E2 pages
0
0 0 0
0 0 0
0 0 0 0 0
Z 0 2 0 2× 8
and
0
0 3 3
0 0 0
0 0 0 0 0
Z 0 2 0 2× 8
Only the latter of these is consistent with the lower bound H4(Co3) ≥ 3, and provides the desired
upper bound on H4(Co3)(3).
To complete the proof we must verify that H4(Co3)(2) ≤ 2. The 2-Sylow in Co3 is contained
in three maximal subgroups: one the form 24 ·GL4(F2), one of the form 2 · Sp6(F2), and one of
order 210.33. Lemma 4.1(2) and Lemma 4.2 give
H4
(
24 ·GL4(F2);Z
)
= Z/2⊕ Z/4⊕ Z/3,
H4(2 · Sp6(F2);Z) = Z/2⊕ Z/8⊕ Z/3.
By Lemma 2.1, H4(Co3)(2) is a direct summand of both Z/2 ⊕ Z/4 and of Z/2 ⊕ Z/8, which
forces H4(Co3)(2) ⊆ Z/2. �
Third Homology of some Sporadic Finite Groups 23
6.4 McLaughlin group
HAP is able to directly compute
H3(McL) = 0
by using the permutation representation of degree 275. HAP is unable to directly compute
H3(3McL) because the smallest faithful permutation representation of 3McL has degree 66825.
Lemma 2.4 only provides the upper bound H3(3McL) ≤ 3. Nevertheless, with some human
involvement, we do have:
Theorem 6.5. H3(3McL) = 0.
Proof. The computer calculation of H3(McL) leaves only the 3-part of H3(3McL) to be com-
puted. But we can also dispense with the other parts directly. The 2-Sylow in McL is contained
in a maximal subgroup of shape M22, and H3(M22)(2) = 0 by computer calculation (see Proposi-
tion 5.3). The 5-Sylow is contained in a group of shape 51+2 : 3 : 8, and HAP quickly computes
H4
(
51+2 : 3 : 8
)
(5)
= 0.
(The ATLAS contains generators for most maximal subgroups of sporadic groups.) Lemma 2.2
handles the primes p ≥ 7.
Only the prime 3 is left. The 3-Sylow in 3McL is contained in two maximal subgroups,
one of shape 35.M10 and the other of shape 32+4 : 2S5. The latter is more useful, and for the
remainder of the proof we will call it S. The quotient 2S5 is the one listed in the ATLAS under
the names “2S5i” and “Isoclinic(2.A5.2)”; it is the group of shape 2S5 that contains elements
of order 12. This 2S5 has a faithful 4-dimensional representation over F3. The quotient of S
in McL has shape 31+4 : 2S5, and the “central” 3 is not central, but rather transforms by the
sign representation of 2S5. In terms of the 4-dimensional module, it corresponds to a symplectic
form on 34 which is 2A5- but not 2S5-fixed. There is also a symplectic form which is 2S5-fixed,
and the 32+4 subgroup of S extends 34 by both symplectic forms simultaneously.
After a multi-hour computation, HAP reports
H1(S;F3) = H2(S;F3) = 0, H3(S;F3) = 3,
from which we learn that
H1(S)(3) = H2(S)(3) = 0, H3(S)(3) is cyclic.
On the other hand,
H3(2S5)(3) = 3,
and since the extension S = 32+4 : 2S5 splits, H3(S)(3) contains H3(2S5)(3) as a direct summand.
Passing to cohomology, we learn that the pullback map
H4(2S5)→ H4(S)
is an isomorphism.
There is a unique conjugacy class of order 3 in 2S5, and the restriction map H4(2S5)(3) →
H4(〈3a〉) is an isomorphism. Take any element g ∈ 2S5 of order 3 and lift it to an order-3
element g̃ ∈ S. Then the composition H4(2S5)(3)
∼→ H4(S)(3) → H4(〈g̃〉) is an isomorphism. On
the other hand, all conjugacy classes of order-3 elements in 3McL meet the normal subgroup
32+4 ⊆ S, and the composition H4(2S5)(3)
∼→ H4(S)(3) → H4
(
32+4
)
is zero.
Thus the image of H4(2S5)(3)
∼→ H4(S)(3) has trivial intersection with the restriction map
H4(3McL)(3) ↪→ H4(S)(3), and so H4(3McL)(3) = 0. �
24 T. Johnson-Freyd and D. Treumann
6.5 Suzuki group
The Schur cover of the Suzuki group is the beginning of a famous sequence of subgroups of 2Co1
centralizing actions of binary alternating groups on the Leech lattice; 6Suz centralizes an action
of 2A3
∼= Z/6, which corresponds to a “complex structure” on the Leech lattice. In particular,
6Suz has a conjugate pair of 12-dimensional irreducible complex representations, (either one
of) which we will call V throughout this section. The underlying real representation of V is
(Leech⊗R)|6Suz.
The maximal subgroups of Suz are listed in [43]. The 2-Sylow subgroup of Suz is contained
in the centralizer of 2a, a subgroup of shape
21+6 · SWeyl(E6) ⊂ Suz,
where SWeyl(E6) is the index-2 subgroup of the Weyl group that acts with trivial determinant
of the reflection representation. The ATLAS calls this group SWeyl(E6) = U4(2). We write
it as a Weyl group to make the action on 26 transparent: it is the mod-2 reduction of the
action of Weyl(E6) on the E6-lattice. In particular, the quadratic form on 26 associated with
the extraspecial group 21+6 has Arf invariant −1.
Lemma 6.6. Let Eadj
6 denote the compact Lie group of adjoint type E6. There is a homomor-
phism
21+6 · SWeyl(E6)→ Eadj
6 ,
whose kernel is the central Z/2 and which maps 21+6 onto the 2-torsion subgroup of a maximal
torus in Eadj
6 .
Proof. Let T.W be the normalizer of a maximal torus in a compact Lie group, and T [2] for
the 2-torsion of T . A subgroup T [2].W ⊂ T.W is studied in [40] and proved to be nonsplit
for groups of type E6 in [8]. The quotient of 21+6 · SWeyl(E6) ⊂ Suz by the central Z/2 is
also a nonsplit extension 26 · SWeyl(E6), as can be quickly confirmed in GAP. (Indeed, GAP
can easily look up the maximal subgroup 21+6.SWeyl(E6) of Suz in the ATLAS, compute its
quotient 26.SWeyl(E6), and work out that there are no nontrivial homomorphisms into it from
SWeyl(E6).) We use Cohomolo to compute
H2
(
SWeyl(E6), 2
6
)
= Z/2,
so the two non-split extensions of SWeyl(E6) by 26 must be isomorphic. �
Remark 6.7. There is similar relationship between a centralizer in Co1 and the group 28 ·
Weyl(E8) in the E8 Lie group, which has some conformal-field theoretic significance. The E8 Lie
group acts on the E8-lattice VOA, and the simple group Co1 acts on Duncan’s “supermoonshine”
SVOA of [11, 12]. In [12], the latter is constructed out of the former in such a way as to give
a natural identification between subquotients of E8 and Co1 of shape 28 · PSWeyl(E8). (Here
PSWeyl denotes the quotient of SWeyl by the center, which is nontrivial for Weyl group of E8.)
The homomorphism from Lemma 6.6 can be constructed by starting with this identification
and analyzing Z/3-centralizers (in E8 and in Co1). It would be interesting to find a direct
“moonshine” construction of Suz from the E6 lattice making this homomorphism transparent.
Our goal in this section is to prove:
Theorem 6.8. The Suzuki group and its Schur covers have the following fourth cohomology
groups:
H4(Suz) = Z/4, H4(2Suz) = Z/8, H4(3Suz) = Z/12, H4(6Suz) = Z/24.
H4(6Suz) is generated by c2(V ), where V denotes either 12-dimensional complex irrep.
Third Homology of some Sporadic Finite Groups 25
We will split the proof into a series of lemmas.
Lemma 6.9. c2(V ) has order 24 in H4(6Suz).
Proof. Since the underlying real representation of V is Leech⊗R, we have
c2(V ) = −p1
2 (Leech)|6Suz.
Then [28, Theorem 0.1] gives an upper bound of 24 on the order of c2(V ).
The action of 6Suz on the Leech lattice includes elements with Frame shape 38; according
to [28, Theorem 7.1], p1
2 (Leech) has nontrivial restriction to such elements. This gives a lower
bound of 3 on the order of c2(V ).
6Suz contains a maximal subgroup of shape 6A7. As observed in [28, Lemma 4.1], there is
a unique conjugacy class of subgroups D8 ⊆ A6, where D8 denotes the dihedral group of order 8.
Along the standard inclusion A6 ⊆ A7, the 6-fold cover pulls back to the cover 3 × 2D8 of D8,
where 2D8 denotes the binary dihedral group of order 16. This group 2D8 is the one used in
[28, Lemma 4.1], where it is shown that p1
2 (Leech)|2D8 has order 8. This gives a lower bound
of 8 on the order of c2(V ). �
Lemma 6.10. The 3- and 5-primary parts of H4(Suz)(5) vanish.
Proof. The 5-Sylow in Suz is contained in a maximal subgroup of shape J2 : 2. By Proposi-
tion 6.2, the outer automorphism of J2 acts by multiplication by −1 on H4(J2)(5) = 5, and so
H4(Suz)(5) ⊆ H4(J2 :2)(5) = 0.
The 3-Sylow in Suz is contained in a maximal subgroup of shape 35 : M11. There are two
nontrivial 5-dimensional M11-modules over F3. They lead to LHS spectral sequences with E2
pages:
0
0 3 3
0 0 0
0 0 0 0 0
Z 0 0 0 8
or
0
0 0 0
0 3 0
0 0 0 0 0
Z 0 0 0 8
The former is incompatible with H3(Suz)(3) = 3, and the latter immediate gives H4(Suz)(3)
= 0. �
Lemma 6.11. H0
(
SWeyl(E6); H4
(
21+6
))
is either Z/2 or Z/4.
We were unable to determine the exact value of H0
(
SWeyl(E6); H4
(
21+6
))
. We remark that
the order-2 class therein has many descriptions. It arises as c2 of the unique 23-dimensional
complex irrep of 21+6. It also arises as follows. By Lemma 6.6, the nonsplit extension 26 ·
SWeyl(E6) is naturally a subgroup of the compact Lie group of type E6 (adjoint form); in this
realization, 26 is the group of 2-torsion points in the maximal torus. The generator of H4(BE6)
restricts to 26 to the E6 quadratic form living in Sym2
(
26
)
⊆ H4
(
26
)
, and this form pulls back
to 21+6 to the SWeyl(E6)-fixed order-2 class.
Proof. Let us write J for SWeyl(E6) and E for the 6-dimensional SWeyl(E6)-module over F2.
In Section 3.3 we identified H4(2.E) as(
E∗.Alt2(E∗).Alt3(E∗)/E∗
)
.2.
This group has a unique nonzero element which is divisible by 2; it lives in the subgroup
X = E∗.Alt2(E∗).Alt3(E∗)/E∗, and so H0(J ;X) ≥ Z/2. On the other hand,
H0(J ;E∗) = H0
(
J ; Alt3(E∗)/E∗
)
= 0, H0
(
J ; Alt2(E∗)
)
= Z/2.
26 T. Johnson-Freyd and D. Treumann
Indeed, E∗ and Alt3(E∗)/E∗ are simple J-modules of dimensions 6 and 14 respectively, and
the unique J-fixed point in Alt2(E∗) is the underlying alternating form of the quadratic form
defining the extension 2.E. From the long exact sequence H•(J ;A)→ H•(J ;A.B)→ H•(J ;B)→
H•+1(J ;A)→ · · · , we find
H0(J ;X) ≤ Z/2
and
H0(J ;X.2) ≤ (Z/2).(Z/2) = Z/4. �
We may now compute the E2 page of the LHS spectral sequence for the extension
21+6.SWeyl(E6) using HAP and Cohomolo:
2 or 4
0 2 0
0 0 2
0 0 0 0 0
Z 0 0 2 4
From the vanishing of E03
2 , E
12
2 , and E21
2 , we learn that the restriction map
H3(Suz;Z)→ H3(SWeyl(E6);Z)
is an isomorphism on 2-primary parts. It follows that the preimage of SWeyl(E6) in 2Suz is the
Spin double cover of SWeyl(E6) ⊂ SO(6), which we denote by 2SWeyl(E6).
The preimage of 21+6 · (SWeyl(E6)) ⊆ Suz in 2Suz is of shape 21+6 · (2SWeyl(E6)). Using
HAP and Cohomolo, we compute that its LHS spectral sequence has E2 page:
2 or 4
0 2 0
0 0 2
0 0 0 0 0
Z 0 0 0 8
Lemma 6.12. There is a quaternion group Q′ ∼= Q8 ⊆ 21+6.2SWeyl(E6) such that the central
element of Q′ maps to an element of SWeyl(E6) of conjugacy class 2a.
There are two conjugacy classes of elements of order 2 in SWeyl(E6). They can be dis-
tinguished by the orders of their preimages in 2SWeyl(E6): elements in class 2a lift with or-
der 2 (and both lifts are conjugate in 2SWeyl(E6)), whereas elements in class 2b lift with
order 4. So the content of the Lemma is the existence of such a Q′ such that the composition
Q′ ↪→ 21+6.2SWeyl(E6)→ SWeyl(E6) is injective.
Proof. We will find this Q′ by finding a larger group: we will hunt for a binary tetrahedral group
2A4 ⊆ 21+6.2SWeyl(E6), and then take Q′ to be its 2-Sylow. Let us say that copy of 2A4 inside
some extension of SWeyl(E6) is “appropriate” if the central element of that 2A4 maps to class
2a ∈ SWeyl(E6). Then for our search, it suffices to find an appropriate 2A4 ⊆ 26.SWeyl(E6).
Indeed, H1(2A4) = 3 and H2(2A4) = 0, and so any 2A4 ⊆ 26.SWeyl(E6) will lift to a 2A4 ⊆
21+6.2SWeyl(E6).
To find an appropriate 2A4 ⊆ 26.SWeyl(E6), we recognize that
26.SWeyl(E6) ⊆ 26.Weyl(E6) ⊆ (maximal torus).Weyl(E6) ⊆ Lie group E6,
Third Homology of some Sporadic Finite Groups 27
where the group 26 is nothing but the 2-torsion points in the maximal torus. Consider the
standard Lie group embedding F4 ⊆ E6. This leads to an embedding
24.SWeyl(F4) ⊆ 26.SWeyl(E6).
covering an embedding SWeyl(F4) ⊆ SWeyl(E6). Because the Lie group F4 has no outer auto-
morphisms, the Weyl group of F4 contains all automorphisms of the F4 root lattice (isomorphic
to the D4 lattice). There is a standard identification between the F4 lattice and the Hurwitz
quaternions
{
a + bi + cj + dk ∈ H | a, b, c, d ∈ Z
}
t
{
a + bi + cj + dk ∈ H | a, b, c, d ∈ Z + 1
2
}
.
The group of units in the Hurwitz numbers is a copy of 2A4. This provides a subgroup
2A4 ⊆ SWeyl(F4) ⊆ SWeyl(E6), which is easily seen to be appropriate: central 2 ⊆ 2A4
acts by −1 on the F4 lattice, and so with trace −2 on the E6 lattice, and so fuses to class
2a ∈ SWeyl(E6).
Finally, we claim that the extension 24.2A4 splits. Indeed, the action of 2A4 on 24 is the
mod-2 reduction of the action on the F4 lattice, and for this action, H2
(
2A4; 24
)
= 0. Thus we
have found an appropriate 2A4 ⊆ 24.SWeyl(F4) ⊆ 26.SWeyl(E6). �
Lemma 6.13. The pullbacks
H4(2Suz)(2) H4
(
21+6.2SWeyl(E6)
)
H4
(
2SWeyl(E6)
)
(2)
have trivial intersection.
Proof. Let Q′ ⊆ 21+6.2SWeyl(E6) denote the quaternion group found in Lemma 6.12, and let
Q ⊆ 2SWeyl(E6) denote it image. Then Q ∼= Q8 is another quaternion group since the center
of Q′ is not in the kernel of the map Q′ → Q.
We claim that H4(2SWeyl(E6)) → H4(Q) is an isomorphism. Indeed, consider either 4-
dimensional faithful complex representations of 2SWeyl(E6). Class 2a acts on this representation
with trace 0. It follows that this representation decomposes over Q as one copy of the 2-
dimensional irrep plus two copies of the same 1-dimensional irrep, and so c2(4-dim rep)|Q has
order 8. We furthermore learn that c2(4-dim rep) generates H4(2SWeyl(E6)).
We henceforth write α ∈ H4(2SWeyl(E6)) ∼= Z/8 for this generator. Let α̃ denote its pullback
to 21+6.2SWeyl(E6). To prove the proposition, it suffices to show that 4α̃ is not in the image of
H4(2Suz).
The central element of Q′ is an order-2 lift of class 2a ∈ SWeyl(E6). Any such lift fuses
to class 2a ∈ Suz. But 21+6.SWeyl(E6) is the centralizer of an element of class 2a ∈ Suz. It
follows that Q′ is conjugate in 2Suz to some other quaternion group Q′′ ⊆ 21+6.2SWeyl(E6)
whose central element covers the center of 2SWeyl(E6).
Since Q′ is a lift of Q, we find that α̃|Q′ is a generator of H4(Q′), and so 4α̃|Q′ 6= 0. On
the other hand, since the center of Q′′ maps to something central in 2SWeyl, the 4-dimensional
representation of 2SWeyl breaks up over the image of Q′′ as either four 1-dimensional repre-
sentations or two copies of the 2-dimensional representation, and in either case we find that
α̃|Q′′ = c2(4-dim rep)|Q′′ has order at most 4, so that 4α̃|Q′′ = 0. Since Q′ and Q′′ are conjugate
in 2Suz, the class 4α̃ cannot be the restriction of a class in H4(2Suz). �
Proof of Theorem 6.8. H4(Suz)(p) vanishes for p ≥ 7 by Lemma 2.2, and for p = 5 by
Lemma 6.10. Lemma 6.10 gave H4(Suz)(3) = H4(2Suz)(3) = 0, and so Lemma 2.4 provides
the upper bound H4(3Suz)(3) ≤ 3. But Lemma 6.9 provides the lower bound H4(3Suz)(3) ≥ 3,
and so
H4(3Suz)(3) ∼= H4(6Suz)(3) ∼= Z/3,
generated by the 3-part of c2(V ).
28 T. Johnson-Freyd and D. Treumann
We now argue that H4(2Suz) = 8. Lemma 6.9 implies that H4(2Suz) contains an element
of order 8, namely the 2-part of c2(V ), where V denotes the 12-dimensional irrep of 6Suz.
Lemma 6.13 implies that H4(2Suz) has order at most 16. Furthermore, since the 2-part of c2(V )
has order 8, its restriction to 21+6 must be nonzero. On the other hand, for every represen-
tation W of 21+6, c2(W ) ∈ H4
(
21+6
)
has order dividing 2. (Indeed, for the one-dimensional
irreps of 21+6 this is automatic, and for the unique irrep of dimension 23 it is a straightforward
computation.) Thus c2(V ) restricts to the unique class in H4(21+6) which is divisible by 2. From
this we learn that the only way for H4(2Suz) to have order 16 is if c2(V ) is divisible by 2.
Suppose that there were a class “1
2c2(V )”. Since c2(V ) restricts to 2D8 with order 8, this class
1
2c2(V ) would have to have order 16 when restricted to 2D8. The order-16 classes in H4(2D8)
are the ones that have nontrivial restriction to the center of 2D8, which by construction is the
center of 2Suz. But all classes in H4(21+6.2SWeyl(E6)), hence all classes in H4(2Suz), restrict
trivially to the center. This proves
H4(2Suz) ∼= Z/8,
generated by the 2-part of c2(V ).
Finally, we argue that H4(Suz)(2) ∼= H4(3Suz)(2) ∼= Z/4 by repeating the logic from [28,
Theorem 5.3]. Namely, we have a commutative diagram
2D8 6Suz 2Co1
D8 3Suz Co1,
which, upon applying H4(−)(2), gives the diagram
Z/16 Z/8 Z/8
Z/4× (Z/2)2 H4(3Suz)(2) Z/4.
The north-then-west compositions are injective, and so both southern arrows must be injective.
It follows that
H4(Suz) ∼= H4(3Suz)(2) ∼= Z/4. �
7 Pariahs
7.1 Janko groups 1 and 3
Using the permutation representations listed in the ATLAS, HAP is able to compute
H3(J1) ∼= Z/30, H3(J3) ∼= Z/15.
HAP is unable to compute H3(3J3) directly.
Theorem 7.1. H3(3J3) ∼= (Z/3)2 × Z/5. Both H4(J3) and H4(3J3) consist entirely of Chern
classes, and are detected on cyclic subgroups.
Proof. Let V323 denote a choice of complex irrep of J3 of dimension 323. (The two choices
are the characters χ4 and χ5 when listed by increasing dimension; both are real, and are ex-
changed by the outer automorphism of J3.) Then c2(V323) restricts nontrivially to all cyclic
Third Homology of some Sporadic Finite Groups 29
subgroups of order 5 in J3. For the 3-parts of H4(J3), we focus on the conjugacy classes 3a and
9a, and any choice of 1920-dimensional irrep V1920 (there are three such irreps, all real, with
characters χ14, χ15, and χ16). Then c2(V1920) restricts nontrivially to both 〈3a〉 and 〈9a〉.
Choose any lift of 3a ∈ J3 to 3J3, for example 3c ∈ 3J3. (The classes 3a, 3b ∈ 3J3 are central.)
The class 9a ∈ J3 lifts to a single conjugacy class in 3J3, also called 9a. With these new names,
we still have that c2(V1920)|3c and c2(V1920)|9a are nontrivial. Finally, consider the smallest
faithful representation V18 of 3J3, with dimension 18 and character χ22. Then c2(V18)|3c = 0,
but c2(V18) restricts with order 3 to 〈9a〉.
It follows that the image of the map H4(3J3)(3) → H4(〈3c〉)× H4(〈9a〉) is not cyclic. On the
other hand, the HAP computation of H3(J3)(3) together with Lemma 2.4 imply that the domain
has order at most 9. So H4(3J3)(3) ∼= (Z/3)2. �
7.2 O’Nan group
Theorem 7.2. H3(O
′N) ∼= H3(3O′N) ∼= Z/8.
Proof. The p-parts of H4(O′N) vanish for p = 5 and p ≥ 11 by Lemma 2.2. The 7-Sylow is
contained in a subgroup isomorphic to PSL3(7), and a HAP computation gives H4(PSL3(7)) ∼=
Z/16.
The 3-Sylow in 3O′N is an extraspecial group of shape 31+4, and its normalizer N is a maximal
subgroup of shape N = 31+4 : 21+4.D10. HAP computes
H3(N) ∼= Z/4× Z/8× Z/5.
It follows that H4(3O′N)(3), and hence also H4(O′N)(3), vanishes.
The 2-Sylow in O′N is contained inside the nonsplit extension 43 · GL3(2). According to
Lemma 4.1(4),
H4
(
43 ·GL3(2)
)
(2)
∼= (Z/2)2 × Z/8.
The F2-cohomology ring of O′N, including the action of Steenrod squares, was computed by [1].
They find that
H1(O′N;F2) = H2(O′N;F2) = 0, H3(O′N;F2) ∼= H4(O′N;F2) ∼= F2,
but
Sq1 = 0: H3(O′N;F2)→ H4(O′N;F2).
It follows that H4(O′N)(2) is cyclic of order strictly greater than 2. Since H4(O′N)(2) is a direct
summand of H4
(
43 ·GL3(2)
)
(2)
, the only option is H4(O′N)(2) ∼= Z/8. �
7.3 Janko group 4 and Lyons group
The two largest Pariahs are Janko’s fourth group J4 and Lyons’ group Ly. Both have trivial
Schur multiplier [20], and so their cohomologies vanish in degrees ≤ 3. We find that in fact
their cohomologies vanish in degrees ≤ 4. Only one other sporadic group has this property: the
cohomology of M23 vanishes in degrees ≤ 5 [32].
Theorem 7.3. H4(J4) is trivial.
The full cohomology ring of J4 is computed away from the prime 2 in [19].
30 T. Johnson-Freyd and D. Treumann
Proof. The only primes not covered by Lemma 2.2 are 2, 3, and 11. The 11-Sylow in J4 is
contained in a maximal subgroup of shape 111+2 : (5× 2S4). It is easy to check that H4
(
111+2 :
(5× 2S4)
)
(11)
= 0; for example by observing that the central 10 ⊆ 5× 2S4 acts on 112 through
the isomorphism 10 ∼= (Z/11)× and applying Lemma 2.3.
The 3-Sylow is contained in a maximal subgroup of shape 211 : M24. There are two conjugacy
classes of elements of order 3 in M24; the restriction map H4(M24)(3) ∼= Z/3→ H4(〈3a〉) is zero,
whereas H4(M24)(3) ∼= Z/3 → H4(〈3b〉) is an isomorphism [16]. But J4 has only one conjugacy
class of order 3. It follows that H4(J4)(3) = 0.
The 2-Sylow is contained in 211 : M24, and also in a maximal subgroup of shape 21+22.3M22.2
centralizing conjugacy class 2a ∈ J4. This latter subgroup turns out to be more useful. Using
Cohomolo and Proposition 5.3, we find that the E2 page for the LHS spectral sequence reads
≤ 4
0 0 2
0 0 0
0 0 0 0 0
Z 0 2 4 22 × 3
The entry “≤ 4” in bidegree (0, 4) comes from the LES for the extension
H4
(
21+12
)
= 212.Alt2
(
212
)
.
(
Alt3
(
212
)
/212
)
.2
from Section 3.3 and the calculations
H0
(
3M22.2; 212
)
= H1
(
3M22.2; 212
)
= H0
(
3M22.2; Alt3
(
212
))
= 0,
H0
(
3M22.2; Alt2
(
212
))
= 2.
We showed during the proof of Theorem 6.3 that H4(M22 : 2)(2) = 22 is detected by restricting
to the three conjugacy classes of order 2 in M22 : 2. All three of these classes have order-2
preimages in 21+22.3M22.2. But both conjugacy classes of order 2 in J4 meet 21+22. It follows
that the images of H4(M22 : 2)(2) → H4
(
21+22.3M22.2
)
(2)
and H4(J4)(2) → H4
(
21+22.3M22.2
)
(2)
have trivial intersection.
In particular, if H4(J4) 6= 0, then it contains an order-2 class α whose restriction to 21+12 is
the unique element q̃ ∈ H4
(
21+12
)
which is twice some other element. That unique element is
pulled back from H4(212), where it corresponds to the quadratic form q ∈ Sym2(212) defining
the extension 21+12. Choose any pair of vectors v1, v2 ∈ 212 such that q(v1) = q(v2) 6= 0. Their
lifts generate a quaternion group Q8 = 21+2 ⊆ 21+12, and q̃ ∈ H4(21+12) restricts nontrivially
to that quaternion group. (These lifts of v1, v2 have order 4 in 21+12, and so are in conjugacy
class 4a in 21+12.3M22.2.)
Choose g̃ ∈ 21+12.3M22.2 in conjugacy class 4b. The character table libraries confirm that
this g̃ has the following properties: g̃2 is the nontrivial central element in 21+12 ⊆ 21+12.3M12.2;
the image g in 3M22.2 of g̃ is in conjugacy class 2a. In particular, g acts on 212 fixing 8
dimensions. Regardless of the Arf invariant of q, one can find a basis such that q vanishes on at
most one basis vector; thus we can find a vector v1 ∈ 212 with q(v) 6= 0 and vg 6= v. (Following
GAP’s conventions, we write the action of 3M22.2 on 212 from the right.) Set v2 = vg; then
q(v2) 6= 0, and so the lifts of v1, v2 generate a Q8 as in the previous paragraph.
Furthermore, we have arranged for the lifts of v1, v2 together with g̃ to generate a binary
dihedral group 2D8 ⊆ 21+12.3M22.2 extending this Q8. Suppose that H4(J4) 6= 0, and let α
denote its order-2 class. Then α|2D8 has order 2, and so is divisible by 8. But then α|Q8 = 0.
This contradicts the fact that Q8 detected q̃, and so H4(J4) must vanish. �
Theorem 7.4. H4(Ly) is trivial.
Third Homology of some Sporadic Finite Groups 31
Proof. For p > 5, the Sylow p-subgroup of Ly is cyclic, so Lemma 2.2 applies.
G2(5) and 3McL : 2 are subgroups (in fact, maximal subgroups) of the Ly [29, Propositions 2.5
and 5.4]. The 5-Sylow is contained in G2(5), so the H4(Ly)(5) vanishes by Lemma 4.3.
The 2- and 3-Sylows are contained in 3McL : 2. By Theorem 6.5, the only cohomology
of the latter is pulled back from the quotient Z/2, and so is detected on a conjugacy class of
order 2 (specifically, class 2b ∈ 3McL : 2). But Ly has only one conjugacy class of order 2, and
it meets 3McL ⊆ 3McL : 2 (since 3McL has a class of order 2). It follows that the pullback
H4(2)
∼→ H4(3McL : 2) and the restriction H4(Ly)(2) → H4(3McL : 2) have nonintersecting
images. �
8 Monster sections
8.1 Held group
The Held group is small enough to be accessible by the methods of Sections 2.2–2.3.
Theorem 8.1. H4(He) ∼= Z/12. It is spanned by fractional Pontryagin classes.
Proof. The primes not covered by Lemma 2.2 are p = 2, 3, and 7.
The normalizer of a 7-Sylow in He has shape 71+2 : (3 × S3). There are no 7s in its low
cohomology: H4
(
71+2 : (3× S3)
)
= H4(3× S3) = 2× 32.
The 3-Sylow in He is inside a maximal subgroup of shape 3S7, with H4(3S7) = 22 × 4 × 32.
We claim that the inclusions H4(He)(3) → H4(3S7)(3) and H4(S7)(3) = 3 → H4(3S7)(3) have
nonintersecting images, giving an upper bound H4(He)(3) ≤ 3. To show this, we first observe
that if C3 is a cyclic group of order 3 then the two nonzero classes in H4(C3) ∼= Z/3 are canonically
distinguished: one, which we will call t2 ∈ H4(C3), is the cup-square of both nonzero classes in
H2(C3) ∼= Z/3, and the other is not a cup-square. Now consider the class c2(Perm) ∈ H4(S7),
where Perm denotes the defining permutation representation. There are three conjugacy classes
of order 3 in 3S7: the “central” one (not actually central – it is inverted by the odd elements
of S7), and two that act on Perm with cycle structures 1431 and 1132, respectively. It follows
that c2(Perm)|〈1431〉 = −t2 whereas c2(Perm)|〈1132〉 = +t2, meaning that c2(Perm) takes different
values on these two classes. However, these two classes merge to the same class in He, and so
c2(Perm) cannot be the restriction of a cohomology class on He.
To establish the lower bound H4(He)(3) ≥ 3, we observe that the smallest irrep of He has
dimension 51, and conjugacy class 3b ∈ He acts with trace 0, and so c2 of this irrep, when
restricted to 〈3b〉, does not vanish.
For the prime p = 2, we use the 2-Sylow-containing maximal subgroup of shape 26 : 3S6.
The E2 page (localized at p = 2) of the LHS spectral sequence for this extension reads
E04
2
0 2 22
0 0 0
0 0 0 0 0
Z 0 2 2 22 × 4
with
E04
2 = H0
(
3S6;V.Alt2(V ).Alt3(V )
)
, V =
(
26
)∗
.
Since
H0(3S6;V ) = H0
(
3S6; Alt2(V )
)
= 0, H0
(
3S6; Alt3(V )
) ∼= Z/2,
32 T. Johnson-Freyd and D. Treumann
we find that
E04
2 ≤ 2.
We claim that the inclusions H4(S6)(2) ∼= H4(3S6)(2) → H4
(
26 : 3S6
)
and H4(He)(2) →
H4
(
26 : 3S6
)
have trivial intersection. To establish this claim, we first study H4(S6)(2). There
are four 5-dimensional complex irreps of S6, corresponding to the characters χ3, χ4, χ5, and χ6.
Their second Chern classes, when restricted to the conjugacy classes 2b, 4a, and 4b in S6, are
2b 4a 4b
c2(χ3) 0 −t2 +t2
c2(χ4) 0 −t2 −t2
c2(χ5) 1 −t2 −t2
c2(χ6) 1 −t2 +t2
Our notation is the following. The two classes in H4(〈2b〉) ∼= Z/2 are “0” and “1”. If C4 is
a cyclic group of order 4, the two generators of H2(C4) have the same cup-square in H4(C4),
which we call “+t2”; the generator of H4(C4) which is not a cup-square is called “−t2”.
The images of {c2(χ3), c2(χ4), c2(χ5), c2(χ6)} in H4(〈2b〉)×H4(〈4a〉)×H4(〈4b〉) ∼= Z/2×(Z/4)2
together span a group isomorphic to (Z/2)2 × Z/4. It follows that these four classes span
H4(S6)(2) ∼= (Z/2)2 × Z/4 and that the restriction map H4(S6)(2) → H4(〈2b〉) × H4(〈4a〉) ×
H4(〈4b〉) is an injection.
On the other hand, the subgroup 26 ⊆ 26 : 3S6 ⊆ He meets both conjugacy classes of order 2,
and the order-4 preimages in 26 ⊆ 26 : 3S6 of the classes 4a, 4b ∈ S6 are He-conjugate to
preimages of order-2 elements in S6. It follows that the image of H4(S6)(2) ∼= H4(3S6)(2) →
H4
(
26 : 3S6
)
does not intersect H4(He).
We have so far established the following upper bound on H4(He)(2): it is a group of order at
most 4. The last ingredient needed is a cohomology class of order divisible by 4. Let V denote
the irreducible He-module with character χ19: it is real and of dimension 7650. Consider the
conjugacy class 4a ∈ He. It squares to 2a, and
χ19(4a) = 6, χ19(2a) = 90.
Therefore 4a acts with spectrum 11938i1890(−1)1932(−i)1890, and so the total Chern class of V ,
when restricted to 〈4a〉, is
c(V ) = 11938(1− t)1890(1− 2t)1932(1 + t)1890 = 1− 2t2 + · · · (mod 4t).
In particular, c2(V )|〈4a〉 6= 0. But V is a real representation, and therefore Spin (since He has
trivial Schur multiplier). So it has a fractional Pontryagin class, twice of which is c2(V ). It
follows that p1
2 (V ) has order divisible by 4, and so H4(He)(2) ∼= Z/4. �
8.2 Harada–Norton and Thompson groups
We were able to obtain partial results about the Harada–Norton and Thompson groups HN
and Th.
Theorem 8.2. H4
(
HN;Z
[
1
2
]) ∼= Z/3. At the prime 2, H4(HN)(2) has order at most 16 and
exponent at most 8.
Proof. Lemma 2.2 handles the primes p ≥ 7. The 5-Sylow in HN is contained in a maximal
subgroup of shape 51+4.21+4.5.4; the LHS spectral sequence gives H4
(
51+4.21+4.5.4
)
(5)
= 0.
Third Homology of some Sporadic Finite Groups 33
The 3-Sylow is inside a maximal subgroup of shape 31+4 : 4A5, where by “4A5” we mean the
“diagonal” central extension 2.(A5 × 2). HAP can directly compute
H4
(
31+4 : 4A5
)
(3)
= 32.
We claim that the generator of H4(4A5)(3) = 3, when pulled back to 31+4 : 4A5, is not the
restriction of a class on HN. Indeed, that generator has nontrivial restriction to the elements of
order 3 in 4A5, and so its pullback has nonzero restriction to some elements of order 3. But both
conjugacy classes of order 3 in HN meet 31+4 ⊆ 31+4 : 4A5. Finally, we claim that H4(HN)(3) 6= 0.
There is a unique conjugacy class of order 9 in HN, and its traces on either 133-dimensional
representation, which characters χ2 and χ3, are χ2(9a) = 1 and χ2
(
9a3
)
= χ2(3b) = −2. From
this one can compute that c2(χ2)|〈9a〉 6= 0.
The 2-Sylow in HN is contained in the centralizer of conjugacy class 2b, which has shape
21+8
+ .(A5 o2). The E2 page of the corresponding LHS spectral sequence provides an upper bound
of
∣∣H4(HN)(2)
∣∣ ≤ 26. Let E = 28 ∼=
(
28
)∗
and J = (A5 o 2); then
H0(J ;E) = H1(J ;E) = H0
(
J ; Alt3(E)
)
= 0, H2(J ;E) ∼= Z/2,
while
H0
(
J ; Alt2(E)
) ∼= Z/2, H0
(
J ; Alt2(E)/2
) ∼= Z/2; H1
(
J ; Alt2(E)/2
) ∼= (Z/2)2.
Therefore the E2 page of the LHS spectral sequence, after localizing at 2, reads
2
2 22
0 0 2
0 0 0 0 0
Z 0 2 2 22
As usual, the image of H4(A5 o 2)(2) → H4
(
21+8
+ .(A5 o 2)
)
(2)
does not intersect H4(HN): the
former is detected on elements of order 2, but both conjugacy classes of order 2 in HN meet
21+8
+ ⊆ 21+8
+ .(A5 o 2). �
Theorem 8.3. H4
(
Th;Z
[
1
3
]) ∼= Z/8.
Proof. Lemma 2.2 handles the primes p ≥ 7. The 5-Sylow is inside 51+5 : 4S4, and Lemma 2.3
implies H4
(
51+5 : 4S4
)
(5)
= 0. The 3-Sylow does not live in any nice maximal subgroups, and
so we cannot compute H4(Th)(3).
The 2-Sylow in Th is contained in the Dempwolff group of shape 25 · GL5(2). According to
Lemma 4.1(3), H4
(
25 ·GL5(2)
)
(2)
= 8, and the proof of that lemma established that c2(V ) 6= 0,
where V denotes the 248-dimensional irrep of 25 ·GL5(2). This irrep V extends to the defining
248-dimensional irrep of Th, and so the restriction map H4(Th)(2) → H4
(
25 · GL5(2)
)
(2)
is
nonzero. Since the image of that restriction map is a direct summand, we must have H4(Th)(2) ∼=
H4
(
25 ·GL5(2)
)
(2)
∼= Z/8. �
8.3 Fischer groups
The Fischer groups Fi22, Fi23, and Fi24 are a “third generation” version of the Mathieu groups
M22, M23, and M24. Specifically, the 2-Sylow in FiN , for N ∈ {22, 23, 24}, lives in a sub-
group of shape 2dN/2e−1.MN , making the calculation of H4(FiN )(2) systematic. The extension
2dN/2e−1.MN splits for N = 22 and does not split for N = 23 and 24.
The 3-Sylows in all cases are contained in orthogonal groups over F3. We were able to handle
the 3-parts of H4(G) for G = Fi22 and 3Fi22, but not for the larger Fischer groups.
34 T. Johnson-Freyd and D. Treumann
Theorem 8.4. The Fischer groups have the following cohomologies away from the prime 3:
1) H4
(
Fi22;Z
[
1
3
])
= 0;
2) H4
(
2Fi22;Z
[
1
3
])
has order 2 or 4;
3) H4
(
Fi23;Z
[
1
3
]) ∼= Z/2;
4) H4
(
Fi′24;Z
[
1
3
])
has order 2 or 4.
Proof. Lemma 2.2 handles all primes p ≥ 5 except for H4(Fi′24)(7). But the 7-Sylow in Fi′24 is
inside a copy of Held’s group He, and H4(He)(7) = 0 by Theorem 8.1.
We now inspect the LHS spectral sequences for the extensions 2dN/2e−1.MN ⊆ FiN . The
three cases are:
0
0 0 2
0 2 0
0 0 0 0 0
Z 0 0 12 0
0
0 2 2
0 0 0
0 0 0 0 0
Z 0 0 0 0
0
0 2
0 0 0
0 0 0 0 0
Z 0 0 0 12
210 : M22 ⊆ Fi22 211.M23 ⊆ Fi23 211.M24 ⊆ Fi24
The first proves claim (1) and provides the upper bound for claim (2). The second provides the
upper bound for claim (3).
Let α ∈ H4(M24)(2) ∼= Z/4 denote a generator, and let α̃ denote its pullback to 211.M24.
Consider the conjugacy classes 4b and 4c in M24. The formula given in [16, Section 3.3] provides
α|〈4b〉 = 0 whereas α|〈4c〉 has order 4 in H4(〈4c〉). These classes admit preimages in 211.M24
living in conjugacy classes 8e and 8a respectively, and so α̃|8e = 0 whereas α̃|8a has order 2.
(The pullback map H4(C4) → H4(C8) along a surjection of cyclic groups C8 → C4 has image
of order 2.) Both 8e and 8a fuse in Fi′24 to class 8a, and so α̃ cannot be the restriction of
a cohomology class on Fi′24. Together with the above spectral sequence, we find the upper
bound in claim (4).
All that remain are the lower bounds. Let ω\ denote the “gauge anomaly” of the Monster
CFT, studied in [27]; cf. Section 8.4. The McKay–Thompson series for class 4b in the Monster
group M has a nontrivial multiplier (of order 2), and so ω\|〈4b〉 is nonzero. But 4b meets
2Fi22 ⊆ Fi23 ⊆ 3Fi′24, and so ω\ restricts with order at least 2 to all of these groups. �
Theorem 8.5. H4(Fi22) = {0} and H4(3Fi22) ∼= Z/3.
Combined with Theorem 8.4(2), we find that H4(6Fi22) has order either 6 or 12.
Proof. Given Theorem 8.4(1), we must only compute H4(Fi22)(3) and H4(3Fi22)(3). But the
3-Sylow in Fi22 is contained in a maximal subgroup isomorphic to Ω7(3), and H4(Ω7(3))(3) = 0
by Lemma 4.4. By Lemma 2.5, the inclusions H4(Ω7(3))(3) → H4(3Ω7(3))(3) and H4(Fi22)(3) →
H4(3Fi22)(3) have the same cokernel; this cokernel is Z/3 by Corollary 4.5. �
8.4 Monster
The main result of [27] is that H4(M) contains a class ω\, arising as the gauge anomaly of the
Moonshine CFT, of exact order 24. It is reasonable to conjecture that ω\ generates H4(M). The
calculations in this paper allow us to come close to proving that conjecture:
Theorem 8.6. H4(M) ∼= Z/24⊕X, where the Z/24 summand is generated by ω\ and where the
order of X divides 4.
Third Homology of some Sporadic Finite Groups 35
Proof. The primes p = 11 and p ≥ 17 are covered by Lemma 2.2. To calculate H4(M), we
must study the p-Sylows for p = 2, 3, 5, 7, 13. For these p, the p-Sylow is contained in the
normalizer N(pb) of an element of conjugacy class pb. These normalizers are all of shape
N(pb) = p1+m.J,
where m = 24/(p− 1) and J ⊆ Co1. Specifically
p J
2 Co1
3 2Suz.2
5 2J2.4
7 3× 2S7
13 3× 4S4
The extension p1+m.J splits for p ≥ 5. When p = 3, 31+12.2.Suz.2 does not split, but the
quotient 312 : 2Suz.2 does split. When p = 2, the quotient 224 · Co1 does not split.
The center of the group J in all cases has order p−1, and acts by a faithful central character
on pm. Applying Lemma 2.3, we find that Hi
(
J ; Hj
(
p1+m
))
= 0 for 0 < j < p. Combined
with the HAP calculation of H4(2J2) from Section 6.2, the known value H4(3 × 2S7)(7) = 0
(which follows by the methods of Lemma 2.2), and the trivial result H4(3 × 4S4)(13), we find
that H4
(
M;Z
[
1
6
])
= 0.
At the prime 3, central character considerations imply that the only potentially nonzero
entries of total degree 4 on the LHS spectral sequence for 31+12.2Suz.2 are, in the notation of
Lemma 3.3:
H0
(
2Suz.2; Sym2
(
312
))
, H1
(
2Suz.2; Alt2
(
312
)
ω
)
, H4(2Suz.2;Z)(3).
The first vanishes because 312 is antisymmetrically but not symmetrically self-dual as a 2.Suz
module. Actually, as a 2Suz.2 module, 312 is not self-dual at all: the symplectic pairing changes
by a sign via the surjection 2Suz.2→ 2. The last vanishes by Theorem 6.8.
Therefore H1
(
2Suz.2; Alt2
(
312
)
ω
)
gives an upper bound for H4(M)(3). The group 2Suz.2
is too large for Cohomolo to handle directly, but its 3-Sylow-containing maximal subgroup of
shape 35 : (M11 × 2) is not, and Cohomolo computes
H1
(
35 : (M11 × 2); Alt2
(
312
)
ω
) ∼= 31.
This provides an upper bound for H1
(
2Suz.2; Alt2
(
312
)
ω
)
, and so
H4(M)(3) ≤ 3.
On the other hand, [27, Lemma 3.2.4] shows that H4(M)(3) ≥ 3.
The p = 2 part of H4(M) is more subtle.
We first claim that the Z/8 ⊆ H4(M)(2) generated by ω\ is a direct summand. Equivalently,
we claim that ω\ is not divisible by 2. Consider the subgroup N(3b) = 31+12.2.Suz.2 ⊆ M. Since
its quotient 312 : 2Suz.2 splits, we can find a copy of 6Suz ⊆ 6Suz.2 ⊆ M. The central 3 ⊆ 6Suz
is generated by class 3b by construction, and the central 2 ⊆ 6Suz ⊆ M is of class 2b. It follows
that this 6Suz is (conjugate to) a subgroup of the normalizer N(2b) = 21+12 · Co1, where it
lives over a copy of 3Suz ⊆ Co1. As observed in the proof of Theorem 6.8, the corresponding
6Suz ⊆ 2Co1 contains the group 2D8 used in [28]; thus the 6Suz ⊆ M contains a 2D8 such that
the central element is of class 2b ∈ M. This is the 2D8 ⊆ M used in [27, Section 3.3] to show
that the order of ω\ is divisible by 8. It follows in particular that the order of ω\|6Suz is divisible
36 T. Johnson-Freyd and D. Treumann
by 8. But H4(6Suz)(2) = 8 by Theorem 6.8, and so ω\|6Suz is not divisible by 2, proving the
claim.
The last step of the proof is to study the LHS spectral sequence for the extension 21+24 ·Co1:
H0
(
Co1; H4
(
21+24
))
H0
(
Co1; H3
(
21+24
))
H1
(
Co1; H3
(
21+24
))
H0
(
Co1; H2
(
21+24
))
H1
(
Co1; H2
(
21+24
))
H2
(
Co1; H2
(
21+24
))
0 0 0 0 0
Z 0 0 2 4
The 2 in the bottom row is H3(Co1), and the 4 in the bottom row is H4(Co1)(2); the latter result
is due to [28].
From Section 3.3, we know
H2
(
21+24
) ∼= 224, H3
(
21+24
) ∼= Alt2
(
224
)
/2 ∼= 2275,
while
H4
(
21+24
) ∼= 224.Alt2
(
224
)
.
(
Alt3
(
224
)
/224
)
.2 ∼= 22300.2
has exponent 4.
Without much difficulty, GAP can compute
H0
(
Co1; 224
)
= H0
(
Co1; Alt2
(
224
)
/2
)
= H0
(
Co1; Alt3
(
224
))
= 0.
and
H0
(
Co1; Alt2
(
224
)) ∼= Z/2.
The groups Hi
(
Co1; 224
)
for i = 1, 2 were calculated by Derek Holt and reported in [26,
Lemma 1.8.8]. They are
H1
(
Co1; 224
)
= 0, H2
(
Co1; 224
) ∼= Z/2.
A presentation for Co1 is given in [35]. Using it, [27, Section 3.5] calculates
H1
(
Co1; Alt2
(
224
)
/2
) ∼= Z/2.
These calculations almost fill in the E2 page. The remaining missing ingredient is E04
2 =
H0
(
Co1; H4
(
21+24
))
. Using the above calculations together with the long exact sequence for
cohomology with values in an extension, one finds:
H0
(
Co1; 224.Alt2
(
224
)
.
(
Alt3
(
224
)
/224
))
= H0
(
Co1; 22300
) ∼= Z/2.
It follows that H0
(
Co1; H4
(
21+24
))
is isomorphic to either Z/2 or Z/4. We suspect the former,
but were unable to compute it.
Although we do not know whether E04
2 = 2 or 4, we claim that E04
∞ = 2. To prove this we
quote two facts. First, according to [27, Section 3.5], ω\|21+24 has exact order 2, and so provides
an element of order 2 in E04
∞ . Second, we showed above that ω\ is not divisible by 2 in H4(M).
Since the map H4(M) → H4
(
21+24.Co1
)
is an inclusion onto a direct summand, it follows that
ω\|21+24.Co1 is not divisible by 2. But H4
(
21+24.Co1
)
surjects onto E04
∞ , and sends ω\ to a nonzero
value. So the image of ω\ in E04
∞ cannot be divisible by 2, proving that E04
∞ 6= 4.
All together, we find an E∞ page of the following form
2
0 ≤ 2
0 0 ≤ 2
0 0 0 0 0
Z 0 0 2 4
It follows that H4(M)(2) has order at most 32, completing the proof. �
Third Homology of some Sporadic Finite Groups 37
Acknowledgements
This project began when John Duncan asked us to compute H4(O′N;Z). The referees provided
detailed and valuable corrections. We also thank Graham Ellis, Jesper Grodal, Geoff Mason,
and Robert Wilson for numerous comments and hints about finite groups and cohomology. This
work was supported by the US NSF grant DMS-1510444 and by the Government of Canada
through the Department of Innovation, Science and Economic Development Canada and by the
Province of Ontario through the Ministry of Research, Innovation and Science.
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1 Introduction
1.1 Motivation
2 Preliminaries
2.1 Notation
2.2 General methods
2.3 Characteristic classes
3 Elementary abelian and extraspecial p-groups
3.1 Elementary abelian groups
3.2 Extraspecial p-groups for p odd
3.3 Extraspecial 2-groups
4 Dempwolff groups, Chevalley groups and their exotic Schur covers
4.1 Dempwolff and Alperin groups
4.2 A few exotic Chevalley groups
5 Mathieu groups
6 Leech lattice groups
6.1 Higman–Sims group
6.2 Janko group 2
6.3 Conway groups
6.4 McLaughlin group
6.5 Suzuki group
7 Pariahs
7.1 Janko groups 1 and 3
7.2 O'Nan group
7.3 Janko group 4 and Lyons group
8 Monster sections
8.1 Held group
8.2 Harada–Norton and Thompson groups
8.3 Fischer groups
8.4 Monster
References
|
| id | nasplib_isofts_kiev_ua-123456789-210236 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:51Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Johnson-Freyd, T. Treumann, D. 2025-12-04T13:06:41Z 2019 Third Homology of some Sporadic Finite Groups / T. Johnson-Freyd, D. Treumann // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20D08; 20J06 arXiv: 1810.00463 https://nasplib.isofts.kiev.ua/handle/123456789/210236 https://doi.org/10.3842/SIGMA.2019.059 We compute the integral third homology of most of the sporadic finite simple groups and of their central extensions. This project began when John Duncan asked us to compute H⁴(O'N; Z). The referees provided detailed and valuable corrections. We also thank Graham Ellis, Jesper Grodal, Geoff Mason, and Robert Wilson for numerous comments and hints about finite groups and cohomology. This work was supported by the US NSF grant DMS-1510444 and by the Government of Canada through the Department of Innovation, Science and Economic Development Canada, and by the Province of Ontario through the Ministry of Research, Innovation and Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Third Homology of some Sporadic Finite Groups Article published earlier |
| spellingShingle | Third Homology of some Sporadic Finite Groups Johnson-Freyd, T. Treumann, D. |
| title | Third Homology of some Sporadic Finite Groups |
| title_full | Third Homology of some Sporadic Finite Groups |
| title_fullStr | Third Homology of some Sporadic Finite Groups |
| title_full_unstemmed | Third Homology of some Sporadic Finite Groups |
| title_short | Third Homology of some Sporadic Finite Groups |
| title_sort | third homology of some sporadic finite groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210236 |
| work_keys_str_mv | AT johnsonfreydt thirdhomologyofsomesporadicfinitegroups AT treumannd thirdhomologyofsomesporadicfinitegroups |