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.


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 H 4 (G; Z), the Pontryagin dual of H 3 (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 = H 2 (G) -and have left empty spaces where G = n.G. 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 of order dividing 4. Further partial results for the groups HN, Th, Fi 23 , and Fi ′ 24 are listed in §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 [CCN + 85]. With F 2 -coefficients, the entire cohomology rings of many of the smaller sporadic groups are listed in [AM04]. The Mathieu entries are reviewed in [DSE09]. Significantly, H 3 (M 24 ) was first computed in that paper using Graham Ellis's software package "HAP," which we have found can also determine H 3 (G) for G ∈ {HS, 2HS, J 2 , 2J 2 , J 1 , J 3 , McL} using the permutation models given in the ATLAS. For the larger groups G, although HAP cannot calculate H 3 (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 H 4 (BG; Z) ∼ = Z. If G is a finite group of Lie type, split and simply connected and defined over the field F q , then Jesper Grodal has shown that with finitely many exceptions, H 4 (G; Z) ∼ = Z/(q 2 − 1). Part of our motivation has been to see whether we could discern any patterns in H 4 (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 H 4 (G; Z)) can explain and predict some features of moonshine [Gan09,Gan16,CdLW16,GPRV13]. Such a cocycle can arise as the gauge anomaly of a group 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.
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:

Methods
In this section we review some standard constructions in group cohomology, which we return to repeatedly in the following sections. The first of these is computing the p-primary part of H 4 (G; Z), which we denote by H 4 (G; Z) (p) , one prime at a time. An upper bound for the p-primary part is provided by the following lemma: 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 → α| S : H k (G; Z) (p) → H k (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 §3.1) of H 4 (Z/p; Z) and H 4 (Z/p × Z/p; Z), allow us to dispose of many of the larger primes, at least for sporadic groups: Lemma 2.2. Let p ≥ 5 be a prime and let G be a sporadic simple group whose Sylow p-subgroup has order p or p 2 . Then either (p, G) = (5, J 2 ) or else the p-part of H 4 (G; Z) vanishes.
Proof. Let H ⊂ G be a Sylow p-subgroup. Suppose first that H has order p. Then G has strictly fewer than (p − 1)/2 conjugacy classes of order p. (We checked this by inspecting the ATLAS's tables: it is not true for every non-sporadic group, e.g. SL 2 (F 32 ) has 15 conjugacy classes of order 31.) It follows that for a generator h ∈ H (or indeed for any element h of order p in G), there is an x ∈ G with xhx −1 = h a , where a is neither 1 nor −1 mod p. Conjugation by such an x acts trivially on H • (G; Z) but nontrivially on H 4 (H; Z), so the image of the restriction map H 4 (G; Z) → H 4 (H; Z) must vanish. After Lemma 2.1, H 4 (G; Z) must vanish as well.
For all sporadic groups, if the p-Sylow has order p 2 then it is isomorphic to Z/p × Z/p. (Note this is not true for a general simple group, e.g. the 3-Sylow in SL 2 (F 16 ) is cyclic of order 9). Furthermore by inspecting the tables one sees that such a sporadic group has strictly less than (p − 1)/2 conjugacy classes of order p, unless p = 5 and G = J 2 . With this single exception, we conclude that H 4 (G; Z) → H 4 ( g ; Z) is zero whenever g has order p -in particular H 4 (G; Z) → H 4 ( g ; Z) is zero whenever g is an element of H ∼ = Z/p×Z/p. But when p is odd, H 4 (Z/p×Z/p; Z) is detected on its cyclic subgroups.
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 [Wil17] for a survey of maximal subgroups of finite groups.) Sometimes we know H 4 (J; Z), either by induction or by computer. The Lyndon-Hochschild-Serre (LHS) spectral sequence gives an upper bound for H 4 (S; Z) (and therefore for H 4 (G; Z)), in terms of the cohomology groups (with twisted coefficients) We describe the groups H j (E; Z) for j ≤ 4 as Aut(E)-modules in §3. We used extensively Derek Holt's software package "Cohomolo" to determine the groups H 1 (J; −) and H 2 (J; −), but sometimes the following vanishing criterion can be employed instead: Proof. In that case the trivial J-module and H j (E; Z) are in different blocks.
The LHS spectral sequence allows us a comparison between the cohomology of a group and of its Schur cover. Recall that a finite group is called "perfect" if its abelianization is trivial.
Lemma 2.4. Let G be a finite perfect group and choose a cyclic subgroup n ⊂ H 2 (G). Then the pullback map H 4 (G; Z) → H 4 (n.G; Z) is an injection. The cokernel has order dividing n if n is odd, or dividing 2n if n is even.
Proof. We treat the case n = H 2 (G) for clarity; the general case is no harder. Consider the LHS spectral sequence for the extension n.G. Its E 2 page, in total degree ≤ 5, reads: The rows are filled in using the universal coefficient theorem. We have indicated some of the multiplicative structure: we labeled a generator of H 3 (G; Z) ∼ = Z/n by the name "x" and we labeled a generator of H 2 (n; Z) ∼ = Z/n by the name "y." In particular, y 2 and xy generate (Z/n)s in degrees (i, j) = (0, 4) and (3, 2) respectively.
The even differentials in the spectral sequence vanish automatically, so E ij 3 = E ij 2 . The d 3 differential takes y → x, and so d 3 (y 2 ) = 2xy. It follows that if n is odd, then d 3 : E 04 3 → E 32 3 is injective, and if n is even, d 3 : E 04 3 → E 32 3 has kernel of order 2. Thus the E 4 page reads: depending on the parity of n, where X ⊂ (Z/n) is the kernel of d 3 : E 22 3 → E 50 3 . Corollary 2.5. Let p be an odd prime such that H 1 (G; Z/p) = 0 and H 2 (G; Z/p) = p. Let pG denote a nontrivial central extensions of G by the group Z/p. Suppose that S ⊂ G also has H 1 (S; Z/p) = 0, and that the central extension pG, when restricted to S, is nonsplit. Then the pullback map H 4 (pG; Z) → H 4 (pS; Z) induces an injection This injection is an isomorphism if S contains the p-Sylow of G.
As we have mentioned, each page of the LHS spectral sequence provides an upper bound for H 4 (G) (p) . We can improve this upper bound whenever we can show that the images of the two maps have trivial intersection -in that case only the groups (2.1) can contribute to H 4 (G; Z).
With the improved upper bound in hand, the last step is to give a lower bound for H 4 (G; Z). In almost all cases these come from the characteristic class of a representation G → K, where K is a Lie group. For K = U(n), resp. Spin(n), this characteristic class is c 2 , resp. p 1 2 . In two cases we appeal to K = E 6 and E 8 . For some of the Monster sections, it is not possible for such representations to give a strong enough lower bound, and we instead appeal to the construction of [JF17].

Elementary abelian and extraspecial p-groups
3.1. Elementary abelian groups. Lemma 3.1. Let E = p n be an elementary abelian p-group and let E * := Hom(E, µ p ), where µ p denotes the group of pth roots of unity in C * .
(2) If p is odd, we have isomorphisms of GL(E)-modules If V is an elementary abelian p-group, we regard it as an F p -vector space in the obvious way. We may identify E * with the usual dual F p -vector space to E by fixing at the outset an isomorphism µ p ∼ = Z/p. We use Sym n (V ) and Alt n (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: • Sym n (V ) := H 0 (S n ; V ⊗n ) are the coinvariants of V ⊗n by the symmetric group action • Alt n (V ) is the quotient of V ⊗n by the subspace spanned by tensors with a repeated tensorand (tensors v 1 ⊗ · · · ⊗ v n with v i = v j for some i = j). Though Sym n (E * ) and Sym n (E) * are not isomorphic as GL(E)-modules if p ≤ n (instead the dual of Sym n (E * ) is the space of divided powers of E), let us record: where (−1) σ denotes the sign of the permutation σ, is GL(V )-equivariant and descends to a perfect pairing between Alt n (V ) and Alt n (V * ).

3.2.
Extraspecial p-groups for p odd. If p is prime, E = p n 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 z i t v (where i ∈ Z/p and v ∈ E) by the formula If ω is bilinear, this multiplication is associative, z 0 t 0 is a two-sided unit, and z −i+ω(v,v) t −v is the two-sided inverse to z i t v : 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(v + w) − j(v) − j(w) for some function j : E → Z/p -in particular if p is odd then ω(v, w) and determine isomorphic groups, so when p is odd we may as well assume that ω ∈ Alt 2 (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 = p 1+2m is a copy of the extraspecial pgroup of exponent p. (The extraspecial group of exponent p 2 comes from a non-bilinear cocycle ω : E × E → F p . The extraspecial groups of order 2 1+2m will be treated in §3.3; the group (p.E) ω that we have defined is always elementary abelian when p = 2). The automorphism group of p 1+2m is E : GSp(E, ω), where E acts by inner automorphisms , ω(gv, gw) = aω(v, w)} acts by (g, a)(z i t v ) = z ai t gv . The scalar a = a(g) is determined by g.
Let L ω ⊂ Alt 2 (E * ) denote the line spanned by ω. It is a one-dimensional GSp-submodule by construction, and if ω is nondegenerate then E * ∼ = E ⊗ L ω . Provided p is odd, we have a splitting Lemma 3.3. Let p be an odd prime, let E = p 2m be an elementary p-group and let ω ∈ Alt 2 (E * ) be a nondegenerate symplectic form. Then if m ≥ 2, as GSp 2m -modules. If m ≥ 3, a possibly nontrivial extension of Alt 2 (E * ) ω by Sym 2 (E * ).
Proof. We consider the action of GSp on the LHS spectral sequence We have H 2 (p) = L ω and H 4 (p) = L 2 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; F p ) is the gradedcommutative F p -algebra generated by a copy of E * in degree 1 and a second copy of E * in degree 2; in particular: All together, we have on the E 2 -page: The d 2 differential vanishes and the d 3 differentials L ω → Alt 2 (E * ), E * ⊗ L ω → Alt 3 (E * ), and L 2 ω → Alt 2 (E * ) ⊗ L ω are the injections discussed above. It remains to understand d 3 : (Alt 2 (E * ) ⊕ E * ) ⊗ L ω → H 5 (E). We claim that this map is an injection when m ≥ 3, and that when m = 2 its kernel is Alt 2 (E * ) ω ⊗ L ω ⊂ Alt 2 (E * ). Note also that when m = 2, the map E * ⊗ L ω → Alt 3 (E * ) is an isomorphism. In this range of degrees, the sequence stabilizes after page 4, and so on the E ∞ page we see

3.3.
Extraspecial 2-groups. The case of the extraspecial 2-groups may be treated as follows. If E is elementary abelian then any central extension 2.E is determined up to isomorphism by the function 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 [Gri73]. But O(Q) still acts on the cohomology of 2.E. The LHS spectral sequence begins: We first wish to describe the d 3 differential. To do so, recall first that H • (E) injects into H • (E; F 2 ) ∼ = Sym • (E * ) as the subalgebra in the kernel of the derivation Sq 1 : In particular, it sends the generator of the 2 in degree (0, 2) to Sq 1 (Q) ∈ Sym 3 (E * ). The image of Sq 1 : Sym 2 (E * ) to Sym 3 (E * ) is isomorphic to Alt 2 (E * ), and under this isomorphism Sq 1 takes Q to its underlying alternating form Let us suppose that Q is nondegenerate and E = 2 2m . Then in particular B Q = 0, so that This cannot happen when m ≥ 2, and so d 3 : E * → E * . Alt 2 (E * ). Alt 3 (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: The d 3 differential emitted by the Sym 2 (E * ) in degree (2, 2) always has kernel -Q itself -and nothing more provided m ≥ 2. Finally, if m ≥ 3, then d 5 : E 04 5 → E 50 5 is nonzero, and the E ∞ page looks like This can be simplified slightly.
To complete the proof of (3), it suffices to give an element of H 4 (2 5 · GL 5 (2)) whose order is divisible by 8. There is a famous embedding of 2 5 · GL 5 (2) into the compact Lie group E 8 . Let us write e for the generator of H 4 (BE 8 ). We will prove that the restriction e| 2 5 ·GL 5 (2) is such an element.
For the remainder of the proof, let V denote the 248-dimensional adjoint representation of E 8 . The dual Coxeter number of E 8 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 H 4 (BG): In particular, c 2 (V ) = −60e. Since 60 is divisible by 4, to show that the order of e| 2 5 ·GL 5 (2) is divisible by 8, it suffices to show that the order c 2 (V )| 2 5 ·GL 5 (2) is divisible by 2.
We will do so by finding a binary dihedral group 2D 8 ⊂ 2 5 ·GL 5 (2) such that c 2 (V )| 2D 8 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 2 5 · GL 5 (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 2 5 · GL 5 (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 2 5 · GL 5 (2). There are four conjugacy classes of order-8 elements in SmallGroup(64, 151), and GAP quickly checks that all four merge in 2 5 · GL 5 (2) to conjugacy class 8c.
Finally, SmallGroup(64, 151) contains a copy of the binary dihedral group 2D 8 of order 16. Since 2D 8 is a finite subgroup of SU(2), its cohomology is easy to compute: in particular, H 4 (2D 8 ) is cyclic of order |2D 8 | = 16 and is generated by c 2 of the "defining" two-dimensional representation. As in [JFT18, Section 6], let us index the irreps by: In particular, V 0 is the trivial representation, V 6 is the "defining" two-dimensional irrep, V 5 is the other faithful irrep, V 4 is the two-dimensional real irrep of D 8 , and V 1 , V 2 , and V 3 are the nontrivial one-dimensional irreps.
Character table constraints provide a unique fusion map 2D 8 → 2 5 · GL 5 (2) sending the elements of order 8 to conjugacy class 8c. Along this map, the 248-dimensional irrep V of 2 5 · GL 5 (2) decomposes as: [JFT18] gives a formula for the second Chern class of any representation of 2D 8 in which the representations V 1 and V 2 appear with the same coefficient. That formula is: where we have identified H 4 (2D 8 ) = Z/16 by identifying 1 ∈ Z/16 with c 2 (V 6 ). Applying this formula to the 248-dimensional representation V gives: In particular, c 2 (V ) is nonzero in H 4 (2D 8 ). As explained above, this implies that H 4 (2 5 · GL 5 (2)) contains an element of order divisible by 8 (namely, the restriction of the generator of H 4 (BE 8 )), and so must be isomorphic to Z/24.

A few exotic Chevalley groups.
For the most part, any central extension of a finite Chevalley group G(F q ) is the group of F q -points of a central extension of the algebraic group G. In particular if G is of simply connected type then the multiplier H 2 (G(F q )) 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.
Proof. We computed these using HAP. The computation of H 3 (Sp 6 (F 2 ) is fast, but computing H 3 (2 · Sp 6 (F 2 )) took many hours. Two of the faithful permutation representations of 2 · Sp 6 (F 2 ) have degrees 240 and 276 (the latter coming from the embedding 2 · Sp 6 (F 2 ) ⊂ Co 3 ). Our machine 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
Jesper Grodal has shown that H 4 (G 2 (F q )) is cyclic of order q 2 − 1 if q = p r with either p or r sufficiently large. The computations in the theorem show that this holds also for q = 5, but not q = 3 or q = 2.
The group called "O n (q)" in the ATLAS is not the n × n orthogonal group over F q . Rather, it is the simple subquotient of the orthogonal group. To avoid confusion, we will follow Dieudonné and write "Ω n (q)" for this simple group. When n ≥ 5 and q are odd, Ω n (q) is the commutator subgroup of SO n (F q ) = Ω n (q) : 2, and is the image of Spin n (F q ) in SO n (F q ), and is the kernel of the "spinor norm" Proof. The criterion in Lemma 2.2 applies for the primes p ≥ 5. The 2-Sylow is contained in Sp 6 (F 2 ), giving an upper bound of H 4 (Sp 6 (F 2 )) (2) = 2 × 4 for H 3 (Ω 7 (3)), and an upper bound of H 4 (2Sp 6 (F 2 )) (2) = 2 × 8 for H 3 (Spin 7 (3)), both from Lemma 4.2. It is straightforward to check that second Chern class of the 78-dimensional real representation of Ω 7 (3) restricts with order 4 to conjugacy classes 4a, and so the fractional Pontryagin class of this representation has order 8.
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 SO 7 , corresponding to the Dynkin subdiagrams B 2 ⊂ B 3 and A 1 × A 1 ⊂ B 3 . These lead to two maximal subgroups of Ω 7 (3) that contain the 3-Sylow: There is one more maximal subgroup of Ω 7 (3) containing the 3-Sylow, corresponding to the Dynkin diagram inclusion A 2 ⊂ B 3 , which we will not use in the present proof, but will use in the proof of Corollary 4.5. The spectral sequence for 3 5 : SO 5 (F 3 ) has E 2 page: The bottom line was computed in HAP, and the middle entries in Cohomolo. The entry E 04 2 = 3 corresponds to the symmetric pairing on 3 5 .
We claim that the maps To see this, first note that SO 5 (F 3 ) ∼ = Weyl(E 6 ) has a 6dimensional irrep, on which the conjugacy class 3b acts with trace 3. It follows that elements, and so is not the restriction of a class in H 4 (Ω 7 (3)).
The other maximal subgroup we consider is the one of shape 3 1+6 It is the normalizer of conjugacy class 3a. GAP can work with Ω 7 (3) by using its degree-351 permutation representation, and quickly find this subgroup. In particular, GAP finds that the action of (2A 4 × A 4 ).2 on 3 6 is generated by the following three matrices: The Chevalley group Ω 7 (3) is has an exceptional cover: its multiplier is 6, whereas the multiplier of Ω 7 (q) is generically 2 = π 1 (SO(7, C)). Proof. It must calculate H 4 (3.Ω 7 (3)) (3) . To do so we use the third maximal parabolic of Ω 7 (3), of shape 3 3+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.Ω 7 (3) restricts to a nontrivial central extension 3.S. One can show, for instance by running a LHS spectral sequence, that H 1 (S; 3) = 0 and H 2 (S; 3) = 3. In particular, there is a unique nonsplit extension 3.S.
The smallest complex representations of Ω 7 (3) and 3.Ω 7 (3) have dimensions 78 and 27 respectively, equal to the smallest representations of the simple Lie group E adj 6 (C) and its simply connected cover E sc 6 = 3.E adj 6 . However, Ω 7 (3) does not preserve the Lie bracket on the 78-dimensional representation. It does preserve a lattice, and preserves the Lie bracket "modulo 2": in fact, Ω 7 (3) embeds into the twisted Chevalley group 2 E 6 (2) ⊂ E adj 6 (F 4 ). The finite subgroups of the Lie group E adj 6 (C) were classified in [CW97]. In particular, E adj 6 (C) and Ω 7 (3) intersect along the subgroup S, lifting to the nonsplit extension 3.S ⊂ E sc 6 (C). Both H 4 (BE adj 6 ) and H 4 (BE sc 6 ) are infinite cyclic, but the restriction map H 4 (BE adj 6 ) → H 4 (BE sc 6 ) is not an isomorphism: its cokernel has order 3. By Corollary 2.5, this forces the inclusion H 4 (S) → H 4 (3.S) to have cokernel of order 3.

Mathieu groups
The low-degree homology groups of all Mathieu groups can be computed in HAP, and are listed in [DSE09]; that paper was the first to compute H 3 (M 24 ). In [GPRV13] it is shown that the restriction map H 4 (M 24 ) → H 4 ( 12b ) is an isomorphism, and that H 4 (M 24 ) is generated by the "gauge anomaly" of "M 24 moonshine." In [JFT18, Theorems 5.1 and 5.2] we gave direct proofs of the results H 4 (M 23 ) = 0 and H 4 (M 24 ) = 12 following the method outlined in §2, and we also recognized that the generator of H 4 (M 24 ) from [GPRV13] is more simply described as the fractional Pontryagin class of the defining degree-24 permutation representation. We remark that the same holds for M 11 : Proof. Let Perm denote the permutation representation of M 11 . The two conjugacy classes of order 8 in M 11 have the same spectrum on Perm: they act by diag (1, 1, 1, ζ The total Chern character of Perm, restricted to a cyclic group of order 8, is therefore In particular, c 2 (Perm) has order divisible by 4. But Perm is a real and therefore Spin representation, and so p 1 2 (Perm) has order divisible by 8.
The Schur cover of M 12 is studied in [CdLW16]; they compute H 4 (2M 12 ) = 8 2 × 3 with HAP, and show that the map H 4 (2M 12 ) → g∈2M 12 H 4 ( g ) has kernel of order 2. To fully describe H 4 (2M 12 ) requires moving slightly beyond cyclic groups, and also requires some notation. Let Perm denote (a choice of either) degree-12 permutation representation of M 12 , and write V 12 for the unique 12-dimensional faithful irrep of 2M 12 . Then V 12 is real, and hence Spin, as a 2M 12 -module (since 2M 12 is has no central extensions). Perm ⊗ R is not Spin as an M 12 -module, but is automatically Spin as a 2M 12 -module. Write p 1 2 (Perm) and p 1 2 (V 12 ) for their fractional Pontryagin classes. The group 2M 12 has two conjugacy classes of elements of order 3: class 3b acts on Perm with cycle structure 3 4 . 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 1 2 2 1 8 1 ; classes 8c and 8d differ by the central element and act with cycle structure 4 1 8 1 . Finally, there is a unique conjugacy class of quaternion subgroups Q 8 ⊂ 2M 12 in which the center of Q 8 maps to the center of 2M 12 .
We remark that the outer automorphism of 2M 12 switches the two degree-12 permutation representations and also switches 8ab with 8cd.
Proof. We choose the following generators of H 4 (Q 8 ) and H 4 ( 8a ) ∼ = H 4 ( 8c ) ∼ = H 4 (C 8 ) and H 4 ( 3b ) ∼ = H 4 (C 3 ): the generator of H 4 (Q 8 ) 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 H 4 (C n ) which is a cup square (it is unique by what Conway and Norton call "the defining property of 24").
The covers of M 22 are not directly computable by HAP, since they do not have sufficiently small permutation representations. The smallest permutation representations of Janko's second group J 2 (also called the Hall-Janko group HJ) and of its double cover 2J 2 have degrees 100 and 200 respectively, and HAP computes: Proof. Given the traces where ζ is a primitive 5th root of unity, it is easy to compute that c 2 (V ) has nontrivial restriction to the cyclic subgroups 2a , 3b , and 5a . The latter two restrictions immediately give that c 2 (V ) has order divisible by 15. Furthermore, by inspecting the proof of Lemma 2.4, we see that any class in H 4 (2J 2 ) with nontrivial restriction to the center must have order divisible by 8.
6.2. Conway groups. In [JFT18, Theorems 0.1 and 5.3] we showed that H 4 (Co 1 ) ∼ = Z/12, H 4 (2.Co 1 ) ∼ = Z/24, and that these groups are generated by the fractional Pontryagin classes of the 276-and 24dimensional representations, respectively. Let us denote the 24-dimensional real representation of 2.Co 1 by the name Leech. The second and third Conway groups Co 2 and Co 3 are subgroups of 2.Co 1 , and so Leech restricts to representations of each (where it splits as a trivial representation plus a 23-dimensional irrep).
Theorem 6.2. H 4 (Co 2 ) ∼ = Z/4 is generated by the restriction of p 1 2 (Leech). Proof. In [JFT18, Theorem 7.1] we gave a formula for p 1 2 (Leech)| g for all elements g ∈ 2.Co 1 in terms of the Frame shape of g in the Leech representation. The conjugacy class 4g ∈ Co 2 has Frame shape 4 6 , and so p 1 2 (Leech) restricts with order 4 to this conjugacy class. This gives the lower bound H 4 (Co 2 ) ≥ 4.
For the upper bound, Lemma 2.2 handles the primes ≥ 7. For the primes 3 and 5, we note that Co 2 contains a subgroup isomorphic to McL, which in turn contains the 3-and 5-Sylows. Since H 4 (McL) = 0 (see §6.3), we learn that H 4 (Co 2 ) (p) = 0 for p odd.
It remains to give an upper bound for the 2-part of H 4 (Co 2 ). The 2-Sylow in Co 2 is contained in a subgroup isomorphic to 2 10 : (M 22 : 2), where E = 2 10 is an irreducible (M 22 : 2)-module over F 2 . The subgroup E contains elements with Frame shape 2 12 . By [JFT18, Theorem 7.1], p 1 2 (Leech) restricts nontrivially to such an element, and so p 1 2 (Leech)| E ∈ H 0 (M 22 : 2; H 4 (E)) is nonzero. There are two irreducible 10-dimensional (M 22 : 2)-modules over F 2 , which we will call V a and V b , where the letters "a" and "b" match the notation in the ATLAS. They enjoy: Choose also an order-2 liftg of g in E : M 22 : 2, and letα ∈ H 4 (E : M 22 : 2) denote the pullback of α. Theñ α| g = α| g = 0. But Co 2 has only three conjugacy classes of order 2, distinguished by their traces on Leech, and all three classes meet E. Sinceα| E = 0, we find thatα takes different values on conjugate-in-Co 2 elements, and so cannot be the restriction of a class in H 4 (Co 2 ). This completes the proof that H 4 (Co 2 ) ∼ = Z/4. Theorem 6.3. H 4 (Co 3 ) ∼ = Z/6 is generated by the restriction of p 1 2 (Leech). Proof. The conjugacy class 6e ∈ Co 3 has Frame shape 6 4 in the Leech representation. It follows from [JFT18, Theorem 7.1] that p 1 2 (Leech)| 6e has order 6, giving the claimed lower bound H 4 (Co 3 ) ≥ 6. Lemma 2.2 handles the primes ≥ 7, and Co 3 contains a copy of McL, which contains the 5-Sylow.
The 3-Sylow in Co 3 is contained in a subgroup of shape 3 11 : (2× M 11 ). There are two irreducible 11-dimensional representations of M 11 over F 3 , dual to each other. They lead to LHS spectral sequences with E 2 pages Only the latter of these is consistent with the lower bound H 4 (Co 3 ) ≥ 3, and provides the desired upper bound.
Only the prime 3 is left. The 3-Sylow in 3McL is contained in two maximal subgroups, one of shape 3 5 .M 10 and the other of shape 3 2+4 : 2S 5 . The latter is more useful, and for the remainder of the proof we will call it S. The quotient 2S 5 is the one listed in the ATLAS under the names "2S 5 i ′′ and "Isoclinic(2.A 5 .2)"; it is the group of shape 2S 5 that contains elements of order 12. This 2S 5 has a faithful 4-dimensional representation over F 3 . The quotient of S in McL has shape 3 1+4 : 2S 5 , and the "central" 3 is not central, but rather transforms by the sign representation of 2S 5 . In terms of the 4-dimensional module, it corresponds to a symplectic form on 3 4 which is 2A 5but not 2S 5 -fixed. There is also a symplectic form which is 2S 5 -fixed, and the 3 2+4 subgroup of S extends 3 4 by both symplectic forms simultaneously.
After a multi-hour computation, HAP reports from which we learn that On the other hand, and since the extension S = 3 2+4 : 2S 5 splits, H 3 (S) (3) contains H 3 (2S 5 ) (3) as a direct summand. Passing to cohomology, we learn that the pullback map is an isomorphism. There is a unique conjugacy class of order 3 in 2S 5 , and the restriction map H 4 (2S 5 ) (3) → H 4 ( 3a ) is an isomorphism. Take any element g ∈ 2S 5 of order 3 and lift it to an order-3 elementg ∈ S. Then the composition H 4 (2S 5 ) (3) ∼ → H 4 (S) (3) has trivial intersection with the restriction map H 4 (3McL) (3) ֒→ H 4 (S) (3) , and so H 4 (3McL) (3) = 0. 6.4. Suzuki group. The Schur cover of the Suzuki group is the beginning of a famous sequence of subgroups of 2Co 1 centralizing actions of binary alternating groups on the Leech lattice; 6Suz centralizes an action of 2A 3 ∼ = 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 .
Unlike the case of the Conway groups, the 2-Sylow subgroup of Suz is not contained in an extension of shape (elementary abelian).(smaller). It is, however, contained in an extension of shape (extraspecial).(smaller); specifically, of shape 2 1+6 · SWeyl(E 6 ), where by "SWeyl" we mean the index-2 subgroup of the Weyl group consisting of even reflections. The extension does not split. Two other groups with this feature are: Conway's group Co 1 , whose 2-Sylow lives in a nonsplit extension of shape 2 1+8 · SWeyl(E 8 ); and the Monster group M, whose 2-Sylow lives in 2 1+24 · Co 1 . From the point of view of this paper, the difference between Co 1 and the other two is that the former also contains a 2-Sylow-containing subgroup of shape (elementary abelian) : (simple), making it comparatively computable.
Our goal in this section is to prove: Theorem 6.5. The Suzuki group and its Schur covers have the following fourth cohomologies: We will split the proof into a series of lemmas.
Proof. Since the underlying real representation of V is Leech⊗R, we have c 2 (V ) = − p 1 2 (Leech)| 6Suz . Then [JFT18, Theorem 0.1] gives an upper bound of 24 on the order of c 2 (V ).
The action of 6Suz on the Leech lattice includes elements with Frame shape 3 8 ; according to [JFT18, Theorem 7.1], p 1 2 (Leech) has nontrivial restriction to such elements. This gives a lower bound of 3 on the order of c 2 (V ).
6Suz contains a maximal subgroup of shape 6A 7 . As observed in [JFT18, Lemma 4.1], there is a unique conjugacy class of subgroups D 8 ⊂ A 6 , where D 8 denotes the dihedral group of order 8. Along the standard inclusion A 6 ⊂ A 7 , the 6-fold cover pulls back to the cover 3× 2D 8 of D 8 , where 2D 8 denotes the binary dihedral group of order 16. This group 2D 8 is the one used in [JFT18, Lemma 4.1], where it is shown that p 1 2 (Leech)| 2D 8 has order 8. This gives a lower bound of 8 on the order of c 2 (V ).
Proof. The 3-Sylow in Suz is contained in a maximal subgroup of shape 3 5 : M 11 . There are two nontrivial 5-dimensional M 11 -modules over F 3 . They lead to LHS spectral sequences with E 2 pages: The former is incompatible with H 3 (Suz) (3) = 3, and the latter immediate gives H 4 (Suz) (3) = 0.
As mentioned above, the 2-Sylow in Suz is contained in an extension of shape 2 1+6 .SWeyl(E 6 ), where by SWeyl(E 6 ) we mean the even subgroup of the Weyl group. The ATLAS calls this group SWeyl(E 6 ) = U 4 (2). We will write it as a Weyl group to make the action on 2 6 transparent: it is the mod-2 reduction of the action of Weyl(E 6 ) on the E 6 -lattice. In particular, the quadratic form on 2 6 defining the extension 2 1+6 has Arf invariant −1.
Proof. Let us write J for SWeyl(E 6 ) and E for the 6-dimensional SWeyl(E 6 )-module over F 2 . In §3.3 we identified H 4 (2.E) as (E * . Alt 2 (E * ). Alt 3 (E * )/E * ).2. This group has a unique nonzero element which is divisible by 2; it lives in the subgroup X = E * . Alt 2 (E * ). Alt 3 (E * )/E * , and so H 0 (J; X) ≥ Z/2. On the other hand, We were unable to determine the exact value of H 0 (SWeyl(E 6 ); H 4 (2 1+6 )). We remark that the order-2 class therein has many descriptions. It arises as c 2 of the unique 2 3 -dimensional complex irrep of 2 1+6 . It also arises as follows. The nonsplit extension 2 6 ·SWeyl(E 6 ) is naturally a subgroup of the compact Lie group of type E 6 (adjoint form); in this realization, 2 6 is the group of 2-torsion points in the maximal torus. The generator of H 4 (BE 6 ) restricts to 2 6 to the E 6 quadratic form living in Sym 2 (2 6 ) ⊂ H 4 (2 6 ), and this form pulls back to 2 1+6 to the SWeyl(E 6 )-fixed order-2 class.
We may now compute the E 2 page of the LHS spectral sequence for the extension 2 1+6 .SWeyl(E 6 ) using HAP and Cohomolo: 2 or 4 Let 2SWeyl(E 6 ) denote the Spin double cover of SWeyl(E 6 ) ⊂ SO(6, R). From the above spectral sequence, we learn: Lemma 6.9. The 2-Sylow in 2Suz is contained in a maximal subgroup of shape 2 1+6 .2SWeyl(E 6 ).
There are two conjugacy classes of elements of order 2 in SWeyl(E 6 ). They can be distinguished by the orders of their preimages in 2SWeyl(E 6 ): elements in class 2a lift with order 2 (and both lifts are conjugate in 2SWeyl(E 6 )), 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 ′ ֒→ 2 1+6 .2SWeyl(E 6 ) → SWeyl(E 6 ) is injective.
To find an appropriate 2A 4 ⊂ 2 6 .SWeyl(E 6 ), we recognize that where the group 2 6 is nothing but the 2-torsion points in the maximal torus. Consider the standard Lie group embedding F 4 ⊂ E 6 . This leads to an embedding covering an embedding SWeyl(F 4 ) ⊂ SWeyl(E 6 ). Because the Lie group F 4 has no outer automorphisms, the Weyl group of F 4 contains all automorphisms of the F 4 root lattice (isomorphic to the D 4 lattice). There is a standard identification between the F 4 lattice and the Hurwitz quaternions The group of units in the Hurwitz numbers is a copy of 2A 4 . This provides a subgroup 2A 4 ⊂ SWeyl(F 4 ) ⊂ SWeyl(E 6 ), which is easily seen to be appropriate: central 2 ⊂ 2A 4 acts by −1 on the F 4 lattice, and so with trace −2 on the E 6 lattice, and so fuses to class 2a ∈ SWeyl(E 6 ).
Lemma 6.11. The pullbacks have trivial intersection.
Proof. Let Q ′ ⊂ 2 1+6 .2SWeyl(E 6 ) denote the quaternion group found in Lemma 6.10, and let Q ⊂ 2SWeyl(E 6 ) denote it image. Then Q ∼ = Q 8 is another quaternion group since the center of Q ′ is not in the kernel of the map Q ′ → Q. We claim that H 4 (2SWeyl(E 6 )) → H 4 (Q) is an isomorphism. Indeed, consider either 4-dimensional faithful complex representations of 2SWeyl(E 6 ). 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 c 2 (4-dim rep)| Q has order 8. We furthermore learn that c 2 (4-dim rep) generates H 4 (2SWeyl(E 6 )).
The central element of Q ′ is an order-2 lift of class 2a ∈ SWeyl(E 6 ). Any such lift fuses to class 2a ∈ Suz. But 2 1+6 .SWeyl(E 6 ) is the centralizer of an element of class 2a ∈ Suz. It follows that Q ′ is conjugate in 2Suz to some other quaternion group Q ′′ ⊂ 2 1+6 .2SWeyl(E 6 ) whose central element is central covers the center of 2SWeyl(E 6 ).
Since Q ′ is a lift of Q, we find thatα| Q ′ is a generator of H 4 (Q ′ ), and so 4α| Q ′ = 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 representations or two copies of the 2-dimensional representation, and in either case we find thatα| Q ′′ = c 2 (4-dim rep)| Q ′′ has order at most 4, so that 4α| Q ′′ = 0. Since Q ′ and Q ′′ are conjugate in 2Suz, the class 4α cannot be the restriction of a class in H 4 (2Suz).
In particular, if H 4 (J 4 ) = 0, then it contains an order-2 class α whose restriction to 2 1+12 is the unique elementq ∈ H 4 (2 1+12 ) which is twice some other element. That unique element is pulled back from H 4 (2 12 ), where it corresponds to the quadratic form q ∈ Sym 2 (2 12 ) defining the extension 2 1+12 . Choose any pair of vectors v 1 , v 2 ∈ 2 12 such that q(v 1 ) = q(v 2 ) = 0. Their lifts generate a quaternion group Q 8 = 2 1+2 ⊂ 2 1+12 , andq ∈ H 4 (2 1+12 ) restricts nontrivially to that quaternion group. (These lifts of v 1 , v 2 have order 4 in 2 1+12 , and so are in conjugacy class 4a in 2 1+12 .3M 22 .2.) Chooseg ∈ 2 1+12 .3M 22 .2 in conjugacy class 4b. The character table libraries confirm that this g has the following properties:g 2 is the nontrivial central element in 2 1+12 ⊂ 2 1+12 .3M 12 .2; the image g in 3M 22 .2 ofg is in conjugacy class 2a. In particular, g acts on 2 12 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 v 1 ∈ 2 12 with q(v) = 0 and vg = v. (Following GAP's conventions, we write the action of 3M 22 .2 on 2 12 from the right.) Set v 2 = vg; then q(v 2 ) = 0, and so the lifts of v 1 , v 2 generate a Q 8 as in the previous paragraph.
The normalizer of a 7-Sylow in He has shape 7 1+2 : (3 × S 3 ). There are no 7s in its low cohomology: H 4 (7 1+2 : The 3-Sylow in He is inside a maximal subgroup of shape 3S 7 , with H 4 (3S 7 ) = 2 2 × 4 × 3 2 . We claim that the inclusions H 4 (He) (3) → H 4 (3S 7 ) (3) and H 4 (S 7 ) (3) = 3 → H 4 (3S 7 ) (3) have nonintersecting images, giving an upper bound H 4 (He) (3) ≤ 3. To show this, we first observe that if C 3 is a cyclic group of order 3 then the two nonzero classes in H 4 (C 3 ) ∼ = Z/3 are canonically distinguished: one, which we will call t 2 ∈ H 4 (C 3 ), is the cup-square of both nonzero classes in H 2 (C 3 ) ∼ = Z/3, and the other is not a cup-square. Now consider the class c 2 (Perm) ∈ H 4 (S 7 ), where Perm denotes the defining permutation representation. There are three conjugacy classes of order 3 in 3S 7 : the "central" one (not actually central -it is inverted by the odd elements of S 7 ), and two that act on Perm with cycle structures 1 4 3 1 and 1 1 3 2 , respectively. It follows that c 2 (Perm)| 1 4 3 1 = −t 2 whereas c 2 (Perm)| 1 1 3 2 = +t 2 , meaning that c 2 (Perm) takes different values on these two classes. However, these two classes merge to the same class in He, and so c 2 (Perm) cannot be the restriction of a cohomology class on He.
To establish the lower bound H 4 (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 c 2 of this irrep, when restricted to 3b , does not vanish.
We have so far established the following upper bound on H 4 (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 χ 19 : it is real and of dimension 7650. Consider the conjugacy class 4a ∈ He. It squares to 2a, and In particular, c 2 (V )| 4a = 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 c 2 (V ). It follows that p 1 2 (V ) has order divisible by 4, and so H 4 (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.  (HN) (2) has order at most 16 and exponent at most 8.
All that remain are the lower bounds. Let ω ♮ denote the "gauge anomaly" of the Monster CFT, studied in [JF17]; c.f. §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 2Fi 22 ⊂ Fi 23 ⊂ 3Fi ′ 24 , and so ω ♮ restricts with order at least 2 to all of these groups.   [JF17] is that H 4 (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 H 4 (M). The calculations in this paper allow us to come close to proving that conjecture: Theorem 8.6. H 4 (M) ∼ = Z/24 ⊕ X, where the Z/24 summand is generated by ω ♮ and where the order of X divides 4.