Nikolaus 2004 The 2 modular characters of the Conway group Co1
 Title: The 2 modular characters of the Conway group ${\mathrm{Co}}_{1}$ Author: Jon Thackray Version: 1 Date: 20041209 Status: Draft

# Contents

## Section 1 : Purpose of the talk

To describe present a set of results giving the 2 modular character table of Conway's first group, suitable for inclusion the the forthcoming Atlas of Brauer Characters Part 2.

## Section 2 : Background

Conway's first group is described in [Conway]. Throughout this paper, it will be denoted Co1. The character table is given in [Conway et al] The representations used as a starting point are taken from the Birmingham Atlas of Finite Group Representations [Wilson].

At the point of starting on the problem, some small representations of Co1 were already known, as well a block of defect three containing three irreducibles. These are considered known, and will not be described in this talk. The talk will address only the principal block, in which we are seeking twenty six irreducibles.

At this stage the results are incomplete. The determination of the composition factors of one tensor condensation is ongoing. There are seven other results which are condensation only. There are a further three results which combine condensation and character theory to yield an existence proof (the character theory implies a decomposition which the condensation proves to be the most that can occur). The remaining fifteen irreducibles have all been constructed and their indicators computed, and for seven of these of indicator plus the orthogonal group has also been determined. Two further indicators are known, as two of the three ireducibles proved by condensation constitute a complex conjugate pair.

It is hoped that the final condensation conjecture will be known by around the middle of 2005.

## Section 3 : Demonstration of results

Our first character ϕ1 is the trivial character 1. Co1 has a double cover 2.Co1 with a 24 dimensional real representation, which reduces modulo 2 to a 2 modular irreducible of the simple group. This is our second character ϕ2 Since this representation is real, its skew square 276 has a trivial submodule, and hence a composition factor 274, which can be seen to be irreducible by restriction to Co2 and to 3.Suz.2. This gives ϕ3

ϕ2ϕ3 decomposes as 2000 ⊕ 4576. This can be demonstrated theoretically using symmetrised power decomposition. But the meataxe proves the result anyway. This gives two new characters ϕ5 and ϕ6 The skew fourth of ϕ2 has composition factors 2.ϕ1 + 2.ϕ3 + 1496 + 8580. We thus have two further characters ϕ4 and ϕ7 ϕ2ϕ4 has composition factors 17952 and 17952 ¯¯¯¯¯ , giving ϕ8 and ϕ9. These characters have complex values on elements of orders 23 and 39, which can be computed by taking ranks of suitable sums of element powers.

The skew square of ϕ3 has composition factors 3.ϕ1 + ϕ2 + 2. ϕ3 + ϕ4 + ϕ7 + 26750. 26750 is ϕ10. The skew sixth of ϕ2 has composition factors 8.ϕ1 + 2.ϕ2 + 8.ϕ3 + 3.ϕ4 + 2.ϕ7 + 2.ϕ10 + 57200. 57200 is ϕ11.

ϕ2ϕ7 has composition factors ϕ6 + ϕ8 + ϕ9 + 165440. 165440 is ϕ12.

Analysis of the condensed permutation representation of Co2 on 3.Suz.2 yields two new characters ϕ13 and ϕ16. Uncondensing reveals the degree of ϕ13 to be 218800. ϕ16 can then be discovered to have degree 420256. This representation has not been constructed.

Using condensed tensor analysis, we find ϕ3ϕ4 has composition factors 2.ϕ1 + 2.ϕ2 + 4.ϕ3 + 2.ϕ7 + ϕ10 + 2.ϕ11 + 250448. 250448 is ϕ14. Also ϕ3ϕ5 has composition factors 2.ϕ2 + 4.ϕ6 + 3.ϕ8 + 3.ϕ9 + ϕ12 + 256496. 256496 is ϕ15. ϕ14 and ϕ15 are the last characters to have been constructed.

Using condensed tensor analysis, we find ϕ2ϕ11 has composition factors ϕ17 + ϕ18 + ϕ19. Character theory shows that at least this much decomposition occurs, hence each of thes is irreducible. Further, again from character theory, ϕ17 and ϕ18 are a complex conjugate pair. We obtain an irreducible subspace of the condensed tensor product, and analyse it using condensed generators of Co3 to determine the degree of ϕ17, and hence of the other two. These are the last characters currently known to exist. The remainder of these results are from tensor condensation only. They therefore assume that the condensation algebra is equal to the full Hecke algebra.

Decomposing the condensed tensor product ϕ3ϕ10 we obtain 26.ϕ1 + 14.ϕ2 + 28.ϕ3 + 6.ϕ4 + 4.ϕ5 + 4.ϕ6 + 6.ϕ7 + 3.ϕ8 + 3.ϕ9 + 7.ϕ10 + 8.ϕ11 + ϕ12 + 4.ϕ13 + 2.ϕ14 + 2.ϕ16 + ϕ20. From this we conjecture an irreducible ϕ20 of degree 4100096.

Decomposing the condensed tensor product ϕ4ϕ7 we obtain 28.ϕ1 + 12.ϕ2 + 26.ϕ3 + 6.ϕ4 + 3.ϕ5 + 2.ϕ6 + 6.ϕ7 + ϕ8 + ϕ9 + 7.ϕ10 + 6.ϕ11 + 4.ϕ13 + ϕ14 + 2.ϕ16 + ϕ20 + ϕ21. From this we conjecture an irreducible ϕ21 of degree 6120022.

Decomposing the condensed tensor product ϕ3ϕ11 we obtain 16.ϕ1 + 6.ϕ2 + 12.ϕ3 + 2.ϕ4 + 2.ϕ5 + 2.ϕ6 + 2.ϕ7 + 2.ϕ8 + 2.ϕ9 + 2.ϕ10 + 4.ϕ11 + 4.ϕ13 + 3.ϕ16 + ϕ21 + ϕ22. From this we conjecture an irreducible ϕ22 of degree 7025950.

Decomposing the condensed tensor product ϕ5ϕ7 we obtain 38.ϕ1 + 34.ϕ2 + 40.ϕ3 + 14.ϕ4 + 16.ϕ5 + 30.ϕ6 + 14.ϕ7 + 15.ϕ8 + 15.ϕ9 + 7.ϕ10 + 5.ϕ12 + 2.ϕ14 + 8.ϕ15 + 5.ϕ17 + 5.ϕ18 + 7.ϕ19 + ϕ25. From this we conjecture an irreducible ϕ25 of degree 4634432.

Decomposing the condensed tensor product ϕ4ϕ8 we obtain 44.ϕ1 + 36.ϕ2 + 50.ϕ3 + 16.ϕ4 + 13.ϕ5 + 24.ϕ6 + 18.ϕ7 + 14.ϕ8 + 13.ϕ9 + 11.ϕ10 + 4.ϕ11 + 2.ϕ12 + 2.ϕ14 + 6.ϕ15 + 3.ϕ17 + 3.ϕ18 + 4.ϕ19 + ϕ23 + 2.ϕ25. From this we conjecture an irreducible .

Decomposing the condensed tensor product ϕ4ϕ8 we obtain 44.ϕ1 + 36.ϕ2 + 50.ϕ3 + 16.ϕ4 + 13.ϕ5 + 24.ϕ6 + 18.ϕ7 + 14.ϕ8 + 13.ϕ9 + 11.ϕ10 + 4.ϕ11 + 2.ϕ12 + 2.ϕ14 + 6.ϕ15 + 3.ϕ17 + 3.ϕ18 + 4.ϕ19 + ϕ23 of degree 9144846.

Decomposing the condensed tensor product ϕ4ϕ11 we obtain 144.ϕ1 + 72.ϕ2 + 142.ϕ3 + 32.ϕ4 + 22.ϕ5 + 30.ϕ6 + 38.ϕ7 + 20.ϕ8 + 20.ϕ9 + 32.ϕ10 + 29.ϕ11 + 6.ϕ12 + 16.ϕ13 + 6.ϕ14 + 6.ϕ15 + 8.ϕ16 + 2.ϕ17 + 2.ϕ18 + 3.ϕ19 + 2.ϕ21 + 2.ϕ24. From this we conjecture an irreducible ϕ24 of degree 27621392.

The condensed tensor product ϕ5ϕ11 has not been complete created yet. The decomposition so far, consistent with character theory and tensor products is 72.ϕ1 + 49.ϕ2 + 82.ϕ3 + 24.ϕ4 + 17.ϕ5 + 27.ϕ6 + 30.ϕ7 + 15.ϕ8 + 15.ϕ9 + 19.ϕ10 + 8.ϕ11 + 5.ϕ12 + 4.ϕ14 + 4.ϕ15 + 4.ϕ17 + 4.ϕ18 + 7.ϕ19 + 4.ϕ23 + 3.ϕ25 + ϕ26 From this we conjecture an irreducible ϕ26 of degree ≤ 51741858.

## Section 4 : Some further thoughts

The tensor condensations in the above section have resulted in some very large matrices, which take a long time to produce and are hard to reduce. For example, the condensed tensor product ϕ5ϕ11 has degree 470296 and size around 27.5Gb. Each matrix takes around 90 days to produce on a P4/3400, and a correspondingly long time to split. I wish to examine some possibilities for reducing the size of the matrices, and hence as a consequence the split time.

The degree of the condensed matrices is roughly equal to the uncondensed degree divided by the condensation subgroup order. So why don't we just pick a larger condensation subgroup? The answer is that the larger subgroups (of McL at least) are unfaithful, ie there is an irreducible representation which condeses to zero. In this case the culprit is 1496.

But, if we are merely interested in the multiplicities, and we had an alternative means to determine the missing multiplicities, we could then make use of a larger condensation subgroup. For example, the full Sylow 3 subgroup of McL is faithful on every representation except 1496, and would lead to condesned matrices of degree around 160000.

Condensation relies on looking at the subspace fixed by the condensation subgroup, under the action of a suitably smaller algebra. If M is an irreducible of the group algebra kG, and e the central idempotent formed from the sum over the elements of the condensation subgroup H, then our condensation space is Me acted on by ekGe. But Me is just the subspace in the direct sum decomposition of M as a kH module all of whose summands are isomorphic to the trivial representation.

So, what would happen if we chose a different central idempotent f? The first obvious consequence is that we no longer have control of the multiplicity of the trivial representation. But, if we also condensed using the original e, we could regain this information. In terms of our original example, we would gain a factor of nine, only to lose a factor of two on the matrix construction. But the reduction process is cubic in the degree, so here we would gain twenty seven divided by two.

However, if we choose an idempotent associated with a non-linear character, we could well lose any possible advantage due to higher multiplicity of the irreducible direct summand, coupled with large degree. So, initially I'd prefer to consider only non-trivial linear characters. A further complication is that condensing over GF(2), the trivial character is the only linear character. So, we might also have to consider moving to GF(4) (which for the Sylow 3 subgroup would give a number of non-trivial linear characters). We'd still get a factor of about two (or possibly better, since only the non-trivial character condensation needs to be done in the field extension) on overall space requirements, and a factor around six on reduction speed. With larger odd order subgroups greater savings might be made.

The next question that needs to be answered is how do we make these non-trivial character condensations? I think that for permutation condensations, we need to compute the relevant idempotent (I believe there are formulae from Schur for these). For tensor condensation, I think we need a permutation of the symmetry basis for the right hand component. Essentially, instead of taking the basis in the order of the duals of the irreducbles of the condensation subgroup, I believe we now need dual ⊗ λ, where λ is our non-trivial linear character. Note that in this case, the tensor product is irreducible (another advantage of sticking to linear characters).

Food for thought!

## Section 5 : Status of results

 Degree Proved Constructed Indicator Group sign Conjectured Incomplete ϕ1 1 ✓ ✓ + ϕ2 24 ✓ ✓ + + ϕ3 274 ✓ ✓ + - ϕ4 1496 ✓ ✓ + + ϕ5 2000 ✓ ✓ + + ϕ6 4576 ✓ ✓ + + ϕ7 8580 ✓ ✓ + - ϕ8 17952 ✓ ✓ o ϕ9 17952 ✓ ✓ o ϕ10 26750 ✓ ✓ + + ϕ11 57200 ✓ ✓ + ϕ12 165440 ✓ ✓ + ϕ13 218800 ✓ ✓ + ϕ14 250448 ✓ ✓ + ϕ15 256496 ✓ ✓ + ϕ16 420256 ✓ ϕ17 378016 ✓ o ϕ18 378016 ✓ o ϕ19 616768 ✓ ϕ20 4100096 ✓ ϕ21 6120022 ✓ ϕ22 7025950 ✓ ϕ23 9144846 ✓ ϕ24 27621392 ✓ ϕ25 4634432 ✓ ϕ26 ≤ 51741858 ✓ ϕ27 40370176 ✓ + ϕ28 1507328000 ✓ + ϕ29 313524224 ✓ +

## Section 6 : The (partial) decomposition matrix for the principal block

 ϕ1 ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 ϕ7 ϕ8 ϕ9 ϕ10 ϕ11 ϕ12 ϕ13 ϕ14 ϕ15 ϕ16 ϕ17 ϕ18 ϕ19 ϕ20 ϕ21 ϕ22 ϕ23 ϕ24 ϕ25 ϕ26 ϕ27 ϕ28 ϕ29 χ1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ2 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ4 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ5 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ6 2 2 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ7 2 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ8 2 1 3 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ9 3 2 3 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ10 6 2 5 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ11 3 1 3 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ12 4 1 4 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ13 1 2 3 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ14 12 5 11 3 2 2 2 1 1 3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ15 6 4 7 2 1 1 2 1 1 2 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ16 12 4 11 2 1 1 2 1 1 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ17 4 1 4 0 0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 χ18 4 1 4 0 0 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 χ19 6 3 6 2 1 0 1 0 0 2 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 χ20 8 4 8 2 1 1 2 1 1 2 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 χ21 10 5 11 2 1 1 3 1 1 3 3 0 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 χ22 19 7 18 4 2 2 3 2 2 5 3 1 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 χ23 16 8 17 4 2 3 4 2 2 5 4 1 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 χ24 20 14 24 7 4 8 8 5 5 6 3 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 χ25 11 9 12 5 4 3 3 1 1 3 3 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 χ26 17 10 17 5 4 4 5 2 2 4 4 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 χ27 11 7 11 3 3 4 4 2 3 2 2 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 χ28 11 7 11 3 3 4 4 3 2 2 2 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 χ29 24 14 27 7 4 7 8 5 5 7 5 1 2 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 χ30 25 16 28 8 5 7 8 5 5 7 6 1 2 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 χ31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 χ32 39 24 44 12 7 12 13 8 8 11 7 2 3 3 3 1 1 1 1 0 0 0 0 0 0 0 0 0 0 χ33 36 20 37 10 7 10 10 6 6 9 6 2 3 2 2 2 1 1 2 0 0 0 0 0 0 0 0 0 0 χ34 15 7 13 3 2 2 4 1 1 3 4 0 2 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 χ35 24 10 21 5 3 4 5 2 2 5 5 1 3 1 0 2 0 0 1 0 1 0 0 0 0 0 0 0 0 χ36 28 12 28 6 3 4 6 3 3 7 6 1 3 1 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 χ37 36 20 35 9 7 10 11 6 6 8 7 1 4 2 2 3 1 1 2 0 1 0 0 0 0 0 0 0 0 χ38 67 43 73 22 15 25 22 12 12 18 10 6 3 5 5 2 3 3 5 1 0 0 0 0 0 0 0 0 0 χ39 94 55 99 27 19 31 28 17 17 24 13 7 6 6 6 3 4 4 6 1 0 0 0 0 0 0 0 0 0 χ40 63 31 63 15 9 13 16 9 9 15 13 2 7 3 2 4 1 1 1 0 1 1 0 0 0 0 0 0 0 χ41 62 32 62 16 10 14 16 8 8 15 12 3 6 3 2 4 1 1 2 0 1 1 0 0 0 0 0 0 0 χ42 55 29 54 15 9 12 15 6 6 13 9 3 3 2 2 2 1 1 2 0 1 0 1 0 0 0 0 0 0 χ43 61 30 59 15 9 15 17 8 8 14 8 3 4 2 2 2 2 2 3 0 1 0 1 0 0 0 0 0 0 χ44 74 44 75 22 16 24 21 11 11 18 12 6 5 5 4 4 3 3 6 1 1 0 0 0 0 0 0 0 0 χ45 68 36 68 18 11 17 19 10 10 16 10 3 5 3 3 3 2 2 3 0 1 0 1 0 0 0 0 0 0 χ46 78 44 79 21 15 26 23 13 13 19 11 6 6 5 4 3 4 4 6 1 1 0 0 0 0 0 0 0 0 χ47 87 52 90 25 18 30 27 16 16 21 13 6 6 6 6 3 4 4 6 1 1 0 0 0 0 0 0 0 0 χ48 36 20 38 9 6 6 10 5 5 8 9 1 4 2 2 2 0 0 0 0 0 0 0 1 0 0 0 0 0 χ49 48 24 47 10 7 10 14 7 7 10 10 1 6 2 2 3 1 1 1 0 1 0 0 1 0 0 0 0 0 χ50 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 χ51 127 69 127 34 23 36 35 19 19 30 22 8 11 7 6 7 4 4 7 1 2 1 0 0 0 0 0 0 0 χ52 57 34 62 15 10 15 18 10 10 14 13 2 5 3 3 3 1 1 1 0 0 1 0 1 1 0 0 0 0 χ53 168 94 172 46 30 49 50 27 27 41 26 10 12 10 9 6 6 6 9 1 2 0 1 0 0 0 0 0 0 χ54 170 110 186 53 36 58 56 34 34 45 33 9 12 13 12 7 6 6 9 1 1 1 0 0 2 0 0 0 0 χ55 90 48 92 21 14 21 25 15 15 21 19 3 10 4 4 5 2 2 2 0 1 1 0 1 1 0 0 0 0 χ56 111 61 117 27 17 29 34 21 21 27 22 3 12 6 6 5 3 3 2 0 1 1 0 1 1 0 0 0 0 χ57 185 115 199 56 37 62 59 35 35 48 31 11 11 12 12 6 7 7 10 1 1 1 1 0 2 0 0 0 0 χ58 192 121 208 59 39 64 63 37 37 50 34 10 12 13 13 7 7 7 10 1 1 1 1 0 2 0 0 0 0 χ59 221 135 234 65 44 72 70 41 41 56 38 12 15 14 14 9 8 8 12 1 2 1 1 0 2 0 0 0 0 χ60 210 124 220 60 39 66 66 38 38 52 34 10 14 12 12 8 8 8 11 1 2 1 2 0 2 0 0 0 0 χ61 140 80 148 37 23 38 42 25 25 34 27 5 12 7 7 6 4 4 4 0 1 2 1 1 2 0 0 0 0 χ62 226 132 235 64 41 69 70 39 39 56 38 11 16 13 12 9 8 8 11 1 3 1 2 0 2 0 0 0 0 χ63 215 125 226 58 39 64 66 39 39 53 41 10 19 13 12 10 7 7 9 1 2 2 0 1 2 0 0 0 0 χ64 232 132 242 63 40 62 70 39 39 57 43 9 18 12 12 10 6 6 8 0 2 1 2 1 2 0 0 0 0 χ65 256 144 262 69 46 71 75 41 41 61 45 13 21 14 13 12 8 8 12 1 3 1 1 1 1 0 0 0 0 χ66 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 χ67 322 186 336 86 58 92 98 57 57 78 61 14 28 18 18 15 10 10 13 1 3 2 1 2 3 0 0 0 0 χ68 180 109 192 48 38 55 58 35 35 43 38 11 16 11 11 8 6 6 9 1 1 2 1 2 3 1 0 0 0 χ69 270 153 277 72 53 76 81 45 45 64 50 16 22 15 14 13 8 8 14 1 3 2 2 1 2 1 0 0 0 χ70 256 141 263 65 46 74 78 46 46 61 41 14 22 13 13 10 10 10 13 1 3 1 3 1 2 1 0 0 0 χ71 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 χ72 282 161 293 75 53 83 87 51 51 68 49 15 23 15 15 12 10 10 14 1 3 2 3 1 3 1 0 0 0 χ73 308 179 321 85 60 91 95 54 54 75 55 18 23 17 17 13 10 10 16 1 3 2 3 1 3 1 0 0 0 χ74 305 178 323 82 57 92 96 59 59 76 58 15 27 18 17 14 10 10 13 1 3 3 2 1 4 1 0 0 0 χ75 276 159 290 72 53 78 84 50 50 67 56 15 26 16 15 14 8 8 12 1 2 3 1 2 3 1 0 0 0 χ76 558 324 580 158 103 170 170 96 96 137 92 28 40 32 30 24 20 20 29 2 6 4 4 0 4 0 0 0 0 χ77 558 324 580 158 103 170 170 96 96 137 92 28 40 32 30 24 20 20 29 3 6 4 4 0 4 0 0 0 0 χ78 386 225 402 106 75 118 119 69 69 94 68 22 30 22 21 17 14 14 21 2 4 3 3 1 4 1 0 0 0 χ79 425 249 446 118 82 130 132 77 77 105 75 24 33 25 24 18 15 15 22 2 4 3 3 1 4 1 0 0 0 χ80 335 194 353 87 64 95 103 62 62 81 67 17 31 19 19 16 10 10 14 1 2 3 1 3 4 1 0 0 0 χ81 416 245 436 115 81 129 130 76 76 102 75 23 32 24 23 18 15 15 22 2 4 4 3 1 5 1 0 0 0 χ82 428 264 458 125 88 138 138 81 81 108 80 24 30 27 26 18 15 15 23 2 3 4 3 1 6 1 0 0 0 χ83 457 264 472 126 88 136 139 78 78 110 79 26 34 25 24 20 16 16 25 2 5 3 4 1 4 1 0 0 0 χ84 438 253 458 118 82 125 134 77 77 107 84 22 36 24 23 20 13 13 19 1 4 4 3 2 5 1 0 0 0 χ85 536 321 562 153 108 173 170 98 98 131 91 31 38 33 32 22 21 21 32 3 5 3 4 1 5 1 0 0 0 χ86 451 264 475 120 86 134 140 84 84 110 88 23 41 27 26 21 15 15 20 2 4 4 1 3 5 1 0 0 0 χ87 497 286 519 133 92 147 153 90 90 121 90 25 42 28 27 22 17 17 23 2 5 4 3 2 5 1 0 0 0 χ88 584 341 608 163 113 177 180 102 102 143 103 32 44 34 32 26 20 20 31 3 6 4 4 1 5 1 0 0 0 χ89 574 333 596 157 110 172 175 100 100 139 103 32 46 33 31 26 20 20 30 3 6 4 3 2 5 1 0 0 0 χ90 514 294 540 136 97 150 158 95 95 127 94 29 44 30 28 22 17 17 24 2 4 4 4 2 5 2 0 0 0 χ91 522 304 550 140 99 154 164 99 99 128 98 26 44 29 29 23 17 17 23 1 5 5 5 2 7 2 0 0 0 χ92 538 312 565 143 103 158 167 100 100 131 101 29 47 31 30 24 18 18 25 2 5 4 4 3 6 2 0 0 0 χ93 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 χ94 728 432 769 202 141 228 231 138 138 180 130 39 57 43 42 30 27 27 37 3 7 6 6 2 9 2 0 0 0 χ95 722 417 753 194 138 215 223 131 131 175 130 39 60 41 40 32 25 25 36 3 7 5 5 3 7 2 0 0 0 χ96 742 432 779 201 141 221 230 136 136 182 137 39 62 43 41 33 25 25 35 3 7 6 5 3 8 2 0 0 0 χ97 736 423 771 195 136 211 227 134 134 179 139 36 64 41 40 33 23 23 31 2 7 6 5 4 8 2 0 0 0 χ98 784 460 825 215 150 237 245 144 144 193 142 42 62 45 44 33 27 27 38 3 7 6 6 3 9 2 0 0 0 χ99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 χ100 908 532 952 252 174 277 283 165 165 224 160 49 70 53 51 38 32 32 46 4 9 6 7 2 9 2 0 0 0 χ101 900 524 945 246 168 267 279 163 163 222 164 46 72 51 49 38 30 30 41 3 9 7 7 3 10 2 0 0 0

## Section 7 : References

[Conway] J.H.Conway: A group of order 8315553613086720000. Bul LMS (1969) 1, 79.
[Conway et al] Conway, Curtis, Norton, Parker, Wilson - An atlas of finite groups
[Wilson] The Birmingham Atlas of Finite Group Representations

## Section 8 : Acknowledgements

I'd like to thank Richard Parker, Rob Wilson, Gerhard Hiss (and LDFM), Juergen Mueller, Frank Lubeck and Klaus Lux for putting me up to all this sort of stuff.