Contents

Section 1 : Purpose of the talk

Section 2 : Background

Section 3 : Demonstration of results

Section 4 : Some further thoughts

Section 5 : Status of results

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

Section 7 : References

Section 8 : Acknowledgements

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 in 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.

The results are as complete as they can be subject to condensation. There are eight 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.

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.

ϕ₂ ⊗ ϕ₃ 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 ϕ₂ ⊗ ϕ₄ 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.

ϕ₂ ⊗ ϕ₇ 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.

Using condensed tensor analysis, we find ϕ₃ ⊗ ϕ₄ has composition factors 2.ϕ₁ + 2.ϕ₂ + 4.ϕ₃ + 2.ϕ₇ + ϕ₁₀ + 2.ϕ₁₁ + 250448. 250448 is ϕ₁₄. Also ϕ₃ ⊗ ϕ₅ 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 ϕ₂ ⊗ ϕ₁₁ has composition factors ϕ₁₇ + ϕ₁₈ + ϕ₁₉. Character theory shows that at least this much decomposition occurs, hence each of these 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 ϕ₃ ⊗ ϕ₁₀ 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 ϕ₄ ⊗ ϕ₇ 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 ϕ₃ ⊗ ϕ₁₁ 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 ϕ₅ ⊗ ϕ₇ 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 ϕ₄ ⊗ ϕ₈ 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 ϕ₂₅ of degree 9144846.

Decomposing the condensed tensor product ϕ₄ ⊗ ϕ₁₁ 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 ϕ₅ ⊗ ϕ₁₁ is now complete created yet. The decomposition is 120.ϕ₁ + 78.ϕ₂ + 130.ϕ₃ + 38.ϕ₄ + 22.ϕ₅ + 42.ϕ₆ + 44.ϕ₇ + 22.ϕ₈ + 22.ϕ₉ + 30.ϕ₁₀ + 12.ϕ₁₁ + 4.ϕ₁₂ + 6.ϕ₁₄ + 6.ϕ₁₅ + 6.ϕ₁₇ + 6.ϕ₁₈ + 8.ϕ₁₉ + 4.ϕ₂₃ + 4.ϕ₂₅ + ϕ₂₆ >From this we conjecture an irreducible ϕ₂₆ of degree 43122168.

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 ϕ₅ ⊗ ϕ₁₁ 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 condensed 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	Condensed degree	Proved	Constructed	Indicator	Group sign	Conjectured	Incomplete
ϕ1	1	1	✓	✓	+
ϕ2	24	4	✓	✓	+	+
ϕ3	274	6	✓	✓	+	-
ϕ4	1496	4	✓	✓	+	+
ϕ5	2000	12	✓	✓	+	+
ϕ6	4576	42	✓	✓	+	+
ϕ7	8580	38	✓	✓	+	-
ϕ8	17952	72	✓	✓	o
ϕ9	17952	72	✓	✓	o
ϕ10	26750	118	✓	✓	+	+
ϕ11	57200	210	✓	✓	+	+
ϕ12	165440	690	✓	✓	+	+
ϕ13	218800	882	✓	✓	+	+
ϕ14	250448	1044	✓	✓	+	+
ϕ15	256496	1038	✓	✓	+	+
ϕ16	420256	1696	✓	✓	+	+
ϕ17	378016	1508	✓	✓	o
ϕ18	378016	1508	✓	✓	o
ϕ19	616768	2486	✓	✓	+	+
ϕ20	4100096	16894					✓
ϕ21	6120022	25160					✓
ϕ22	7025950	28734					✓
ϕ23	9144846	37358					✓
ϕ24	27621392	113428					✓
ϕ25	4634432	19004					✓
ϕ26	43122168	177320					✓
ϕ27	40370176	165972	✓		+
ϕ28	1507328000	620460	✓		+
ϕ29	313524224	1289640	✓		+

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.