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 six results which are condensation only. Nineteen of the remaining twenty irreducibles have all been constructed and their indicators computed, and for the fifteen of these of indicator plus the orthogonal group has also been determined. There are no characters of indicator minus. A twentieth irreducible has been proved but not constructed.

Section 3 : Demonstration of results

A lot of use of condensation has been made in this talk. I am not going to explain here what condensation is. But, I will point out the subgroup used. In GAP, for groups such as McL, one can obtain a table of marks. The condensation subgroup used is element 245 thereof, which is a subgroup of order $3^5$. Using the information on the QMW atlas, one can restrict from ${\mathrm{Co1}}_1$ to McL, and then using information from GAP one can obtain generators for this subgroup. Tensor condensation is described in Lux and Wiegelmann.

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 restriction to ${\mathrm{Co}}_3$ and 3.Suz.2.

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. Getting the character values of ϕ₁₃ is easiest achieved by restriction to subgroups.

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.

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 to have been constructed, though one more has been proved. 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. But, in fact, this is the sole constituent of the ordinary irreducible χ₃₁, and hence is known to exist

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 ϕ₂₃ 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.

Decomposing the condensed tensor product ϕ₅ ⊗ ϕ₁₁ we obtain 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 (update, on an i7/11700 (about 10 times faster than the P4/3400) with a vastly improved tensor condese program (better order, caching, greasing) it now takes around 10 hours), and a correspondingly long time to split (update using Meataxe64, it takes around a day). 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	228	138	240	62	43	70	72	42	42	54	42	10	16	13	13	8	8	8	10	1	1	2	1	2	4	1	0	0	0
χ69	318	182	325	86	58	91	95	52	52	75	54	15	22	17	16	13	10	10	15	1	3	2	2	1	3	1	0	0	0
χ70	304	170	311	79	51	89	92	53	53	72	45	13	22	15	15	10	12	12	14	1	3	1	3	1	3	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	330	190	341	89	58	98	101	58	58	79	53	14	23	17	17	12	12	12	15	1	3	2	3	1	4	1	0	0	0
χ73	356	208	369	99	65	106	109	61	61	86	59	17	23	19	19	13	12	12	17	1	3	2	3	1	4	1	0	0	0
χ74	353	207	371	96	62	107	110	66	66	87	62	14	27	20	19	14	12	12	14	1	3	3	2	1	5	1	0	0	0
χ75	324	188	338	86	58	93	98	57	57	78	60	14	26	18	17	14	10	10	13	1	2	3	1	2	4	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	434	254	450	120	80	133	133	76	76	105	72	21	30	24	23	17	16	16	22	2	4	3	3	1	5	1	0	0	0
χ79	473	278	494	132	87	145	146	84	84	116	79	23	33	27	26	18	17	17	23	2	4	3	3	1	5	1	0	0	0
χ80	383	223	401	101	69	110	117	69	69	92	71	16	31	21	21	16	12	12	15	1	2	3	1	3	5	1	0	0	0
χ81	464	274	484	129	86	144	144	83	83	113	79	22	32	26	25	18	17	17	23	2	4	4	3	1	6	1	0	0	0
χ82	476	293	506	139	93	153	152	88	88	119	84	23	30	29	28	18	17	17	24	2	3	4	3	1	7	1	0	0	0
χ83	505	293	520	140	93	151	153	85	85	121	83	25	34	27	26	20	18	18	26	2	5	3	4	1	5	1	0	0	0
χ84	486	282	506	132	87	140	148	84	84	118	88	21	36	26	25	20	15	15	20	1	4	4	3	2	6	1	0	0	0
χ85	584	350	610	167	113	188	184	105	105	142	95	30	38	35	34	22	23	23	33	3	5	3	4	1	6	1	0	0	0
χ86	499	293	523	134	91	149	154	91	91	121	92	22	41	29	28	21	17	17	21	2	4	4	1	3	6	1	0	0	0
χ87	545	315	567	147	97	162	167	97	97	132	94	24	42	30	29	22	19	19	24	2	5	4	3	2	6	1	0	0	0
χ88	632	370	656	177	118	192	194	109	109	154	107	31	44	36	34	26	22	22	32	3	6	4	4	1	6	1	0	0	0
χ89	622	362	644	171	115	187	189	107	107	150	107	31	46	35	33	26	22	22	31	3	6	4	3	2	6	1	0	0	0
χ90	610	352	636	164	107	180	186	109	109	149	102	27	44	34	32	22	21	21	26	2	4	4	4	2	7	2	0	0	0
χ91	618	362	646	168	109	184	192	113	113	150	106	24	44	33	33	23	21	21	25	1	5	5	5	2	9	2	0	0	0
χ92	634	370	661	171	113	188	195	114	114	153	109	27	47	35	34	24	22	22	27	2	5	4	4	3	8	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	824	490	865	230	151	258	259	152	152	202	138	37	57	47	46	30	31	31	39	3	7	6	6	2	11	2	0	0	0
χ95	818	475	849	222	148	245	251	145	145	197	138	37	60	45	44	32	29	29	38	3	7	5	5	3	9	2	0	0	0
χ96	838	490	875	229	151	251	258	150	150	204	145	37	62	47	45	33	29	29	37	3	7	6	5	3	10	2	0	0	0
χ97	832	481	867	223	146	241	255	148	148	201	147	34	64	45	44	33	27	27	33	2	7	6	5	4	10	2	0	0	0
χ98	880	518	921	243	160	267	273	158	158	215	150	40	62	49	48	33	31	31	40	3	7	6	6	3	11	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	1004	590	1048	280	184	307	311	179	179	246	168	47	70	57	55	38	36	36	48	4	9	6	7	2	11	2	0	0	0
χ101	996	582	1041	274	178	297	307	177	177	244	172	44	72	55	53	38	34	34	43	3	9	7	7	3	12	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
[Lux and Wiegelmann] - Tensor condensation
[Wilson] The QMW Atlas of Finite Group Representations

Section 8 : Acknowledgements

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