By John Horton Conway

ISBN-10: 0198531990

ISBN-13: 9780198531999

This atlas covers teams from the households of the category of finite uncomplicated teams. lately up to date incorporating corrections

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**Sample text**

He has also computed a large number of modular character tables, intended for a later A lr ILA§ publication. A§ groups. He has greatly increased the usefulness of this A If D.. A§ by adding this and other information, and over the last few years has cheerfully shouldered the enormous task of gathering and transforming our untidy heaps of material into a form fit for ppblication. xxxiii My own function was to initiate and control the entire project, to collaborate with each of the above, and (eventually) to write this Introduction.

2. Ordering of classes and characters Many table compilers arrange the conjugacy classes in an order which to some extent reflects the power maps. , having powers in the original class. The merits of such arrangements are usually more evident to the compiler than the user, who is seldom properly informed about the principles (if any) of the arrangement, and so cannot use the implied power map information. It is better to choose a simpler arrangement, and explicitly indicate the power maps. A§, the conjugacy classes within a given coset are arranged firstly, by increasing succession of n, the order of their elements; secondly, for elements with the same n, by decreasing succession of N, their centralizer order; thirdly, for elements with the same nand N, by increasing succession of d, the degree of the algebraic number field generated by their character values (so that rational elements come first); and fourthly, for elements with the same n, N, and d, in a manner which seems best compatible with the p' parts, so that, other things being equal, we prefer to arrange elements of order 10 in the same succession as the elements of order 5 that are their odd parts.

An extremely valuable test which we recommend when the above simple rules of thumb have failed is to compute the skew squares of one or more characters X, and check their inner products with selected irreducibles. One should also check the consistency of restriction maps from groups to subgroups whose tables are also known. Of course it can be a good idea to check a doubtful point against another published table, which might well be the one from which ours was originally derived. A disagreement probably means that we believed we had found and corrected an error in the source table, but of course we could be wrong, or could have inadvertently introduced some error ourselves.

### Atlas of finite groups: maximal subgroups and ordinary characters for simple groups by John Horton Conway

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