Abstract
We compute the 3-modular character table of the group \(\mathrm{O'N}.2\). Much of the table is deduced character theoretically from the known 3-modular character table of the sporadic simple O’Nan group \(\text {O}'\text {N}\). We finish the remaining questions module theoretically with an application of condensation.
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Appendix: Brauer Trees for \(\text {O}'\text {N}.2\) in Characteristic 5
Appendix: Brauer Trees for \(\text {O}'\text {N}.2\) in Characteristic 5
There are six blocks with Brauer trees for \(\text {O}'\text {N}.2\) in characteristic 5 and only the \(4^{th}\) block in the GAP labeling of all blocks is not straightforward. It contains the following ordinary irreducibles:
Its tree is a stem, more precisely, an unfolding of a tree for \(\mathrm{O'N}\). The middle node is the ordinary irreducible of degree 51832 and its neighbors are the two irreducibles of degree 52688. Consider the following permutation of conjugacy classes of \(\text {O}'\text {N}.2\): \(\pi = 15(ab)16(ab)8(bc)10(bc)24(ab)30(ab)56(ab)56(cd)\). We note that \(\pi \) is an automorphism of the ordinary character table that switches the 7th and 8th ordinary irreducibles of degree 26752 while fixing the other nodes of the Brauer tree under consideration. Accordingly, we may suppose that the Brauer tree has the following form.
We further observe that the restriction of \(\pi \) to the 3-regular classes is an automorphism of the 3-modular character table of \(\text {O}'\text {N}.2\), which means that any relabeling of conjugacy classes that are necessary to ensure that the 5-modular Brauer tree is as given above does not result in a change to the 3-modular character table computed in earlier sections. In other words, our descriptions of the 3-modular and 5-modular character tables of \(\text {O}'\text {N}.2\) are compatible. A detailed discussion of character table automorphisms as label permutations and compatibility of character tables is available in Sect. 10 of [12].
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Lux, K., Ryba, A. (2019). The 3-Modular Character Table of the Automorphism Group of the Sporadic Simple O’Nan Group. In: Feldvoss, J., Grimley, L., Lewis, D., Pavelescu, A., Pillen, C. (eds) Advances in Algebra. SRAC 2017. Springer Proceedings in Mathematics & Statistics, vol 277. Springer, Cham. https://doi.org/10.1007/978-3-030-11521-0_12
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