Linear representation theory of 2-Sylow subgroup of symmetric group

This article describes the linear representation theory of groups arising as the 2-Sylow subgroup of symmetric group.

All these groups are rational groups, i.e., any two elements of such a group that generate the same cyclic subgroup are conjugate. Hence, all the characters in characteristic zero are rational-valued. In fact, all these groups are rational-representation groups: all the representations in characteristic zero can be realized over the rational numbers.

Particular cases
In the table below, we list only even values of $$n$$. This is because for even $$n$$ ,the 2-Sylow subgroup of $$S_n$$ is isomorphic to the 2-Sylow subgroup of $$S_{n+1}$$.