Linear representation theory of 2-Sylow subgroup of symmetric group

This article gives specific information, namely, linear representation theory, about a family of groups, namely: 2-Sylow subgroup of symmetric group.
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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 $S_{n+1}$.

$n$	2-Sylow subgroup of symmetric group$S_{n}$	Order of the group	Number of irreducible representations (= number of conjugacy classes)	Number of irreducible representations of degree one	Number of irreducible representations of degree two	Number of irreducible representations of degree four	Linear representation theory page
0	trivial group	1	1	1	0	0	linear representation theory of trivial group
2	cyclic group:Z2	2	2	2	0	0	linear representation theory of cyclic group:Z2
4	dihedral group:D8	8	5	4	1	0	linear representation theory of dihedral group:D8
6	direct product of D8 and Z2	16	10	8	2	0	linear representation theory of direct product of D8 and Z2
8	wreath product of D8 and Z2	128	20	8	6	6	linear representation theory of wreath product of D8 and Z2