# 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.
View linear representation theory of group families | View other specific information about 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 $S_{n+1}$. $n$ 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