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.
In the table below, we list only even values of . This is because for even ,the 2-Sylow subgroup of is isomorphic to the 2-Sylow subgroup of .
|2-Sylow subgroup of symmetric group||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|