Difference between revisions of "Linear representation theory of 2-Sylow subgroup of symmetric group"

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| 2 || [[cyclic group:Z2]] || 2 || 2 || 2 || 0 || 0 || [[linear representation theory of cyclic group:Z2]]
 
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| 4 || [[dihedral group:D8]] || 8 || 5 || 4 || 1 || 0 [[linear representation theory of dihedral group:D8]]
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| 6 || [[direct product of D8 and Z2]] || 16 || 10 || 8 || 2 || 0 || [[linear representation theory of direct product of D8 and Z2]]
 
| 6 || [[direct product of D8 and Z2]] || 16 || 10 || 8 || 2 || 0 || [[linear representation theory of direct product of D8 and Z2]]

Latest revision as of 02:32, 6 November 2011

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 groupS_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