Linear representation theory of projective general linear group:PGL(2,7)

Contents

This article gives specific information, namely, linear representation theory, about a particular group, namely: projective general linear group:PGL(2,7).
View linear representation theory of particular groups | View other specific information about projective general linear group:PGL(2,7)

Summary

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,1,6,6,6,7,7,8,8
number: 9, maximum: 8, lcm: 168, sum of squares: 336

Family contexts

Family Parameter values General discussion of linear representation theory of family
projective general linear group of degree two over a finite field, denoted $PGL(2,q)$ for field size $q$ $q = 7$, i.e., field:F7, so the group is $PGL(2,7)$ linear representation theory of projective general linear group of degree two over a finite field
outer linear group of degree three over a finite field, denoted $OL(3,q)$ for field size $q$ $q = 2$, i.e., field:F2, so the group is $OL(3,2)$ linear representation theory of outer linear group of degree three over a finite field

GAP implementation

The degrees of irreducible representations can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(PGL(2,7));
[ [ 1, 2 ], [ 6, 3 ], [ 7, 2 ], [ 8, 2 ] ]

The character table can be computed using GAP's Irr and CharacterTable functions:

gap> Irr(CharacterTable(PGL(2,7)));
[ Character( CharacterTable( Group([ (3,8,7,6,5,4), (1,2,6)(3,4,8) ]) ),
[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Group(
[ (3,8,7,6,5,4), (1,2,6)(3,4,8) ]) ), [ 1, -1, 1, -1, 1, 1, -1, 1, -1
] ), Character( CharacterTable( Group([ (3,8,7,6,5,4), (1,2,6)(3,4,8)
]) ), [ 6, 0, 0, 0, -1, -2, 0, 2, 0 ] ),
Character( CharacterTable( Group([ (3,8,7,6,5,4), (1,2,6)(3,4,8) ]) ),
[ 6, 0, 0, 0, -1, 2, -E(8)+E(8)^3, 0, E(8)-E(8)^3 ] ),
Character( CharacterTable( Group([ (3,8,7,6,5,4), (1,2,6)(3,4,8) ]) ),
[ 6, 0, 0, 0, -1, 2, E(8)-E(8)^3, 0, -E(8)+E(8)^3 ] ),
Character( CharacterTable( Group([ (3,8,7,6,5,4), (1,2,6)(3,4,8) ]) ),
[ 7, -1, 1, -1, 0, -1, 1, -1, 1 ] ), Character( CharacterTable( Group(
[ (3,8,7,6,5,4), (1,2,6)(3,4,8) ]) ), [ 7, 1, 1, 1, 0, -1, -1, -1, -1
] ), Character( CharacterTable( Group([ (3,8,7,6,5,4), (1,2,6)(3,4,8)
]) ), [ 8, -1, -1, 2, 1, 0, 0, 0, 0 ] ),
Character( CharacterTable( Group([ (3,8,7,6,5,4), (1,2,6)(3,4,8) ]) ),
[ 8, 1, -1, -2, 1, 0, 0, 0, 0 ] ) ]