Linear representation theory of projective special linear group:PSL(3,4)

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This article gives specific information, namely, linear representation theory, about a particular group, namely: projective special linear group:PSL(3,4).
View linear representation theory of particular groups | View other specific information about projective special linear group:PSL(3,4)

Summary

Item Value
degrees of irreducible representations over a splitting field (such as or ) 1, 20, 35, 35, 35, 45, 45, 63, 63, 64
grouped form: 1 (1 time), 20 (1 time), 35 (3 times), 45 (2 times), 63 (2 times), 64 (1 time)
maximum: 64, quasirandom degree: 20, number: 10, sum of squares: 20160

Family contexts

Family name Parameter values General discussion of linear representation theory of family
Mathieu group degree 21, i.e., the group linear representation theory of Mathieu groups
projective special linear group of degree three over a finite field field:F4, i.e., the group linear representation theory of projective special linear group of degree three over a finite field

GAP implementation

Degrees of irreducible representations

The degrees of irreducible representations can be computed using GAP's CharacterDegrees, CharacterTable, and PSL functions: 

gap> CharacterDegrees(CharacterTable(PSL(3,4)));
[ [ 1, 1 ], [ 20, 1 ], [ 35, 3 ], [ 45, 2 ], [ 63, 2 ], [ 64, 1 ] ]

Character table

The character table can be computed using GAP's CharacterDegrees, CharacterTable, and PSL functions: 

gap> Irr(CharacterTable(PSL(3,4)));
[ Character( CharacterTable( Group([ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19), (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12)
     ]) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Group([ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19),
      (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12) ]) ), [ 20, -1, -1, 0, 0, 0, 4, 2, 0, 0 ] ), Character( CharacterTable( Group(
    [ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19), (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12) ]) ),
    [ 35, 0, 0, 0, 0, 3, 3, -1, -1, -1 ] ), Character( CharacterTable( Group([ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19),
      (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12) ]) ), [ 35, 0, 0, 0, 0, -1, 3, -1, -1, 3 ] ), Character( CharacterTable( Group(
    [ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19), (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12) ]) ),
    [ 35, 0, 0, 0, 0, -1, 3, -1, 3, -1 ] ), Character( CharacterTable( Group([ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19),
      (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12) ]) ), [ 45, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4, 0, 0, 1, -3, 0, 1, 1 ] ),
  Character( CharacterTable( Group([ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19), (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12)
     ]) ), [ 45, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6, 0, 0, 1, -3, 0, 1, 1 ] ), Character( CharacterTable( Group(
    [ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19), (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12) ]) ),
    [ 63, 0, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3, -1, -1, 0, -1, -1 ] ), Character( CharacterTable( Group(
    [ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19), (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12) ]) ),
    [ 63, 0, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4, -1, -1, 0, -1, -1 ] ), Character( CharacterTable( Group(
    [ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19), (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)(5,18,9,19,21,17,12) ]) ),
    [ 64, 1, 1, -1, -1, 0, 0, 1, 0, 0 ] ) ]
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