Linear representation theory of Janko group:J2

This article gives specific information, namely, linear representation theory, about a particular group, namely: Janko group:J2.
View linear representation theory of particular groups | View other specific information about Janko group:J2

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as ${\overline {\mathbb {Q} }}$ or $\mathbb {C}$)	1, 14, 14, 21, 21, 36, 63, 70, 70, 90, 126, 160, 175, 189, 189, 224, 224, 225, 288, 300, 336 maximum: 336, lcm: 151200, number: 21, sum of squares: 604800

GAP implementation

The degrees of irreducible representations can be computed using the CharacterDegrees and CharacterTable functions:

gap> CharacterDegrees(CharacterTable("J2"));
[ [ 1, 1 ], [ 14, 2 ], [ 21, 2 ], [ 36, 1 ], [ 63, 1 ], [ 70, 2 ], [ 90, 1 ], [ 126, 1 ], [ 160, 1 ], 
[ 175, 1 ], [ 189, 2 ], [ 224, 2 ], [ 225, 1 ],   [ 288, 1 ], [ 300, 1 ], [ 336, 1 ] ]

The full character table can be printed out using Irr and CharacterTable: [SHOW MORE]

gap> Irr(CharacterTable("J2"));
[ Character( CharacterTable( "J2" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "J2" ),
    [ 14, -2, 2, 5, -1, 2, -3*E(5)-3*E(5)^4, -3*E(5)^2-3*E(5)^3, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, 1, -1, 0, 0, E(5)+E(5)^4,
      E(5)^2+E(5)^3, -E(5)-E(5)^4, -E(5)^2-E(5)^3, -1, 0, 0 ] ), Character( CharacterTable( "J2" ), [ 14, -2, 2, 5, -1, 2, -3*E(5)^2-3*E(5)^3,
      -3*E(5)-3*E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, 1, -1, 0, 0, E(5)^2+E(5)^3, E(5)+E(5)^4, -E(5)^2-E(5)^3,
      -E(5)-E(5)^4, -1, 0, 0 ] ), Character( CharacterTable( "J2" ), [ 21, 5, -3, 3, 0, 1, -3*E(5)-4*E(5)^2-4*E(5)^3-3*E(5)^4,
      -4*E(5)-3*E(5)^2-3*E(5)^3-4*E(5)^4, -2*E(5)-2*E(5)^4, -2*E(5)^2-2*E(5)^3, -1, 0, 0, -1, E(5)+E(5)^4, E(5)^2+E(5)^3, 0, 0, 1, -E(5)^2-E(5)^3,
      -E(5)-E(5)^4 ] ), Character( CharacterTable( "J2" ), [ 21, 5, -3, 3, 0, 1, -4*E(5)-3*E(5)^2-3*E(5)^3-4*E(5)^4, -3*E(5)-4*E(5)^2-4*E(5)^3-3*E(5)^4,
      -2*E(5)^2-2*E(5)^3, -2*E(5)-2*E(5)^4, -1, 0, 0, -1, E(5)^2+E(5)^3, E(5)+E(5)^4, 0, 0, 1, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ),
  Character( CharacterTable( "J2" ), [ 36, 4, 0, 9, 0, 4, -4, -4, 1, 1, 1, 0, 1, 0, 0, 0, -1, -1, 1, -1, -1 ] ), Character( CharacterTable( "J2" ),
    [ 63, 15, -1, 0, 3, 3, 3, 3, -2, -2, 0, -1, 0, 1, -1, -1, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "J2" ),
    [ 70, -10, -2, 7, 1, 2, -5*E(5)-5*E(5)^4, -5*E(5)^2-5*E(5)^3, 0, 0, -1, 1, 0, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3, 0, 0, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ] )
    , Character( CharacterTable( "J2" ), [ 70, -10, -2, 7, 1, 2, -5*E(5)^2-5*E(5)^3, -5*E(5)-5*E(5)^4, 0, 0, -1, 1, 0, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4, 0,
      0, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ] ), Character( CharacterTable( "J2" ), [ 90, 10, 6, 9, 0, -2, 5, 5, 0, 0, 1, 0, -1, 0, 1, 1, 0, 0, 1, -1, -1 ] ),
  Character( CharacterTable( "J2" ), [ 126, 14, 6, -9, 0, 2, 1, 1, 1, 1, -1, 0, 0, 0, 1, 1, -1, -1, -1, 1, 1 ] ), Character( CharacterTable( "J2" ),
    [ 160, 0, 4, 16, 1, 0, -5, -5, 0, 0, 0, 1, -1, 0, -1, -1, 0, 0, 0, 1, 1 ] ), Character( CharacterTable( "J2" ), [ 175, 15, -5, -5, 1, -1, 0, 0, 0, 0,
      3, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0 ] ), Character( CharacterTable( "J2" ), [ 189, -3, -3, 0, 0, -3, -3*E(5)-3*E(5)^4, -3*E(5)^2-3*E(5)^3,
      -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, 0, 0, 0, 1, E(5)+E(5)^4, E(5)^2+E(5)^3, E(5)+E(5)^4, E(5)^2+E(5)^3, 0, 0, 0 ] ),
  Character( CharacterTable( "J2" ), [ 189, -3, -3, 0, 0, -3, -3*E(5)^2-3*E(5)^3, -3*E(5)-3*E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4,
      -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, 0, 0, 0, 1, E(5)^2+E(5)^3, E(5)+E(5)^4, E(5)^2+E(5)^3, E(5)+E(5)^4, 0, 0, 0 ] ), Character( CharacterTable( "J2" ),
    [ 224, 0, -4, 8, -1, 0, 3*E(5)-E(5)^2-E(5)^3+3*E(5)^4, -E(5)+3*E(5)^2+3*E(5)^3-E(5)^4, 2*E(5)+2*E(5)^4, 2*E(5)^2+2*E(5)^3, 0, -1, 0, 0, 1, 1, 0, 0, 0,
      -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), Character( CharacterTable( "J2" ), [ 224, 0, -4, 8, -1, 0, -E(5)+3*E(5)^2+3*E(5)^3-E(5)^4,
      3*E(5)-E(5)^2-E(5)^3+3*E(5)^4, 2*E(5)^2+2*E(5)^3, 2*E(5)+2*E(5)^4, 0, -1, 0, 0, 1, 1, 0, 0, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ),
  Character( CharacterTable( "J2" ), [ 225, -15, 5, 0, 3, -3, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "J2" ),
    [ 288, 0, 4, 0, -3, 0, 3, 3, -2, -2, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "J2" ), [ 300, -20, 0, -15, 0, 4, 0, 0, 0, 0,
      1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0 ] ), Character( CharacterTable( "J2" ), [ 336, 16, 0, -6, 0, 0, -4, -4, 1, 1, -2, 0, 0, 0, 0, 0, 1, 1, 0, -1, -1
     ] ) ]

External links

• J2 on the Atlas of Finite Groups