Linear representation theory of Janko group:J1

This article gives specific information, namely, linear representation theory, about a particular group, namely: Janko group:J1.
View linear representation theory of particular groups | View other specific information about Janko group:J1

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as ${\overline {\mathbb {Q} }}$ or $\mathbb {C}$)	1, 56, 56, 76, 76, 77, 77, 77, 120, 120, 120, 133, 133, 133, 209 maximum: 209, lcm: 175560, number: 15, sum of squares: 175560

GAP implementation

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

gap> CharacterDegrees(CharacterTable("J1"));
[ [ 1, 1 ], [ 56, 2 ], [ 76, 2 ], [ 77, 3 ], [ 120, 3 ], [ 133, 3 ], [ 209, 1 ] ]

We can use Irr and CharacterTable to print out the whole character table:[SHOW MORE]

gap> Irr(CharacterTable("J1"));
[ Character( CharacterTable( "J1" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "J1" ),
    [ 56, 0, 2, -2*E(5)-2*E(5)^4, -2*E(5)^2-2*E(5)^3, 0, 0, 0, 0, 1, E(5)+E(5)^4, E(5)^2+E(5)^3, -1, -1, -1 ] ), Character( CharacterTable( "J1" ),
    [ 56, 0, 2, -2*E(5)^2-2*E(5)^3, -2*E(5)-2*E(5)^4, 0, 0, 0, 0, 1, E(5)^2+E(5)^3, E(5)+E(5)^4, -1, -1, -1 ] ), Character( CharacterTable( "J1" ),
    [ 76, 4, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 0, 0, 0 ] ), Character( CharacterTable( "J1" ), [ 76, -4, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 0, 0, 0 ] ),
  Character( CharacterTable( "J1" ), [ 77, 5, -1, 2, 2, -1, 0, 0, 0, 0, -1, -1, 1, 1, 1 ] ), Character( CharacterTable( "J1" ),
    [ 77, -3, 2, E(5)+E(5)^4, E(5)^2+E(5)^3, 0, 0, E(5)+E(5)^4, E(5)^2+E(5)^3, 0, E(5)+E(5)^4, E(5)^2+E(5)^3, 1, 1, 1 ] ),
  Character( CharacterTable( "J1" ), [ 77, -3, 2, E(5)^2+E(5)^3, E(5)+E(5)^4, 0, 0, E(5)^2+E(5)^3, E(5)+E(5)^4, 0, E(5)^2+E(5)^3, E(5)+E(5)^4, 1, 1, 1 ] ),
  Character( CharacterTable( "J1" ), [ 120, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, E(19)+E(19)^7+E(19)^8+E(19)^11+E(19)^12+E(19)^18,
      E(19)^2+E(19)^3+E(19)^5+E(19)^14+E(19)^16+E(19)^17, E(19)^4+E(19)^6+E(19)^9+E(19)^10+E(19)^13+E(19)^15 ] ), Character( CharacterTable( "J1" ),
    [ 120, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, E(19)^4+E(19)^6+E(19)^9+E(19)^10+E(19)^13+E(19)^15, E(19)+E(19)^7+E(19)^8+E(19)^11+E(19)^12+E(19)^18,
      E(19)^2+E(19)^3+E(19)^5+E(19)^14+E(19)^16+E(19)^17 ] ), Character( CharacterTable( "J1" ), [ 120, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0,
      E(19)^2+E(19)^3+E(19)^5+E(19)^14+E(19)^16+E(19)^17, E(19)^4+E(19)^6+E(19)^9+E(19)^10+E(19)^13+E(19)^15,
      E(19)+E(19)^7+E(19)^8+E(19)^11+E(19)^12+E(19)^18 ] ), Character( CharacterTable( "J1" ), [ 133, 5, 1, -2, -2, -1, 0, 0, 0, 1, 1, 1, 0, 0, 0 ] ),
  Character( CharacterTable( "J1" ), [ 133, -3, -2, -E(5)-E(5)^4, -E(5)^2-E(5)^3, 0, 0, E(5)+E(5)^4, E(5)^2+E(5)^3, 1, -E(5)-E(5)^4, -E(5)^2-E(5)^3, 0, 0,
      0 ] ), Character( CharacterTable( "J1" ), [ 133, -3, -2, -E(5)^2-E(5)^3, -E(5)-E(5)^4, 0, 0, E(5)^2+E(5)^3, E(5)+E(5)^4, 1, -E(5)^2-E(5)^3,
      -E(5)-E(5)^4, 0, 0, 0 ] ), Character( CharacterTable( "J1" ), [ 209, 1, -1, -1, -1, 1, -1, 1, 1, 0, -1, -1, 0, 0, 0 ] ) ]

External links

• J1 on the Atlas of Finite Groups