Linear representation theory of double cover of alternating group:A7

GAP implementation
Although the group can be constructed and manipulated explicitly as PerfectGroup(5040,1), this is a time-consuming process. Instead, we can use the stored information about the group using the "2.A7" symbol for it.

Degrees of irreducible representations
These can be accessed using the CharacterDegrees, and CharacterTable functions as follows:

gap> CharacterDegrees(CharacterTable("2.A7")); [ [ 1, 1 ], [ 4, 2 ], [ 6, 1 ], [ 10, 2 ], [ 14, 4 ], [ 15, 1 ], [ 20, 2 ], [ 21, 1 ], [ 35, 1 ], [ 36, 1 ] ]

Character table
This can be computed as follows:

gap> Irr(CharacterTable("2.A7")); [ Character( CharacterTable( "2.A7" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     1, 1, 1, 1 ] ), Character( CharacterTable( "2.A7" ),    [ 6, 6, 2, 3, 3, 0, 0, 0, 0, 1, 1, -1, -1, -1, -1, -1 ] ), Character( CharacterTable( "2.A7" ), [ 10, 10, -2, 1, 1, 1, 1, 0, 0, 0, 0,     1, E(7)+E(7)^2+E(7)^4, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6,      E(7)^3+E(7)^5+E(7)^6 ] ), Character( CharacterTable( "2.A7" ),    [ 10, 10, -2, 1, 1, 1, 1, 0, 0, 0, 0, 1, E(7)^3+E(7)^5+E(7)^6,      E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4, E(7)+E(7)^2+E(7)^4 ] ), Character( CharacterTable( "2.A7" ), [ 14, 14, 2, 2, 2, -1, -1, 0, 0, -1,     -1, 2, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ),    [ 14, 14, 2, -1, -1, 2, 2, 0, 0, -1, -1, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ), [ 15, 15, -1, 3, 3, 0, 0, -1, -1, 0,     0, -1, 1, 1, 1, 1 ] ), Character( CharacterTable( "2.A7" ),    [ 21, 21, 1, -3, -3, 0, 0, -1, -1, 1, 1, 1, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ), [ 35, 35, -1, -1, -1, -1, -1, 1, 1, 0,     0, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ),    [ 4, -4, 0, -2, 2, 1, -1, 0, 0, -1, 1, 0, -E(7)-E(7)^2-E(7)^4,      E(7)+E(7)^2+E(7)^4, -E(7)^3-E(7)^5-E(7)^6, E(7)^3+E(7)^5+E(7)^6 ] ), Character( CharacterTable( "2.A7" ), [ 4, -4, 0, -2, 2, 1, -1, 0, 0, -1, 1,     0, -E(7)^3-E(7)^5-E(7)^6, E(7)^3+E(7)^5+E(7)^6, -E(7)-E(7)^2-E(7)^4,      E(7)+E(7)^2+E(7)^4 ] ), Character( CharacterTable( "2.A7" ),    [ 14, -14, 0, 2, -2, -1, 1, E(8)-E(8)^3, -E(8)+E(8)^3, -1, 1, 0, 0, 0, 0,      0 ] ), Character( CharacterTable( "2.A7" ),    [ 14, -14, 0, 2, -2, -1, 1, -E(8)+E(8)^3, E(8)-E(8)^3, -1, 1, 0, 0, 0, 0,      0 ] ), Character( CharacterTable( "2.A7" ),    [ 20, -20, 0, -4, 4, -1, 1, 0, 0, 0, 0, 0, -1, 1, -1, 1 ] ), Character( CharacterTable( "2.A7" ), [ 20, -20, 0, 2, -2, 2, -2, 0, 0, 0,     0, 0, -1, 1, -1, 1 ] ), Character( CharacterTable( "2.A7" ),    [ 36, -36, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 1, -1 ] ) ]

Here is a command to display the character table in a more user-friendly fashion:

gap> Display(CharacterTable("2.A7")); 2.A7

2 4   4  3  3  3  1  1  3  3  1   1   2   1   1   1   1      3  2   2  1  2  2  2  2  .  .  .   .   1   .   .   .   .      5  1   1  .  .  .  .  .  .  .  1   1   .   .   .   .   .      7  1   1  .  .  .  .  .  .  .  .   .   .   1   1   1   1

1a 2a 4a 3a 6a 3b 6b 8a 8b 5a 10a 12a  7a 14a  7b 14b 2P 1a 1a 2a 3a 3a 3b 3b 4a 4a 5a  5a  6a  7a  7a  7b  7b 3P 1a 2a 4a 1a 2a 1a 2a 8b 8a 5a 10a  4a  7b 14b  7a 14a 5P 1a 2a 4a 3a 6a 3b 6b 8b 8a 1a  2a 12a  7b 14b  7a 14a 7P 1a 2a 4a 3a 6a 3b 6b 8a 8b 5a 10a 12a  1a  2a  1a  2a

X.1     1   1  1  1  1  1  1  1  1  1   1   1   1   1   1   1 X.2     6   6  2  3  3. . .  .  1   1  -1  -1  -1  -1  -1 X.3     10  10 -2  1  1  1  1. . .   .   1   B   B  /B  /B X.4    10  10 -2  1  1  1  1. . .   .   1  /B  /B   B   B X.5     14  14  2  2  2 -1 -1. . -1 -1   2   .   .   .   . X.6     14  14  2 -1 -1  2  2. . -1 -1  -1   .   .   .   . X.7     15  15 -1  3  3. . -1 -1 .   .  -1   1   1   1   1 X.8     21  21  1 -3 -3. . -1 -1 1   1   1   .   .   .   . X.9     35  35 -1 -1 -1 -1 -1  1  1. . -1   .   .   .   . X.10     4  -4. -2 2  1 -1  .  . -1   1   .  -B   B -/B  /B X.11    4  -4. -2 2  1 -1  .  . -1   1   . -/B  /B  -B   B X.12    14 -14. 2 -2 -1 1  A -A -1   1. .  .   .   . X.13    14 -14. 2 -2 -1 1 -A  A -1   1. .  .   .   . X.14    20 -20. -4 4 -1  1  .  .  .   .   .  -1   1  -1   1 X.15    20 -20. 2 -2 2 -2  .  .  .   .   .  -1   1  -1   1 X.16    36 -36. . .  .  .  .  .  1  -1   .   1  -1   1  -1

A = E(8)-E(8)^3 = ER(2) = r2 B = E(7)+E(7)^2+E(7)^4 = (-1+ER(-7))/2 = b7